^/^.
\
:
A SYNOPSIS
ELEMENTAIIY RESULTS
PUKE MATHEMATICS
CONTAINTNO
PROPOSTTTONS.'FORMUL^, AND METHODS OF ANALYSIS.
WITH
ABRIDGED DEMONSTRATIONS.
ItY AN InDEX TO THE PaI'ERS ON Pi HE JIaTHEMATU S -WHUJI AKE TO
FOVNn IN THE PHINCIPAL JoiBNALS AND TllANSACTlONS OF LEAUM I) Socill lEP,
poTH English and Foreion, of the present centuky.
SlTPLEMFNTFI)
BE
G.
S.
CARR, M.A.
LONDON
FRANCIS noi)(;soN,
farkin(tDon street, e.c
CAMBRIDGE: MACMILLAN & BOWES.
80
1886.
(AU
rights reserved
)
:
C3
neenng
Librai:'^
LONDON
PRINTED BY
C.
F.
HODGSON AND
GOUGH SQUARE, FLEET STREET.
76.
6j6
SON,
N TVERSJT?
PREFACE TO PART
Tin: work, of which
now
part
tlio
I
issued
a
is
first instal-
ment, has been compiled from notes made at various periods
and
of the last fourteen years,
chiefly during* the
engagements
!Many of the abbreviated methods and mnemonic
of teaching.
rules are in the
form in which
I originally
my
wrote them for
pupils.
The general object
the compilation
of
indicates, to present within a moderate
is,
as
the
title
compass the funda-
mental theorems, formulas, and processes in the chief branches
of pure
and applied mathematics.
Tlie
work
is
intended, in the
supplement the use of
arranged
witli tlie
tl,ic
To
to follow
ordinary text-books, and
view of assisting
revision of book-w^ork.
place,
first
this
tlie
end
and
it
is
student in the task of
I have, in
many
cases,
merely indicated the salient points of a demonstration, or
merely referred to the theorems by which the proposition
proved.
I
am
convinced that
it
is
more
is
beneficial to tlie
student to recall demonstrations with such aids, than to read
and re-read them.
The
Let them be read once, but recalled often.
difference in the effect
upon the mind between reading a
mathematical demonstration, and originating cue wholly or
PEEFACE.
IV
partly,
is
very great.
It
may be compared
to tlic difference
between the pleasure experienced, and interest aroused, when
in the one case a traveller
passively conducted through the
is
roads of a novel and unexplored country, and in the other
case he discovers the roads for himself with the assistance of
a map.
In the second place, I venture to hope that the work,
when completed, may prove
an aide-memoire and book
useful to advanced students as
of reference.
The boundary
of
mathematical science forms, year by year, an ever widening
circle,
and the advantage
having at hand some condensed
of
statement of results becomes more and more evident.
To
the original investigator occupied with abstruse re-
searches in some one of the
many branches
work which gathers together
may not
sitions in all,
of mathematics, a
synoptically the leading propo-
therefore prove unacceptable.
Abler
hands than mine undoubtedly, might have undertaken the task
of
making such a digest
be more
;
but abler hands might
usefully emj^loycd,
less hesitation in
is
it
suggestions or criticisms which
it
!
have their
With
have the
may be regarded
The degree of success which
Ic:
perhaps,
effect
The design
somewhat comprehensive, and
the present essay
it
also,
this reflection I
commencing the work myself.
which I have indicated
relation to
—
and with
may meet
may
in
as tentative.
with, and the
call forth, will
doubt-
on the subsequent portions of the work.
respect to the abridgment of the demonstrations, I
may remark,
that while
some diffuseness
of explanation is not
only allowable but very desirable in an initiatory treatise,
conciseness
is
one of the chief reciuiremcnts in a work intended
V
PREFACE.
and rcfiTeiico only.
In order,
liowever, not to sacrifice clearness to conciseness,
much moro
for
tlio
piii-i)OSos
la])our has
of the
of revision
been expended upon this part of the subject-matter
book than
will at first sip^ht
be at
all
The only
evident.
])alpal)le I'esult lK'in<^ a
compression of the text, the result
so far a neji^ative one.
The amount
is illustrated in
of compression attained
the last section of the present part, in which
moro than the number
treatises
is
of
propositions
usually
given in
on Geometrical Conies are contained, together with
the figures and demonstrations, in the s})ace of twenty-foui*
pages.
The foregoing remarks have a general
work
as a whole.
earlier sections
tliat,
With the view, however,
more acceptable
in those sections,
been more
application to the
to beginners,
making the
of
it
will
be found
important principles have sometimes
fully elucidated
and more
illustrated
by exam})les,
than the plan of the work would admit of in subsequent
di\isions.
A
feature to which attention
form system
of reference adopted
may
be directed
throughout
all
is
the uni-
the sections.
AVithtlie object of facilitating such reference, the articles have
been numbered progressively from the
large Clarendon figures
;
commencement
in
the breaks which will occasionally
be found in these numbers having been purposely made, in order
to leave
room for the
insertion of additional matter,
be re(piired in a future edition, without distui-bing
numbers and
references.
With the same
if it
should
tlie oi'iginal
object, demonstrations
and examples have been made subordinate
to enunciations
and
formidie, the former being [)rinted in small, the latter in bold
rREFACE.
Vi
By
type.
more
tliese aids, tlic
reudily sliown,
Ijetwecn theorems
intordcpcndoncc of propositions
and it becomes easy
to trace the
is
connexion
in different branches of mathematics, with-
out the loss of time which would be incurred in turning to
The advantage thus gained
separate treatises on the subjects.
will,
however, become more apparent as the work proceeds.
The Algebra
section
not quite correspond
named
the particulars
was printed some years ago, and does
mth
the succeeding ones in some of
above.
Under the pressure
of other
occupations, this section moreover was not properly revised
before
f>'oing
On
to press.
that account the table of errata
will be found to apply almost exclusively to errors in that
section
;
but I trust that the hst
Great pains
is e:s.haustive.
liave been taken to secure the accm^acy of the rest of the
volume.
I have
Any
now
intimation of errors will be gladly received.
to
acknowledge some
the present part has been compiled.
of Equations,
of the sources
and Trigonometry sections,
debted to Todhunter's
well-known
completeness of which
it
from which
In the Algebra, Theory
am
I
treatises, the
would be superfluous
largely in-
accuracy and
in
me
to dwell
upon.
In the section entitled Elementary Geometry, I have added
to simpler propositions a selection of theorems
send's
Modern Geometiy and Salmon's Conic
from Town-
Sections.
In Geometrical Conies, the line of demonstration followed
agrees, in the main, with that adopted in Drew's treatise on the
subject.
I
am
author cannot be
inclined
to think that the
much improved.
It is
method
of that
true that some im-
portant properties of the ellipse, which are arrived at in
Vll
I'KKI'ACR.
Drew's (\)nic
tions, can be
tlirou^^h ccitnin iiitcnncdialc jn'oposi-
Si^ctioiis
once from
dcMliicod at
not on that account be dispenscnl w
]\Ioreov(>r, tlie
Ili(>
hyperbola
lias
first
itli,
for they are of value in
of projection applied to
nietliod
not so successful; because a
is
metliod of
But the intcM-mediate propositions can-
orthogonal projection.
th(Mns('lv(\^.
tlie circle ])y tlic
to bo proved
true in the case of
which
pi-oj)erty
the equilateral
hyperbola, might as will be proved at once for the general case.
I
have introduced the method of projection but spanngly,
alwavs giving prefei'ence to a demonstration which admits of
being
n])])li(Ml
in the
identical
form to the
The remarkable analogy
the hyperbola.
the two curves
same
is
The account
and to
ellipse
subsisting between
thus kept prominently before the reader.
of the C. G. S. system of units given in the
preliminary section, has been compiled from a valuable contribution on the subject
by Professor Everett,
published by the Physical Society of London.*
and the tables
of
of Belfast,
This abstract,
physical constants, might perhaps have found
a more appropriate place in an after part of the work.
however, introduced them at
of the great importance of the
of
measurement Avhich
and from a
belief
is
I have,
the commencement, from a sense
rcfonu in the selection of units
embodied
that the
in
the
C
G. S. system,
student cannot be too
early
familiarized with the same.
The Factor Table
wliich folluAvs
rei)rint of Burckluirdt's " Tnhlr.^
is,
ilrs
to its limited extent, a
diviscurs,'' published in
* "Illustrations of the Centimetrc-Grammc-Sccoiid System of Units."
London
:
Taylor and Francis.
1875.
riJEFACE.
Vlll
1814-17, which give the least divisors of
In a certain sense,
to 3,036,000.
it
numbers from
1
be said that this
is
all
may
the only sort of purely mathematical table which
is
indispensable, because the information which
gives cannot
it
be supplied by any process of direct calculation.
rithm of
Not
arrived at through
can only bo
most deterrent kind
a liigh integer.
it is
in continuation of Burckhardt's has
and
similar to
of a
recently been constructed for the fourth million
Glaisher, F.R.S.,
who
eighth, and ninth millions
were
and Rosenberg, and pubhshed
The
in
is
in
com-
factors for the seventh,
calculated previously
by Dase
1862-05, and the
The
said to exist in manuscript.
formation of these tables
by J.^Y. L.
now engaged
I believe is also
pleting the fifth and sixth millions.
is
loga-
the tentative process of successive divisions
the subject of
A table
These
prime factors.
so its
by the prime numbers, an operation
million
The
a number, for instance, may be computed by a
formula.
when
absolutely
tenth
history of the
both instructive and interesting.*
As, however, such tables are necessarily expensive to purchase, and not very accessible in any other
of persons,
way to the majority
seemed to me that a small portion
it
would form a useful accompaniment
I have, accordingly, introduced the first eleven
ardt's tables,
which
integers nearly.
give the least factors of
Each double page
* Sec " Factor Table for the Fourth
London: Taylor and Francis. 1880.
Pt. IV.,
and Nature, No. 542,
p. 402.
of
them
to the present volume.
pages of Burckhtlie first
100,000
of the table here pi-iiUed is
JifiJltoii."
Also Camh.
By James
J'hil. Soc.
Glaisher, F.R.S.
Proc, Vol. TIL,
an exact rcpi'otluctiuii, iu
I)ai^n:
It
may
tal)le
ji
l)e
Hiii^^e
;i
(|iiar(o
work.
noticed licro that Prof. Lebesquo constructed
to a))out
extent, on
tliis
of seven,
n)ulti|)les
))y
all l)iit tlie tyi'e, of
of JJm-c'kluirdt's great
and
about one-sixtli.*
tlius
tlio
re<lucins^^
])lan
tlie
of omlttinj^ tlio
of tlie tabic
size
]3ut a small calculation
re(|uired iu
is
using the table which counterbalances the advantage so gained.
The
values of the
Gamma-Function, pages 30 and 31 have
,
been taken from Legendre's table in his ^'Excrciccs de Calcnl
Tome
Infnjral,''
Volume, but
all
I.
it is
The
the numerical tables of
In addition to
treatises
table belongs to Part II. of
tlie
Volume
and Lefebure de Fourcy
;
I.
in the
autliors already
have been consulted
Higher Algebra
tliis
placed here for the convenience of having
same
named,
section.
tlie
Snowball's Trigonometry
;
following
—Algebras, by AYood, Bourdon,
;
Salmon's
the Geometrical Exercises in Potts's Euclid
;
and Geometrical Conies by Taylor, Jackson, and Renshaw.
Articles 2G0, 431, o69,
are original.
The
latter
and very nearly
all
the examples,
have been framed with great care, in
order that they might illustrate the propositions as completely
as possible.
G. S. C.
Hadi.ry,
^Iiiii>i,i;.sKx
;
AIu)/ 23, 18ba.
* "Tables divcrscs pour
premiers."
la
decomposition des nombrca en leura HietLurs
Par V. A. Lebesque.
Paris.
b
18G-ii.
EIUiATA.
Art. 13,
——
TABLE OF CONTEXTS.
PAUT
I.
SECTION I.— MATHEMATICAL TABLES.
The
Introduction.
C. G. S. Systkm of Units
Notation and Definitions of Units...
Physical Constants and Formulas
Table
I.
II.
III.
Pap,
—
...
...
...
...
1
...
...
...
2
— English Measures and Equivalents in C. G. Units
— Pressure of Aqueous Vapour at different temperatures
S.
4
4
— Wave lengths
and Wave frequency for the principal'
lines of the Spectrum ...
...
...
...
...
CoeffiTheir Densities
The Principal Metals
Expancients of Elasticity, Rigidity, and Tenacity
Specific Heat
Conductivity
Rato
sion by Heat
—
lY.
4
;
;
;
;
Sound
of conduction of
Resistance
The Planets
V.
...
;
Electro-magnetic Specific
...
...
of Orbits
...
— Powers and Logarithms of
TT
and
e
...
...
...
...
...
5
...
...
...
G
...
6
Vll.^Square and Cube Roots of the Integers 1 to .30
Logarithms
of
VIII. — Common and Hyperbolic
Prime numbers from 1 to 100...
...
...
IX.
—
Factor Table
Explanation of the
The Least Factors
X.
Talilo
..
...
numbers from
Values OF THE Gamma-Function
of all
SECTION
5
...
— Their Dimensions, ^Masses, Densities,
and Elements
VI.
;
...
G
...
7
to 00000...
S
...
1
the
30
IT.— ALriEBRA.
x.^of
Ailiilo
Factors
...
...
...
...
...
Newton's Rule for expanding a Binniuial
Multiplication and Division
...
...
Indices
...
Highest Common Factor
...
...
...
...
...
1
...
...
...
12
...
...
...
2S
...
...
...
20
30
CONTENTS.
xii
No.
of
Artu-lo
33
Lowest Common Multiple
—
Evolution
Square Root and Culic Root
...
Useful Transformations
...
...
...
...
...
...
...
...
IN
ONE Unknown Quantity.— Exam
Maxima and Minima by
58
59
...
•••
•••
•••
Duplicate and Triplicate Ratios
...
...
...
...
Compound Ratios
/.•
Variation
...
...
^^
70
Theorem
...
^2
...
...
...
...
...
••
'^
...
...
•••
•••
••
•••
7G
Arithmetical Progrkssion ...
Geometrical Progression ...
Haumonical Progression
Permutations AND Combinations
Surds
SO
54
rr,i:s
a Quadi-atic lv[uation
Simultaneous Equations and Exam it, i:a
...
...
Ratio AND Proportion
The
38
45
Quadratic Equations
Theory OP Quadratic Expressions
Equations
3.5
...
...
...
...
•-•
...
...
...
...
...
...
...
...
79
83
87
94
108
...
Simplification of -/a +
Simplification of
Va-^
\/fe
and Va-^\/h
121
124
V~h
Binomial Theorem
Multinomial Theorem
125
137
142
149
160
...
•••
Logarithms ...
Exponential Theorem
...
...
Continued Fractions and Convergents ...
...
...
General Theory of same
To convert a Series into a Continued Fraction ...
A Continued Fraction with Recurring Quotients
•.•
...
...
...
Indeterminate Equations ...
To reduce A Quadratic Surd to a Contini-kd Fkaction
...
To form high Convergents rapidly
...
...
...
...
General Theory
...
...
...
...
167
182
186
188
195
197
...
...
199
Equations
...
...
...
...
—
Special cases in the Solution of Simultaneous Equations,
..
211
Method by Indeterminate Multipliers
^liscellancous Equations and Solutions
...
...
...
21.S
...
...
...
"J
On Symmetrical
...
...
...
...
iM'.l
...
...
...
...
2"J;?
...
...
...
...
...
...
...
...
...
...
...
...
2:5;»
...
...
...
...
2li>
Imaginahy Expressions
E.xpressions
...
...
^IeT1K)D OF InDETEKMINATI; CoEl'FiiTlN
Method
OK Proof BY
I
NDiCTioN
IS
...
—
F(tuR Cases
Parfial FuACTiONS.
CONVEHGENCY AND J')lVEK(iENCY OF SeKIES
General Theorem of ^
(a:)
...
11
'I'.Vl
233
235
U
CONTENTS.
XI
N... <.f
Expansion of
a
FKAnmN
-1-H
...
...
...
...
...
...
...
...
...
...
...
...
Iiiia^Miiary Koots...
...
...
...
258
203
SlMMATIOX OF SkRIKS BY THE ^[Ernon OF DlFFIMiKNCE.S
...
...
Interpolation of a term
...
...
...
2r,t
...
...
2t;7
DiRKCT Fact(M{ial Skrik.s
Invkusk Factorial Serifs
...
...
...
2(58
270
272
271
Ui;cLKi:iN(i
Skhiks
-•'''I
Tho General Term
Case of Quailratic Fuctiir with
Lagrann^e's Rule
...
...
Su.MMATiON
IJY
...
Paktial Fractions
CoMi'OsiTE Factorial Series
Miscellaneous Series
—
...
...
...
...
...
...
...
...
...
...
...
...
...
•••
Sams of tho Powers of the Natural Numbers
Suraof (i+(a + J)r+(a + 2(Z);-Hctc
...
...
Sum of n'' — n (7i — l)''4-&c. ...
...
...
POLYOONAL Nu.MI!ERS ...
...
...
FuiURATE Numbers
Hypergeometrical Series
Proof that
Interest
c" is
...
incommensurable
...
...
••
-">7
27G
27'J
...
...
2H.5
287
289
21>1
...
...
...
...
21*5
••.
•••
•••
•••
2'JG
•••
•••
••
'^^^2
Annuities
PROr.AniLITIES...
Inequalities
...
...
...
Arithmetic ^Ican > Geometric 'Mean
Arithmetic Mean of ?h*'' powers > m^^ power of A.
^l.
...
...
-..
Scales OF Notation ...
Theorem concerning Sam or Difference of Digits
Theory OF NuMRERS
Highest Power
^*^^
...
330
332
...
334
•••
•^•i2
•••
...
...
...
3i7
3^9
...
...
305
Format's Theorem
...
...
...
....
...
...
3r)9
Wilson's Theorem
...
...
...
...
...
••
...
...
...
...
...
371
374
of a
Prime
Number
(livisil.lebv2» + l
Divi.sors of a
S,
ji
contained in \m^
380
SECTION TIT.^THHOKY OF EQX'ATTON.S.
...
...
Factors OF AN EyuATiox
To compute /(a) numcricallv
...
Di.scriminati(m of Roots
...
...
-..
•••
...
•..
...
...
4i»3
of/ (.}•) ...
To remove an assigned term
To transform an equation ...
DEiiiVED Functions
Ei^TAL Roots OF an Equation
Pra-tical Rule
...
^O'J
...
..
-..
••
...
...
...
...
...
...
..
••
410
424
428
430
...
...
••
•••
4.32
Descartes' Rule OF SicNs
The
'^'"^
•^^''
.
CONTEXTS.
XIV
No.
of
Article.
448
Limits OF THE Roots
Newton's Method
Rolle'fi Tlieorem
Newton's Method OF Divisors
RECiPROC-\Ti
...
...
...
...
...
Equations
Binomial Equations
=
Solution of .^"±1
Cubic Equ.viions
Cardan
s
bj Do Moivre's Tlicorcm
...
...
...
...
...
...
...
...
...
...
...
...
Method
Trigonometrical Method
452
454
459
4GG
472
480
483
484
489
—
Biquadratic Equ.ations
Descartes' Solution
Ferrai'i's Solution
...
...
...
...
...
...
...
...
...
...
...
...
492
496
499
502
Euler's Solution
Commensurable Roots
Incommensurable Roots
—
Sturm's Theorem
Fourier's Theoi-em
...
...
...
...
•••
...
...
...
...
...
...
...
...
...
...
...
518
525
527
...
...
...
528
Lagrange's Method of Approximation
Newton's Method
of
506
...
...
Approximation
Fourier's Limitation to the same
...
Newton's Rule for the Limits of the Roots
5.30
Theorem
Horner's Method
532
Sylvester's
..
...
...
...
—
Symmetrical Functions of the Roots of an Equation
...
...
...
^ms of the Powei'S of the Roots ...
...
Symmetrical Functions not Powers of the Roots...
The Equation whose Roots are the Squares of the
Differences of the Roots of a given Equation
Sum of the m"' Powers of the Roots of a Quadratic
Equation
Approximation
534
538
545
Root of an Equation through the
Sums of the Powers of the Roots
E-KPANsioN of an Implicit Function of a;
Determinants
533
to the
548
551
—
Definitions
55-i
...
Analysis of a Deterniinaut ...
Synthesis of a Detcrminiuit
556
564
568
569
Product of two Determinants of the 7i"' Onlei
Synimetrioal Determinants ..
Reciprocal Determinants ...
Partial and Ci>niplcnu'ntary DcierininnntH
570
574
575
576
General Tlieory
To
raise the
Order
of a
Dotonninant
CONTENTS.
XV
\u.
Aril.
Theorem
of a Partial Ik-ciprocal Dclonuiiiunt
...
...
Product of DiU'ercuce.s of /i Quantitius
...
...
...
Product of Squares of UiiTereiices of samo
...
...
Rational Algebraic Fraction expressed as a Ucleniiinant
Eli.mi.n.mio.n
—
Solution of Linear Ecjuations
Orthogonal Transformation
Theorem
of the
?i.—'2"'
Power
llu;iii:sT
578
579
5sl
...
...
...
r»s2
...
...
...
...
r)Sl.
of a Deteniiinant...
...
r).s5
5SG
Method
...
...
...
...
...
...
...
...
...
...
...
587
5H8
...
...
5'J3
^lethod by Symmetrical Function.^
Eliminatiox BY
I.'.
...
Bezout's Method of Elimination
Sylvester's Dialytic
..f
.')77
Cu.MMON Factou
SECTION IV.— PLANE TRIGONOMETRY,
Angular Measuremext
...
...
...
...
...
Trigonometrical Ratios
...
...
...
...
...
Formula) involving one Anglo
...
...
...
Formula; invoMng two Angles and Multii)lc Angles
Formula-" involving three Angles ...
...
...
Ratios OF 45°, G0=, 15°, 18°, <fcc
Properties of THE Triangle
The s Formula) for sin ^^, &c.
The Triangle and Circle
Solution of Triangles
...
...
...
...
...
...
G13
...
G27
...
G7-1
GOO
700
...
—
...
...
...
...
...
...
...
...
...
...
...
...
Quadrilateral in a Circle
Bi.sector of the Side of a
718
720
859
733
Triangle
738
Bisector of the Angle of a Triangle
Perpendicular on the Base of a Triangle
Regular Polygon AND Cikcle
Subsidiary Angles
70-i
709
—
Right-angled Triangles
...
Scalene Triangles.
Three cases
Examples on the same
...
COO
GOG
...
...
...
...
...
...
71-2
711i
74G
...
...
...
...
...
749
Limits of Ratios
753
De
75G
758
Moivre's Theorem
Expansion of cos ?i0, &c. in ])owcrs of sin and cosO ...
Expansion of sinO and cos in jiowers of
...
...
Expansion of cos"
ami shi" U in cosines or sines of
multiples of
(^
...
...
...
...
...
Expansion of cos 7i0 and sin7j8 in powers of biuO
Expansion of cosn9 and sin «0 in powers of cos
Expansion of cos nO in descending powers of cos
Sin a -f- c sin (a + /3) + <tc., and similar series
7G4
...
772
...
775
...
...
779
780
783
CONTENTS.
XVI
No.
of
Artiol.;.
Gregory's Scries for
powers of tauO
calculation of
71>1
...
..
...
792
...
...
...
...
795
+ a.). — Series
Sina;
=
Sum
of sines or cosines of
(a;
for
...
...
...
Angles in A. P.
...
...
a;
Exi^ansion of the sine and cosine in Factors
and cosnf expanded in Factors
in Factors involving (^
Sin(? and cos
e' — 2cos6 + e"' expanded in Factors
De
Sin
...
...
is
sin
...
TT
tlio
Proof that
?i
..
iucomracnsural)lc
Formulas for
TT
in
1/0
Moivre's Property of the Circle
Cotes's Properties
Additional Formulae
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
Properties of a Right-angled Triangle
...
...
...
Properties of any Triangle...
...
...
...
...
Area
...
...
...
...
of a Triangle
...
...
between a Triangle and the Inscribed,
Escribed, and Circumscribed Circles
...
...
Other Relations between the Sides and Angles of a Triangle
790
800
807
808
815
817
819
821
823
832
835
838
Relations
Examples
of the Solution of Triangles
...
...
...
841
850
859
SECTION v.—SPHERICAL TRIGONOMETRY.
Introductort Theorems
—
870
871
Definitions
Polar Triangle
Right-angled Triangles
Napier's Rules
—
881
Oblique-angled Triangles.
Formula) for cos a and cos A
The (S Formula) for siniJ, sinJa,
sin^
—
sin JB
sin a
_
sin b
cos 6 cos
(7
=
sin
sin
cot a sin
Napier's Formula)
895
—
Circumscril)cd Circles
898
900
—
Spherical Excess
of Sphei'ical
Cagnoli's
sin
89G
897
...
Inscribed and Escribed Circles
Area
882
884
894
i— cot ^1
Gauss's Formulas
Areas
...
c
Spherical Triangle and Circle
Si'UERiCAL
<tc.
Polygon
Theorem ...
Theorem
Lhuillior's
...
902
903
904
905
XV 11
CONTKNTS.
N... o<
ArtirUi.
PoLTiiEnnoNs
The
...
...
...
...
...
...
...
RcgulaT Solids
Tlio Aiii^'lo hot woiMiAdjiuvnt Faces
Kadi i of Iii.siTil)L'd and Circiim.scriheil Spheres...
i'06
1>07
five
i'OO
...
'JlO
SECTION VI.— ELEMENTARY GEOMETRY.
Miscellaneous Propositions
—
...
...
...
920
at Buccussive surfaces
...
...
i.'2I
Reflection of a point at a single surface
do.
do.
Relations between the sides of a triangle, the segments
of the base,
and the
line
drawn from the vertex
...
i'22
...
923
024
Equilateral triangle Yli'C; P.'P + Pi>" + i'0"
Sum
of squares of sides of a quadrilateral
...
Locus of a point whose distances from given
points are in a given ratio
To
...
...
...
...
...
...
Locus of vertex
...
...
divide a triangle in a given ratio
Sides of triangle in given ratio.
Harmonic
division of base
lines or
...
...
...'
...
Triangle with Inscribed and Circum.scril}ud circles
TuE
TUE Tangencies
Tangents and cliord of contact, fty
To find any sub-multiple of a line
...
PRor.LE.Ms OF
Triangle and three concurrent lines
Inscribed and inscribed circles
=. u...
Three cases
;
—
...
...
.
...
92(3
930
032
933
935
037
0-48
...
950
...
951
&c.
...
...
953
...
...
...
954)
CoNSTULCTiox OK Tkiano.les
...
...
...
...
...
Locus of a point from which the tangents to two circles
have a constant ratio ..
...
...
...
...
CoLLiNEAR AND CoxcuKUENx Systems
900
Ni.N'E-PoixT Circle...
...
...
;
/?,
s
...
t/,
.
003
007
Triangle of constant species circum.scribcd or inscribed
to a triangle
Radical Axis
—
...
...
...
...
...
...
Of two Circles
Of three Circles
Inveksiox
\)77
l*8i
il'j7
—
Inversion of a point
loOU
...
...
...
...
...
do.
circle
...
...
...
...
...
lii(,»0
do.
right line
...
...
...
...
...
Iiil2
...
...
\u\C,
...
...
Iu21
Pole and Polar ...
...
...
...
...
Coaxal Circles
...
...
...
...
...
Centres and Axes of Similitude
Homologous and Anti-homuldgons pi )ini.s
—
do.
do.
chord.s
C
...
...
Iu37
...
...
Iu38
CONTENTS.
xvin
No. of
Article.
1U43
1045
1046
1049
1052
1058
1066
1075
1083
1087
1090
Constant product of anti-siniilitudc
Circle of similitude
Axes
of similitude of three circles
Gergonne's Theorem
Anharmoxic Ratio and Pencil
HoMOGUAi'Hic Systems of Points
Involution
...
Projection
...
On
Drawing
Perspective
...
Orthogonal Projection
Projections of the Sphere
—
Additional Tueorems
Squares of distances of
circle
P
from equidistant points on a
1094
1095
...
Squares of perpendiculars on radii, etc. ...
Polygon n-ith inscribed and circumscribed
of perpendiculars on sides, &c....
circle
Sum
1099
SECTION VII.— GEOMETRICAL CONICS.
"
—
Sections of the Cone
Defining property of Conic
PS
= ePM
1151
Fundamental Equation
Projection from Circle and Rectangular Hyperbola
Joint Properties of the Ellipse and Hypereola
—
Definitions
...
CS:CA:CX
F8 ±PS'=AA'
CS"
= AC ^ PC
^ QSP
be a tangent PSZ is a right angle
Tangent makes equal angles with focal distances
SZ bisects
If
PZ
Tangents of
focal
chord meet in directrix
CN.CT =AC'
as :PS =e
NG NC =^
:
PC'
:
117;i
QN = PC :AC
P^- AN. NA'
:
= L'C-
1174
:
JO'-
1 1
Cn.a = PC'
sy.s'Y'
PP =
1160
1162
1103
1164
1166
1167
1168
1169
1170
1171
1172
AC
Auxiliai-y Circle
J'N:
1156
1158
= PC'
1178
1179
1180
A(!
To draw two tangents
Tangents subtend equal angles
7t;
1177
at the focus
Ubl
—
TON TK NTS.
No. of
Asymptotic
ok Titr
PHOi'F.nTir.s
RN^-IW =
TTviM;i;itor,\
—
11^:^
IK"
rn.p>=n("
n-i-
...
('!:=
AC
J'l>
i-Mnillcl (o
i.s
Arti.l.'.
ll-<'
the
1I>^7
A.svni|.l..t.-
Qh'=r-
"^^
'
= n an.i (>v = <iV
Qn.Qx= Plr= UV'--(iVArn.TK = cs'
n^i^
rrj
ll'-'l
•••
Joint Propkrtiks of Ellipre and HYi-r.Kr.oLA rksi'mkh.
jniATK DiAMETKKS
ii-'"-i
^
Cox-
—
QV-.FV.VF'^CD'' :Cr^
11^.'^
rF.cn = AC .no
rF.ra = nc" ami ff-fo'^zAC
nin.
PG.FG' =
ii
Diameter bisects parallel cliord.s ...
Supplemental chord.s
Diameters arc mutually conjugate
...
...
...
(72)^
l-"-5
OQ
.
Oq
= ^1 dfc
l--^0.-S
^F}r-
=
FC-
\
i7
1-^-5
OQ' O'i
.
SR:QL =
'2'
1
I'-^H
7>C'''
= cn'
FS.PS'
11'''^
1201
CV.CT=r!F^
CN=dR, CB = pN
CN^ ± CK' = A C\ FF-
CP±
''•'»
ll'-'7
Cr)'
=
CD-
:
121t
CF-
1-'1'»
e
•••
••
1-''
Properties of Parabola deduced from the Ellipse
...
121'.»
•.•
1'2"20
Director Circle
...
•••
...
••
The PAHAnoLA
Defining property I',S' = P.!/Latus Rectum = 4JN
If FZ be a tangent, I'SZ is a right angle
Tangent
bisects Z ,ST.U
ST=SF =
1
l--'^
focal fliord intersect at right anglos
...
...
...
...
.••
in
l'--*^
••
A^ = AT
NG = 2AS
FN"- = iAS.AN
SA:SY:SF
To draw two
l--^27
l^^*^
l^^O
1-'^^
tanj^ents
SQ:SO:SQ'
Z
OSQ = OSQ'
DiAMF.TK.nS
...
...
The diameter
-•^3
l—'i
fe';^ .If
Sa
Tangents of a
directrix
and
^'-^'^
••
l--^-^
and QOQ'
=
...
bi.sects itanilh'l
{ QSQ'
...
chords
1
^'-i-
...
••
••
l-'^'^
...
...
...
I'-^^S
XX
CONTENTS.
No. of
Article.
QP=4P.9.Pr
0(2.0q :0(/
:
Oq
1239
1242
1244
1245
1252
1253
1254
1258
=rS iFS
Pai-abola two-thirds of circnmscribing parallelogram
Methods OF Drawing A Cuxic
To find the axes and centre
To construct a conic from the conjugate diameters
Circle OP Curvature
Chord
of curvature
=
QV^
-^
PV ult
CT)''
.
1
1
c
,
—
bemi-cliords
-—
ot curvature,
•
,
C-P
OD'-
CL^
rf^,=r,
-77-r
PJf
In Parabola, Focal chord of curvature
do.
Common
Kadius
of curvature
= 2SP"'
-^
find the centre of curvature
MiSCKLLAM;OUS TlIKOKLMS
...
12G0
...
ST
chords of a circle and conies are equally
clined to the axis
To
1259
AC
= 4SP
12G1
i
1263
1265
1267
.
INDEX TO TROPOSITIONS OF EUCLID
REFERRED TO
Tho
references to Euclid are
made
in
IX THIS
Koinan and ^Vrabic numerals
BOOK
I.
4.
—
Triaui^'los arc
an<^le of
I.
5.
1.
0.
I.
8.
;
e.g.
(VI. 19).
T.
equal and similar
if
two
sides
and the included
each are equal each to each.
— The angles at the base of an isosceles triangle are equal.
— The converse of
sides of
— Triangles are equal and similar
5.
if
ecjual
I.
WORK.
IT).
eacli arc
tlie tliroe
each to each.
— The exterior angle
of a triangle is grojiter than the
interior
and opposite.
I.
20.
I.
26.
— Two sides of a triangle are greater than the third.
— Triangles are equal and similar two angles and
if
ponding side
I.
27.
— Two straight
one corres-
of each are equal each to each.
lines are parallel
if
tlicy
make equal
alternate
angles with a third line.
I.
29.
I.
32.
— The converse of 27.
— The exterior
angle of a triangle is cqiial to tho two interior
and opposite; and tlic three angles of a triangle are equal
to two right angles.
C'riK.
1.— The interior angles of a ]-)olygon of n sides
=
(«-2)7r.
— The exterior angles =
— Parallelograms or triangles upon
C'oK. 2.
I.
35 to 38.
27r.
bases and between tho
I.
la.— The conq)lements
of
same
tlie
same or equal
parallels are equal.
the parallelograms about the diameter
of a parallelogram are c([ual.
T.
M — Tlio
square on the hypotenuse of a right-angled triangle
equal to the scpiares <m the other sides.
I.
48.
— The converse of 47.
is
—
INDEX TO PTtOrOSITIOXS OF EUCLID
XXll
BOOK
II.
4.— If
II.
arc the two parts of a riglit line, {a +
a, h
and
If a right line be bisected,
externally, into
II. 5
and
6.
iy
=
a" -\-1nh-\-h-.
also divided, internally or
two nnequal segments, then
— The rectangle of the
unequal segments
on half the
difference of the squares
II.
;
is
line,
eqnal to the
and on the
line
between the points of section; or (a + i) (a-h) = a? — l/.
The squares on the same unequal segments are together
9 and 10.
double the squares on the other parts or
—
;
II.
11.
— To divide
of the
a right line into two parts so that the rectangle
whole
line
and one part may be equal to the square on
the other part,
II.
12 and 13.
— The square on
sum
the base of a triangle
on the two sides
of the squares
is
equal to the
or mimis (as the
lolus
under
upon it
vertical angle is obtuse or acute), twice the rectangle
either of those sides,
or
a-
and the projection
= h- + c''-21jccosA
(702).
BOOK
III.
3.
—
If
III. 20.
III.
a diameter of a circle bisects a chord,
it
:
of the other
it is
perpendicular to
and conversely.
— The angle at the centre of
a circle
is
twice the angle at the
circumference on the same arc.
III. 21.
III. 22.
ITT. 31.
111. 32.
— Angles in the same segment of a circle are equal.
— The opposite angles of a quadrilateral inscribed in
—
— The angle betAveen a tangent and a chord
contact
111.
a circle
are together equal to two right angles.
The angle in a semicircle is a right angle.
33 and 34.
is
from the pcint of
equal to the angle in the alternate segment.
— To describe or to
cut
of
ti
segment
of a circle
which
shall contain a given angle.
III.
35 and 30.— The rectangle of the segments of any chord of a
circle drawn through an inta'ual or external point is eiiual
to the square of the semi-chord perpendicular to the
diameter through the internal point, or to the square of the
III.
37.— The converse
tangent from the external point.
tlic lino
of 3G.
If the rectangle be equal to the .scpiare,
which meets the
circle touches
it.
iJKKRI.'KKn T(i IN
lUJUK
IV.
IV.
IV.
IV.
IV.
1\'.
WoKK
IV.
— To
a
a trian^k' of yivcii form
— To describu same about a
— To inscribe a circle in a triangle.
— To describe a circle about a triangle.
10. — To construct two-iiftlis of a right angle.
— To construct a regular pentagon.
2.
in
insoi-ilif
3.
i-irclc
circle.
tlic
4.
5.
11.
BOOK
VI.
THIS
VI.
— Triangles and parallelograms of
proportional
their bases.
—A right
parallel to the side of
1.
the
same
altitude
arc
to
VI.
2.
line
sides proportionally
VI
3 and A.
;
a triangle cuts the other
and conversely.
— The bisector of the
interior or exterior vortical
angle of
a triangle divides the base into segments proportional to
the sides.
VI.
4.
VI.
5.
VI.
0.
Eipiiangular triangles have their sides proportional honiologously.
— Tlie converse of
— Two triangles are equiangular
-i.
and the
VI.
sides about
— Two triangles are equiangular
7.
if
they have two angles equal,
them proportional.
and the sides about two
if
they have two angles equal
other angles proportional,
provided
that the third angles are both greater than, both less than,
or both equal to a right angle.
VI
6.
— A right-angled triangle
is
divided by the perpendicular from
the right angle upon the hypotenuse into triangles similar
to itself.
VI 11 and
l.'i.
— Equal
lyaralldoijrams,
or trianjlcs which
have two
angles equal, have the sides about those angles reciprocally
jiroportional
jiDi-tion,
VI.
ll*.
— Similar
;
and conversely,
if
the sides are in
tliis i)ro-
the figures are eciual.
triangles are in the duplicate latio of their
homo-
logous sides.
VI. 2".
VI. 23.
— Likewise similar
— E(|uiangular parallelogi-auis are in
jjiilyg'ons.
the ratio compoundetl of
the ratios of their sides.
VI. B.
— The rectangle of the sides of a (riangle
is
ccjual to the s(]uare
of the bisector of the vertical angle i>lus the rectangle of
the segments of the base.
INDEX TO PROPOSITIONS OF EUCLID.
XXIV
VI. C.
— The rectangle of
the sides of a triangle is equal to tlie rectangle under the perpendicular from the vertex on the
base and the diameter of the circumscribing circle.
VI. D.
— Ptolemy's
Theorem.
The rectangle
quadrilateral inscribed in a circle
of the diagonals of a
is
equal to both the
rectangles under the opposite sides.
BOOK
XI.
4.
—A
XI.
5.
— The converse of
XI.
right line perpendicular to two others at their point of
intersection is perpendicular to their plane.
If the first line is also perpendicular to a
4.
fourth at the same point, that fourth line and the other
two are in the same plane.
XI.
6.
XI.
8.
XI.
20.
perpendicular
same plane are
—Right
perpendicular
a plane, the
— one of two
other
angle are
— Any two three plane angles containing a
greater than the
than
angle are together
— The plane angles any
lines
parallel.
to tlie
parallel lines is
If
to
is also.
of
solid
third.
XI. 21.
of
four r'nAit ano'les.
solid
less
TABLE OF CONTENTS.
PART
II.
SECTION VIIT.— DIFFERENTIAL CALCULUS.
Introduction
...
...
...
•••
•
••
...
...
...
...
1405
...
...
...
...
1407
...
Successive differentiation
Infinitesimals.
Differentials
No. of
Article.
1400
Differentiation.
1411-21
Methods
SoccEssivR Differentiation'
—
Leibnitz's theorem
Derivatives of the
...
...
?ith
...
order (see Index)
...
...
...
...
14(30
14G1-71
1480
1483
1488
Partial Differentiation
...
...
...
Theory of Operations
Distributive, Commutative, and Index
—
...
...
laws...
...
Expansion of Explicit Functions
Taylor's and Maclaurin's theorems
...
...
...1500,1507
Symbolic forms of the same
...
...
...
...
f(x + h,y + k),&c
Methods of expansion by indeterminate
1520-3
1512-4
coefficients.
•••
...
...
...
••
Four rules...
Method by Maclaurin's theorem
...
...
Arbogast's method of expanding ^ (2)
...
...
...
...
Bernoulli's numbers
Stirling and Boole
Expansions of ^ (j; + /0 — ^ (•'')•
•
Expansions of Lmplicit Functions
1527-31-
1524
15:3i>
1.">:'.".'
.
—
151G-7
Lagrange's, Laplace's, and Burmann's theorems, 1552, 1550-03
Cayley's series for
--—
Abel's series for if>{x-\-a)
...
...
...
••
...
...
...
.••
...
•••
••
•
...
...
...
...
...
1-'"-
1580
Indeterminate Forms
Jacodians
1555
...
...
Modulus of transformation
1^*^^
lOUt
CONTENTS.
XXVI
No. of
Article.
1620
QiAViics
Euler's theorem
...
..
...
...
...
1621
...
...
...
...
...
...
...
1626-30
1635-45
1642
1648-52
1653-5
...
...
...
...
1700
...
...
...
...
1725
...
...
...
1737
...
...
...
1760
...
...
...
...
...
...
...
...
...
...
...
1794
1799
1813
1815
...
...
...
...
1830
...
...
...
...
1841
Three or more independent variables...
..
...
...
...
Eliruinant, Discriminant, Iuvariant,Covariant, Hes.sian
Theorems concerning discriminants
Notation ^ = 6c-/, &c
Invariants
Cogredients and Emanents
Implicit Functions
—
One independent
Two
variable
independent variables
...
w independent variables
Change of the Independent Variable
Linear transformation
...
Orthogonal transformation
Contragredient and Contravariant
Notation z„=p,&:,c.jq,r^s,t...
—
Maxima and Minima
One independent
Two
variable
independent variables
Discriminating cubic
Method
of
...
...
...
...
undetermined multipliers
Continuous maxima and minima
1852
1849
...
...
...
1862
...
...
...
1866
SECTION IX.— INTEGRAL CALCULUS.
1900
Introduction
Multiple Integrals
Methods of Integration
By
1905
—
Parts,
Substitution,
Division,
Standard Integrals
...
...
Various Indefinite Integrals
Circular functions
—
...
Rationalization,
...
...
...
1908-19
...
...
...
...
1921
...
...
...
...
...
...
Partial fractions. Infinite sei'ies
Exponential and logarithmic functions
Algebraic functions
...
...
...
...
Integration by rationalization...
...
...
...
...
...
...
...
...
...
...
...
...
Integrals reducible to Elliptic integrals
Elliptic integrals
approximated to
Successive Integration
Hyperbolic Functions cosh
Inverse relations
...
...
a-,
sinh-i;,
...
Geometrical meaning of tanh
tanh.r
...
...
...
•
<S
...
...
...
...
...
...
Logarithm of an imaginary quantity
1954
1998
2007
2110
2121-47
2127
2148
2180
2210
2213
2214
CONTENTS.
XXVll
No. of
Article.
DkFINITK iNlKfiRALS
Summation
'I'liroKKMS
of series
...
...
...
KESrECTINU LiMITS OF iMECiliATION
Methops ok evaluating Definite
Change
in the
method
tliis
...
...
...
iNTEiiBAi-s {Eiyitt rules)
Differentiation under the sign of Integration
Integration by
...
...
order of integration
...
...
...
...
...
...
...
...
...
...
...
2261
2262
...
...
22'.'4
...
...
2317
...
...
2341
...
23i>l
...
2451
Approximate Integration hy Beunoulm's Series
The Integrals j;(Z,m) and r(»)
logr(l + M) in a converging series ...
...
Numerical calculation of r (x)
...
...
Integration of Algebraic Forms
Integration of Logarithmic and Exponential
Integration of Circular Forms
2'J3U
2233
2245
2253
2258
...
2280
F(>i;.m.s
...
...
Integration of Circular Logarithmic and Exponential Forms
Miscellaneous Theorems
Frullani's, Poisson'.'^, Abel's,
Kummer's, and Cauchy's
2700-13
formula)
Finite Variation of a Parameter
Fourier's formula
The Function \P{.c)
Summation of
4/ (.«)
2571
—
...
series
2714
...
...
...
...
...
...
...
...
...
...
2726
2743
2757
...
27»)<>
by the function
«/'
(.t)
as a definite integral independent of \p{l)
Nu.merical Calculation of log r{x)
2771
Change of the Variables
Multiple Integrals-
2774
in a
Definite JMl'ltiple I.ntegral
Expansions of Functions in Converging Seriks
—
Derivatives of the nth order
...
...
...
...
Miscellaneous expansions
...
...
...
...
...
...
...
...
Legendre's function
Expansion of Functions
X„
in
Trigonometrical Series
Approximate Integration
^lethods by Simp.son, Cotes, and Gauss
...
...
...
2852
2911
2936
2955
2991
2992-7
SKCTION X.— CALCULUS OF VARIATIONS.
Functions of one Independent Variable
Particular cases...
...
...
maxima and minima
Geometrical applications
...
...
...
...
...
3028
3033
3045
••
••
...
...
...
3051
...
...
...
...
3069
...
...
...
...
3070
...
Other exceptional cases
Functions of two Dependent Variables
Relative
...
CONTENTS.
XXVlll
No. of
Article.
Functions of two Independent Variables
Geometrical applications
Appendix
—
...
...
...
...
...
...
...
Oil the general object of the Calculus of Variations...
Successive variation
...
Immediate integrability
SECTION
...
...
...
...
...
...
...
...
XL— DIFFERENTIAL
...
...
..
Definitions and Rules...
...
...
...
...
...
...
...
...
First Order Linear Equations
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
Integrating factor for il/dc+iVfZi/
Riccati's Equation
...
=
...
First Order Non-linear Equations
Solution by factors
...
Solution by difFei'entiation
Higher Order Linear Equations
Linear Equations with Constant Coefficients
Higher Order Non-linear Equations
...
...
...
...
...
...
...
Depression of Order by Unity...
Exact Differential Equations
...
...
...
...
Miscellaneous Methods
...
Approximate solution of Differential Equations by
Taylor's theoi'em
...
...
...
...
...
Singular Solutions OF Higher Order Equations
Equations with more than two Variables...
order P. D. Equations
first
Non-linear
order P. D. Equations
first
Non-linear
first
of Reciprocity
Symbolic Methods
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
Solution OF Linear Differential Equations BY Series
...
Solution by Definite Integrals
...
...
...
...
more than two Independent Variables
...
Differential Resolvents of Algebraic Eqlaitons
P. D. Equations with
3236
3237
3238
3251
3262
3270
3276
3289
3320
3340
3380
3381
3399
order P. D. Equations with more
than two independent variables
Second Order P. D. Equations
Law
...
3150
3158
3168
3184
3192
3214
3221
3222
3.301
Simultaneous Equations with one Independent Variable...
Partial Differential Equations
Linear
3084
3087
3090
EQUATIONS.
Generation of Differential Equations
Singular Solutions
3075
3078
3409
3420
3446
3470
3604
3617
3629
3631
XXIX
CONTENTS.
Xo. of
Article.
SECTION
XII.
— CALCULUS
OF FINITE
DIFFERENCES.
'^7(^6
F0KMri,.K KOR FlKST AND uth UlKKKKKNCF.S
li/.^O
Expansion by factorials
...
...
...
•••
Gcnemting functions ...
The operations 1/, A, and
...
...
...
••
3732
...
...
...
•••
373-5
...
...
...
•••
3/.)7
...
...
...
...
...
...
...
...
...
...
...
...
•••
...
Herscbel's theorem
A
(/.r
...
theorem conjugate to Machiuriu's
Interpoi-ation
Lagrange's interpolation formula
MhXHANlCAL QUADRATURK
Cotes's and Gauss's formula3
...
Laplace's formula
3759
37G2
370H
^^^72
3777
o/lH
'^781
Summation of Series
Approximate Summation
3820
SECTION XIII.— PLANE COORDINATE GEOMETRY.
Systems of Coordinates
—
Cartesian, Polar, Trilinear, Areal, Tangential, and
4001-28
Intercept Coordinates
ANALYTICAL CONICS IN CARTESIAN COORDINATES.
^032
4048
Lengths and Areas
Transformation of Coordinates
The Right Line
Equations of two or more right
4(>
lines
General Methods
Poles and Polars
^^^G
4U»1
The Circle
Co-axal circles
The Parabola
The Ellipse and Hyperbola
Right
line
and
ellipse
Polar equations of the conic
Conjugate diameters
The
4110
4114
4124
Determination of various angles
Hyperrola referred to its Asymptotes
The rectangular hyperbola
'^^^^
4250
4310
433t.
4o4b
4375
4387
439-
.
CONTENTS.
XXX
No.
of
Article.
4400
4402
4417
4430
4445
4464
4487
4498
4522
The General Equation
The ellipse and hyperbola
Invariants of the conic
The parabola
Method without transformation
of the axes
.
.
Rules for the analysis of the general equation
Right line and conic with the general equation
Intercept equation of a conic
Similar Conics
Circle of Curvature
...
—
4527
Contact of Conics
4550
CoNKOCAL Conics
ANALYTICAL CONICS IN TRILINEAR COORDINATES.
4601
The Right Line
Equations of particular
and coordinate
lines
ratios
of particular points in the trigon
Anharmonic Ratio
The complete quadrilateral
The General Equation of a Conic
Director- Circle
...
Particular Conics
Conic circumscribing the trigon
Inscribed conic of the trigon
...
Inscribed circle of the trigon
...
General equation of the
circle...
Nine-point circle
Triplicate-ratio circle
...
Seven-point circle
Conic and Self-conjugate Triangle...
On lines passing through imaginary points
Carnot's, Pascal's, and Brianchon's
The Conic referred
Contact
—
Related conics
...
Theorems
to two Tangents and the
Chord of
...
Anharmonic Pencils of Conics
Construction of Conics
Newton's method of generating a conic
Maclaurin's method of generating a conic
The Method or Reciprocal
Tangkntial Coordinates
Abridged notation
Polars...
4628
4648
4652
4656
4693
4697
4724
4739
4747
4751
4754
47546
4754e
4755
4761
4778-83
...
4803
4809
4822
4829
4830
4844
4870
4907
XXXI
CONTKNTS.
No. of
Article.
On
Tin:
1m kksection
ok two
Conics—
Geonictricftl^mcanin^ of v/(- 1)
The Methop
...
'I'-^l^'
...
'^•'-l
of Pkojection
41K5G
Invariants anp Covakiants
To
find the foci of the general conic
...
...
..
5008
THEORY OF PLANE CURVES.
Tangent and Normal
Radius of Curvature and Evolute
Inverse Problem and Intrinsic Equation
Asymptotes
Asymptotic curves
Singularities of Curves
...
Concavity and Convexity
-''l^'*^
-"il-^^
51G0
51G7
r.l72
—
...
...
...
5174'
Points of inflexion, multiple points, &c.
...
...
5176-87
...
...
Contact of Curves
Envelopes
Integrals of Curves and Areas
Inverse Curves
Pedal Curves
Roulettes
...
...
...
•••
...
...
...
...
5212
...
...
5220
5229
5230-5
...
...
•••
5239
...
...
...
...
...
...
•••
...
...
...
•••
•••
•••
•••
Area, length, and radius of cui'vature...
The envelope
of a carried curve
Instantaneous centre
...
...
...
...
••
Holditch's theorem
...
...
...
...
•••
...
...
...
...
...
•••
•••
to the cycloid...
...
...
...
...
...
...
...
Epitrochoids and hypotrochoids
...
...
...
Trajectories
Curves of pursuit
•••
Quetelet's theorem
Transcendental and other Cuhves
—
Epicycloids and hypocycloids...
Catenary
5247
5-49
5250
Prolate and curtate cycloids
The
The
The
The
The
The
The
o243
5244
^24G
52-*8
Caustics
The cycloid
The companion
5188
5192
519G
...
5258
52G0
52G
GG
5273
Tractrix
5279
Syntractrix
5282
Logarithmic Curve
Equiangular Spiral
5284
5288
Spiral of Archimedes
5296
Hyperbolic or Reciprocal Spiral
5302
XXXn
CONTENTS.
No. of
Article.
The Involute of
tlie
Circle
...
TheCissoid
The Cassinian or Oval of Cassini
The
The
The
The
The
The
The
The
...
...
...
...
...
5313
5309
...
...
...
...
...
6317
...
...
...
...
..
5320
5327
Versiera (or Witch of Agnesi)
...
...
...
Quadratrix...
...
...
...
...
semi- cubical parabola
...
...
...
...
5359
folium of Descartes
...
...
...
...
5360
...
...
...
...
Lemniscate
Conchoid
Lima9on
...
...
Cartesian Oval
Linkages AND LiNKwoRK
Kempe's
...
five-bar linkage.
Eight cases
Reversor, Multiplicator, and Translator
Peaucellier's linkage
The
The
The
The
The
A
A
six-bar invertor
...
...
...
...
...
...
...
...
...
...
...
eight-bar double invertor
Quadruplane or Versor Invertor
...
Pentograph or Proportionator
...
...
Isoklinostat or Angle-divider
...
...
...
...
linkage for drawing an Ellipse
5335
5338
5341
5400
5401-5417
5407
...
5410
...
5419
...
5420
...
5422
...
5423
...
5425
...
5426
...
linkage for drawing a Lima9on, and also a bicir...
...
...
5427
for solving a cubic equation
...
...
5429
...
...
...
...
...
...
cular quartic
A linkage
On
-5306
...
...
...
three-bar motion in a plane
The Mechanical Integrator
The Plauimeter
...
5430
5450
5452
SECTION XIV.— SOLID COORDINATE GEOMETRY.
Systems of Coordinates
The Right Line
...
...
...
...
5501
5507
5545
5574
...
...
...
...
...
5582
...
...
...
...
...
...
...
...
...
...
The Plane
Transformation of Cooi;niNATES
...
...
The Sphere
The Radical Plane
Poles of similitude
Cymxdrical and Conical Surfaces
Circular Sections
...
...
...
...
...
5585
5587
5590
...
...
...
...
5596
...
...
...5590-5621
Ellipsoid, HvriiRUOLOiD, and Paraboloid
CONTENTS.
N... of
Article.
Centrai,
Qcapiuc Surfack—
Tangent and
diainotnil plaiu's...
Eccentric values of the coordinates
...
...
...
CoNKOCAL QuAi'urcs
Reciprocal and Enveloping Cones
Thk Genkral Equation of a Quadric
...
...
...
...
...
...
...
...
...
...
...
...
5026
5038
•''^•^0
56G4
5073
Reciprocal Polars
•'''""
Theory of Tortuous Curves
The Helix
General Theory of Surfaces
•'"^^•Jl
575i;
—
...
...
5780
...
...
5783
...
...
...
5806-9
...
-.
...
...
General equation of a surface ...
Tangent line and cone at a singular point
The
^^795
Indicatrix Conic
Eulei-'s
!•
and Meunier's theorems
Curvature of a surface...
...
Osculating plane of a line of curvatnri!
5826
5835
...
-..
...
.•
Integrals for Volumes and Surfaces
...
...
...
5837-48
^856
5871
...
...
•.
587i)
Geodesics
...
...
...
Invariants
Guldin's rules
...
...
...
5884
5903
5925-40
Centre of Mas.s
Moments and Products of Inertia
Momcntal ellipsoid
Momental ellipse
Integrals for
moments
of inertia
...
...
...
5953
5978
Perimeters, Areas, Volumes, Centres of Mass, and Moments
of Inertia of various Figures
—
Rectangular lamina and Right
The Circle
The Right Cone
Frustum of Cylinder
The Sphere
The Parabola
The Ellipse
Solid...
...
...
6015
<'019
0043
...
...
Fagnani's, Grinitli's, and
...
...
...
0048
6050
G067
0083
li;uid)ert's tlu'orenis
The Hyperbola
The Paraboloid
The Ellipsoid
Prolate and Oblate Spheroiils
...0088-0114
''llS
Ol"'^<3
<>11--
0152-05
;
rKKFACK TO VAUT
ArOLOGlES for the noii-completioii of
its
must form the
In
the
tliis
voliniie
due to friends and enquirers.
earlier period are
involved in
Jl
;iii
;it
The hd)our
production, and the pressure of other duties,
autlior's excuse.
compilation
of
following works have been
Sections
made use
VIII.
of
:
to
XIV.,
the
—
Treatises on theDifFerential and Integral Calculus, by liertratid,
Hymer, Todhunter, Williamson, and Gregory's Examples
on the same subjects; Salmon's Lessons on Higher Algebra.
Treatises on the Calculus of Variations, by Jellett and Todhunter
Boole's Differeutial Equations and Supplement
;
Carmichtiers Calculus of Operations
Boole's Calculus of
;
Finite Differences, edited by Moulton.
Salmon's Conic
Sections;
Ferrors's Trilinear Coordinates;
Kompo
on Linkages {Fruc. of Roij. Soc, Vol. 23)
Frost
aud Wolstenholme's Solid Geometry Salmon's Geometry
;
;
of Three Dimensions.
Wolstenholme's Problems.
The Index which concludes the work, and which,
hoped, will supply a
o2
serial publications
felt
:
five
is
want, deals with 890 volumes of
of tliese publications, thirteen belong
to Great Britain, one to
four to France,
it
to
Xew
South Wales, two
Germany, three
to
to
America,
Italy,
two
to
Russia, and two to Sweden.
As
the volumes
only date
from
the
year
180(j,
the
XXXVl
PREFACE.
important contributions of Euler to the " Transactions of
the
Academy,"
Petersburg
St.
excluded.
centur}^
last
complete classified index to Euler's papers,
ver^-
and others
as well as to those of David Bernoulli, Fuss,
the same Transactions, already
The
of
titles
are
unnecessary to include them,
was, however,
It
because a
the
in
exists.
Index, and of the
this
in
works of Euler
therein referred to, are here appended, for the convenience
of those
Tableau
who may wish
general
St.
des
to refer to the volumes.
I'Academie
de
publications
de
Imperiale
Pek'i-sbourg depuis sa fondation. 1872. [B.M.C.:*
11. Ii.
2050, e.]
I.
II.
Commentarii Academias Scientiarum Imperialis Petropolitanae.
1726-1746; 14 vols. [B.M.C: 431,/.]
1747-75, 1750-77; 21 vols.
Novi Commentarii A. S. I. P.
[B.
III.
M.
Acta A.
C.
:
431,/. 15-17,
S. I. P.
1778-86;
g.
1-16,
h. \, 2.]
[B.M.C:
12 vols.
431, /i.;3-8; or
T.C. 8,a. 11.]
IV.
Nova Acta A.
A.
9-15,
S. I. P.
LIS;
1787-1806;
15 vols.
[B.
M.
C.
V. Leonliardi Euler Opera minora coUecta, vel Commentationes
meticte collectse
;
2 vols.
1849.
[B.
M.
VI. Opera posthuma mathematica et pliysica
8534,/]
VII. Opuscula analytica
Analysis infinitorum.
:
431,
or T.G. 8, a.23.]
;
1783-5; 2
[B.M.C:
vols.
;
rj.
:
853
2 vols.
[B.M.C:
J-,
Aritli-
ee.]
1862.
[B.M.C:
50,/. 15.]
529,6.11.]
G. S. C.
Endslkigh Gakdkns,
London, N.W., 1886.
British
Museum
Ctitiilogui
i
[Correct tons nhich
1,
•Art.
mi important
are
marknl with an
imtirink.)
MATHEMATICAL
TAJiLKS.
INTRODUCTION.
The Ccnthnctrv-Grammc-Second system of units.
— The decimal measures
of length are the kilometre, decimetre, centimetre,
miUimetre. The same prefixes are used with the litre and
gramme for measures of capacity aud v o lume; ^'^''f^
Also, 10' metres is deuouiinated a metre-><ei:en; 10"^ metres,
a seventh-metre ; lO^f grammes, a gramme-fy'tcen ; and so on.
o;rainme-imttwH is also called a megagramme ; and a
milliontli-gramme, a microgramme ; and similarly "with other
Notation.
metre^
hectometre,
decametre,
A
measures.
—
The C. G. S. system of units refers idl pliymeasurements to the Centimetre (cm.), tlie (Jramme (gm.),
and the Second (sec.) as the units of length, mass, and time.
Definitions.
sical
Tlie quadrant of a meridian is approximately a metre3-28U8Gyi feet
89-370-i-j2
More exactly, one metre
=
seven.
=
inches.
The Gramme is the Unit of mass, and the weight of a
gramme is the Uiiit qftveight, being approximately the wciglit
of a cubic centimetre of water; more exactly, 1 gm. =
15-432:U1) grs.
is
The jL/f;v' is a cubic decimetre: but one cubic centimetre
the C. G. S. Unit of volume.
'22000(37 gallons.
-035317 cubic feet
1 litre
=
The
=
the Unit of force, and is the force wliicli,
in one second generates in a gramme of matter a velocity of
one centimetre per second.
The 7'i/v/ is the Unit of worlc and energg, and is the work
done by a dyne in the distance of one centimetre.
The absolute Unit (f atmospltcric pressure is one meg:ulyno
})er square centimetre
71«'0G1 cm., or 2!>"."j14 in. of mercurial
y8ri7 dynes.
column at 0" at London, where ^
l)i/ne (dn.) is
=
=
Volume = l\ is the pressure per
upon a body divided by the cubic dilatation.
Elasticitij of
15
unit
area
MATHEMATICAL TABLES.
Rigidity
=
the shearing stress divided hj the angle
n, is
of the shear.
Young's Modiihifi
= M, is the longitudinal
by the elongation produced, = 9nh -h (3/j + ??).
stress divided
Tenacity is the tensile strength of the substance in dynes
per square centimetre.
The Gramme-degree
of water at or near
the Unit of heat, and is the amount
1° C. the temperature of 1 gramme
is
of heat required to raise
by
0°.
Thermal capacifi/ of a body is the increment of heat
When the incrcdivided by the increment of temperature.
raonts are small, this is the thermal capacity at the given
temperatu"e.
Specific heat is the thermal capacity of unit
at the given temperature.
mass
of the
body
The Electrostatic iinit is the quantity of electricity which
repels an equal quantity at the distance of 1 centimetre mth
the force of 1 dyne.
3 X 1 0^'' electroThe Electromagnetic unit of quantity
=
static units approximately.
The Unit of potential is the potential of unit quantity at
unit distance.
The Ohm is the common electromagnetic unit of resistance,
\(f G. G. S. units.
and is approximately
10^
The Volt is the unit of electromotive force, and is
C. G. S. units of potential.
The Weber is the unit- of current, being the current due to
an electromotive force of 1 Volt, with a resistance of 1 Ohm.
It is
-j^ C. G. S. unit.
Resistance of aWiTe
Specific resistance X Length -r- Section.
=
=
=
=
Physical constants and Formulce.
= 3:2-1908t' feet per second.
= l>!^ri7 centimetres per second.
In latitude X, at a liciglit h above the sea level,
— -OOOOO:]/;) centimetres per second.
cos
g = (98O-0U56 —
Seconds ])endaliim = (iJ9-85G2 — -2536 cos 2\ — -0000003 h) centimetres.
THE 7';.17i"i7f.— Semi-polar axis, 20,854890 feet* = G-3;.4ll x lO^centims.
3782t x 10"
20,9_'G202 „ * =
Mean semi-equatorial diameter,
39-377780 x 10' inches* = TOOOlOO x lO" metres.
Quadrant
meridian,
In the lutitudc of London,
cj
2-.'')028
2/\
oi"
Volume, r08279 cubic centimetre-nines.
JIass (with a density 5g) = Six gramme- twenty-sevens
*
These dimensions nro
liikcn frjiu C'larko'a
nearly.
"Geodesy," 1880.
MATIIEMA TICAL TA BLES.
Velocity in orbit
=
2033000 ccntims per
=
Ohii.iuity, -2:f 27' lo".»
sec.
13713.
-01079.
I':cCentricity, e
Prounession of Apse, U"-2.').
Precession, t>0"'20.*
Centrifugal Ibrce of rotation at tlio equator, ;>-3'.)12 dynes per «^nunnio.
Force of sun's attraction, -0839.
Force of attraction upon moon, -2701.
2H0.
Katio of (/ to centrilu^ral force of rotation, g rw*
'.»'.*
Aberrat i<.n, 20"-ll to 20"70.*
Sun's horizontal parallax, h"7 to
Aiii^ular vclucily of rotation
1 -=-
=
=
:
Semi-diameter at earth's mean
Approximate mean
Tropical year,
Sidereal year,
distance,
1»>'
l"b2.*
distance, 1>2,UUOUUO miles, or l"'i8 centimetre-tliirleens.t
3Go2422l6
days, or 31,550927 seconds.
305-250374
„
Anomalistic year, 305-259544 days.
31,558150
„
Sidereal day, 8010
!•
seconds,
=
=
0-98 10^ granimes.
Earth's ma.'^s X -011304
TJllJ M0UN.—Uas8
Horizontal parallax. From 53' 50" to 01' 24".*
month,
29d. 12h. l-lin. 2878.
Sidereal revolution, 27d. 7h. 43m. 1 l-40s. Lunar
Greatest distance from the earth, 251700 miles, or 4U5 centinieire-tens.
225000
303
„
Annual regression of Nodes, 19° 20'.
Hulk. {The yt'ar+l)-^19. The remainder is the Guhlen Number.
The remainder is the Ejiact.
{Tlie Uulih'ti Number— 1) X 11-^30.
Least
„
„
Inclination of Orbit, 5° 9'.
—
GRAVITATION.— Attraction
between masses
m, m' at a distanco /
)
~
mm
clvues
x l-54o x It/
The mass which at unit distance (1cm.) attracts an eijual mass with unit
force (1 dn.) is = v/(l-543x 10^; gm. = 3iV28 gm.
Tr.rr^/i!.— Density at 0°C., unity
Volume
at 4°,
1
;-'
0000l3 (Kupffer).
elasticity at 15°, 2-22
Compression
Descamps).
1
for
proportional to
<
sq.
cm.,
to raise tlie temperature of a
The heat required
i° is
;
X iV".
megadyue per
j
+ -00002/" + OiJ00003i*
4-51x10-* (Amaury and
mass of water from
0° to
(Regnault).
=
1-4-273.
for 1° C, -003065
Spccitic heat at constant pressure
i.jAq
G'yI-8'ii'iS'.— Expansion
_
volume
0° with Bar. at 76 cm. = -0012932 gm. per
Specific heat at constant
Density of dry air at
(Regnault).
=
At
unit pres. (a megadyne) Density
Density at press, p = jtx 1-2759 X 10"'.
Density of saturated steam at t°, with j) taken)
from Table 11., is approximately
j
SOUND. — Velocity
Velocity in dry air at
Velocity in water at
LIGHT. — A'elocity
=
=
U' =
t°
in
cb.
cm.
-0012759.
_
-7931 .0 9^;)
(i
-|-
OUoOOO
lO"*
\/(elasticify of mediuvi -^ detusifij).
3o2 10 ^(1 -+--00300/) centimetres per second.
14;:U(J0
„
„
a medium of absolute refrangibility
3004 X lO"'-^^ (Coruu).
/i
=
If I' be the pressure in dynes per sq. cm., and / the temperature,
(Biot & Arago).
^i-l
29(K; X lU-''i'-4-(l-|--OO30i;/)
=
• These fliita iire from the "Nautical .Mmniiack" for 1S8:{.
t Inuisil ol' Vuuus, IbTt, " Aalruin. S c. XuI.lcs,'' Vols. 37,
'<i8.
MATHEMATICAL TABLES.
Table
I.
Various Measures and their Equivalents in C. G. S. units.
Dimensions.
1 inch
1 foot
1
mile
1 nautical do.
= 2-5400
= 30-4797
= 160933
= 185230
Pressure.
cm.
„
„
„
= 6'451G sq. cm,
= 929-01 „
Pq. yard = 83G1-13
„
sq.mile = 2-59 X 10^°,,
cb. inch = 16387 cb. cm.
cb. foot = 28316
cb. yard = 761535
gallon
= 4541
„
= 277-274 cb. in.
1 sq.
1
gm.persq.cm.= 981 dynes per sq. cm,
per sq. in.
76 centimetres-)
1 lb.
of mercury [
)
at 0° C.
inch
1 sq. foot
]
1
1
1
1
1
^^^^- P^^
gms. per
„
„
„
„
1
ounce
1
pound
1 ton
1
1
1
kilogramme
pound Avoir
pound Troy
= -06479895 gra.
= 28-3495
= 453-5926 „
= 1,016047 „
= 2-20462125 lbs.
= 7000 grains
= 5760 „
479
68971
=
1,014,000 „
=
70-307
„
=
cm.
^
-014223
1
cramme
1
grain
1 oz.
=
=
dynes
981
6fi-^)Cj777
=2-7811x10*
1
1
kilogram-metre
1
foot-grain
lbs.
Mas^.
grain
sq.
=
= 4-4497 X
1 cwt,
=4-9837x10'
= 9-9674 X 10»
1 ton
WorJc (^ = 981),
gramme-centimetre = 981
or the voof water
at 62° Fall., Bar. 30 in.
1
^q- ^"-
=
Force of Gravity.
upon
.,
lume of 10
pcrsci.foot
1 lb.
10-^
1 lb.
foot-pound
1 foot- ton
1
=
=
1 'hor.se po-wer' p. sec.
erofs.
981 X
10-"^
1-937
xlO\,
W
1-356x10" „
3-04 X
„
7-46x10®
ITeat.
^'cIoclfl/.
1 mile
per hour
1 kilometre
„
= 44704 cm. per sec.
= 27'777
„
Table
II.
Pressure of Aqueous Vapour in
dynes per scquare centim.
Teinj).
1
gramme-degree
1
pound-degree
pound-degree Fah.
1
C,
=
42 X 10"
ergs,
=191x10-
„
106 x lU*
„
=
M.
Magn.
Specific
Resistance
C.
0=
at
Elect.
I
THEMA TFCA L
TA BLES.
MATHEMATICAL TABLES.
Table
BURCKTIARDT'S FACTOR TABLES.
For all
Explanation.
number from
1
M.'.Mr.KKS FJiOM 1
to 9'JOOO.
— Tlic tables give
up to 99000
:
the least divisor of cvciy
but numbers divisible by 2, 3,
All tlie digits of the number whoso
or 6 are not printed.
divisor is sought, excepting the units and tens, will be found
in one of the three rows of larger figures. The two remaining
digits will be found in the left-haiid column. The least divisor
will then be found in the column of the first named digits, and
in the row of the units and tens.
If the number be prime, a cipher is printed in the place of
its least divisor.
in the first left-hand column are not conseThose are omitted which have 2, 3, or 5 for a divisor.
Since 2"-. 3. 5"^ = 300, it follows that this column of numl)er3
will re-appear in the same order after each multiple of 300 is
The numbers
cutive.
reached.
—
Mode of using TnE Tables. If the number whose prime
factors are required is divisible by 2 or 5, the fact is evident
The
upon inspection, and the dinsion must be effected.
quotient then becomes the number whose factors are required.
If this number, being within the range of the tables, is yet
not given, if is dirisihle by 3. Di\'iding by 3, we refer to the
tables again for the new quotient and its least factor, and so on.
—
Required the prime factors of 3101-55.
This number is within the range
Dividinrr by 5, the quotient is G2031.
of the tables.
But it is not found printed. Therefore 3 is a divisor of it.
Dividing by 3, the quotient is 20G77. The table gives 23 for the least factor
Dividing by 23, the quotient is SW.
of 2ftr)77.
The table gives 2i» for tlie least factor of H'.tO. Dividing by 20, the quo3.5.23.20.31.
Therefore 31015-3
tient is 31, a prime number.
Again, roipiired tiie divisors of 02881. The table gives 203 for the least
Referring to the tables lor 31 7,
divisor. Dividing by it, the quotient is 317.
Ex.\Mrr,ES.
=
a cipher is found in tbe place of the least divisor, and this signifies that 317
is a prime nundjer.
Tlitrefore 02S81
203 X 317, the product of two primes.
It may be remarked that, to have resolved 02881 into these factors without the aid of tiio tables by the method of Art. 3G0, would have iuvolved
fifty-nine fruitless trial divisions by prime uumbei-a.
=
1-9000
O
r^
•o
I-
—
9000-18000
—
n
^o<o— o-oo-i~
P50i^ <»)Ot-r--a>
«
ui
c«
^lor^cro c^oooo
CO
1
::.^-^j:;-
°-22;: °5°^i;i^
18000-27000
O -
1-
O C
l~ 1- 1-
O
O
1-
o o 2 U
r;-„
-t^-(NCO
—
OCO
— f
00
CO
— -
27000 — 36000
-o« ..o^o^o Mj^2::°°
^
36000 — 45000
-
= ^
CO
45000 — 54000
Ol-O
0-
— mi^t-
ooot^oo
^
54000 -ei^OOO
rH
63000 — 72000
on- 0-CI--0
003JI-1-0
^
72000-81000
n
w
—
a>
—
— —
CO
e^trc*
_
1
-
en
CO
05
j
CJ
-
<M
:r
CO
I-
CO
CO
-
Cl
CO t- t^
- O
O
I-
O -
I-
-
I-
O O
1^
M c
m 000 -90000
o
s
('
C-- -gooo-
5 = 2 =
°"
90000 — 99000
30
ALGEBRA.
FACTORS.
1
a'-h'= (n-b)
2
3
<r-lr
a'-\-i/'
And
6
=
— b" =
a"-\-b" =
8
Gr+^O G^*+&)
4
(a—b)
5
a"
{(i
9
10
11
12
(a-b)
(«+6).
{(r-\-(tb-irb-).
(a-\-b) {(r-ab-\-b').
generally,
«"-.ft"
ft
=
=
+
(«"-^
b) {a"-'
{a-\-b)
(^<'
+ «"--/>+ + //-')
— a"--b+...—b"-')
...
(«"-^-a»- -6+...+6"-')
+ =
c)
.^•^^+(^f
+ +
/>
+ o)- = (r-\-'2(W-\-b-.
{(i-b)- = ir-'lab-\-}r.
+ =: a'-^\\irb^?uib-^-b^ =
{a-bf = a'-[\(rb-{-{\alr-lf' =
/
'
/\-'
I
.1
I
(r^
/
I
7'
6)''
(^/
iilwiiys.
if
n be even.
if
n be odd.
r') .r-^
-\-(ihc.
'i-(bc-{-('a-\-(ih).i
V
I
I
'
/
a'^b'-^i\ab
.
{a-\-b).
d'-li'-Wab {a-b).
Generally,
{a±by=a'±7(i'b-]-2\(vb-±'^'m'b'-\-:irui'b'±2](tW-^^
Newton's Rule ior forming- the
coefficients
:
Miiltiphj (inn
index of the leading qnantifi/, and divide bij
the number of terma to that place to obtain the coefficient of the
term next following
Tims 21xr)-^3 gives 35, the following
See also (125).
coefficient in the example given above.
coefficient by the
.
To square
a polynomial
:
twice the 2)roduct of that term
Thus,
=
rt-
Add
to
the square of each term
that folloivs it.
and every term
{a-\-b-^r-]-f/y
+ 2rt(ft + r-+f/) + //-+2/>(r+^/)+r- + 2rr/+^/-.
'
ALQEBUA.
34
=
{a--^ah-\-h-) {ir-ab-\-¥).
13
a'-f a-6"+6'
14
a'+b'
15
(„.+iy=,.Hi+2, (.+iy=.'+-L+3(,,.+i).
=
{a'+ab
16
{a-\-b-^cY
17
(a-^b + cf
V2-\-I)')
(a'-ab V2+6^).
= (r-\-b--]-c'+2bc-\-2('a-\-2ab.
= a'-\-b'-[-c'-\-',^ {b'c^bc--\-c'a-\-ca'
^a-b+a¥)-\-6abc.
Observe that
in an algebraical equation the sign of any
changed throughout, and thus a new formula
obtained, it being borne in mind that an even power of a
negative quantity is positive.
For example, by changing the
sign of c in (16), we obtain
letter
may
be
{a-\-b-cf
= a^ + + c'-2bc-2ca + 2ab.
h'
=
=
(«+6+c) {a+b-c)
18
a'+b'-c'+2(ib
19
20
= a'-(b-cy = (a + b-c) {a-b-\-c).
a'-\-b'-\-c'-^abc = {a-^b+c) {a'-\-b'-\-c'-bc-ca-ab).
21
bc'+b'c-\-ca'+chi+ab'-\-(rb-{-a'-\-b'-\-e'
{a-^b)--c'
^y
(1).
(r-b'-r-\-2bc
25
= {a-^b+c^((r+b'-^r).
bc'-\-b-c-\-c(r+c'a + ab''-^a-b + :\(ibc
= {a-\-b-\-c){bc-}-ca-\-ah).
bc'-\-b'c + m''-\-c'a-\-ab'+(rb + 2abr={b-{-c)(c-\-((){a-j-b)
— h' — r
b(r + b'c + cd' + c'n + ab- + (rb — 2a be —
= {b^c-a) {c^(i-b) {(i^b-c).
bc^-b^c + ca'-c^a-irab'-irb = (b-c) (c-a) (a-b).
26
2b'c'-\-2c'a'-Jr2a'b~-a*-b'-c'
27
.rH2.t%+2.r/ + //
22
23
24
({'
=
{(i
+ b-]-c) ib-\-c-a) (c + a-b) (a-\-b-c).
= (,,+//) + + /^).
Generally for the division of
see (545).
C^-•
{x
+
//)"
.^//
— {x"
-\- //")
by
.r-
+ xi/
-\-
y-
.
MULTIPLIOATIOS AXD
J >I
35
VISION.
MrLTirLTCATTON AXD DTVTSTOX,
15YTHE MKTIKil) OF DETACH HI) roKFFICI KNTS.
28
Ex.
1
(a*-SaV + 2ab' + b*) x (a'-2a6»-26').
1+0-3+2+1
l+U-2-2
1+0-8+2+1
-2-0+6-4-2
-2-0+6-4-2
1+0-5+0+7+2-6-2
Result
Ex. 2:
a^-5aV- + 7a%* + 2o:'b'-Gab"-2b'
(x^-bx' + 7x' + 2x'-6x-2)-r-(x*-Sx' + 2x + l).
1+0-5+0+7+2-6-2(1+0-2-2
-1-0+3-2-1
0-2-2+6+2-6
+2+0-6+4+2
-2+0+6-4-2
+2+0-6+4+2
1+0-3+2+1)
Result
a;»-2.i— 2.
S(/ntlift}r
Ex. 3: EmployiTig the
-0
+3
-1
Re.sult
]a.st
Dici.sinn
example, the work stands thus,
1+0-5+0+7+2-6-2
0+0+0+0
+3+0-6-6
-2+0+4+4
-1+0+2+2
1+0-2-2
[See also (248).
Note that, in all operations with detached coefficients, the result mn.st he
written out in successive powers of the quantity which stood in its successive
powers in the original cxpre.-^sion.
ALGEBRA.
36
INDICES.
29
Multiplication:
a'
a}x
c(^
=
a}'^^
=
a^,
or
^^a^;
EVOLUTIOS.
37
—
C
To form the H.
F. of two or more
32 Otherwise.
algebraical expressions
Sfpanite the e.rprcHtiions into their
simplcd fdctorx.
The 11. C. F. will be the product of the
factors co)nmoiL to all the exjyrGSsionSy taken in the loivest
powers that orcur.
:
LOWEST COMMON MULTIPLE.
33
The L. G. M. of two quantities
hi/ the E. G. F.
equal
is
to their
product
dicided
—
34 Otherwise. To form the L. C. M. of two or more
algebraical expressions
Separate them into their simplest
The L. G. M. will he the product of all the factors
factors.
that occur, taken in the highest powers that occur.
:
Example.— The H.
and the
L. C.
M.
is
and aXh-xfc'e
C. F. of a\h-xfchl
a'{b — xY'c'de.
is
a=(6-.r)V
EVOLUTION.
To
extract the Square Root of
S\/a
powers
and reducing
.,
Arranging accoi'ding
expression becomes
35
Detaching the
41a
3a \/a
"'-—^
to
of a,
2-
,
16a2-24;a'-|-41a-24a5
16
coefficients, the
work
,
,
+ 16-+'-
is
to one denominator, the
+16
as follows
:-
16-24 + 41 -2-4 + 10 (4-3 + 4
16
8-3
-3
-24 + 41
24- 9
8-6 + 4
'
D
Result
li.
32-24 + 16
-32 + 24-16
— — =a — ^v/a+l
4a — 3o* + 4
'
T
/
,
1
ALGEBRA.
38
To extract the Cube Root
37
Sx'
of
- 36a;'' ^ij + 66x'y - 63xhj ^y + 33,cy - 9^- ^y + y\
The terms here contain the successive powers of .r and \/y
detaching the coefficients, the work will be as follows:
—
therefore,
III.
II.
I.
6-3)
-6)
;
8-36 + 66-63+33-9 + 1(2-3 + 1
12
-18+
9^
EQUATIONS.
=
{^-f/Y =
42
(.r-//)-'+4r//.
0^'+.y)'
43
44
39
(.r+//)'-4f//.
Examples.
2
y g^ - b' + ^6' - x' _ 3 y/g^ - + y/r- //-
v/c^-o^
^Z'^
y/c^-rf^
,[38.
9(r-a;^) =4(r-tZ-),
a^
To simplify
a
compound
= y^c'+w
fraction, as
'
a^
— ab + h-.,+
1
a-
+ ah +
li-
1
1
a*— a6 + 6*
a*
+ ab +
b''
multiply the numerator and denominator by the L. C. M. of
denominators.
(a^ + ab + b') + (a'-ab + b')^a- + lr
Result
ab
(a- + ab + b')-(a--ab \-b')
all
the smaller
QUADKATIC EQUATIONS.
'2(1
46
If
«cr-+2^>( -fr
an even number,
.i'
=
(I
that
;
is, if
=
.
Method of solution without
47
Divide by
= U.
= 0.
x'— -—x+
2.r— 7« + 3
Ex.:
2,
the coefficient of
7
3
2
<j
-
the formula.
,r
be
—
,
40
ALGEBRA.
w
n
Complete
^1
the square,
7
2
x^
/7\-
,
x-\-
{
2
Take square
re—
root,
4
\
—
4
49
3
25
= —-
16
2
16
= ±—
4
= ^^ =
"4
3
or
-2*
Rule for "completing the square" of an expression
48
—
49
=
/
.r
33^
.
fci'
:
Add
The
"
the square of half the coefficient of
solution of the foregoing equation, employing formula (45),
=
_fc±v/fe2_4^^
2a
7±5
= 7^y49_24
4
= -^ =
o
^
like
x.
is
1
"'
2-
THEORY OF QUADRATIC EXPRESSIONS.
If a,
/3
50
be the roots of the equation ax--\-hx-\-c
a.v'-]-kv-{-c
51
Sum
52
Product of roots
=
a+/8
of roots
a/3
= 0,
then
a {a—a){.v-S).
=—
=
-.
a
-.
a
Condition for the existence of equal roots
b^—4<ac must vanish.
53
54 The solution of equations in one unknoAvn quantity may
sometimes be simplified by changing the quantity sought.
Ex.(l):
2.+ «»L-l+
Sx + 1
6.^-
+ 5.g-l ^
3a;
+l
l^Lte
+ bx—l
=14
(1).
dx'
+ 1) ^
+ bx — l
6(3j;
6ar
-^^^^
j^
(^)-
.
EUi'ATloXS.
thus
- = i^-
+
y
41
y
y having been dcUrininetl from this quadratic, x
55
Ex.2:
a;H
is
=
x-\
ij,
(2).
+X+-- =4.
-,
(..!)%(.. l) =
Pat
afterwards found from
and solve the qnadi-atic
c.
in y.
X
56
l'^>i-
x'
'^
+ x+^^2x' + x + 2
2x-
lx^
Put
v2x*-\-x + 2
57
Ex.^:
=
y,
=
-iy
+1
+ X + S v/2.«* + x + 2 =
2,
=
4.
+ x + 2 + 3y/2.x^ + x + 2
and solve the quadratic
^^"+3|v.=
'i
''
2
,
+ 3
'
?
¥-'
Iti
=
3
2.1
A
quadratic in
y
Tojind Md.vitnd
58
Q
Ex.— Given
to find
what value
(ind MininKt rahn'.s
1
1
(I
hi/
menus of a
(h'd t ic Ju/ uatiou.
;/
of x will
=z x^
=
make
3.r
y a
+ G.c + 7,
maximum
or mitiimnm.
Solve the quadratic equation
3.c'
+ 6a; + 7-y =
0.
^^ -3±y3y-12
Tl,>,s
,.45
o
In order that .r may be a real quantity, we must have '^y not less than 12
therefore 4 is a minimum value of //, and the value of x which makes y a
;
minimum
is
— 1.
O
ALGEBBA.
42
SIMULTANEOUS EQUATIONS.
General solution
unknotvn quantities.
icith tu-o
Given
59
(ti^v-\-b,ij
=
Cil
^^.
—
c,b,
a.a-[-b,i/=eJ'
— cA
^
^
c,a,-c,a,
h^a.—b^a^
a^h.—a,h^
'
General solution with three unknown quantities.
60
= (U\
chA^-{-b,y-^c,z
Griven
= dj
+ (h (brC,-b,e,)
a..v-\-biy-^c.z
oe
_
d,(h,c,-hc.?i
+
d, {b,e,-b,Cs)
niid
— b.eyY
^
«i {b-2Cz—bsCo)-{-ao {hc^—b,Cs)-\-a3 {biC.
symmetrical forms for y and
z.
Methods of solving simultaneous equations bettceen two
unknown
x and y.
quantities
—
Find one unhioivn in terms of the
I. By substitution.
other from one of the tivo equations, and substitute this value
Then solve the resulting equation.
in the remaining equation.
61
Ex.:
.r
+ 52/
77/
From
(2),
y
=
4-.
Substitute in (1)
.i-
IL By
62
Ex.:
Eliniinitc
l)y
+ 20
the
=
(1)]
{-I)]'
thus
;
.r=3.
23,
method of Multipliers.
+ 5y =
36
2x-:hj=
5
'6x
.»
= 23
= 28
(1) \
(2)V
multiplyincr oq. (1) by 2, and (2) by 3; thus
6x + l0y
6x—
=
57,
.'/=
3;
I9y
,T
=
=
72,
% = 15,
7,
by subtraction,
by substitution
in
Lt[.
(2).
.
I'Uil'ATJONS.
63
Til.
4'.\
clKm^,in^: thr (/unnfitirs
/>//
x-y=
2
+ + =
:iO
Ek. 1:
.r-2/-
Let
+
.(•
v/
a-
=
//
(1))
(-1))'
— =
x
",
.sou;iIif.
i\
if
Substitute tliese valiius in (1) and (2),
uv
+u
=
30
n
=
=
10
x-\-i/
From which
64
^*
x
2 -Ltl
Ex.2:
=6
+
X — 1J
Substitute
;:
for
^^^' in (1)
x-y
;
10,
and
=
//
9
(1)
+ 7>r =
0i
(2)
;
2.-^1^
,.
=
2-^-92+10
From which
=~
z
or
—!^ = 2
x-y
From which
Substitute in (2)
;
or
65
Ex.3:
'.iy
.'/
=
2
^
~
77?
3.j;
Divide each (juantity by xy
and
a-y
3.vy
x
—
»/.
= 6,
(1)
)
(2))-
;
^
«
=1
V
and (I) by
*'
3,
(^))
f
- + ^=3
2,-
—
7
^"'^ '*^~ ~^'^'
=
^+
y
Multiply (o) by
0.
2,
or
or
+ 5^=
2x + 7y
0;
=
I
—
x
tlius
-i.
=
10 ^l^JL
x+y
z'
)
•
(I,
I
and by subtraction y
i.s
eliminated.
ALGEBRA.
44
66
%
IV.
tfhen the equations are
tx,
in the terms tvhich contain
+ 7.ry = 52/^
^x-^ = n
h^x' + ltx' = 6fx'
(1),
bx-Stx
and, from (2),
=
=
a'
and
y.
(1)7
(2)5"
52x^
Ex.1:
From
=
y
substituting
homogeneous
(3)
j
(4))'
17
6t\
52 + 7t
(3) gives
substituted
a quadratic equation from whicli t must be found, and its value
in (4).
tx.
X is thus determined and then y from y
=
;
Ex.2:
67
From
from
2x''
+ xy + '3y' =
3y-2x=
(1)
|
(2)3'
4
=
tx,
by putting y
x'(2 + t + 5t')
(1),
a.
(2),
squaring,
a;'
=
16
(3))
(3^-2)=
4
(4)3
(9^^-12^ + 4)
9t'-12t + 4^
a quadratic equation for
t
l6
beino-
found from
.
'
= 16
= 2 + + Sf,
;
t
t.
tliis,
equat'on (4) will determine x
;
and
finally y
— tx.
RATIO AND PROPORTION.
68
\i
a\h
v.
e
\
d\ then ad
a-\-b __c-\-d
d
~~b
69
a
^
'
= be,
— b _ e—d
b ~ d
" T = 17 = 7 = '^"
"""
'
and
—= —
a-^b
a—b
^
'
;
_ e-\-d
c—d
T - i+</+/+&c.-
General theorem.
70
:= kv. = k say,
^b = 4d = 4
J
If
.
^
^
pa''
then
+ f/c"-]-re" + Szc.
} I
lpb"-\-qd"-\-rf"-^&G.))
where
/),
in (71).
q, r, Sic. arc
any quantities ^vhatever.
Proved as
6
.
AND ritOPOHTlON.
72/1770
45
—
Rile. To verify any equation between such proportional
quantities: Suhsfifufe for d, r, c, (Jv., their eqiiicalenta kh, Inly
kj\ ^'c. respect Ivcl ij, in the given equation.
71
Ex.
— If
a
'.h
c
'.:
show that
to
d,
:
y/g —
_
\/a
^c — d
Fni
kb for
and kd
a,
for c
— \/b
Vc—
-/d
thus
;
^/a^>
x/c-d
^/kb-b
s/by/k-i
s/b
-^kd-d
Vds/k-l
^d
Va-K/ h ^ Vkh-Vb _
v/6
Vc-Vd
~VdWk-\)
s/kd-Vd
(
x/k-1)
^ v^
-/d'
Identical results being obtained, the proposed equation
72
li a
a
then
:
di:
\^a
:
a'
:
::
d
I
cr
c
The
73
c
:
:
a
Also
b
:
:
a^
:
must be
e &c., forming a continued proportion,
'.
fr,
the duplicate ratio of a
I
b,
b^,
the triplicate ratio of a
I
b,
^^h is
the subduplicate ratio of a
:
6,
is
the sesquiplicate ratio oi a
:
h.
h^
true.
fraction -^
is
made
and so on.
to approach nearer to unity in
by adding the same quantity to the numerator and
denominator. Thus
value,
-—!—
6
IS
nearer to
—
than
1
—
is.
f)
aj
Def. The ratio
the ratio ac Id.
74
is
+
compounded
of the ratios a
:
b
and
c
:
d
:
75
li a
:
b
c
::
ing ratios, aa
:
d and
:
,
bl/
::
cc'
:
a'
b' ::
:
c'
:
d'
;
then, by
compound-
dd'.
VARIATION.
76
77
•
•
rt
If
^ Gc^ 7
^
and
78
and
If
If
oc c
-
[
coed)
<t
i.u
,
Ij
a c,
then ac
cc^) ^ve
then (a + b)
cc
LI and1
bd
a
and \/ab a
c.
— —
'^
"'
oc
c
may assume
cc c
d
= hib, where m is some constant.
r1
r
.
ALGEBEA.
46
ARITHMETICAL PROGRESSION.
General form of a series in A. P.
a + 2f/, « + ;W,
=
first
term,
a
d = common difference,
= last of n terms,
s = sum of n terms
then
79
a,
a-\-{n
a-\-(I,
— l)d.
/
;
80
I
=a-[-{n-l)d.
81
*
=
82
s={2a-\-{n^l)d}^.
Proof.
—By
(«
+ I
/)
writing (79) in reversed order, and adding both series
together.
GEOMETRICAL PROGRESSION.
General form of a series in G. P.
83
a, ar,
a
r
I
s
84
ar,
«r" "^
ar^,
= first term,
= common ratio,
= last of n terms,
= sum of n terms
= ar"-\
;
then
l
85
s
If r
=a —
r
be less than
1,
s=
86
Proof.
series
— (85)
is
or
1
and n be
-i^,
I—
a
—
infinite,
since
r"
= 0.
obtained by multi]>lyiiig (83) by
from the other.
r,
and
siil)(racting
one
—
PEIiMUTATlONS AND COMIHSATIOXS.
47
HARMONICAL PROGRESSION.
87
cf,
i_
1_
-,
,
a
1
— 1
-y,
a
r
Or when a
88
:
b
a
::
-b
b
:
between any three consecutive
term
89
«^''
90
Approximate sum
ft
+
rf'
the reciprocals
&G. are in Aritb. Prog.,
,
b
Harm. Prog, when
&c. are in
b, Cy d,
a
of the series
+ 2d'
a-^Sd
,
=
of
—c
rehition subsisting
tlie
is
terras.
r^
-.
;
[87, 80.
{n-l)a-{n-2)
n terms of
&c., wlien d
the
Harm.
Prog.
small compared with a,
is
_ {a-{-(l)"-a"
Proof.
— By takintr instead the G.P.
2
1
,
+ 7— —77-0 +
91
Arithmetic mean between a and h
= —^^.
92
Geometric
do.
=
\/ab.
93
Harmonic
do.
=
—
The
;
r^TK
+
•••
•
—r.
<t-\-h
three
means are
in
continued proportion.
PERMUTATIONS AND COMBINATIONS.
94
Tlie nui'iber of permutations of v things taken all at a
thne
=
n{u-\){n^'>)
—
...\\.'lA
=
n\ or
;i"".
ho true for n things.
Proof hy IxnucriON. Assume the
Now take ?i + l things. After eaeh of those the remaining n things may bo
arranged in n ways, making in all nX n\ [that is (»t + l) !J permutations of
See also (23."^) for the mode of proof by
w + 1 things; therefore, &c.
!
Induction.
foniiiila
to
ALGEBRA.
4S
95
The number
time
is
P {n,
P {n,
Proof.
—By (94)
P (n,
therefore
;
-i-
96
The number
time
is
G
tions; therefore
G {n,
is
G {n,
r)
r) is greatest
of n things
^
things taken r at a
...(n-r-j-l)
_
= 7T
n^^^
r
=
things admits of r
!
permuta-
P{ii, r) -^ r!
when
r
= ^u
or
i{n +
\),
according
even or odd.
The number
98
;
—
— r) = C {n, n r).
For every combination of
97
7i
1.2.3. ..r
r\ {n
as n
out of each pei-mutation
(n, r).
^
'
n(^>.
of factors.
- n{n-l)in-2)
~
r)
^
number
left
=
.
of combinations of
denoted by
r r,,
{n—r)l
(w-r+l)
...
(«—r) things are
for
= nl
r = the
r)
Observe that
r).
= n (w-1) {n-2)
r)
n things taken r at a
of permutations of
denoted by
^
'
*
is
of
homogeneous products
denoted by H(y,
^
When
=
1.2...r
r
is
>
n, this
•
V\
reduces to
(>-+l)(>'+2)...(/^
99
of r dimensions
r).
+ >— 1)
(V-I)!
—
PrOOK. Jl{n, r) is equal to the number of terms in the pi-otluct of the
expansions by the Bin. Th. of the n expressions (1— a.i')~\ (1 — Z/.j)"\
(1
— cr)"',
&c.
Pnt a=-h
(1-a:)-".
=c=
(128, 129.)
&c.
=
1.
The number
will
be the coenTicIeut of
x'^
in
ri'lUil UTA
TIONS AND COM I'.LXA TIOXS.
VJ
The niimbor of perimitations of n tliin<j:s tjiken all towhen a of them are alike, h of them alike, c alike, &c.
100
gether,
a
For, if the a things were
permutations where there
all
is
...
&c.
would form a!
So of b, c, &c.
different, they
now but
one.
The number of combinations of n things
any^ of them will always be found, is
101
in
^1 c!
!
r at a time,
which
=
C(n-p, r-p).
For, if the p things be set on one side, we have to add to them
r—j) things taken from the remaining n—2) things in every
possible way.
C(n-\, /— 1) + C(;t-1,
Theorem:
102
Peoof by Induction
or as follows
;
:
>•)•
Put one out
of
n
G{u — l,r) combinations of the reTo complete the total
at a time.
there are
letters aside;
=
C{u,
r)
maining 71 — 1 letters r
C(n, /•), we must place with the excluded letter all the combinations of the remaining n—l letters /*— 1 at a time.
be one set of P things, another of Q things,
the number of combinations
things, and so on
FQIl ... &;c., the
formed by taking one out of each set is
product of the numbers in the several sets.
103
If there
another of
ii
;
=
P things will form Q combinations with
second of the P things will form Q more
combinations and so on. In all, PQ combinations of two
Similarly there will be PQE combinations of three
things.
This principle is very important.
things; and so on.
For one
the
Q,
of the
A
things.
;
104 On the same principle, if p, 7, r, &c. things bo taken
out of each set respectively, the number of combinations will
be the ])roduct of the iiuniberR of the separate combinations
;
that
is,
=
C{rp)
.
('{Qr/)
.
C{Rr)
...
Sec.
60
ALGEBRA.
The number of combinations of n things taken m at a
when p of the n things are alike, q of them alike, r of
them alike, &c., will be the sum of all the combinations of
105
time,
each possible form of
coefficient of
(l^-.^'
x'"
in the
m dimensions, and this
expansion of
is
equal to the
+ ,T2+•••+.^'')(l+c^' + cT'-h...+a?'')(l^-T^-.^''+•.+.^'•)••••
The total number of possible combinations under the
same circumstances, when the n things are taken in all ways,
106
1, 2,
3
at a time,
... 7i
=
(p+l){g+l){r+l)...-l.
The number
107
m
!
and the
when they are taken m
be equal to the product of
in the expansion of
of permutations
ways
at a time in all possible
coefficient of
x'"
will
&c.
SURDS.
108 To reduce >/2808. Decompose the number into
prime factors by (360) thus,
;
= y2\ 3M3 = 6 Vl3,
h' c' h c- = a' h"
c =
V28iJ8
^a'"
109
6'" c^
= a'»
To briug 5^3
110
To
y^ z'
c'
Vh6'
an entire surd.
to
= yi875,
= a;- y^ z^" = V^z^.
5y;3
a;»
fV'
b'^"
=
vo'. 3
rationnllse fractions hnvinf:; .surds in their
drnnminators.
j_^
y?.
1
^
^49
^
y4o
its
.
SUIiVS.
111
61
.J3^o=^^r-'<"^^'^°''
since
(9
^^^
- ^80)
l+2y8-v/2
(l
^
+ ^80) =
(9
(1
81 - 80, by
( 1)
11 + 4/3
+ 2y3)'--J
+ 2/3+^2) (11-4/3)
73
1^3
Put
„
3*--i*"
?/3-v/2
3*
=
a-,
2'
=
—
.
thereiore
ij,
and take G the L.C.M. of the deriominafors 2 and
1
-=
3'
3, tlien
+ 3«2' + 352* + 3'2» + 3«2« + 2*
z
3i-2*
3^-2'
= 3 y9 + 3 y72 + 6 + 2 yG48 + 4 y3 + 4 v/2.
114.
^^^
if
Here the
.
y3+/2
result will be the
same
as in the last exainplo
the signs of the even terms be changed.
[See
5.
115 A surd cannot bo partly rational that is, y/a cannot
rrovcd by squaring.
be equal to >^''h + c.
;
116
117
he product of two unlike squares
'J
^7 X
y/'^
The sum
or
=
is
irrational;
^/2], an iri-atioual (piantity.
difference
produce a single surd;
that
of
is,
two unlike surds cannot
\/a-\-x/h cannot be equal
15y S(jnaring.
to \/c.
118
If "
-\-
\/m
=
J'i-^'^n; then a
Theorems (115)
119
If
=
h
and w
=
n.
to (118) are i)roved indirectly.
^/a+
then
By squaring and by (HH).
W>=
^.c-\- s^ij,
—
52
ALGEBRA.
To express
120
in
two terms \/7 + 2V6.
Let
+ 2v/6= ^x+^tj;
v/7
+y =
then
x
and
X-2J
by squaring and by (118),
7
= ^7'-{2^6y = a/49-24 =
= 6 and y =
.•.
5,
by (119)
;
1.
ic
ye+i.
Result
General formula for the same
\/a±^b=\/i{a-{-x/a'-b)±\/i{a-x/a'-b).
121
Observe that no simplification
perfect square.
To
122
Vb
\/a-\-
Let
c
must be found by
x
=
= x-{-
and
3.t;
is
a perfect cube.
= x+y7j.
.-.
;
y9v/a — 11n/2
9 v/3
a^—h
c= ^49- 50= -1.
x=\, y
+ = 7
Result
Ex. 2:
= a,
effected unless
is
\^y.
from the cubic equation
V7 + 5^2
Ex.1:
Cubing,
a
= cv'^—c.
7/
simplification
4.(;*
is
y/a^—b.
trial
4cr^— SccV
No
—b
v/a+ Vb.
simplify
Assume
Then
effected unless a'
is
- 11 v/2 =
liy2
.-.
.r
=
=
a; v/.x'
9v/3
.-.
=
=
."{
1
+
v/2.
v/-T+ v/y, two different surds.
+ 3.« y?/ + Sy ^x ^y^y,
(.T
+ %)ya;-)
(:3x-
and
+ 2/)y2/)
?/
=
2.
.
'
.^^o^
^
^
lUSOMlAL THEOREM.
123
To
simplify
Assume
v/(12
53
v/(12 + 4y3 + '4yr) + 2yi5).
+ 4v/3 + 4v/5 + 2yi5) = ^x^ ^y-\-
^z.
Square, and equate corresponding surds.
Result
v/3+yi+-/5.
To express \/A + B
in the form of two surds, wliere A
are one or both quadratic surds and n is odd.
Take (/
such that q (A^—B-) may be a perfect n^^ power, say />", by
124
and
B
Take
(361).
s
and
the nearest integers to V'y
t
(^4
+
/?)''
and Vq{A--B)\ then
2Vq
Example:
y89y8 + lU9y2.
A =89^3, B = 109^/2,
To reduce
Here
A''-B'
=
vq (A + By = 9+f
l;
.-.
f
\
p=l
^qiA-By=l-f\'
Result
and q
=
I.
being a proper fraction
;
.-.8=9,1=1.
i(^9 + l + 2±v/9 + l-2)
=
y3+v/2.
BINOMIAL THEOREM.
(n+by
125
126
General or
(/•4-l)^"
=
term,
r!
127
or
,
a"-n/
''[,
,
if
If b
n be a positive integer.
be negative, the signs of the even terms
will
be changed.
—
,
ALGEBRA.
54
If
n be negative
expansion reduces to
tlie
{a+br^
128
=
General term,
129
v\
Elder's proof.— Let the expansion of (1 +.'«)", as in (125),
be called /(7i). Then it may be proved by Induction that the
=f{m + n)
f{'>n)Xf{n)
equation
(1)
m
and n are integers, and therefore universally
true when
because the form of an algebraical product is not altered
true
by changing the letters involved into fractional or negative
is
;
Hence
quantities.
/(m + ?i+j9 + &c.) =fim)Xf{n)Xf(p),
Put
= 71= = &c.
7n
theorem
to
2^
is
Jc
terms, each equal
—, and
the
proved for a fractional index.
—n for m in (1) thus, whatever n may be,
Again, put
;
f{-^i)Xf{n)=f{0)
= l,
for a negative index.
which proves the theorem
130
&c.
For the greatest term in the expansion of (a-^-by, take
{n-l)b
^
c {n^-l)b or
of
-—f'
°
a — of—
a-{-b
...
r = the mtegral part
-,
^^
^^
according as n is positive or negative.
But if b be greater than o, and n negative or fractional,
the terms increase without limit.
Examples.
Required the 40th term
Hero
By
r
(127),
= 39
ilio
_42!
a
;
term
/
y!3i)!\
_
=
will
1
;
b
(
1
= -
—
'-
;
n=
12.
he
2..\-_
:W
(.f
_
^-^
.
iU-i^. (2.,-y«
1-2.3
\sl
(^(3^
^
'
lilNOMIM. TllbUiUKM.
Roqniied the Slst term of (a — .r)"*.
Here
By
r
= 30 h=-x\
^~^^
7i
;
(129), the term
= — 4.
is
^ 4.5.6...80.:U.:V2.:i1
"
1.2. 3... 30
(
^_ 31 .82. 33
- 1.2.3 -a"
«•'"
,>.
-^^
IleqairoJ the greatest term ia the expansion of
131
—
=
Here n
(l-|-.r)"'.
=
-6
a
thert'foro
a
^^
=
h
1,
_ 5x|> _
(n-l)fc
=
231
—
''^
^^
— when
a
x in the formula
.
1-H-
r = 23, by (130), and the greatest term
24 .25 .26 .27 lU^
,.o3 5.6.7...27 /14\'»^
_
~^ ^ 1.2.3... 23\17/
1.2.3.4 \17/'
,
132
We
fore
Find
r>Jt +
l
negative term in the expansion of (2a + 3&)'*'.
tlie fir.st
must take
r the first integer
= V +1 =
6|
14
8
17
11
which makes n — r-\-\ negative; there= 7. The term will be
therefore r
5
8
2.
1"\
C
,
(2(7)-»(36/ by (126;
17. 14.11
85^2^ "1 J/_
.
7!
133
Required the
ooeffioient of
.?;"
in the
(2«)5'
expansion of
—
i-
-^
j
.
g±M=(2.3..V(2-a.r'=C^)'(i-^)-'
the three terms last written being tliose which produce
by the factor (l-|-3a;-|- Jx*) for we have
.r'*'
after niultiplyii
;
33(|)%3„.x3^(^;:-)%lx35(|)^'
giving for the
The
coeflicit
nt of
J'" iu
the result
coefficient of j" will in like jnanjicr be
Ibi
/'
i
!]
".
ALGEBRA.
56
134
To write
the coeflBcient of
The general term
(2jt
+ l-r)!r!
Equate 4n—4r + 2
in the expansion of (x-
x^'"*^
^)
is
to
(2«+l-r)!r!
x"-
3m+l,
thus
_4n — Sm + l
.
Substitute this value of r in the general term; the required coefficient becomes
(2n+l)\
[i(4n + Sm + 3)]\ [i{4n-Sm + iy
ae
The value
of r sho'
shows that there
is
no term in
x^'"*^
——
"^."^
unless
An
135
l-\-7iXy
136
the
is
an
**
integer.
Ex.
first
approximate value of (1+a?)", when x is small,
x^ and higher powers of x.
is
by (125), neglecting
— An approximation to
two or three terms
(1000-1)*
=
\/y99 by Bin. Th. (125)
is
obtained from
of the expansion of
10-1
.
1000-5
=
10- 3^^
= mi-
nearly.
MULTINOMIAL THEOREM.
The general term
expansion of (a-{-hx-\-cx--\-&G.y
in the
«(»-!) (»-2)...(p+l) ^^„ j,^,^^,
j3,
ql rl si
where
is
,^„«r..,..
...
+ r + 5+&c. =
j;-f-r/
n,
and the number
of terms p, q, r, &c. corresponds to the
of terms in the given nndtinomial.
]> is integral, fractional, or negative, according as n is one
or tlie other.
number
If
138
n be an integer, (137) may bo written
]'
,
pi
a' h"
,
(/
.
r\
(••
</,
.
.
.
.r''+2''+3.,
,
.V
I
I
neduccd
fi-om Mio
Tliii.
Thoor.
MUI/VISOMIA L TIIKOUEM.
Kx.
1.
— To write
Hero put )/=
1(».
,.
f
=
lie
enofficient of a'fec" in tlieoxpanfiinii of {n +{>
1, i>
=
q=\,
'A,
10!
Kesi.lt
:^!
Ex.
2.
— To obtiiin
57
till'
*-
=
=
r.,
s
=
()
-f
in (IMS).
7.8.0.10.
5
CDcflSc-icnt of
.r*
in tlio cxiciusion ol
(l-2,« + 3.i'»-4a;')'.
Here, compariiii,' with (l;{7),
we
q
1
liave
+
'2r
ii
+
=
;is-
\, h
=
= —2, c =
8,
:i,
</
= — 4,
r
+ i/)"
:
ALGEBRA.
58
Employing formula (137), the remainder
of the
work stands
as follows
2M-^})(-l)'"'^"^-*''(-'>"=
iT(-i)(-|-)(-|)l-^^=(-4)'(-)»=
Result
139
The number
nomial
{a-\-h-{-r--\-
The greatest
m terms)",
Proof.
notation
n
15
22f
of terms in the expansion of the multin terms)'' is the same as the number of
to
homogeneous products
and (98).
to
"
of n things of
dimensions.
expansion of
coefficient in the
being an integer,
/'
See (97)
{a-{-J>-\-<^-{-
is
— By making the denoniiuator in (138) as small as possible.
is
The
explained in (96).
LOGAEITHMS.
142
log,,
Def. — TJie
^^'
=
•^'
signifies that a'
= N,
or
logarithm of a nmiiber is the power
base must he raised to produce that number.
143
144
=
log MN =
log—- =
log.«
=
log:;/ j/ =
log (3/)"
l,
log
1
to irliirJt tin
= 0.
log 3/4-log N.
log
.1/
— log N.
«log3/.
J- log
J/.
[li'^
;
EXPONENT! A L
145
'"f^"'
—
^
Til IKlU EM.
59
n;;77/
In miij Imsr is niiml fo
nuMibcr dlrulitl hi/ fhc /ni/(irifhiii of the
hdur, the two last named logaritlnns being taken to any tlie
Tliat
(he
is
'riir
loijnvitJnn-
same
of a
lixjiirilhiii
of
innnhrr
flu'
l^ase at ])leasui'e.
Pi;(iOl'.
—
c
Let, log,. u
= a" =
=
.)'
and
h";
.'.
\i)<y,.l,= i/\
a
= b",
ihvn a
that
=
c-'',
is,
b
= r".
\o<r,^a
=
Eliiiiiiuitc
y
146
l<>K/.<^
=
,—^-
,og,„.v
147
•'>'<
'•=--"
ii'
<.
-'
y.
e. d.
(1 ^5)-
= ;2gby(l4n).
called the modulus of the common system of logarithms
that is, the factor which will convert loi>arithms of nnml^ers
calculated to the base e into the corros]~)ondino; loo-nrit1ims to
See (154).
the base 10.
is
EXPONENTIAL THEOREM.
149
where
*'
c
= + r.r + '-^ + '^ + etc.,
1
=
{a-])-\ ('(-\Y + \ (n-\y-Scc.
=
fl
+ (a — l)j^
Ivxpand tliis Ijy JJinomial Thcoiviii, ami
thus c is obtained. A.ssnme r.,, c„ Arc, as the
coefficients of the succeeding poweis of ;r, and with this assumption write
Form the product of the first two
ont the expansions of a"", a", and a'"*".
Therefore equate
series, which product must be equivalent to the third.
In
the coefficient of .c in this product with that in the expansion of a'*".
the identity so olitained, equate the coefficients of the successive powers of
y to determine Cj, f,, &c.
PiJOOF.
n^
collect the coedicients of
.c
;
.
ALGEBRA.
60
Let
be that value of a which makes
e
150
''^^
=
Proof.
152
=
=
a
154
\og,n
a;
1
=1
\og{l-.v)
158
for
and
>
2/^
+
lot,^ (//
^
,t
= c in (150), we
= log^ a.
obtain
Therefore by (149)
.,.-£-
+ :;^_±-+&c.
= -.v~:^-^-'^-&c.
,r
»»
[154
4
+ :^ + 4^+&c.^-.
in (157);
tluis,
)^ +|(!^y'+i(^f+&c.(
+
(m-\-l
Put ^
159
is, c
l..sl±£=2J,,
=
(2;»5).
in (1-^0).
—
l..g»,
[See
{a-l)-i(a-iy-hi(a-iy-&G.
156
Put
then
^
that
e"" ;
l«g(l+.r)=
.-.
,
+ + ^ + ^ + ^^-
in (149)
155
157
1
'2-718281828...
— B}^ making x =
By making
=
= i + .,.+ |l4.i_4.&c.
,.'
151
c
for
,r
in
,\\m-\-\'
thus,
(157);
^
1
+ !) -loi^M*
;)\m
l'
)
CUNTIS
ri:i>
FRA
<
"I'lOXS.
<;i
CONTINUED FRACTIONS AND CONVERGENT S.
160
ill
tlu"
'l^> liii'l foiiV('rt<( Ml
rule for
100000
99113
II. ('.
F.
ts to
3-M.i:)9
=
;n4ir,<>
roceea hh
d
ALGEBBA.
62
The
convergent
/i"'
is
(lnqn-\-\rqn-l
7»
The true value
therefore
of the continued fraction will
be expressed
by
p^anPn-r-^Pn^.^
163
(f„q»-i-\-qn->
in
which
fraction
ct'^
is
the complete quotient or value of the continued
commencing with
164
a^.
Pnqn-i—Pn-iqn
= ±1
alternately,
by (162).
The convergents are alternately greater and less than the
and are always in their lowest terms.
original fraction,
165
The
difference
continued fraction
between F^ and the true value of the
is
<
>
and
qniqn+qn+i)
quqn+i
Pkoof.— By taking
Also
F
witli a less
tlie
^«
difference,
- Y"^'"^^"
b'irst class of
continued
[
Second
otlier fraction
Fnfrfion.s.
class of
contimu
fraction.
fraction.
1/
_
^1
<'i
hi, itc.
(163)
-
according as F„
F'^
Grucral Theory of Vontlnucd
167
increases.
is nearer the true value than any
denominator.
166 l'\iFn+\ is greater or less than
greater or less than i^,,+i.
ill,
n
this difference therefore diminishes as
and
—
are taken as positive (piaiitities.
^'2
^h
''•—
''.{
— &c.
is
CONTINI'JJJ) Fh'AcriONS.
'.
(ion.
Sec.
'''
,
tlir
It'
IVactiou
is
tcfiiKMl (•(nii/ioiKii Is of the coiit iiiiicd
;ii"('
compoiicnt
he
s
in
iiitiiiilc
iiiiiiihrr,
the coiit
_f'\
1
lit'
iiiiicil
1'-i
.
^1
Pa
.
_
tloiiotod l)v
ki
^^2
^^i
and
.
;
168
IViK'-
said lo bo iuliniU-.
Let the successive conver<ients bo
^'i
{]^
law of formation of the convergents
For
/',
I'^or
so oi
is
r,
I
Pn
•/«
=
=
f. P„
<f„
Un
1
-
I
+
+
'^, />«
f'n
.'
7« -
-1
{
Pn
(
V"
=
—
"
ftn
pn
^^<
qn-l—f^H Hn-l
1
'>„ />«
.'
[Proved by Induction.
The
between the successive differences of the
by (108),
relation
converg-ents
is,
l^_lK^
169
7»+i
Take the
171
—
h„..U„U !K_1K^\
sign for
\q.
qn^\
Un
7'',
and
The odd convergents
th(>
for
i^,
+
v.-i
^ov
^,
Vi
I
'.
^'•\
decrease, and the oven convero-onts, '-, ^-,
'/•'
S:c.,
continually
qi
i^^'O
increase.
ovorv even confollowing convergents.
(lGi>)
Every odd convn'oHMit
vergent
&c., continually
V.t
is less,
than
all
is
—
greater, and
l^i.i'.
If the diffei-onco between consecutive convergents diminishes without limit, the infinite contiiuied fraction
if the same difference tends to a fixed
is said to l)e dcjiiilfr.
value greater than zero, the infinite continued fraction is hidrfinifr ; the odd convergents tending to ..nc value, and the
172
even converu'ents to another.
ALGEBRA.
64
F is
173
Proof.
174
if the ratio of every (|uotioiit to
greater than a fixed quantity.
definite
component
is
— Apply
F
tlie
(1(39) successively.
incommensurable when tlie romponents
infinite in number.
is
next
arc*
all
V
are
proper fractions and
Proof.
175
all
(/„
— Indirectly, and
if a
by (168).
be never less than
/>
+ l,
the convero'ents of
positive proper fractions, increasing in magnitude, Pn and
also increasing with 7^.
By (167; and (168).
ill this case, V be infinite, it is also definite, being
a always =h-\-\ while h is less than 1, (175); and
being less than 1, if a is ever greater than h-{-\.
By (ISO).
176
= 1,
If,
if
177 V is incommensurable when it is
components are all proper fractions and
V (167), we have
180
If in the continued fraction
always; then, by (168),
'p,^=^ hy-\-hih.2-\-h]^h.2,b-i-{-
...
less than
infinite in
and q^z=
to n terms,
and the
number.
1
,
a„
= h„
-\-
1
p^^-{-\.
181 Ifj iu the continued fraction F, a,^ and h,i are constant
and equal, say, to a and h respectively then ^,;„ and (/„ are
respectively e(pml to the coefiicients of x!'"'^ in the expansions
;
/.
h
-
of
1
Proof.
—
p,i
and
— ax —
2
,
q^ are the
a-\-hv
andT
1
J?
w*'' tei-uis
of
— ^^c —
.,.
hxr'
two recurring
See (IGS)
seiies.
and (251).
182
/''>
The
is
(I
Scries info a dnitiuucil Fractiini.
i + ^ + :!l +
series
to a continued fraction
('(jual
poneiits
v(»n'('i't
;
second, and
th(^ first,
1
ir,v
u
iii-{-i(.r
...
+—
1^ 0^'^'^),
with
//-t-l"' coinj)on('iits
u'i
«,,
;/
-|- 1
com-
being
,.r
+ ""
-i<^'
[Proved by Induction.
COSTISUFA) FUACTIOXS.
183
Tlio scries
-+ — + -^+...+
rr^r.
is
05
o(|ual to
ncnts, the
;i
1
r
r
rr,
T
coiitiimod fraction
first,
second, and
//
+ !"'
l\r
J_
'Hie sio-n of w
in
f'''\i'- ... ''»
(1C»7),
with
components
!JI_1±,
may
184
ments
.r"
r'-
'
be changed in
//
+
coiiipo-
1
beini^-
[Proved
eitlicr of
l)v
In.lucfinn.
the state-
(182) or (18;3).
185 Also, if any of these series are convergent and iidinite,
the continued fractions become infinite.
find fhr rahir of a rnntinurd fraction with
To
186
rrrnrrinii: qnitfirnf.s.
Let
tlie
continued fraction be
where
?/
=
--
-
Form the |/"' conso tliat there are m recurring quotients.
'Vhvu, by substituting the
vergent for X, and tlie m^^' for //.
complete quotients a„-\-i/ for a„, and a,,,.,,, -f-// for */„,,„ in (lC),s),
two equations are
obtaincMl of the
from which, by (eliminating
is obtained.
terminimr
//,
a
forms
(iii;i(lr;it ic
;''
1R7
i)i'
!^^
Tf
a colli iinicd
I
nii't
^h—
ion, ;ni<
El
7i'
I^
(/"
(miumI ion
foi'
(h--
ALGEBRA.
66
tlie correspoiKliiii^
by
first
(1<J8), ]iroduces tlie
1
convorgents
vi
'"
"\ developed
continued fraction
hn
(In
then
;
/>„_!
+ (ln-l-\-
(ln--2-\-
+
'"
h,
b.
fh
+ «1
quotients being the same but in reversed order.
tlie
INDETERMINATE EQUATIONS.
Given
188
=
aa-\-hii
c
integral values of x and // wiiich
satisfy the equation, the complete integral solution is given by
from fractions, and
free
a, /3
.r= a — ht
y
where
t
is
Example.
Then X
= fi+at
any integer.
— Given
= 20,
y
=
Src
4<
=
y=
v.alues of x
and y may be
•
.l.
+ r,M
exhiliited as niuler:
2
.S
-i
5
6
20
17
14
11
8
5
2-1
-1
-4
!)
11-
1!)
24
29
7/=-G
For solutions
tliat is, t must be
I
in positive integers
0, 1, 2, 8, 4, 5,
or
/
6,
must
If the equation be
=
solutions are given by
,v—
lie
84
between \"
7
89
= 6j;
and
—t
giving 7 positive integral solutions.
(u—hjl
tlie
'2.
= -2 -I
x= 26 28
t
189
1 1
;
20-:U\
x
The
=
+ -h/
are valnrs
a-\-ht
c
;
—
INDETEnMIXA TE
EXAMPLK
Here X
=
4c -8//
:
=
10,
//
=
~
= 7+
'
,/
The simultiinoous vahu'S
-i
-2
-9
.,
,j
of
;
fiinii.Nli :ill
U
und
.r,
/,
-:i
TIONS.
li>.
satisfy ilie equaticjii
7
t=-l)
= -5
= -rA
FA^J'A
will he as follows
//
--2
-I
1-
7
10
3
7
1
solutions.
tilt'
*
-5-1
:
-J
:;
l:{
10
I'.'
11
1.-)
19
1
positive integral solutions is infinite, ami the least positive
integral values of x and ij are given by the limiting value of /, viz.,
The number of
t>-\that
mast be —1,
is, t
0, 1, 2, 3,
and
t>-\-'
or greater.
190 It" two values, a and /3, cannot readily be found
inspection, as, for example, in tlie equation
17.t'
dlridr
In/
tions to
t,
f/ic
an
+ 13// =
huisf roi'ffirient,
intc/jer;
14900,
and equate
4a— 2
;
=
l-.it.
thus
4
Pat
the re iiiaiiiliKj frac-
thus
*+'+ if ="«+!;,
Repeat the process
by
4
'"
—
ALGfEBBA.
68
Here the number
integers
^
X.
of solutions in positive integers
7
—
,
lymg between
,
and
~ Tq
or
1137
-—
-
is
equal to the
number
of
;
^^^ ^^Tf
t^^t
;
67.
is,
—
191 Otherwise. Two values of x and y
the following manner
may be found
in
:
Find the nearest converging fraction
This
is
—
By
.
(1G4)
to
17
y^.
33
[By
(160).
we have
17x3-13x4 =
-1.
Multiply by 14900, and change the signs;
17
+ 13
(-44700)
which shews that we may
^ take
=
=
,
,
( /5
and the general solution
may be
=
(59600)
a
^^^
59600
written
=
-44700 + 13/,
y=
59600-17^.
x
14900
-44700
This method has the disadvantage of producing high values of a and
192
The values
of x and
equation ax + bi/
satisfy the
positive integers, which
form two Arithmetic Pro-
in
//,
=
c,
gressions, of which h and a are respectively the
See examples (188) and (189).
differences.
193
common
Abbreviation of the method in (169).
Example
Put X
194
y8.
:
= 92,
To
ll.i;— 18;/
and divide by 9
;
=
63.
then proceed as before.
ohld'ni iiifriinil s(thitioN.s'
nf
(H-\-f)t/-\-rz
=
(I.
Write the equation thus
ax -{-III/
Put successive integers for
;:,
= —
(J
cz.
and solve for
.r,
//
in encli cnse
ItEDUGTION OF A QtlADJiATfC srUD.
A QUA1>HATI(^ SLIHI) TO
CONTINUED FRACTION.
TO
195
GO
Iv'KDlTCF]
EXAMI'I-K
:
^29=
5+v/29-r,
=
y29 + 5_ ^, v/29-:5^
'^^
4
.
~
^
5
v/29 +
"
2_
.
,
~
5
/29 + 3_
—4
-
"^
5
^
5
^^29 + 2'
v/29-3_
^
5
V
,29 + 5'
"^^ ,'29 3'
+
v/29-5
,
'^
5-h
4
_
+ "—4—-
g
^
+ 5 = 1U+
^/29
A
29
-5 =
4
,
^V^29 +
2,
:>'
•
^
+
10
+
v^29+.y
v/29 +
5'
Tlio (iiiotients 5, 2, 1, 1,2, 10 arc the gTcatest integers
contained
The quotients
now
tlie
in the quantities in the first cohimu.
recur, ami the surd \/29 is equivalent to
continued
fraction
1_ 1_ 1_
5+
2+1+1+
The convcrgents
5
T'
11
2'
1_ 1_
J
2+ 10+ 2+ 1+ 1+
to v/29,
27
10
3'
5'
1
1
formed as
70
727
13'
135'
]
2
in (IGO), will
1524
283'
2251
418'
+ c^c.
be
3775 9801
701' 1820'
Note that the last quotient 10 is the greatest antl twice
the first, that the ><i'ron(l is the first of the recurring ones, and
that the recurring quotients, excluding the last, consist of
pairs of equal terms, <[iu)tients e(|ui-distant from the first and
These properties are universal. (See 204
last being ecjual.
196
-210).
To
197
Suppose
for)) I
m.
the
liiii'h
ro)H'C)'i»'r)ifs rnpUllji.
number
of recurring (piotients, or
any
ALGEBRA.
70
multiple of that number, and let the m^^' convergent to \/Q be
represented by F„,; then the 2??^"' convergent is given by the
formula
198
i'or
quotients.
=i
F„„
.Jf;,+ -^^- by (203) and (210).
example, in approximating to \/29 above, there
m 2x5 10 therefore, by
=
=
Take
i^,y
=
^-—
—
the
,
ai-e five
recurring
;
convergent.
10^'^
1820
Therefore
F,,
=
{||^^
the 20^'' convergent to \/29
convergents is saved.
;
1820
)
)801
)
+ 29x^
^
and the labour
192119201
35675640^
of calculating the interveninf
GENERAL THEORY.
199
'J'he
may bo
process of (174)
=
Tli(!
Tl
v/
(juotients
tions on
g
'/,, rr,, a.^,
tlie left.
=
f
/
iScc.
,
:-
a.
^±^ = ..+
200
exhibited as follows
''Q + r„,-
1
1
+ a, -h
a,
1
+
a^-irScv.
are the integral parts oF the
fi'ac-
A
Ri'iDUCTios OF A
201
•'•I'intions coinicct
'I''"'
r,
r.,
Tlie
r=
(I.
=
^/,.
substitute for
(/„
of
:
—
All the (juantities
205
'I'Ik'
206
No
207
n
208
I'^or all
>i
/„
'Hi'-
The
last
The
first
(7o,
7-0,
r.,
greatest
or
/•
= 1,
c is
^^''^^^t
[By
tliis
r„
values of
number
(luotient
becomes
^
'
",
Tndnctioii.
wIh-ii
of wliicli
to (203) tlie following theorcius
(/,
r.,,
r,
and
and
c,
r ai'e
we
<i„
ai'*^
positive integers.
= a^.
can be greater than
then
iii-c
This gives
'.)<))
204
ii'S
bo
the complete quoticDt
relations (1
tlie
i^
it
^=:^
^iiLZ!zL-_Lii!±l2_
v'^^
only the integral part.
demonstrated
(|ii;iiit
—
,-.=
,r
r()iivorn:(Mit to \/(? will
?/*''
The tnu> value
209
tlic rnniiiiiiii^-
>:5
^=
By
iii^-
71
h' srii'it.
i>i:.\'r
)'.,—(
202
is
(,n'
2'/,.
= a,.
n groat(>r tlian
1,
rt—r„
is
<
>•„.
cannot be greater than 2a'l
and after that the terms repeat.
of (juotients
is
2(i,,
complete quoti(>nt that
commence each
is
repeated
is
cycle of re}»eated terms.
^
\
'\ and
ALGEBRA.
72
210
then
be the last terms of the first cycle
respcctivcly equal to rto, r.,, c-2', «,«-2j ''m-2j
rp^ (187).
are equal to a.^, r.^, r.^, and so on.
I^et
</,„,
<*»n-ij '''m-ii
c„,_2
r,„, r,„
Cm-\
;
^^^'c
EQUATIONS.
Special Cases in the Snlntion of Simidtaneous Equations.
211
First, witli
a^e-\-hyii
If the
=
two unknown
i\\
_
^
denominators vanish,
^=
«2
'\
quantities.
_
cA—Co bi
w^e
and X
^1^2
— ^2 ^1
have
=
cc,
?/
= 00
;
b.,
unless at the same time the numerators vanish, for then
a._h,_c,,
0.
c./
'
a,
~h~
'^
~
0.
'
and the equations are not Indepmdent, one being produced by
multiplying the other by some constant.
212 Next, with three
the ecpiations.
If
(l^,
(L,
(I,;
unknown
all vanisli,
quantities.
See (60) for
divide each equation by
two
have three equations for finding the
ratios
-'-
.v,
and
and we
•
,
two
only of which equations are necessary, any one being deducible from the otlier two if the three be consistent.
213
^«
solve simultaneous equulions
bt/
huleterntnuitr
Multipliers.
Ex.
— Take the equations
+ lip
+ 2// +
+ r,//+ 7,v+
bx + 8y-\- 10,v - 2/r
7x + 6y +
+ iw
,/'
;lv
//•
'5.'c
5,^
=
=
=
=
27,
-l-S,
05,
53.
MISCELLANEOUS KQJWTIONS.
7:\
Multiply the first by A, the second by /?, the third l)y C,
one C(inution nnnniltiplied and then add the results.
leaviiiLi;-
Thus
;
+ (2.l-h'")/>'4-8^' + (;)//
+ (;5J +7/>' + l()C'-f-5) .v-[-(4J + /;-26'^-4) w
= 27-l + 18Zy + GoO + 5;].
(J+3//4-5r7-{-7)./'
To determine either of the unknowns, for instance .f,
equate the coefficients of the other three separately to zero,
and from the three equations find A, B, G. Then
*
^ 27J+48/y + G50 +
o:]
A + W-\-hG-\r7
'
MISCELLANEOUS EQUATIONS AND SOLUTIONS.
^''±1
214
Divide by
x^,
and throw into
=
0.
factors,
by
(2)
or
See also
(o).
(480).
.r'-7d-i)
215
=
i).
=
—1 is a root, by inspection; therefore ''+1
X
Divide by x-\-l, and solve the resulting (quadratic.
aH n; =
216
I'
x*-\-lC).>-
=
+
2
•'
.i;"
Rrr-E.
factors,
— Divide
if
=
l-j.'),/'
is
a factor.
!'>''>.
=
-""^
(')<)
X
7,/',
2'
= 7.
/i''
the absolute* terra
(here
possible, such that one of them,
of the other, equals the coefficient of x.
solution of a cubic equation.
I.
455) into two
minus the
scpian*
^^ee (483) for i^cMicral
;
74
ALGEBRA,
217
= 145(>0,
= «+v and
.r*-i/
'*"t^^
.f
Eliujiiiate
c,
and obtain a cubic
218
.i^-/
= 8.
= z—v.
.r-v
2/
in 2,
= 3093,
which solve as
in (216).
= 3.
ci— 1/
Divide the first equation by the second, and subtract from
result the fourth power of x—y.
Eliminate {x^-\-if)j and
obtain a quadratic in xij.
tlie
On forming Symmetrical
219
Expressions.
Take, for example, the equation
(y-c){z-h)
'Vo
= aK
form the remaining equations symmetrical with
this, write
the corresponding letters in vertical columns, obser\dng the
circular order in which a is followed by h, h by c, and c by a.
So with
X,
;?/,
and
z.
Thus the equations become
0/-rj {z-h)
[z-a) Gr-e)
{.v-h){y-a)
To
= a\
= b\
= c\
solve these equations, substitute
x
= h + c,-\-x\
y
= c-\-a +
y',
and, ]nulti})lying out, and eliminating
::
//
=a
and
-{-
;:,
b
-\-
::'
we obtain
^^ho{b + r)-a(lr-hr)
hc
niid
tlicrefore,
— ca — ab
by symmetiy, the values of y and
;:,
by the
niK> just given.
+ + //- = ^r
:^-^-.r-\-x.v = fr
'H/ + .*//-r^
220
•••
//
:5(//.r
+
.v.'
+ ,o/)-=r
.^'
:l/n--\-2r',r-\-2'rlr-a'-b'-c*
(1),
(2),
(3);
(4).
—
iMAniXAUY
Now
add
(1), (2),
and
From
(3),
+
// -|(4) and (5), (,r
(o) are readily solved.
and
221
and
;:)
(2)
Result
by
X"
=
«'^
(I),
jr-z^=fr
(2),
z^-.n/=f-
05).
(3./'//^
- Jf -if-
X
_
/A.2_ft^
Obtain
o])tain
obtained, and then (1), (2),
and subtract
(:>),
X
av(>
is
.,.-^^;/~
Mnltiply
75
i<jxi'i:j:ssi()xs.
square of
tlie
::'')
(1).
= h'<- -n\
y
^
(-V--/.*
a'b''-r'
by proportion as a fraction
witli
^^^'
^
numerator
= x^ — yz = a^.
.v=n,-\-bz
222
,^
z
Eliminate a between
(2)
=
=
and
(1),
az^c.i
(2),
Lv-^(a/
(3).
and substitute
(3),
tlie
value of
X from equation (1).
—
XieSUlC
-r-"
r^
'„
j:
;,-
IMAGINARY EXPRESSIONS.
223
'Hic following are conventions
:
—
equivalent to a^^{—li); that a y,/{
\)
vanishes wlien a vanishes; that the symbol a,
1) is sub\'(— 1) is denoted
ject to the ordinary rules of Algebra.
That v/(-'f-)
is
y(—
l)y
/.
ALOEBEA.
76
224
225
duct
If a
«
=
?'/3
=
and
//3
7
a
i^
-f-
—
/'fS
then
;
=y
a
and
(i
=
B.
are conjugate expressions
;
tlieir
pro-
a- 4-/3-.
The sum and
226
both
+
+
real,
227
The modulus
228
If the
229
If
two conjugate expressions are
])roduct of
but their difference
imaginary.
is
-\-x/a^+^^.
is
modulus vanishes,
and
a
must vanish.
/3
two imaginary expressions are equal, their moduli
are etiual, by (224).
sions
of the product of two imaginary expresequal to the product of their moduli.
The modulus
230
is
Also the modulus
quotient of their moduli.
231
of
the
quotient
is
equal to
the
METHOD OF INDETERMINATE COEFFICIENTS.
+
+
+
=
+
+
.4'
be an equa=.
B'x -\- G'x'
232 If A Re a«tion which holds for all values of .^', the coefl&cients .1, B, &c.
A',
that is,
B\ C
G', &c.
not involving ;r, then
.
.
.
=
B=
A—
. -_.
;
Proved by
the coefficients of like powers of x must be equal.
See (234) for
putting X
0, and dividing by x alternately.
=
an example.
METHOD OF PROOF BY INDUCTION.
233
Ex.
— To prove that
5
Assume
1
+- +
•»
-\-
...
+ir
=
-
-"-.
;
iwirriAL FiLurnoxs.
= »(» + !)
+6
(2n + l)
6
^
(« +
(n
+ 1)- ^
(m
{n (2n + l)
-l-G
r«-H)}
(5
lU» + 2)(2u + 3) _
n (n'+\){ 'l n+l)
'
6
G
where
+ 1)
'
n' is
written for
«+l
o
It is
thns proved that i/ the formula he true for n
But the formula
true when n
4
true when >t
is
therefore it is
therefore universally true.
trial
;
=
=
;
it is
also true for
n+
1.
2 or 3, as may be shewn by actual
therefore also when n
5, and so on
=
;
—
234 Ex. The same theorem proved by the method of Indetermiuate coefficients.
Assume
1^2^ + 3-+.. .+n^
.-.
1
=A + Bn
+ 2- + 3-+. ..+»' + (« + !)-
therefore,
=
.-l+5(/^
+Du^
+Cn^
+&c.;
+ l) + (7(« + l)- + D(n +
l)''
+ &c.;
by subtraction,
«H2n + l = B + C(2n + l) + D{3n' + Sn+l),
which contain higher powers of n than the
highest which occurs on the left-hand side, for the coefficients of such terms
may be shewn to be separately equal to zero.
Avriting no terms in this equation
Now
equate the coefficients of like powers of n
3jD=
;
thus
1
1,
,
and
^
=
0;
2
;
ALGEBRA.
78
—
235 First. When there are no repeated factors in the denominator of the given fraction.
Ex.
—To resolve
^''^""^
3a;—-—
into partial fractions.
"^
^-2 "^ x-3
(a-l)(J"-2)(«-3) " ^:il
= A(x-2)(!c-S)-\-B(x-S)(x-l) + C{x-l)(x-2).
'
Sx-2
C do not contain
then
and
Since A, B,
of X, put x
=l
;
3-2 = ^(1-2)
Similarly,
—
:r—,
{x—l){x—2){x—o)
x,
and this equation
B = -4
.-.
;
and, putting
a;
3a;
-2
Secondly.
Ex.
—Eesolve into
These forms are necessary and
7x'-lOx' + 6x
A
\'
7
a;-2
2 (a;-3)'
a repeated factor.
fractions
partial
^
lx^-\Ox'^^x
(-^=i?(.:t^
.
A^«"^«
is
=
^
2(a!-l)
—When there
G
.-.
^ __1
(a;-l) (a!-2) (a;-3)
236
values
x be put
if
H ®°°®
all
A = ^.
(1-3), from whicli
= 2, we have
6-2 = i? (2-3) (2-1)
= 3,
9-2 = 0(3-1) (3-2);
true for
is
i-
—
B
.
j!
-.,„'
(a;— 1)^(33
+ 2)
-
C ^ D
= (^^» "^ (^::ir "^ ^^ "^ ^rr2sufficient.
Multiplying up,
we have
= A ix-\-2) +B (x-1) ix + 2) + C (x-iy(x + 2) +D (x-iy
(I).
Makea;
= l;
Substitute this valne of
7x'-lOx' + 5x-2
Divide by
a;
—1
;
X
=
1
= ^(1 + 2);
A
thus
in (1)
;
.-.
^
=
1.
^ B (x-].){.v i2) + C (x-iy(x + 2)+D (x-iy.
thus
7x'-Sx + 2
Make
7-10 + 6
.'.
=
B(x + 2) + C(x-l)(x + 2)+D(x-iy
7
again,
Substitute this value of
B
-3 + 2 =
in (2),
J? (1
+ 2)
;
.-.
B=
(2).
2.
and we have
= G (x-l) (x + 2) +D (x-iy.
+ 2 = G (x + 2)+D (x-l)
time,
7 + 2= C (1+2);
C = 3.
7a;*-5a;-2
Divide by .T-l,
Put
a;
=1
a thiwl
7a;
.-.
(3).
79
I'AirriAL FRACTIONS.
Ijastl}^
make
a;
= —2
in (3),
-14 + 2 = X>(-2-l);
1
Result
7
z—j.
(a— 1)'
+
TTT H
7
is
roots not repeated.
Ex.— Resolve ,t—tw^2-.
i.
4
,
«-l
(a;-l)
Thirdly.— When there
237
D=
.-.
3
2
H
a;
2
+rii"
a quadratic factor of imaginary
—
into partial fractions.
ix
,
Here we must assume
Cx + D
Ax-{-B
+ + l)
(a5»+l)(j!»
a;^
a!
+l
x'
+ x+l'
x-i-l and X- + X + 1 have no real factors, and are therefore retained as
denominators. The requisite form of the numerators is seen by adding
too'ether
°
Multij)l}iiig up,
1
we have
=
a;-
Substitute this value of
=
+l =
repeatedly in (1)
x^
coefficients to zero
;
thus
= -Cx'-Dx =
(G-D)x + C-l=0.
= 1,
thus
^
;
1
XT
^^'"'^^
(..^
Fourthly.
_
=
—When
there
of imaginary roots.
40.^'
Cx + C-Dx-
'
+ l)(x^ + * + l)
Rv —"Resolve
;
.r
1)=
238
thus
;
{Cx + D) i-x)
or
Equate
z.
repeatedly
+ + l=0;
x-=-x-l.
ar
Substitute this value of
(1).
.'.
;
.-.
=
+ l)
= Ax' + Bx = -A + Bx
Bx-A-l = 0.
5 = 0,
^ = -1.
let
1
0;
(x'
x^ = —I.
(Ax + B) X
coefficients to zero
Again,
r~,-
x+d
+ x + l) + {Cx + D)
(x'
x- in (1)
or
Equate
x+b
the equation
(Ax + B)
Let
1
——- ^
fractions, such as
two simple
^
1.
''+1
.tHx + 1
is
— 103
_.
^
a-Hl
a repeated quadratic factor
^
i)artiiil
fractions.
—
;
ALGEBRA.
80
Assume
40.7;
(x +
-103
_Cx±B _
Ax + B
^
iy {x'-4x + Sy
(.r2_4x
+ 8y
(.'?;--4a;
(.r+l)-
=
a;--'-4^
+8
-^ + -^;
+
4-
40.t;-103
Ex + F
+ 8)-
l'
a;
{iAx + B) + {Cx + D)ix-—ix + 8) + iEx + F)(x--4:X + 8y}
+ {G + H(x + 1)} (x'-4x + Sy
to determine A and B, equate rt;-— 4a; + 8
{x
+ l)(1).
In the first place,
to zero
thus
a;2=4a;-8.
Substitute this value of x- repeatedly in (1), as in the previous example,
until the first power of x alone remains.
The resulting equation is
40a;
Equating
-103=
coefficients,
we
(17.4
;
+ 65) -48^ -75.
a?
obtain two equations
17^ + 65= 40
48^ + 75 = 103)'
)
..
,
f
^^«--^^«^^
A =
2
B=
l.
Next, to determine
and D, substitute these values of A and 5 in (1)
the equation will then be divisible by a;^— 4a; + 8.
Divide, and the resulting
equation is
=
2x + l3+{Cx + B+(Ex + F)(x'-4x + 8)] (x +
iy
+ {G + H(x + l)]{x'-4x + 8y
(2).
— 4a; + 8 again to zero, and proceed exactly as before, when
Equate
finding A and B.
Next, to determine E and F, substitute the values of (7 and D, last found
divide, and proceed as before.
in equation (2)
Lastly, G and
are determined by equating a' + l to zero successively,
a;'-
;
as in
Example
H
2.
CONVERGENCY AND DIVERGENCY OF
SERIES.
(7„, a„+^ auy two
convergency may
be applied. Tlie series will converge, if, after any fixed term
(i.) The terms decrease and are alternately })0sitive and
239
Let
ai-{-a.^-\-a;i-\-&c.
consecutive terras.
be a
scries,
The foUowino-
and
tests of
negative.
(ii.)
Or
if
"(' n 1-1
greater tlian unity.
is
always
(j
renter
than
some
(piantity
SERIES.
(iii.)
Or
if
——
'''1
ill
known
a
(iv.)
Or
tity greater
(v.)
Or
+
i.s
never
coiivei\u:ing series.
if
l-^—n)
241
The
tjreafrr
than some (juan-
[% -
than unit3^
if
l^-^—ii — l]\og)i
The conditions
(i.)
always
is
always
is
tl'
i/rrdfcr
and
tlian
unity.
of divergency are obviously the converse
to (v.).
series
converges,
ai-^a.,x-\-a.iX^-{-&c.
always less than some quantity p, and x loss than
if
^^
1
[By 239
242
To make
(iiiantitv /s
iii.
^
some quantity greater than
of rules
less tluui tlic corrcspoiidiii^ I'atio
1
V^'»j+i
240
81
the
make
,v
sum
(ii.)
of the last series less than an assigned
less than
,
,
I'
hvincr the o^reatest co-
efficient.
Grnrral Tltcnron.
243 If ('") be positive for all positive intec^ral values of .r,
and continually diminish as <> increases, and if )n be any positive integer, then the two series
•/>
<^(l)
+ (^(2) + (^GJ) + ^(l) +
<li{l)-\-m(f>{m)-\-m-4>{m-)-\-m''(t>{m')-Y
arc either both coiivern-ent
244
is
Ajiplication of
tliis
oi*
diverofent.
theorem.
diverjT^ent or convero-init
when p
41
To
asc<'rtain whotlier the
is «^i-eater
than unitv.
ALGEBRA.
82
Taking
m = 2,
tte second series in (243) becomes
1.2,4,8,0
^
^ ^
2'^
8^
4p
a geometrical progression whicli converges
245
series of
I'lie
which --:-
^,-
(log ny'
convergent if j9 be greater than unity,
not greater than unity.
is
therefore the
;
the general term
is
?i
The
246
series of
and divergent
if
p be
[By (243), (244).
which the general term
is
1
n\{ii)X'{n)
where \
(n)
V{n){r^'(n)}^'
signifies \ogn,X'^{n)
signifies log {log ()i)},
and
convergent if ^ be greater than unity, and divergent
[By Induction, and by (243).
be not greater than unity.
so on,
if j)
247
is
The
series ai
+ cu + SLC.
ncu log (n) log2 {n)
is
convergent
if
log''(7i) {log,^i
{n)y
greater than unity
always finite for a value of p
here signifying log (log iz), and so on.
is
;
log'' (7;)
[See Todhunter's Alr/ehra, or Boole's Finite Bijjcrences.
EXPANSION OF A FRACTION.
248
A
fractional expression such as
:,
——
\
42.'
—--'.
10a3
—--
bx-\-l\x^
—
may
Q>x,
be expanded in ascending powers of x in three different ways.
First, by dividing the niiraerator by the denominator in
tlie ordinary way, or by Synthetic Division, as shewn in (28).
Secondly, l)v the metliod of Indeterminate Coefficients
(2:32).
Thirdly, by Partial Fractions and the Binomial Theorem.
—
SERIES.
To expand by
Assume
'^''^
method
tlie
proceed as follows
83
of Indeteriiiiii:ite CoefficiLiits
:
=
~\^'^'
,
.
4x-lUr = .1+
,
-1
+
^'•'-
+ C.>- + J).c' + E.v' + & c.
llx+
Cx--\-
nx''+
Ex'+
I<\r''+...
— OAx—
GBx-—
GC'u;*-
Gi*./;'—
OiiV'-...
+
ll.lj;-4-ll/^a;'
Et|uate cocUicients of like powers of
.1
C-
JJ- 6A
6B + IIA
x,
=
=
+ liac*+llA/'+...
6Bx*- G^.V-...
G.-Lc'-
thus
U,
=
4,
.-.
J!
=-lO,
.-.
C=
0,
.-.
1)= 40;
U,
.-.
E=UO;
0,
.-.
i''=;30-i;
D-6C+UB-GA=
E—6D + IIC— OB =
F-6E + 11D-6C=
I
;
li;
The formation of the same coefficients by synthetic division
exhibited, in order that the connexion between tlio two processes
clearly seen.
The division of 4a; lO.r l)y 1— ('..i'-f U.r-G,/' is as follows:—
is
now
may
be
—
+ 4-10
+
24 + 84 + 240 + GGO
6
-44-154-440-1210
+ 24+ 84+ 240 + GGo
-11
+
6
^l
If wc>
;i04.j;''—
stop
'JTO/^
:it
sum
the term llO.r', then the undivided remainder will
and the complete result will be
lie
+ CtiOi/,
4.r
249
+ 4+14 + 40 + 110 + 304+
F
n C D E
+ 14x- + 40..H110. +
^_^^^,fZ:^-
Here the conchidiiig fraction may be regarded as the
to inlinity after
original expression
is
four terms of the series, just as the
considered to be tlie sum to inlinity of
the whole series.
Tf the general term be reipiired, the method of exSee (257),
pansion by partial fractions must be adopted.
wliere tlie Lrcneral term of the foregoing series is oljtained.
250
.
ALGEBRA.
84
RECURRING SERIES.
&c. is a recurring series
connected by the relation
a^^-\-(ii.i'-]- Uod'- -\-ayv'^-\-
efficients are
251
(In
The
= Ih «« +
-
1
Scale of Relation
7>2
«« - 2
+ + Pm
•
•
.
if
the co-
(in - m-
is
1 -PI^V -JhO^ —... —lhn^V''\
252
The sum
[The
253
of n terms of the series
first
m
is
equal to
terms
— l terms + the last term)
m— 2 terms + the last 2 terms)
—IhJC^ (first m— 3 terms + the last 3 terms)
—piV
(first tn
—p^x^
(first
-~i>i«-i'^'""^ (first
—p,nX"' (the last
term + the last m — \ terms)
m terms)] -^ [l—p^.v—pocV^— —
/>,„cr"'].
...
If the series converges, and the sum to infinity
quired, omit all " the last terms " from the formula.
254
255
Example.
— Required
term, and the apparent
4'c
sum
+ 14r + 40,v^ +
Observe that
six arbitrary
is
re-
the Scale of Relation, the general
to infinity, of the series
110,ii'^
+ 304^^-8o4/+
terms given are
sufficient to
...
determine a Scale
form l—px — qx' — rx^, involving three constants p, q, r,
by (251), we can write three equatious to determine these constants
The solution gives
110= 40p4- 14(2+ 4r\
namely,
of Relation of the
for,
;
304
854
= llOp + 402 + 14r k
= 304;j+110g + 40rJ
Hence the Scale
The sum
Pi
= ^,
Pi
of Relation is
1
p
=
P3
The
=6,
first
— 6xthe
-I-
1 l.r
7
=
- 1 1,
=
6.
found from (254), by putting
m = 3.
th ree terms
two terms
X the first term
first
r
— 6.« + ll.r — 6.r^.
of the series without limit will be
= — 11»
G,
= 4,c + 1 4.r +
= —24^-— 84a;*
=
+ 44a;'
40.i-''
4«-10a:-
RE CURBING SERIES.
4.r-10x'
^^ -
G.i;
1
tlie meaning i)f
powers of x, the
256
which
is tliat,
terms
first six
To obtain more terms
tlius the
+ 1 Ix* -<;.(•»'
fmction bo expiunluil in asuentling
bo those given in the question.
this
if
will
of the series,
we may
use the Scale of Relation
;
7th term will be
(6
X 854- 1 1 X 30i + 6 X 1 10)
=
a:^
2440a;^
find the general term, S must be decomposed into
fractions; thus, by the method of (2'35),
To
257
l)artial
_
4.B-10a;-
l-6.c +
By
the Binomial
,
1
Theorem
^,
Sx
—
r-=-r
l—zx
—
1
2
8
1— 2x
1-a;
,
l-Sx
ll.j;'-(3a;-'
(128),
=
l+3.c + 3-.r +
=
-
+S''x'\
+ 2=.c + 2^r +
+ 2" *
=-3-3.c-3.r -
-Sx".
Hence the general term involving
'.c",
x" is
+ 2''»'-3)x''.
(;3'>
And by this formula we can write the " last terms" required in (2.")3), and
Also,
so obtain the sum of any finite number of terms of the given series.
by the same formula we can calculate the successive terms at the beginning
expeditious
than
In the present case this mode will be more
of the series.
that of employing the Scale of Relation.
into partial fractions for the sake
If 5 in decomposing
of obtaining the general term, a quadratic factor ^vitli imaginary roots sliould occur as a denominator, tlie same method
258
-<S'
must be pursued,
for the imaginary quantities will disappear
In this case, however, it is more conSup[)ose the fraction
a general formula.
in the final result.
venient to employ
which gives rise to the imaginary roots to be
L-\-Mx
<i+b,r
p and
+
x"-
_
~
L-\-Mx
{p—r){q-x)'
q being the imaginary roots of ri-{-hx-{-x'
Suppose
j)
q
=
= a—ij^y
= 0.
(i-\-i)^,
where
i
=v
—1.
,
ALGEBRA.
86
If, uow, the above fraction be resolved into two partial
fractions in the ordinary way, and if these fractions be expanded separately by the Binomial Theorem, and that part of
tlie general term furnislied by these two expansions written
out, still retaining j) and r/, and if the imaginary values of p
and q be then substituted, it will be found that the factor will
disappear, and that the result may be enunciated as follows.
The
259
coeflQcient of
^
a;"
in the expansion of
L + iU.r
be
will
^g^-^[Ha«-'^-C(«,3)a"-^HC(«,o)a»-^'-...l
^^"+^^
260
+C(«-l,5)«"-«j8»-...}.
With the
aid of the
known expansion of sin nO
term may be reduced
Trigonometry, this formula for the
6
in wliich
If n
=
tan
—
<^
=
ti^'*
tan
be not greater than 100, sin
(viO
in
to
^
L + 3/a
— ^) may
be obtained
from the tables correct to about six places of decimals, and
accordingly the «*'' term of the expansion may be found with
corresponding accuracy. As an example, the 100*'' term in
the expansion of ^
,
to be
To
261
41824
—
^
.,
is
readily found by this
method
-^- x^.
„9
(/cfrrniinc
whether a
may
Scries
i.s
rernrri)i;ii- (tr not.
terms only of the series be given, a scale
be found which shall produce a recurring
If certain tirst
of relation
i>;ii'cn
BECUBRINQ
87
Sl^RII'JS.
The method is
whose first terms arc those given.
The innnber of niikiiowii coefficients
exemplified in ('255).
must be
j), </, r, &c. to be assumed for the scale of relation
equal to half the number of the given terms of the series, if
If tlie nund^er of given terras be odd,
that number be even.
it may he made even by })refixing zero for the first term of
the series.
series
Since this method may, however, produce zero values
one or more of the last coefiicients in the scale of relation,
it may be advisable in practice to deteruiine a scale from the
first two terms of the series, and if that scale does not jn-oduce
the following terms, we may try a scale determined from the
first four terms, and so on until the true scale is arrived at.
If an indefinite nnmber of terms of the series be given,
we may find whether it is recurring or not by a rule of
Lagrange's.
262
for
263
Let the series be
S= A + ]Jx + G.r -h Dx'
-f-
&c.
Divide unity by S as far as two terms of the quotient, wliicli
will be of the form p-\-qx, and write the remainder in the form
/S'V, S' being another indefinite series of the same form as S.
Next, divide S by S' as far as two terms of the quotient,
and write the remainder in the form S"x-.
Again, divide /S" by S'", and proceed as before, and repeat
this process until there is no remainder after one of the
The series will then be proved to be a recurring
divisions.
series, and the order of the series, that is, the degree of tlie
scale of relation, will be the same as the number of divisions
which have been effected in the process.
—
To determine whether the series
recurring or not.
Introducing x, we may write
KxAMi'LK.
45,
...
1, o,
(i,
10, 15, '21, 28,
is
S
Then we
=
.shall
l+3.r + 6.r+10.rH15.):' + 21,/' + 2,V' + :](u''+45..:\..
have
'
= 1 — ;lf4-...
6x'
TlH-iir..io
S'=
^vith a
+ 8.r' + 15.1!* + 24.c''
3
-f-
rcniaimler
obx''
+ itc.
+ 8.j;+15.7;-4-2kr» + .'3.V + A-c.,
.s"
;;
9
.
.'U),
ALGEBBA.
88
with a remainder
Therefore we
may
Lastly -^„
+ 8a;H
-^ (.r
=
take S"
= 3—
.13
().)'
1 +8.1-
+
10.x-'^
+ &c.
...)•
+ 6.r + 10a* + &c.
without any remainder.
Consequently the series
is
a recurring series of the third order.
It
is,
in
the expansion of
fact,
SUMMATION OF SERIES BY THE METHOD OF
DIFFERENCES.
—
264 Rule. Form successive series of differences until a series
Let a, b, c, d, &c. be the first
of equal differences is obtained.
terms of the several series then the 7^^'' term of the given
;
series is
265
a+i.-m+ ("-^(r'^ e+ ("-^("-f-^ .l+
The sum
of
n terms
=„« + !^M4 + ^iIi=ii§^<- + &c.
266
Proved by Induction.
5 + 15 + 35 + 70 + 126 +
+ 10 + 20 + 35 + 5G +
a...l+
Example:
h ...4
c
...
...
6 + 10
(Z...4+
e
The
lOOti'
term of
...
1+
tlie first
^
The sum
1
+
...
series
1.2
^
1.2.3
^
1.2.3.4
of 100 terms
,,,.,^100.09
==
+ 15 + 21+...
5+ 6 + ...
100+
,
.100. 99. 98«j^ 100. 99. 98.97 ,, 100.99.98.97.96
^^
1.2.3.4T.r1.2.3.4
-^^4+-^-2r3-^+
FAiToh'fAl,
267
ililTtTeiices.
— Given log 71,
Form
SO
/•;>'.
interpolate a term bclwecn two terms of u series by
'^'-^
the motliod of
Ex.
^
,s7';/.7
log 72, log 73, log 74,
it is
required to find
log 71
6
...
r
...
(Z ...
log 72
l-8ol2".83
log 73
('itJG),
log 74
l-8573:i2')
l-8G:i3229
-0060742
•00r)9904
-0059088
--0000838
—00000-22
- -0000816
a...
7"2o4.
I<>g
tbo scries of diBerences from the given logarithms, as in
1-8G02:U7
coii.sidered to vimish.
Log 72'o4 mnst be regarded
as an interpolated term, the
number
of
its
place being 2-54.
Therelore put
2-.'')l.
for n in
formula (265).
log 72-54
Result
=
1-8605777.
DIRECT FACTORIAL SERIES.
268
5.7.9 + 7.0.11
Ex.:
d
=
m=
n
a
>*'•
269
term
To
=
=
common
number
first
find the
.1:3.15
+
..
of terms,
factor of
sum
first
term —d.
+ n + i</)
(^/
+ + //<;/
r/).
I
of u terms.
the last term with the ne.rt hitjlirsf fnrtor take
next loiiwst factor,
and dlrlde
Ixj
{m
+
1 )
'/.
— By Induction.
Thus the sum
n
+ 11
miml)er of factors in each term,
thefi'i\st ti'Dii ir'ilh flie
Proof.
<.i.n .1;]
difference of factors,
== (fi-\-n(/) {a
Rule. — Fnnii
-I-
= 4, a = 3,
of 4 terms of the above series will be, putting d
^,
h
=
= 2,
vi
— 3,
11.13.1 5.17-3.5.7.9
^j^^,
Proved either by Induction, or by the
ractliod of
.
Indeterminate
Coefficicnt.s.
)
ALGEBRA.
90
INVERSE FACTORIAL SERIES.
Ex.:
270
Defining
;}tii
+ 7^9_ii + 9.11,13 + 11.13.15+---
5^7^9
d, in,
oi,
a as before, the
1
=
term
+ n-\-l(I)
(n-]-ud) {a
To
271
factor,
its last
—
d.
of 4 terms of the above series will be, putting d
272
13.15
5.7
= ^'" = ^^
= 2, m = 3,
1
1
Proof.
example.
— id)
Rule. From the first
of n terms.
factor take the last term wanting its first
and divide hy (m — 1)
Thus the sum
^^
{a-^n-\-m
sum
find the
term wanting
...
(3-1)2
•
— By Induction, or by decomposing the terms, as in the following
To sum the same series by decomposing the terms into
Take the general term in the simple form
Ex.:
fractions.
partial
(r-2)r(r + 2)
Resolve this into the three fractions
its
Substitute 7, 9, 11, &c. successively for
equivalent the three series
Ij
1+
8 1
5
2^ C
_
+
_
_
9
9
of n
2
1
2«
7
is
+5
+ 1
_
2
+3
_
2_ )
2n + 5)
2
2» + 3
13
+ _i_ + _l_ +^1-1,
7)*
13
2»i
+3
2u + 5
2»
+_l_l=Jil
2n + 7)
4(5.7
1_
(2» + 5) (2«
—
•^
-t-
(2n
+
seen, Jiy inspoction, to he
taking -—a result ol)taiiicd at once by the rule in (271),
^
'^
term, and
series has for
+_1_|
2u
13
11
11
11
terms
1(1_1_
8(5
2
+i
1 + 1 +1_+
j
8l
and the sum
1
9
7
'7
8 I
1
2
1+
and the given
r,
.--
+ 3)(2n + 5)(2n + 7)
5.7.9
for the
»i"'
or last term.
+ 7}
]
'
3
for the first
G
;
FACTOh'IAL SERIES.
273 Analogous
and (270), or else
shewn in (272).
terms
-J_+_i_+__7__+_10L.+
Ex.:
1.2.3
has for
may be reduced to tlie types in (268)
may be decomposed in the manner
series
tlie
91
its
2.3.4
3.4.5
5.
-i.
<;
general term
__1
3»-2
n(n + l)(n-\-2)
and we may proceed as
,_5
n+1
,i
in (272) to
4_
n+2
.......
,
'^ '^""^^>''
Hnd the sum of n terms.
The metliod
of (272) includes the method known as "Summation bySubtraction," but it has the advaut;igo uf being more general and easier of
application to complex series.
COMPOSITE FACTORIAL SERIES.
274
If the
two
series
M
N-5
i^r _l5.G .,^5.6.7 3^5.6.7.8 ,^
/I
^-3
1,..
,3.4
..
3.4.5
3
,
3.4.5.0
t
,
be multiplied together, and the coefficient of .f* in the product
be equated to the coefficient of x*" in the expansion of (1 —x)'^^
we obtain as the result the sum of the composite series
5.6.7.8xl.2 + 4.5.C).7x2.3 + :K4.5.6x3.4
+ 2. 3. 4. 5X4. 5 + 1. 2. 3. 4X5.
275
Generally,
if
=
,.
(r
r,.= (?i— r)
and
the
(^)^
sum
of n
—l
7! 4!
the given series be
Aa + Aa.+ --+A,.
where
4! 2.11!
+ 1)
+ 2)
(//-/• + 1)
(/•
...
..
(r
,(?„-!
+ v- 1),
{n- ri-p-l)
terms will bo
/>!7!
(;,4-;,
+ r/-l)!
(I),
,
ALGEBRA.
1)2
MISCELLANEOUS SERIES.
Sum
276
of the powers of the terms of an Arithmetical
Progression.
1+2+;)+...+//
=
+2»+y +...+«'=
1
H o.+y+
+ „. =
...
5
^^^j'
"("
+ 1) (2« + l)(3»-+3» -1) ^
[By the method
A
1.2.3
of
+
^,
Indeterminate Coefficients (234).
sum of
way is
general formula for tlie
... n, obtained iu the same
>•
=s.
powers
7-"'
tlie
of
!
wliere Ji, A.,, &g., are determined by i)uttiug
successively in the equation
j>
=
1, 2, 3,
&c.
1
2 0^
+ 1)!
~(;?+2)!"^r(/>)!'^r(r-l)(i>-l)!^"'"r(r-l)...(r-7>+l)
«"'
277
^
(>,
+ !)
Proof.
278
+ +
+
^,'"
(^/
^/)"'
+ + 2^/)"'+... + +
V/+.S,(' (m, 2i
—By Binomial Theorem and
Summation of a
//</)"'
(^/
(^/
,s^,,,^f'"-
«'"-->/-
(276).
scries parthj Arithmetical
and
pa rill/ Geometrical.
Ex.\m?lt;.
terms.
— To hud the sum
of the series
1
+3,i'
+ 5.r + to
n
—
Let
s
S.V
.'.
by
=
=
l+;?.r
+ 5.r' +
7.i:»
+...
.i+3.r + oj;'+
...
+ (2/1-1)
+ {2n-:i)
x"-',
j""'
+ (*Ju- 1)
x",
Kiiblractioii,
6-
( 1
-.,•)
= + ±v + + + + 2.t" - (2« - 1
r""'
= l+2a^^- —-^--(2H-l).r",
2.r^
1
-
2./.»
.
'
. .
)
x"
1
1
•
279
A
X
l_(2n-l).r"
'-
1-x
general formula for
^-
2x(l-.r''-')
+
tlio
(I-../--
sum
+
\^'
of n terms of
0-ry
Obtained as in (278).
Rule.
—
Mi(Itij)Ji/
Inj
the
and subtract
ratio
the resulting
series.
280
281
r-'—
Tj-^,
=
l+.r+.»"+.t'+...+cr"-'+-pi
=
l+2.r+;ja-+lr''+...
,
n-i
,
(n-{-\)r'*~)i.v"^'
(l-cf)-
282
(^<-l).r+(/i-2).rH('<-.'i).''^'+...
=
283
+ -V-H'i'""'
(^^-i);;-"^;+->""
.
B,(253).
i^,,^W//^--])^;/(»-])0/-2)_^^^.^.^^^,„^
Hv making 4^-=^
in (12r»).
.etrrat*.
ALGEBRA.
94
The
284
series
(n-4)(n-5)
»-,-j
,
^~"T""^
^
,
^y.,
(
=
/S'
S
=
S
=
>S^
=
Proof.— By
285
The
terms, and the
or
^^
,
4!
0i-r-l)(n-r-2)...(n-2r-[-l)
^^~
consists of
_ {n-6){n-r,){n-7) ^*
;5!
—
sum
is
if 71
be of the form 6m-\-S,
if
?i
be of the form
if
?i
be of the form 6m,
n
6/?i
+ l,
n
—
if 7i
n
be of the form 6m±_2.
(545), putting;)
series
= x^-y,q = xy,
7i^-n («-!)'•+
n\,
and applying (546).
Hd^l^ {n-2y
o
!
^n{n + l)\
takes the values
0,
according as r
<n, =n, or =)i + l.
Proof.
given by
is
—By expanding
(e^
— 1)",
in
two ways:
first,
by
tlie
Exponential
secondly, by the Bin. Th., and each term of
the expansion by the Exponential. Equate the coeflicients of a;** in the two
Theorem and Multinomial
;
results.
Other results are obtained by putting
when
divided by r\,
The series (285),
the coefiBicient of aj'" in the expansion of
r
= n-{-2,
is,
n-\-o, &c.
in fact, equal to
POLYGONAL NUMBERS.
286
By
I'unctiou
05
exactly the same process we may detlucc from the
f'*}" the result tliat tlie scries
{t"*'
—
n'-n (N-'2y+
"^"~^^
(,,_.l.)'-_&c.
=
n;
or 2'*.?^!, according as r is < n or
takes the values
being e<j[ual to the coefficient of x''
this scries, divided by r
in the expansion of
!
,
f
^.3
yi
,-,.5
POLYGONAL NUMBERS.
Tlie n^^' term of the r'^' order of polygonal numbers is
equal to the sum of n terms of an Aritli. Prog. Avhose first
term is unity and common difference r 2 ; that is
287
—
=1
288
I'hc
{2+{n-l)(r-2)]
sum
__
of
It
= n + ln
terms
uiu-^l)
2
u{u-l)(n-\-l)(r-2)
"^
«
By
Order.
{n-l){r-'2).
resolving into two scries.
.
ALGEBRA.
96
—to
form, for instance, the 6*'' order of polythe first three terms by the formula,
and form the rest by the method of differences.
la practice
gonal numbers
— write
16
Ex.:
[r-2
= 4]
4
28
lo
13
9
5
4
i
66
45
17
4
120
91
25
21
4
29
4
...
...
...
FIGURATE NUMBERS.
289
The
n"'
term of any order
is
the
sum
of n terms of the
preceding order.
The
n^^
term of the
r*^'
order
is
njn+l)_^in +r-2)
(r-1)
290
The sum
of
n terms
/j („
,.
^
_j
is
n{n-\-l)...{n
Order.
=
+ r-l) _
H{n,r).
)
[By
98.
,
97
iiYi'h'iiCKoM irrincAi. si:riks.
IIYPEUGEOMETKICAL SERIES.
,+^_^,,. + ^(^+lMim),,.=
1. -2.7(7+])
l.y
291
"^
if
x
convergent
if
.r
is
<
1
if
x
is
>
1
=
1,
^^,
^,
1.2.;$.y(y+l)(7 + 2)
and divergent
is
and
a(a+1)(a + 2)/3(/3+1)(^ + 2)
the series
(-•5'* ''•)
;
is
convergent
if
divergent
if
and divergent
if
—a—
—a—
7—a—
-y
/3 is
positive,
y
/3 is
negative,
(239
iv.)
/3 is
zero.
(239
v.)
Let the liypergeometrical series (291) be denoted by
F{a, ft, y) then, the series being convergent, it is shewn by
;
induction that
292
ria,ft-^\,yjyi}
]
concl.ulin-
l-/r,
I-/.-,.
1-c^c.
^vlH"^c
/.',,
I:,,
Jr.,
Sec with
.
.-:,,,,
are given
...
^vitl.
,
\-k,,z,..
l)y tlic foniiiilie
+ r-1)(y+)— 1-y8).r
+ 2>—2j(y + 2>— 1)
_ ()8 + r)(y+r-a).r
_
(a
(y
(y+2;-l)(y+2r)
f'(a-f-r,
/8
+ r,
yH-2r)
Tlie continued fi-action may be conchuhMl at nny point
When r is infinite, r/o^ 1 and tlic continurfl
k-ir^Ur'
=
with
fraction
is infinite.
o
.
ALGEBRA.
98
293
Let
_!^ +
+"l.y
+ 1-^
+ '^^•
1.7
1.2.V
v+1) 1.2.,{.v(v4-l
1.2.y(y+l)
2.;{.y /"^i 1,/(-/+2)
'^'
f{y)
::^
1
_1_.>X
'
'
the result of substituting
/3 =:=
«
=X
Tlien,
.
by
/(y + 1)
—
for
last, or
_
Ay)
in
ic
1_^ p^_ P2_
1+1+1 +
In this result put y
(291),
and making
independently by induction,
with
294
(y+1) (7+2)
.
= ^ and
'Pj!]_
...
j),„
+ I+&C.
=
—
for
-^
(y+m
,r,
1) (y+/>«]
and we obtain by
Exp. Th. (150),
Or
the continued fraction may be formed by ordinary division
by the other.
of one series
295
('"'
is
the last and
incommensurable,
(17-1),
m and
by putting x
=
n being integers.
From
'
INTEREST.
If r
be the Interest on £1 for 1 year,
11
the inimber of years,
/'
the
A
the auiouut in
I'l-incipal,
296
At Siiuple Interest
297
At
Compound
Interest
n.
A =
A =
years.
Then
P{l-^)n').
/*(t+r)".
%
(-^-i)-
1:
.
i\Ti:i:i:sr
298
If
AM)
if the payments of
")
Interest be made 7 C
times a year
)
But
A
be an amount due in
worth of
-1.
99
.l\.\ ///'//vS.
.„
A
=
I'
h + -j
^
^^
years' time, and
11
the ])resent
/'
Theu
299
At Simple
300
At Compound
Interest
Interest
Discount
301
7*
=
/'
=
-j-^
.
.
,
By
(-200).
By
(297).
= A- P.
ANNUITIES.
302
The amount of an Annuity of £1 in n years,
1
[
,,(,,_])
=
nA~
=
,tA-hi{N-\)r
>'•
.^
^y
(82).
at Simple Interest
303
304
1
'resent value
Amount
at
of
same
Compound \
Interest
Amount when
the pay- ^
value of
(
j
1
.
,
_!1
Y"'
_
/
\
=
'*
i
1
(/
V'
'
'b'
iw
~
(-J^'*^*)-
1
_ ___7_
J
same
,^-.
''
'
1
Present
n^(09,j).
^
'
.
of
annum
1+r)" —
— ~(l+r)-l
Interest/
are made q times -|)er C
ments
(
—
"(l+r)-l
)
Present worth of same
305
_
'^yr
'
1
(2:is).
ALGEBRA.
100
306
Amount
wlicn the payof the Annuity
times per
are made
meuts
)
f
m
_
annum
( 1
'
-f >•)"—
I
i
m
[
J
l-(l+ r )-
^
Present value of same
~
\{l-{-r)'-—l}
m{(l-fr)i-l}
307
Amount when
terest
is
the
In-
paid q times
and the Annuity
times per annum
m
...
"^
J
V\
qi'
Present vahie of same
m
(i+v)-
PROBABILITIES.
If ^11 tlie ways in which an event can happen be m
number, all being equally likely to occur, and if in n of
these m ways the event would happen under certain restrictive
309
in
then the probability of the restricted event hapconditions
pening is equal to n-^m.
Thus, if the letters of the alphabet be chosen at random,
any letter being equally likely to be taken, the probability of
a vowel being selected is equal to -i^q. The number of unrestricted cases here is 26, and the number of restricted
;
ones
5.
m
events are not equally probable,
all the
divided into grou{)s of ccpially probable cases.
The probability of the restricted event happening in each
group separately must be calculated, anel tlie siun of these
probabilities will be the total })robability of the restricted
event liappening at all.
310
they
Ifj
however,
may be
I'UOHAIIILiriHS.
ExAMPLK.
lol
— Tlicro are three bags A, B, and G.
A
contains 2 white and
B
„
3
„
C
„
4
„
black balls.
.'}
t
5
drawn from
A
bapf is taken at random and a hull
bability of the ball being white.
=
Hero the probability of the bag A being chosen
probability of a white ball being drawn
l-
Required the pro-
it.
=
J,
and the
Therefore the [jrobability of a white ball being drawn from
~
3
And
1'
X 3
3
7
~
drawn from
J*
3 ^ 9
B
l'
7
drawn from G
the probability of a white ball being
-1
~
.1
15-
5
Similarly the probability of a white ball being
~
8ub.sc([nonfc
i
~ 27'
Therefore the total probability of a white ball being drawn
If a
and
J)
1
4
15
7
27
be the number of ways in
tlie
number
of
^
.j2_
ways
401
945'
an event can liappen,
can fail, then the
wliicli
in wliieli it
311
rrobabilitv of the event lia])penin2r
312
l'rol)al)ihty of
Thus
the event failing
Certainty
=
=
r.
=
1.
If p, p' be the respective probabilities of two iudcpcndcnt
events, then
=
313
rrol)al)ility
314
)}
of not
/yo//i
315
))
of one
happening and one faiHng
316
„
(^f
both liappening
pp'.
happening == i—pp'.
of l)o(h failing
=
(!—/>)
(1 —/>').
-
ALGEBlLi.
102
If the probability of an event happening in
of its failing q, then
one
trial
be
j>,
and the probability
317
Probability of the event happening r times in n trials
=
318
C{n, r)2fff-\
Probability of the event failing r times in n trials
=
319
n
[By
{n, r) ^j" "''(/''.
induction.
Probability of the event happening at lea><t r times in
the sum of t\iQ fivHt n—r-[- 1 terms in the expansion
trials
of 0;
C
+
=
ry)".
Probability of the event failing at least r times in n
sum of the last n r-[-l terms in the same expansion.
320
trials
= the
—
The number of trials in which the probability
same event happening amounts to j/
321
of the
_ iog(i-;/)
log (!-;>)'
From
the equation (1
—
=
j^)*"
1 —])'
—
322 Dei'inition. AVhen a sum of money is to be received if
a certain event happens, that sum multiplied into the probability of the event is termed the expectation.
—
Example. If three coins be taken at random from a bag
containing one sovereign, four half-croAvns, and five shillings,
the expectation will be the sum of the expectations founded
upon each way of drawing three coins. But this is also equal
that is,
to the average value of three coins out of the ten
-i^ths of 35 shillings, or 10s. Qd.
;
The probability that, after r chance selections of the
numbers 0, 1, 2, 3 ... 7i, the sum of the numbers drawn will
323
be
6',
is
equal to the coefficient of
.t'*
in
the expansion of
,
ri;i)r.M:iLiiii:s.
324
'I'lie
lo:
probability of the existence of a certain cause of
an observed event out of several known causes, one of wliieli
vi^ist liave produced the event, is proportional to tlie a jn-iarl
probability of the cause existinu: multiplied by the probability
of tlie event happening from it if it does exist.
Thus, if the a priori probabilities of the causes be /',, /'.
... Sec.
and the corresponding probabilities of tlie event happening from those causes (^),, (/_, ... itc, then the probal)ility
of the ?•"' cause having produced the event is
J
X{1'Q)
325 If A'> P-i ••• &c. be the a jfrlori probabilities of a second
event hap])ening from the same causes respectively, then,
after the first event has happened, the probability of the
second happening
For
this
is
the
is
sum
t {PQ)
POP'
of such probabilities as
(i
\^''//,
which
is
the probability of the r^'' cause existing multiplied by the
probability of the second event happening from it.
Ex.
1.
— Suppose there are
and
A
4 vases containing each 5 wliite and (i l)l:ick
2 vases containing each ^ white and 5 black
'2 white and 1 black
1 vaso containing
white ball has been drawn, and the probability that
of 2 vases
is
P,
= ^
]'..
Therefore, by ('S-l), the pn)l)ability
2.
came
Irani tiie
group
required.
Here
Ex.
it
ball.s,
balls,
ball.
— After
=
'f
P,
=
!
i-c(iuii-fd is
4,:.^
2.;^
L2
7.11
7.8
7.3
427
the white ball has been drawn and ro])laced. a ball
of the ball being lilack.
drawn again; required the probability
104
ALGEUltA.
He-o
The
P;
probability,
= A,
p;
= |,
p.
=
|
by (325), will be
4. 5.6
7.11.11
4.5
7.11
2.3.5
1.2.1
7.8.8
7.3.3
2.3
1.2
7.8
773
I£ the probability of the second ball
of P{P'.P'z.
58639
112728'
being white
is
required, QiQ^Qi
must be employed instead
The probability of one event at least happening out of
number of events whose respective probabilities are a, h, c,
326
a
&G.,
P1-P2 + P3-P4+&C.
is
where
P^
and so
on.
is
the probability of 1 event happening,
For,
by
(316), the probability
l-(l-a) (l-h) (l-c)
327
The probability
and no more out
of
01
...
=
is
y.a-^ah + ^ahc-,
...
of tlie occurrence of r assigned events
events
is
where Q„. is the probability of the r assigned events ; Q.,.+i the
l events including the r assigned events.
probability of r
+
be the probabilities of the r events, and
a, 1/, c ... the probabilities of the excluded events, the required probability will be
For
ii a, h, c ...
=
328
ahc
...
{l-a'){l-h'){l-c')
ahc
...
{l-1.a-{-'^aU-:ia'l/c'-\-...).
...
Tlu^ probability of ani/ r events hapjiening
Note.— If
a
= h = c = &c.,
tlien
^Q,
= C (n,
and no more
r)
Q,,
cl'c.
INEQUALITIES.
105
ii\i<:QLiALrrib;s.
330
^^'+^^-+- •"^'^"
'>-.>+ ••• +^'»
/>i
+
the fractions i^,
^,
lies
...
between the -Teatest
uiul least of
-^, the (leiiominators being
all
of
the same sign.
PuoOF.
— Let
k be the greatest of the fractions, and
Substitute in this
ar<kbr.
way
for each
a.
-
Similarly
any other; then
if
k be the least
frapCtion.
^
331
> V"0.
«.+«.+ ...+»„ > y„,7^~^,;
332
71
Arithmetic mean
or,
> Geometric mean.
—
Proof. Substitute both for the greatest iind least factors their ArithRepeat tlie process
metic mean. Tiie product is thus increased in value.
indefinitely. The limiting value of the G. M. is the A. ]\I. of the quantities.
q:^'
333
m
excepting when
„'"
PlJOOK
diere
.»
=
"
a
334
-'.
+
is
?,'"
>
{^l+!i)'\
a positive pi'opei' IVaet ion.
=("t'')"'[(l+.r)'"
+ (l-..)"'},
Kinploy Hin. Tli
+b
":'+""'+..+":
excepting ^vhen
///
is
> ^'.+".+ +".. y\
a positive proper fraction.
I"
—
;
Ahdi'JiiUA.
106
The ArUhmetic mean of the m"' poivers is
Otherwise.
greater than the m"' power of the Arithmetic mean, excepting
is a positive proper fraction.
when
m
Pkoof.— Similar
left side,
336
If
-''
and
m,
are positive, and x and
then
(l
K
337
by
Substitute for the greatest and least on the
to (332).
employing (333).
,1^
mx
less
than unity
+ ciO-'"> l-mx.
(125, 240)
m, and n are positive, and n greater than
enough, we can make
ra
;
then,
taking' x small
For X maybe diminished
is > (l-\-xy\ by last.
338
If ^
until
be positive,
If X
be positive and
>
1
If X
be positive and
<
1,
339
When n becomes
,
the
first
...
vanishes,
log (l+.r)
> <r- ^.
^ogj^^, >
<'•
.^^^^^
'
second
and
this
(150)
os.
(155, 240)
(^''^^^
two expressions
o.-^.7...(2y^
2. 4.
2yi
-J
> {l—mx)'^, and
<
infinite in the
the
product Ues between
is
log {l-\-x)
1.3.5... (27Z-1)
2.-4.(3
l^nx
becomes
+ l)
6. ..2m
infinite,
and their
1.
Sliewii by adding 1 to each factor (see 7o), and multiplying the result by the original fraction.
340
II'
'"
he
>
II,
and
ii
>
<(,
—
;
SrA LES OF NOT. TloX.
I
341
If ",
f>
1)0 |)ositiv(>
SinulaHy
a''
(
i7
quantities,
is
>
!,'.'>
{^'
„V/'
1
('i+I'f"'.
f"*'
+ +
','
'
These and similar theorems may be proved hy takinf^ lou^arithms of each side, and employing the Expon. Th (loH), Sec.
SCALES OF NOTATION.
iVbe a whole number
342
It"
of
scale,
tlie
=
^<'
JJ,,'"
of h-{-
and
digits,
I
+I>u-xr"-^ +p„_,r"-'-+
...
/•
the radix
-^j>ir-\-p,,,
where
j^„, 2>„_i, ...2>„
343
Similarly a radix-fraction will be exju'essed by
where
2>i, p.,, cVe.
ExAMi'LKS
:
are the digits.
are the digits.
o-12l!
in the scale of 7
•104o in the same scale
344
Kx.— To
== 3
=
7
7''
.
+
+
•!
^.,
r
.
7-
+
+-,-
•
7
+
7'^
+ (J
'*
.
7'
transform :U2G8 from the scale of 5 to the
scale of 11.
JJii'ide sucrcssiiu-Jij hi/ tlw iinr radix.
RuLK.
11
11
3426S
1348 -<
111-40-3
1
1—9
Result
\[)'3t,
in
which
t
stands fm-
Id.
1
;
ALGEBRA.
108
Ex.— To
345
of
7, e
transform -tOcl from the scale of 12 to that
standing for 11, and f for 10.
Rule.
—Multiply successively hy
the neio radix.
•tOel
7
5-i657
1
6-1931
7
1-0497
7
0-2971
346
Ex.
— In what scale
Result
-5610
docs 2f7 represent the number 475
in the scale of ten ?
Solve the equation
27--
+ 10;- + 7 =
Result
>•
=
475.
[178
13.
The sum of the digits of any number divided by —
same remainder as the number itself divided by
347
?'
leaves the
r— 1
;
being the radix of the
r
scale.
(401)
The difference between the sums of the digits in the
even and odd places divided by r-\-l leaves the same remainder as the number itself when divided by r + 1.
348
THEORY OF NUMBERS.
349
If
ft
is
prime to
b,
is
,
in its
lowest terms.
b
Proof.
— Let
=—
i,
b
a fiaction in lower terms.
1)^
Divide a by
«,,
by
6,,
h
and
remaimler
remainder
n.,
(juotieiit 5,,
h.,
quotient
7,
the H. C. F. of a and a„ and of b and
Let «„ and b„ be the highest eoninion factors thus determined.
so on, as in
liiidinii-
i,
(see 30).
THEORY OF NUMBERS.
—=
Tlu'ii, hocaiise
'
6
and
so on
T hero
any
if
thus
;
and
u
ft )!(.'
''
/'
=
-
=
'
'•'=',
— Q\''\ "i
=
.-.
,
^,
•
=
~
txc
arc C(|uimnlti|)k's of a„ and b„
b
7
"
It"
prime to
is
Pkoof.
and
A,
(I
(t
-^=
a and
p
is
351
— Let
- reduced
now prime
If
'^^ is
T
II
terms be ^
to its lowest
and
to 7,
that
;
by
divisible
(lb
,
prime to
a
c,
prime to
is
352
If
and a
354
••
'
not
;
to
;M-ft,
,
and,
'>
that
^
is
Therefore, &c.
it.
then
h
must
be.
ah
i)rime
=
c
is
by
l>
a multiple of
be each of them prime to
if
h, c,
&c.
is
divisible
must
p be prime
by
c.
r,
is
to
Therefore, if a"
if j) be a prime
If
tower of
Also,
"
is
least of the
all
but one of the factors, that factor
(:ir.l)
is divi.sihle
it
prime to
by ^^
must divide
h,
i>
cannot be jirime to
a.
any power of
<i
is
prime to any
It.
if
prime to each otlier, the product
prime to any other protluct of their
a, 0, c, &c. are
any of their powers
powers.
by a prime, one at
by it.
also be divisible
j>.
and
355
of
Jind h
is
a
=7;
therefore, by last, b
If abed...
Or,
}
;
it is (.Hjual
=—
1
folhiws,
it
^-
[By (351).
divisible
;
Then
.
r.
factors a,
'/
<^f
c
/',
is,
c
is
and //arc
n'
1
Pi;nOl'.— Let
353
b
/>.
neither greater nor less than
to
not prime to
is
then
'.-;
b
'(
a
is,
h
('(jiiiimiltiples of
Hut
•
that
;
(70;
fraction exists in lower terms.
350
since
100
is
ALGEBRA.
110
No
356
expression with integral coefficients,
can represent primes only.
A + B,e + Cx' -h
— For
Proof.
.
.
.
as
siicli
,
by x
divisible
it is
if ^1
=
and
;
if
by A,
not, it is divisible
when x=: A.
357
'^^^G
nnmber
of primes is infinite.
—
Pkoof.
Suppose if possible p to be the greatest prime. Then the product of all primes up to p, plus unity, is either a prime, in which case it
would be a gieater prime than p, or it must be divisible by a prime but
no prime up to p divides it, because there is a remainder 1 in each case.
Therefore, if divisible at all, it must be by a prime greater than p. In
;
either case, then, a prime greater
358
If
(I
Proof.
h,
— Assume ma — nb = ma —
""
-
b
A
359
exists.
h, and the quantities a, 2a, oa,
the remainders will be different.
be prime to
(b—i) a be divided by
thanp
)ib,
iii
only.
360
To
less
than
b,
Then by
m — in
number can be resolved
way
and n being
"~"'
into
prime factors
resolve 6040 into
the
hij
its
(350).
in
[By
Rule. — Divide
...
one
(353).
prime factors.
prime numbers
snccessiveli/.
2x51 5040
2 504
1
21
252
126
7 63
2
1
1
3L9_
Thus
3
5040
=
361 Required the least multiplier of 4704
the product a perfect fourth power.
By
4704
(19G),
Then
2'. 3'. 7-
the indices
5,
1,2
8, 4,
x
4 being the
=
2'. 3». 7-
2". 3'. 7-
=
3584
is
wliicli will
make
2'. 3. 7-.
=
2». 8'. 7'
least multipli's
i)f
=
I
I'espcctively.
Tlius
2'. 3-..5. 7.
the multiplier reciuired.
84',
which
:ire
not less than
2
UF
TllEUliY
362
mmilHTS :uv
All
..f
one of
Ill
MJMUl'JIiS.
tin-
forms
<.|-
2//
2//-f
•2n(n'2n
„
,,
,,
,,
,,
,,
,,
,,
—
1
\
',\n
ov:\n±\
1//
or l//il
ly/
or
oi-
I//
+2
ly/±l or 4/<—
r)/M)i- ."iz/il
or r>//i-
aiul so oil.
AH
363
l'i;00K.
all
square numbers are of
— By squarino;
tlii"
lonns
tlie
.">»
o/t,
±
form
1,
I'tuzt-.
or .u/ilwliii-h ciniiitrelieiHl
numbers whatever.
364 All cube numbers are of the form
And similarly for other powers.
i)roduct
III
or 7^'dzl-
m
m
m
1>
F
V
factors in
'
//^
;
ExAMi'LK.
3,
<3,
— The
hitrliesfc
will divides
'.,
which
The successive
so on.
b}'
Heio the
of 8 which will divide 29!.
Their ntinilier is
27 can be divided by 'A.
power
12, 15, 18, 21, 24,
l»,
p
which p
!
and
will divide a second time
divisions are eejuivalent to dividing
it
factors
in
,
!
For there are
" =
"Jn
The highest power of a prime jh which is contained
is the sum of the integral parts of
365
tlu>
'"
.w<
I
"'_
(the
!>
inte,t,M-al pai-t).
27 can be divided a second time.
Till'
factors
=
(the integral part).
;5
One
9-f
it,
factor, 27,
18,
is
divisil)le a third time.
3+ =
13; that
'I'hc
[)roduct
366
1
is,
3"
of
is
"7;^
=
1
Their nmnlx'r
(intei^'ral i):irt).
the highest power of 3 which will divide 29!.
:mv
/•
consecutive integers
is divisiljlc
byr!.
PkooK:
«("-!)
•••
is
(»-'•+!)
is
necessarily an inteu'cr, by
('.'<;).
ALCIEBEA.
112
If ^^ be a prime, every coefficient in the expansion of
{a-\-hy\ except tlie first and last, is divisible by n.
By last.
367
368
If "
pansion of
Proof.— By
369
p
;
a prime, the coefficient of every term in the exexcept a", &", &c., is divisible by n.
l)t>
{(i-\-h-\-c ...)",
Put
(367).
/3
for (6
+ c+
...).
Frrmaf.s Theorem.— U p be a prime, and
then
Proof
iV^"^
—1
W={l +
:
is
\
divisible
N
prime to
by p.
+ ...y = N^Mp.
By
(368).
—
370 If V be any number, and if 1, r/, ^>, c, ... (p 1) be all
and if n be their
the numbers less than, and prime to p
number, and x any one of them then ,if 1 is divisible by p.
;
—
;
Proof.
—If
x, ax, hx ... (p — l)x be divided by p, the remainders will be
and prime to^ [as in (358)] therefore the remainders will be
(p — l) therefore the product
all different
1, a, b, c
;
...
;
x"ahc...
371
l
Wilson's
+ (^— 1)
Put
!
is
—1
j)
ip—l)
= ahc
...
{p — \)+Mp.
Theorem.— If p be
divisible
for r
by
and n
a prime, and only then,
p.
in (285),
and apply Fermat's Theorem
to each term.
372
hy
If
i'
be a prime
=
2y«
+ l,then
{vlf + {-iy
is
divisible
p.
—
By multiplying? together equi-di.'^tant factors of
Proof.
Wilson's Theorem, and putting 2n + l for /).
373
\,Qt
N =((''
tegers, including
Proof.
— The
Similiirly in
//",
1
&c.
— 1)
;
,
!
in
the number of in'• in prime factors
which are less than u and prime to it, is
!>''('''
number
/•'•,
(j»
of intogcrs
Take the
piimo
to
N
contained
])r()duct of those.
in
ri"
is
n"-
TUEORT OF NUMBERS.
113
tlio miinhcr of intcfjfors less tliaii mikI ])i-imo to
^fxScc.) is the ])roduct of the coiTcspoiuliiig miiuhcrs
for X, ^[, &c. separately.
Also
(Xx
The number
374
is
=
(y
+ l) (v 4-1)
N, incliidiiif^
For it is equal
of divisors of
(/'
+ !)
...•
1
and
to the
^V itself,
number
of terms in the product
+ ...+r7'')(l+/.-h...+^")(l+r+...+'")---<-'tc.
(l+./
The number of ways of resolving
number of its divisors (374).
375
N into two
half the
S(juare the
as
two
two equal factors must,
If the factors of
7, r,
377
The sum
a-l
Proof.
If
of the
379
in this case, be reckoned
of the divisors of ^V is
a^^-'-l
Ex.
is
each pair are to be prime to each other,
&c. each equal to one.
376
is
factors
number be a
divisors.
put j;,
378
If the
c^-^'-l
h'^^'-^
'
— Bj the product
h-1
'
in (-374),
c-1
'"
and by
(85).
be a prime, then the j? — 1*** power of any number
form mj^ or )fij)-\-l.
By Fermat's Theorem (3(30).
7^
— The
To
12"*
power of any number
is
find all the divisors of a
I.
of the form
number
;
13fM.
or 12m-\-l.
for instance, of 50-1.
—
—
ALGEBRA.
114
The divisors of 504 are now formed from tlie numbers iu column
placed to the right of that column in the following manner
II.,
and
:
Place the divisor 1 to the right of column II., and follow this rule
Multiply in order all the divisors which are written down by the next number
in column II., which has not already been used as a multiplier : place the first
netv divisor so obtained and all the folloiving products in order to the right of
column
380
II.
Proof
sum
tlie
>^r
numbers
is
+]
1, 2,
is
3
...
r^^'
is
powers
(x"
+ S2X-"-'-
(«
— 1)
n natural
- 2")
...
.
.
.
- »0
S,,.,x-'
Multiply out,
Thus, using (372),
re-
+ (-iy([ny = M(2n + l).
in succession for x,
Sr
(»'
by 2w+l, by (36G).
to be less than 2ji-|-l.
of the form
of tlio first
2}i-\-l.
factors divisible
x'"-S,x"'-'
Put
by
x {x"- V)
:
constitutes 2»
jecting X, which
equations
of the
divisible
and the solution of the (n—
= M{2n + l).
1)
THEORY OF EQUATIONS.
FACTOKS OF AN EQUATION.
iicncral form of a rational integral equation of the
The
left
side
will
be designated
/(,/•)
w^''
degree.
in the following
summary.
401 If /(')
By assuming
402
If «
divided by «
the
404
—
rt,
tbe remainder will be /(")•
= P {x — a)-\-lL
be a root of the equation /(.i;)
To compute
403
and
bt-'
/(,(;)
numerically;
/(rt)
= 0,
tlien/(rt)
divide f{.r)
by x
— a,
[101
remainder will bef{a).
Ex^^MPLE.— To
= 0.
+ l2x'-x'' + 10 when x = 2.
4-3 + 12 +0 -1 +0 +10
8 + 10 + 4-i + 88 + 174 + 34.8
4 + o + 22 +
+ 87 + 174 + 358 Thus /(2) = 358.
find the value of 4x''-2x'
-lrl
If
a,h,c...h be the roots
then, by (401) and
405
/Or) =7>o
(
of the
eciuation
f {(r)
=
;
1U2),
(.t-^/) G^-'>)
O'-O
...
('^-/O.
multiplying out the last cijuation, and equating coefficients with
:—
cquatiou (100), cousidcriiig j'u
1, the following results lu-o obtained
By
=
THEORY OF EQUATIONS.
116
406
X
1
^
(
^'"
=
_
~
~~i^
_
~
—ih
_
y
/ir —
of all tlie roots of /(a?).
of the products of the roots taken
two
\
(
the
sum
(
the
sum
at a time.
of the products of the roots taken
three at a time.
\
of the products of the roots taken
time.
rata
"^
— l)"j?,j =
product of
The number
407
sum
sum
the
the
(
all
of roots of
the roots.
/(.^') is
equal to the degree of
the equation.
408
Imaginary roots must occur in pairs of the form
a
The quadratic
+ /3v/^, a-i3\/^.
factor corresponding to these roots will
for it will be
;
then have real coefficients
x'-2ax + a-+ii\
If /('«) be of an
of the opposite sign to
409
Thus
410
a;^
—1 =
If /('O
odd degree,
it
[405, 226
has at least one real root
p,j.
lias at least
one positive root.
^Q of an even degi^oe, and jhi negative, there
and one negative root.
is
at least one positive
Thus
411
a;'*—
1 has
+1
—I
for roots.
terms at the beginning of the equation are of
If several
one sign, and
and
the rest of another, there
one, positive root.
Thus
412
x^
all
+ 2x*-\-Sx^ + — 5x—4< =
If all the
x'^
is
has only one positive root.
terms are positive there
is
no positive
root.
If all the terms of an even order are of one sign, and
the rest are of another sign, there is no negative root,
413
all
one, and only
414
Thus **— a;" + «'—;« + 1
=
has no negative root.
7
DISCRIMINATION OF ROOTS.
1 1
415
If :>11 tlio indices are even, aiul all tlie terms of the same
there is no real root; and if all the indices are odd, and
the terms of the same sign, there is no real root but zero.
sijjfii,
all
=
=
has no real root, and x'^ + x^ + x
+ x^ + l
has no real rout
In this last equation there is no absolute term, because such a
terra would involve the zero power o( x, which is even, and by hypothesis is
wanting.
Thus
but
x*
zero.
DESCARTES' RULE OF SIGNS.
416
In the following theorems every two adjacent terms in
which have the same signs, count as one " continuation
of sign"; and every two adjacent terms, with different signs,
count as one chanw of siefu.
/(.r),
/(<')' multiplied by (/. — a), has an odd number of
changes of sign thereby introduced, and one at least.
417
418
./*
cannot have more positive roots than changes of
more negative roots than continuations of sign.
(c)
sign, or
the roots of f{.r) are real, the number of
equal to the number of changes of sign in
and the number of negative roots is equal to the number
f{.r)
of changes of sign in/(— .r).
Wlien
419
positive roots
all
is
;
420
Thus,
it
being
known
that the roots of the equation
a;*-10.t''
+ 3o.c2-50.c + 24
=
number
of positive roots will be equal to the number of
changes of sign, which is four. Also f(—x) =x*+lOx'^ + 3bx- + 50x + 2i!
0,
and since there is no change of sign, there is consequently, by the rule, no
negative root.
are
all real
;
the
=
If the degree of /(./•) exceeds the number of changes of
sign in f{x) and /(—a;) together, by /t, there are at least
421
/j.
imaginary roots.
If, between two terms in /(,/) of the same sign, there
be an odd number of consecutive terms wanting, then there
must be at least one more than that number of imaginary
roots ; and if the missing terms lie between terms of different
422
1
:
TEEORY OF EQUATIONS.
118
sign, there is at least one less than the
same number
of
imaginary roots.
Thus, in the cubic equation
x^
+ 4x — 7 =
0,
there must be two imaginary
roots.
And
x°—l
in the equation
=
there arc, for certain, four imaginary roots.
423 If an even number of consecutive terms be wanting in
f{x), there is at least the same number of imaginary roots.
Thus the equation x^ + 1 —
has four terms absent and therefore four
;
imaginary roots at
least.
THE DERIVED FUNCTIONS OF f{.v).
Rule for forming the derived functions.
424
Multiply each term hy the index of x, and reduce the
index by one; that ^s, differentiate the function with respect
to X.
Example.
—Take
(«) =
f (x) =
f(x) =
f(x) =
f(x) =
f (x) =
/
/' (*)j /'
('''')y
<^c.
x^+
5x'+
a;*+ a;'-
x'-x-l
+ dx"" - 2a; 20x' + 12x' + 6x-2
60a;2 + 24« +6
4x^
120x+24.
120
are called the
first,
second, &c. derived functions of/ (a;).
425 To form the equation whose roots differ from those of
f{x) by a quantity a.
Put x y-\-a infix), and expand each term hy the Binomial
TJieorem, arranging the results in vertical columns in the foU
lowing manner
=
—
/(a + 2/)
=
(«
+ 2/)'+(a + 2/)H(a + 7/)»-(a + y)=-(a + 7/)-l
a"
—
;
TBANSFOJIMATION OF AN EQUATION.
('om paring this result with that seen in (421),
426
80
tliat
tlie
equation
is
generally of
?/''
in tlie transformed
^
^
'
L±
LI
lA
coefficient
seen that
=/(")+/'(")//
/•("+.'/)
\A
it is
119
,
r
To form tlie equation most expeditiously when a has a
a, and the
miinerical value, diride f{,i') continuously hi/ x
succcssli-e remainders ivlU furnish the coefficients.
427
—
ExAiirLE.
— To expand f(y +
f(x)
=
2)
Z'
when, as
Divide repeatedly by x — 2, as follows
1
+
1
+
2
+
1
+
+
-f
2
1
1
1
1
+
-f 7 +
+10 +
3
1
14
13
31
+
+
+
:
1-1
-
2G
+50
+ 49=/(2)
+ 5 +17 + 47 + 119 = rC2)
+ 2 +14 + 02
+ 7+31 + 109 = (i)
11
+ 2 +18
+
9
+ 2
+ 11
1
-
-
I
in (425),
+ x' + x'-x'-x-l.
=
+49 _/'(2)
|3
I
That these remainders
are the required eoeflicients
is seen by inspecting the
form of the equation (420)
for if that equation bo direpeatvided by x a
II
edly, these remainders aro
— =
14
f(^\
obviously produced
2.
a
roots are each less by 2 than the
0.
49
?/+ll?/' + 49(/"'+109//'+ 119//
Thus the equation, whoso
proposed equation,
when
=
|5
is
+
i-oots of
the
=
To make any assigned term vanish in the transformed
equation, a must be so determined that the coeffieient of tliat
term shall vanish.
428
Example.
(420),
— In
order that there
we musi have /*(a)
may be no term
involving
if
in equation
= U.
Find /'(a) as in (424);
.-.
a = -\.
120« + 24 = 0;
The equation in (424) must now be divided repeatedly by a; + | after the
manner of (427), and the resulting equation will be minus its seeoud term.
thus
THEORY OF EQUATIONS.
120
429
Note,
= 0,
y(-t')
tliat
tlic
to
remove the second term
requisite value of a
coefficient of the
=—
is
of tlie equation
^
^
;
the
tliat is,
second term, with the sign changed, divided hy
and hy the numher expressing
the coefficient of the first term,
the degree of the equation.
To transform /'(''') "^^o an equation in y so that // - <p {.<),
(j)~'^{y), the inverse function of y.
a given function of x, 2^ut x
430
Example.
=
— To obtain an equation whose roots are respectively three times
— Gx + 1 = 0.
^ — -^ + 1 =
Here
the roots of the equation x^
and the equation becomes
431
To transform
coefficient
/(,v)
of the first
=
0,
2/
= 3a;
;
therefore x
?/— 54^ + 27
or
= —,
= 0.
into an equation in which the
be unity, and the other
term
shall
coefficients the least possible integers.
Example.
— Take the equation
288x^ + 24>0x''-176x- 21
Divide by the coeflacient of the
equation becomes
Substitute -^ for
x^
x,
+
first
5
—• x^—
and multiply by
Next resolve the denominators
3
'^
57^
= 0.
term, and reduce the fractions
—x —
11
h^
;
we
7
—
KB
;
the
^'
get
into their prime factors,
life'
,
"^2.3^
_
2.3'^^
yic"
2^3
^Q
The smallest value must now be assigned to h, which will suffice to make
12,
each coefficient an integer. This is easily seen by inspection to be 2'-. 3
0,
and the resulting equation is i/ + 10if — 88y l26
the roots of which are connected with the roots of the original equation by
=
—
the relation
y
=
I2x.
EQUAL ROOTS OF AN
By
by
+
=
EQUATIOli^.
i^^ powers of ;• by (i05),
ox})audiiin^ 7X''^
^')
and c(iiiating the coefficients of z in the
(1'2()),
and also
two ex-
EQUAL nOOTS.
pansions,
tVoiii
all
it
is
provctl
wliicli result it
uTUMiual,
tliat
appears
,*•.
h,
f{^v)
then
/(..)
=
t
roots equal to
when
= iK{,'-ay {.v-hy {.v-i'Y
^
+
+
;^;!
(-ia^
<'f/((')
&c. arc
ind,
...
+
*..;
and
/'(,<)
will
bo
{.v-(iy-' {.v-hy-' {.v-cY-\..
= a, /(..),
= b, &c.
.v,
.v
/'{,'),
.•/'-'Oi')
Lft
/ (.0
= A', X: a1 X\ Xl
A',
= product of
A1=
Xl=
all
.
.
.
vanish.
all
Prdctiral mctlnnl offfiH/in^' the
445
b, r,
&c., so that
r,
and the greatest common measure
444
When
roots a,
tliat, if tlie
and /'(,») can liave no common measure
however, there are r roots each equal t<^
/'(.*')
volving
If,
s roots e(|ual to
433
121
(-([Udl nKits.
where
X;::,
the ftictors like {x
„
<^^,
(.-«)».
„
Find the greatest comraon measure of
—
{x-a)\
„
„
Similarly
and /'(f)
/(j-)
=
I'\ (.')
say,
= F,(x),
F:(x) = F3(x),
I'\(x)aud'F;Cv)
F^(x) And
Lastly, the greatest
common measure
Next perform the
of F,„.i(x) and
f(x) -i- F,(.r)
F,(x)-^F,(')
And,
iinully,
/''„,-i(*)
=
I'\,0')
=
1-
divisions
0,
(.,')
-4-
<Pi{x)
i''«.-.C'-)=V'..C'-)
1:
=
<p,(.r)
.say,
='P,{.r),
=
=
A',,
A',,..
[T. 82.
THEORY OF EQUATIONS.
122
=
=
Xj
0, &c. will furnish all the
0, X,
those which occur twice being found from X.^; those which
occur three times each, from X^ and so on.
The
solution of the equations
of/ (a;)
roots
;
;
446
If /('^) lias all its coefficients
commensurable, X^^X^^X^,
&c. have likewise their coefficients commensurable.
Hence, if only one root be repeated r times, that root must
be commensurable.
the following theorems, unless othermse stated,
understood to have unity for the coefficient of its first
447
III all
/(-/•) is
term.
LIMITS OF THE KOOTS.
If the greatest negative coefficients in /(,/')
448
be j) and q respectively, thenp + 1 and
the roots.
449
If
and
/( — aO
x''-''
and
x''-'
—(^ + 1)
and /( —
are the highest negative terms in
respectively, {l
+
<^')
are limits of
\/p) and —{l-\-^q) are
/(<i')
limits
of the roots.
450
If
then —
/-^
be a superior hmit to the positive roots of /(
will be
an
—
j
»
inferior limit to the positive roots of /(-f)-
If each negative coefficient be divided by the sum of all
the preceding positive coefficients, the greatest of the fractions
unity will be a superior Hmit to the positive
so formed
451
+
roots.
Neivtons tnethod.—Viit x=h-\-i/ in /(./•)
452
Take
then
453
//
is
fW yf'W
;
then, by (420),
^^^
so that /•(/')' /(^O^
a superior limit to the positive roots.
//
^11
positive;
According as /{a) andf{b) have the same or different
number of roots intermediate between a and b is
signs, the
even or odd.
.
123
INTEGRAL BOOTS.
real root of the eriuation f (r)
every two adjacent real roots of /(./')•
455 ('OH. I.—/(•'•) cannot have more than one root gi-eatcr
than the greatest root in /(./); or more than one less than
llic least root in/'(,r).
real roots, /'•(') has at least
456 CoR. 2.— If f(.r) has
Rollrs Thcorrm.—Ono
454
lios iH'lwoeii
m
—/
III
real roots.
457
f(u')
lias
/t
imaginary roots,
f{.r)
umber
is
a, /3, y ... K be the roots of /(.<)
of changes of si^-n in the series of terms
CoK. -1.— If
458
11
3.— If
CoH.
has also
at least.
/i
;
then the
f{^), /W, /(«, /(7).-/(-^)
number of roots oif{.c).
equal to
tlie
NEWTON'S METHOD OF DIVISORS.
discover the integral roots of an equation.
To
459
ExAMTLE.
— To ascertain
if
5
5 be a i-oot of
«*-6x» + 80x*-i;6.i- + 10-.
If 5 be a root
next
will divide 105.
it
the quotient to the
If 5 bo a root it will divide
the next coefficient and so on.
105
— 1/b
^
—\55
)
—'Si
-155.
Add
_86
the quotient to
5
;
If the
)
21
0.
—155.
Result,
coefficient.
Add
=
tried be a root, tlic divisions will be effectiblo
the last quotient will bo -1, or —}>o, if/u he
)
to the end,
not unity.
and
55
U
number
>-
n
—6
_e
2_Z_.
-1
In employing this method, limits of the roots
be found, and divisors chosen between those limits.
460
may
first
Also, to lessen the number of trial divisors, take any
integer
then any divisor a of the last term can be rejected
if a
m does not divide /(///).
461
—
m
;
-f-1 and — 1
any of the roots determiiK'd as above are
repeated, divide f{x) by the factors correspijndiTig to them,
and then applv tlie method of divisors to the resulting ecpiation.
In practice take
To
///
find wliether
=
THEORY OF EQUATIONS.
124
Example.
— Take the equation
Putting X
= 1,
1,
The values
of n
—m
0,
3,
(since
1,
2,
rii
—24.
G,
4,
8,
= l)
5,
3,
numbers only 1,
9 are the only divisors of 144
Of these
=
we find/(l)
2,
divisors of 144 are
12,
IG,
24,
15,
23,
&c.
are therefore
7,
2, 3,
last
The
9,
11,
8,
and 8
&c.
will divide 24.
wliich it is of use to try.
and
3.
roots of the equation will be found to be
±
±2
462
If
/('^')
Hence 2, 3, 4, and
The only integral
and F{X) have common roots, they are concommon measure of /(a:) and F{X).
tained in the greatest
463
If /('') l^as for its roots a,
then the equations /(.*)
roots a and
=
(p
{a), h,
and/[(^(A')j
(jt
{b)
=
amongst others
have the
;
common
h.
But, if all the roots occur in pairs in this ^vay, these
equations coincide.
464
For example, suppose that each pair of roots, a and b, satisfies the equation
We may then assume a — b = 2z. Therefore/ (2 + r) = 0. This
h = 2r.
equation involves only even powers of z, and may be solved for z'.
a+
465
=
Let a&
Otherwise:
— 2rx + z. Perform the
form Px + Q, where P and Q
=z
x^
The equations
from a +
b
= 2r,
P = 0, Q =
ab
=
—
—
then /(;«) is divisible by {x a){x l)
division until a remainder is obtained of the
;
only involve
determine
z.
z,
by (462)
;
and a and
I arc
found
z.
EECIPROCAL EQUATIONS.
A
466
a,
—
;
reciprocal equation has
also the relation
its
roots in pairs of the form
between the
Pr =Vu-r^ OY else p,
coefficients is
= —Pn-r-
A reciprocal equation of an even degree, with its last
term jwsit ire, may be made to depend upon the solution of an
equation of half the same degree.
467
;
BINOMIAL EQUATIONS.
468
is
a
KxAMiM.K
125
•l-,/''^-24,r''-f57r'-7o./-''-f :)7./--2-1../
:
of
rocii)rtic;il ('(lualioii
cncii dcLi-i'd', with
:iii
its
+ =
t
last
term
positive.
Any
form may bo
term l>e posit Ire
(UkI, 1/ till' lust term he neijafirc, l>i/ diriduuj hi/ ,r— 1 or r' — ],
so us to liriiiij thi' t'tjiiiition tn mi
rm. tlrr/ree.
Then proceed
reciprocal equation w
reduced to
it
diridiiuj
hi/
hi/
is 7iot
liicli
-f
,/
I
the
//"
of this
la.sf
<
manner
in tlie
foHowinn'
469
First brin<^ toovther e(|nidistant ti'rins, and
equation by
,/'"*;
By
,v -\
})utting
relation
,/'-
a cubic in
equation.
]"*ut ;)
470
j)„,
for
471
P,u
=
=.
'I'he
the
;ind
//,
by making- repeated use of
tlie
i,r
-\
)
"~ ->
^^'<^'
eipration
is
reduced to
the degree being one-half that of the original
.r
-|
,
Tlie relation
may be
diviih'
tlius
-|
?/,
:
and
p„,
for
x„^-\
.
between the successive factors of the form
by the e(juatioii
exi)ressed
equation for
= p'-nip'"
_L
-+
f_i V
ji,,,,
\
in
,,
^" f»/
terms of
'-
p'"
— >•— 1)
...
^>, is
...
(in
I>y
— '2r-\-\)
(^')i")),
„,_..^,
putting 7
=
1.
BINOMIAL EQUATIONS.
472
If a
where
m
473
If "
is
be a root of
.r"
— 1 = 0,
then
a"* is
likewise a root
any positive or negative integer.
be a root of
./"
+ 1 = 0,
then
a-'"^' is
likewise a root.
THEORY OF EQUATIONS.
126
474
If
^'''
ai^d
be prime to each other,
^2-
x'"'
—l
and x^—1
common root but unity.
iim — qn = 1 for an indirect proof.
have no
Take
475
If
n be a prime number, and
the other roots are a,
These are
476
may
If
a^, a^
...
if
a be a root of
ct;"
— 1 = 0,
a'\
by (472). Prove, by (474), that no two can be equal.
all roots,
be not a prime number, other roots besides these
The successive powers, however, of some root
^i
exist.
will furnish all the rest.
477
If r*/'— 1
prime factors
;
(l+a + a^+
=
=
has the index n
m2)q; m, ]}, q being
then the roots are the terms of the product
+a-^)(H-/3 + /3'--[- ... +/3''-^)
X(l + 7 + 7'+
...
where
a is
/3
7
but neither
478
n
If
a, /3,
nor 7
= m^
= 1.
x^-l,
„
»
»''-!'
Proof as in (475).
be a root of x""—!
7
then the roots of x''—l
x^>^-a
„
»
(i
...
+7'"')>
and
a
(l+« + a^+
-
a root of «'"— 1,
= 0,
= 0,
r.--|3=0;
=
will
be the terms of the product
+„-^)(l+/3 + /3-^+...+r-^)
X(l + 7 + 7^"+... +7""')-
=
a^" + 1
may be treated as a reciprocal equation, and
depressed in degree after the manner of (468).
479
480
The complete
solution of the equation
.1
is
obtained by
The
71
-
-1 =
De Moi\Te's Theorem.
different roots are given
0?
=
±
cos
V
(757)
by the formula
—1
sill
n
which r must have the successive values
71
in
concluding with ^
,
if
n be even
;
and with
0, 1, 2, 3, &c.,
-~
,
if
//
be odd.
—
7
CUBIC EQ UA TIONS.
481
1
2
Similarly the n roots of the ofiuatiuu
.r"
+
1
=
are given by the formula
n
u
r taking the successive values 0, 1, 2, 3, &c.,
up to
~^
'
,
if
'
n be even
482
is
7/
;
'I'he
and up
to
number
if
,
)i
be odd.
of different values of the product
equal to the least
are integers.
common
multiple of
m
and
n,
when m and
CUBIC EQUATIONS.
483
To
solve the general cubic equation
a;^
Remove
the term
j^at^
formed equation be
484
Cardan
s
of this equation
+ jj.r + qx -f = 0.
/•
by the method of (429).
.v'^-\-q,r-\-r
mcfhoiL
— The
=
x
=
0.
complete theoretical solution
by Cardan's method
Put
is
Si/::
+q =
0;
as follows
.'.
:
(i.)
i/-\-:i
yH,v^ + (3v.v + 7)(y + ,v)
Put
Let the trans-
^
+r =
0.
= - 3^
Substitute this value of //, and solve the resulting quadratic
The roots are equal to 1/ and .r* respectively ; and we
//^.
have, by (i.),
in
485
r
{-iWf+j^r+i-^-vj+f;}'
THEOBY OF EQUATIONS.
128
Tbe cubic must have one
real root at least,
Let »i be one of the three values of
of the three values of
486
Let
Then, since
y^,
Viu^ = my
+1-
if in
X
"^
and
9"
(400).
"*"
\/ TT
'^
^'
(
^°^ " "°®
'
[
=-
ci'
am-ta'-n,
ii,
1-
- L yZs.
[472
the roots of the cubic will be
I,
m+
Now,
\/
^
j
"^
be the three cube roots of unity, so that
1, n, a-
a=-l
487
^
j
lij'-
u'in-\-n)i.
the expansion of
^V
2
I
4
^
273
by the Binomial Theorem, we put
= the sum of the odd terms, and
= the sum of the even terms
and « = ^ — v;
m= +
m = + v/ — 1, and n = — y v— 1
fx
V
then we shall have
or else
/u
according ^^ \/ 'T
By
;
y,
/u.
'^
fji
§^
substituting these expressions for in and
488
(i-)
If
V"
^
+
(ii.)
If
—/i + >'%/— 3,
97 ^^ negative, the roots
"^
+
If
4
t^
27
=
489
is
now
'/^/'^'
equal to
when -p
4
+ 77=
27
But
<6'
yv—o.
will be
/J
-m;
—:?/!,
Trigonometrical method.
solved in
Assume
—
fi.
.1'^
may be
appears that
the roots are
0,
2ot,
m
it
3
,2
since
in (487),
— — vn/S.
+ y^S,
—fx
2/1,
(iii.)
—fi —
cj^
-r
n
^® positive, the roots of the cubic will be
2/^,
r-
;
^^ imaginaiy.
i^ ^^^^
+
r/.r
following manner, by Trigonometry,
negative.
is
=
tlie
— The equation
+r =
?;
cos
Divide the equation by n^; thus
a.
cos'' a
+
-
-?
cos« a
4
-2-
cos a
cos a
-
^
-\
=
^^
-^ =
4
0.
By
((357)
I
TilQVADRATW EQUATIONS.
Equato
n
coefficients in tlic
must now be found
two
^villl
(M|n;itions
the result
;
the aid of
129
is
Trigonometrical
tlie
tabh's.
The
490
roots of the cubic will
491
tive.
Observe
l)e
n cos (^tt— a).
n cos (jTr+a),
n cos a,
that, according as
.2
3
--
+ ^-
is
positive or nega-
Cardan's method or the Trigonometrical wall be practiIn the former case, there will be one real ami two
cable.
hnaginary roots ; in the latter case, three real
roots.
BIQUADRATIC EQUATIONS.
492
Descartes' Solution.
.v'
the term in
Assume
.r'
— To solve the equation
+ qA' +
r.v -\- s
=
(i.)
having been removed by the method of
(.t.-+(u'+/) {.i--i\v-\-^-)
('t29).
=
(ii.)
Multiply out, and equate coefficients witli (i.) and t1ie
lowing equations for determining /", g, and e are obtained
ir+/='/ + ^'%
493
/=A
.ir-
^.«4.2rye'+(r/-4v)e2-j-
;
fol-
Kf=''*'
i'''')
=
(iv.)
The cubic in t? is reducible by Cardans method, when the biquadratic
a
//5 and
two real and two imaginary roots.
For proof, take «
y as
Form the sum of eucb pair
the roots of (i.), since their sum mu.st bo zero.
for tlio values of e [see (ii.)]. and "PP'Y t^° ^^^^^^ ^" ('i88) to the cubic in e*.
1/ the biquadratic has all its roots real, or all imaginary, the cubic will liave
all its roots real.
iy for four imaginary roots of (i.),
Take n
//3 and
and form the values of e as before.
494
±
495
will he
— ±
±
hits
If' «'>
f^f y' ^e the roots of
_i(„+/5 + y),
—a ±
tfie
cubic in
l(a+/3-y),
4(/9
e',
tlie
roots of
+ y-n),
^ (y
tlie
biquadratic
+ a-/3).
THEORY OF EQUATIONS.
130
For
sum
proof, take w, x, y, z for the roots of the biquadratic; then, by (ii.), the
Heuce, we have only to solve the
of each pair must give a value of e.
symmetrical equations
1/
+2 =
Z
-I- iC
/3,
w + ,7;=— a,
= — /j,
?o +
?/
y,
w-\-z
a-
=
+ =
CI,
Ferraris solution.
496
;/
— To the
= —y-
left
member of the equation
= 0,
x^+lKv^-\-qx^-\-ra:-\-s
add the quantity
+ bx + —,
ax^
and assume the result
= (,.+|,+,„y.
Expanding and equating coefficients,
is obtained
cubic equation for determining
497
the
following
m
8m^—4iqm^-\-{2j)r—8s)m-\-4!qs—2^^s
Then x
i).
given by the two quadratics
is
^
498
—r =
The cubic in
2
,
+
m is
»
2
,
'*'
+
2(Lv-{-b
= ± -TvTT
,
'*'
reducible by Cardan'' s method ivhen the biquadratic
Assume a, /3, y, B for the roots of the
has two real and tioo imaginary roots.
biquadratic then aft and yB are the respective products of roots of the two
in terms of aftyS.
quadratics above. From this find
;
m
Elders
499
solution.
have
500
and
.V'*'
Assume
x
— Remove
+
the term in x^; then
q.v"^ -\-r.v-\-s
= i/-\-z-\-u,
and
it
=
we
i).
may
be shewn that
i/, z^^
u^ are the roots of the equation
fA.±f-4-'^"'~^'t-^=0
^ ^ ^
^
2
501
The
six values of y,
restricted by the relation
Thus X
=
10
//-!-;.
+
/<
z^
yzii.
(>i
and
//,
thence obtained, are
=
will take four different values.
COMMENSURABLE ROOTS.
131
COMMENSURABLE ROOTS.
502
'1'" t'""l
commonsuraUlo roots
tlio
First transform
tlic
form
liaving
503
and
j>o
by pultinjr
it
.r"+/>i.r"-'+;>.>.r"--+
= 1»
iiii^l
tlio
•''
=
of an ofiiiation.
"^^^
'ir
=
+/>„
...
^^^^
equation of
0,
remaining coefHcients integers.
(l-Gl)
This ecpiation cannot have a rational fractional root,
roots may be found Ijy Newton's method of
tlie inte<]:ral
Divisors (451)).
These roots, divided each by h, will furnish the
surable roots of the original equation.
504
Example.
— To find the comrnensurablo roots of
8 1 /'
- 20 7x' - 9x» + 89.r + 2,r - 8 =
Dividing bj 81, and proceeding as in (431),
tion to bo
a;
The transformed equation
commen-
equation
tlie
0.
\vc find the requisite substitu-
= -^.
is
,/_23/-V + 801)/- + lG2//-5832 = 0.
The roots all lie between 24 and —34, .by (451).
The method of divisors gives the integral roots
G, —4. and 3.
Therefore, dividing each by 9, we find the commensurable
and |.
equation to be
3, — f
roots of the original
,
505
To
the roots
G,
obtain the remaining roots
—4, and
3, in
diminish the transformed equation by
manner (see 427)
:
-4
is
therefore
if
-
18(/
-
81
=
0.
which arc
(1+ \/2) and 9 (1— v/2)
incommensuiable roots of the proposed equation are
The
roots of
—
1_23- 9 + 801 + 1G2- 5832
6— 102— GGG + 810 + 5832
1-17-111 + 135 + 972
- 4+ 84 + 108-972
-21- 27 + 243
3_ 54-243
1_18- 81
6
The depressed equation
;
the following
;
and, consequently, the
V'l and 1— v/2.
1+
—
—
THEORY OF EQUATIONS.
132
INCOMMENSURABLE ROOTS.
—
506 Sturm's Theorem. lff(x), freed from equal roots, bo
divided by /('>')> and the last divisor by the last remainder,
changing the sign of each remainder before dividing by it,
until a remainder independent of x is obtained, or else a remainder which cannot change its sign; then /(a^), /'('<-')' ^^^
the successive remainders constitute Sturm's functions, and
are denoted by
f{a-)
,
/
(<r)
,
f^
{.v)
,
The operation may be exhibited
M^r)
507
Note.
rejected,
—Any
=
&c
/« Gv)
as follows
•
:
q,f,{.v)-f,{.v),
constant factor of a remainder may.be
set down for the corres-
and the quotient may be
ponding function.
An
inspection of the foregoing equations shews
That /„, (ft') cannot be zero; for, if it were, /'('^O ^^^
would
/i (x) would have a common factor, and therefore /(<^')
have equal roots, by (432).
(2) Two consecutive functions, after the first, cannot
vanish together for this would make/*„ (x) zero.
(;3) AVhen any function, after the first, vanishes, the two
adjacent ones have contrary signs.
508
(1)
;
509
If,
as X increases, f(x) passes through
the
value zero,
StnrrrC s Junctions lose one change of sign.
For, before ./'(-O tf^kes the value zero, /(a-) and/, {x) have contrary signs,
and afterwards tliey have the same sign; as may be shewn by making h
small, and changing its sign in the expansion off{x + li), by (420).
510 If d-nij other of Sturm's functions vanishes, there is
neither loss nor gain in the number of changes of sign.
This will appear on inspecting the equations.
511
Ri:sui;r.
equal
to
— The nnmhvr of roots
the difference in
Sturm's functions, when x
off(,v) hrtu'cen a and h is
number of cluoiges of sign in
and when x = b.
the
=a
—
INCOMMENsuit ABLE ROOTS.
512
<'"";•
—
Tlio total
iiiiin])ci- ui"
mid h =
then bo the same as that
by taking
will
a
=
\VTien tho
-\-
cc
number
—
cc
133
roots of /(,*) will he foiiiid
tho aU^n of each fuiictioii
;
of its first term.
of functions exceeds the degree of f{x)
by unity, the two following theorems hold
:
513 If thr jir^t trrins hi all flii'fdnrtionii, aj'trr
/losifirc ; all flu' roots off{j) arc real.
t
lie
first ^ are
514
If the first terms are not all positive ; then, for every
of si(j)i, there will be a pair of imaginart/ roots.
For the proof put .r = + co and — oo, and examino the number of
r/iiiiK/c
changes of sign in each
case,
applying Descartes' rule.
(-tl6).
and if <p (x)
(•'') ^^^ no factor in common with /(<>'),
0; then the rest of
take the same sign when /(,<)
Sturm's functions may be found from f{.f) and <p (</;), instead
For the reasoning in (509) and (510) will ajjply to
of /'(./).
the new functions.
515
and
If ^
=
_/"(,/')
If Sturm's functions be formed without first removing
equal roots from /(,/'), the theorem aWII still give the number
of distinct roots, without repetitions, between assigned limits.
516
if/(.r) and /, (x) be divided by their highest common factor (see 444),
the quotients be used instead of /(./) and/, (.r) to form Sturm's functions
then, by (olo), the theorem will ajjply to tho new set of functions,
which will dilfer only from those formed froin/(,c) and/, (./.) by the absence
of the same factor in every term of the series.
For
and
if
;
517
Example.
— To find the position of the roots of the equation
x'-4x^ + x- + Gx + 2
Sturai's functions, formed according to
the rule given above, are here calculated.
The first terms of the functions are all
positive ; therefore there is no imaginary
root.
The changes of
in the funcas X passes
integral
through
exhivalues, are
sign
tions,
bited in the adjoin-
ing
table.
There
are two changes of
sign lost while x
passes from
1 to
—
and two more
lost while X passes
from 2 to 3. There
0,
= 0.
f(x)=x*—4a^+ x^+ Gx+
A(^')
=
2
THEOEY OF EQUATIONS.
134
—
and
\
and two roots also between
are therefore two roots lying between
2 and 3.
These roots are all incommensurable, by (503).
Fourier's Theorem.
518
lowing quantities
f{x),
;
—Fourier's
f'{.v),
functions are the
fol-
/"(cv).
f'i^v)
—
As x increases,
Properties of Fourier's functions.
Fourier's functions lose one change of sign for each root of
the equation /(.f)
0, through which x passes, and r changes
of sign for r repeated roots.
519
=
520
of
If
any
of the other functions vanish,
changes of sign
an even number
is lost.
—
Results.
The 7iumher of real roots of f{x) hetween a
cannot he more than the difference hetween the numher
a, and the
of changes of sign in Fourier's functions when x
numher of changes ivhen x
j3.
521
and
j3
=
=
522
roots
When that
is
odd,
difference
is
and therefore one
odd, the number of intermediate
at least.
523 When the same difference is even, the number of intermediate roots is either even or zero.
Descartes' rule of signs follows from the above for the
are the signs of the
signs of Fourier's functions, when x
oo, Fourier's functions are all
terms in /(«); and when a3
524
=
=
positive.
525
Lagrange's method of approA'wiating to
mensiirahle roots of an equation.
the incom-
Let a be tlie greatest integer less than an incommenDiminish the roots of /(,r) by a. Take
surable root oif{x).
the reciprocal of the resulting equation. Let h be the greatest
Diminish
integer less than a positive root of this equation.
the roots of this equation by h, and proceed as before.
526 Let a, h, e, &c. be the quantities thus determined
an approximation to the incommensurable root oif{x)
the continued fraction
x
= a + -—
h-\-
-,f-l-
;
then,
will
be
INCOMMENSURABLE ROOTS.
527
AV/c/o/<'.v
a little less than
=
+
mcfluK/ <tf(ii>pr(uinnfti<ni.- If r, hv a ((uaiitity
0, so
one o£ the roots of the eciuatioii fU)
0; tlieu c^ is a first approximation to the
=
//)
that /(r,
value of tlie root.
/(,-,
and
In
Co,
h
is
+
but
/,)
]3i
Also because
=/(,.,)
sniall,
+ /,/{,-,)
+^'/"(.-,)
+ &c
(KG),
a second approximation to the root will be
tlie same way a third ap]:»roximation may be obtained from
and so on.
—
To ensure
Fourier's limitation of Newton s method.
&c. shall successively increase up to the value
for all values
Ti + h without passing beyond it, it is necessary
528
that
Ti, ^2, Cg,
of X
between
c^
and
Ci-\-h.
Thatf{;ii) andf'{,v) should have contrary signs.
Thatflx) and f" {x) should have the same sign.
(i.)
(ii.)
Fia.
Fia. L
2.
A proof may be obtained from the figure. Draw the curve
y=f(^x). Let OX be a root of the equation, and ON = Ci;
draw the successive ordinates and tangents NPy PQ, QBy &c.
Then OQ = c^, OS = Cg, and so on.
Fig. (2) represents Tg > OX, and the subsequent ai)proximations decreasinof towards the root.
the Roots.— hot the cobe respectively 'divided by the Binomial
coefficients, and let ^o, (T,, a, ... o„ be the quotients, so that
530
Newton
s
Rule for Limits of
ethcients of /(./)
f{x)
=
ao-'j"
+ uai,/;"- + ''t^"''
+
...
+ » ",.-!»;+«
—
TUEOUY OF EQUATIONS.
13G
Lot ^Ij, ^lo, ^Ig
An be formed by tlic law A^ = a^— ri^_i(X^+i.
Write the first series of quantities over the second, in the fol.
loA\dng
manner
.
.
:
«0,
rti,
('2,
((3
(^n-l,
««,
ylo,
Ai,
^2,
ylg
.-l,,_i,
A,,.
"Whenever two adjacent terms in the first series have the
sign, and the two corresponding terms below them in
the second series also the same sign; let this be called a
double 2Jermanence.
AVhen two adjacent terms above have
different signs, and the two below the same sign, let this be
known as a variation-permanence.
same
—
Rule.
The number of double 'permanences in the associated series is a superior limit to the number of negative roots
531
The number of variation-permanences is a superior limit to
number of positive roots.
The number of imaginary roots cannot be less than the
number of variations of sign in the second series.the
532
Sylvesters Theorem.
— Let /(f +
X)
be expanded by
(426) in powers of x, and let the two series be formed as in
Newton's Rule (530).
Let P (X) denote the number of double permanences.
Then P (X) ~ P [fi) is either equal to the number of roots
of /(ft'), or surpasses that number by an even integer.
—
The first series
Note.
then stand thus,
/-^X),
/«(X),
The second
series
where
533
G,
(X)
[2/«-2(X),
may
On{^),
may be
multiplied
by
[n_,
and
will
[l/"-^(^)-h/W-
be reduced to
r/„_,(x),
r/„_,(x)...rr(x),
= {/'• (X)}^ - '^y^^ f-' (A) f^' (X).
—
Horner's Method. To find the numerical values of the
Take, for example, the ecpiation
roots of an equation.
x'-4t^ + x^-\-6x-\-2
and
find limits of the roots
= 0,
by Sturm's Method or otherwise.
INCOMMENSUnAIiLi: HOOTS.
It has
been shewn
137
—
138
THEORY OF EQUATIONS.
ti'ying 5 instead of 4.
This gives
— Y?
with a minus sign, thereby pi-oving the
A.,
24 and
existence of a root between
2-5.
The new
coetficieuts are
A.^, B.^, C.,, D.^.
gives 1 for the next figure of the root.
Affix ciphers as before, and diminish the roots
coefficients as A^, B^, C^, D^.
by
1,
distinguishing the
new
Note that at every stage of the work A and B must preserve their signs
unchanged. If a change of sign takes place it shews that too hirge a figure
has been tried.
—
After a certain number of
the calculation pi'ocecd thus
have been obtained (in this example four), instead of
adding ciphers cut ott" one digit from B^, two from C'^, and three from D^.
This amounts to the same thing as adding the ciphers, and then dividing
each number by 10000.
To abridge
:
figures of the root
Continue the work with the numbers so reduced, and cut off digits in like
manner at each stage until the D and C columns have disappeared.
Aj and Bj now alone remain, and six additional figures of the root are
determined correctly by the division of A^ by By.
To find the other root which lies between 2 and 3, we proceed as follows
:
This gives Jj
After diminishing the roots by 2, try G for the next figure.
negative; 7 does the same, but 8 makes J., positive.
That is to say, f{2'7)
is negative, and/ (2'8) positive.
Therefore a root exists between 2*7 and
2-8, and its value may be approximated to, in the manner shewn.
Throughout this
Observe also that the
each
stagfe
^ of the
A
will preserve the negative sign.
for the next figure of the root given at
last
calculation
trial
number
process
the formula
by
I
y
too great, as in the former case
—
f(c)
,
{, will in this
/(c)'
was always too small.
^
it
case be always
^
SYMMETEICAL FUNCTIONS OF THE HOOTS OF
AN EQUATION.
Notation.
/(••'•)
= 0.
Let
.s,„
— Let
denote
a'"
a, h, c
+
//"
+
...
.
,
.
.
bo the roots of
tlic
sum
of tlie
tlie
equation
powers of
7/^^''
the roots.
Let .s,„,p denote a"'h^' + lr(iP-\-a'"<'''-{permutations of the roots, two at a time.
...
througli
all
Similarly let .9,„,p
denote (rh'\'''-\-a'"h^\V'-\- ... taking
the permutations of the roots three at a time ; and so ou.
,,
,
the
all
SYM}n:rRICAL FUNCTTONS of TIIF
534
SUM>^
wliere
m
OF
nil': /vmi7;a'.s'
the degree
is less tliau n,
h'OOTS.
of tuf
13!)
norrrs.
oiJ'{.i).
Obtained by expanding by division each term in the vahio of/'(.i) given
at (432), arranging tlie whole in powers of .r, and equating coeiricieiits in llie
result and in the value ofy^^r), found by differentiation as in (1-21).
535
If "i
^0 greater than
Obtained by multii)lying
a, h,
c,
By
//,
/(./•)
forimihi will be
tlie
=
by
.'""",
substituting for
.i;
the roots
&c. in succession, and adding the results.
these formula?
.<„
s.,,
.«„
&c.
may be
calculated successively.
To find the sum of the negative powers of the roots, put
equal to n—1,
2, ?^
3, &c. successively in (535), in
order to obtain s_i, s_o, s_3, &c.
536
m
)i
To
537
Rule
s,.
i'i
n
^'^"J
calculate
:
=—
independently.
.'?,.
r
—
—
X
'^''•^''''ii'^''>t'J
rorjjicieut
I^OlV^^iS
of
the
in
x~''
e:q)amlo)i
of .V.
Proved by taking f(x) = {x~ a) (x-h)(x — c) ... dividing by
expanding the logarithm of the right side of the equation by (loO).
,
SYMMETniCAL FUNCTIONS WTUGU
NOT POWERS OF THE BOOTS.
538
of
x",
and
AJ^E
These are expressed in terms of the sums of ])o\vers of
the roots as under, and thence, l)y (531), in terms of the routs
explicitly,
539
'V„,,^,.,,
The
."f,
;
=
.V,,,*,,*,/
last equation
and expansions
—
.V„,
+ /, A-,,
may be
—
*•„,
+ ,,*,,
]iroved
—
A',
,+ ,.?,„
+2.V„, + ^ +
by mnlti})lying
of other symmetiical functions
,.
,
.<„,.,,
by
may be
obtained in a similar way.
If
('•) be a rational integi'al function of ,r, then the
symmetrical function of the roots of ./'('), denoted by
540
</>
—
•
TEEOUT OF EQUATIONS.
140
+
&c., is equal to
^ (a)-{-(j> (h) -{-<!> (()
the remainder obtained by dividing
coefficient of
tlie
<{>
0*')
/'('')
x''''^
in
^7 /('*')
Proved by multiplying the equation (432) by yji, and by
tlieorera (401).
To find tlie equation wliose roots are tlie squares of
differences of tlie roots of a given equation.
541
tlie
Let F{^) be the given equation, and S,. the sum of the r*''
powers of its roots. Let/(/c) and s^ have the same meaning
with regard to the required equation.
The coefficients of the required equation can be calculated
from those of the given one as follows
:
The coefficients of each equation may he connected tvith the
sums of the ])oivers of its roots hy (534) and the sums of the
;
poivers of the roots of the
tivo
equations are connected hy the
formula
542
2^,
= nS,,,-2rSA._,-\- ^''('^''-^) S,S,,_,-
...
+«S,,
—
Rule. 2s,. is equal to the formal expansion of {S—Sf'' hy
Binomial Theorem, with the first and last terms each mulAs the
tiplied hy n, and the indices all changed to siffixes.
equi-distant terms are equal we can divide by 2, and take half
the series.
the
Demonstration.
— Let
Let
a, h, c
<l>(x)
...
be tbe roots of
i^(a;).
= (x-ay-' + (x-hy+
(i-)
right by the Bin. Theor., and add, substituting
In the result change x into a, b, c ... successively, and add the n
Si, S2, &c.
equations to obtain the formula, observing tliat, by (i.),
Expand each term on the
(a)
If
r?
be
degree of
tlic
+0 (&) +
7^
(,*•),
...
=2.v
then
'k)i(ii
— l)
is
of /(.').
543
The
where
a, ft, y,
term
last
of the equation
7" ('')
=
is
the degree
%
(UG).
equal to
n^T{a)FmF{y)...
that
544
...
are the roots of F'
F'{a)F'{h)
If -^(''O
have imaginary
^^^s
...
{•>').
Proved by shewing
= n"F{a)F(ft)
...
negative or imaginary roots,
roots.
f(.i)
must
SYMMETRICAL FUNCTIONS OF THE ROOTS.
545
The sum
of tlic
.*-— />.r+7
ratic ecjuation
.v„,
powers of
v«"'
=
;>'"
—
,„
ni]>"'
.,
+
,
-(/
By
546
By
547
sum
'I'he
is n if ni
(537)
;
m^^'
<!>{.,•)
sum
—
.v=
where
a, /3, y,
-j
.*»)
p
,„_t
(|u:i(l-
-
(/-...
(io") expaudiug the logarithm by (15G).
powers of the roots of
and zero if it be not.
</y*
—1 =
)i,
= a,-\-a,x-\-n,x- + &c
of the selected
^-illbe
= 0.
—
in {ni
roots of the
expanding the logarithm by (15G).
If
then the
of tlie
be a multiple of
tlie
141
(i.):
terms
;a"-"'<^(a.r)+y8"-"'<^()8./0+y"""'*(r'O
&c. are the
n^^'
+
^^'C.}
roots of unity.
For proof, multiply (i.) by a""'", and change x into
and add the resulting equations.
a.v;
so with
/?,
y, &c.
To approximate to the root of an equation by means of
the sums of the powers of the roots.
548
By
taking
m
large enough, the fraction
tuill
^vill
approx-
imate to the value of the numerically greatest root, unless
there be a modulus of imaginary roots greater than any real
root, in which case the fraction has no Umiting value.
549
Similarly
tlie
fraction
•^''"•^'"^•-~'^'"":.'
increases, to the grrnfr.^f product of
api)roximates, as
any pair
m
of roots, real or
imnginary excepting in the case in which the product of tho
pair of imaginary roots, though less tlian the product of tho
two real roots, is greater than the scpiare of the least of tliem,
for then the fraction has no limiting value.
;
/
THEORY OF EQUATIONS.
142
550
Similarly
tlie
'
fraction
'"
'"'^y '""^^
'"'^^
^'m'^'m + 2
approximates,
^m + 1
m
increases, to tlie sum of tlie two numerically greatest
roots, or to the sum of the two imaginary roots with the
as
greatest modulus.
EXPANSION OF AN IMPLICIT FUNCTION OF
Let
r{A,v''+)+!f{By-]-)^...-\-fiS.r^+)
=
.r.
(1)
be an equation arranged in descending powers of y, the coefficients being functions of x, the highest powers only of x
in each coefficient being written.
It is required to obtain y in a series of
of
descending powers
X.
First form the fractions
a
a
—b
—
—
Let
a
a
a
y8'
—n
—
-
t
—c
— y'
a
a
—d
—S
n
a
—s
—a
,c)\
be the greatest of these algebraicalh^, or
and greater than the rest, let it be the
Then, with tlie letters corresponding to these
equal and greatest fractions, form the equation
if
several are equal
last of such.
-^Ku'^i)
Au''-^
(3).
Each value of in this equation corresponds
commencing with ux^.
to a value of
?/
Next
select the greatest of the fractions
/.•
K
_
— -——
K
V
/.
Let
v/,
—
— Ar»
Ix—m
K
—
/x
— I'—s
K—
/jx
v^V
cr
^,,
'
:=
t'
be the last of the greatest ones.
the corresponding equation
Kh"-\- ...-\-Nh''
=
(I
•
Form
(5).
Tlien each value of u in this equation gives a cori'cspondiiig
value of y, commencing with »./.
—
EXPANSION OF AN
Proceed
is
in tliis
way
IMI'LICIT Fi'M'TIDN.
143
until llie last tVactiun of tlio series (2)
reached.
To
obtain the second term in the exjjansion of
=
,/
.,.'(//+//,)
Ill
//,
put
03),
(1)
and //, /"
cmplojnng tlie dilTerent vahies of n, and again of
and ti, &c. in succession; and in each case this substitution
will produce an equation in // and x similar to the original
/'
equation in
//.
new
^vith the
Eepeat the foregoing process
observing the following additional rule
eqnation in
y,
:
Wien all the values of t, t\ t", ^c. have heeu ohtained, the
negative ones only must be employed informing the equations
(7).
in n.
obtain y in a series of ascending powers of /.
(1) so that a, /3, y, &c. may be in asccndliKj order of magnitude, and a, h, c, &c. the lowest powers
in the respective coefficients.
of
To
552
Arrange equation
.t'
Select f, the greatest of the flections in (2), and proceed
exactly as before, with the one exception of substituting the
vford jmsitive for negative in (7).
553
Example.
{x^
It is
2/
4.7;
+ 7x- + a;") >f - >/ =
required to expand y in ascending powers of
The
The
— Take the equation
+ x') + ( 3.^^ - hx^) + ( -
The
two being equal and
Eqnation (3)
and from
is
—
and
u
=
^
— ^,
7/,),
ij,
tir.st
arrange in ascending powers of
^^'
'
^'
^'
^"^^ ^'
'
•
/
=
1.
t'
= ^.
.tV, or
a;*(-4^
+ i/,).
of tlieso values, in the originul equation,
y,
thus
-4x* + (-rjx'-{-)y,+ (-4.c' + )yl
The lowest power only
^
= 1.
according to (G), cither
a:(-i + J/,),
the
with
t
= 0,
and —4*, with
to sub.stitute for
= x (1 +
-
—4ir—u''
^(1 + y,),
Tut y
=1
is
this
have now
—
=i=
\+3n- 4ir = 0,
?/
Equation (5)
we have
gi-catest,
fractions (4) reduce to
which gives
"NVe
- -'^. "qZo' ~ ^E
fractions (2) are
first
0.
x.
-
10x'y\
of x in each cocflicient
is
- bx'y\ -
.c'y]
here written.
=
0.
and
—
.
THEORY OF EQUATIONS.
144
The
now become
fractions (2)
4-3
4-3
_4-5
4-5
0-1'
0-2'
0-3'
0-4'
h
1,
From
/=
tlieso
1,
and equation
(3)
— 4— 5?t =
Hence one
of the values of y^
Therefore
y
Thus the
first
is,
5'
-h -h -h
becomes
0;
.•.
as in (G),
= x {1 + (-f + 2/2)}
«;
=—
= x (— f +
= aJ-f'»'+
«
4..
yi
2/2)-
...
two terms of one of the expansions have been obtained.
DETERMINANTS.
554
Definitions.
— The
determinant
is
b,
to a^h-i—dtbi-,
a.^
b^
63
Ci
C.2
C;
Another notation
The
called a determinant of the second order.
the third order is
^Tifl is
A determinant of
=
a^
b,
equivalent
hi
letters
rti
{Ihc^
63^2)
+ a. %c^ — b^c^) +
a.^ {b,c.,
- b.,c^)
2 ± (lyh.^c^, or simply {a^h.yC-^.
named constituents, and the terms are
The determinant is composed of all the
is
are
called elements.
elements obtained by permutations of the suffixes
1, 2, 3.
determinants of
the next lower order, and are termed minors of the original
Thus, the first determinant above is the minor
determinant.
of ^3 in the second determinant. It is denoted by C^. So the
minor of % is denoted by vli, and so on.
The
555
of the
coefficients of the constituents are
A
determinant of the
forms below
b,
b.
7^*^
order
may be
wintten in either
,.
«,.
...
«„
...
«„,
..
6,
...
b.
...
a.,.
In the latter, or double suffix notation, the first suffix indicates
The former notation
the row, aud the second the column.
will be adopted in these pages.
—
^^
DETERMTXAXTS.
A Compoiiift' ilotrrminant is one in which the ft a., «!
h, h., b.
Tuimber of cohimns exceeds the innnber of rows,
and it is wntten as in the annexed example.
Its vahie is the sum of all the determinants obtained \)\ takiiif^
a number of rows in every possible way.
A Simple ih'tormlnnnt lias single terms for its constituents.
A Compound drtrrmuinut has more than one term in somo
See (')70) for an exam[)le.
or all of its constituents.
I
For th(^ definitions of Si/mmetrical, Rcripronil, Parfial,
and Complcmciitari/ determinants; see (574), (575), and (576).
General Theory.
The number of constituents is n^.
Tlie number of elements in the complete determinant is
556
[?^^.
Any
The first or leading element is aih.,<\ ... /„.
element may be derived from the first by permutation of the
557
suffixes.
The sign of an element is + or — according as it has
been obtained from the diagonal element by an even or odd
number of permutations
Hence the following
of the suffixes.
rule for determining the sign of
an
element.
Take the suffixes in order,
RuT.E.
Let
their places in the first element.
of places passed over
Ex.
— To
the
find
Move
the
;
then
sign
(
— 1)"*
"'^7/
In
all,
tlio
detei-minanfc
«4
h
^s
'h
'^i
.suffix 1, tlircc pliiccs
1
4
3
''>
2
three places
3, one phwje
1
2
2
4
3
')
3
4
5
„
seven places; therefore (—1)'
n
give the sign required.
of the element n^h^CrJ\c.^ of
2,
„
and put them hach to
he the whole number
m
= —1
1
gives the sign required.
any element be transi)o.^ed, the sign
changed.
Half of the elements are plus, and half are minus.
558
two
suffixes in
of the element
559
is
The elements
are not altered
])y
changing the rows into
columns.
If
two rows or columns are transposed, the sign
U
of tho
TUEOUY OF EQUATIONS.
146
determinant
is
clianged.
Because each element changes
its
sign.
If
two rows or columns are
determinant
identical, the
vanishes.
560 If all the constituents but one in a row or column
vanish, the determinant becomes the product of that constituent and a determinant of the next lower order.
—
A
\ successive
cyclical interchange is effected by n
561
transpositions of adjacent rows or columns, until the top row
has been brought to the bottom, or the left column to the
right side.
Hence
A
cyclical interchange changes the sign of a determinant
of an even order only.
The
r*^
row may be brought
to the top
by
r— 1
cyclical
interchanges.
562
If
each constituent in a row or column be multiplied
factor, the determinant becomes multiphed by it.
by the same
If each constituent of a row or column is the sum of
terms, the compound determinant becomes the sum of
simple determinants of the same order.
m
m
Also, if every constituent of the determinant consists of
terms, the compound determinant is resolvable into the sum
of m^ simple determinants.
m
563 To express the minor of the
a determinant of the n V^ order.
r"'
row and ¥^ column
as
—
I*ut all the constituents in the r^^ row and /.-^^ column equal
to 0, and then make r— 1 cyclical interchanges in the rows
and L 1 in the columns, and multiply by (_iy'-+^)(«-i).
—
r._.
564
To express a determinant
minant
as a deter-
of a higher order.
Continue the diagonal with constituents
of " ones," and fill up with zeros on one side,
and with any quantities whatever (o, /3, y, &c.)
on the other.
_
f_Y\(r-\*k-\){n-l)^
1
a
1
fi
e
V
C h
h
r)
{)
a h
-
h
/•
<>'
f
c
U7
vi:TEnMTy.iNTS.
each constituent of a
another given cohimn
And the same is true if we read row' instead of
is zero.
*
Thus, referring to the determinant in (555),
cohimn.'
Tlio sura of tbe products
565
column by
tlio
of
corresponding minor
in
'
Taking
For
in
/)'''
tlie
and
r/^'
cohimns,
Taking tho
a
and
r
rows,
each case we have a dotcnuinant with two columns identical.
I" i^ny row^ or column tlie sum of the pnxhicts of each
That is,
constituent by its minor is the determinant itself.
566
Taking the
The
567
Also,
j)^^
hist equation
if
Or taking the
cohimn,
i'lpCq)
c
may be expressed by ^- p rp
'
express the determinant
a„ a
row.
= A.
then
represent the sum of all tlu^ determinants of the
second order wdiich can be formed by taking any two columns
out of the a and r rows. The minor of {dp, (\) may be wi'itten
(Ap, Cy, and signifies the determinant obtained by suppressThus
ing the two rows and two columns of Op and c,.
or
three
when
notation
similar
a
And
{Ap,
S
{dp,
Cg)
Cq).
A
^
more rows and columns are selected.
'2{apr,^) will
=
Analyji'is
568
Rule. — To
of a determinant.
resolve into its elements a determinant of thn
Express it as the sum of n determinants of the
n*^ order.
(n—iy'' order by (5G0), and repeat the process with e ach oj
the
new determinants.
EXAMl'LE
«l «J «3 "i
c,
q
r,
c^
«/,
(/j
d^
ll^
Again,
:
=
a,
b,
b,
b,
——
:
TUEOUY OF EQUATIONS.
148
569
Si/nthesh of a determinant.
The process
rules.
by making use of two evident
Those constituents which belong to the row and
column
of a given constituent a, will be designated
is facilitated
*'
a's conAlso, two pairs of constituents such as a^, c, and
forming the corners of a rectangle, will be said to be
stituents."
dg, c^,
*'
conjugate" to each other.
Rule
Rule
No
I.
with one of
its
constituent
he
loill
found
in the
same term
oion constituents.
The conjugates of any two constituents a and
and Us constituents.
II.
b
will he cuinnion to a's
Ex.
— To write the following terms in the form of a determinant
uhcd + Ifyl +fh^ + ledf+ cghp
+ 1 ahr + elpr
—fhpr—ahlr—ach^ — Ifhg — bdf^ — efhl — cedp.
The determinant will be of the fourth order and since every term must
contain four constituents, the constituent 1 is supplied to make up the
number in some of the terms. Select any term, as ahcd, for the leading
;
diagonal.
Kow
a
ia
1)
is
apply Rule
I.,
not found with e,f,f, g,p,0...(l)not found with e, h, h,p,l,0 ..(2).
c is
not found with
/,/,
dis not found with
I,
r, 1,
0... (^).
g, h,h,l,r,0...(-i).
Each constituent has 2 (»— 1), that is, 6 constituents belonging to it, since
n =?'4. Assuming, therefore, that the above letters are the constituents of
a, b,
c,
and
d,
and that there are no more, we supply a sixth zero constituent
in each case.
Now
apply Rule
to
II.
and b are
a and d g,
to a
—
e,
— The constituents common
to a and — /;
p
to
and d —
c
;
to b
-/,
h
;
h,
li,
The determinant may now be formed.
being
abed,; place
in the
e,
p, the eoiijugatus of
and
and
c
—
d-
1,
/,
;
/•,
0.
Tlie diagonal
a and
b,
either as
diagram or transposed.
Then /and/, the conjugates
and
of a
and
the conjugates of b and c,
1 is one of y/s constituents, sini-e it is
must therefore be in the second row.
1
to c
;
0,
Similarly
tlie
places of y
and
0,
c,
may be
written.
must be placed
and of
as indicated, becau.so
found in any term with
nt)b
Z
and
r,
/i,
and
are assigned.
In the c-ase of b and d we have h, h,
from wliieh to clioose the two
conjugates, but, we see that
is not one of them because that would assigu
two zero constituents to b, whereas b has but one, which is already placed.
By
similar reasoning the ambiguity in selecting the conjugates
I,
r
is
removed.
The foregoing method
is
rigid in
the case of a complete determinant
DETEKMJXAXTS.
1
to
having different constituents. It becomes uncertain when the zero conKtituonts increase in number, and when several coustitueuts are identical.
Ihit even then, in tlie majority of cases, it will soon afford a cluo to the
required arrangement.
I'RODUCr OF TWO DETERMINANTS OF
570
Tin:
C)
{Q)
(I.,
...
a,
h, h,
...
h,
</,
L
(lUDEU.
„"•
a,
a>
...
.U A,
a„
/;,
X,
/..
L, L,
X,
Tlie values of A^, By ... L, in
tUo first column of S arc annexed. For the second column
write //s in the place of as.
For the tliird columu write f's,
and so on.
.
Ai
=
...
li
J„
ii,,
...
L,
ay ai-\-(L,a.,-\-
Ly r= aiXi4-^^X.+
.
...
+(?,.X,
For proof substitute the values of Ay, By, &c. in the determinant S, and
then resolve ,b' into the sum of a number of determinants by (oC>2), and note
the determinants which vanish through having identical columns.
—
Rule.
To form the determinant S, which is the product of
two determinants P and Q. First connect hij plus siijus the
constituents in the ro2VS of both the determinants F and Q.
and
Nou) place the first row of F upon each row of Q in turn,
let each two constituents as thnj touch become jn-oducts.
column of S.
same operation upon Q with the second rou- if
P to obtain the second column of S ; and again with the third
row of F to obtain the third column of S, and so on.
Tliis is the first
Perform
the
of columns, both in F and Q., be n, and
rows r, and if n be > r, then the determinant
S, found in tlie snme way from F and Q, is equal to the sum
of the C{ii, r) products of prirs of determinants obtained by
taking any r columns out of F, and the corresponding r
columns out of Q.
571
the
If the
number
But
if
number
of
n be
<r
the determinant
For in that case, in every one of
be two columns idcutical.
th
S
vanishes.
component determinants, there
will
THEORY OF EQUATIONS.
150
572
The product of tlie determinants P and Q may be
in four ways by changing the rows into columns in
formed
either or both
P
and
Q.
Let the following system of n equations in XicV,.^ ... x^
be transformed by substituting the accompanying values of
573
the variables,
The ehminant of the resulting equations in ^^ ^.,
^» is
the determinant S in (570), and is therefore equal to the
product of the determinants P and Q. The determinant Q is
then termed the modulus of transformation.
•
•
•
574 A Symmetrical determinant is symmetrical about the
leading diagonal. If the E's form the r^'' row, and the K's
the k^^ row ; then B;,
K^ throughout a symmetrical determinant.
=
The square
of a determinant is a symmetrical determinant.
^ Reciprocal determinant has for its constituents the
minors of the original determinant, and is equal to its
?i — l'^ power; that is,
575
fii'st
A,
...
Proof.
A,
h
576
...
4
— ^Multiply both
sides
of the equation by the original
determinant (o55j. The constituents on the left side all
vanish excepting the diagonal
of A's.
Partial and Complementary determinants.
rows and the same number of columns be selected
fi'om a determinant, and if the rows be brouglit to the top,
and the columns to the left side, without changing their order,
then the elements common to the selected rows and columns
form a Partial determinant of the order r, and the elements
7wt found in any of those rows and columns form the Complementary determinant, its order being n — r.
If r
DETERMINANTS.
151
Ex.— Let the selected rows from the dctcrmiiiaTit («,''3r,(/^^J be tlio
Becond, third, and fifth and the selected columns bo the third, fourth, aud
The orijjinal and the transformed determinants will be
fifth.
;
(J,
THEORY OF EQUATIONS.
152
equal.
by
Therefore
their product.
it is
And
divisible
equation are of the same degree
579
by
;
on the left therefore
seen to be unity, for both sides of the
^n {n 1).
eacli of the factors
the quotient
is
viz.,
—
'
The product of the squares of the|
same n quantities
j
;
'
_
~
differences of the
...
S,;
—
Proof.
Square the determinant in (578), and
write Sf for the sum of the r"' powers of the roots.
580
With the same meaning
minant taken
of
an order
r,
less
for
s-^,s.2...,
than
the same deter-
n, is equal to the
sum
of the products of the squares of the differences of r of the n
quantities taken in every possible way ; that is, in C {n, r)
ways.
Ex.:
^1
—
:
ELIMINATION.
153
ELIMINATION.
582
Solution of
11
li)i('(ir
rquntious in u rnriahlrs.
The equations and the values
of the varialjles are
arranged
below
«,.r,
+ f/,.r,+ + r/„.r„ =
^„
...
where
&c. are
To
A
is
.r,A
=
J^^,
+
/;,
f,+
the determinant annexed, and J^, B^,
minors.
+
...
^„
/.,
«,
...
/i
...
a
its first
find the value of one of the
unknowns
x^.
/.
HuLE. Multiply the equations respectively by the minors
of the r''' column, and add the results, x^ will he equal to the
?•'*
fraction whose numerator is the determinant A, with its
column replaced by Hj, ^2 ••• s„, and whose denominator is A
itself.
If SI, ^2 ••• ^« ^^''tl A all vanish, then x^, Xo ... x^ are in
the ratios of the minors of any row of the determinant A.
583
For example,
in
the ratios
The eliminant
584
C^: Cj
:
C^
...
:
of the given equations
is
:
€„.
now
A=
0.
Orthoi^onal Transformation.
If the
two
sets of variables in the
n equations (5y2) be
connected by the relation
.»•,
+4+
...
+.<•;=
?;
+ f: +
...
+
f;,
(D.
then the changing from one set of variables to the other, by
substituting the values of the Ts in terms of the xs, in any
function of the former, or rice versa, is called orthogonal
transformation.
When
I.
equation (1)
The determinant
is satisfied,
A = ± 1.
X
two
results follow.
THEORY OF EQUATIONS.
154
II.
A
Each of
the constituents of
else to minus that
equal
is
to
the corre-
minor according as
sponding minorj or
A
is
or
—
Proof. Substitute the values of 4i, l^ ... 4„ in terras of x^, x^ ... a;„ in
equation (1), and equate coefficients of the squares and products of the new
We get the n^ equations
variables.
a\^-V
+
fliCj
h^h^
+ =
+ =
1
a.,ai
a\
+
+
hjb^
+ =
hl
+ =
1
a-
a-,an
+
hl
+
=
1
hA),, -f
Form the square of the determinant A by the
rule (570), and these equations show that the
product is a determinant in which the only constituents that do not vanish constitute a diagonal
A
Also
+
of
'
Therefore
ones.'
A'^
=
aiA
A = ±l.
= ^i +^,04-/ls0+ = .4i
a^\
= AS) + AS}-\-A^ +=^^3
and
1
Again, solving the first set of equations
for ai (writing a, as a-^^a^, &c.), the second
set for ao, the third for a.^, and so on, we
have, by (582), the results annexed which,
proves the second proposition.
;
—
&c.
&c.
—
Theorem. The 7i 2*'' power of a determinant of the
order multiphed by any constituent is equal to the corresponding minor of the reciprocal determinant.
585
n^^
—
Let p be the reciprocal determinant of A, and /3,. the minor of i?^
"Write the transformed equations (582) for the xs in terms of the 1%
and solve them for l^. Then equate the coefficient of x^ in the result with its
coefficient in the original value of .^0.
Proof.
in p.
Thusp^3= A
586
To
(/3ia;i+...+ /3,.a;,.
A/3,
.-.
=
p&,
=
+ ...),
and
i,
A"-'&, by (575);
=
.-.
+
6.r"'-'
+
ca?'^-'
+
...
aV+/>V-^+cV--'H-...
If it is desired that the equation
and y
/3,
...
=
+Mr+
•••
;
A'-'^fc,.
eliminate x from the two equations
«.r'"
in X
&>i+
;
put
—
instead of
following methods will
still
«,
and
=
=0
(1),
(2).
should be homogeneous
clear of fractions.
be applicable.
The
155
ELIMINATION.
Bezant's Method.
1.
— Suppose m >
n.
—
Rule. Bring the (ujuatiom to the same degree by multiThen multiply (1) by a\ and (2) by a, and
plying (2) by «"•-".
subtract.
Again, multiply
by a\r-\-b\
(1)
and
by
(2)
{'i,r-\-b),
and
subtract.
Again, multiply (1) by ax--\-b'x-\-c\ and (2) by {a£'-\-bx-\-c),
and subtract, and so on until n equations have been obtained.
Each null be of the degree m—l.
Write under these the
plying (2) successively by x.
equations obtained by multim equations
The eliminant of the
required.
is the result
Ex.— Let
m—n
the equations be
(
(
a
(
a'x^
a;'
a;"'
+ 6h + c + d ax' + ex +f = 0,
=0.
+ h'x' + cx +d'
re"
ai'
a;'
fZ
-
five equations obtained by the method, and their eliminant, by (583),
writing capital letters for the functions of a, b, c, d, e, f,
The
are,
A,x*
+ B^a^+G^x^ + D,x + E, =
A^* + B.^+C^' + D^ + Ej
A^x' + B,x' + (7,a:^ + I)^^E,
a
x'
a x^^-b' x^-\-c x^-d'
Should the cqaations be of
metrical determinant.
II.
Rule.
and
=0
+ h' x^-\-c' x' + d' X
587
(2)
A.,
Bn C, D, E,
A,
Z?3
G,
A
a'
b'
c
d'
=
m— l
times
— To eliminate x from
The
m+n
I'X-
->t
qx
'px^-\-qx--'rrx
a.i "
+ 6x' + c.r + =
px + qx + r =0
\-
r
tZ
+ ra?
+ 6«' + car+c/.c
=0
I
I
arc?
=
=
=
aa?-\-b3?-\-cx + d=
oa;*
be a sym-
n — l times;
and eliminate x from the m-\-n
(1) successively by x,
;
equations and their eliminant
px^^-qx'^
will
Silvester's Diahjtic Method.
resulting equations.
Ex.
E,
d'
b'
the same degree, the eliminant
—Multiply equation
and equation
A^ B, C, D, E,
'
=
<»
i>
and
p
q r
q
r
q r
a
a
2^
b c
b c
d
d
THEORY OF EQUATIONS.
156
588
Method of elimination by Symmetrical Functions.
III
Divide the two equations in (586) respectively by the
terms, thus reducing them to the
coefficients of their first
/Gr) =.r- + /)i^^-^+
forms
<l>
Rule.
(cv)
—Let
=
a, b, c
the equation <p{a)
(p
+
.v''
...
(b)
<{>
</i cr'*"^
+
...
.
. .
-\-p^=
0,
+ 7n = 0.
Form
represent the roots of f{x).
This will contain sym0.
(c)
.
.
.
=
metrical functions only of the roots a,b,c....
and
Express these functions in terms of
the equation becomes the eliminant.
pi, p2
"-by
(538), ^'c,
—
Reason of the rule. The eliminant is the condition for a common root of
(a), ^ (6) ...
the two equations. That root must make one of the factors
vanish, and therefore it makes their product vanish.
589
of
The ehminant expressed in terms
and the roots o, )3, y ... of ^ {x),
/(,7'),
(,,_«)
(a-^) (a-y)
...
of the roots a, b,c
will
(/.-«) (6-/3)
...
be
{h-y)
...
&c.,
being the product of all possible differences between a root of
one equation and a root of another.
is a homogeneous function of the coequation, being of the n^^ degree in the
coefficients of f{x), and of the m^^^ degree in the coefficients
590
The eliminant
efficients
of
.^
either
of
(a').
591 The sum of the suffixes of p and q in each term of the
ran.
Also, if p, q contain z ; if p.y, q., contain z'
eliminant
if 2?s, qs contain :^, and so on, the eliminant will contain z"'".
=
;
Proved by the
the roots
592
may
a, b, c
fact that p^ is a
...,
j^z •••i ^Iv Q2 •••
-
Elimination by the Method of Highest
Let two algebraical equations
^=
of r dimensions of
If the two equations involve x and y, the elimination
be conducted with respect to x ; and y will be contained
in the coefficients p^,
593
homogeneous function
hy (40G).
and
i?
= 0.
in x
Common
Faetor.
and y be represented by
:
:
ELIMINATION.
It
required to eliminate
is
J
157
x.
according to descending powers of .i\
Arrange
and, having rejected any factor which is a function of // only,
proceed to find the Highest Common Factor of J and B.
and
Ji
The process may be
c^A =
+'"i^''i'l
c B = (hR +ro7?.>
_ T^
<^3^i — (l^^h -r r^li^
c^Bo = QiRs +
(jxli
^
"
-({ \
,
ri
j
exhibited as follows
Ci^ *'i a^6 the multipliers re^'2J
''i»
quired at each stage in order to avoid
and these must
fractional quotients
^^ constants or functions of y only.
q^^ q^^ q^^ q^ are the successive quo;
tients.
r^^Bi, r.jB^, rji^,
r.^
are the successive remainders
;
i\, rg, rg, i\
being functions of y only.
as soon as a remainder is obtained
a function of y only ; i\ is here supposed to be such
The process terminates
which
is
a remainder.
Now,
we
the simi)lest factors having been taken for
Ci,
c-,,
c^
see that
The values
which
of x
and
v/,
simultaequations
the
neously
A {) and B Q, are those
obtained by the four pairs
of simultaneous equations
satisfy
=
=
following
The
final
which gives
values,
equation in y,
all admissible
is
1^
I
tlio
th(>
f
it should happen that
remainder i\ is zero,
equasimultaneous
THEORY OF EQUATIONS.
158
594
To
find infinite values of x or y whicli satisfy the given
equations.
Put X
=
~.
z
Clear of fractions, and
make
2
= 0.
the two resulting equations in y have any common
oo, satisfy simultaneously
the equations proposed.
If
roots, such roots, together with
Similarly
we may put
y
=
.13
—.
=
.
PLANE I^IMTJOXOMETRY.
ANGULAR MKASrHK.MKXT
600
'I'Jx'
the radius.
is a liadiari, and is tlic
which subtends an arc equal to
Circular mcasui'o
of
ii'iit
aiiulr at (lie centre of a circle
Ileiiee
measure
=
—
601
Cireulai-
602
Circular measure of two right angles
of an anole
^^
radius
'I'he unit of Centesimal measure
oni'-liundredth part of a right angle.
603
is a
604
is
'i'he unit of Sexagesimal measure
the one-ninetieth part of a right angle.
To change degrees
vice vprni'i,
employ
^"^
=
H'T II
.
=7r.
is
the
is
into grades, or circular measure, or
the ihrvQ equations included in
OTie of
90~100""7r'
Let
OA
revolving line
round 0.
be fixed, and
OP
Draw
dicular to AA'.
tions of OP,
let the
describe a circle
FN always perpenThen,
in all
])osi-
PN =
the sine of the angle AOI',
ON z=
the cosine of the angle
PN =
the tangent of the angle JO/'.
-
.
a Degree, and
THIGONOMETRICAL RATIOS
-
.
Grade, and
where D, G, and G are respectively the nundjers
grades, and radians in the angle considered.
606
50
AOP,
of degrees,
TL'IG GNOME TBY.
PLANE
160
607
608
If
P
If
P be
It"
If
y
P
AOP
is
positive.
the line AA', sin ylOP
is
negative.
be above the line
helo2o
AA\
sin
AOP
lies to
the right of
J5j5',
cos
lies to
the
BB\
cos ylOP
of
/t//
is positive.
is
negative.
Note, that by the angle AOP is meant the angle through whicli
has revolved from OA,
initial position
and this angle of revolution
may have any magnitude. If the revolution takes place in the oppos'te
direction, the angle described is reckoned negative.
609
OP
it-,.-;
610
The secant
of
or
;
an angle
cos
611
The cosecant
of
A
is
sec
the reciprocal of
=
A
an angle
612
The cotangent of an angle
tan
or
A
cot
is
1.
the reciprocal of
is
=
siu^ cosec^
or
=
its
sine,
1.
the reciprocal of
^
its cosine,
its
tangent,
1.
Relations hctirrcn the trii>onomet7'ical functions of the
same un^le.
613
siuM+cosM
614
sec'^
615
cosecM
616
tan
If
A
^
=
[1.47
l.
l+tmiM.
=
l+cotM.
= ^i^.
[606
.
[606
cos^
"
tail^
rrz
h
617
sin
A
=
s/(r-\-/r
cosy4
/?io
618
•
sin.1I
=
=
tally!
.
,
vlH-iair/l
('()Syl=:
1
-.
x/1+taii-i
^,_
[617
,
.
TIUOOSOMETliICA L HATIOS.
619
620
621
622
The Coniploment
Tlie
Supplement of A
is
=
=
^0^-A.
180'-yl.
= (OSy|,
taii(lM) -.l) = eot.l,
sec (1)0 —^) = c'osec A.
sill
(18(r~^)=
cos
(180°-^)
siiiyl,
= -cos ^,
= -tan .4.
the figure
Z QOX. =
1 80'
y/,
[607, 608
-•/.
cos(
The
sine,
—
= ~sm^.
— ^) = cos^.
sin(— ^)
623
624
—
.1 is
sin(00 -y1)
tan (180 -.4)
Ill
of
IGl
By Fig., and (607), (608).
and cotangent of 180° — ^, and of
secant, cosecant,
will follow the
same
and tangent.
rule as their reciprocals, the cosine,
[610-612
To reduce any ratio of an angle greater than 90° to the
ratio of an angle less than 90°.
625
—
Rule. Determine the sign of the ratio by the rules (007),
and then substitute for the given angle the acute angle formed
by its two bounding lines, produced if necessary.
Ex.— To
find all the ratios of 600°.
Measuring 300° {= 660°— 360°) round
the circle from ./ to P, we find the
acute angle AOP to be 60°, and F lies
helow AA\ and to the rUjki of BB'
Therefore
sin 660°
=r- sin
cos 660°
=
60'
2
cos 60°
=
and from the sine and cosine
found by (610-616).
Inverse Notation.
by
sin"'
— The
'
J
all
the remaining ratios
angle whose sine
,/•.
Y
is
mav
l)e
x isdenoted
PLANE TRIGONOMETRY.
162
All tlie angles which have a given sine, cosine, or tangent, are given by the formula
626
sin-^^= nir+^-iye
cos-Vr =:
2mT±e
U\\\\r=
mr-\-6
(1),
(2),
(3).
In these formula} 6 is any angle wliich has x for
tangent respectively, and n is any integer.
its
sine,
cosine, or
Cosec"'.T, sec"' a;, cot"' a; have similar genei'al values, by (610-612).
These formulae are verified by taking A, in Fig. 622, for 0, and making
n an odd or even integer successively.
FORMULAE INVOLVING TWO ANGLES, AND
MULTIPLE ANGLES.
= sin A cos 5 + cos ^
— sin A cos^— cos^
voii(A-\-B) = QosA cos ^— sin A
coh{A — I}) = cos ^ cos J5 + sin A
627
628
629
630
Proof.
sin
{A-\-B)
sill
(A—B)
— By
(700) and (701),
sin B,
sin
B,
sin i^,
sin
B.
we have
= sin A cos B + cos A siu B,
sin G = sin (A+B), by (622).
sin
and
C
{A — B) change the sign of B in (627), and employ (623),
cos(^ + B) = sin {(00"-^)-^j, by (621).
Expand by (628), and use (621), (628). (624). For cos (-1-5) change the
To
obtain
.sin
(624),
sign of
B
in (629).
631
tan (.4+/?)^ ^^'"^+^^'"-;^
632
tan (/!-/>')
633
cot(.l
634:
1-1
=
+ /i) =
.
MI'I/ni'LE AXOLES.
635
163
,
PLANE TRIGONOMETRY.
164
654
^
/A-o
X
655
^
A\
— tan A
1— tau^
1
[631, 632
= 3sm A—4< sinM,
cos 3^ = 4 cosM — 3 cos ^
3tan J— tanM
tan 3^ =
l-3tanM
siu
656
657
658
3A
By
putting
B = 24
in (627), (629),
and (G31).
= sin" 4 — sin" B
= co^-B — cosM.
cos (^ + IJ) cos (^ - i^) = cos^ ^ - siu^ B
= cos"-B — sin" A.
659
(A + B)
sin
660
sin
{A — B)
From
sin
661
= ± v/l +
Y+
— — cos — = ± \/l —
c<^® -9"^
662
sin
sin
664
^osA^-^
--
^.
[Proved by squaring.
sinyl.
4 = 4 {\/l + sin^->/l-sin^}.
663
when
sin
(627), &c.
{
v/1
+ sin ^ + yi - siu ^}
between —45° and +45°.
lies
In the accompanying diagram the
signs exhibited in each quadrant are the
signs to be prefixed to the two surds in
665
the value of sin^y according to the quad-
rant in which
For cos
-
—
lies.
change the second sign.
L
Proof.— By examining
ibe cbangosof sign in (UtH) and (662) by (607).
—
—
—
.
.
MULTIPLE ANGLES.
666
667
668
669
105
+ /i) + sin (.1 2 cos^l sin /i = sin (A + 7^) - sin {A - li).
2 cos
cos /i = cos (.1 + U) + cos (J - B).
lAwA sin y^=z cos(.l-/;)-cos(.l + 7y).
2 sin
A
cos
B=
sin
(A
/?).
.1
[G27-G30
670
sin .1
+ sin B =
l sin
l-/y
li+Z^ cos -i_
671
sni yl
— sni B =
2 cos
——
672
cos ^
+ cos jK =
2 cos
-
01 o
cos B
— cos A =
2
Obtained by cbaugiug
It is
thus
674
676
into
——
.-
,
—-—
sin
—
— cos
-
—— ——
-
and
-
sni
B
~
into
-
,
in (G66-669).
advantageous to commit tbe foregoing fonnula; to memory, in words,
= sin sum + sin difference,
= sin sura — sin difference,
= cos sum + cos difference,
= cos difference — cos sum.
sin first
+ sin second = 2 sin half sum cos half difference,
— sin second = 2 cos half sum sin half difference,
sin first
cos first
+ cos second = 2 cos half sum cos half difference,
cos second — cos first
= 2 sin half sum sin half difference.
2 sin cos
2 cos sin
2 cos cos
2 sin sin
sin
+ 6+ C)
= sin A cos B
cos
C + sin B
C
cos
A
(J
-\-
675
A
sm
;^
sin
cos
C
cos
A
B — sin A
sin
B
cos
sin C.
+ 7J+C)
= cos A cos B cos C — cos A sin B sin C
— cos B sin C sin A — cos C sin A sin B.
Um{A + n-\-C)
_ tan A + tan B -f tan C — tan A tan B tan C
1 — tan 7^ tan C — tan C tan A — tan ^ tan li'
cos(.l
Prouf.— Put
B+C
fur L' in (G27), (02l>),
and (Gol).
PLANE TRIGONOMETRY.
166
If
677
A +B+C
180°,
A
+
sill
B + sill C =
sm A
+
sill
B — sill C
=4
siii
— sin — cos—.
fos^
+ cos7iH-cosC = 4
sill
— sin —
sill
678
=
4 cos— cos— cos—.
ABC-+
£t
680
681
682
1.
^
+ COS 1^ — COS C = 4 cos— cos— — —
tan A + tan B + tan C = tan A tan ^ tan C\
cot 4 + cot
I + cot ^ = cot 4 cot ^ cot ^.
sin 2A + sin 2i? + sin 2C = 4 sin ^ sin B sin C.
cos2^H-cos2ii + cos2C = — 4cos^ cos^ cosC— 1.
cosyl
679
siii
Jd
siii
1.
General formulae, including the foregoing, obtained by
applying (666-673).
and n be
A + Bi-G =
nC
nB
nA
sm -^
683 4 sm -rp sm -;j^
^
If
-rto
TT,
4
.
'
any integer,
.
Ji
/»o>i
684
4^ COS
nB
nC
— COS—
cos-^
^^A
cos (!^-n^)+cos(^-.ii)
liA-\-B + C
635
= 0,
^=^ ^=
4 sin '-^ sin '-^ sin
2i
686
4 cos ^^4 cos
2t
+ cos(^-..c) + cos^|r.
sin
uA - sin
yi
j5
- sin nC.
2i
2i
cos
2i
Rule.— 7/; informulce
cos;iy/
+ cos?iZ)'+cosyiO+l.
2t
(683)
^o
(686), two factors on the left
be cliaiujed hy ivriting sin for cos, or cos for sin, then, on the
right side, change the signs of those ter)ns n-hirJi <1o not run tain
the angirs of the (dtpveit factors.
—
—
COMMON
JxSUlJ-h^.
I<i7
Thus, from (083), wc obtain
687
=
sill
J
-
cos
-— COS ^
z
^
_sm(^'^'^-„.l) + sm(^"2"-„y;) + sin('^''-«c) + siM".
A
Formula for the construction of Tables
of sines, co-
sines, &c.
688
sin
where a
689
=
(?i
10",
uu = sin «rt— sin (ii—l)u —A- sin 7ia,
= 2 (1 -cos a) = -0000000023504.
+ l)a — sin
and A
Formulae for verifying the tables
sin,l
+ sin(72° + ^)-8in(72"-yl) = sin (3G^ + ^)-sin
+ yl) +COS (72^ + ^1) = cos (36^ + J) +cos
sin (60° + ^) -sin (60°-^) = sin^.
cos yl+cos (72°
(36"-^!),
(36^-^1),
RATIOS OF CERTAIN ANGLES.
690
sin 45°
= cos 45° = -i^,
691
sin 60^
=
^^,
cos 60'
tan 45°
=
Li
692
-15' =
sml8^
^,
-1-^^
694
r^-
table
By
taking- tlie
^'
5
r
sin54=^^^~-,
i>-ives tlie
=
= \/^-^|^.
\/5-|-l
tan54^
695
= ^'6.
cosl8"=^'^ + 2'^^
^
•
tan 60°
= 2-y3^
= 2 + v/'>3'
= ^^,
tan]8^
«/Nj
,
'-t
tanl5°
cotI5'
693
I
= 1.
4'?
cos 54
\/5
= vS—
2^2
= V'^^^-
complements
ratios ol
'
^Jn",
of these ani^Hcs, llie
75', 72", and :'<>
.
same
PLANE TRIGONOMETRY.
168
696
Proofs.— sin 15°
697
sin 18''
698
sin 54^ from sin
is
obtained from sin (45°-30°), expanded by (628).
from the equation sin 2x
3.7;
=
3 sin
—
cos
-4 sin^r,
a;
3.i-,
= 18°.
= 18°.
where x
where x
the ratios of various angles may be obtained by taking the sum,
difference, or some multiple of the angles in the table, and making use of
699
And
known
formulae.
Thus
12^
= 30°-18°,
7^°
= ^,
&c., &c.
PROPERTIES OF THE TRIANGLE.
700
701
702
c
= aQOsB-{-bcosA.
a
sin
«'
c
h
A
sin
B
= 6'+c'— 26c cos A.
Proof.— By Euc.
11.
704
12 and 13,
cos
703
If g
C
sin
= ^H-6 +
g
^
^^^
a?
= b' + c'-2c.AD.
A
=
^
denote the area
26c
ABC,
.u4=^^I^^i-^, ^o4 = ^^^.
[641,042, 703.0, 10,
A_J
7-V
A
tan
705
706
sin
707
A=
708
A
=^V
1.
— L\ (M — r
{s-b){s-c)
I{k'
.V
be
(.y— a)
.v(.y_„)
(s—b)
{s
•
— c).
^ sin A = Vs (s-a) (s-b) (.v-c),
=
1
[635, 704
[707, 706
\^'2b-i'''-^2(-a--^'2irb--(i'-b'-r\
.
rifOl'EUTlh'S
nr riilASULKS.
Thr Trianii/r tun I
Let
r
= radius of inscribed circle,
r^=
radius of escril)ed circle
touching
tlie
side a,
B = radius of circuinscribiu'
circle
709
l
b roiii h
i«^.,
A
=
.
710
^
-1-
n
^
.
+
-^
c
r =
cos
[By
711
712
a
= r cot It + r cot
-
>
-
.
('irrlt
iC'.t
A
PLANE TRIGONOMETRY.
170
SOLUTION OF TRIANGLES.
Right-angled triangles are solved
by the formula?
e^=rt2+6-^;
718
la^e siu A
719
h,
=
\a
z=:
,
ecos^,
A
h tau
,
&c.
Scalene Triangles,
Case
720
I.
— The equation
a
sin
h
A
siu
[701
B
determine any one of the four
quantities A, B, a, h Avhen the remaining three are known.
will
721
When,
The Ambiguous Case.
I., two
and an acute angle
opposite to one of them
are given, we have, from
in
Case
sides
the figure,
.
sin
^=
C
e sin
A
-C arc
Then C and 180
Also
h
because
722
'C
.
=
=
Wlicn an angle
cvo»
the values of
± v'a- —
c-
C and C, by
siu-
A
(622).
,
A 1) + DC.
1> is
to
be determined from the equation
...
.sill
/;
/,
=:
.
.
sin
.1,
a
and
\o
''
is
u small fracLiou
a
by putting
sin
(U^C)
;
tlic
i'or
fiirular
sin
.1,
measure
of
B may
and using theorem
be appi-oximatcd
(rOC)).
,
.
SOL UTIOX OF TU I A M
723
./
TF.
('\^i'
are known,
— AVluMi two sides
side a
tlie tliird
(r=
when logarithms
h-
is
1
by
priven
tlic
7
1
inclndcMl aiigl(!
formula
+ (--'2hrvosA,
[702
are not used.
Otlierwise, eni])loy
tlie
followin<r formula with lo-^^ai-ithms
i^-c
/j_f;
724
L ES.
and the
r
/>,
i
A
,
^^»-^V-^-,TX7/*'^^>
Obtained from
('~''
= ^!°^^---^'A^, (701).
sm/Z + smG
h-{-r
and then applying
(670) and (07 1).
the above eiiuation, and
havinir l)oen found from
2
—"trl
i?
being equal to 90'—-''-,
and G having been determined, a can be found by Case
726
If the logarithms of b
log{b—c) and
$
we have
= tan"'—,
log(/>
+ c) may
I.
and c are known, the trouble of taking out
be avoided by employing the subsidiary angle
and the formula
c
727
Or
tan
else the subsidiary angle
tan i
728
If a
=
X(B-C)
-
tan («
= cos"'
(B- G) =
'''
,
^
)
cot
[C55
^
and the formula
tan' ^ cot
^J
be ri(|uirod without ealcuhit iug the aiiglis
/>'
[04:3
mid
/
',
we may
use the
formula
(^<
729
**
~
+ (jMii
cos ^
[From
,1
the figure in
drawing
^
^^''^™
{u-c)
^
a
9(i0,
by
|)eijteiidiculnr
'^ ^'^' pi'^'^i^^'ed-
If a be required in terms of ?», c, and A alone, and in a form adapted
to logarithmic eomputatiun, employ the subsidiary aiiglo
730
=
and the
f..i-muhi
sin"'
a
=
(
(/-
,'
"
.COS-
+ .) cus«.
'
),
[702, G37
PLANE TRIGONOMETRY.
172
Case
III.^
— AVhen
the three sides are known,
tlie
angles
may be found without employing logarithms, from the formula
co^A=!^±StzlL\
731
[703
'2oc
732
are to be used, take the formulae for
If logarithms
sin^, cos^, or tan—-; (704) and
(705).
QUADRILATERAL INSCRIBED IN A CIRCLE.
,o.li
733
= "'+'>'-/
-f
2{ab-\-cd)
= a» + b''-2ab cos B =
(702}, and i^ + D = 180°.
From AC2cdcosB, by
^\nB
734
735
and
c-
^
.
+ (P +
= -^S_
ab-\-cd
Q
=
=
x/{s-a)(s^b){s-c){s-d)
area of
A BCD,
=l{a-{-h + c-]-d).
Area = lah sin B + ^cd sin B substitute sin B
(ae+hd)i„,l+he)
s
;
736
"
AC^=
'
from
_
last.
^,,,_ ,33
{ab-\-cd)
lladius of circumscribed circle
= -L ^\ab-\-cd) {ac-^bd) {ad-\-bc).
737
4^)
If
AD
bisect the side of the triangle
tan 51)^
733
739
740
743
4A
=
b'-c''
= 2cot^ + cotJ?.
+ c' + 2bc cos A) = hAb' + c'-ia').
cot7?^I>
AD'
=
i
(b'
If ylD bisect the angle
742
ABC in D,
tan
of a triangle
yl
/;;>.
1
ABC,
+ c,__
^^^- = ^^
tan
= cot B^C^b
2
b-
JD=^cos^
A
[713, 734, 736
.
isriisn>iAuy angles.
If
AD
III-
]H>i{H'n.licular to
hi'
AD
744
BC
sin
A
//''
745
li'
—
G + c'
sin
r'^
sin
Radius of circumscribing
Radius of inscribed circle
J}-tnnC
tan
7) -h
-
cosec
of
=
748
TT
n
2
Area
CIRCLE.
= —a cot —
,
r
(/
r.
of sides
=^
n
tan
= R.
=
= a.
= n.
circle
Side of polyo:on
746
7?
tan
REGULAR rOLYGUN AND
Number
173
Polygon
\na- cot
— =
ItilV
sill
^^
—
nr- tan
—
USE OF SUBSIDIARY ANGLES.
To adapt a±_h
749
=
Take
750
i^'or
a-h
to logarithmic computation.
tan"'
take ^
/
then
;
h
= tan"
a
—h
a^h =
til
a sec'
6.
us
av/2 cos (0 + 45°)
COS0
To adapt
751
Take
=
a cos
tan"' ^
;
C±h sin C
to logarithmic computation.
then
b
a cos
C±6
sin
= v/(a^ + i-) sin (9 ± C).
For similar instauces of the use
To solve a
752
of
[By 617
a subsidiary angle, see (72G) to (730).
quadratic equation by employing a subsidiary
angle.
If
x-—22)X + i2
=
l)e
the equation,
[
Hy lo
.
PLANE TRIGONOMETRY.
174
Case
I.
—
< ^r,
If a be
put
^=
sin''
B
then
;
P
=
x
Case
II.
— If q be
>»", put
X
Case
III.
—
If q
2pcos'^. and
=p
'^.,
= sec-0;
(Izki tan
be negative, put
x
= Vq cot
2^5
0),
siV
[639,640
f-.
then
[614
imaginary roots.
^=
tan*
and
— y^ tan —
2
then
;
[644, 645
2
LIMITS OF RATIOS.
-g-=-r =
753
when
'
9 vanislies.
For ultimately
=
^=4i
AP AP
e
.
n sin—
755
(co^~)
— Put
[601,606 q
= ^ when n
754
Proof.
l-
(
—
1
when
l-siu"—
]
^,
is infinite,
n
gy
putting -- for
in last.
is infinite..
and expand the logarithm by (156).
DE MOIVRE'S THEOREM.
756
a+/
(t'os
where
i
sin a) cos /8+/ sinyS)
= cos (a+)8+7+
= V — 1.
...)
Proof.
— By
Induction, or
«
=
cos
by putting
Expansion of cosnO,
^iu
&c.
...
(a+^+7+
•••).
[Proved by Induction.
(cos 6-\-i sin ^)"
757
+
tj-c,
a,
n6+}
/5,
sin
&c. each
>i^.
=
in iwwcrs sinO
758
c'osM^
=
cos"^-C^(/J, 2) cos"-'^^ sin-^
759
sin
n0
=
n cos""' ^ sin 6—C{n,
^ in (756).
and cosB.
4-C(»,4)cos"-^^sin^^-ctc.
I'liOOF.
parts.
— Expand
Ji)
cos"'^ sin*^+&c.
(757) by Bin. Th., and o<iuatc real and imaginary
TliiaOXOM ETHICAL
760
170
ShJlilES.
>Hang-(-(»,:i)lM.r-g+Ac.
Unu,e=
In series (758, 7.59), stop ut, aiul cwiliulc,
n.
Note, n is here an integer.
all
fmns
willi indices grciiter
than
Let
c^c.
s^
= sum
of the G{n, r) products of tana, tau/3, tany,
to n terms.
+ ^+y+c^'C.) = cosa cosyS
762 i'Os(a-{-fi+y-\-Scc.) = cosa cos/S
real and imaginary
Pkooi'. — By
761
sill
(a
...
(.v,-.v,
...
(1
+ .v,-.^c.).
-6',+.s-.U.).
jiarts in (7o6j.
e(|u:itinf;
Exjmnsions of the sine and cosine in powers of the angle
764
(.OS^=l-^ + -[^+el'C.
sill^=^--^+|:^-&C.
Proof.
— Put
(9
-
for 6 in (757) and n
= eos^+f siii^.
766
e'<'
768
c''-\-r-'"
=
2 cos
of
^.
av^(Z
ro.s-"
l
sin,''
+ »'tau(9 _
6
midtiplcs of
2"
772
'
+ C'(/',2)
773
^\'lleu
//
cos"^
is
774
And
and (755).
By
(150)
,^
in cosines or sines of
('(//,;{)
cos
(/<
l9
-(I) ^
+ .Vc.
even,
2) cos
wlicn
(7'>ir)
0.
cos n9-\-n cos(//-2;
cos (;/-!) ^4-
2"-i (-1)*"
+ (;(//,
=
,
= cos 6—i sin 0.
t'''-e-" = 2/ sin ^.
^
Expansion
x employing
e''
itaii^-^—
770
=
7/
.sill"
^ =: cos
n0-n
(n—l) e-C{n,
is
;{)
cos (/<-2) ^
COS {u-(\)e-\-kQ.,
odd,
2" '(_1)"2 si,i«^
=
siu«^— «
^^(///i) sin(//- l-)^~r'
.sin
(/I— 2)^
(;/,;{)
sin (//-(I) ^
+
,<;.'c.
—
!
.
PLANE TRIGONOMETRY.
17(3
Observe that in these series the coefficients are those of the Binomial
Theorem, with this exception If nbe even, the last term mud he divided by 2.
The series are obtained by expanding (e" ± e'")" by the Binomial Theorem,
collecting the equidistant terms in pairs, and employing (768) and (769).
:
Exjpansion of cosnO and sinnO in ijowers of sin
AYheii n
775
cos
= — —n'— sm- 6
n
1
116
o /J
•
1
When n
is
!
^>^\''
When
n
sin
is
nO
!
even,
r
=
n cos
sm^ 0—
Ll
siu 6
)
^
'-^
—
/
When
^mnv
=
Proof.
n
— By
/^ siu 6-\'
^-^
^
sin^ 0-\- &c.
[
)
!
odd,
—
- "("-!) oy
n
sin^—
^
^
siir6/
5
in
+—
^
-i
o
(>r-y)
g
,i„,
siii^
I
^ ^tc.
we may assume, when n is an even integer,
l+A^sin-6 + yl^siu*e+...+^^sin""^+..,.
(758),
cos«^
Put
is
2''
j^2
——
!
778
4
odd.
b
i
.
^
«^-l i„.fl^(»^-l)(«-3^)=:.»,
"' sia'e
= C08g n -!t^s\n'0
+ y"
cosng
777
— 2") sm* a
n- (n^-—
,
-{
b
776
9.
even,
is
—
—
and in cos nd cos ?j.c sin nO sin n.v substitute for cos nx and
their values in powers of iix from (764). Each tei-m on the right is of
the type ^^.^ (sin 6 cos a; cos ^ sin.t)"''. Make similar substitutions for cos.«
and sina; in powers of x. Collect the two coeHicients of .r'^ in each term by
the multinomial theorem (137) and equate tlium all to the coeUicient of .t"
on the left. In this equation write cos"^ for 1 sin" 6^ everywhere, sind then
equate the coefficients of sin'-''^ to obtain the relation between the successive
quantities A.,^ and A.^^^^ foi" the series (775).
To obtain the series (777) equate the coefficients of instead of those of .c''.
When n is an odd integer, begin by assuming, by (7o9),
d-\-x for 6,
sinn.T;
+
—
.i'
sin
»/y
=
/I,
sin ^
+ yl,
sin^G +
itc.
177
TniriOXOMhJIlilCAL SERIES.
779 The expansions of cos nO and sin nB in powers of cos
are obtained by chanr^ing 6 into ivr— in (775) to (778).
780
Expansion of cos vO
=
2 COS nO
(2 cos
ey- n
drsrrvdlng poirrrs of cos
in
(2 cos ^)"-
^
B.
+ !!i^^ (2 cos $)"
,..(.-. -l)0^-r-2)...(.-2r+l)^.,_p,„
"
'-
..._^
r\
up
to the last positive
power
Pkoof. — By expanding each
of 2 cos
=
log(l-z.r)+log(l-^)
by (156), equating
783
_
~
log{l-.-(.r
+
l-z)}
and substituting from (768).
coefficients of -",
sin a-\-c sin {a-\-^)-^(r sin (a+2)3)
+ &c.
to
n terms
sina-rsin(a-ff)-c "sin(a+/<y8)4-^"^'si» {a-\-n-l^}
^
l-2ccos;8+6-'
be
If c
<
1
and n
_
'°*
785
=
B.
term of the identity
infinite, this
sill
g-c
l-2ccos)8+c^
cos
787
when
Similarly
c is
<
*
+ &c.
to n terms
sin into cos in the
numerator.
a+c cos (a+^) + c-
a similar result, changing
786
becomes
sin (g—ffl
cos (a4-2i8)
1
and n
infinite.
of summation. — Substitute for
Method
tlie
sinrs
or
Sum the two resulting
cosinrs their exponential valiirs (768).
geometrical series, and substitute the sines or cosines again for
the exponential values
788
(760).
csin(a + iS) + ^sin(a-h2/8)+^sin(a4-.'^/3)
=
infinity
789
bij
('
<•<)>
(
a
+
/3)
infinity
Obtained by
tlie
—
c'"'""^
sin
(a+c
sin
/3)
c'
^'"'^
cos
(a-j- r
2 A
sin ^)
to
— sin a.
+ ^ cos (a + 2y8) + ^ cos (a +
rnle in (7J?7).
+ &c.
— cos a.
;?)Sl
+ cVc. to
PLANE THIGOXOMETRY.
178
790
If, in the series (783) to ( 789), /3 be changed into
the alternate teims will thereby be changed.
Expansion of
791
e
The
powers of tan
6 in
(Gregon/s
B
>
n,
the signs of
series).
1.
— By expanding the logarithm of the valne of
e"'*
lor the calculation of the value of
Formulje
f
= tan e-^j}^+ ^-^ -&c.
series converges if tan 6 be not
Proof.
/>
in (771)
by
(158).
w by Gregory*s
series.
^ = tau-^i + tau-^i - 4tan-^i-taii-^;i^
792
[791
= 4taii^l-taii-^-iT
't^/U +
99
o
Proof. — By employing the formula for tan (A±B), (631).
794
^''^^^
To prove
795
that
Convert the value of tan
a continued
may be
ir is
6 in
incommensurable.
terms of
from (764) and (765) into
B
-^
fraction, thas tanO = —
1— 3— -z5— 7— (EC.
-77
.
_ «! «:
tany ~ 3— 5— 7—
Put
for 6,
and ussumc that
tt,
=
,
VI
may
Hence
p":
and therefore -
(tc.
-,
commensurable.
is
Let
4
2
4
or this result
obtained by putting id for y in (294), and by (770).
6__
""
;
and n being integers.
J
we
lien
shall
have
i
7i
= .— — — —-^
dn 5)i
tn— &c.
-
,
But unity cannot be
The continued fraction is incommensurable, by (177).
equal to an incommen.^niable quantity. Thcrelore t is not commensurable.
796
^^
sin
797
^ f
^''^" ^'
j;
,1-7/.
=
+
Proof. — %
•where vx
= nsin
—
"
(a;
*'^"
+ a),
2/'
^
iU
=
7)
sin
— — '" sin
.'/
0+
-
+
-|j
2//
sin
2a + -^ sin3a + tfec.
sin 4?/— ^- sin 6^
+ ic,
.
u
1
suhs'itnt'ing the ea-pmential values of the sine or tangent
and (770), and
798
('oeniciont of
in
a;"
where a = r cos 6 and
For proof, substitute
Z<
o
= r cos^,
h
(769)
then eliminating x.
=
=
the cxi.ansion of
c"-^
cos
6.1;
=
^
cos n^,
rsinO.
for cos
/^.r
from (768); expand by (150); put
and employ {1'>1).
T h'n\6 in the coellicicnt of x",
.
rindoxoMirnnt -al
W lion
799
'^^
where
e is
<
1
,^^-^^ =
I— e COB 6
,
1
+ '2h
,s7;a7/';.s.
+
cos
1
cos 20
'2'r
+ 2//
71)
cos :iO+...
l>
l+v/l_e*
For proof, put
feries of
c
=
"
ami 2 cos ^
.,
= +
.r
-
-,
expiind the fraction in two
powers of x by the mcbhoJ of (257), and substitute from (768).
+ sin(a+-i)8) + + siii;a+(;/-l)/3j
sin(a+^^)si.4^
800
siiia+siiMa4-/8)
...
-^
sin
801
(•()sa
+ ('()s(a +
+ ('()s(a + 2^) + ...+C'()s;a + (/J-l,)^;
/8)
terms in tliese series liuve tlie signs
alternately, change \i into /3-f-7r in tlie resnlts.
802
If the
Proof.
— Multiply the series by
= — in
803
If
804
Generally, If
/3
niulti[)lc of n,
the
,
—
apply (669) and {QQQ).
/3
sum
=
^'^,
of tlie
if
and
r^''
r
if
an integer not a
r be
powers
of the sines or C(^sines
he odd; and
if
/•
he even
it is
— Dcnotinp^ the sniu of the scries
805
then
^ and
and
(800) and (801), each series vanishes.
in (800) or (801) is zero
General Theorem.
2 sin
+
c
ccosa + c,eos^«
+ r,x + c,r + ...+
r,..."
by
+ />j + ...+r„cos(a + H/3)
F (.r)
=
;
^ {e''F(c'')+e-*'F(>r"')],
and
806
fsina
+ r,sin(«+/5)
Proved bv substituting
(7Gt"'),
Ac.
f
...
+c-„sln(a
for the sines
\-
n, I)
= }-.{,'' F {e'')-c-'' Fie-'')]
and cosines
their exponential values
.
PLANE TRIGONOMETRY.
180
Expansion of
807
=
'f
^"
the sine
—
2ci?" ?/"
and
cosine in factors.
cos n ^
+
y-""
1^-- 2.r//cos^+/] \x'--2.vijQ0^{e^^-Vf
o
to
n factors, adding
—
to the angle successively.
j^
—
Proof. By solving theqaadraticon the left, wegefcic=i/(cos?i^ + isinn0)".
The n values of .r are found by (757) and (626), and thence the factors. For
the factors of a'"±y" see (480).
808
sin
?i<^
= 2^^-^
sill <^
siu
(^<^
+ ^) sin (^^ + -^ j
.
.
as far as n factors of sines.
Pboof.— By putting x =
ij
=\
and
^
=
in the last.
2</>
be even,
809
If
sin
= 2""^ sini^ cos</»(sin'— — sin^<^j
nij>
810
'^
If
(^sin'
——
sin-</)j
&c.
be odd, omit cos and make up n factors, reckoning
for each pair of terms in brackets.
From (808), by collecting equidistant factors in pairs, and
^^
(/>
two factors
—
Proof.
a pplying (659).
811
COS
n<i>
= 2"-' sin U + ^J sin U-Jr*^)...
Proof.— Put
812
cos
Also,
n<l>
813
If
=
'i
if
d)
+
to n factors.
-^ for ^ in (808),
zn
n be odd,
2"-' cos
<^
/siii'^
be even, omit cos
— sin'c^j l*^^^^'^ ~ sin'c^j
...
(^.
PiiOOF.— As in (809).
n
814
Proof.
815
=
2-^ sin^
—Divide (8o9) by sin
(p,
sin
and make ^ vanish
(^i=ll^.
;
then apply (754).
-^ = ''Sl-(-^)lS-(.01S>-(0
PnooF.-Put ^
intinite.
sin^ sin '^...
=
^ in (SOD)
and (812)
;
divide
by (8M-) and make n
,
AJ'hITinXAL FOKMUL.r..
817
181
e'-2cos^ + e'-'
Proved bv
making n
sulistitutiiiu'
intinitc,
+ -^-,
..•=!
1/
= 1-
*^"^^
--f"'
series of factors to 4
aud reducing one
"
De Moirrrs
— Take
^
of the
any point, and
FOB = d any angle,
JWC = COD = &c. =
819
'"
^"^
n
,T
821
Coles's properties.
.r
v«~r'
.r^+r"
822
cot
824
c<)secLM+cot2J
829
&c.
—
If 6
...
&c.
= ——
= PB.PC.PD
= Pa.Ph.Pc
...
...
&c.
&c.
ADDITIONAL FORMUL/IJ.
yl+tnii A = 2 eosot- 2 1 = sec A cosec A.
823
828
= a- - 2.n- cos 5 + r,
= r, IV siiii|^ = PB.PC.PD
H
827
factors.
(807) and (702), since Pi?"
820
826
;
r.
;
= Plf PC- PD- ...ion
By
jr
=
OB =
— 2a "r" cos w^ 4- r-"
OP
J
^
Proprrfi/
7'
by putting
hiu'^
z=0.
Circle.
^ •" (^<J").
''o'"
co=*-^
iaii.l
tan
=
=
cot.4.
sec
1+taii
-
taiirJ-V+^V
+tMiiyi
= tan tail B.
+ cot Ji
cosec/1 = sec. 1-f cosec* ^1.
.1
.
1
.1
cot .l
sec"^
=
— sill*-!-.
cos'—
+ sec.l =
J
.
,,
J
taii^i.
+
PLANE TRIGONOMETRY.
182
830
n ^+1^+6 =
tan B tan
831
If
832
-|,
C+tan C Urn ^4 H-tau A
A-{-B-\-C =
cot i^ cot C+ cot
cot ^4
eos2Z^
833
tani^
In any
838
ABG, G being
ii+»^
=V(^6)-
= i («+«»)•
= 'i^cosiC.
cosi(^-i?)
= ^siniC.
tan^^+tanj^
a'-b'
,.
c
UmiA-UmiB a-b'
^') = ^^ ^os ^ + ^" ^^^ ^ + "^ ^^^ ^•
(«' + ^'
Area of triangle ABC = ^bc sin A
siu^ + /i
c'
'
2
sinS sinC
i
= _2abc_ ^^g i^
840
= i {a+b-^cY tan i^
With the notation
843
the right angle,
sini(^-J^)
839
842
= -J
tan2B=-^.
= g;.
^inA-B _
1
841
+ cot yl cot B = l.
triangle,
835
837
l.
tan-i- + tan-i
T=|-
In a right-angled triangle
QQa
^^^
=
Tr,
«J»"y + ^i»
834
tan B
r
/
^^g iy^
sin
;o^
o
^.^^
tan
1
\B
A
sin iJ
C\
tan ^C.
of (709),
= i{a + b + c) Um^A tan \n tan \C.
A = v/rr,nr,.
=
i^/^»'
,''('''.
.
^/cosl+^y cos/Z + r (M)sr:=
l/»*
sin.l
sin/isinC.
•
ADDITIONAL FORMUL/E.
183
=
=
sum of per844 R-irr
\ {a cot A-\-h cot li-\-p cot C)
pendiculars ou the sides from centre of circumscribing circle.
Tliis
on
R
may
shown by applyinf:^ Enc. VI. D.
and the quadrilateral so formed.
also bo
as diainetcr
845
>*a
n >\ =
=
<*os I
A
cos }yB cos
1
C.
+ ^ (r. r„) + v/(n. n)^_
1=1 + - +—. taii.M==V—
846
r
847
v/(r,
>•
r..)
>*6
>'a
849
fi^^t^
to the circle described
*'6''c
>'c
be the centre of inscribed
If
0^ =
—
circle,
cos
,
A.
I
,
a-\-o-{-c
850
rt (^>
851
'f
853
cos
854
cos
If s
1
^1
C—c QosB) =
cos7i+c cos C
/>
852
cos
.1 -\-b
cos 7i+c
¥—c'.
= c cos {B-C).
cos C = 2a sin 2^ sin C.
+ cos B + cos C
sin /i
'1(1
=l-\r
= \{a + h + c),
—COS" a — COS" h — COS"
c-\-'l
cos a cos h cos c
= 4 sin.v sin {s — a)
855
sinC
+ 6+c
a
— h)
sin
— c).
— 14-('Os-r/ + cos-/>4-<'os"^*+2 cosa cos 6 cose
= 4 cos COS {s—a) cos {s—h) cos (.v—
sin {s
(-v
c).
A-
4 COS — COS — cos —
856
=
COS
.y
4- COS
(.V
.
— + cos — + cos {s — c).
(.v
^/)
.
4 sni
/>)
— sm —h sni —
(I
.
.
(•
= — sin* + sin (*—«) + sin (*— 6)+sin (*--c).
858
..
Proof.
= .(i+^+^ + ...).s(i-,.!^U +
— Equate cocfScients of
(81")) or of
cos^ by
(7t".'>)
0^ in
and (SlG).
the expansion of
'
-
...).
by ("Oi) and
;
184
PLANE TRIGONOMETRY.
859
Examples of the Solutions of Triangles.
Cask II. (724).— Two sides
1
and the included angle 47° 25', to
Ex.
feet,
of a triangle b, c, being 900
find tlie remaining angles.
:
tan ^^=^
2
=
—
"^
b+c
therefore
log tan ^
therefore
i tan^ (J5-0)
(J5
cot
and 700
cot 23° 42' 30"
4=1
8
2
— C) =
log cot
— —log 8
;
= i cot 23° 42' 30" -3 log 2,
10 being added to each side of the equation.
.-.
L cot 23° 42' 30" =
3 log 2
.-.
itan|(B-(7)
=
=
10-3573942*
/
^
.-.
(5- 0) =
j
(
|(5 + C)
and
.-.
Ex. 2: Case III. (732).— Given the sides
=
a, b, c
to fiud the angles.
A
.
_ /4.3 _ /2
-Vl2:5-Vl0'
/( s-b)(s-c)
.
sis-a)
Ltau^ =
1<^+^ (log
9 respectively,
7, 8,
.
*""-2=V
therefore
15° 53' 19-55"*
= 66° 17' 30"
5 = 82° 10' 49-55"
9-4543042
And, by subtraction, G = 50° 24' 10-45".
-9030900
2-1)
=
96505 15;
2
U=
therefore
\B
found
is
in
24°
5'
41-43".*
a similar manner, and
= 180°— J.— ^.
G
—
c
=
6953 and
nearly a right angle,
it
cannot be
Ex. 3. In a rigbt-angled triangle, given the hypotenuse
a side 6 = 3, to fiud the remaining angles.
Here
cos
A = —^
.
But, since
A
is
6953
determined accurately from log cos A.
.
A
ll-cosA
''"^^V"""^
therefore
L
sin
^=
therefore
^=
_
/3475
-V6953'
10 + ^ (log 3475 -log 6953)
^=W
therefore
Therefore take
'o9'
89° 58' 31 04" and
=
9-8493913;
ro-o2"*
i;
=
0°
1'
28-96".
* See Chambers's Mathematical Tables for a concise explanation of the
of obtaining these figures.
method
SPHERICAL TRIGONOMETRY.
INTRODUCTORY THEOREMS.
—
Planes through the centre of a sphere
Definitions.
intersect the surface in ^yr^f circles; other planes intersect
Unless otherwise stated, all arcs are
it in small circh's.
870
measured on great
circles.
The poles
of a great circle are the extremities of the
diameter perpendicular to its plane.
The sides a, 6, c of a spherical triangle are the arcs of
great circles BG, CA, AB on a sphere of radius unity; and
the angles .1, B, C are the angles between the tangents to the
sides at the vertices, or the angles between the planes of the
The centre of the sphere will be denoted by 0.
great circles.
The ])olar triangle of a spherical triangle ABC has for its
angvdar points A\ B\ C\ the poles of the sides /?C, CA, AB
of the primitive triangle in the directions of A, B, C respecThe sides of
tively (since each great circle has two i)oles).
A'B'C are denoted by a\ h\ c.
The
871
sides
and angles of the
polai- triangle are respectively the
supplements of the angles and
of the
sides
that
primitive triangle
;
is,
n'-\-A
Ltt
=
EC
C'A' in a,
=
//+/;
r'-\-('
^
n,
pi'oduc-cd cut the sidi'S A'H',
11.
therefore 1!II
=
11
y.
is
tJif
pole of
Siniilarly C'li
\,
A'C\
=
^,
therefore, by adilition, a -|- CJ[=ir and GII=.A\ because A' is the pole of BC.
The polar diai.'r;im of a spherical i»nIytron is formed in the same way, and
tlie same relations subsist between the sides and au'^les of the two Hy^ure.*'.
2 u
SPHERICAL TRIGONOMETRY.
186
—
Rule. Hence, any equation between the slides and angles
of a spherical triangle jyroduces a siqjplementary equation by
changing a into tt — A and A into tt— a, ^c.
The centre of the inscribed circle, radius r, is also the
centre of the circumscribed circle, radius R\ of the polar
872
triangle,
and
v-^ll'
= ^tt.
—
Pkoof. In the last figure, let
be the centre of the inscribed circle of
ABC; tlun 01), the perpendicular on BC, passes through A\ the pole of BG.
Also, OD
r; therefore OA'=h'r—r. Similarly 0B'= 0G'=Itt—7-; therefore
is the centre of the circumscribed circle of A'B'C, and r-\-B'= ^tt.
=
873 The sine of the arc joining a point on the circumference
of a small circle with the pole of a parallel great circle, is equal
to the ratio of the circumferences or corresponding arcs of the
two
circles.
For
it is
the sphere
;
equal to the radius of the small circle divided by the radius of
that is, by the radius of the great circle.
874
Two
sides of a triangle are greater than the third.
[By XI. 20.
875
The
sides of a triangle are together less than the cir-
cumference of a great
876
The angles
[By XI.
circle.
of a triangle are together greater than
21.
two
right angles.
For ir—A + TT — B + TT—C
877
if
two
is
<
27r,
If
[By the geomctric;il proof
two angles
[By the polar triangle and (877).
greater angle of a triangle has the greater side
'I'he
opposite to
PnooF.
Z
A
lil>
880
it.
— If
= .1,
J? be > ^, di-aw tie arc ItD mooting AC in P, and make
therefore
but
+ nG>BC, therefore AC>BC.
BB = AD
The greater
opposite to
in (89-i).
of a triangle are equal, the opposite sides
are equal.
879
triangle.
sides of ajriangle are equal, the opposite angles
are equal.
878
by (875) and the polar
it.
;
BD
side of a triangle has the greater angle
[By tho
jx.lar triangle
and (879).
—
OBL IQ i'E-A S UL ED
Tit IA
NO L ES.
187
RIGHT-ANGLED TlUAN(i LKS.
—
Napier s Rules. Tn the triangle J BO let C be ;i li^Hit
r), (Jtt
angle, then «, QTr
./), and//, are called
7?),
Taking any part for middle part,
the five circular jxn-/^.
Napier's rules are
jn-odiirt of faiKjfiifs ifaJjdcciif jvtrfs.
I. sineofun'diUrjiarf
product of cosines of opposite parts.
II. sine of middle part
In applying the rules we can take J, B, c instead of their
complements, and change sine into cos, or vice vers/i, for those
parts at once.
Thus, taking b for the middle part,
881
—
—
(W —
=
=
= tan a
sin b
Ten equations
in all
cot
.7
= sin B
sine.
are given by the
rules.
—
Proof. From any point P in OA, draw
perpendicular to OC, and EQ to OB;
therefore I'liQ is a right aiigle
tlierefore OB
is perpendicular to PR and Q1i\ and therefore
to PQ. Then prove ai.y (oruiuhi by proportion
from the triangles of the tetrahedron OPQR,
which are all right-angled. Other\vise, prove
by the formulas for oblique-angled triangles.
PR
;
OBLIQUE-ANGLED TRIANGLES.
cos a
882
= cos b
cos c-fsin h sin c cos
.1.
—
Pkook. Draw tangents at A
to the sides c, b to meet OH, OG
Express VE'^ by
in D and E.
(7<i2) applied to each of tho
triangles DAE and 1)0E, and
subtiact.
and. I'' are both >
If .1/?
J,
them fo meet in ,1', the
pole of A, and employ the Iriaugle A'BC.
]trodiicc
If
duce
AB
7>.l
The
883
alone be
to
>
^.
pro-
meet BC.
sup})leineutary formula, by (871),
<*<>^l =r
—cos
/;
cos r-f sill
/>'
is
>in
r
<•<)>//.
—
—
.
^
,
8PHEEICAL TRIG0XOMETR7.
188
A
^m^=yjt
/sill
884
—
(.s
//)
sill
(.V
b
8111
c
8ill
= J ^"if-^i"(/-^^)
cos4
885
^
oo/?
886
A
i
^
/sill
(.V
tan—-=\/
Proof.
—
A———
sill
/>)
sill
A-
.
r*
(5——-L
r)
1/
= i{a-{-h-\-c).
/
\
whevQ
fi
Substitute for cos
/I
from (872), and
by
(673).
Sill ys
— sin^— = \ (1 — cos^l).
throw the nnmerator
sill
/>
T
^
^^
^
'I
sill
—c
I
I
a)
of the whole expression into factors
Similarly
for cos -.
or
The supplementary formulae are obtained
rule in (871).
They are
in a similar
way,
by the
cos4 = J<^<^^{s-'iU'<^HS-C)
Y
887
ooo
888
sill
ooft
889
I
890
sm B
2
.
tan
a
2
-n-
t^
2
/— COS
=
V^
.S
•
siu
cos
B
/>
•
t*iu
C
(^'
—
sin
/.
C
_
—
—
t'os 8 COS
—— ^——- ^)'—
= d^ — —
Qos{S — B) cos(.S— C)
where S = i (A-[-B+C).
(^'
/
= \/siu.s* sin (* — sin (s — b) sin — c)
Let
= ^ \/l + ^ cos a cos 6 cos c— cos' a — cos- — cos'
(T
(a'
//)
/>
Then the supplementary form, by
891
S
=
=
1^
892
sin
8'
cos
I
^
—
2(r
.
.
sin
/>
The following
(884 to
I.
is
[S—A) cos [S — B) cos (,N — C'j
\/ 1—2 cos^ cosii cusC— cos
— cos- Z^— cos-C.
a/— cos
—
sinrt=
sin c
[By sin^
893
(871),
c.
rules
=
will
*^S
H sm />
sm fi
C
.
2 sin fy cos
•
^
•
and (884, 885), &c.
produce the ten formula^
892)—
Write sin before each factor
In
the s
cahics
o/sin—
—
—
.
OULIQUE-AKCLEI) TRIASCLES.
cos
\^
^
tan
,
A, and A,
sin
,
707),
Phinr
///
cor I'cf^pouding
fo chta'ni flif
189
Trii/->iioinetri/
(701—
fannul tn in Spherical Trijo-
iionirfri/.
11. To ohfain the svpplementarij forms of the five resultn^
transpose lanje and small letters everyivhere^ and transpose
sin and cos everywhere hut in the denominators, and write
minus before cos S.
A _
sill
n(\A
sin
sin a
PuooF.— By
cular to
hy
T.
47,
B_
sin
sin
C
sin c
fj
Otliei-wise, in the figure of 882,
(8S-2).
HOC, and NR, XS
to
UB, 00.
PN = OP sin c
and therefore
Prove
sin L'
PRO
= OP
draw
and
PN perpendi-
FSO
sin ^ sin
right angles
(J.
COS 6 cos (7 = cot a sin 6— cot A sin C
895
To remember tins formnla, take any four consecutive
angles
J), and, calling the first and fourth the
extremes, and the second and third the middle parts, employ
the following rule
Rule. Product of cosines of middle parts
cot extreme
cot extreme angle X sin middle angle.
side X sin middle side
and sides
(as a, C,
h,
:
—
=
—
Pkoof.
cos
c,
— In
and
the formula for cos a (882) substitute a similar value for
for sin c put sin
896
—
C—
-
smA
NAPIER'S FonmrL.E.
tan i (A
(1)
-B) =
^-^-)
sni-o-(r<
/o\
(-^
i
1
'
<
'""^^-'
I
+
in
^^)
i'O^l
((t
-^ cot —
+ 6)
— h)
= .....i (»+/,)
.
2
.
*•"*
C
2-
(.3)
(,.„i("-/')=jii^Tpqr7]y*""i7-
/ «\
A
1
/
tan.U''
(4)
+
I
/\
'')
=
—
—
<*os
(A — li)
r
r|:i:p7j^t»»^.
i-
,
Rule. /// ///c rahic of tan (A — 13) change sin to cos /o
o&^«m tan^(A + B). To obtain (3) «?«(/ (-4)/rc)??t (1) and (2),
transpose sides and angles, and change cot to tan.
Proof.
J-
— In the values
m sin b for sin ^
-^
m = sinj4±sinZ>
sill
a
±
.
—
sill
6
of cos.l and cos 7^, by (883), put msina and
and sin B, and add the two equations.
Then put
r
/.>nA r»i-T.i\
and transform by (()70-d72).
,
,
.
i
—
190
SrUERICAL TRiaOKOMETR Y.
897
GAUSS'S FORMULAE.
+ 7J) _
smi(.4
(1)
(io^\{a-h)
cos ^6^
(2)
C0S-2C
cos^C
sin^c
cosi(^ + i^)
(3)
_ cosi(rt+&)
sin^C
cosl^c
cosi(^-/J) _sini((< + ^)
sill
^
C
sin ^c
From any of these formulae the others may be obtained
by the following rule
—
:
Rule. — Change,
the sign of the letter B {large or small) on
one side of the equaJion, and ivrite sin for cos and cos for sin
on the other
Proof.
side.
—Take
substitute the
s
sin-^-
(^ + 7?)
values by
(88-i,
=
sin-^x4 cos }jB
+ coslA
sin iZ>,
885), and reduce.
SPHERICAL TRIANGLE AND CIRCLE.
Let r be the radius of the inscribed circle of ABG ; r« the radius of
the escribed circle touching the side a,
and B, Ba the radii of the circumscribed
898
circles
(1)
;
then
tan r
= tan ^A sin (v— «) =
sni a
(3)
SI
(4)
n^
2 cos
cusS+cos
Pkook.
— Tlio
(^^
^
.
sin?r^l siiioTi siiioC
^A
cos
^B
cos I ^
— yl) + cos (S — B)-\-liic.
first
value
is
found from
ri^lit-auglcd triangle OAF, in which
The otliei- vahies I)y (881-892).
tlie
AF = s — a.
spjiEuicAL
899
i;ni r„
(1)
=
'nnASiii.i-: asi> ciikli:.
tail
,\/l
sin.v
=
111 (.V
^
(3)
-f'''[' sin
sin
A
=T
(>)
M
\A sini/i
i.s
IB
cos
—
l'.»l
</)
\C
cos
sin ^(7
oV
_(.os.S-c()s(N-.l) + c<)s(N-yi)+t'<>'^i''>'-^-')
Proof.— From tlie right-angled triangle O'AF', in which AF'= s.
NoiK.— The first two values of tan r„ may be obtained from those
tan r by interchanging s and s—a.
900
tiiii«
(1)
tan
=
—cos S
Tift
S
c()s(.S-.l)
_
(3)
of
sill \<i
sin
_
A
2 sin
cos yj cos ^c
^^< sin
^j sin ^c
(4)
.ill
+ sin
(,v
— (i) + sin
(.v
— A) +
c^'C.
—
The fir.st value from the right-angled
OBD, in which Z OTID = S — A. ^Tlie other
Proof.
triangle
.v
values by the formulie (887-892).
901
h.nR„
(1)
05)
=.sin A
(1.)
=
(5)
=
Proof.— From
tan^n _oos(S— ^)
= —
cds.S
Sin tU<
sin lb sin U'
2 sin hn
<*os
;in
(.V
^b cos Ic
— rO + sin
the right-angled triangle
(.v
— + .^in — c)
o7>7^
/>)
in
(.v
which z
O'/;/^
=
tt-.S.
SPHERICAL TRiaONOMETBY.
192
SPHERICAL AREAS.
ABC = (A-^B+C-tt) r- = Er
E = ^+7i + C— the spherical excess.
area of
902
wliere
tt,
Proof.
— By adding the three lunes
and observing that
get
(
ABF =
A+]l+^]
TT
CDE,
27rr'
=
27rr
+ 2ABC.
IT I
IT
AREA OF SPHERICAL POLYGON,
903
n being the number
Area
The
GAFB,
BGEA,
ABDG,
=
=
=
of sides,
{interior Angles
— (» — 2)
{277— Exterior Angles}
tt]
r'
r^
{27r— sides of Polar Diagram} r.
last value holds for a curvilinear area in the limit.
—
Proof. By joining the vertices with an interior point, and adding the
areas of the spherical triangles so formed.
GagnolVs Theorem.
904
•
sin
1 L^
c> tj
_
—
— u)
\/ {si" ^ ^^^ { s
-T
2 cos
^\n(s
— h)
r-j
\
-^a
sin(.y
— r)y
\
cos ^b cos ^c
Proof. -Expand sin \_\{A + B)~},{Tr-G)'] by (628), and transform by
Gauss's equations (897 i., iii.) and (669, 890).
LlhulUier's Theorem.
905
UuiE = y [tan k
—
tan
i (.s— ^0
ian
I {.s^-h)
tan
i
{s-c)}.
Multiply numerator and denominator of the left side by
Proof.
{A+B)
2 cos 1 (A + B-G + n) and reduce by (6G7, 668), then eliminate i
by Gau8.s'.s foniiulio (S!i7 i., iii.) Tnuisfonii by (()72, 673), and substitute
from (886).
'
roLYIIHIili-ONS.
\\y.\
Vi)\M\VA)\{()NH.
Let
tlic
and
N,
/•',
/>'
;
tlu-n
=
//+.S
906
Pi;,),,i.-.
number
— Project
|)olvhc(lron
till'
=
its area =
of sides",
and
s
formed. Then
polygons, and equate to
angles, and cdircs, of any
of fiiccs, solid
iiuiiilx'i-
hr
|)(.lylicdr<.n
sum
[s
E+'2.
Let vi =
Jin internal splicre.
one of tlie spherieal poIy-,'on.s so
.Sum this for all the
r. by fOn.'i).
upon
of anj^les of
— {»i — 2) tt]
4n-?-*.
THE FIVE REGULAR SOLIDS.
TiOt VI
he the niimher of sides
of ]ilane angles in each solid angle
))iF= nS
907
in
;
=
each face,
ii
the nninher
therefore
'2E.
Fi-om these equations and (OOd), Hnd I\ S, and
in
I'J
terms of
ni
ajid
«, thus,
1
F
^w
'2
/
1
V nt
1_
1\
n
2
.1
/
'
In order that F, S, and
a rehition
which admits
^
N
E may
»
2
(
V
1
1
^_
n
vi
1=
_1\
2
/
'
E
be positive, we must havi-
1
+
m
^
h
+
_
1
-1
>
whole numbers, corresponding
of five solutions in
the five regular solids. The values of hi, », F, S, and
solids are exhibited in the following table
:
—
E
to
for the five regular
.
;
SPHEBIGAL TRIGONOMETB Y.
194
909 If I be the angle between two adjacent faces of a
regular polyhedron,
smi/
=
cos
•-
sill
n
—m
=
a be the edi^e, and *S'
Proof.— Let J*(^
the centre of a face, T the middle point of
inscribed and circumcentre
of
the
the
PQ,
scribed spheres, ABC the projection of PST
upon a concentiic sphere. In this splierical
triangle,
=
C
(881,
I.
ii.),
cos
that
PST.
rii
STO
Also
Now, by
=
B
and
n
2
A
=
sin
B
cos
BG
cos — = sui — sin ^1.
m
IS,
n
Q.
e. d.
If r, B be the radii of the inscribed and circumscribed
spheres of a regular polyhedron,
910
r
Proof.
OS
= 4 tan ^I cot —
— In
= PT cot
5in
-'^
above
the
tan
AG =
J
J.
tan
Also
BG
« =
,
figure,
OS
-^ tan
=
cot
A
=
OP =
r,
OP = PT cosec
cot }J cot
^I tan ^.
B.,
PT =
AG, and by (881,
;
n
therefc)re, &c.
-[^
i.),
;
and
ELEMENTARY GEOMETRY.
MISCELLANEOUS PROPOSITIONS.
920 To find the point in a given line QY, the sum of whose
distances from two fixed points <S', S' is a minimum.
Draw SYB
making
QY in
=
r7i'
at right angles to
TS.
Then
P.
P
QY,
Join BS\ cntting
will be the required
point.
—
Proof. For, if D be any other point
PR.
on the line, SD = Dli and SP
But BD + US' is > BS'; therefore, (to.
B is called the reflection of the point i9,
and SP.S" is the path of a ray of light
=
reflected at the line
QY.
QY are
not in the same plane, make SY, YB equal perpenlast in the plane of S' and QY.
Similarly, the point Q in the given line, the diSerence of whose distances
from the fixed points 8 and B' is a maximum, is found by a like construction.
If ^, S'
and
diculars as before, but the
The minimum sum
(^7^
And
the
maximum
of distances
from 8,
S' is
given by
+ ^-7^)-= SS"'+4^SY.S'Y'.
difference
from S and R'
is
given by
{SQ-R'QY= {SHy-4^SY.IVY'.
Proved by VJ.
D., since
SBB'S' can be
Hence, to find tlic
shortest distance from P
to Q en route of the hncs
AB, BC, CD', in other
words, the path of the ray
reflected at the successive
921
surfaces
AB, BG, CD.
Find P, the reflection of
P
at
surface; then Pj, the
reflection of 2', at the second surface next Pj. the reflection of P,
at the third surface and so on if
,
the
first
;
;
inscribed in a circle.
—
;
ELEMENTARY GEOMETRY.
196
Lastly, join
there be more surfaces.
CD in a. Join aPj, cutting BG in b.
PcbaQ is the path required.
Q
with P,, the last reflection, cutting
Join hP^, cutting AB in c. Join cP.
The same construction will give the path when the surfaces are not, as
in the case considered, all perpendicular to the same plane.
922 If the straight line d from the vertex of a triangle
divide the base into segments _p, q, and if h be the distance
from the point of section to the foot of the perpendicular from
the vertex on the base, then
[II. 12, 13.
The following
cases are important
When p = q,
(i.)
b'+c'
:
= 2q'-^2d'
the sum of the squares of tiuo sides of a
triangle is equal to twice the square of half
the base, together ivith tivice the square of the
i.e.,
bisecting line
(ii.)
(iii.)
drawn from
the vertex.
When p = 2q, 2b'-]-c' = 6q'+M\
When the triangle is isosceles,
b'=
923
P
c'
(II.
= 2)q + (P.
be the centre of an equilateral triangle
If
any point in space.
= 2PD' + 2BD\
= 60D" + SPO\
B0 = 20D;
Proof.— PB' + PC
PA' + 2PD'
and
ABC
and
Then
FA'+PB'-^PC =
Also
12 or 13)
3 {PO'-\-OA').
(922,
(922,
i.)
ii.)
therefore, &c.
—
Hence, if P be any point on the
CoR.
surface of a sphere, centre 0, the sum of the squares of
And if r, the radius
its distances from J, B, G is constant.
of tlie sphere, be equal to OA, the sum of the same squares is
equal to
Or".
MISCELLANEOUS PROPOSITIONS.
The sum
924
of
lo:
squares of
\\\c
the sides of a (]uadrihitcral is equal
to the sum of the squares of the
diagonals plus four times the square
of the line joiuinpr the middle points
of the diagonals.
(9-J2, i.)
925
Cor.
—The sum
of the squares
of the sides of a parallelogram is
sum
equal to the
926
of the squares of the diagonals.
In a given line
from a point
P
AG,
to find a point
have a given
ratio to its distance in a given
direction from a line AB.
shall
Through P draw BPC parallel to the
given direction. Produce AP, and make
CE in the given ratio to CB. Draw PX
parallel to EC, and
two solutions when
points.
To
927
find
whose distance
to
BC
CB.
There are
AP
cuts
two
in
[Proof.— By (VI.
2).
a point X in AC,
XY from AB parallel
XZ from BG
AE
Draw
CE
have a given ratio to
shall
distance
XY to
parallel to
parallel to
its
AD.
BC, and having
AD
BE
AG
to
the given ratio.
Join
cutting
\n
X, the point required.
[Proved by (VI. 2).
To
928
X
find a point
on any
curved, whose
distances XY, XZ, in given directions from two given lines AP, AB,
shall be in a given ratio.
straight
line,
P
Take
or
any point
in
the
first
line.
Draw PB
and
PB
BC
parallel to the direction of XY,
parallel to that of XZ, making
have to
cutting
AB
BC the
in
I).
given ratio.
Draw
DE
Join PC,
parallel
to
CB. Then AE produced cuts tlie line in
X, the point required, and is the locus of
such points.
[Proof.
— By (VI.
2).
X
whose distance
ELEMENTARY GEOMETRY.
198
XY
through a given point P so that
To draw a line
the sogmeuts XP, FY, intercepted by a given circle, shall be
in a given ratio.
929
Divide the radius of the circle in that ratio,
and, with the parts for sides, construct a triangle
upon PC as base. Produce GD to cut the
PDC
X
circle in
Draw
XPY and
GY.
PD + DC = radius
PD = DX
Then
therefore
;
CY=CX;
But
thereforePDisparallelto(7r(I.5, 28) therefore
&c., by (VI. 2).
;
From
930
a given point
which
P
draw a
side of a triangle, to
in the
line
PX
shall divide the area of the tri-
angle in a given ratio.
EG
Divide
draw
D
in
AX parallel
to
in
PD.
the given ratio, and
will be the line
PX
required.
ABD ADG =
:
API)
= XPI)
931
To
(I.
the given ratio (VI.
therefore, &c.
37)
and
ABG
XY drawn
in a given ratio by a line
divide the triangle
parallel to any given line AE.
Make BD to BG in the given
make PY a mean proportional to
and draw YX parallel to EA.
Proof.
(VI. 1).
1),
;
—AB
divides
ABG
ratio.
BE
Then
and BB,
in the given ratio
Now
ABE XBY
:
::
BE
::
ABE: ABD;
or
XBY =
therefore
:
BD,
(VI. 19)
ABD.
and exterior vertical angles at P of the
be bisected by straight lines which cut the base
in G and D, then the circle circumscribing GPD gives the
locus of the vertices of all triangles on the base AB whose
sides AT\ PP m-o in a constant ratio.
932
If the interior
triangle
APB
—
MISCELLANEOUS PROPOSITIONS.
Proof.
The
Z
199
—
CPD = i(APB + BPE)
=
a right angle
;
therefore P lies on the circumference of
Also
the circle, diameter CD (III. 31).
AP PB
.S,
AC GB
::
:
(VI.
:
and A.), a
::
AD DB
:
fixed ratio.
AD
DB :: AG I GB
divided harmonically in B and G i.e.,
one extreme part as the other extreme part is to the middle
part.
If we put n, b, c for tlie lengths AD, BD, OD, the proportion is
expressed algebraically hy a h :: a c c b, which is equivalent to
933
AD
is
;
:
;
or, the ivhole line is to
—
:
—
:
'+! = a
c
934
AP BP = 0A: OG = OG OB
Also
:
:
AP'
and
:
BP'
= OA
:
(VI. 19)
OB,
AP'-AO' GP" BP'-BC\
:
(VI.
:
3, <k
B.)
935 If Q be the centre of the inscribed circle of the triangle
ABG, and if AQ produced meet the circumscribed circle,
radius E/\nF; and
cular to BG then
FOG
if
bo a diameter, and
AD
perpendi-
;
FC=FQ = FB='lR^\n:^.
ill) Z.FAI) = FAO=},{B-C),
and /_CAG = \{B^C).
(i.)
Proof of
(i.)
Z.FQG= QGA-\-QAC.
But QAG =z QAB = BGF {\\\. 2\)
FG = FQ.
FQG = FGQ;
FB = FQ.
Similarly
Also Z GCF is a right angle, and
FOG = FAG = \A; (III. 21)
;
.-.
.-.
.-.
936
FC = 2Esin4.
If -K, r
be the radii of the circumscribed and inscribed
circles of the triangle
centres; then
Proof.
(see last figure),
— Draw QH perpendicular to ^C
triangle AOF, OQ'
by similar triangles
therefore
AUG
and 0,
Q,
the
()Q^=IV—2Hr.
= li'-AQ.QF
(922,
;
then QII
iii.),
=
AQ QII OF FG
AQ.FC = OF.QU = 2 Rr,
QFC, AQH,
:
::
r.
By
QF = FG
and
:
;
the isosceles
(935,
i.),
and
—
ELEMENTARY GEOMETRY.
200
The problems known as
Given
937
in position
any three
the Tangencies.
of the following nine data
—
three points, three straight lines, and three circles,
it is
required to describe a circle passing through the given points
and touching the given lines or circles. The following five
principal cases occur.
viz.,
938
I.
Given two points A^
Analysis.
GD
touching
circle,
J9,
and the straight
CD.
line
— Let ABX
be the required
Therefore
in X.
GX'=^GA.GB.
(III.
36)
X
can be found, and the
Hence the point
centre of the circle defined by the intersection of the perpendicular to (yD through
and the perpendicular bisector of AB.
There are two solutions.
X
Otherwise, by (926), making the ratio
one of equality, and DO the given line.
Cor.
— The point X thus determined
which the distance
solution of (941)
Q
is
the point in
II-
at
is
subtends the greatest angle. In the
a similar point in the circumference GD.
(III. 21,
939
GD
AB
Given one point
A
and two straight
lines
&
I.
16)
DG, DE.
In the last figure draw AOG perpendicular to DO, the bisector of the
angle D, and make OB
OA, and this case is solved by Case I.
=
940
line
DE, and
the
AGF.
—
PEF
Let
be the required
touching the given line in E and the
Analysis.
circle
Given the point P, the straight
III-
circle
circle in F.
Through
draw
the centre of the given
perpendicular to DE.
be the centre of the other circle.
Let
Join 11K, passing through Z'', the point of
contact.
Join AF, EF, and AF, cutting
Then
the required circle in X.
circle,
K
IF,
AHGD
^DHF = LKF;
UFA = KFE
therefore
angles) therefore
;
and
AF.AF
A circle
AF,
FE
Then, because
straight line.
:c=
AG .AD
(1.27)
(the halves of equal
are in the same
AX.AP = AF.AE,
by similar
(III. 36)
must then be described through
P
and
X
^X
can bo found.
to touch the given line,
triangles, therefore
;
THE PROnLEMS OF THE TANOE^X'rES.
201
by Caso I. There are two solutions with exterior contact, as appears from
Case 1. These are indicated in the diagram. There are two more in wliich
the circle J C lies witliin the described
analogous, C taking the place of A.
IV. Given two points Ay
941
the circle
B
The
circle.
construction
quite
is
and
CD.
Draw any circle through A, B, cutting tho
and DC,
Draw
required circle in C, D.
Draw FQ to touch
anil lot them meet in F.
Then, because
the given circle.
AB
FC.FD = FA FB = FQ\
.
(TIL 30)
pass through
A, B therefore a circle drawn through A, B, (^
must touch FQ. and therefore the circle CJ),
in Q (III. 37), and it can be described by Caso
There are two solutions corresponding to
I.
to the circle CD.
the two tangents from
and the required
circle
is
to
;
F
942
V. Given one point P, and two
circles, centres
A and
—
B.
F
Let FFO be tho required circle touching the given ones in
Analysis.
Join the centres QA, QB. Join FG, and produce it to cut tho
0.
and //, and the lino of centres in 0. Then, by the isosceles
circles in
therefore AE, BG are
triangles, the four angles at F, F, G, II are all equal
is a
Hit, and
BO ::
parallel, and so arc .1/'', BII\ therefore AO
211 IjK, and
Again, Z IIJiK
centre of similitude for tho two circles.
IILK= IIGK (III. 21) there(HI. 20) therefore
are similar; therefore OF OG
fore the triangles OFN,
OF OK. ON. Thus
therefore, if OF cut the required circle in X,
the point A' can be found, and tho problem is reduced to Case IV.
to touch tho given circles.
Two circles can be drawn througli F and
One is the circle FFX. The centre of the other is at tlie ]K)int where EA
and //.
and JIB meet if produced, and this circle touches the given ones in
and
E
;
AF
:
FAM = 2FNM
:
=
FNM =
;
;
OKG
= OK OM
.
OX
.
.
=
X
F
943
An analogous construction, employing the internal centre of similitude 0', determines the circle which passes through F, and touches one given
See (1017-1)).
circle externally and the other internally.
The centres of similitude are tho two points which divide tho di.stanco
between the centres
in the ratio of the radii.
2d
See (1037).
_
ELEMENTAUY
202
C'EOMETRY.
—
to all circles which touch
944 Cor. The tangents from
the given circles, either both externally or both internally,
are equal.
For the square of tbe tangent
is
always equal to
OK. ON"
or
OL OM.
.
945 The solutions for the cases of three given straight lines
or three given points are to be found in Euc. IV., Props. 4, 5.
In the remaining cases of the tangencies, straight lines
By drawing a circle concentric
with the required one through the centre of the least given
circle, the problem can always be made to depend upon one
of the preceding cases the centre of the least circle becoming
one of the given points.
946
and
circles alone are given.
;
947
Definition.
—A cenb-e
of similarity of firo jjlane curves
line being drawn through it
to cut the ciirccs, the segments of the line intercejited between
the ]Joint and the curves are in a constant ratio.
v\s
any straight
a point such that,
948
If
AB,
AG
touch a
any straight
C, then
circle at
AEDF,
line
B
and
cutting
the circle, is divided harmonically by
the circumference and the chord of contact
BG.
Proof from
AE AF = AB'.
= BD.DG+AD\
BB.BC=EB. BE.
(III.
.
AB'
and
3G)
(923)
(III.
35)
949 If " /^j 7> in the same figure, be the
perpendiculars to the sides of ABG from
any point E on the circumference of the
5
=
a".
then
/3y
Draw the diameter BII=d
right angle. Similarly EG^ = yd.
But
circle,
Prooi'.
950
BG
—
EE
If
then EB'' =
EB EC=ad
;
.
be drawn parallel to the base
and
of a triangle,
if
EB, FG
intersect
in 0, then
AE AG
:
By
VI.
2.
Cou.— If
::
EG OB
:
Since each ratio
AC =
n
.
FO
= FE
::
AE, then
BE =
(n-j-l)OE.
:
:
OG.
BG.
ft(l,
because
BEH
(Yl. D.), therefore
is
a
etc.
;
;
MISCELLANEOUS riiOrOSlTIOKS.
203
T1h> tliroc lines drawn from the ann^les of a tii;in<^^l(' to
the niiihUe jioints of tlie opposite sides, intersect in tl»e same
point, and divide each other in the ratio of two to one.
951
For, l>y tlio last theorem, any ono of these lines is divided by each of tho
others in tho ratio of two to ono, measuring from tho same cxlrcniity, and
nuist therefore bo intersected by them in tho same point.
This point will bo referred to as the ccntruid of the triangle.
The perpendiculars
952
frora the angles upon tho opposite
same point.
Bides of a triangle intersect in the
Draw BE, CF
perpendicular to the sides, and let
Let AO meet IW in J>. Circles
and BFEC, by (III. 31)
will circumscribe
them
A
intersect in O.
AEOF
therefore
therefore
;
(III.
FAO = FEO = FCB
Z BDA = BFC = a right angle
Z
;
21)
AO \h perpendicular to BC, and therefore the
perpendicular from A on BC passes through 0.
i.e.,
called the orthocentre of the triangle
is
ABC.
— The
perpendiculars on the sides bisect the angles
is therefore the centre
the point
of the inscribed circle of that triangle.
Proof. From (III. 21), and the circles circumscribing OEAF and OECD.
CoTi.
of the triangle
DEF, and
—
ABC touches the
D, E, F and if tlie^ escribed circle
and if
to the side a touches a and h, c produced in D', TJ', F'
If the inscrilied circle of a triangle
953
sides a,
h, c
in the points
;
;
then
BF'
and
and
= nD'= CD = s-c,
= A F' = s
AE'
;
similarly with respect to
the other segments.
—
Proof. The two tangents from
any vertex tocithercirclfbeingoqual,
r=
half tho
folif)\vs that ll) +
perimeter ol' A Bt\ whicli is made up
of three pairs of equal segments
it
;
CD = s —
therefore
c.
Also
AE'+ A F'= A r + Cl> +
=
therefore
,
1
7?
2--
AE'
AF'
=
s.
+ Bjy
ELEMENTARY QEOMETBY.
204
The Nine-Point
The Nine-point
Circle.
through
E, F, the feet of the perpendiculars on the sides of the
It also passes through the middle points of
triangle ABG.
the sides of ABG and the middle points of OA, OB, OG ; in
all, through nine points.
954
circle is the circle described
I),
—
Proof. Let the circle
cut the sides of ABG
again in G, H, K; and
OA, OB, OG
in L,
/.EMF=EDF
M, N.
(III. 21)
= 20DF
Cor.);
(952,
therefore, since OB is the
diameter of the circle cir-
cumscribing
OFBD
(III.
31), ill is the centre of
that circle (III. 20), and
therefore bisects OB.
Similarly OG and OA
and L.
are bisected at
N
Again, Z
MGB = MED
scribing OEGD.
and
Similarly
(III. 22)
Therefore
H
K bisect
MG
GA
is
=
OGD, (HI.
parallel to
21), by the circle circumOG, and therefore bisects BG.
and AB.
955 The centre of the
nine-point circle is the
middle point of OQ, the
line joining the orthocentre and the centre
of the circumscribing
circle of the triangle
ABG.
For the centre of the N. P.
the intersection of
the perpendicular bisectors
of the chords DG, EH, FK,
perpendiculars
and these
bisect OQ in the same point
(VI.
N, by
2).
circle is
956 The centroid of
the triangle jUiG also lies on the line
2BQ.
so that OB
=
B
Pkook.— The
fore
A
centroid,
= 2GQ
and
it
triangles
;
QUG, OAB
tluM-eforo Oli
= 2h'(2
OQ and
are similar, and
;
and Ali
divides 0(2 as stated (051).
= 2ii'c;
divides
AB =: 2nG
;
therefore
;
B
it
in
thereis
the
CONSTIiUCTION OF
"201
line joining the centres of the circumscribed
Hence the
957
TillANGLES.
and nine-point circles is divided harmonically in the I'atio of
1 by the centroid and the orthocentre of the ti-ian<^le.
2
These two points are therefore centres of similitude of the
and any line drawn
circiimscri])ed and nine-point circles
through either of the points is divided by the circumferencca
:
;
in the^-atio of 2
:
See (1037.)
1.
FD intersect the sides of AJiC in
958 The
the radical axis of the two circles.
lines
BE, EF,
For, if EF meets BC in F, tlieu by the circle circumscribing FCEF,
FE FF = FC FB therefore (III. 3G) the tangents from F to the circles
.
.
;
are equal (985).
The nine-point
959
circle
touches the inscribed and escribed
circles of the triangle.
—
be the orthocentre, and 7, Q
Proof. Let
the centres of the inscribed and circumscribed
Produce AI to bisect the arc 1>G in T.
circles.
Bisect AG in L, and join GL, cutting AT in .S'.
The N. P. circle passes through G, V, and
L (9o-i), and !» is a right angle. Therefore
QA
GL is a diameter, and is therefore
Therefore GL and QA are parallel.
(957).
QT, therefore
But QA
= R=
=
GS
= GT = CT sin A
,A
o.,
2Ii sin.
(935,
i.)
ST =2GS cos 6
GST = GTS).
Also
(e being the angle
N
being the centre of the N. P. circle, its
^R; and r being the radius of
radius
the inscribed circle, it is required to shew that
= NG =
NI = NG-r.
NP = SN'-{-SP-2SN. SI cos 0.
SN=IR-GS;
SI = TI-ST = 21i sin 4 -2GS cos
Now
Substitute
and
GS
=
2F
lA,
;
to prove the proposition.
J be the centre of the escribed
shewn in a similar way that NJ =
If
is
sin''
(702)
circle
touching BC, and
NG +
r^.
r„ its radius, it
construct a triatif^lefrom certain data.
960 When amongst Wxg data we have the sum or difference
of the two sides AB, AC; or the sum of the segments of the
base made by Ad, the bisector of the exterior vertical angle;
or the difference of the segments made by AF, the bisector of
To
ELEMENTA li Y GEOMETRY.
206
le
interior vertical
tlie
;
tlie
followinf]f
construction will
lead to the solution.
Make AE = AD = AC. Draw
DII parallel to AF, and suppose
EK drawn parallel to AG to meet
the base produced in K; and
complete the figure. Then BE
the sum, and IJD
euce of the sides.
is
EK is the
segments
sum
is
the differ-
of the exterior
of the base,
and
Till is
the difference of the interior seg-
BDH = BEG = iA,
ZADG = EAG =i(B+C),
L DCB = \BFB = i (0-7)).
ments. Z
961 When the base and the vertical angle are given the
locus of the vertex is the circle ABC in figure (935) and the
locus of the centre of the inscribed circle is tlie circle, centre
and radius FB, When the ratio of the sides is given,
se e (932).
;
;
F
To construct
962
Analysis.
a triangle
from a point
of its vertices
when
its
form and the distances
A' are given.
— Let ABG be the required triangle.
Oa
A'B make the triangle A'BG' similar to ABG, so that
AB A'B :: GB G'B. The angles ABA', GBG' will
BG :: AA' GC, which
also be equal; therefore AB
Hence
gives GG', since the ratio AB BG is known.
:
:
:
:
:
the point G is found by constructing the triangle
A'CG'. Thus BG is determined, and thence the trifrom the known angles.
angle
ABG
To find the locus
of a point P, the tangent from which to a
given circle, centre A,
has a constant ratio to
its distance from a given
963
point B.
AK
Let
be the radius of
the circle, and jj q the
given ratio.
On A JJ take
AC, a third propoitional to
:
AB
and AK, and make
AD:DB=]r
With
ccnti-e
:
.j\
D, aud a radius
TANGEXTS TO TWO
mean proportional betwcon
bo the required locus.
/>/>
Cfjunl to a
Prook.
A,
J?,
— Suppose
and
C,
r
207
ClRCLEFi.
and DC, describe a
Join
bo a point on the rcMpiirod locus.
to
It will
circle.
7' witli
7>.
Describe a circle about PBC cutting Al' in
cutting P77 in (1, and join AH and UF. Tlien
TK- = AF'-A
=
A'-
J7^--7U AC
Ai'.vF (II. '1)
=
and anotlicr about AliF
l'\
(by constr.)
= A V'-TA
=
(HI.
.
ur.rjj
.
AF
(III. Z^\)
:3G).
Therefore, by h^'pothe.sis,
2>-
=
f/
:
(VP
Z Z)7V?
therefore
.
rn
DB
964
CoE.
bisector of
If
BC,
l'>
as
:
T)B (by constr.)
:
=
22)
PC//
DF
are simihir; therefore
construction.
two given
is
;
(III. 21).
a
mean
pro-
Hence the
and DC.
—
= GP rn = A D
2) = PZ''7>' (III.
(VL
=q
tlie
locus becomes the pcrpciulicular
shown in (1003).
is otlicr\Ndse
find the locus of a point P, the tangents
To
965
to
PP'^
DPP, BGP
Therefore the triangles
portional to
:
= PGA
circles shall
have a given
from wliich
(See also
ratio.
10;3(3.)
B
be the centres, a, h the radii
q the given ratio. Take c, so
that c h
p q, and describe a circle with
va' — c'. Find the
centre -1 and radius ^l.V
locus of P by the last proposition, so that
the tangent from P to this circle may have
the given ratio to FB. It will be the required locus.
Let A,
(rt
> fe), and p
=
:
:
:
=
— By hypothesis and construction
Proof.
q'
Cor.
FT-
— Hence the point can
which the tangents
966
to
To
ir'-o^ +
BF'
FT' + U'
h-
to
two
_ AF'-AX'
BP"
be found on any curve from
have a given ratio.
circles shall
find the locus of the point
two given
,^-
from which the tangents
circles are ecpial.
Since, in (965). wo have
simplifies to the following
now p
= q,
and
therefoi-e c
= h,
the construction
:
Take
AN=
y(a--6'-), find in ^IP take
AB AN
:
:
AC.
The perpen-
But, if the circles inter.sect,
dicular bisector of PC is the required locus.
then their common chord is at once the line required. See Radical Axis
(985).
ELEMENTARY GEOMETRY.
208
and Concurrent systems
CoU'niear
and
nfj)oints
lines.
—
967 Definitions. Points lying in the same straight line are
Straight lines passing through the same point are
colUnear.
concurrent^ and the point is called the focus of the pencil of
lines.
Theorem.
—
If the sides of the triangle
ABG,
or the sides
produced, be cut bj any straight line in the points a, 6, o
respectively, the line is called a transversal, and the segments
of the sides are connected by the equation
{Ah
968
Conversely,
if
hC) {Ca
:
aB) (Be
:
:
cA)
=
1.
this relation holds, the points a, h, c will
be
collinear.
— Througli
Proof.
Ab
:
= AD
bG
:
A
any vertex
parallel to the opposite side
transversal in D, then
Ca and
Be,
draw
BG,
to
cA
=
:
AD
meet the
aB
:
AD
(VI. 4), which proves the theorem.
Note. In the formula the segments of the
sides are estimated positive, independently of
direction, the sequence of the letters being prepoint may be supposed to travel
served the better to assist the memory.
from A over the segments Ab, bC, &c. continuously, until it reaches A again.
—
A
By
969
the aid of (701) the above relation
may be put
in
the form
(sin
ABb
:
sin
bBC) (sin C^a
:
sin
aAB) (sin BCc
:
sin
cCA) = l
If
be any focus in the plane of the triangle ABC, and
AG, BO, CO meet the sides in a,b,c; then, as before,
{Ab bC) {Ca aB) {Be :cA) = l.
970
if
:
Conversely,
be concurrent.
Proof.
if
— By the trans-
Bb to the triangle
AaG, we have (9G8)
{Ab bG) {CB Ba)
versal
:
:
x{aO
:
0.1)
= 1.
And, by the transversal
Cc to the triangle AaB,
(Bc:cA)(AO: Oa)
x(aG: CB)
= \.
Multiply these equations
together.
:
this relation holds, the lines
Aa, Bh, Cc will
;;;
COLLINEAR AND COX('rnili:XT
200
SYSTIJ^fS.
Tf J>r, ca, ah, in tlic last figure, be produced to meet the
sides of .l//Oin 1\ Q, R, then eac^h of the nine lines in the
tiu-nro will be divided liannoiiically, and tlie points J\ (,',
971
R
be collinear.
^vill
—
Proof.
(i.)
Take hP a transversal
{Cr
by
tlu-rcforo,
CP
Be) (cP
:
(AB
PC a
070),
:
pb) (bC
:
= ep
Pb
(.la
:
aO) (OG
(Ap
:
pO) (OC
Aa
therefore
all tlio lines
1
CA)
=
1.
CA)
=
1
:
:
by (^08),
:
pb.
AOc\
transversal to AOe, and b a focus to
and
Thus
=
uB.
Pb) (bC
:
Be) (ep
:
cP
Take
therefore,
hC)
:
for focus to Abe,
therefore
&
:
:
(AB
(iii.)
ABC;
(Ab
a transversal to Abe, therefore
But, by (070), taking
(0G8
to
cyl)
:
CP PB = Ca
(1170),
Take
(ii.)
PB) (Be
:
:
Cc) (cB
:
Cc) (eB
:
=
aO
ylj)
:
:
BA)
:
I?-l)
=
=
therefore,
by
1,
1
i'O.
are divided harmonically.
=
AQ QC the harmonic
In the equation of (070) put Ab bC
ratio, and similarly for each ratio, and the result proves that P, Q, R aro
(iv.)
:
collinear,
Cor.
by (008).
—
Proof.
same figure qr, rj), pq be joined, the three
through P, (^, li respectively.
If in the
lines will pass
— Take
harmonic division
972
of
:
as a focus to the triangle abc, and employ (070) and the
show that the transversal rq cuts be in P.
of be to
a transversal intersects the sides AB, lUl, CD, &c.
in the points a, h, c, &c. in order, tlien
If
any polygon
{Aa
:
aB) {Bb
:
bC) {Cc
;
cD) {Dd
:
(IE)
...
kc =
1.
—
Pkoof. Divide the polygon into triangles by lines drawn from one of the
angles, and, applying (908) to each triangle, combine the results.
Let any transversal cut the sides of a triangle and
three intersectors AO, HO, CO (see figure of V70) in tbe
points A', B', C, a, h' c respectively; then, as before,
973
tlieir
,
(J7/
:
IjC)
,
{Ca
:
a
B)
{/)','
:
c'A')
=
1.
forms a triangle with its intorsector and the transfour remaining linos in smvissioi for transversals to each
trianMe, applying CJOS) symmctricallv, and ciMidiinr the twelve equations.
2 E
Phoof.— Each
Take the
versal.
.'^ide
—
ELEMEXTAllY GEOMETRY
210
974 If the lines joining corresponding vertices of two triangles ABC, abc are concurrent, the points of intersection
of the pairs of corresponding
sides are collinear, and conversely.
—
Let tho concurront lines
Proof.
Take be,
Aa, Bh, Cc meet in 0.
transversals respectively to
ca, ah
the triangles OBG, OCA, OAB, applying (9G8), and tlio product of the
three equations shows that P, E, Q
lie on a transversal to ABC.
975
Hence
p
follows that, if the lines joining each pair of
corresponding vertices of any two rectilineal figures are concurrent, the pairs of corresponding sides intersect in points
which are collinear.
The figures in this case are said to be in pprspectlce^ or in,
homology, with each other.
The point of concuiTcnce and
the hne of collinearity are called respectively the centre and
axis of perspective or homology.
See (1083).
it
—
976 Theorem. When three perpendiculars to the sides of a
triangle ABC, intersecting them in the points a, b, c respectively, are concurrent, the following relation is satisfied ; and
converse^, if the relation be satisfied, the perpendiculars are
concurrent.
Afr-bC'+C(r-aB--\-Bc'-cA''
— If the perpendiculars
&c.
47).
Examples. — By the application
Proof.
meet
in
=
0, then Ab''
0.
— bC-=
AO'-—OC-,
(I.
of this theorem, the concurrence of the
three perpendiculars is readily established in the following cases:
(1) When the perpendiculars bisect the sides of the triangle
(By employing I. 47.)
(2) When they pass through the vertices.
(3) The three i-adiioflhe esci-ibed circles of a triangle at the points of
So also arc; the radius of the
contact between the vertices are concui-rent.
inscribed circle at the point of contact wilh one side, and the radii of tho two
escribed circles of the remaining sides at tho points of contact beyond the
included angle.
In these cases employ the values of the segments criven in (953).
(4) The pei'j>endiculars equidistant from the vertices with three concurrent perpendiculars are also concun-cnt.
(5) When the three perpendiculars from the vertices of one triangle upon
the sides of the other are concurrent, then tho perpendiculars from the
vertices of the second triangle upon the sides of the tirst are also concurrent.
Proof.
— If A, B, G
angles, join
tiou witlj
and A\
If,
C
are corresponding vertices of the triapply the theorem in conjuuc-
AB\ AC\ BC, BA\ CA, CR, and
(I.
47).
TRiAXiiLES rinrvMscjiunxt}
a rniAXi;rr:.
of rnnstant species ptrctimsrribrd
Trianii'lr.s'
to
a
211
trian<j;le.
Let J//C' be any triano-l(^, and ^ any point; and lot
AOli, BOO, C(K\. The circumferences
will be the loci of the vertices of a triangle of constant form
whose sides pass through the
977
circles circumscribe
points
^1, //,
C.
PkOOF. — Draw any line hAr
fi-oni
circle
meet in n.
The angles AOD, CO A arc supplements
of tlie angles c and b (III. 22) therefore
BOC is the supplement of a (I. 32) thereAlso, the
for© a lies on the circle OUC.
to circle,
and produce
1>C, cli
to
;
;
being constant, the angles
angles at
are constant.
978
Tlie triano-le ahe is a
pendicular to OA, on, 00.
a, b, c
maximum
wlien
its
sides are per-
—
PuooF. The triangle is greatest when its sides are greatest. But the
sides vary as Oa, Ob, Oc, which are greatest when they are diameters of the
circles; therefore &c., by (111.31).
979 To construct a triangle of given S])ccies and of given
limited magnitude which shall liave its sides passing through
three given points .1, B, 0.
Determine
by describing circles on the sides of ABC! to contain angles
Construct
equal to the supplements of the angles of the speciticd triangle.
the figure nb'O independently from the known sides of ahe, and the now
OBC\ &c. Thus the leJigths Oa, Ob, Oc
0A(\ OnC
known angles ObC
are found, and therefore the points a, b, o, on the circles, can bo determined.
The demonstrations of the following propositions will now be obvious.
=
=
Triant^les of constant species inscribed to a
980
Let
ahr, in the last figure,
trian<;^le.
be a fixed triangle, and
O
any point. Take any point .1 on l>r, and h^t tlie circles circumscribing OAr, OAb cut the oilur sides in /?, O. Then
.47:?(J will be a triangle of constant form, and its angles will
have the values A =. Oha + Ora, &c. (III. -Ji.)
981
OA,
The
01),
triangle
00
are
ABO
will evidently be a
drawn perpendicular
minimum
to the sides of
wht^n
ah''.
982 To construct a triangle of given form and of given
limited magnitude having its vertices u})on three fixed lines
6r, ca, ah.
ELEMENTARY OEOMETIiY.
212
Construct the
fitjure
ABCO
ABG
independently fi-om the known sides of
and the angles at 0, which are equal to the sui)i)lenients of the given angles
Thus the angles OAG, &c are found, and therefore the angles ObC,
a, b, c.
From these last angles the point
&c., equal to them (III. 21), are known.
can be determined, and the lengths OA, OB, 00 being known from the independent figure, the points A, B, C can be found.
may be taken, the angles AOB, BOO
Observe that, wherever the point
COA are in all cases either the supplements of, or equal to, the angles
respectively; while the angles aOb, bOc, cOa are in all cases equal to
c,
a, b
C zh
c,
A±a, B±b.
—
Note. In general problems, like the foregoing, wliicli
admit of different cases, it is advisable to clioose for reference
a standard figure which has all its elements of the same affecIn adapting the figure to other cases, all that
tion or sign.
is necessary is to follow the same construction, letter for
letter, observing the convention respecting positive and
negative, which applies both to the lengths of lines and to
609).
the magnitudes of angles, as explained in (607
983
—
Radical
Ad is.
—
984 Definition. The radical axis of two circles is that
perpendicular to the line of centres which divides tlie distance between the centres into segments, the difference of
whose squares is equal to the difference of the squares of the
radii.
Thus, A,
B
being the centres,
a, h
the radii, and
IP
the
the radical axis
985
It follows that,
is their
common
chord
if tlie
;
and
circles intersect, the radical axis
that, if they do not intersect, the
radical axis cuts the line of centres in a point the tangents
from which to the circles are equal (I. 47).
To draw
the axis in this case, see (960).
nADfCAL AXIS.
Otliorwisc: let the two circles cut
Tcsptctivcly.
any
I)i'sciil>c
ciri-lo
tlie line
tliroii^Mi
('
r' iind //, intersectiiif^ the former in IJ and
will cut the central axis in the required \>o\ut
—
Proof.
IC. ID = IE. 1F= IC. 11/
from I to the circles are equal.
986
T}ii-<n'inn.
— The
21:^
P
of centres in C,
and
/''.
anrl
and another
/',
Tlieir coniniou
C,
7>'
throii},di
chord
Ij
F
f.
(111. 'M)
;
therefore the tangents
dift'orencc of the squares of
tangents
from any })oint /' to two circles is equal to twice the rectangle
under the distance between their centres and the distance of
the point from their radical axis, or
Proof.
by
PK' -FT' = (Ar--Br-)-(a--V) = (AC/'-mr) - (AP-FP),
47) & (t»84). Bisect AB in C, and substitute for each ditference
(I.
of
squares, by (II. 12).
987
Cor.
1.— If r
be on the circle whose centre
is 7?,
then
PK' = 2AB.PN.
Coif. 2.— If two
circles in A', A", Y, >"
988
7^X
chords be drawn through
;
tlien,
by
P
to cut the
(III. 30),
PX- P Y. P Y =2AB. PN.
.
If Ji variable circle intersect two given eireU^s at constant angles a and ft, it will intersect tlieir radical axis at a
constant angle and its radius will bear a constant ratio to
Or
the distance of its centre from the radical axis.
989
;
PN PX =
:
rt
cos a
— 6 cos
fi
:
AB.
;
ELEMENTARY GEOMETRY.
214
Proof.
if
— In the same
PX=PY
be
its
figure, if
be
therefore
.
(I
:
ft
a constant ratio if the angles
is
centre of the variable circle, and
tlie
FX (XX'- YY') = 2An
= 2a cos and YY' =
PX PX = a cos a — & cos
XX'
But
which
P
radius; then, by (088),
(i,
TN.
2h cos
:
ft
;
AB,
are constant.
ft
FX PN = the cosine of the angle at which the
This angle is
radius PX cuts the radical axis.
990
Also
circle
of
:
therefore constant.
—
991 Cor, A circle which touches two fixed circles has its
radius in a constant ratio to the distance of its centre from
their radical axis.
or Ott.
This follows from the proposition by making a = ft =
If
992
P
be on the radical axis
(i.)
;
P
The tangents from
FK =
or
993
(ii-)
P
through
then (see Figs.
to the
two
1
and 2
of 984)
circles are equal,
FT.
(986)
The rectangles under the segments of chords
FX
are equal, or
.
FX'
= FY
.
FY'.
(988)
Therefore the four points X, X', T, Y' are con994
cychc (III. 36); and, conversely, if they are concychc, the
chords XX' J YY' intersect in the radical axis.
(iii-)
—
995 Definition. Points which
a circle are termed coney die.
996
(iv.)
If
P
lie
be the centre, and
of a circle intersecting the
two
on the circumference of
if
PX = PY
be the radius
circles in the figure at angles
=
Z> cos pJ
a and /3; then, by (993), XX'=YY', or a cos a
that is, The cosines of the angles of intersection are inversclj/
as the radii of the fixed circles.
997
The
radical axes of three circles (Fig. 1046), taken two
called their radical centre.
and two together, intersect at a point
PimOF.
in
— Letyl, B,
(7
b/
t'lc
which the radical axes cut
centres, a,
JJC,
tion ('J84) for each pair of circles.
A
the radii, and X, Y, Z the points
tlie equation of the definithe results, and apply {iUO).
h, c
CA, AB.
Add
Wrire
circle whose centre is the radical centre of three
other circles intersects them in angles whose cosines are
inversely as their radii (996).
998
i.\\i:iiSit>.\'.
Henco.,
thogonally,
999
if
this fourtli circle cuts
it
cuts
them
all
one of the others or-
orthogonally.
whicli intersects at angles
'I'he circle
a, ft,
whose centres are .1, li, C and radii a,
centre at distances from the radical axes uf the
circles,
y three fixed
A, r, has its
iixed circles
proportional to
cos
/>
—C
BO
cos y
ft
C
'
—
(( COS a
cos y
CJi
(/
COS
'
—h
AB
a
COS
/3
And therefore the locus of its centre will be a straight line
passing through the radical centre and inchned to the three
radical axes at angles whose sines are projiortional to these
fractions.
Proof.
for each
— The result
p-.iir
is
obtained immediately by writing out equation
('J89)
of fixed circles.
The Method of Inversion.
Any two points F, V
—
1000 Definitions.
diameter of a fixed
situated on a
circle
and radius
is
so that 01\0r'= /r, are
called immerse itoints with respect to the circle, and either
point is said to be the inverse
whose centre
A-,
of the other.
its
The
circle
and
centre are called the circle
and centre of
incersion, and.
k the constant of inversion.
1001 If every point of a plane figure be inverted Avith
respect to a circle, or every point of a figure in space witii
respect to a sphere, the resulting figure is called the inverse
or image of the original one.
Since OB k 0B\ therefore
:
1002
:
OP
:
OP'
=
OP'
:
fr
= A^
:
0P'\
1003 Let D, jy, in the same figure, be a pair of inverse
In the perpendicular bisector of
points on the diameter 00'.
VD\ take any point Q as the centre of a circle passing through
1), I)\ cutting the circle of inversion in R, and any straight
Then, by (III. 3r.),
line through
in the points P, B.
OB
on
OR- (1 000). Hence
.
OB =
.
OU =
ELEMENTARY GEOMETRY.
216
1004
two
(i-)
i^i
^
'"ii'e
inverse points; and, conversely, any
on a circle.
pairs of inverse points lie
The
circle cuts orthogonally the circle of inverand, conversely, every circle cutting anotlier
orthogonally intersects each of its diameters in a pair of
inverse points.
1005
(ii-)
sion (III. 87)
1006
(iii-)
from which
;
The
line
IQ
is
to a given circle
the locus of a point the tangent
is equal to its distance fi^om a
given point D.
—
1007 Def. The line IQ, is called the axis of reflexion for
the two inverse points D, D', because there is another circle
of inversion, the reflexion of the former, to the right of 1(^,
having also D, I)' for inverse points.
1008 The straight hnes drawn, from any point P, within
or without a circle (Figs. 1 and 2), to the extremities of any
passing through the inverse point Q, make equal
chord
angles with the diameter through FQ. Also, the four points
QO QP.
0, A, B, P are concyclic, and QA QB
AB
.
—
=
.
In eitlicr figure OR
OA OQ and OR OB OQ (1000),
by similar triantrles, Z OR A = OAR and ORB = OR A in figure
But OAB = OBA (I. 5), there(1) and the supplement of it in figure (2).
Pkoof.
:
:
:
:
therefore,
fore, &c.
Also, because Z
OR A = OR A,
the four points 0. A, B,
and therefore
(^.4
each case
(III. 21),
1009
The inverse
inversion
is
.
QR = QO QR
.
P
lie
on a
circle in
(III. 35, 3G).
of a circle is a circle, and the centre of
See
the centre of simihtude of the two figures.
also (1087).
—
be the point where the common
PnoOF. In the figure of (l<»lo), let
tangent RT of the two circles, centres A and R, cuts the central axis, and let
any other line through
cut the circles in P, Q R\ Q'. Then, in the demonh\ a constant
OQ OP
stration of (942), it is sliown that 01' OQ,'
quantity.
Tliercfore either circle is the inverse of the other, k being the
=
j'adius of the circle of inversion.
.
=
INVERSION.
1010
luaki'
'r"o
circles.
Rule.
— Take
the taiKjeiits
217
inversions of two
tlio
cii-clos
o-iveii
0(iu:il
the centre of inversion so that the squares of
it to the given circles may he irroportioual to
from
their radii (965).
PKOOK.-(Fig. 1013)
Therefore OT^
:
AT =
AT
Jc^
:
:
= OT
=
07" P, since
07?,
therefore L' A' remains constant
7?7i'
lili,
:
:
OT
if
OR.
OT^ <x AT.
/.:
:
:
1011 lleiiec three circles may be inverted into equal circles,
for the reciuired centre of inversion is the intersection of two
can be drawn by (965).
circles that
The
1012
inverse of a straight line
is
a circle passing
through the centre of inversion.
Proof.
Q\
—Draw OQ perpendicular
and take
line,
P
any
otlicr point
F be the inverse points.
OQ-OQ';
Then
on
to the
Let
it.
OP
0F=
.
by similar triangles,
is
right angle and
is tho
constant, therefore the locus of
circle whose diameter is OC^.
Z
OFQ'
1013
therefore,
=
OQr, a
Example.
;
OQ
F
—The inversion of a poly-
gon produces a figui'e bounded by circular
arcs which intersect in angles equal to tho
corresponding angles of the polygon, the
complete circles intersecting in the centre
l//>'
of inversion.
1014 If tlie extremities of a straight line V'Q' in the last
figure are the inversions of the extremities of l'(,^, tlien
pq pq =
:
v/(op oq)
.
:
^^{0P
FQ FQ = OF
OQ
Proof.—By similar triangles,
OF. Compound these ratios.
:
:
.
oq').
OQ' and
FQ
:
F(2'
=
:
1015 From the above it follows that any homogeneous
equation between the lengths of lines joining pairs of points
PR .QT SfJ, the same
in space, such as ]H} RS TU
points appearing on both sides of the equation, will l)o
true for the figure obtained by joining the corresponding
pairs of inverse points.
=
.
.
.
For the ratio of each side of the equation to the corresponding side of tho
equation for the inverted points will bo the same, namely,
y'iOF.OQ.OU
...)
:
^/{()F .OQ'.Oh'-
...).
.
ELEMENTARY GEOMETRY.
218
Pole and Polar.
—
1016 Defixitiox. The iMar of any point
to a circle is the perpendicular to the diameter
drawn through the inverse point
P with respect
OF (Fig. 1012)
F
1017
It follows that the polar of a point exterior to the
tangents fron the point;
the hne joining their points of contact.
circle is the cliord of contact of the
that
is,
FQ
Also,
is the polar of F with respect to the circle,
In other words, 0iy
centre 0, and FQ is the polar of Q.
point P hfhig on the polar of a point Q', has its oivn polar
alivays passing through Q'.
1018
The line joining any two points P, p
the point of intersection of their polars.
1019
Q',
is
the polar of
—
its
Proof. The point Q' lies on both the lines P'Q', i^'Q'y and therefore has
polar passing through the pole of each line, by the last theorem.
1020 The polars of any two points F,p, and the line joining
the points form a self-reciprocal triangle witli respect to
the circle, the three vertices being the poles of the opposite
sides.
The centre of the circle is evidently the orthocentre
of the triangle (952).
The circle and its centre are called the
polar circle and j^olar centre of the triangle.
If the radii of the polar and circumscribed circles of a
triangle
be r and P, then
ABC
= 4iIV cos A
r^
—
Proof. In Fig. (052),
described round AB(\BOC,
BOO
and
r^
is
the supplement of
vl
= OA OD = OA OB
.
through B, and
.
cos
B
cos
C
the centre of the polar circle, and the circles
COA, A0J3 are all equal; because the angle
&c.
Tlierofore 27i' OD
OB 00 (VI. C)
00 -^ 2/i'. Also, OA 2Zi' cos.l by a diameter
is
;
.
.
=
.
=
(III. 21).
Coa.val Circles.
—
1021 Definition. A system of circles having a counnon
line of centres called the central axis, and a coninion railical
axis, is termed a coaxal system.
1022
If
be
tlie
variable centre of one of the circles,
and
COAXAL CinCLES.
07v
its rjuliuP,
the whole system
is
included in
210
tlie
equation
()l--(>K-= ±8",
where
1023
S is a
111
constant length.
the
first
species (Fig. 1),
OP-OK' =
S\
and S is the length of the tangent from I to any circle of the
system (985). Let a circle, centre I and radius B, cut the
Central axis in D, D'.
When is atZ) or //, tlie circle whose
radius is OK vanishes. "When
is at an infinite distance,
the circle developes into the radical axis itself and into a line
at infinity.
The points D, D' are
1024
called tlie
J
In the second species (Fig. 2),
on-- 01- =
and
imiting points.
half the chord R]i
8-,
common
to all the circles of
the system.
Tliese circles vary between the circle with
centre I and radius S, and the circle with its centre at infinity
as described above.
The points 7?, R' are the common points
of all circles of this system.
The two systems are therefore
distinguished as the timiting jwints sjjccies and the common
2>oints sjxjcics of coaxal cii'cles.
S
is
ELEMENTARY GEOMETRY.
220
1025 There is a conjugate system of circles having R, B! for
limiting points, and D, 1)' for common points, and the circles
of one species intersect all the circles of the conjugate system
of the other species orthogonally (1005).
Thus, in figures (1) and (2), Q is the centre of a circle
of the opposite species intersecting the other circles orthogonally.
1026 In the first species of coaxal circles, the limiting
points D, D' are inverse points for every circle of the system,
the radical axis being the axis of reflexion for the system.
Proof.— (Fig.
OP-P =
OB' =
1)
01)
therefore
.
0K\
OIP,
(II. 13)
therefore D, D' are inverse points (1000).
1027 Also, the points in which any circle of the system
cuts the central axis are inverse points for the circle whose
centre is I and radius S.
[Proof.— Similar to the last.
—
1028 Problem. Given two circles of a coaxal system, to
describe a circle of the same system
(i.) to pass through a
given point; or (ii.) to touch a given circle ; or (iii.) to cut a
given circle orthogonally.
—
1029
I. If the system be of the common points species, then, since the
required circle always passes through two known points, the first and second
cases fall under the Tangencies.
See (91-1).
1030
points,
circle,
To solve the third case, describe a circle through the given common
and through the inverse of either of them with respect to the given
which will then be cut orthogonally, by (1005).
1031
II. If the system be of the limiting points species, the problem is
solved in each case by the aid of a circle of the conjugate system.
Such a
circle always passes through the known limiting points, and may be called a
conjugate circle of the limiting points system. Thus,
1032
To
—
solve case (i.)
Draw a conjugate circle through the given point,
it at that point will be the radius of the required circle.
and the tangent to
—
1033
To solve case (ii.) Draw a conjugate circle through the inverse
of either limiting point with respect to the given circle, which will thus be
cut orthogonally, and the tangent to the cutting circle at either point of
intersection will be the radius of the required circle.
1034
To
solve case
(iii.)
— Draw a conjugate
and the common tangent of the two
1035
cut
circle to touch the given one,
will be the radius of the required circle.
Thus, according as we wish to make a circle of the system loncli, or
the given circle, wo must draw a conjugate cii'cle to ctd
<iii/in</(i)i(iUi/,
orlhiHjvnalli/, ur
iuuch
it.
CENTRES AND AXES OF SIMILITUDE.
1036
drawn
221
be coaxal, the squares of the tanp^cnts
from a point on tlie tliii-d are in
distances of the centre of the third circle from
If three circles
to
any two of
tlieni
the ratio of tlie
the centres of the other two.
PuoOF.
from
— Let A,
a point
P
bo tho centres of the circles PK,
centre (,', to tho other two
C
I),
on tho
;
circle,
dicular on the radical axis.
PA"-
;
By
.
PK'
:
PN
the tanr^onta
tho perpeu-
(080),
= 2AG PN
thereforo
FT
and
PT'
PT'
= AC
:
= 2nG
.
PN,
BG.
Centres and (Lies of similitude.
—
Let 00' be the centres of sirailitudo
1037 Definitions.
(Def 1* 1-7) of the two circles in the figure below, and let any
Then the
cut the circles in r, Q, P\ Q.
line tlirough
OQ OQ' is called the ratio of
constant ratio OP OP'
similitude of the two figures ; and the constant product
is called the i)roduct of anti-si inilitade.
OQ
OP .OQ'
.
:
=
.
=
:
OF
See (942), (1009), and (1043).
or Q, Q' on the samo
The corresponding points P,
are termed liomoltxjoas, and P, (/ or
straight line through
Q, P are termed anti-homologous.
V
any other line Opqp'q be drawn through 0.
any two points i', p on the one figure be joined, and
if P', j/, homologous to P,}) on the other figure, be also joined,
But if the points
the lines so formed are termed liomolotjous.
Avhich are joined on the second figure are anti-liomologous to
those on the first, the two lines are termed anti-homulojous.
Thus, Pq, l/p' are anti-homologous lines.
1038
Tlien,
1039
Lt^'t
if
0, and wliose radius
square root of the product of anti-similitude,
Tlie circle Avhose centre is
e(iual to the
is
is
called the circle of anti-similitude.
The four pairs of homologous chords Pp and Fp'y
Qq and Q'q\ Pq and P'q, Qp and (/p of the two circles in the
And in all similar and similarly situated
figure are parallel.
figures homologous lines are parallel.
1040
Proof.— By
(VI.
'2)
aud the
dciiuitiou (917).
;;
ELEMENTA E Y GEOME TB Y
222
The four
pairs of anti-liomologous cliords, Pj) and
JPq ^^^ QP) Qp ^^^ ^^I'i of the two circles
meet on their radical axis.
Proof.—
OP OQ' = Op Oq = ]c\
1041
^/'Z
'
Q'l
^^^ Pp'i
.
.
wliere k is the constant of inversion therefore P, -p, Q', q' are concyclic
therefore Pp and Q'q' meet ou the radical axis.
Similarly for any other pair
of anti-homologous chords.
;
—
From this and the preceding proposition it
104:2 Cor.
follows that the tangents at homologous points are parallel
and that the tangents at anti-homologous points meet on the
radical axis.
For these tangents are the limiting positions of
homologous or anti-homologous chords.
(IIGO)
1043
two
Let 0,
D
be the inverse points of
^^^th respect to
A and JJ then the constant product of
circles, centres
;
anti-similitude
OF OQ
.
or
JH^
OQ OF = OA OD
.
.
or
OB
.
OC.
CEXTHES
Proof.
AXES OF
SlMU.lTfhi:.
223
— By similar right-angled triangles,
OT OC and OB OH OD;
OA OD = Oli OC
0.1 OG = OT = OP 0(J,
OB OD = Olf' = or .OQ';
OA OB OC OD = OF OQ OL*' OQ',
0-1
therefore
and
AXJ)
:
:
also
(1),
.
.
and
:
:
.
(III.:;.;)
.
.
therefore
.
.
.
.
.
.
therefore &c., by (1).
1044
The foregoing
definitions
and properties
(10.S7
to
have respect to the external centre of simihtudo
0, hold good for the internal centre of simihtiide 0\ with the
usual convention of positive and negative for disitances
measured from 0' upon lines passing through it.
1U4;5), wliicli
1045 Two circles will subtend equal angles at any point on
the circumference of the circle whose diameter is 00\ where
This circle is
are the centres of simihtude (Fig. 1043).
0,
also coaxal with the given circles, and has been called the
O
circle of shnllitude.
—
Proof. Let A, B be the centres, a,
diameter 00'. Then, by (D3"2),
b the radii,
and
K any point on the
circle,
KA KB = AO BO =
:
:
by the definition
(O-io)
AO'
BO'
:
=
a
:
h,
;
a
therefore
:
KA =
h
:
KTI
;
that is, the sines of the halves of the angles in question are equal, which
are iu the ctnstant
pioves the first part. Also, because the tangents from
ratio of the radii a, h, this circle is coaxal with the given ones, by (lUoG, 1)34).
K
1046
The
six centres of similitude
circles lie three
Qpr, Rpq, called axes
of simUitude.
Proof.
—
P,p, Q,
and three on four straight
Taking
any
three of the sets of points
named, say P, q, r, they are
shewn at once to be collinear by the transversal
theorem (*JGs) applied to
the triangle ABC.
For the segments of its
made by the points
sides
P, 'i, r are in the ratios of
the radii of the circles.
q,
R, r of three
lines I'ijli, I'qr,
—
—
ELEMENTARY GEOMETRY.
224
From tlie investigation in (0 12), it appears that one
touches two others in a pair of anti-homologous points,
and that the following rule obtains
1047
circle
:
Rule. — The
right line joining the points of contact imsses
through the external or internal centre of similitude of the tivo
circles according as the contacts are of the same or of different
kinds.
—
1048 Definition. Contact of curves is either internal or external accordiug as the curvatures at the point of contact are in the same or opposite
directions.
1049
Gergo7ine''s
method of descrihing
the circles ichlch touch
three given circles.
Take Pqr, one of the four axes of similitude, and find its poles o, /3, y
with respect to the given circles, centres A, B, G (lOlG). From 0, the
radical centre, draw lines through a, /3, y, cutting the circles in a, a', b, b',
Then a, b, c and a', b', c will be the points of contact of two of the
c, c.
requii'ed circles.
PjiOOP.
E,
F
a, b,
P
c,
Analysis.
touch the
a', b',
c.
— Let the
Let
be, b'c
circles
G
in
meet
in
A, B,
circles
and ab, a'b' in r.
Regarding E and F as touched by
A, B, G in turn, Rule (1047) shews
that Art', bb', cc' meet in 0, the centre
of similitude of jE/ and F and (1041)
shews that P, q, and r lie on the
radical axis of E and F.
Regarding B and G, or G and A,
or A and B, as touched hy E and F
in turn, Rule (1047) shews that P, q, r
are the centres of similitude of ]> and
G, G and A, A and B respectively;
and (1041) shews that
is on the radical axis of each pair, and is therefore
the radical centre of A, B, and G.
Again, becansc the tangents to E and F, at the anti-homologous points
a, a', meet on Bqr, the radical axis of E and F (1042)
therefore the point
;
ca, c'a' in
ci
;
;
;
of meeting
the pole of <(«,' with respect to the circle .1 (1017).
Therefore
through the pole of the line Pifr (1018). Similarly, bb' and cc'
pass through the poles of the same line J'qr with respect to J> and G.
Hence
is
aa' ])as.ses
the construction.
1050
In the given configuration of the circles A, B, C, the
shews that each of the three internal axes of
(leinoiistration
(,>rp, Jlju/ (Fig. 104()) is a radical axis and
connnon chord of tAvo of the eight osculating circles which
can be drawn. The external axis of similitude Vi^li is the
similitude P(/r,
AXirAIDlOXIC JiATIO.
radical axis of the two remaining circles which touch J,
and C eitlier all externally or all internally.
7/,
of the three p^iven circles is also
of similitude of the four paii-s of
Therefore the central axis of each pair
osculating circles.
passes through 0, and is peri)endicular to the radical axis.
passes thi-ough 0, and is })erpenThus, in the figure,
1051 Tlie radical centre
common internal centre
tlie
EF
dicular to Pqr.
Anharmonic Ratio.
1052
Dkfixitiox.
— Let a pencil
of
be cut
four lines through a point
by a transversal in the points A, B,
(', D.
The anharmonic ratio of the
])encil is any one of the three fractions
A B CD
AB.rn
.
or
AD.BC
The
1053
AC.BI)
AC. liD
between these three different
relation
ratios is
obtained from the equation
AB
.
CD-\^AD BC
.
Denoting the terms on the
i-atios may be expressed by
p
The
:
by
left side
p :p + q,
q,
= AC
and
2'
q
.
BJ).
the three anharmonic
</,
:p + q.
are therefore mutually dependent.
merely of the anharmonic ratio in any two sj'stems
immaterial which of the three ratios is selected.
ratios
Hence,
is
if
the identity
to be established,
it is
1054 In future, when the ratio of an anharmonic pencil {0, AHl^'D] is
mentioned, the form All. CI) AD. I!C will be the on<? intended, wliatever
the actual order of the points ,1, 7), <\ D may be. For, it should be observed
that, by making tlie line 01> revolve about 0, the ratio takes in turn eacii of
the forms given above. This ratio is shortly expressed by the notation
{0,AI!('1J}, or simply {AUCD}.
:
1055
(>D, the
unity.
to
AB
:
If the transversal be drawn parallel to one of the lines, for instance
two factors containing 7) become infinite, and their ratio becomes
They may therefore bo omitted. The anharmonic ratio then reduces
liC.
Thus, when
IJ is at infinity,
{0, ^17)'rx}
1056
The anharmonic
.
we may
.47?
:
write
BC.
ratio
CD _
AD.BC
An
=
sin
Aon
siuAOJ)
2g
sinrO/>
y^'niBifC
•
ELEMENTARY GEOMETRY.
226
and
its
value
is
therefore
tlie
same for
all
transversals of the
pencil.
Proof.
—Draw
cular from
substitute
A
i>
upon
.
07? parallel to the transversal, and let p be the perpendiMultiply eacli foctor in the fraction by p. Then
Oli.
OA.OB sin AOB, &c. (707).
AB =
The anharmonic ratio (105(3) becomes harmonic when its value is
See (933). The harmonic relation there defined may also be stated
four points divide a line harmonically when the jjroclud of the extreme
1057
unity.
thus
:
segments
is
equal
to the
proih(ct
<
if
the ivhole line
and
the middle segment.
Homograpliic Systems of Points.
—
If x, a, h, c be the distances of one
1058 Definition.
variable point and three fixed points on a straight line from a
on the same and if x, a\ b', c be the distances of
point
then the variable
;
similar points on another Hne through
points on the two hues will form two homographic si/stems
when they are connected by the anharmonic relation
;
mKQ
J-WO»
(cT'-r/)
(h-c)
_
(.i^-c) {(i-h)
(x-a)
{b'-c)
{.v'-c)
{a-b'y
Expanding, and writing A, B, C,
becomes
D
for the constant coeffi-
cients, the equation
1060
From which
A.t\v'-^B.v-\-av-\-D
^r>«^
C.r-\-D
-,
= 0.
Bd-\-D
'
—
1062 Theorem. Any four arbitrary points .i\, x.,, x^,a\on
one of the lines will have four corresponding points x[, x.>, x'^, x^
on the other determined by the last equation, and the tiro sets
of points ivlll have equal anliarmonic ratios.
Pkoof.
—This may be shown by actual substitution of the value of each x
in terms of
by
a'',
(1
00 1), in the harmonic ratio
|
^vVV^t }
1063 If the distances of four points on a right line from a
upon it, in order, are a, a, /3, /3', where a, li\ a\ (5' are
point
the respective roots of the two quadratic equations
(Lv''-\-2lur+b
= 0,
the condition that the two
aa--\-2h\v+b'
])airs of
points
roii'/iujtitc is
1064
=
may be
;
liarmonicalh/
,
(ib-\-a'b
=
2hh.
IXVOLFTIOy.
PiJOoK.
— The liannonic relation, by
(a -a') (rJ-l'/)
227
(\0o7),
is
=
(a-ly) (<t'-/5).
the sums and produrta of the roots of
Multiply out, and substitute for
quadratics above iu terms of their coefficients by
1065
(.Jl,
tlio
."i'i).
II,
bo the quadratic expressions in (10G3) for two pairs of
represent a third pair harmonically c<iMJiiirate with », and ii.,,
then the pair of points it will also be hai-nionically conjugate with every pair
jjfiven l)y the equation u^ + \ll^
0, where A is any eonsiant.
For the condition (10G4) applied to the last equation will be identically satisfied.
jioiiits,
If
and
»,.
if h
=
Inrolufion.
—
1066
Defixitioxs.
Pairs of inverxi' points 77'', (.>(/, &c., on
the simie right line, form a system in involution, and the relation between them, by (1000), is
OP OF = OQ OQ =
.
.
&e.
F Q
A
^1
I
= k\
n
Q'
P,
^
I
[
Tlie radius of the circle of inversion is k,
i) is
called the centre of the sijstein.
and the centre
Inverse points are also
termed conjiujtite points.
AVhen two inverse points coincide, the point
is
called a
forns.
= h^ shows that there are two foci
from the centre, and on opposite sides
of it, real or imaginary according as any two inverse points
lie on the same side or on opposite sides of the centre.
1067
J
,
/>'
The equation OP-
at the distance h
1068 If the two homogi^aphic systems of points in (1058)
be on the same line, they will constitute a system in involutiun
B = C.
— Equation (lOCO) maj- now be written
when
PiiOOF.
^.r.c'
a constant.
Therefore
—A
-
+ 7/ (a- + «')+/? =
is
0,
the distance of the origin
from the ccntro
of inversion. Measuring from this centre, the equation becomes
representing a system iu involution.
l^'
=
A',
1069 Any four points whaferrr of a system in involution on
a right line have their anharmouic ratio equal to that of their
four conjugates.
—
.
ELEMENTARY GEOMETRY.
228
Proof.
— Let
j/; q, i/;
j:>,
r;
r,
be the distances of the pairs of inverse
s, s
points from the centre.
In the anharmonic ratio of any four of the points, for instance {]'q'i's},
Ic'-^q, &c., and the result is the anharmonic
substitute 2^-=h^-^]j\ q
=
{p'qr's'].
ratio
Any two
1070
inverse points P,
F are in harmonic relation
with the foci A, B.
Pkoof.— Let
pp
^f
=
that
p,
p be
^h
k\ therefore
'
F
B
F
A
the distances of P, P' from the centre
= ——
Jc
,
+ -—~
—
= h+p
k—p
p —Ic
h
p'
o
^,
-thereiore -^
therefore
p
p)
0; then
-
-
k
;
|f = If'
is,
^^^'^
If a system of points in involution he given, as in
(1068), by the equation
1071
and a pair
of
Axic^E{x^-x)-\-B =
conjugate points by the equation
a:i?-\-2hc-\-h
the necessary relation
between
1072
Ah^Ba =
;
—
The roots of equation (2)
therefore substitute in (1)
Proof.
(1)
x+x
1073
Cor.— A system
=
(1);
=
a^ h,
(2);
and
h is
2Hh.
must be simultaneous values of
and
xx'
a
in involution
=—
x,
x in
(51)
Li
may
be determined from two given
pairs of corresponding points.
Let the equations for these points be
Then there
and
ax^ + 2]ix + h =
two conditions (1072),
Ab + Ba = 2Hh and
are
aV" + 2hx
+ h'
xW + Ba =
=
0.
2Uh',
B
can be found,
A geometrical solution is given in (985). C, D C, B' ai'C, in that conin involution
struction, pairs of inverse points, and I is the centre of a system
Each circle intersects the
defined by a series of coaxal circles (1U22).
whose centre
central axis in a pair of inverse points with respect to the circle
from which A,
II,
;
is
and radius
2.
The relations which have been established for a system of coUincar
of
points may be transferred to a system of concurrent lines by the method
(105G), in which the distance between two points corresponds to the sine of
the angle between two lines passing through those points.
1074
oor)
'METHODS OF PnOJECTWX.
The Method of Projection.
1075
DkI'INITIOXS.
— Tlic
pntjictioit
of
any
j)(>iiil
/'
in Sj)aco
point p in wliicli ;i ri^Hit line ()l\ drawn
called the rcrfcr, intersects a fixed plane
from a fixed point
called the phnie of projection.
If all the points of any fio-ure, ])lane or solid, he thus ])rojected, the figure obtained is called the ^injccfiou of the
original fia'ure.
(Fi_i>-.
of
1()7'.>) is tlio
—
1076 Projective Propertie.'i. The projection of a right line is
a right hne. The projections of parallel lines are parallel. The
projections of a curve, and of the tangent at any point of it aro
another curve and the tangent at the corresponding point.
,
The anharmonic ratio of the segments of a right lino
not altered by projection for the line and its projection are
(105G)
but two transversals of the same anharmonic pencil.
1077
is
;
Also, any relation between the segments of a lino
similar to that in (1015), in which each letter occurs in every
term, is a projective propertij.
[Proof as in (1056).
1078
1079
Tlieorem.
— Any quadrilateral PQRS may be projected
into a parallelogram.
CONSTWrCTION.
Produce PQ, SR to
meet in A, and PS,
Qlh to meet in B.
Then, with any
point
project
for vertex,
upon
lateral
plane
quadri-
the
j'xib
any
parallel to
GAB. The
projected
figure jhps will be a
parallelogram.
—
Proof.
The
OPQ, OPS in-
planes
tersect in OA, and
they intersect the
plane of projection
which
is
parallel to
0-1 in the lines pq,
r$. Therefore pq and
rs are parallel to
and therefore
to
OA,
each
Similorlj, j)"?,
qr are parallel to OB.
other.
"'
—
ELEMENTARY GEOMETRY.
230
—
1080
The opposite sides of the parallelogram ji^/rs meet in two
Cor. 1.
and
points at infinity, which are the projections of the points A, R
itself, which is the third diagonal of the complete quadrilateral FQRS, is
projected into a line at infinity.
;
AB
Hence, to project any figure so that a certain line in it may pass to
Take the jilane of lirojcdion 'parallel to the plane which contains the
given line and the vertex.
1081
infinity
—
1082
it is
Cor. 2.
To
only necessary to
make the projection of the
make AOB a right angle.
On
quadrilateral a rectangle,
Perspective Draifing.
1083
Taking the parallelogram pqrs, in (1079), for the original figure,
Suppose this
the quadrilateral PQRS is its projection on the plane ABab.
Let the planes OAB, pah, while
plane to be the plane of the paper.
remaining parallel to each other, be turned respectively about the fixed
In evei'y position of the planes, the lines Oji, Oq, Or,
parallel lines AB, ah.
Os will intersect the dotted lines in the same points P, Q, R, S. When the
planes coincide with that of the paper, pqrs becomes a ground pilan of the
parallelogram, and FQRS is the representation of it in perspective.
AB is then called the horizontal line, ah the picture line, and the plane of
both the picture plane.
To
1084
plan.
Rule.
to pb, to
P,
tJce
find the projection of
—Eraw pb
to
any point p
in the
ground
any point b in the picture line, and draw OB parallel
B Join Op, Bb, and they tvill intersect in
meet the horizontal line in
point required.
In practice, ph
is
drawn perpendicular
The point
cular to AB.
and
the station point.
B
is
to ah,
then called the
and
jjoi;;/
OB
therefore perpendiof sight, or centre of vision,
1085 To find the projection of a point in the grojind plan,
not in the original plane, but at a perpendicular distance c
above it.
— Take
a new picture line parallel to the former, and at a distance
coseca, ivhere a is the angle hctween the original plane a)id the
plane of p)rojccti(»i. For a plane through the given ptnnt, parallel to the
original plane, will intersect the plane of projection in the \w\\ picture line
so constructed.
Thus, every point of a figure in the ground ^dan is transferred to the
Rule.
above
it
=c
drawing.
1086
The whole theory of perspective drawing is virtually included in
the fuicgoing propositions.
The original plane is conmionly horizontal, aud
1, and the height of
the plane of pnjiction vertical. In this case, cosee n
iho. pidure line for any point is equal to the height of the jiuint itself above
the original plane.
The distance BO, when B is the point of sight, may be measured along
A B, and bj) along ah, in the opposite direction} for the lino Bb will continue
to intersect Oj^ in the point I'.
=
.
Orth oii;on a I
— In
1087
I* rojrct io n
Di:riNiTioN.
pi-ojection are parallel to eaeh other,
oi-tli()^n)iial
The vertex
plane of iirojectiou.
dered to he at infinity.
in
])rojectioii
and
tlic
linos
of
|)erj)en(licular to the
this case
may
be consi-
1088 The projections of pai-allel lines are jjaiallel, and the
piDJected segments are in a constant ratio to the oi-iginal
seu'nients.
1089
Areas are
in a
constant ratio to their projections.
For, lines parallel to the intersection of the original pinne and the plane
of projection arc unaltered in length, and lines at right angles to the former
This ratio is the ratio of the areas, and is
are altered in a constant ratio.
the cosine of the angle between the two planes.
Projections of the Sphere.
lu Strreograjyhic projection, the vertex is on the surface of the sphere, and the diameter through the vertex is
])erpendicular to the plane of projection which passes through
the other extremity of the diameter. The projection is therefore the inversion of the surface of the sphere (lUl2),and the
diameter is the constant /r.
1090
1091 Ii^ (Uohular projection, the vertex is taken at a distance from the sphere equal to the radius -i- \/2, and the
diameter thi'ough the vertex is perpendicular to the plane of
I)rojection.
1092
In Gnomon ic projection, which
tion of sun-dials, the vertex
is
is
used in the construc-
at the centre of the sphere.
Mercator's projection \\h\ch is employed in navigation,
in maps of the world, is not a projection at
]\Ieridian circles of the sphere are
all as defined in (1075).
represented on a plane by parallel right lines at intervals
Th(> pai'allels of latieqnal to the intervals on the equatoi-.
tude are represented by right lines jierpendicular to the
meridians, and at increasing intervals, so as to preserve the
actual ratio between the increments of longitude and latitude
1093
,
and sometimes
at every point.
With
r for the radius of the sphere, the distance, on the chart, from the
r log tau (40*^ + JX).
is A, is
equator of a point whose latitude
=
ELEMENTARY GEOMETRY.
232
Additional Theorems.
1094 The sum of the squares of the distances of any point
r from n equidistant points on a circle whose centre is and
=
+
OP'-)
n (r'^
radius 7'
Proof.— Sum the values of FB-, PG\ &c., given
This theorem
is
.
in (819),
and apply
(803).
the generalization of (923).
1095 In the same figure, if P be on the circle, the sum of
the squares of the perpendiculars from P on the radii OP, 0(7,
&c., is equal to ^wr^.
PuooF.
Describe a circle upon the vaclius through P as diameter, and
—
apply the foregoing theorem to this
1096
tween
— The
circle.
of the squares of the intercepts on the radii be(I. 47)
the perpeudiculai's and the centre is also equal to -g-ur.
Cor
].
sum
—
The sum of the squares of the perpendiculars from the
Cor. 2.
equidistant points on the circle to any right line passiug through the centre
is also equal to \nr'^.
Because the perpendiculars from two points on a circle to the diameters
drawn through the points are equal.
1097
—
The sum of the squares of the intercepts on the same
Cor. 3.
right line between the centre of the circle and the perpendiculars is also
1098
equal to
(I-
^nr\
47)
If the radii of the inscribed and circumscribed circles of a
regular polygon of n sides be r, R, and the centre 0; then,
1099
is e(]ual
1100
The sum
I.
to
of the perpendiculars from any point
P
upon the
sides
'iir.
n.
upon any right line, the sum
from
from the vertices upon the same line is equal to nj^.
Ifj) be the perpendicular
of the j)erpendical;irs
P
on the
1102 IV. The sum of the squares of the perpendiculars from the
upon the right line is = n (p' + ^R').
vertices
1101
yides
is
III.
Proof.
r—Or
of the squares of the perpendiculars from
The sum
= n{r-^-\OP-).
— In
cos
theorem
(0+ -^),
I.,
the values of the perpendiculars are given by
with successive integers for m.
values, and apply (803).
Similarly, to prove II.
;
Add
together the u
take for the perpendiculars the values
-'"" \
7>
/n
2>-Rcos[0+-—).
I
To prove III. and IV., take the same expressions for the perpendiculars;
square each value; add the results, and apply (803, 804).
For additional ])ro])ositions in the subjects of this section,
see the section entitled rianc Coonlinaic Ucomctri/.
a
GEOMETRICAL CONICS.
THE SECTIONS OF THE CONE.
1150
AP
Definitions.
—A
Conic Section or Conic
is
the curve
which any plane intersects the surface of a right cone.
A right cone is the soHd generated by the revolution of
one straight line about another which it intersects in a fixed
in
point at a constant angle.
Let the axis of the cone, in Fif^. (1) or Fif^. (2), be in the plane of the
paper, and let the cutting plane
be perpendicular to the paper. {L'ead
either the acce)ited or unaccented letters tJtrouijhout.)
Let a sphere be inscribed
and touching the cutting plane in
in the cone, touching it in the circle
interthe point S, and let the cutting plane and the plane of the circle
sect in XM. The following theorem may be regarded as the dfifmbuj properfij
of the curve of section.
PMXN
EQF
EQF
—
1151 Theorem. The distance of any point P on the conic
from the point S, called the focus, is in a constant ratio to
FM, its distance from the line XM, called the directrix, or
FS
:
FM =
FS'
:
PJ/'
= AS AX =
:
e,
the eccentricity.
\_See next
1152
CoK.
either focus
just proved.
page for
the Proif.']
— The
*S',
conic may be generated in a plane from
N', and either directrix XM, X'M' by the law
,
1153 The conic is an EJIips^e, a Parabola, or an JFi/jjrrhoItt,
according as e is less than, equal to, or greater th.-m unity.
That is, according as the cutting plane emerges on both sides
of the lower cone, or is parallel to a side of the cone, or intersects both the upper and lower cones.
AH sections made by parallel planes are similar; for
inclination of the cutting })lane determines the ratio
1154
tlie
AI-]
:
1155
circle
AX.
—
The limiting forms of the curve are respectively
when e vanishes, and two coincident right lines when
becomes
infinite.
2h
e
GEOMETRICAL CONIC S.
234
—
Proif ok TiiEOKKM 1151. Join P, 8 and P, 0, cutting the ciicular s-ction
and draw FM parallel to NX.. Because all tanjjcnts from the same
or A, to either sphere are equal, therefore HE = I'Q = I'S and
point
Now, by (VI. 2), HE NX = AE AX and NX = PM;
AS.
therefore PS P^f = AS AX, a constant ratio denoted by e and called
in Q,
AE=
:
:
:
:
the eccentric ill/ of the conic.
Referring the letters either to the ellipse or the hyperbola in the
any other point on
subjoined figuie, let C be the middle point of A A' and
Let Djy, h'li be the two circular sections of the cone whose planes pass
it.
the plane of the
with
the intersections
through C and A^; PCP' and
conic* In the elii|)si', J!<' is the common onlimitc of the ellipse and circle
but, in the hyperbola, PC is to be taken equal to ihe langeut from C to the
circle DD'.
N
PN
;
1156
The fundamental equation
FN''
is
Proof.— PiV^
= NP
.
:
of the ellipse or hyperbola
AN. NA' = BC~ AC.
PC = CD CD' (III.
:
NR' and
NR
.
= AN AC
CD
similar triangles (VI. :!, G),
Multiply the last e(juatiuns together.
:
I
and NE'
35, 36).
:
CD'
Also, by
= A'N
:
A'C.
I
.
Cor.
1157
and
ICLLirSE
Tlir:
235
UYl'F.nV.OLA.
1.— P.Ylms
values at two points
e(|ui- distant from AA'
e(|ii;il
Hence
tlie
curve
is
sym-
metrical witli respect to
.-LI and inr.
These two lines are
called the Dinjor and luitwr a.rcs, otherwise the
transverse and conjuijate
axes of the conic.
AVhen the axes are
equal, or BC
AG, the
ellipse becomes a circle,
and the hyperbola beor
comes reef angular
=
./
equilateral.
1158 Any elHpse or hyperbola is the orthogonal projection
of a circle or rectangular hyperbola respectively.
PiioOF.
tlie
—
tliuorem
NP, mcnsnrc NP' = AN NA' tliercfore by
AC. Therefore a circle or rectangular hyperand having its plane inclined to that of the
Aloiifr the ordinate
PN
:
P'N
= PC
.
;
:
having AA' for one axis,
PG-^AC, projects orthogonally into the
conic at an angle whose cosine
See Note to (1-JOl).
ellipse or hyperbola in question, by (108'J).
bola,
=
1159
known
TTence any projective j^roperfi/ (107G-78), which is
to belong to the circle or rectangular hyperbola, will
also be universally true for the ellipse
and hyperbola respec-
tively.
THE ELLIPSE AND HYPERBOLA.
J(Hnf propcrtirs of the Ellipse
1160
Dfi'INITIons.
Dfi'initions.
—The
and
Ili/pcrhola.
to a curve at a point P
adjacent
iiu
im^-Mij^n cm
an cn4|<n.»
Ulcl\>ll through
drawn
PQ,
^»',
tanrjeuf
(Fi
right line
nut;
is the rigllU
110(3) IS
point
(?,
/
in its ultimate position
when Q
is
made
to coincide
with P.
The normal is the pori)endicular to the tangent through
the point of contact.
GEOMETRICAL CONICS.
236
In (Fig. 1171), referred to rectangular axes tlirougli the
is called
the length
centre G (see Coordinate Geometry)
the ordinate; PT the tangent; PG the northe abscissa;
the sulnormal.
S, 8' are
the suhtangcnt ; and
mal;
the foci; XM, X'M'the directrices; PS, PS' the focal distances, and a double ordinate through S the Latus Rectum.
The auxiliary circle (Fig. 1173) is described upon AA' as
;
CN
PN
NG
NT
diameter.
diameter parallel to the tangent at the extremity of
another diameter is termed a conjugate diameter with respect
to the other.
The conjugate hyperhola has BG for its major, and AG for
A
its
minor axis (1157).
1161
The following theorems (1162)
property
PS
to (1181) are deduced from the
obtained in (1151).
The propositions and demonstrations are nearly identical for the ellipse
and the hyperbola, any difference in the application being specified.
1162
TJf
:
PM = e
CS
:
CA
:
CX, and the common
ratio is
e.
THE ELLIPSE AND TIYPERBOLA.
= AC'-nr^
CS- = AC--^BC~
CS-
1164
in tlio ellipse.
[For nS
ill
= AC,
the liyperbola.
[Oy assuming JlC.
BC'=:SL.AC.
SL SX = OS CA,
SL.AC=CS.SX=CS(CX^CS) = CA'^CS'
by (11G3).
Sco
(117(1).
1165
Proof.— (Figs,
.-.
1166
of 11G2)
(1151, 1102)
:
:
= 7?C^
(1104).
P, Q, two points on the conic,
If a right line tlirougli
meets the directrix in Z, then
(11G2)
SZ bisects
QSE.
the angle
.17/
PnooF.—By similar triangles,
therefore by (VI. A.)
1167
If
PZ
ZP
:
Z(2
= MP NQ = SP SQ
:
:
FSZ
be a tangent at P, then
(Uol),
and FS'Z' are
right angles.
Pkoof.
1168
—Make Q coincide
M'iih.
P
in tlic last theorem.
The tangent makes equal angles with the
focal dis-
tances.
Proof.— In (1100), PS PS' = PM P-V (1151) = PZ PZ'; therefore,
when PQ becomes the tangent at P, Z SPZ = S'PZ\ by (1107) and (VI. 7).
1169
The tangents
:
:
:
at the extremities of a focal chord inter-
sect in the directrix.
ZR
then, if
Proof.— (Figs, of 1100). Join
for nir>7) proves h'SZ to bo a right angle.
PKOOF.-(Figs. 1171.)
therefore
ZP
is
a tangent,
ZR
is also,
CN.CT=AC'.
1170
therefore
;
-^^'
TS'+TS
=
(VI.
-pl
_ NX'+NX
3,
A.)
=
2rT
^^^^- (1151)
= ^,
_ 2GX
TS'-TS~ NX'-NX' ^ 2CS
CN.CT- CS.CX = AC^.
'2CN'
(1102)
GEOMETRICAL CONIC S.
238
1171
If
^'^-*'
be the normal,
GS: PS=^
Proof.— By
(11 G8)
and (VI.
GS _GS'
PS
1172
A^G
PS' + PS
j:)/ its
Proof.— (Figs. 1171.)
.
.
otp.i
the tangent,
^,
we obtain
CN^CG _
(11G2)
2GA
to viinus.
NC =
:
PS'=e.
A.),
The subnormal and the
the axes, or
:
_ GS'+GS _2CS
F8'
But, for the hyperbola, change
3,
GS'
abscissa are as the squares of
BC
:
AC\
Exactly as in (1170), taking the normal instead
CN
GG = -~r,
-—
-
CA^^aS^
ON
'
GA'
GN_GX_CA'
7^~ T^S'^
,
.
^yT
.
or^=i^
NO AG^
^^'^
^<^'
....ry^
v-'--'-"-'/'
(11G4).
^
1173 The tangents at P and Q, the corresponding points on
the elUpse and auxihary circle, meet the axis in the same point
But in the hyperbola, the ordinate TQ of the circle being
T.
drawn, the tangent at Q cuts the axis in N.
is
=
CQ"" (11 70)
Then CN. CT
Join TQ.
ellipse
right angi
angle (VI. 8) tlicrclbre QT is a tangent.
a rigut
Proof.
OF.— For the
>,T
fore C(J,T
For the liyperbola
:
;
:
Interchange
N and
T.
;
thcre-
THE ELLTPSE AXD
IIYrEKJiOLA.
239
u
PN QN =
1174
liC: AC.
:
PuooF.— (Figs.
TluTefuro NLf
:
1
1
NG NT =
73).
NC =
and CN NT = QN\ (VI. 8)
by (1172).
(115H), and sliows tliat an ellipse is tlio
.
I'N^
:
QN-;
is equivalent to
projection of a circle equal to
^'oiJ-
.
tl.crcforo,
This proposition
nrt]io<,'onal
1175
I'N',
— The area of the
tiie
auxiliary circle.
ellipse is to that of tlie auxiliaiy
iy^:. 10 (1089).
circle as
PN' AN.NA' = BC AC\
QN^ = AN. NA' (III. 35, 3G).
1176
:
Proof.— By
(
1
:
174), since
1177
Cii
ri:uoF.
rt
_
CT -
— (Figs.
PiV.
Wf
.
'
•
An
The construction
cnt proof of this theorem is given in (11-">G).
tlie hyperbola in (llGl) is thus verified.
indopendfor JJC iu
= nc\
Ct
1173.)
Cn.rt
_
FN'
CNTcT - CNTnT
_PN'
ai.Ct:
Therefore, by (1170) and (117G),
_ FN"- .„. _ ...
^~ AN NA' ^"^^ ^"'
ACP = BC AG^.
.... o.
~W^
'^^^•
.
:
1178 If SY, S'Y' are the perpendiculars on tlie tangent,
then Y, Y' are points on the auxiliary circle, and
SY,SY' = BC\
SY
= TW
in W.
Then FS
(11G8).
Tliereforo
(11G3). ALso, .Sr= YW, and ,S('
r.S".
Therefore
iS
AC. Similarly CY"
AC. Tlierefoie 1', 1" are on the circle.
Hence ZY' is a diameter (III. 31), and tiu'refore SZ
S'Y', by similar
triiintrles
therefore 6Y SZ
SA SA' (111. 3."., '.W^
CS' ^ C.V (11. 5)
IW- (llGi).
Proof.— Let P.S meet
=
S']V=AA'
=
=
W
CY=
=
=
.
;
=
1179
Cuii.— If C'^ be drawn
PE=
then
1180
Tkoblem.
— To
=
.
i)arallel
CY =
t^)
the tangent at J\
AC.
draw tangents from any point
to
an
elhpse or hyperbola.
—
Construction.
(Figs, 1 181.) Describe two circles, one with centre O and
radius OS, and another with centre S' and radius
..LI', intersecting in M,
=
GEOMETRICAL CONICS.
240
jr. Join MS', M'S'. These lines will intersect the curve in P, F', the
For another method see (1204).
points of contact.
S'M by construction. ThereAA'
Pkoof.— By (11G3), PS'±PS
(1. 8), therefore OP is a tangent
PM, therefore Z OPS
fore PS
=
=
= 0PM
=
by
(1168).
Similarly
1181
P'S
= P'M\
and OP'
is
a tangent.
The tangents OP, OP' subtend equal angles
at either
focus.
Proof.— The
by
(I. 8),
angles
as above
OS'M', and
(I. 8).
;
OSP, OSP'
and these
are respectively equal to OMP, OM'F,
by the triangles OS'M,
last angles arc equal,
Similarly at the other focus.
Asymptotic Projyerties of the Ilyperhola.
The asymptotes of the hyperbola are the
1182 Defdiagonals of the rectangle formed by tangents at the vertices
A, A, B, B'.
from any point R on an
1183 If the ordinates RN,
asymptote cut the hyperbola and its conjugate in P, P', P, P*,
—
EM
li
—
;; :
iiYPERnoLA.
t;//;
24.1
then either of ihe following pairs of equations will define both
the branches of each curve
RS--p\' =
Proof.
—
r>r'= /*\--/?.v-
(1),
RM--DM'= Ar'= inP-H)P
{'!).
by
By
prove (1):
Firstly, to
ENC, OAC, wo have
=
-^.^-
proportion from the similar triaugles
-^ =
'
cN'-XG'
= CN'-AG\
^~^A~f-- = 4^.
By
AN. NA'
(llTiV), since
Tberefuro
EN'-FX' =
thereforo
(II. 6)
^J the theorem (G9)
BL'-.
Also, by (117G), applied to the conjugate hyperbola, the axe.s being
"
"
reversed,
^^
''°''^''''
now
*"^"Sles
Tn''
liC'
jyN^-BC'
P'N'-BC = EN' or P'N'-EN^ = BC\
therefore
Secondly, to prove (2) By proportion from tlie triangles EMC, OBC, we
I)}iP
EiP _ Af^ _
^""^^
CM' ~ BC ~ CM'-BC'
by (11 rC), applied to the conjugate hyperbola, for in this case we should
BM MB'' = C^P - BC\
have
:
.
= -^S
^^^,J^
x»C
BC
Therefore
Also, by (117C), since
CM, D'J/
/-(ira
-'r
DM'- AC'
I)'M'-AC-
therefore
= EM'
EM'-BM' = AC\
tl^ereforo
5
are equal to the coordinates of D',
=
—
=
AC
7)7''
or
-r
C^P
-,-r7j
EM-'
-
—
FR.Rr = nC1185
^''">IJ-
2.
—As
tinually diminish.
and
^
RN,
If the same ordinates
CoE. 1.
other asymptote in r and /•', then
1184
^y similar triangles
°
B'M'-EM' = AC.
I)R.Dr
EM
meet the
= AC\
(II. o)
conR recedes from C, /'/i' and
Ilencc the curves continually ap})roach
/>>/?
the asymptote.
1186
If
^E be
Proof.—
1187
I^D
:
CE =
is
:
EN
:
AC,
CS and CS
-
CO.
parallel to the asymptote.
7,'.V*
Therefore
the directrix,
CE CO = CX CA = CA
:
US'- PS'
BC'
PA"
=
PIT
:
DM;
.,,e^.
therefore,
r.V
by (VI.
,
"2).
(11G4)
GEOMETBTCAL CONICS.
242
1188
The segments
of
any
riglit line
Proof,
—
and
1189
Con.
1190
Cor.
QTi
:
Qr
:
between the curve and
Qli = qr.
the asymptote are equal, or
QU =
Qu
=
qB,
:
gZ/'")
qr
:
qu
)
Compound
tlie
and employ
'
l.~PL = PI and QV = qV.
2.—CH = HL. Because PD is
ratios,
(1184).
parallel to 10.
(1187)
1191
QP
.
Vkoof.— Qli
Qr
and
= PL' = RV'-QV'= Q V--R V.
QU=PL: FE Compound theratios. Therefore, by (11 8i),
Qu = El Pe
QR Qr = PL.P1 = PL" (1180).
Qr
:
:
)
and
.
4PH.PK=
1192
P]iGOF.—
3
:
Pn PE =
PK Pe =
:
:
GO
Go
:
On
:
Go)
")
CS\
.-.
PH. PK PE.Pe= CC
=
GS^
:
:
:
Oo^
AEG''; tberefore, by (1184).
Joint Properties of the Ellipse and Hi/perhola resumed.
If PGP' be a diameter, and QV an ordinate parallel to the
conjugate diameter CD (Figs. 1105 and 1188).
1193
QV: PV. VP =
Tliis is tlic fuiidruncntal
most important form of
it.
CD'
:
CP\
equation of tbc conic, equation (1170) being the
:
;
THE ELLirSE AND HYPERBOLA.
243
Otlierwisc
In the
QV: (P-iV = CD-.VPK
ellipse,
Tutliolivpcrbola,
QV'
and
Q
:
F-
:
CV'-CP-= CIX
C F-+ 67'-
Proof. —
=
.
CP^;
CD' CP\
:
—
{EUlps^e. Fig. 1195.)
By orthogonal projection from a circle.
If G, r, P', D, Q, V are the projections of c, p, p, d, q, v on the circle
The proportion is therefore trae in the case of
id cdcp
pv vp
qv'
Therefore &c., by (1088).
the circle.
=
.
(Ilijperhvla.
CTP_
CF'
Fig.
1188)—
_ nv^ _ PL' _ nv'^pu ^ qf'
~ CV- ~ CP'~ CV'±CP' GV'-CP^
qt^
(1101)
CV^+CF''
The parallelogram formed by tangents at the extremiconjngate diameters is of constant area, and therefore,
being perpendicular to CD (Figs. 1195),
1194
ties of
i'i'
Proof.
rF.CD = AC.BC.
— (Ellipse.) —By orthogonal projection from the
{ILiperhola.
fore,
Fig. 1 1 88.)
=
by (VI. 15), ALGl
If
1195
PF intersects
FF.PCm
— GL.Cl = APU PK =
OGo
.
CO
.
circle (1089).
Co (1192)
;
there-
= AG.BG.
the axes in
= EC
and
and G\
FF.FG = AC\
t
1197
Con.—
FG PG = CD = FT. Ft.
.
By
(1194)
GEOMETRICAL CONICS.
244
The diameter
1198
bisects all chords parallel to the tangent
at its extremity.
Proof.
— {Ellipse.
QV=VQ'.
1199
Fig. 1195.)
{Hyperhola.)
CoE.
1.
By
— By
from the
projection
circle
(1088)
(1189.)
— The tangents
at the extremities of
meet on the diameter which bisects
any chord
it.
—
Proof. The secants drawn through the extremities of two parallel chords
meet on the diameter which bisects them (VI. 4), and the tangents are the
limiting positions of the secants
when
the parallel chords coincide.
the
— the tangents from a point are
diameter through the point must be a principal
1201 CoR. — The chords joining any point Q on the curve
1200
Cor.
2.
equal,
If
axis.
(I.
8)
3.
with the extremities of a diameter PP' , are parallel to conjugate diameters, and are called supplemental chords.
FQ
For the diameter bisecting PQ is parallel to
diameter bisecting P'Q is parallel to PQ.
1202
(VI. 2).
Similarly the
Diameters are mutually conjugate If CD be parallel
CP will be parallel to the tangent at D.
;
to the tangent at P,
—
— Observe
Proof.
(EUijJse.
Fig. 1205.)
—By projection from the
circle (1088).
with its ordinates and
tangents be turned about the axis Tt through the angle cos"^ (PC -^ AC), it
ordinates
corresponding
its
with
becomes the projection of the auxiliary circle
]SI"OTE.
that, if the ellipse in the figure
and tangents.
(Hi/perbola. Fig. 1188.)— By (1187, 1189) the tangents at P, D meet the
asymptotes in the same point L. Therefore they ai-e parallel to CD, CP (VI. 2.)
If QT he the tangent at Q, and
to the tangent at any other point P,
the ordinate parallel
CV.CT=CF'.
1203
Pkoof.— CP
QV
bisects
PQ
Therefore, by (VI. 2),
(1199).
Therefore
CV CP =
:
CW
:
CE
PT7 is
= CP
parallel to
:CT.
QP.
N
ELLIPSE AND nYFERBOLA.
Tni'J
1204:
(T
—
Hence, to draw two tangents from a point T, wo may find
above equation, and draw QVl^ parallel to tho tangent at 1' to
Cor.
tVoni the
245
dL'tennine tho points of contact Q, (/.
Let FN, I)N be the ordinates at the extremities of conjugate diameters, and PT the tangent at P. Let the ordinates
and I\ in the clhpse, but at T and C in the hyperbohi,
at
meet the auxiUary circle in p and <l ; then
N
CN=
1205
en =:pN.
(in,
—
Proof.
(Ellipse.) Gp, Cd are parallel to the tangents at d and p (Note to
Therefore pCd is a right angle.
Therefore pNC, CRd are equal
right-angled triangles with CN=^dli and CR
pN.
1'20"2).
=
ON CT = AG'
(Eijperbola.)
.
DR.CT= 2ACDT = 2CDP = AG BG (119-i)
GN DR GR
AG pN ...^,.
= -FN= IN ^^^"^^"^
irR=JfG = FN (^^^^^'
,^:V
and
.
CN
•
(1170),
•
;
.
.
.
.,
,
.
.
,
'"^"°^"^^-
•
•
But
^=^^(VL8);
2)N
1
.-.
GR = pN.
Also Cp
= Gd;
GpN, dCR are equal and similar; therefore
parallel to pN.
angles
1206
Cuij.—
Proof. — (Ellipse.)
triangles,
we have
DR
By
(in
:
(117-1).
= BC
:
and
tri-
dR
is
AC.
(Hijperbola.)
By
the similar right-angled
dR fN = GR TN = DR PN;
:
:
:
dR:DR= pN PN= AG BG
therefore
therefore tho
CN = dR
:
:
(1174).
In the same figures,
1207
{Ellipse.)
1209
{Uiipcrhola.)
Proof.
= A (" /> A'-'+ PS' = BCK
CN'-CR' = AC; DW-PX' = BC\
CX'-j- Cn'
;
— Firstly, from the right-angled triangle CNp in which
^'^V
= CR
(li2<.:.).
=
BC- AC^,
Secondly, In the ellipse, by (1174), PIi'- + PA'' dR' + pN'
and dR- +pN'— A C, by ( 1 205) For the hyperbola, take difference of squares.
:
.
:
GEOMETRICAL COKICS.
246
1211
1212
{Ellip.^e.)
CP'-^CD'
{Ihjperhula.)
CP'-CD'
= AC'+BC\
= AC'-BC\
By (1205—1210) and
Proof.— (Figs. 1205.)
CNP, CRD.
(1.47), applied to
tlie
triangles
The product of the focal distances is equal to the square
of the semi-conjugate diameter, or
PS PS = CD\
1213
.
—
Proof.
=z4^AC'-2CS'-2CP'
=
(922,1)
(Ryperhola.)— Similarly with
2PS PS' = Pg' + PS''-(PS'-PSy =
.
&c.
of the segments of intersecting chords
are in the ratio of the squares of the diameters
them, or
Q0(/, (}'0q
parallel to
OQ.Oq
—
OQ'.Oq
:
=
CD"
:
CD\
By
projection from the circle (1088)
true for the circle, by (111. 35, 3C),
Proof.
is
.
The products
1214
tion
2P8 PS' = (PS + PSy - PS'- PS"
2(AC' + BC'-CP') (1104) = 2CI>-(1211).
Fig. 1171.)
(Ellipse.
(Ellipse.)
Let
(Byperlola. Fig. 1188.)
to Ee, meeting the asymptotes in
parallel
;
= QR.Qr
OR.Or-OQ.Oci
for the propor-
Draw lOi
be any point on Q^.
then
i
I and
;
= PV
(11.5)
(1191)
(1).
PI
OR Or _ PL' _ CD' ,,.on
.Or
^°^ OI = p¥'^^'^ 07=Pi' '-dTOi-pETPe-^C'^^^''^^OQ.Og _ CD'
.
OR.Or-PU CD' _, n^
^,
Therefore
„
PL
on
,
=
^iToTII^O^
^
''^'
'
.
^^^'
^^
0/. OZ-i^'C"^
"
W
Similarly for any other chord Q'O'/ drawn through 0.
OQ.Oq:
Therefore
OQ' Oq
.
=
CD'
Cor.— The tangents from any
1215
CD''.
I
point to the curve are
in the ratio of the diameters parallel to them.
is without the curve and the chords become tangents, each
For, when
product of segments becomes the square of a tangent.
FT
drawn to any
If from any point Q on a tangent
conic (Fig. 1220), two perpendiculars (}R, QL be drawn to the
respectively ; then
focal distance PS and the directrix
1216
XM
SIl
Proof.— Since
:
QL =
:
— By applying
Cor.
of (1181)
is
obtained.
by (VI.
2),
:
:
therefore
c.
QU is parallel to ZS (1107), therefore,
SE PS =QZ:PZ= QL PM;
SR QL = PS PM =
:
e.
the theorem to each of the tangents from Q, a proof
1217
'^'/"'
iJii'crfur Cirrlr.
— The
mi'innioi.A.
17
locus of the point of iiitcris a circle
two tangents always at right angles
the Dlnrfor Circle.
Btcrioii,
called
axd
i:LLirsi:
777/;
7',
of
—
Proof. Perpendiculars from S, S' to tlic tancfents meet them in points
Therefore, by (II. 5, ('>) and
y, Z, y, Z', which lie on the auxiliary circle.
TC ~ AC =
(III. 35, 3G),
Note.
TZ.TZ'
W=
Therefore
—Theorems
(1170),
= SY
.
A(T ± BCT;
(1177), and
S'Y'
= BC^.
(1178)
a constant value.
(1203)
may
also
be
deduced
at once for the ellipse by orthogonal projection from the circle; and, in all
such cases, tlie anahigous projieity of the hyperbola may be obtained by a
similar projection from the rectangular hyperbola if the property has already
been demonstrated for the latter curve.
n
the points A, S (Fig. 1102) be fixed, while C is
to an infinite distance, the conic becomes a parabola.
Hence, any relation which has been established for parts of
1218
moved
the curve which remain finite, when
qrlJJ he a property of the j^arahoht.
AC
thus becomes infinite,
relating to the elli])se may generally bo
by eliminating the (juantities which
become infinite, cmi)loying the ])rinciple thntjlulfe ililj'rreiirea
may be neglected in consideriiKj the ratios of injhiite quantities.
1219
Theorems
ada])ted to the parabola
Example.
in (1213)
— In (1193), when P'
FS' becomes
=
2C'P.
-pF=7t
Therefore
QV
;
\rS .rV
is
at infinity,
IT* becomes
Thus the equations become
""'
'•'
=
in the parabola.
2CP-
2CP: and
GEOMETRTCAL CONICS.
248
THE PARABOLA.
8 be
If
on
tlie
pert ij
and
j^j
P any point
curve, the defining -pro-
is
PS =
1220
and
e
=
PM
\.
(1153)
Hence
1221
AX =
1222
AS.
The Latus Rectum
1223
If
PSZ
=
2 AS.
be a tangent at P, meeting the directrix in Z,
a right angle.
in (1167)
The tangent
Proof.— PZ
/.PSZ = PMZ
1225
(1220)
PZ
is
Proof.— As
1224:
= 4^AS.
SL = SX
Pkoof.—
then
XM
the focus,
tlie directrix,
is
;
tbeorem (1166) applying equally
at
common
P
to the parabola.
8PM, 8ZM.
PMZ PS = PM
bisects the angles
to the triangles
PSZ,
;
Cor.—
ST = SP =
SG.
(1. 20,
1226 The tangents at the extremities of a focal chord
intersect at right angles in the directrix.
Proof.
(ii.)
—
(i.)
They
and
(122;J).
They
6)
PQ
intersect in the directrix, as in (1160).
SZM, SZM'
bisect the angles
(1224), and therefore include a
right angle.
1227
The curve bisects the sub-tangent.
ST = SP (1225) = PM = XN, and
Proof.—
The sub-normal
Pkoof.— ST = SP = SG
1228
is
AN = AT.
^IX
half the latus rectum.
and
TX =
,S'.V
(1227).
=
AS.
KG = 2 AS.
Subtract.
PAUAnoLA.
Tin:
rN'= iAS.AN.
1229
=
Puoor.—FN'
Nd
TX.
The
1230
FT
P
and
(1227)
=
•l-.l.'^
.
AN (1228).
See (1210).
.-IC infinite.
each bisect S}f, the latter
right angles.
Proof.— (i.) The tangent
(ii.)
making
;
taii.cronts at .1
l)isi'eting it at
= AN.2N(}
(VI. 8)
Otherwise, by (117(;) imd (11G5)
/.
10
bisects
SPY = MPY.
SM
by
at A,
(\fl. 2), since
by
at right angles,
SA
SY
1231
C.m.-
1232
To draw tangents from
:
(I.
SP.
:
AX =
4),
AS.
since
[By
PS =
PM
and
similar triangles.
to the parabi^la.
a point
—
CoNSTKUcnoM. Describe a circle, centre
and radius OS, cutting the diiectrix in
M'. Draw MQ, M'Q' parallel to the axis,
meeting the parabola in (2, Q'. Then OQ,
0(/ will be tangents.
M,
=
Proof.— OS, SQ
OQ
OQ'
Oif,
MQ
(1220);
OQS = OQ^f,
therefore, by (I. 8), Z
fore
is a tangent
there-
Similarly
(1221).
a tangent.
is
Otherwise, by (1181). When S' moves
to infinity, the circle MM' becomes the
directrix.
1233
C.R.
1.— The
triangles
SQ
Proof.—
z
:
SQO, SOQ' are
SO
:
SQ'O
=
and
SQ'.
SQO = MQO = SMM' =
Similarly
similar,
SOQ'.
(III.
20)
SOQ.
—
1234 Cor. 2. The tangents at two points subtend equal
angles at the focus and they contain an angle equal to half
the exterior angle between the focal distances of the points.
;
Proof.—
z QOQ'
Also
1235
Def.— Any
z OSQ = OSQ\ by (Cor. 1).
= SOQ + SQO = n-OSQ = '^QSQ'.
line parallel to the axis of a parabola is
called a iViamctcr.
The chord of contact QQ! of tangents from any point
bisected by the diameter through O.
1236
O
is
Proof.
— This proposition and the corollaries are included
by the principle
iu (1218).
Au
independent proof
2 K
is
in
(1108-1200),
aa follows.
GEOMETRICAL CONIC S.
250
The construction being
ZM = ZM'
wo have
;
as in (1232),
therefore
(^
_,
QV^VQ'
(VI. 2).
1237
liW
Coi^
at
P
is
1.
— The
tangent
;
and
diameter
RW.
parallel to Q(2'
OP = PV.
Proof.
— Draw
QW= WP
Q'R' = R'O.
Similarly
1238
the
therefore
2.— Hence,
Cor.
meter through
If
QR = RO
QFbe
(VI.
2).
tlie dia-
P bisects all chords parallel to the
tangent at P.
a semi-chord parallel to the tangent at P,
QV'
1239
= 4^PS.PV.
the fundamental equation of
the parabola, equation (1229) being the
most important form of it.
This
is
—
Let QO meet the axis in T.
similar triangles (1231),
Proof.
By
Z.
SRP = SQR = STQ = FOR
;
and ZSPJ^= OPB (1224). Therefore
PB''= PS .PO = PS .PreindQV=2PR.
Otherwise: See (1219), where th(
equation is deduced from (1193) of the
ellipse.
—
1240 CoE. 1. If V he any other point, either within or
without the cm-ve, on the chord QQ' and iw the corresponding
,
vQ
diameter
.vQ'
= 4/>8 .jyv-
(11.
5)
—
1241 CoK. 2. The focal chord parallel to the diameter
through P, and called the parameter of that diameter, is equal
For PV \n this case is equal to PS.
to 4>ST.
The products of the segments of intersecting chords,
Q'Oq', are in the ratio of the parameters of the diameters
which bisect the chords ; or
1242
(lO<i,
OQ.Oq
Proof.—By
:
OQ'.Oq
(1240), the ratio
is
= PS
equal to
Otherwise: In the ellipse (1214), the ratio
PS when
PS'
PS.
:
4PS .pO
is
=
4P'/S
:
CIP
-
„
.
j^O.
PS PS'
= 1jy.,'
1 o
,,.',
is
at iutinity
(1213)
o
L JJ
and the carve becomes a parabola (1219).
.
>S"
ON CONSTRUCTINO
CONIC.
Till:
251
—
1243 (^oiL The squares of tlie taiip^eiits to a parabola from
any point are as the focal distances of the points of contact.
Proof.— As
ift
(121o).
The area
124:4
Otherwise, by (1233) and (VI. 19).
of the parabola cut off
by any chord Q(/
is
two-tliirds of the circumscribed parallelogram, or of the tri-
angle formed by the chord and the tangents at
(j,
(/.
—
PROor. Througla Q, q, q', &c., adjacent points
on the curve, draw right lines parallel to tlio
diameter and tangent at P. Let the secant Qq
Then, when q coincides
cut the diameter in 0.
with Q, so that Qq becomes a tangent, we have
=z PV (1237).
Therefore the parallelogram
= 'lUq, by (I. 43), applied to the parallelogram
of which 0(^ is the diagonal. Similarly vq = 2iiq',
OP
17y
Therefore the sum of all the evanescent par&c.
is equal to twice
allelograms on one side of
and
the corresponding sum on the other side
these sums are respectively equal to the areas
PQ
;
PQV, PQU.—(NE\\Wii,
Sect.
I.,
Lem.
II.)
Fracfical methods' ofcotisfntcrmg the Conic.
To draw the Ellipse.
1245
Fix two pins at S, S' (Fig. 11G2). Place over them a loop of thread
having a iierimeter SPS' = ,S',s" + .Lr. A pencil point moved so as to keep
the thread stretched will describe the ellipse, by (llGo).
1246
axes in
Othprir;,f>.—(Vig. 1173.)
II,
K.
PK = AC
and
K
Draw
PR = EC
PHK
parallel to
(1174).
slide along the axes,
P will
moves so that the points
II,
1247
To draw the Ilijperbola.
QC, cutting the
PHK
if a ruler
describe the ellipse.
Hence,
the pin S' (Fig. 11G2) serve as a pivot for one end of a bar of any
convenient length. To the free end of the bar attach one end of a thread
whose length is less than that of the bar by A A' and laslen the other end of
the thread to the pin S.
A pencil ])oinL moved so as to keep the thread
stretched, and touching the bar, will describe the hyperbola, by (11G3).
Make
;
1248
Otherwise:
— Lay
(Fig. 1188), starting and
and negative directions.
off any scale of equal parts along both asymptotes
numbering the divisions from C, in both positive
Join every pair of points L, I, the jimdiicf of whose distances from C is
the same, and a series of tangents will be formed (irj2) which will detiue
the hyperbola.
See also (12b9).
GEOMETRICAL CONICS.
252
To drcnv the Parahola.
1249
Pi'oceed as in (1247), with this difference: let the end of tlie bar, before
attached to S', terminate in a " T-square," and be made to slide along the
directrix (Fig. 1220), taking the string and bar of the same length.
—
1250 Otherivise: Make the same construction as in (1248), and join
every pair of points, the algebraic sum of whose distances from the zero
point of division is the same.
—
Peoof. If the two equal tangents from any point T on the axis (Fig.
1239) be cut by a third tangent in the points Ji', r then EQ may be proved
equal to rT, by (1233), proving the triangles SBQ, SrT equal in all respects.
;
1251
Cor.
— The
triangle
SRr
is
always similar to the isosceles triangle
SQT.
To find tlte axes and centre of a given central conic.
Draw a right line through the centres of two parallel chords. This
1252
(i.)
by (1198) ; and two diameters so found will intersect in
the centre of the conic.
(ii.) Describe a circle having for its diameter any diameter PP' of the
Then PQ, P'Q are parallel to
conic, and let the circle cut the curve in Q.
the axes, by (1201) and (III. 31).
line is a diameter,
Given two conjugate diameters, CP, CD, in
and magnitude : to construct the conic.
1253
On CP take PZ = Glf'-^CP measuring
in the ellipse, and towards C in the
;
from C
hyperbola (Fig. 1188). A circle described
through the points C, Z, and having its
centre
on the tangent at P, will cut the
tangent in the points where it is intersected
by the
axes.
—
Proof. Analysis: Let AC, BC cut the
tangent at P in T, t. The circle whose
diameter is Tt will pass through C (III. 31),
and
will
make
CP. P.^ = PT. Pi (III. 35, 3G)
Hence the
= CD' (11 97).
construction.
Circle
and Radius of Curvature.
—
Tlie circle wliicli lias the same tangent
Definitions.
with a curve at
(Fig. 1259), and which passes through
another point Q on the curve, becomes the circle of curvature
when Q ultimately coincides with F; and its radius becomes
the radius of curvature.
1254
P
CIRCLE OF cunvATirnE.
253
—
1255 Otherwise. The curie of rurratitre is the circle which
passes through three coincident points on the curve at /'.
Any chord PIT
1256
elionl
oj'
of the circle of curvature is called a
niri'dfure at P.
Tlirough Q draw PQ' parallel to Plf, meetinpr the
tangent at P in P, and the circle in Q' and draw C^F parallel
to pp.
J?Q i& called a .subtense of the arc PQ.
1257
,
1258
Theorem.
—Any chord
of curvature
PII
is
equal to
PQ
divided by the
the nl ti unite' iml lie uf the square of the arc
parallel to the chord : and this is also equal to
subtense
HQ
Proof.— 7?Q'
becomes
PH;
= RP-^RQ
And when Q
(III. 36).
and RP, PQ, and
QF become
move.s up to P, RQ'
equal because coincidoit Uuci^.
1259 In the ellipse or hyperbola, the semi-chords of curvature at P, measured along the diameter PC, the normal
FF, and the focal distance PS, are respectively equal to
err
err
CP'
PF
err,
AC
'
'
the second bcimr the radius of curvature at P.
Pkoof.— (i.) By
limit
when
T'P'
PU = ^^
PI' = 2l'l'.
vr
(1258),
becomes
.
cr>'
CP'
(ll'J3)
UP
PFC (III. 31), wc Lave
PU.PF = CP PU = 2CD\ by (i.)
(iii.) By the similar triangles PIU, PFE (I1G8), wo have
PI.PE = PU.PF = 2C'D-, by (ii.) and PE = AC (1179).
(ii.)
By
the similar triangles I'lfU,
.
;
tho
—
GEOMETRICAL CONTCS.
254
In tlie parabola, tlio chord of curvature at P (Fig.
1250) drawTi parallel to the axis, and the one drawn through
the focus, are each equal to 4SF, the parameter of the dia-
1260
P
meter at
(1241).
The cliord parallel to the axis =QV^-^PV= 4PS
(1258).
(1239) and the two ehoi'ds arc equal because they make equal angles with
the diameter of the circle of curvature.
Proof.— By
;
1261
Cor.
(Fig. 1220)
—The radius of curvature
is
equal to
Proof.— (Fig. 1259.)
2SP'-
of the parabola at
P
^ SY.
^PU = ^PI sec IPU = 2SP BecPST
(Fig. 1221).
1262 The products of the segments of intersecting chords
are as the squares of the tangents parallel to them (1214-15),
(1242-43).
1263 The common chords of a circle and conic (Fig. 12G4)
are equally inclined to the axis and conversely, if two chords
of a conic are equally inclined to the axis, their extremities
are concyclic.
;
Proof.
—The
products of the segments of the chords being equal (HI.
them are equal (1262). Therefore, by (1200).
35, 36), the tangents parallel to
1264
The common chord
of
any conic and
of the circle of
curvature at a point P, has the
same inclination to the axis as
the tangent at P.
Proof.
—Draw any chord
Qq
parallel
The circle circumto the tangent at P.
Bcribing PQq always passes through the
same pointy (1263), and does so, therefore,
when Qq moves up
circle
becomes the
1265
to P,
and the
circle of curvature.
PrvOBLEM.
To find the centre of curvature at any given
^oint of a conic.
—
Draw a chord from the point making the
(Fig. 1261.)
First Method.
same angle with the axis as the tangent. The perpendicular bisector of the
chord will meet the normal in the centre of curvature, by (1264) and (III. 3).
Second Method.— Draw the normal PG and a perpendicular to it
meeting either of the focal distances in Q. Then a perpendicular to
the focal distance drawn from Q will meet the normal in 0, the centre of
1266
from
il,
curvature.
MISCELLANEOUS THEOREMS.
By
Vroov.— (Ellipse or Hyperbola.)
the radius of curvature at V
=
l^F
For
For
(1259),
= ^$ ^^ ^^^''^^ = TF^ ^^
= PG sec' S'PG = PO.
AC = PE (by 1170).
(Parabola.)
2SP
=
By
(rJOl).
255
^^^•'^•>
The radius of curvafn
= ^SP'^SY = 2SP sec -S'PG = PO.
because SP = SG (11-25).
PQ,
Miscellaneous Theorems.
1267
lu the Parabola (Fig. 1239)
QL^
tlien
let
QD
be drawn perpendicular to
= 4A.S.PV.
PV,
(1231,1239)
1268
Let liPIt be any third tangent meeting the tangents OQ, OQ' in
SQO, SPR', SOQ,' are similar and similarly divided by
SR, SP, SR' (1233-4).
Ji',
ii'';
the triangles
1269
1270
With
OR.On;
Cor.—
Also, the triangle
PQQ;
= RQ.R'Q'.
= 20RR'.
OQ: OQ'=^RQ.RP' r:Q'.PR.
1271
1272
1273
Also the angle
1274
In any conic (Figs. 1171),
(1215,1243)
:
RSR'
=
f|
Cor.— If
bisects the angle
=
^^
f
= "^
the iangent
(VI. 3).
SP
(69,
PT
(1181)
IQSQ'.
Hence, in the Parabola, the points 0, R,
Proof—
1275
(12-il)
the same construction and for any conic,
\
S,
AN AT.
= |^ (1170) = j^
ST =
11G3)
E are concyclic, by (1234),
:
meets the tangent at
A
in R, then
SR
PST
1276 In Figs. (1178), SY', S'Y both bisect the normal PO.
1277 The peipendicular from S to PG meets it in CY.
1278 If Ci) bo the radius conjugate to CP, the ptrpcndicular
upMii CY is equal to PC.
1279 SY and CP intersect in the directrix.
1280
(09).
PN
from
D
If every ordinate
of the conic (Figs. 1205) be turned round N,
the plane of the figure, through the same angle PNP\ the locus of P' is
also a conic;, by (ll'J3).
The auxiliary circle then becomes an ellipse, of
which AC and PC produced arc the equi-conjugatc diameters.
If the entire figures be tlius deformed, the points on the axis AA' remain
fixed while PN, IJR describe the same angle.
Hence CP remains parallel to
PT. CP, CD are therefore still conjugate to each other.
in
GEOMETRICAL CONICS.
256
Hence,
tlic
relations in (1205-G)
universally,
jugate radii.
Thus
1281
1282
FN: GB = DB: CN
then
P meets
If the tangent at
FT. FT'
is
when CA, CB
subsist
still
are any con-
FN. CN = DB CB.
or
.
any pair of conjugate diameters
constant and equal to
in T, T',
CD\
—
Proof. Let CA, CB (Figs. 1205) be the conjugate radii, the figures
being deformed through any angle. By similar triangles,
FT CN = CD
:
•
^'B ]
^^^refore
'
FT. FT'
Therefore
1283
PT. Pr'
=
FN. CN
:
=
CD^
:
DB
.
CB.
Clf, by (1281).
P
meets any pair of parallel tangents in T, T,
= CI)\ where CB is conjugate to CF.
thou FT.
Peoof. Let the parallel tangents touch in the points Q, Q'. Join PQ,
PQ', CT, Cr. Then CT, CT' are conjugate diameters (1199, 1201). Therefore FT.FT'
CB' (1282).
If the tangent at
FT
—
=
1284
Cor.— ar. Q'r
= CB\
where CFf
is
the radius parallel to QT.
1285
To draw two conjugate diameters of a conic to include a given
Proceed as in (1252 ii.), making FF' in this case the chord of the
segment of a circle containing the given angle (III. 3B).
angle.
1286 The focal distance of a point P on any conic is equal to the length
QN intercepted ou the ordinate through P between the axis and the tangent
at the extremity of the latus rectum.
Proof.— (Fig.
1220).
QN NX = LS SX =
:
e
:
and
SF
:
NX =
e.
In the hyperbola (Fig. 1183). CO CA = e.
(11G2, 1164).
If a right Hue FKK' be drawn parallel to the asymptote CB, cutting the
one directrix
in
and the other in K' then
1287
:
XE
SF =
1288
Proof.— From
1289
Cor.
K
;
PK = e. CN-AC;
CB = e CN (1287)
S'P = FK'
and GE
= e.CN+AG.
= AC (1186).
.
— Hence
the hyperbola
may
be drawn mechanically by the
method
of (1249) by merely tixing the cross-piece of the T-square at an
angle with the bar equal to JJCO.
1290
1291
make
Definition.
— Confocal
The tangents drawn
conies are conies which have the same
to
any conic from a point
T on a confocal
foci.
conic
equal angles with the tangent at T.
Proof.
— (Fig. 1217.)
Let
T
be the point on the confocal conic.
SY SZ =
S'Z'
S'Y' (1178).
Therefore ST and S'T make equal angles with the tangents TF, TQ and
they also make equal angles with the tangent to the confocal at T (1168),
:
:
;
therefore &c.
1292
In (ho construction of (1253),
cui-vature at I' di-awn thi'ough the centre
FZ
C
is
equal to half the chord of
(]25;»).
DIFFERENTIAL CALCULUS.
IXTRODUCTIOX.
1400
Ff()irfiou.<i.
— A quantity
log.v,
n-^,
a^-{-ax
+
wliicli
depends for its value
Thus, sin ,r,
called afiinctiou of
x' are all functions of x.
upon another quantity x
is
y =/{,!') expresses generally that y
is a particular function.
is
</'.
The notation
a function of
./.
y
= sin«
called a confinuoiis function between assigned
indefinitely small change in the value of x
always produces an indefinitely small change in the value of
1401 /CO
limits,
is
when an
/OO-
A
is
not purd^-
algebraical, such as the exponential, logarithmic,
and circular
transcendental function
functions
If /(.c)
If /(,,)
a"",
is
one which
&c.
the function is called an even function.
log.r, sin.'B, cos,/',
=/( — a?),
— — /( —
ir),
it is
called
an odd function.
Thus, ./- and cos j- are even functions, while a;' and sin x are odd functions
of x; the latter, but not the former, being altered in value l)y changing the
sign of X.
1402 Diffrrcniinl Coefficient or Der'wnth'e.—Lct y be any
function of x denoted by /(-r), such that any change in the
value of X causes a definite change in the value of // ; then x
is called the Independent variable, and // the depend mt rarialde.
Let an indefinitely small change in x, denoted by dx, produce
a corresponding small change
'///
in//;
then the ratio
--•
,
in
the limit when both dy and dx are vanishing, is called the
derivatin', of y with respect
dlffrre)itlal
coefficient,' or
to X.
2 L
—
'
mFFERENTTAL CALCULUS.
258
1403
of
TiiEOEEM.
— The
and generally
<?',
ratio
c///
dr
:
is
definite for eacli valuo
different for different values.
—
ON
Pkoof.
Let an abscissa
(Defs. 1160) be measured from
equal to .r, and a pcrpendicalar ordinate
equal to y. Then, wliat-
NP
may
ever
he the
form of the function
y =f(x), as X varies, the locus of
P will be some line PQTj. Let
//' be values of x
x',
and 7/ near to the former values.
Let the straight line QP meet the
axis in T; and when Q coincides with P,
MQ =
OM =
.
let
the final direction of
QP
cut the
SP
vanish,
axis in T'.
Then
II
they vanish in the
definite ratio at
Let
1404
when h
= |^.
ratio of PN
And, ultimately, when
or --^f^
:
NT'.
Therefore
new
and
,
a
each point of the curve, but different at different points.
NM,
the increment of
x,
be denoted by
^ = /C^+ZQh -/(.Q ^
vanishes,
iLv
a
QS
= =— = tan PT'N,
dx
NT
'-^
^,
»/
li
;
then,
..
\
/'
function of. a', called also the first derived function.
of finding its value is called differentiation.
The process
1405
Successive differentiation.
—
If
^^
or/'(<6')
be
differ-
dcC
entiated with respect to x, the result is the second differential
coefficient oif(a'), or the second derived function ; and so on
to any number of differentiations. These successive functions
may
be represented in any of the three following systems of
notation
:
(Ijl
(Pjl
cJ^
(l^
d.v
da''
fAr^'
dd''
/G^O, /"CO,
y^^
/'"Ci"),
Vtx,
/^(.^O,
i/s^;
//la-,
iV^
^/>
/"GO;
//«...*
The operations
of differentiating a function of x once,
twice, or n times, are also indicated by prefixing the symbols
or,
jL
jH
da?'
d.i^'
more
d
j!;;_
•"
concisely,
d.e^'
"'^
d,, d.^,
d,v'
...
(dv
W7' d„^.
* See note to (1487).
(dy
\dr)
'
TXTnoDrr'Tinx.
1406
nj'trr (liffoivutiatin^
It',
zero in the result,
tlio valiio
following Avoys
-j'~,
:
f'W^
function for
x
.r,
may bo
indicated by.
v/.,.^
-7—
//.ro>
([uantities dx^ thj are called
and
//,
niado
^^.o-
?
in
?/^,
the
.
Infniteslmals and Differentials.
1407
])e
bo indicated in any of tho
any other constant a be substituted for x
If
result
to X
;i
may
2't\}
—
eviinescent
Tlio
and, with respect
<lr, (I'-y are the second
iujinito^linah
;
they are called diffcrentiah.
and ij ; dx^, d^y tho third, and so on.
differentials of x
The successive
1408
,ty
differentials of
by the equations
=/' (.r) fhv d'\f/ =f"{.v) dv'
of d
>j
are expressed in terms
I'
;
and
&c.,
;
=/"(.r) d,v\
d'\,/
Since /"(<r) is the coefhcient of dx in tlie value of di/, it lias
therefore been named the differential coefficient of y or /(.'0-*
For similar reasons /' {x) is called the second, and /" {x) tho
n^^ differential coefficient of f{x), &c.
Two infinitesimals aro of the same order when their
neither zero nor infinity.
If dx, dij are infinitesimals of the same order, dx', dy^^, and
dxdy will bo infinitesimals of the second order with respect to
dx, dy; clx^, dx'dy, &c. ^vill be of the third order, and so on.
1409
ratio
is
dx, dx^, &c. are
sometimes denoted by
x, x,
&c.
—
1410 Lemma. In estimating the ratio of two quantities, any
increment of cither which is infinitely small in eonji)arison
with the quantities may be neglected.
Hence the ratio of two infinitesimals of the same order is
not affected by adding to or subtracting from either of them
an infinitesimal of a higher order.
Example.
—
,
the ratio ^-[K
dx
by
^~
=^
=
—dx -^, for dx is zero
dx
ax
dx
dx,
Thus, in Fig. (1403), putting P*S
=
QR =
FN _ Wl _(hi-hd^_ chi_
ultimately,
(12.'')8),
Jcdx^,
.^^
NT'
I'S
QR
dx
is,
A-
is
^j^^ j.^^^.^^ ^^^ ^j^^
comparison with
QS
= d>j
a constant.
;
we have
Therefore
principle just enunciated;
dx
vanishes in eonqmrison with
selves are iufinitehj small.
that
where
in
PS
* The name is slightly misleading, as
some sense a coefficient of /'(•'")•
it
or
QS
even loh-n those lives them-
seems to imply that
/'(•'')
is
in
/
.
DIFFERENTIAL CALCULUS.
260
DIFFERENTIATION.
DIFFERENTIATION OF A SUM, PRODUCT, AND QUOTIENT.
Let
be functions of
u, V
x,
then
du
dv
d(uv)
+w ^—
= V -^
-T—
ax
d.v
ax
du _ _^\
/ u \ _ (
tAtn
1412
,
(/
1413
Proof.
—
(i.)
and, by (1410),
=
+ du)
(u
1
d
(ill-)
/
u\
[
\
Hence,
(ii
by (1410)
if
=
+ dH + v + do) — (u + v) =
(v
+ dv)—uv
;
udv
r—
= vdu——dv)v
u
;
V
{y
+
,
vdv vanishiBg in comparison with
u be a constant
!iM=o4l
dx
1414
du + dy.
= vda+udv — diidv,
in the ultimate ratio to dx.
u + du
v-{-dv
/
i;
2
_:_
dx J
dx
\
dudv disappears
,... .
therefore &c.,
=
d(n + v)
d(Hv)
(ii.)
V /
\
</ct'
=
c,
^l£)
= -^^.
dx
dx \v
and
dx
v^.
L-
DIFFERENTIATION OF A FUNCTION OF A FUNCTION.
If
7/
be a function of
z,
and
z
a function of
dy__dti_
dx
dz
1415
x,
dz_
dx
'
—
Proof. Since, in all cases, the change dx causes the change dz, and the
change dz causes the change dj thcrefoi'e the change dx causes the change
dy in the limit.
Differentiating the above as a product, by (1412), the successive differ;
ential coefBeients of y can be formed.
the Rake of reference.
Observe that
1416
1417
1418
1419
The
(^j)j:
first
four are here subjoined for
= l/a^x'
= y.z^.
= //..^:.H-//.^2...
=
+ ;{//..-.c^...+//.-Uuu- = Uu^l + %..4;^.c +
(»5^L +
7A
//..,
//:,.
//:,,-:.
//..
A^..-...)
+ U^-^^'
:
'
DIFFEBENTTATION.
2 CI
DIFFERENTIATION OF A COMFOSITE FUNCTION.
If
n and v be
explicit functions of
^^"^^
TT^T dx'
dTv
.i',
so tluit a
"^
=
(^
and
(./•)
iLv
(//'
Here f77^ in tlic first term on tlio riglit is the cliangc in
F{u,v) produced by da, the change in u; and dF in the
second term is the change produced by dv, so that the total
change dF{i(j
v)
may be
dF,
written as in (1408)
+ dF,= ^dii
+ ^du,
do
die
DIFFERENTIATION OF TEE SIMPLE FUNCTIONS.
Since
-^ ^ /('^ + ^^)~/W ^hen
dx
h vanishes, we have tho
lb
following rule for finding
its
value
—
1421 Rule. Expand f(x+h) hy some hiown theorem in
dliido hi/ h; and
ascending poivers of h; subtract f (x)
tlic result imt h equal to zero.
The differential coefficients which follow are obtained by
the rule and the theorems indicated.
m
;
1422
PKOOF.-IIero
^=
^i'"-
"^^""'•
=
-^
•
(12o)
1423
=
y
1
1^
= nx"-^-]-C(n,
2) x"-'h+...
=
)ix"-\
Cou.-
^=k-
i^=«(»-l)...(M-r+l),,--'.
1424
z/
PKOOP.-By
J
Expand
,.„^
1425
wliou h vauishcs.
7^ = ,7k^-
= l<'g..'-;
(145).
^
"
'"''-^-
+ ;-^-'"'^-' =
It
^
X lug,
at\
log. (l
\
+ {)
X I
-^
I
}
7X
the logarithm by (15-5).
,,
U...-
-^=
il-,/
(-1)"-'1«-1
-
a.'\oga"~
rat« = -lin(l.i23)
andr = ,.-l.
262:
1426
DIFFERENTIAL CALCULUS.
—
;
.
sucCESSivr: inFVKRv.xriATjns.
Sonio ilifTeivniiations nro rciulorcd
For example,
by
cnsioi'
2r.:]
lo^'arithm of
liikinj,' tlio
tlio
function.
1445
y
=
^/ J~l.
1
.
P
tlierctoro
,--
r
thcrofore
X,
1446
_
— —1
'''/
-2.«
y.
f?!/
(Zo!
1-x*
=
(si"
7
^y
log
_
=
77-;
;
.
jT
>
-2.rr2-x'')
r-
:,
(H-a;^)»(l-x-)*
= » log sin x
?/
— = log sin a:+ -^
cos
siu «
thereforo
i\-x-)-l log(l + x')
1x
^
theroforo log
;
\
3
.rr
-2.>-(2-3'-)
-/-=:?/
2/
=
tbcrcforo logy
;
V*
;
(1 il 5, '24, '28)
a;
•
1/
therefore
1/,
=
(sin
Otherwise, by (1420), y,
(log siu aj+a; cot x).
a-)'
= x (sin a-)'"' cos + (sin
= (siu
cot + log sin
a;
a;)'
a;
.r)"" (.7;
log sin x
(1420)
a;)
SUCCESSIVE DIFFERENTIATION.
Leibnitz s Theorem.
1460
...
PnooF.— By Induction
C
(«,
'•)
—
If
n be any integer,
+C(«,
»•)//(„_,),;:;,,+
...
4-.V^«x.
Differentiate the two consecutive terras
(233).
i/(,.-r)x2«+
C
(?l,
r
+ 1)
2/(,.-r-l)x^(r.I)x,
and four terms are obtained, the second and third of which are
=
This
C{n, r) ?/(,._„, «(r.nx+C(?i, r + l) .'/(,.-.) x ^/r.i)*
[C{ih r)-^G{n, r+l)}y(,.-r)x2(r.i)-= C{n + l, r+l) ,m-7n
?/
x
2 ..1,,^
Similarly,
the general term of the series with n increased by unity.
all the terms the whole series is reproduced with n in-
is
by differentiating
creased by unity.
DIFFERENTIAL COEFFICIENTS OF TUE
1461
(s»W «'0«x
1462
(cos
1463
1464
(U')„,.
=
=
«" siu {(LV-\-lmr).
where, in the expansion
be rephiced l)y
//,..c.
=
l)y
By
a" cos {fLv-^UTr).
{('"''),..
{ff\f/)„..
n"' 01^ DEE.
=
Indnotion
^"^'^
^^•^-'^•
(1^'^G)
(i"t''-''
c-'-{a
+
(r,.y',/,
the Binomial Theorem,
d'^i/
is
to
(1100, '03)
DIFFERENTIAL CALCULUS.
264
1465
cos
{e""
= r cos
a
wliere
Proof.
—By luduction.
r"e'"'
71
is
= rV'^ cos (6.r + w<^),
and 6 = r siu
(j).
<^
we
Differentiating once more,
obtain
—
h sin (bx + n(p)}
{a cos (bx-\-n(p)
__ yn + igO'lcos^ cos (hx + n^)
—
Thus
6cr)„^
/''len'cos (tx
— Bm<{>
sin {hx
+ n<}>)}
+ n + l^).
increased by one.
1466
log ^)n.
i^'"'-'
(tan-i
1468
=
liLzJ.
^)nx={- 1)""' 1^-1
^ == cot"^
where
Proof.
—By
-^
siu**
^ sin n0,
a.\
Differentiating again,
Induction.
(1-^60), (283)
^^'.
we
obtain (omitting the
coefficient)
(>i
Since,
cos 9 sin nd + n cos nd sin" 6) 6^
sin""'
= w sin""^0 (sin«0 cos0 + cos?i0 sin0) ( — sin-0).
= -(l + x')-' = -sin^O.
= (—1)" l«_sin""' d sin (n + l) d,
by (1437),
0,
(tan-^
Therefore
«),,..!) «
n being increased by one.
^
1469
(
1470
(j^)
\l-rct'
,,
=
)
(n
+ 1) 6-
=(-l)"|;^sin'^^^^cos(>i
+ l)^.
(-1)'^ I^sin'^^i ^
sill
(1-^36,
1468)
//jj;
PuoOF.-By
Then by
(14G0),
(i|^)„^=
^
(xi:.)^^" (r^O.-./
(1469).
Jacohi's Formula.
1471
c/(.-a).
(1-^r)"'^
Pkoof.
—Let
(^"- i)„.
Expand
)j
=
=
(-1)'*"' 1
= l—x^;
-(2h+1)
eacli of these vahics
•
3
...
12/1-1) sin {n cos-\r) -h
«.
therefore
(.r7/"-i)(„.„x.
Also
(//"-i),.
by (1460) and eliminate
=
(.'///"-•O-.x.
(y'"^),„-2}x, the^dcriva-
(M7U
true
Now assume
Call the result equation (1).
Diilerentiate and substitute the result, and also (1471) on
the right side of equation (I) to obtain a proof by Induction.
tivc of lowest order.
for the value n.
OF OriJRATlOXS.
TIinOPiY
1472
Tlii'orcm.
—
If y,
are finictions of
::
2(15
.r,
and
;/
a positive
iiiri\u-('t',
= (//^),.r-"
PnooK. — By
~//«.r
('/^..•)(«-l).r+^' («»
DilTcrontiato for
Tiulnction.
rip;ht its viiluo
by
tlio
i^) (//^2.r)(«-
formula
2) .r
•••+(- 1 )" //--..r-
substituting for
a;,
on
z^y,,^
tlio
itsolf.
PARTIAL DIFFERENTIATION.
''
=/X'^'» ?/) ^^"^ ^ function of two imloppiulinit varidifforcntiation of n with respect to x rrr/irlres tJuif ij
should be considered constant in that operation, and rice versa.
1480
If
ables,
any
Thus, -'— or
be differentiated succes-
Wjx signifies that n is to
sively twice with respect to x^ y being considered constant.
1481
Tlie notation
^
.,
or
7/2^3^*
signifies that
u
is
to bo
successively twice for x, y being considered
constant, and the result three times successively for y, .c being
considered constant.
differentiated
1482
'I'lio
order of the differentiations does not -affect tho
final result, or
=
7/^,^
Vy,..
Froof.— Let u=f(xy); then
"-
=
,h>^
=• f (x + Ik
-.ij
7/
'
4-
/.-)
y^ =fJ^:±]i:^!lzdL(j^J!l in limit.
—
-
/'
(> •. 7/4.
/,•)-/
f.r
+ /^
y)-^f(x,y)
'
—hir^
.
(1181)
, .
if Vy bad boon first formed, and then iiy^, tbe same result would
been obtained. Tlio proof is easily extended. Let u^-=.v;
and so on.
v^y = v^,j
i\^
then
iivis
Now,
=
=
.
'" •'"''*•
liavo
;
THEORY OF OPERATIONS.
1483 Let the symbols <^, ^, prefixed to a quantity, «lenoto
operations upon it of the same class, such as nudtiplication or
differentiation.
Then the law of the operation is said to be
distrUjutice,
when
<i>(.r+//)
=
-i>GO+^K//);
* Sec note to (1487).
2
.M
DIFFEnEXriAL CALCULUS.
266
that is, tlie operation raay be performed upon an undivided
quantity, or it may be distributed by being performed upon
parts of the quantity separately with the same result,
1484
The law
that
the order of operation
is,
is
when
said to be commutative
upon ^x producing the same
may be changed, ^
result as
^
operating
operating upon <^x.
m
^""^ denotes the repetition of the operation 4>
times,
operations. This definition
equivalent to <t>* ... a? to
involves the index law,
1485
and
m
is
which merelv asserts, that to perform the operation n times in
succession upon x, and afterwards m times in succession upon
the result, is equivalent to performing it m-\-n times in succession upon X.
1486 The three laws of Distribution, Commutation, and the
law of Indices apply to the operation of multiplication, and
Therefore any
also to that of differentiation (1411, '12).
algebraic transformation which proceeds at every step by one
or more of these laws only, has a valid result when for the
operation 'of muUijjlication that of differentiation
is
sub-
stituted.
1487
111
making use
employed
ferentiation
of this principle, the
or simply
—r-,
is
respect to x
is
prefixed to the
d^
d^
•^
dx^
rt.r
repetition of the opera-
d^
'-,
-,
differentiating with
of
The
to be pei-formed.
indicated by -7^, —r^,
is
dif-
ux
quantity upon which the operation
tion
r/,,
symbol of
^
,
&c., prefixed to the
dx^dif
An
abbreviated notation is d^., d.y^^, d.^, (/o.,.,,^, &g.
in the symbolic operation of multiplication,
it will be requisite, in transferring the operation to differentiation, to change all such i)idiccs to suffixes when the abbreviated
notation is being used.
function.
Since
d^Xd^
Note.
ever, the
•writing,
= dl
—The notation
17,,
y^,
recommendations of
and of space
w^^a^,
in printing.
fourteen distinct types, while
cZa^rSy,
&c.
is
an innovation. It lias, howand economy of time in
defiuiteness, simplicity,
its
The expression
equivalent
if;,^.;,^
.^
requires at least
requires but seven.
For
THEOIiY OF OPEHATloXS.
2Ci7
I have introduced tho shorLur notation experimonUiUy in tliuse
pages.
„
All such abbreviated forms ofdiffbrential coefiicienta as y'y"y"'... or y // //...,
thou<?h convenient in practice, are iucoaiplete expressions, because the independent variable is not specified.
The operation (f^^iv^ n^d tho derived function »i^s^, would bo more
accurately represented by (./;)J and ("l)], the index a.s usual indicatinj^ tho
JJnt the former notation is simpler, and it has
repetition of the operation.
the advantage of separating more clearly the inilex of dillerentiation from tho
such reusous
,
,
index of involution.
In the symbols
index in each case: in tho
?/' and y^ji t'lo figure 2 is an
shows the degree of involution in tho second, the onfrr of differentia-
first, it
;
The index
tion.
write
)/
and
is
omitted
when
the degree or the order
is
unity, since
wo
//,.
The sTilUx takes precedence of the superfix. yl means the square of y,.
djc(,y*) would be written (//), in this notation.
As a concise nomenclature for all fundamental opoi-ations is of great
assistance in practice, the following
"V
/or *," as
recommended
is
an abbreviation of the phrase, "the
71
with respect
for, or
forms
y,"
j/ix,
and so
"xy, «ix3v,
Similarly,
to, .t."
niay be read
''
:
J^ or
may
be read
(/
7S
)2
-^, -j—, TTTl'
y for two x,"
y,
ditfereutial coefficient of
''ii
°^
^^^®
shortened
for xy," ^^u for two
x, three
forth.
The distinction in meaning between the two forms y,,^ and y„ is obvious.
The first (in which n is numerical and ahoays an inleijer) indicates n successive diflerentiations for x; the second indicates
for the variables x and z.
The symbols -^^ or
y^,
and
-
'{-
two successive
or y^^,
may
ditierentiations
be read, for shortness,
"y for X zero," "y for two x zero"; d.^^^^ (^ {xy) can be read "fZ for two x
(xy)."
three y of
Although the notation r, is already employed in a tot.illy dilferent sense
in the Calculus of Finite Ditferences, my own experience is that tho double
signification of the symbol does not lead to any confusion: and this for the
very reason that the two meanings are so entirely distinct. Whenever tho
operation of differentiation is introduced along with the subject of Finite
</>
Differences, the notation
^
must of course alone be employed.
Thus, in differentiation, wo liavo
1488
Tin: i.isTKir.iTivi:
LAW
(I_^(n-\-r)
=
(/_,i(
1489
The COMMUTATIVE
L.\w
i1^{d^u)
—
(l^{(l^u)
d,,u
=
(ly.u.
</>/:«
=.(lT'u,
ifm^d.^u
=
or
1490
that
is,
The
iNi.KX
LAW
+ (i.r.
(f (,„+„). r^f-
(Mil)
(11^2)
(l-^^^')
DIFFERENTIAL CALCULUS.
268
1491
Example.
—
{ch-d,y = (ch-d,) (d^-d,_,) = d^d^-dAly-d,d^ + d,d, = d,-2d,, + d,,.
(U89)
Here d^d^—dyd^ or d^y = d^^, by the commutative law.
(l4i»U)
(?A = f^2x by the index laio.
=
—
=
'ld^yU + d^yU, by the distributive law.
di^u
Also {d.,-'ld^y-\-d.y) u
2d^yU + d2y^t.
d.:,u
Therefore, finally, {d^—d^Yu
Similarly for more complex transformations.
—
1492 Thus d^ may be treated aa quantitative, and operated upon as such
by the hiws of Algebra d" being written d,,^, and factors snch as d^dy, in
which the independent variables are different, being written d^^, &c.
;
EXPANSION OF EXPLICIT FUNCTIONS.
TAYLOR'S THEOREM.— EXPANSION OF f{x + h),
1500
where 9 is some quantity between zero and unity, and n
any integer.
Proof.— (i.) Assume f(x + ]i)
= A + Bh+CP + &G.
Differentiate both sides of this equation,
equate coefficients in the two results.
—
is
—
first for x,
and again
for /;,— and
—
Lemma. Iff(x) vanishes when a; t:= a, and also
f(x) and f'(x) are continuous functions between the
same limits then f'(x) vanishes for some value of a; between a and h.
For /'(') must change sign somewhere between the assigned limits (see
proof of 1403), and, being continuous, it must vanish in passing from plus to
1501
(ii.)
Cox
when x—h, and
s
Proof.
if
;
minus.
1502
Now, the expression
f(a + x)-^fia)-xf'(a)-
-
j^/"
00
-^'=[/(-+/0-/(«)-V''(«)— -|^/"(")}
when x = and when x=h. Therefore the differential coefficient
vanishes
and h by the lemma.
with respect to x vanishes for some value of x between
Let dh be this value. Differentiate, and apply the lemma to the resulting
and when .e = 06h. Perform the
expression, which vanishes when x
pame process n+l times successively, writing Oh for Odii, etc., since merely
The ivsult shews that
stands for some quantity less than unity.
=
/""('^+-^0-4^[/(" + /O-/(")-/'/OO---|-^/"('O|
vanishes wheu x
is proved.
= Oh.
Substituting Oh and equating to zero, the theorem
EXl'ANSION OF
The
1503
'/I
L'
XT LICIT
Fi'XrTlUXS.
260
term in (1500)
is called the reinaiii(h>r after
obtained in either of the sii]>ji)ined forms,
being due to Lagrangt*,
terms.
last
may be
It
tlie first
|_^(l-e/'
or
l^f-ir+0h)
•/»(,,
+ «/,).
Since the coelHcient ,— ^ diminishes at last without
1504
n increases (230, ii.), it follows that Taylor's scries
conrergcnt if f " (x) remains finite for all values of n.
limit as
is
1505
If in
any expansion of f{x-]-h) in powers of some
then / (,c) ,/'(<«),/" (.v) &g. all become
//
index of h be
neijatice,
,
infinite.
//
+ 1;
cients
index of
If tbe least fractional
1506
then /"^^
become
Pkoof.
and
(.'•)
all
/(,
coeffi-
infinite.
— To obtain the value of
successively for
between n and
h lies
the following differential
and put
/<
=
expansion
/"(.'), (.lifTercntiato the
7i
times
in the result.
MACLAURINS THEOREM.
Put
07
=
notation of
in (loOO),
and write
d'
for
//
;
1507 fV) =/(0)+.rr(Oj+^/'(0)+...
where
0,
as before, lies
between
Putting y =f(,r), this
1508
1509
//
then, with the
( 1-1-0 G),
= //o+'
777-
may
and
+ ^/"(^^r),
1.
also be wiitten
+ T^ 777^ +
—
1
,,
..
T^ +
^^^•
XuTE. If any function f(x) becomes infinite with a fuiite value of
sec"' (1 +.'),
then fix), /"(x), &c. all become infinite. Thus, if /(.r)
Therefore /"(O), /'"(O), &c. are all
(14o8).
f'(x) is infinite when x
infinite, and /(«) cannot be expanded by this theorem.
X,
=
=
ncrnonlli's Scries.— Vnt h
1510
/(O) =nr)-.rf{.r)
= -.v
in (1500); thus,
+ ^f'\,v)-j^f'\A') + &c.
;
DIFFERENTIAL CALCULUS.
270
1511
If
1> (//
Proof.— Let
+
=
y
=
/'•)
tp-'
and
(x)
therefore
«'
Therefore
y
+ h =/(0)
<p
= f (x)
+h =
(//)
= x;
and
,
(j>(y
let
+
+
7/
+ k) =
=/(.'c)-ac/"(a;)
tlien
/.-
=/(.c + /0
;
0.
p^/'Ca^) -&c., by (1510);
which proves the theorem.
EXPANSION OP f{x + h,
Let
f{xfj)
= u.
y
+ h).
Then, with the notation of (1405),
1512
/(c*'+/j, 7/+A-)
=
u+(hu,+ku,) + ^{hhi,,-\-2hku,,-^k'u,J
The general term
1513
is
given by
,
—
{h(l^-\-kdy)"u,
where, in the expansion by the Binomial Theorem, each index
of d^ and dy is changed into a suffix; and the coefficients
d^, d.^, &c. are joined to 21 as symbols of operation (1487)
thus ul is to be changed into u^.
Proof.
thus,
— First expand f(x +
f(x + k, y +
Jc)
h, y +
k)
+ hf,
+
=f(x, y
^c)
as a function of (x
(ic,
y + k) +
+ h')
^^^ h%, (x,
Next, expand each term of this series as a function of (y
writing u for /(,<'//),
f(x,y + k)=
J>fr(x,y
,-^ f2.
(.^,
y
+ k)=
+
/'•)
= rj
by (1600);
y + k) + &c.
+ k).
u
+
ku,
+
r^k'n,,
+|y^X
hu,
+
hku,,
+
r^hJc'ii,,,
+Xhk'u^,^+
^^'^ir
+
1^'ku.^,^
+
j-^/^'A-».3..
+
,^
——
h-k-a.^^^
Thus,
+n-^^X +
+
I
|^/3.(.^-,y
+ /0=^/^V +
r^/u(A2/ + ^)=^ A'«..+
i±
Li
The law by which the terms of the same dimension in h and are formed,
is Feen on inspect ion.
They lie in successive diagonals and when cleared
of fractious the numerical coefficients are those of the Binomial Theorem.
1:
;
EXrAXSTON OF EXPLICIT
FUNCTIONS}.
271
The theorem may be extended inductively to a function of
Thus, if u =/(.'', //, ;.'), we have
three or more variabk'S.
1514
fU-\-h.
//+A-, z
+ = u + (hn,,+h'n^-\-lu,)
l)
the general term being obtaincnl as before from the expression
UL
1515
C'oR.
—
If
21
= f{xnz)
be a function of several inde-
pendent variables, the term {hn^-\-lcii,j-\-JuS) proves, in conjunction with (1410), that the total change in the value of ?/,
caused bv simultaneous small changes in a', ?/, z^ is equal to
the sum of the increments of u due to the increments of x, ij, z
taken separately and supt'rposrd in any order.
This is known as the principle of the siq)crpositi'on of small
quantities.
To expand f{x, ?/) or f{x, y, z, ...) in powers of x, //,
put x, ?/, z each equal to zero after differentiating in
(1512) or (1514), and write x, y, ... instead of h, k, &c.
1516
t^c.,
1517
tlie last
Observe that any term in these series may be made
by writing x-{-dh for x, y + dh for?/, &c., as in (1500).
sy:^ibolic
The expansion
/Gr+/0
1520
—
PuooF. By the
suffixes (1487),
e'"^'/(')
form of taylor's theorem.
in (1500) is equivalent to the following
=
c-'ri^v).
Exponential Theorem (150), writing the
indices of J, as
= (1 + /,J, + iW,,+ .,.)/(^) =f(x)+],f,(x) + lh%(.r)+..., by (U88).
a /(.r) =fi^v+h)-f{.v) = (e"'''-l)/(.r),
Cor.therefore
A'^ (.r)
and generally A"/ CO
=
=
(f'"''
- 1 Yf(.v)
,
(c'"''-l)"f(.r),
the index signifying that the operation
is
performed n times
upon f{x).
1521
Froof.
Similarly
—
/(./+ A,
= j\.e + h,y +
l:),
//+/.•)
=
e'"''"'VC^^
by (150) and (1512).
//)•
DIFFERENTIAL CALCULUS.
272
1522
And, generally, with any number
Cor.— As
of variables,
in (1520),
1523 If u = f (/', //) = (r, 0) where x = r cos 0,
then
and if x = r cos (0 + ^), y' = r sin (0 + w)
expanded in powers of w by the formula
,
</)
;
Proof.
a;
and
u,
The
(r,
+ 0,) =
e'*"'^
^
= e'^^VC^',
(r, 0)
y are functions of the single variable
= w^«„ +
;
therefore
= u^ — r sin 0) +Uy (r cos 0) = xa^—yu^.
My//9
(
d^
=
;
is
2/).
operation d, will be transformed by the same law (1492)
f(x\y')
= r sin Q
f(x'y')
/(.»',//) = «"""'-"'''/(..•,?/).
— By (1520), r being constant,
^
Now
//
e"'(-^''v-2'^'x) j(,.,
y)
= xdy — yd^;
;
therefore
therefore
= l + w{xu,-yu^) + h''X^''ihy-^^y''h,j + f'h.) + &o.
—
1524 Examples. The Binomial, Exponential, and Logarithmic series for {l-\-xT, a'', and log(l+,^), (125, 149, 155),
as
are obtained immediately by Maclaurin's Theorem (1507)
also the series for sin « and cos a^ (764), and tan~^a3 (791).
The mode of proceeding, which is the same in all cases, is
shewn in the following example the test of convergency
(1504) being applied when practicable.
;
;
1525
tail .V
= +1
.V
.v'
«»
+^
lO
.v'
+
^
a^
»)1.)
Obtained by Maclaurin's theorem, as follows
:
+ J$. .iH &c.
J,r»t)«)
— Let
= « and z^ = 2yz
= tan x = y 1
Therefore
_ gp^jS^ = z
y and being used for shortness.
f = 2 sec^ X tan x = 2yz,
f" {x) = 2 (.^, + ?/..) = 2 (.^ + 2/;;),
+
+ V2) = Q{1yz- + yh),
fix) = 2
f {x) = 8(2z^ + 8yV + Zyh'' + 2y*z) = 8 (2z' + Uy"-z"- + 2y'z),
= 272yz'+
r'(x) = 8 (I2yz' + 22yz'+
r\x) = 272z*+...&c.,
f
f
(a;)
2/,
)
^^^
;
;;
(.b)
(%;>;''
4//2''
...,
...)
the terms omitted involving positive powers of
zero, and which therefore need not be computed
is required.
higher than that containing
.t;^
y,
which vanish when x is
term of the expansion
if jig
iJXPAXSioy OF Expjjcir Frxrrins<i.
Hence, by making
/(0)=0;
Thus
tlio
aj
= 1;
/C<>)
terms up
273
= 0, and tlioreforo y = and 2=1, wo obtain
/'(0)=0; /"(())=2; r(0)=0; /''(0) = 10;
r'(0)=0; r\0) = 27-Z.
to x^
may bo
written by substituting these valued in
(1507).
may
In a similar inainicr,
•^^'^••'-i
1526
Methods
bo obtained
+ 4--'' +
+
in'''
iff i\i pdii.shni Inf lnd('f(
7!^r'''"'
+ ^^'
nuindir
Cor/licicnt.s.
= A-f Bx + Cx"4-&c.
Rl'I'E 1.—Assume f (x)
rntiate both sides of the eqiiatlou.
1527
known theorem^ and equate
determine A, B, C,
Obtained by Rule
coefficients
e.rpniid f'(x)
in the
Diffrrhi/
smne
two results
to
ijv.
^.-K.
Ex.
1528
Then
^^ + ]^^ + ]^< + &o.
= .+
Assume
I.
Therefore, by (1-101),
= A+Bx+Cx' + Dx'' + Ex' + F.c'+
(l-.«-)'^ = i? + 2Cu; + 3DxH4LV + 5F.«' +
But, by Bin. Th.
(1-..^/^
siu-'.r
Equate
&c.
^l
=
(1-28),
coefficients
By
putting
sin-'0
U.
therefore
;
x
= 0,
7?
wo
= 1 + V+ 1^ .^*+ .j^ x«+
=1
seo
;
D=
C = 0;
that
.1
—
1
=/(0)
;
E = 0; F= 13
^^
always.
-
lo^y
c
1530
Rl'I-T'
I
-t-
^'
-1-
pj
I.,
^
1^
(^
-r
it.— Assume the series, as before, with iinhiown
Diffierentiate successireh/ until the function reThen equate coefficients in the two rquindcnt scries.
coefficients.
ap2)ears.
K^-
— To expand
sin.^•
in
powers
of x.
= A + Bx+ Cx^-\-Dx^ + Ex*+ Fx^+
4Ej'-i oFx*+
2Cx+
co3.-e= B+
Differentiate twice,
-sinx = 2C+3.2Dx + 4.3/;.cH5.4Z'V+
in the first two equations; tborcforo A — 0, B=l.
Put x =
Assume
sin
a;
31)..-}
Equate
;
=
In a similar manner, by Rnle
1531
^
In this case
coefficients in the first
and third
series.
1 N
-\-
——
—
;
DIFFERENTIAL CALCULUS.
274
-S.2D
-2G=A, ..G=^0;
Thus
= C,
-4..dE
Tlierefoi-0
,
1.2.3
&c.
\,
as in (7G4).
'S:c->
=
x,
tu
r
.
.,
F=
.-.
f (x) ta-icG
Differentiate the equation y
and combine the results so as to form an
III.
witJi respect
-6AF=D,
= 0;
—
= «- .—^ + 1.2.0.4.5
o
sin.-c
Rule
1532
.•.E
--D=-^;
= B,
equation in y,j^, and
= A + Bx + Cx"^ + &c.
assume y
^^^^-^
j^x-
'Differentiate tiuice, and substitute the three values of j, y^, y.^
Lastly, equate coefficients
so obtained in the former equation.
in the result
— To expand
1533
Ex.
of sin
or cos
These
as follows
determine in succession A, B, C,
to
siu
mO and
mO
in ascending
powers
9.
Tliey
series ai'e given in (775-779).
may
be obtained by Rule III.
:
Put
a;
Therefore
=
sinQ
y^
=
?/2x
= — sm (»i sm
and
(m
cos
y
= smmd =
•
\
'
in"^
\
X)
Let
y
Differentiate twice,
(m
siu"^a;).
(^rtsin~^);),
"tnx
/
,
= A + A^x + A^x''+
(i.)
--IN
x)
+ cos {m sm
— x(l — x")
:,
1
Therefore, eliminating cos
sin
(1434)
sin'' x)
,
.
thus
cos
Si'c.
:—^.
— .r)2
yo,—xy^-\-in^y =
(1
(ii.)
+ A„x"+
and pnt the values of
y,
y„ and
("i-)
?/2^
0=:vr(A + A^x + A,x^ + Ay+...+A,X +
-x{A, + 2A.,x + SA^x-+...+nA„x"-^+
in equation
(ii.)
)
)
+ (\-.-){2A, + 2.SA,x+
+ {u-l)nA,,x"-"- + n{n-\-l)A„,^x"-'+(n + l){)i + 2)A„^2-v"+...}.
Equating the collected
coefficients of x" to zei'o,
"^'""'
A.o=
Now, when
y^
= vi,
by
(i.)
.v
;
= 0,
y = 0;
therefore
and therefore A^
= vi,
^1
by
= 0,
wc
get the relation
A
by
(iv.)
And when
(iii.).
diftcTeiitiating
(iii.).
x
= 0,
The
relation
y,,^,
as in
in succession.
Cos md
1534
is
obtained in a similar way.
Rule IV.
Form
the equation in y, y^,
and
Take the n"' derivative of this equation bij applying
Leibnitz's formula (14G0) to the terms, and an equation in
Jiulr J II.
ExrAxsiox OF ExniniT functions.
''^
ohtn'nirtl.
Piif
y(n+2)x» y(n+i)x» ^'^^^ ynx
i'mploij the rcsnltlinj formnln In riilnihttc
y.uo»
4"''-
1535
ill this; and
succession y 3x0)
in
Machiurins cximnsiou (loOZ).
i^i
('"""
1^-x.
=1+
'
(1
Differentiating n times,
(1— a;')
?/,„*2)xo
(<i*
—
,/-
+
^'
.7..
^
—
.S-)
yix—xy^—a-y
is
= 0.
by (14G0), we get
//(,.. 2)
=
+
fi.r
Writing y for the function, the relation found
Obtained by Rule IV.
Therefore
=
x
r-
C2>t+1)
+ w") y,wroi
.«//(„. i;r
—
('t-
+ n=) !/„x =.0.
which produces the cocflisuccession when y^ and y-,r,) have been
formula
a
cients in iMaclaurin's expansion in
calculated.
ARBOGAST'S METHOD OF EXPAIsDlNG
1536
T^-hcrc
= +
^/
;::
Let
//
=
«/)
+ Y^j-*''+
When x = 0, y = {")
+
^^.»'
(::).
j^,.*''
4){z),
by
tlicreforc,
;
i>
(i.)
^^'C
Maclaiirin's tlieorem (1508),
=
1/
4>{fl)+^^\'/.rn-{- jh^Z/-2.-o4-
Hence, in the values of
put
y^,
= 0.
Now, when x=0, z=a;
<f>"{a),
&c.
1537
Here
;
and
2;.ro»
^2xo» '^u-or
y.,,,
= ^,
(/,= 2r,
</,
+
&c., at (141()),
+ hx + c..r +
= —,
.l..:'
2/^
=—
a
,
y..^,
= GJ,
=
Therefore, substituting in
ij,'
r,
H
a(ii.),
/;
has to bo
+ &c.).
1
(a)
<^"{i,)
1
= --^,
we
a
,
^j^
f
It
It
Therefore
(ii.)
^*^'C
therefore y., y.y., &c. become «|)'(^;,
Hence
&c. become a^, (h, rtg, &c.
KxAMi'Li:.— To expand \os(a
«,
j-V[]//.vo
=
a
^
—
..
<t
H
•
«
obtain as far as four terms,
(a)
=
2
---.
U)
DIFFEREXTIAL CALCULUS.
276
1538
2.~To expand {a + aiX-\-a2x'^+ ...+a^x"y in powers of x.
may be employed otherwise, we may proceed
Assume (a^+a^x ^-o.^x^i- ... a„x"Y = Af, + A^x + A^x^+
Ex.
Arbogast's metliod
follows.
Differentiate for
and equate
divide tbe equation by the result; clear of fractious,
powers of u'.
;
BERNOULLI'S NUMBERS.
= l-|^+C.|-iJ.|+i?„|-&c.,
wliere B2, B^, &c. are known as Bernoulli's numbers.
values, as far as B^^, are
7.
_
^'^^
-P
^^'^~273U'
They
tlie
-^
_^G17
V ''~
''~ 6'
nB,,_,-\-
tlie
Tlieir
_ 4.3867
''~
Proof.— Lot 2/=
formula
= 0,
B^, &c. being all zero.
7^3,
7^-
Then, by (1508),
= y^-^y^^x-^-y-iro
,r"
,—
\A
+
r*
2/3x0
,-7
+
X*
2/4-0
,-7
+
&e.
\±
[1
Now 7/0 = 1 and y^ = -l, by (1587).
by (14G0), differentiating n times,
Here y,,^^{-\y'^B,,.
ye'
= y+x.
'J'lierofure,
e^ {y..
Therefore
Substitute
1>3, J/5,
+ '"y,.-r>. + G(n,
2) y^n-2).+
...
+ny, + y]
=
Also
y,.,.
G (n,2) y^„.o^M+ ...+ny^ + yo = 0.
I)„.i, Bn.2, &c., and we get the formula required.
It may be proved,
&c. will all be found to vanish.
ny^„.,^M\-
J)j..
that this will be the case
1541
*
798
C (n, 2) n„_,+ C {n,V>) B^,_,+ ...
... + C{n,2)B,-in + l
odd numbers
y
J.
510'
are found in succession from
1540
as
coefficients of like
^
1539
.r
;
:
a j^riori,
for
e---T
+ l> -^7i:i-
Therefore the series (1530) wanting its second term is the expansion of the
(1401)
expression on the right. But that expression is an even function of
changing the sign of x does not alter its value. Therefore the series in question contains no add powers of x after the first.
.i'
;
1542 The connexion between Bernoulli's numbers and the sums of the
powers of the natural numbers in (27G) is seen by expanding (I— <?')"' in
povvei-8 off', and each term afterwards by the Expoucntial Theorem (150).
EXrAXSIOX OF EXrUCTT FUNCTIONS.
277
1543
"
1544
— — = —'—-
Pkoof.
^
——— =
fi^d
111
1545
1
Proof.
:^:::
of
-^
,
—
p (1540)
—1
-
,
and by
(1
539).
—.2)1
,,
,
2;/
2
]^
-^
T
'
1
0271
=
+ o-i + :^
j-< + :^
4-« +
—In the expansion
*>
1
T——:
^4.
substitute 2i0 for
a*,
and
it
cot 6 (770). Obtain a second expansion by difFerExpand each
in factors).
entiatinj? the higarithm of equation (815, sin
term of tlie result by the Binomial Theorem, and equate cocCQcicuts of like
becomes the expansion
powers of
of
in the t\vo expansions.
STIRLING'S THEOREM.
1546
4>{.r^h)-4>{.v)
wliere
.4
,„=
= h<t>'{,v)^A,h
(-])"/?,„ -h [2n_
{f (.r+//)-f
and
A.„^,
=
Or)}
0.
—
Proof. ^„ .^Ij, A^, &c. are determined by expanding each function of
x + h by (1500), and then equating coeliicients of like powers of x. Thus
To
since
=
(.r)
r',
obtain the gencial relation between the coefficients: put
Equation (LVIC) then
J.^, &c. are independent of the form of (p.
Jp
—=
prod u ces
—j^
and, by (1539),
we see
A
,,.
,
.
1
— AJi — A.Ji^ — A J
J'
—&.C.;
that, for valuL;s of?! greater than zero,
=
and
A,,.
= (-
1 )"7>\,.
-^
[^.
BOOLE'S TlIEORiaL
1547
t^(.r+/0 -</»(.')
Proof.
—
in Stirling's
A^,
A.,.
A^, Ac. arc
Theorem.
=
AJi
{4>' (.r4-//)
+f
(.r)!
found by the same method as that employed
DIFFEL'EXTIAL CALCULUS.
278
For the general relation between the
and equation (1547) then produces
coefficients, as before,
-f~ = A,h+A,Jr + A,h' + &c.
we
and, by comparing this with (1544),
A. =
and
^,._i
make
^
(.<.')
=
e',
;
see that
=
?^.
(-l)»-'i?,„
EXPANSION OF IMPLICIT FUNCTIONS.
—
=
constitutes y
An equation f{x, y)
function of x. If y be obtained in terms of c« by
solving the equation, y becomes an explicit function of x.
1550
an
Definition.
itiq^licit
Lemma.
and
1551
variables x
—
If
two independent
y be a function of
,^,
— By performing the differentiations, we obtain
Proof.
and
F'(y)yryz + F(y)y,,
which are evidently equal, by (1482).
F'(y)
y.jj,
+ F (y)
y^,..,
LAGRANGE'S THEOREM.
1552
Given
powers
of x is
/(.'/)
— Expand u as a function of
n
of (140G),
7/^
the expansion of
z-\-cV(f){f/),
is
evidently
y^
Therefore
y,.
=
=
<!>
(,
by (1507)
= nQ + xu,o+ ,-^«2xo+
in
</'
•••
;
(y)
(//)
//,
+ •'f (//)
;
=
'Jr
and, since
therefore also
+ ,— ?f«xo+&c.
?/>
=
z-\-a'(p
y,
=
v^ =/'(//) y^
(j)
and
(y) for x
and
form wlicn
Assume
Therefore
that
it,
z in
turn,
wc havo
l+.r^j^' (//) y,.
and
»,
=j"(ij)
y,,
(y) x^
(i-)
The following equation may now be proved by
its
thus, with the notation
/(;.')•
Differentiating the equation y
being
u=f{i/)
=/(«)+.'''^W/W+-+j^,-£^[!'^W}"/W]+
Pjioof.
Here
=
i/
induction, equation
(i.)
= 1.
m,,^.
=
=
«(,..i„
f?(„_i), [{/^ (i/)}"
("•)
"J
[{V (y)}"".] (1482)
(15^1) ='/..[{V'0/)}"*'^'.], by
d^.._,^,ih
= ^.«-.).^.[{t(2/) }•'".]
(i.)
..
IML'LK'IT FUNCTIONS.
EXPANSION OF
270
Thus, n becomes ?i-f-l. But equation (ii.) is true when n = 1 for then
equation (i.) therefore it is universally true.
Now, since in equations (i.) and (ii.) the differentiations on tlie rif^ht are
etlected with respect to z, .r may be made zero 6»;/"';/v diirerLiitiatiii^' instead
;
it is
all
;
when
But,
of after.
tious
and
(i.)
(ii.)
=
».«
1553
'=
Here
x
1-
l'^>^-
>/—aij
f(ij)
a
=
u^
Therefore
»„^=
«....= '^-.)=C{9
+h
=
to
:
= \og>j;
log
J,._„, (.^''-^)
cj>
•'
1554
_,3
::
f(y)
^"
'•^
*=
rt
is
2.
now
=
^'
_._
^ n(n + b)
1
y
=
z+,,j\
1
(2u + l) z^\
...
(1552) becomes
CS'-'Ho»-2)...(^" + l)
1.2
^l+
1.2 ...n
:
to
tr"
expand
i>*
a*
i_
a*"^
,
»('n
powers of
y" in
we
+ 7)0> + 8)
ct"
h"
-^
CI
\
tt^
a
—
find
J_
1.2.3
nr» + 0)0> + 10)(»4-n) _^ J_
'
—
powers of
in
and
and, proceeding as in the last example,
=^ r'^";?^"^
f 1
z,
a
a-
= >/;
(3,^-1) (3u-2)
— Given the same equation
y",
00}"/ CO].
;
= ,ogA + Jll+...+
Ex.
therefore e<iua-
(-').
expand logy
(y)
Therefore, substituting the values of x and
log,
°
=V
(//)
therefore, in Lagrange's formula,
•^-;
= —;
a
u,=j'{u) and ^
9(-0/(--);
+
-
z
= 0,
— Given
>J
= -;
./
give
or uT
a* a-
CAYLEY'S SERIES FOR
.
^.,.
I
a*
ii"
—-.
1555
^=...-^r.^«,..+...+^lfi^[.[f;^)]"-a.+....
wliere
^l
== -, ,-.
cp (U)
Proof.—
^=
dz
/'(r)
Diflercntiato Lagrange's expansion (1552)
L
.
x by
Replace
^
l-.T<?.(i/)
=—
-,
since/
^.
Put
9(^/)
is
an arbitrary function.
/' (y)
=
for
•'
^
z,
;
V'(^)
Then make
noting that
and therefore
y = 0.
.
;
DIFFERENTIAL CALCULUS.
280
LAPLACE'S THEOREM.
To expand
1556
I^ULE.
— Proceed
F
st'dntlii'j
1557
Ex.
(zj
3.
Here
powers
in
/(//)
as in Lagranrje's Theorem, merel)/ suh-
for z in the formula.
— To expand
=
/(^)
In the value of
e"
= logz;
(1552), f (z)
m„^o
f{F(z)} = e''"^'' =
Thus the expansion becomes
1558
Ex.
= g+
y
sin y
=
f(y) = y;
becomes
+
...
Fiz)
=
+ a),
sm-'z;
<l>{F(z)}
=
^
ij
{::
(//)
7/
f^tZ(„_i), (sin log r)".
= x sin (y + a)
sin"^ (x sin y
in (1552)
(j>{z)
z;
sinlog2+
a;
4.— Given
Here
= log + x sin?/).
= siu
becomes f {F(z)} = siulogz;
therefore f'(z) = 1.
powers of x when
in
F{z)
e";
/(z) becomes
e"
of x wlien
expand y
to
:
with
<j>
2
=
(y)
sin (sin"'
z
sin (y
= F (z) = sin-'z; therefore
=
^sin(sin-'2 + a) (l — z-)~^
y
Thus
powers of
x.
+ a).
+ a).
f [F (z)}
f(z) becomes
in
= 0.
F'
(z)
= {l-z')-^.
+ lcK{sin'(sni-'z + a)(l-^r)-''\+ix%J,s{n'(ism-'z + a){l-z')-^\ +
=0
with z put
after differentiating.
^
=
a;
sin
ti
+
1-*^'
The
sin
result
is,
as in (796),
2a + \x^ siu oa + &c.
BURMANN'S THEOREM.
To expand one function
1559
function ^
Rule.
fore
<^
1560
Here
—Put
(y)
=
/(/y)
7/
/(//)
in
powers of another
(ij)
=z
stituted for
x
=
(y-z)
xp
(y)
-^^^ (y)
=/W+V'(//)
in Lagrange's exjiansion,
;
{f^/'(^)
there-
},..+•••
signifies tliat after differentiating
//.
and
therefore
;:
is
to be sub-
EXrANSTOX OF TMriTCTT
h— Since = ^(.y), =
CoH.
1561
.'
becomes, by
\vritiiig
ij
since
variable
tlie
//
immaterial what letter
factor of the oreneral term.
<^'^i?.
2.
—
If
;
therefore (15G0)
f|^{(-^])>0/)}^^+...
changed into
is
/'('/)
z
after differentiating,
written for
is
it is
1562
xp-'{.r)
2^1
for ^(//),
./•
/{,-u.o!=/«+... +
But
FUNCTTOX^^.
he simply
//,
;/
in the second
the equation becomes
'^-(..•)=~-+.-(:y),..+-+,T\|^{(-^y},.+
Cor.
1563
3.
—
If
.-:
= 0,
so that
)j
=
we obtain
X(},{//),
the
expansion of an inverse function,
1564:
formula
Ex.
5.
— The
(1528) for sin"'
scries
Let sin"' ^
sin" X
=
=
'J,
x
therefore
be obtained by this
1565
Ex. (3.-If y
=
theorem (1552), since y
a;
=
sin
y
=
in
\p (>j),
(loGS)
;
therefore
X
= 2 v//,
,~—-^,
=
1 .2 .3 Vsiu'
1 .2 Vsin^ ;//yo
.sin7//,!/=o
Put
may
aj
thus,
;
-^
r,
=^
+ ^J'. we
''\
^^^
/// ayo
tlien,
by Lagrange's
find
thus
ll^^}-::m'=r+,.r'+... + 'll^2^i'-+
\
2
/
r
+r
1566
7i
I
Change the sign of
('±^^M)-=i_,„+...(-,).»i£^r±
1567
This last
or odd,
\
«, thu.s
is
series,
continued to
equal to the
sum
—+1
of the
or
two
theorem.
2 o
— -—
terms, according as u
scries,
is
even
as appears by the Binomial
.
:
BIFFEIiENTlAL CALCULUS.
282
Also, by Lagrange's Theorem,
1568
or,
= iogf + (;^)'+^^.4g(f)%
= 2 \/t,
.ogi-^4i^ = <+...+lgr+
K'i/
by putting x
1569
1570
Ex.
7.
— Given
xy
= logy
The equation can be adapted
=
y
to
expand y
therefore
e'",
in
powers of
xi/
=
may
Ex.
8.
— To expand
Here x
1571
= yi'\
f
in
e""
(jj)
x.
xe"".
=
= y\
x&\ from which, by putting z
therefore y
be expanded, and therefore y.
Put xy
y
;
as follows
powers of
= y ~^ =
=
in (15.52),
yi"-'
e-^\
if
we take
z
= 0.
Therefore
+ «ye^^ + a(a-26)'^' + a(a-86)^^^+
e'"'=l
ABEL'S THEOREM.
1572
be a function (developable in powers of
If ^ (')
+
^-(.,+,.i)
1573
Cor.
+
.w ... )
(i.)
\ AJ-^ -^ A^e^'-' -\and multiply the results respectively by
Then the theorem is proved by equation (i.).
(^
—
.
(//)
A,,^
A^(^'
&c. in (1571),
3,
A^^., A^e^, &c.
ylfl,
=
then
«("-'-^)'-'
J.
Proof.— Let
Put 3/ = 0, 1, 2,
e^\
If
({>
(,*')
= x",
Abel's formula gives
-\-C{n,r)a{a-rby-'{,r+rhy-'--^&c.
INDETERMINATE FORMS.
1580 Forms A,
if
—
X
•
Rule.— 7/' ^44
takes cither of these forms irhen
^„
.
,
i/-
^^ « fraction
irhirh
(x)
x=:a:
tJtoi
the first determinate J raction obtained
^-i^
hij
— ^,)l
or
differentiating
.
IXLETEinflXATK FOn}fS.
283
numerator and rJeuominafor simuIfdnpoiisJi/ and suhxt'ttnthig
for X in the result.
the
Q.
1581 But at any stage of the process tlie fraction may be
reduced to its simplest form before the next differentiation.
See example (1589).
—
Proof.
By
(i.)
Taylor's theorem (1-500), since
+ h) _
\P{u + l,)
(p
wlien
^
4.{;)
^
(<i±Bh)
<!>(a)-\-h<p
+ l4''(<i + 0li)
= =
(a)
+ Oh) _
xp\a + Uh)
tj/Ja
(f>'
(a),
v//
(a)
^//"'(u)'
vanishes.
//
^^
^
which
(a
is
^^
^
^
of the first form,
_
f (a) _^
- {4.(a)Y
^(a)
and therefore
_ {0 (a)Y
- (H-n'
..,,,.
<t>'(a)
^^^'^^
(.)
.;<^
^
,p{a)'
xPia)
;^>)
m,
.
^^'"'"'
9(ny
_ ^J^t)
(a)
<p
^OO "
4^\'-0'
1582 Vanishing fractions in Algebra are of the indeterminate form just considered, and may be evaluated by the
rule, or by rejecting the vanishing factor common to the
numerator and denominator.
^,
^
Ex. — When
U'hen
x = a,
form
—
,
Ox
j"^^
x.
{^)
1*
Forms 0^
1584
a-»-a«
= a;
Rule.
^^
(=^)
—If
=
i"
(^
X
(x)
(=')
"^
;//
Tl
For
form
Form
1586
log
tee
X
oo
-
^
= _3 a.
li
(x) takes this
—
""^"'^^^
^"^
'
takes
form
^f
^^^^
any of
.
X — X. Rule.— If
have
e'''^"^-*^"^
<{>{x)~xf.{x) takes this form,
= ^^^ = 77'
^^"^ ^f
tlie
''*''
'-"^"^ ^^
required value
c.
Otherwise
:
./,(a)-i^(a)
=
./>(a)
U - '^f^]-,
L
form
Scr
'2<i
{<^(x)]*^'>
expression he found to he c, hy (1580),
ivill he
the
+ a.r + <r) =
-'
;
(x-a){.c + a)
Rule.— 7^
x", 1".
case, is of the
^^I's
(.r-a)(.r"^^
x = a, find
irJicii
expression.
when x=a,
=
-
the limit of the to(/arithm of the
the loga r it tun r=: xp(ji") log (pi^n), irhlrh, in eaeh
forms
1585
=
.c--a'
Form
1583
these
x
oo
X
(1583).
(p{-d))
u-/i/c/i
/s
of
DIFFERENTIAL CALCULUS.
284
1587
Ex.
Also, with
L—WitL
a;
X
=0,
y
=
-^ = 1 = ^
(^^^^^
=
= 0,
e'-l-xe'
e'-e'-xe'
-x
1
1-
—
JACOJiFANS.
28.'
JACOBIANS.
1600
The
I^^'t
//,
fi)ll(nviii^
II,
Uy
u,
'•.r
'^
'*.-
r, ir be n functions of n vanables
determinant notation is adopted
(I
•*V
'^'p
"^f-
//»
//<•
//»
a', ?/,
The
= 3).
[unr)
u\ w.
first
determinant
is
and
is
respect to
bv
{n
</(.r//.t)
(i(rtr)
(/
u\^
."
:
.r,
//,
;:,
called the Jacohlan of
also denoted
by
?r,
J{iirir),
r, ir with
or simply
J.
1601
Theokem.—
(/{.n/x)
d
{nrir)
of the two tletenniuants be formed by the rule in
(57U), first changing the cohiuins into rows in the second determinant (669),
the first column of the resulting determinant will be
Proof.— If the product
u^Xu
+ u^y„+v,z„ =
u,,^
110
= 010
1
1
DIFFEBENTIAL CALCULUS.
286
1604
If u, V, w,
n functions
transformed into functions of
X=
z
=
+
c/T)
+
(',C
^^^^^^^
^
M
/=
Uz
^ly
":
_
^^
(J
(urir)
J = MJ,
or
is the determinant
where
transformation (573).
Pkoof.
n variables x, ?/, z (n = ^), be
by the linear substitutions
K, v, I
(/(unr)
(hi-^(hr)-^a,l^
(\$
of
(ai^h.,r.^)
called the
modal us
of
287
QUANVfr^^
—
Pkook. / is of tlie third degree, and therefore J,, .7^, J, are of the second
Hence n, i\ w,
degree, and they vanish because u, v, w vanish, by (1G07).
0.
Jt, «7y, /, form six equations of the form (;r, //, i-)'
=
1609
T^
"
variables
otlier variiibU'S
I,
»;,
l^
a?,
y, «
(7?-
= 3)
are connected with n
by as many equations
d.y (hi
fh
(Jiiirn')
J^ tq
Tie
iI^iyjQ
.
?(
= 0, = 0, =
fl(Krii')
tl
dx
dij
dz
li'd^T^
'
(({^rf/z
t'
7('
;
,
DIFFERENTIAL CALCULUS.
288
The theorem may be extended
to any quautic, the quanon the right remaining unaltered. Thus, in a ternary
quantic u of the n^^' degree,
tities
1624
1625
^Tit^-\-yify-\-^if=
{.r(h+i/d,-]-zd,Yu
=
=
and generally
»itf',
n {n-l)
...
{n-r-^l)
u.
Definitions.
n quantics in n variables is the
function of the coefficients obtained by putting all the quantics
equal to zero and eliminating the variables (583, 586).
The Elhninant
1626
of
The Discriminant of a quantic is the eliminant of its
derivatives with respect to each of the variables (Ex.
1627
first
1631).
An Invariant is a function
equation whose value is not altered
of the equation, excepting that the
the modulus of transformation (Ex.
1628
1629
of the coefficients of
by
an
linear transformation
function
1632).
is
multiplied
by
A Covariant is a quantic derived from another quautic,
that, when both are subjected to the same linear
transformation, the resulting quantics are connected by the
same process of derivation (Ex. 1634).
and such
1630
A Hessian
is
the Jacobian of the
first
derivatives of a
function.
Thus, the Hessian of a ternary quantic u,
derivatives are li^, n.
U,r.
u.
(I.,.
U,„
whose
first
d{u^u„uj)
d {wyz)
1631
Ex.— Take
derivatives are
u^
u,j
the binary cubic u
=
=
Sax^
Sbx^
+ 6bxy + Scy'
+ 6cxy + ody'.
=
ax^
II.,,
+ Sbx^y + dcxy^ + dy\
Its first
3a 66 3c
Therefore (1627) the discriminant of m
is tlie annexed determinant, by (587).
1632 The determinant is also an invariant of u, by (1G38) that is, if u
n'£-|-/3'»/; and, if a
bo transfDrmed into v by putting x
nl^fii) and //
corresponding (h;terniinant be formed with tlie cocilicients of r, the new
determinant will be equal to the original one multiplied by (u/3' a'/3)\
;
=
=
—
—
QUANTICS.
1633
Again,
«2,
=
GtUJ
+ G&//,
Wjy
=
Ccc
289
+ O'/y,
Therefore, by (1630), the Hessian of w is
(ax + hii)(cx + dij)
v.^n,^-nl^
=
=
=
6bx + 6ci/.
- {h.c + r//)»
{ac-lr) x-+(ad-bc) x
1634
u^,^
>/
+ (h<J - r)
if.
for, if u bo transformed into v, as
Anil this is also a corariant
bftcut.', then the result of transforming the Ilessiiin bj the same equations
i%.
See (lL'io2).
will be found to be equal to V2zCiy
;
—
1635
If a quantic,
7(
involving n variables
the second degree in
linear functions of the
=/(.?•,//,:: ...),
can be expressed as a function
oF
Xi, X., ... A",,.!, where the latter ai-e
variables, the discriminant vanishes.
—
u = (f>X\ + \//X,A'j + x-^'A^s + «^c.,
PuooK.
Let
A', = a,a; + h^H + c^z + &c.
where
The derivatives u„ u^, &c. must contain one of the fjiotors XjX, ... X,,., in
every term, and therefore must have, for common roots, the roots of the
u-l equations being
simultaneous equations X, = U, X.^ — 0, ... X,,., =
required to determine the ratios of the n variables. Therefore the dis;
criminant of
a,
whicli (lGl2")
is
the eliminaut of the equations Hr
= 0,
n^
= 0,
&c., vanishes, by (oS8).
1636
<^^<"^-
1-
—
If
'^
binary quantic contains a square factor,
the discriminant vanishes
;
and conversely.
Thus, in E.xample (1G31), if u has a factor of the form {Ax + J)i,')'-, the
deterniinaut there written vanishes.
—
1637 Cor. 2. If any quadric
the discriminant vanishes.
An
independent proof
Let M
= Xy be the
is
is
resolvable into two factors,
as follows:
quadric, where
X= {ax + by + cz+...),
Y= (ax + h'y + c'z+...).
derivatives u^, u^, u. are each of the form i>X-\-qY, and therefore have
0, F=0.
for common roots the roots of the simultiineous equations
U, u^
Therefore the eliminant of u^
0, &c. vanishes (IG27).
The
X=
=
1638
The discriminant
=
of a binary quantic
is
an invariant.
—
Pkoof. A square factor remains a square factor after linciir (ransformaHence, by (1G3G), if the discriminant vanishes, the discriminant of
tion.
the transformed equation vanishes, and must thenforo contain the former
discriminant as a/«r/or (see 1G28). Thus the determinant in (1G31) is an
invariant of the quantic ii.
The discriminant
1639
u
=
of the ternary quadric
aa^'-\-bf/'-{-rz'-]-'2fyz+2gz.r-^
2 p
2hxy
290
is
DIFFEIiENTIAL CALCULUS.
the eliminant of the equations
QUAXTICS.
Proof.
— Let the linear substitutiims (1(528) be
x
Solve for
and
I
Find
»?.
d,
The
result
=
=
y
ai-\-l'>i,
l^, £^,
Vx,
'/.,.
= a'i +
(i.)
h'tj
two equations
""f^ substitiitn in tlie
=
'
d.l^^d,,,,^;
,l,
d.E,
+ d„„r.
is
=
(/,
291
-d,=
+ h(-.J.)]^M;
{aJ^
M = ah' —
{a'dr,
+ h'(-dt)}^M
(ii),
modulus of ransformation. Etjuations (i.) and (ii.)
are parallel, and show tiiat the operations d^ and —d^ can be transformed in
(I, n),
the same way as the (juanlilies x and // that is, if 9 (a;, ?/) becomes
— Jf ) -f- J/", where 71 is the degree of the
then f(d^, — (/,) becomes
(</,,
where
a'h, tlie
i
v/*
;
v/-
,
—
is a function of the coefficients only of
(.c, //)
quantic 0. But ^ (d^,
c7,)
the quantic 0, since the order of differentiation of each term is the same as the
degree of the term ; therefore the function is an invariant, by definition (1628).
—
1649
first
<p
— ax*+bx^i/ + ex!i^+fy*. The quantic must
= ax* + bx^y + cx-y^-\-exy'^+fy\ {c = 0) then
Example. Let
{x, y)
be completed; thus, f (xy)
(d^,
=
—
;
+ cc/2y2x— «^!,ax+/(?4x) 9
y)
= a.2if-h.Ge + cAc-e.{Jb+f.2~ki = 4:(l-2af-dhe + c").
12af-Sbe-\-c- is an invariant of ^, and = {\2AF-^BE + C^)-i-M*,
-J,) f
(x, y)
(uti,,
(a;,
5t?3j,z
Therefore
where A, B, C, E, F are the coefficients
linear transformation.
But
if
of
any equation obtained from ^ by a
the degree of the quantic be odd, these results vanish identically.
1650
Similarly,
(p(,r,y),
if
{x,
^P
ij)
are
two
qualities of tho
same degree, the functions
and
<t>{fI,,-(L.)rl^{.v,i/)
rp{d,,-(l.,)<t>i.r,,,)
are both invariants.
1651
Eyi.—Uc(>
{ad..^
=
ax'
— 2bdr^ + cd.i^)
A
1652
+ 2bxy + cy' and
+ 2b'xy + c'y^) =
(ax''
Hessian
—
is
4.
=
ax--^2b'xy
+ cy'
then
;
ac+ca'-2bb', an invariant.
a covariant of the original quMitic.
Proof.
Let a ternary quantic u be transformed by the linear substituz)
^ (£, »j, ;). The Hessians ol tiio
tions in (1604)
so that u
(p (x, y,
=
=
;
two functions are ^11"^'>3) and
d (xyz)
d(n^u^u^)
d{inO
^
^^d{ n^
u^ u.
'
d[-rj,)
'l^W'il
d
)
Kow
(1^30).
(i»/;')
^
djj^.n^
^^
'
^
^^,
djn, 11^
»,)
'
c/(^<;0
'/'•'y^')
The second transformation is .seen at once from the form of the determinant by merely transposing rows and ci)lumns; the first and third are by
theorem (l()04j.
Theret'ore, by definition (itJ2'J), the llessiau of tt is a
covariant.
—
Variables are cogredient when they are
Coij^rcdicnts.
subjected to the same linear transformation thus, .r, ;/ are
1653
;
DIFFERENTIAL CALCULUS.
292
cogredient witli
=
y —
,r
wlicn
x', y'
a^^-hrj \
('i
<h ^
a'
^^^
+
y
=
—
—
ai'-\-hr)'
=
1654 Emancnts. If iu any quantic 2i
X into x-\-px' and y into y-\-py', where x
Tvith X, y; then, by (1512),
,
|
+ ^h
^'^
we change
{x, ij),
(p
,
_
y'
are cogredient
<l>(.r-\-p.v',y+py')
= n-\-p (x'd,-[-y'dy) u-\-iy (.rV/,+//V/,)- m + &c.,
and the
third,
coefficients of p, p^, p^,
emanents of u.
are called the
...
first,
The emanents, of the typical form {x d^-[-y' dyY
covariants of the quantic u.
1655
all
second,
...
u,
are
—
(.r, y),
we first make the substitutions wliich lead to the
Proof. If, in
emanent, and afterwards make the cogredient substitutions, we change
X into
And
if
x-\-px',
and
this into
a$+ J»; + (a£'+ 6»?').
we change
|0
the order of these operations be reversed,
cc
into al-^-hri,
and
this into
ail-\-pt)+l>
The two results are identical, and it follows
by the same operations in reversed order, the
{r)-\-prjf).
{x, y) be transformed
coefficients of the powers of p
that, if
^
in the two expansions will be equal, since p is indeterminate.
the definition (ltJ21>), each emanent is a covariant.
Therefore,
by
loOO
For definitions of contrngredieiits and cotitravariants, see (1813-4).
Fur other theorems on invariants, see (179-i), and the Article on Invari-
ants in Section XII.
IMPLICIT
FUNCTIONS.
IMPLICIT FUNCTIONS OF ONE INDEPENDENT VARIABLE.
If y and z be functions of <t, the successive application of
formula (1420) gives, for the first, second, and third derivatives of the function ^ (//, ::) with the notation of (1405),
1700
1701
4,A!iz)
<t>:
(!>^^
=
=
^:,!i,
+
j>..z...
'k,,.vl+--i4'...'ir~.+<t>u^:+<t',ir,.
+ 't'.~u-
IMPLWIT FUNCTIoXS.
1702
By
finoiitly
K
d/^)
=
iii:ikiii<j^
•.',.=
I,
*.,//^+;?^..//;~.+'?</»...//x-;+<^:..4
=
;.
^i)'.)
tlic
in
,r
::.,^=0, or else
last
by
tlifcc
formnlji},
tlilTcrciitiating
and
(-(.nsc-
independently,
we obtain
1703
=
<f>J.n,)
1704
<P:.r
K
1705
(>,/)
(,'//)
=
=
<l>„,/,,.
+
cf>_,..
(^,,/y:.+2(^,„,//,.+(^,,.+(^,//,...
(^,,//^+;?(^,,...//:.+i?<^,.,.//.+<^3x.
1706
In these formnlff! tlie notation <f>^ is used where the difforentiation
while ^^(.r, y) is used to denote the complete derivative of (}>(x,y)
with respect to x. Each successive partial derivative of the function ^((/, z)
(1700) is itself treated as a function ol' ij and z, and differentiated as such by
formula (14-20).
Thus, the differentiation of the product cp^y^ in (170(1) produces
is partial,
The
If it should not in fact
function <p^ involves y and z by implication.
contain z, lor instance, then the partial derivative 0„, vanishes. On the other
hand, y^, y,^, &c. are independent of z; and z^, Zjj., &c. are independent of?/,
DERIVED EQUATIONS.
1707
If ^
(•''7/)
= 0,
its
successive derivatives are also zero,
and the expansions (1708-5) are then called i\\Q firt^t, second^
and third derived equations of tlie primitive equation ^(.r//) = 0.
In this case, those equations give, by eliminating //j.,
1708
!hi
= — i^',
^JJL= 2 j>..„
j>.,. (f).,
—
<!>:, <f>;
—
<^,y (t>i
_
Similarly, by eliminating ij^ and >/.,,., equation (1705)
give y^^ in terms of the partial derivatives of ^ (.^7/).
See the note following (1732).
1710
SN'ould
1711
1712
Jf
4>{'n/)
=
^
and
Proof.— Bj (1708),
values of
t/o,
and
i)
y^,.
^,
niHl
^
= 0.
i^=0;
p{=:-hr.
di
dd'
</)^
H,,4.-<I>:A.^
Therefore (1704) and (J70o) give these
+
BIFFEUENTIAL CALCULUS.
201
f^Tid (\>y botli vanisli, 7/.^ in (1708) is indeterminate.
If
In tins cape it has two values given by the second derived
equation (170 I), which becomes a quadratic in y^.
1713
</'a-
1714
If
and
fl>.r,n
<\>-i.ri
also vanish, proceed to the third
now becomes a cubic in y^,
(poy
derived equation (1705), which
giving three values, and so on.
Generally, when all the partial derivatives of
a, y
orders less than n vanish for certain values x
have, by (1512), <}>{a, h) being zero,
1715
(j>
=
{a
i>
+ h, h +
=—
Jc)
+ K)"'^ (A'i/).a,i+
(/<4
{,r,
y) of
= h,
we
terms of
{x, y are
higher orders which may be neglected in the limit,
a,h after differentiation.) Now, with the notation
here put
=
of 1406),
H..a-\-Hv.b
therefore
=
= -t^ =
JL
0;
']!.;
the values of which are therefore given by the equation
1716
ihcl
1717
If
Vjc
+ kd,Y<l>{.r,y),^,,=
becomes indeterminate through x and y vanish-
— ^L
ing,
° observe that -^
dx
the latter fraction
algebraic methods.
and y
If X
equation
of X,
\p
0.
may
in this case,
X
often be
= 0,
y
is
and that the value
of
readily determined
by
[x, y) are connected by the
thereby made an impHcit function
in the function
(x, y)
more
(j)
and we have
1718
1719
M-,!/)
K
{>',
Phoof.— (i.)
(ii.)
.'/)
{
'^^^!^^-
i<i>,„t-'i'^M
Differentiate both f and
Differentiate also,
If ?/,?/, z are
1720
=
=
for x,
iK'i'-<i--^<i>.) r,
by (1703), and eliminate
by (1704), and eliminate
functions of
<t>r
\p
*
(^y^)
=
x,
?/,
and
y^x-
then, as in (1700),
<^. »/.,.+(^,.v.,.+(^.r...
y^.
.
205
iMi-rjrrr functions.
1721
<t>.r
('///-)
=
<!>:,.
+
To
1722
obtain
»/;+(^,,//;+<^,.4
Cf),,
U.,,
+
<l>„ //,.,
+
</).
and f
<p^{i'ir:)
^i r
.
iiiako
j.^ (.''.V"-) ,
//=,/•
in
tlio
abovo cMjnations,
U=
be a function of four variables conLet
(p (.?', //, ;:, I)
0, so tliat one of
0, v
0, iv
nected by three equations u
tlie variables, ^, may be considered independent.
1723
X
1
1
./
,/f
I
-
—
.
s
I
'^'''
'f
^'"'"'^
^/
1
./
'
^//
(/{•n/i)
(li
Observe that V^ stands for
tlie
'partial derivative of the function
^,
where
I
^ - — 715;^
^VOA
^^'^^
=
have, by differentiating for
^^^e
.
=
=
—
J
(Ufn'i(')
_
~ _ (.f..)
r/
J
.
_1_
.
'
./
'
complete and
</)f
for the
U
Proof.— (i.) U^ is found by taking the eliminant of the four equntions,
separating the determinant into two terms by means of the element f^—U^,
and employing the notation in (IGOO).
(ii.)
tions,
Xf,
i/t,
and
z>
are found by solving the last three of the
same equa-
by (582).
IMPLICIT FUNCTIONS OF
TWO INDEPENDENT VARIABLES.
=
alone be given, y may
y, z)
Since .r and z
and
be considered an imj)licit function of
are independent, we may make z constant and differentiate
for /'; thus, for a variation in x only, tlie ecjuations (1703-5)
are produced again with ^ (Xyy,z) in the ])lace of <p (.'',//).
1725
If the
equation
<p {.v,
.r.
;:.
1726 It" ' he made constant, z must replace x in those ecpiations as the independent variable.
=
:r,
first for
Again, by differentiating the equation <p{ryz)
z constant, and the result for z, making x constant,
making
DIFFERENTIAL CALCULUS.
296
vre obtain
1727
From
(/>.,.,+ (^,,.//.-+(^,.-//.,.4-f>.//.//.+<^.//.,-.
and the values
tills
1728
=
f/
and
of y^
~^
^" ^'' "^ ^"' ^" ^''
^"''
"^
^
0.
by (1708),
//,,
''"
=
~^
^' ^'"
''
"^
^_^^
be connected by the
a function of two
therefore make z
;?.
independent variables x
constant, and the values of f^ {.r, y, z) and «^2.r (^\ Ih ^') are
identical with those in (1718, '19) if x, y, z be substituted for
a?, y in each function.
1729
If
relation
3S,
i/-
Ih
tlie
i^^
-
By changing
1730
values of ^,
(,2",
function
= 0,
(,r, //, ::)
//
x into
and
y, z)
<^(,f, //, 2)
may be taken as
A¥e may
and
^o.
the same formula give the
;:
(-f, 7/,
,;•).
On
the same hypothesis, if the value of <^^ {x, y, ,?), in
forming which z has been made constant, be now differentiated for z while x is made constant, each partial derivative
^j., xpy, &G. in (1718) must be differentiated as containing x, y,
and z, of which three variables x is now constant and y is a
function of z.
1731
The
1732
result
is
f.(.r,,v,=:)
=
{(<l>..,<!,-^...,4'M-{hA,-^:,A,HAl
generally easier to apply such rules for
to deduce the result in
a functional form for the purpose of substituting in it the values of the partial
In a particular instance
is
it
example proposed, than
differentiating directly to the
derivatives.
1734
=
Example.
— Let
<p
(x,
?/,
z)
= Ix +
;»// +•
and
uz
X and z being the independent variables <p^ (;<•, y,
required.
Differentiating 0, considering z constant,
1,
;
4>x
= + m -•- = — m —
dx
(x, y, 2)
l
l
;
since
02X
a result which
values
is
Z,
^P.
(.r,
y, -)
y
and
,
dx
— xijr
tr
= x- + 7/ + z^
<p_^ (.c,
=
= -VI -^—^ = -m ^,—
are
;
y
V'y
+
y, z)
x"^
;
otherwise obtained from formula (1719) by substituting the
=
= 2x,
<p^
,
y, z)
(.»•,
z)
-~ = —
y
,
\p
Again, to find
</>y
^,
=
=
'">
^s
22/,
>P.
^^. (x, y, z),
=«
= 2z;
;
<p2x
,/.^
differcutiato for
= = ^2: =
= ^,, = = 2.
?»2j/
;
;/.,,
z,
considering x constant iu
the function
-JL—l-m —
ax
y
;
thus
—-^ =
dxdz
+?» --j-=
y
— w-r-,
y^
since
—= dz
=
\p^
.
y
nn-i.K'iT FrxrrioxFf.
297
=
U <p (;v, ?/, ^, ^, ij) be a function of five variables
by three equations vt
Oy w
0, v
0; so that two
may be considered independent, ]^^aking
of the variables ^,
constant, the e([uations in (172:3), and llie values obtained
1735
T^ct
=
coiniectetl
=
=
»?
?}
for [/f, X(, //f, r^ff, hold p^ood in the ])resent case for the variations due to a variation in S, observing that <f>, a, i\ ir now
stand for functions of rj as well as of ^.
1736 The corresponding values
by changing ^ into
of
Z7,, .r^, y^, z^
arc obtained
rj.
IMPLICIT FUNCTIONS OF n INDEPENDENT VARIABLES.
The same metliod
1737
is
applicable to the general case of
by r e(piations « 0, r 0,
=
a function of n variables connected
w = 0...&c.
=
—
The equations
constitute cuiij n
r of the variables wo
:
let these be ^,r),l....
The remaining r
variables will be dependent: let these be u:,ij, ::...; and let
the function be U=(l>{riy,z ... ^, tj, 2^ ...).
For a variation in ^ onhj^ there will be the derivative of the
function U, and r derived equations as under.
please, indeprndcnf
1738
(^.,,r.+(^,;/.+f,-,
+
+ U^Z^. -f
+ ^^.'A+i'.— +
U^.V^-\-Uj,?/^
«'x.*'f
f
...
...
...
+ ^. =
+ ;/,==
+ 7V =
L^^,
0,
0,
&c.,
involving the r implicit functions x^, v/^,
of the r equations, as in (1724), gives
(/(unr...)
-\MOQ
(ft'
1740
where
The
last value
J=
-ri
1
z^,
(/(unc...)
(///
f
•
Also -rp
being found (wactly as
The
&c.
—
solution
1
n
—g-
-r-
-r-^
in (172:3).
in like manner the
values of x^, i/^, ,v^, U^ and similarly with each of the independent variables in turn.
With
1741
^ re])laced
])y
»j
we have
;
1742
^
('j
.'/,
there
If
.
.
.)
= 0,
be n variables and but one equation
there will be ?? — 1 independent and one depen-
DIFFERENTIAL CALCULUS.
208
dent variable. Let // be flependent. Then for a variation in
X only of tlie remaining variables, the equations (1 703-5)
standing for (^ {x,y,« ...). If x
apply' to the present case,
be replaced by each of the remaining independents in turn,
there will be, in all, n—1 sets of derived equations.
</>
CHAXGE OF THE INDEPENDENT VARIABLE.
ic
If II be any function of x, and if tlie independent variable
be changed to i, and if t be afterwards put equal to y, the
p = t/j,:
following formulas of substitution are obtained, in whichi
!k
1760
(Lv
=
= 2lL
a\
l.
.Vy
we get
differentiating these fractions,
J'
oV
^
^^
=
1766
"
-^
^~
'*
^^
=Pir+inhr
^"^'''Z^'''''
'^;
j.^^:
1
Ex.
—
7AQ
^ '^°
=
^ _~
f/r
If X
r
^<^^
^+^'
<'o^
>',
cos
e-r
sin ^
'^'I'roof.— Wiitiiig
aJi« 2/29
;
7jg
Substituting
t
1770
f/'V
'
iu (17C0)
= r^ cos — r sin
= sin + r cos
fl
7-^
tliese Viilues,
=
;
X2e
;
yie=
iLv'
;
_
~
then
r-H-2>-^
{r,
cos
—
^-
and (17G2), we have
rr..^
'
>•
.sin
to tiud
df
,)•„
y,,
=
('\
»";«
cos
— 2i\ sin — r cos
''•«
sin
+ 2/-^ cos 6 — r siu 0.
from x
employ the formtda
{a-^hrr,/„,
;
the above results are obtained.
l^o cliange tlie variable
"(r/'-^'/A/-)
i^''
for
^
thus,
Xg
'",
= r sin B
cos 6 and y
^'^^
to
/
in
('/
+
= h^{d,-7T^\) K->7=2)
Which, multiplication by
^7^
/>.')
... (^/
by the index law
repetition of the operation d, (1102).
'\v„.r>
where
-1)//..
signifies the
—
CnANGE OF
<M('i +
Pk(K)K.—
Nuw
hj',
thcreforo
</,
L\
Thorefoi-e
and
=
(
i
?/,.
+
r,
Un
^'<^Ti.
—
=
=
h" (J,
F £.-/•'('>
Ijf^fc
c(jnations
tlic
?/
T,,
=
,
.
/.
('/.
-n)
-P.-
by
U,,
f^,..
=
^ (,/,-!)
//J-) .V,
((Z.
(J,
"^^^^
/^
?;„.,
= 1.
=
=
=
hx^ii',
-TT^)
'^'"y^ix
Z,
(,/,_„_ -J) r„
2,
&c.,
ir,.
.
. .
'»i'L'
hij,.
(<?,
- 1)
lu-
transformed
l)y
tlio
wliere x, u aro connected witli ^, ij
It is reciuired to cliaiige the
y to ^, rj in the functions F^. and F,,.
//)>
= 0, r = 0.
independent variables
— To
,d
+
- IT^)
('/+.'')".'/"..•
by
+ t.r)""'/;//,„ + (a + ij;)".V(...i;r}
this, and iluiioto (n + hj-)"i/„,
or
i,
U
Bamo fonuula by putting
1772
.
299
VAIUAlifJ:.
J
T,
13ut
1771
f/',.
(</,-«-!)
finally
Thercfdi-o
{n ("
Substitute
l>.r,.
=
=
'/J')''y„,}
= a+
T.) = «
r'
JSniWENDEyfT
Tin:
<r,
Difforrntiate V, u, v,
respect io x, considerinfj ^, tj functions of the independent variables x, y ; and form the (diminant of the residtiiKj
1773
cflc/i-
Kn>K.
find the value of F^
M"//7i
equations; thus,
I'i
Siruilarly, to find Vy.
DIFFERENTIAL CALCULUS.
300
1780
X, y, z;
Given
and ^,v,l known functions of
V^ are expressed in terms of F,, Fj„ T", by
V = f{x,y,z)
\% F„
the formult^
dV _
dV^ ddVO ^j dV^ di^VQ .J
'*'
d[A'f/z)
di
d{,v,/z)
'
'
dC
dr)
—
Proof. Differentiate V as a function of
^ with respect to independent variables
The annexed equations are the result.
a', y, z.
Solve these by (582) with the notation of (1600).
f,
d
+V^y]^+
^j
[.vt/z)
V^i^^
= V
t],
1781
^,
( T^s
d{^V)
rj,
I
Griven
V = f{x,
y, z),
in three equations
?i
wliere x, y,
= 0,
-y^O,
z
are involved witli
w = 0,
it is
required
to change the variables to ^, rj, ^ in F^j F'^,, and F,.
Applying Rule (1773) to the case of three variables,
n
we have
—
CUANGE OF THE INDEPENDENT
1784
1 X
«MrtK
T
1 r
Sin
1'
•
F.=
1786
1787
1788
1789
T;
V,
1^#
1790
and f;
=
To
+ F,
COS f
n
•
,
I'
,
sill
—
- F^
^
COS
VAUIMlLi:.
</>
r-^.
COSA
-,r
,
'\0\
F.cosO-IV"';;'.
= F.siu cos </)+];.sin sin Y'+F cos 0.
= V^ cos cos ^ + r cos sin — F, r sin 0.
= - r sin sin + F^ r sin cos 0.
)•
F,,
if>
^"x
find V^ directly; solve the equations ii, v,
thus,
;
in (1783), for
?v,
r, 0,
the sulution in this case being practicable
Find r„
larly, Fj,
0^,
=
from these, and substitute in F^
F^.i;,+ Fyy,+ F^v
Also F,.
^^
and
1791
To
Viiluu of
I'j,,
=
F..
SimiF,)-^+ T',0^+ V^'P^.
Similarly, F, and \\.
obtain F^^, substitute the value of F^ in the place of F, in tho
in (1785), and, in difi'erentiating Vr, F^, F^, consider each of
these quantities a function of r, 0, and (p.
To change the variables to r, 0, and (j>, in
the equations (178;3) still subsisting. Result
V2.-^V,,+
1792
=
V,,.-^l
therefore,
Also, since «
T;,cosec^6').
>•"
= p,
r sin
by (1770),
V,,
T;+-V(r>'ot6>+i;,+
r
Proof.— Put
V.^-\-Voy-{- ['^,
1'^+
= r cos
and
]:,.+
so that
F,,
=
Fv
x
+
= p cos
^
i;+
^>
and
jj
= p sin 0,
(i.)-
-\ F,,
sin 0, wo have, by the
p =
]\= V,r+~V,+ \ F.,
>•
same formula,
(ii.)-
^.-
J.
Add
together
(i.)
and
(ii.),
and eliminate F„ by (177(;), which gives
T
•
fl
,
T
COS0
r
If
r
l)r
a function of
the single relation,
1793
ii
vnrial.U'S
d--\-tf-\-z'-\-
»;,•+ 1'.,+ 1'..+^^-
./',
...=
//,
::
...
connected by
>*'"
= ''-+^
(!•)•
»V.
.
.
DIFFERENTIAL CALCULUS.
302
—
by differeutiating
,
(i.),
r
therefore
V-,,
=
V„ - +
\\
'-—^
=
V,r
^- +
F,,
=
F,.
l^
Similarly
F,
„
,
+ F. (i- - -^ )
,
&c.
Thus, by addition,
T2,+
F,,
+ &c. =
+^(^7
-,
T,.
^8
—
j
-
F,,+
-^T,,
LINEAR TRANSFORMATIOK
1794
If
I^
= /('^»
'^)5
i/j
and
the
if
equations u,
v,
w
in
(1781) take the forms
y
= a,i-\-b,7] + c,C
by (582),
of
^1,
then
,
j
being the determinant
Srj
= B,.v+B,,/-}- B,z,
{(h^'2<^z)i
and J^ the minor
&c.
1795
the
A
[
The operations
first set of
d^, dy, d^ will
equations below
and
;
now be transformed by
d^, d^, d^
by the second
set.
d,
d,
= {AM,+B,d,+C.Ak) - A
= {A,d,+B,d^+t\d,) - A^
=
Proof. — By
+ J^C^ and
+
[
d^^^
c^^
values of s„
Xt, &c.,
d^ij^
V^
d^
=
d^x^ + d^y^ + d,z^;
= a,V^ + aJ\ + a,V,.
and by substituting the value
we obtain
= h,d^^h,d,-^h,d^
= c,d^^c,d,+c,dj
r/,
d,
.
and the
from the preceding equations.
1797 From^ (1705),
again upon V^, we have
Vi^,
,
of V^,
Operating
and similarh; with
j'a,,,
the formula^,
1798
V,,
=
b; V,^-\-l/; V,^-\-lK^
V,,-\-2bA]\,+2hA y..r^'2bA
F,.,
[
.
)
cnAxni'j
OF THE iNDErENDr:xT vAj^fAnm.
ORTHOGONAL
1799
and
by (582, 584),
A=l, J^^a^,
etc.;
wo
liuvo
(Mjuatioiis
now become
1800
And
1802
TllAN.SFORMATION".
If the traiisfoi-mntion is oi'tliOLr<)n:il (-^S-l),
sinco,
(1701)
803
^r=a,^-^b,r}+r,C-
equations (1705) become
\(f;\
(i,=:a,d,^-h,(l,^^c,(l;
'aI;
.,
The double
d^
V,
dc
=
f/„
== h,d,-\-h.M„-^h,d, >.
d-
=
(i\d ^-\-<i.d„-\-(i(1.^
(\ </,
-h (',
</,
+
r,
d, J
between x. ?/, z and ^, 7?, t, in the six
equations of (1800-1), and the similar relations in (1802-8)
between dj.dy(L and d^d^J^, are indicated by a single diagram
relations
in eacli case; thus,
1804
i
tt'
V
I
fh
(fn
(k
:
DIFFEUEXTIAL CALCULUS.
304
When Y=f{d\ y,
1810
n.+
Proof.
T^2.+
z)
orthogonally transformed,
is
F,,=
V,,-\-V,,^V,,.
—By adding together equations (1708), and by the rehations
+
Oi
hi
+
=
c'l
1,
and
(fee,
aia„
+
hih„
+
c^Ci
=
0,
<fec.,
established in (584).
If two functions u, v be subjected to the same orthogonal transformation, so that
1811
u = (t>{.v,f/,^)
then
Ex.— Let
let
(f7„
$ ((7^,
But
t'oj.
= + +
i'
a'
cZ^,,
ax''
^^
2=
4/
+?^i',j,
,
2/i'i)^^.
Z/'ug^
(Z^, cZ^)
= 2,
c';
= r}^{.r,i/,-) = ^{tv^Q'^
{d^, d^, d^)
.(;'
Then
and
and v
v = ^ {d^, d^, d^) v.
= + hf + cz^ + 2fyz + 2gzx + 21ixij = ^
= aV -^h'n'-^ cC + 2/ v^ + Ig'Kl + lliiri = *,
v =
+ / + = ;H»r + ^'= and ^.
+ci-,, +2/^^, +2r/y,, +2/(V^y
cZJ v = ar,^
+ c'v^^ + 2/'t?^^+ 25r'v^j +
v = a'v^^ +
= 0, &c. Hence the theorem gives
and
(^
1812
and
= ^{^,y],0
v,,,
in other
words, a-\-h
+c
is
a-\-'b-)rC
an invariant.
—
AVhen the transformation is not
1813 Contrnfiredicnt.
orthogonal, (1795) shows that d^. is not transformed by the
same, but by a reciprocal substitution, in which %, &i, c^ are
replaced by the corresponding minors A-^, B^, Oj. In this case
d^, dy, dg are said to be contragredicnt to a', ^, z.
—
1814 Contravariant. If, in (1G29), the quantics are subjected to a reciprocal transformation instead of the same, we
obtain the definition of a contravariant.
1815
and
2/,
AVhen is a function of two independent variables x
the following notation is often used
;<;
dz
_
dz
dp
dif
^
(hi
d.v
^
dz
^
d.vdij
Lot
(-/', ?/, z) = 0.
pendent variables from
(/)
_
dp _(Px
d.v-
Ty-'^'
Tv^^''
f!l^fll
^^
djf
'
It is required to
.r,
y to
z, y.
=r
da}
=
t,
dif'
change the inde-
The formulas
of trans-
—
—
.
DIFFEREXTIAL CALCULUS.
306
(x) is a maximum or minimum
Rule II. Otherwise
an odd number of consecutive derivatives of (x) vanish,
and- the next is minus or plus respectively.
1833
<|>
ivlien
</>
—
(i.) The tangent to the curve in the last figure becomes parallel
Pj^OOF.
as x increases ; therefore, by (U03),
to the X axis at the points A, B, C, B,
tan 6, which is equal to /(.!•), vanishes at those points, while its sign changes
in the manner described.
(ii.)
a;
= a,
Let
E
/"(,«)
be the
first
derivative of
n being even; therefore, by (1500),
which does not vanish when
/(.?')
= f{a)+
/(a ±70
^-^f'{a^Hh).
The last term retains the sign of /"(a), when is small enontrh, whether h
be positive or negative, since n is even. Therefore f {x) diiiiini.shes for any
small variation of a; from the value a if /"(a) be negative, but increases if
/"(a) be positive. Hence the rule.
7;
—
Before
1834 Note.
maximum or minimum,
applying the rule for discovering a
Ave
may
evidently
(iii.)
any constant factor of the function ;
raise it to any constant poioer, paying attention to sign;
tahe its reciprocal ; maximum becoming minimum, and
(iv.)
vice versa ;
tahe the logarithm of a positive function.
(i.)
(ii.)
1835
reject
Ex.
1.— Let
Also
(j)"(x)
{x)
<p
=
—
= x'-7x*-d5x + l,
= 7 (x'-5) + 1)
= V5 makes ^'(.v)
T.^^- 28a;» - 35
f Oe)
Therefore
7(6x^ l2x^).
therefore
=
f"(x) positive; and therefore makes
(p(x)
(,r^
a;
vanish,
and
a minimum.
Here
<i>(x) = (x-2y'(x-2y\
1836 Ex. 2.— Let
f (ao = u(,x-^y'(x-2y'+u(x-sy\x-2y'={x-sy\x-2y\2r>x-6i),
and we know, by (444) or by (14G0), that, when x =
and 13 is an odd number.
derivatives of ^ (c) vanish
;
either a
maximum
or
minimum when
S,
the
first
Therefore
thirteen
(x) is
= 3.
«
To determine which, examine the change of sign in (^'(x). Now (,r — 3)"
changes from negative to positive as a: increases from a value a little less
than 3 to a value a little greater, while the other factors of <p' (x) remain
Therefore, by the rule, cp (a;) is a minimum when a- = 3.
positive.
Again, as x passes through the vakie 2, 0' (x) does not change sign, 10
[x).
being even. Therefore x — 2 gives no maximum or mininuim value ot
Lastly, as x passes through the value
in <p'(x)
from
+
1837
ofy.
change from (-)
to
—
Ex.
;
(
+ )( — )
to
and, consequently, f
3.— Let
^
(,r,
y)
=
x*
(
(a;)
-*.-
the signs of the three factors
— )( + )( + );
is
that
is, (/.'(•«')
changes
a maximum.
+ 2x-y-i/ =
0.
To
find limiting values
—
Axn mimma.
^r.\xl^^.\
307
Here y i.s <,nvon only as nn implicit fuiicLion of
order to employ ronnulto (17uH, 1711),
=
+
+ Iry,
0, = 2.r - -uf
f =
= 0, wo got
= makes 0^=0. Solviiif,' tins eiiuatioii with f
= =fcl, = — 1 wlieu
vauislu's.
—- = 8, positive therefore, when = ±l,
= — 0., = \l-\And then
,
7/,
a'
•l-.c»
1 '2x-
if-^
•!//,
;
(.c,
y has
—1
2/
=
making
the independent vaiiaMe,
>/
limitino- value of
<^
condition
tlie
obtained from
^
equation
tlic
may be shewn
it
maxinmm and miniinum
has both the
.c
,
A
1838
a;
;
a minimnni value.
for
subject to
^^
-
Similarly, hy
when
i/)
i/r
//
ij..^
is
in
niUi'ieutiiiLiiig,
x.
(.*',
(,r, //)
^,i//,,
=
values
±
that,
v^G.
//),
=
i)
(i.),
^,^x//,
(ii.)
Simultaneous values of x and //, found by solving equations (i.) and (ii.), correspond to a ni ixiuium or minimum
value of
(p.
— By (1718),
Proof.
<t>r(-v'j)
=
1839
Ex.-Let
Kipiation
(ii.)
Solving
thi.s
(j>
being virtually a function of x only
with
and
f{x,y)=X!,
becomes
(i.),
//
(:>//"
we Hnd
—
//*
.')
=
4^
(x,
=x (G-r" —
2x^
and
y)
;
and, by (1832),
= 2x' -,•>, + y' =
(i.)
?/).
x' (I./-—
y^).
Therefore .i;=^\'"2, y=\l''\- are values con-esponding to a, viaxinnim
That it is a uiaxiinum, anil nut a ii-iiiimum, is seen by inspecting
value of
(}).
e(piation (i.)
1840
Most geometrical problems can bo treated in this way, and the
alturuative of maximum or minimum decided by the nature of tho case.
Otherwise the sign of <p,^{xii) maybe examined by formula (1710) for tho
criterion, according to the rule.
and Minima
Ma.i'imn
raJue.s
of a function of two
inilcpcndrnt variohlcs.
1841
Rl'LE
irlirii (p^
find
minus or from
—
.1 faacfioii
(x, y) Is a maximum or minimnm
hofh vanish and changr fhcir si(fns fnun jilns to
minus to plus rrsjirrflrrh/, as x aud y nicrraxr.
I.
<p^.
(j>
^o,. —
1842 Rri'E II. Other wise, 0, ami must ranish ;
must ho positive, ami ^^x or ^^y ^'^''t*'^ '^'' negative for a ma.vimum
and positive for a minimum value of <p.
Pkoof. By (1512), writing A, B, C for i^^,, (p^^, fi,j, we have, for small
(j>y
</>.,
</)';y
—
changes
h,
Ic
in the values of x
f i-c+h, y + k)-(l>
(x,
y)
and
=
y,
%+ %+
! (A}r + 2DUc + CIc') + terms
which may be neglected, by (lilO).
mFFEUEXTJAL CALCULUS.
308
Hence, as in the proof of (1833), in order that changinc^ the sign of/; or
have the elYect of changing the sign of tlie right side of the equaThe
tion, tlie first powers must disappear, tlierefore f^ and ^y must vanish.
next terra may be written, by completing the square, in the form
Tc
shall not
-—
\
iA—-+B\ +AG—B-
i
and, to ensure this quantity retaining its
;
AC—B^ must be positive. ^ will then
sign for all values of the ratio h
be a maximum or minimum according as A in the denominator is negative
or positive.
Hence
It is clear that A and B might have been transposed in the proof.
must have the same sign as A.
/.-,
:
B
A limiting
1843
value of
from
is ol)tainccl
1844
<#>A
1846
or, as
tlie
^
y,z)
(<r,
=
(i.),
two equations
= M,
M. = <^A'
(ii-).
may be
they
y, £),
<^ (.v,
subject to the condition
—
2j:
written,
xi'
V'.r
=
0"-);
i-£
(iv.)
V'.
V*-/
Simultaneous values of ,t, y, ,?, found by solving equations
correspond to a maximum or minimum value of (p.
(i., ii., iii.),
— By
Proof.variables x
and
^^C*',
?/,
being considered a function of two independent
and, by (1729, 1730),
(184-1),
z,
2)
=
gives
and
(ii.),
maximum
(p.^sc,
minimum
z)
ij,
=
gives
(iii.)
(1842) may also be applied
without eliminating y by employing the values of (j>2x and <p-,. in (1719, '30).
The
1847
and
criterion of
Ex.— Let
i^i.r,!/,::)
Equations
(ii.)
r
{.r,
cp
or
//, ;:)
= /- + +
(iii.)
=
,
licre
2fi,::
+ 2r,xx-\-2hvi/-l = 0..(i.)
become
r
'i
=
(i.),
From equations
1848 o,vi-/ni +
—
1849
= Rx^
= By
gx-^fy+cz
= li::)
or
^''"^
-"
.'/'
*/*•
we have
hx-\-hij+/z
r,::
= i, sav, (iv.)
=
.r +
+ =
'.
Therefore, by proportion (70) and by
(iv.)
:-
//-
= a:r + hr-^cz' +
and
in
[
a
li
cj
]t
I
;
l-Ti
//
q
t
{R - a) {R - h) (R - r)
+ 2l]ih
-{R-a)f-{R-h)</-{R-c)
Ir
f
=
0,
c-n
= 0,
or (see
KUl)
— —
.
MAXIMA AXn MIXIMA.
mR
is tlio climinaTit of the three equations in
called the (h'srriiniinitiinj nihlr of the (juadric (i.),
its roots are the I'eeiprocals of the niaxinia and niiniiiia
Tliis cul)ic
(T,
309
It is
//,::.
and
values of
-\-
-^ ::'-
if-
To show
1850
are
x-
that the roots of the disciiminating cubic
all real.
Let
be the roots of
li,
7i'i,
1851
J^^
tlic
,/
M'T^ Ji'-r
>
and
h
R=
and
r,
and
h
c
\
>
quadratic equation
= {j^-b){K-c)-f =
o
(v.)
/?.,.
and tlic result is rerrative, bcinp^ minus a
Thereh'.,, and the result is j)ositive.
pqnare tiuantily, l)y (v.). ^lako 7i*
fore the cubic has real roots between each pair of the consecutive values 4-00
liut since the roots are in order of
that is, three real roots,
Ji'p 7?.,. —00
magnitude, the first must be a maximum value of JJ, the third a minimum,
and the intermediate root neither a maximum nor a minimum.
Intake
in
it',
cubic,
tlio
=
,
;
M(tA''nu(i (lud
1852
1>±r,
I^t^t
«/>2,/'
</>2--.
<{>
Minima values of a fuuctiitn of three
or more variables.
Let
he a function of three variables.
bc dcuotcd bj a, h, r,f, g, h and let
be the corresponding minors of the deter-
{''!/-)
</>,/-->
</>.-.r,
fr;,
;
A, B, C, l\ (r, 11
minant A, as in (1G42).
^(x,y,z) is a maxiimnn or minimvm wlirn
and change their signs from plus to minus
or from minus to plus respccticehj, as x, y, and z inrreasc.
Rule
1853
<^x)
^y>
^7.
I.
^^^ vanish
Other'^'ise
—
TJir first dcriraflrrs
1854
Ri'Ti: II.
and
cocfiirimt iu
of
(j>
nnisf
rani.di
;
A
reciprocal determinant of A nmst Iw
u-ilt tie a maximum or minimum according ax
jwsitire ; and
Or, in the place of .1 and a, read Ji
a is negatire or jiositice.
its
tin'
«/>
and
b or
Proof.
C and
the values of
fix + l,
c.
—Punsning
y
.r,
//,
z.
the method of (181-), let
(lolt),
+ n, 2 + 0-9
=
c>,
t,
»/•
C be small changes in
By
('.
2/'
2)
+ 19, 4 ^9. + i
("i'
+
W+
cr-
+ 1H + 2;/;^ + 11<in) + Ac.
The quadric
For constancy of sign on the right, 9^,
9. must vanish.
may then be re-arranged as under by first completing the square of the term.s
It
in t. and then collecting the terms in C, V. ft»d completing the square.
<p,j,
thus becomes
]
'2.1.1
K.
(ai,
+
Jir}
+ g^) + ^—^^
Li
~^ ^
—-7,—
L
"
—
—
DIFFE Z? ENTTA L CALrUL US.
310
Hence, for constancy of sign for all values of E, »?, i^, it is necessary that G
and liC—F'^ should be positiVe. Tliis makes B also positive. By symmetry,
AB-R-
will all be positive.
evident that A, B, G, BG—F", CA-G',
sign of a in tlic first factor then determines, as in (1842), whether ^ is a
maximum or a minimum.
it is
The
1855
The
BG—F' =
condition
may be
put
Since
otherwise.
by (577), the condition that BG'—F"- must be
positive is equivalent to the condition that a and A must have
the same sign. Hence we have also the following rule
a/^
:
1856 R-ULE III. (j>^, (^y, must vanish ; the second of the four
determinants below must he positive, and the first and third
must have the same sign : that sign being negative for a maximum and posltlnefor a mliilinnm value of (^ (x, y, z).
(f>^
1857
4>..:
>!.
f-
<l>..;,
<#>,,--
i
^f^XT^fA
Tlie ilctcrniinant
axd
nil
.vf.wv.t.
tho elimiuiuit of «lio four r(iii!iiiotis, l)y ^•''.^3), nn<l
A'-^
inothod i)f (1721, Proof, i.), to A'-A/<=0, or u
ig
is
c<|uiv:.U-iit, l)V tl.o
=
A
(Notation of'ir.lG).
To dL'tennine whether this value of ii is a nmximtim or minimnrn, either
of the conditions in (185 1, T.) may be npplicd and since, in this example,
'2b, Ac, tho letters a, h, c, f, <j, h may be considered identical
«2,= )la, Uiy
;
—
with those in the
1862
tion of
rule.
To determine
(,r, //,
a limiting: value of
vai'iaMcs connected by n eijuations '^i
111
R^'j^E,
— Axaiimn
liquations:
+ X.,(u,\+
fv + Xi(ni), + X,(n,\-h
•••
+X„(u„),
...
+A„(ii„),
//o
= 0,
X;;,
Xi,
—
</)x-fXi(Ui)x
a ftnic-
", ...),
= 0,
n inidetcrmlncd mnUiplu'rs
m
iciih the JoUoiriiKj
</>
...
...
^n
= 0,
= 0,
m
m-\-n eqxdfions in
+ n quant if i(\'i, x,y, /, ...
The valves of x, y, z, ..., found from these
equations, correspond to a maximum or minimum value of (p.
mal'ing
and
X,*,
in all
X.,
...
X„.
—
Pkoof. Differentiate f and ?/,. ii„_, ... «„ on the hypothesis that x, ?/, z,...
Multiply the resulting
are arbitrary functions of an independent variable /.
equations, excepting the first, by A„ Aj, ... X„ in order, and add them to the value
to zero, since the
be
equated
now
may
The coefficients of x,, y„ z„ ...
off,.
functions of t are arbitrary, producing the equations in the rule.
1863
Ex.
1.— To
conditions
Here
A.c^
m = 3,
tions in the rule
2x
?!-
find the limiting values of r^
=2
;
+ 2AXx + nl =0
+ fim =
+ 2C\z+ttn
a;2
(2)
=0
+ y» + 2= +
>
.
and we obtain
(3))
(.'1.1-'
+ By' + Cz')
result
+
—f—
Ar' —
is
i
X
= 0,
solve for
;
a-,
+ nz —
lx-\-vuj
in
The
with the
Multiply (1), (2)i (^) respectively by
x, y, z, and add; thus ^ disappears,
(1)-^
Substitute this in (1), (2), (3)
1864
z^
I
become
2y + 2]i\y
2z
= x* + y^ +
and lv + my-\- nz = 0.
and, choo.sing X and /u for the multipliers, the equa-
+ By- + Cz^ =
therefore
y, z,
= -r\
their values
0.
=
— 1 + -^^
Lr —
-"'--JJr'
\
and substitute
0,
a quadratic in
r'.
roots are the maximum and minimum values of the square of the
1 made by
radius vector of a central section of the quadric A.v' + Hy'^A- Cz'
nz
0.
the plane Ix-\-my
The
=
+
1865
Ex.
2.— To
=
find the
subject to the condition
maximum
N=
value of u
a'b^c*.
=
(x
+ 1)
(y
+ l)
(^
+ 1),
'
BIFFFnEXTTAL CALCULUS.
312
This
equivalent to fiuding a
is
maximum
valua of
+ 1) +\og (y+l)+\og (z + 1),
logN = x\oga + y log h + z\og c.
log (x
subject to the condition
The equations
in the rule
JL+Xloga =
become
-L.+\\ogh =
0;
-l--+k\ogc
0;
= 0.
eliminating X, these are seen to be equivalent to equations (\SW)).
!Multip1yingnp and adding the equations, we find X, and thence x + \^ y + 1,
z+l; the values of which, substituted in u, give, for its maximum value^
{log (Nahc) }«-=- 3 log a^ log b'' log c\
u
By
=
Compare
where
(374),
a, b, c
and
x, y, z
are integers.
Continuous Maainia and Minima.
1866
where
If fr
P
and
•/>-/'
and Q may
equation
of X
ai^^^
=
{x, y)
4<
For
y.
all
"1 (18-i-)j liave a
common
also be functions of x
factor, so tliat
and
7/
;
tlien tlie
determines a continuous series of values
these values <p is constant, but, at the same
time, it may be a maximum or a minimum ivith respect to any
other contiguous values of cp, obtained by taking x and y so
that xp (xy) shall not vanish.
1867
In this case, i>2^<p.2y — <t>% vanishes with
Rule II. is not appUcable.
;/.,
so that the
criterion in
Proof.
—Differentiating equation
^,„
= Q,^+(H,
If from these values
the expression.
1868
Ex.
— Take
z"-
^'
=
)
we have
fv
= QA + Q-hi
— ^.y X ^„^,
will
appear as a factor of
= cr-lP + 2by(.c' + y')-;,r-y'
(i.),
we form
z
(i.),
as ^
^2^0^^
(.nj)
-(77^-1)
in the
-^y
equation
-=
-^^
The common factor equated to zero gives xr
Here a is a continuous niaxinmm value of
-\-
y"
2/(
= b', and therefore 2 = ±a...(ii.)
3, and —a a continuous minimum.
Equation (i.) represents, in Coordinate Geometry, the surface of an anchor
its centre at a distance b from
a
having
ring, the generating circle of radius
Equations (ii.) give the loci of the highest and
the axis of revolution Z.
lowest points of the surface.
For the application of the Differential Calculus to the
Theory of Curves, see the Sections on Coordinate Geometry.
INTECxRAL CALCULUS.
INTRODUCTION.
1900 The operations of different laf ion mid integration are
Let /'(/*•) be the dm-ivative of
the converse of each other.
</)(.7;); then «/>(./) is called the integral oif(x) with respect to x.
These converse relations are expressed in the notations of the
Differential and Integral Calculus, by
!Mii
=/(,,.)
and by
\f{,r)
—
,/.,
=^
Theorem. Let the curve y =f(,r) be
(1403), and any ordinates LI, Mm, and let OL
<p(l>)
<p{a).
then the area LMuil
1901
=
—
—
(.,).
drawn
as in
= a, OM=b;
PN
OX
Proof. Let
be any value of x, and
the corresponding value of y, and let the area
ONPQ A then ^1 is sunie/iniction of x. Also,
dA
if NN'=(lx, the elemental area NN'FP
=
= ydx
;
=
in the limit
;
therefore
—ax
= y.
Thus
A
that function of x whose derivative for each
(x) + C,
value of X is y or f{x) therefore A
where C is any constant. Consequently the area
is
;
=
=
(f>
J\,W
M
— («). whatever C may be.
The demonstration assumes that there is onl}' one function f (r) corresponding to a given derivative /(x). TliLs may bo formally proved.
LM)nl
<})
(i)
v>
If possible, let \l^(x) have the same derivative as <{>(-r); then, with tlio
same coordinate axes, two curves may bo drawn so that the areas detined jw
febove, Hke LMinl, shall be (j>{.r) and \p(.i-) respectively, each area vanishing
with x. If these curves do not coiiieide, tiien, lor a given value of x, they
have different ordinates, that is, f (x) and 4^' i-^) are different, contrary to the
hypothesis.
The curves must therefore coincide, that is, 9 (x) and ^ (x) aru
idtutit'al.
INTEOJiAL CALCULUS.
314
1902 Since <p(h) — (\>{a) is the sum of all the elemental areas
NN'P'P included between LI and Mm, that is, the sum of
like
the elements ydx or
a and
h,
/(.-r) d.v
taken for
all
values of x between
this result is ^Titten
f/(.r)rf..
= .^(6)-.^(«).
1903 The expression on the left is termed a definite integral
because the hmits a, h of the integration are assigned.*
Wlien the limits are not assigned, the integral is called indefinite.
1904
By
taking the constant (7=:0 in (1901),
ONPQ =
area
Note.
4> (.r)
=
the
(/(.r) dv.
«/
—In practice,
the result of
we have
the constant should always he added
an integration u'hen no limits are assigned.
to
MULTIPLE INTEGRALS.
1905
Let f{x,
y, z)
be a function of three variables
;
then
r«-2 f*v2 r*^2
the notation
is
j
J.ri
\
Jyi
/(.r,
\
*.
;/,
z) d.v
dy dz
zx
used to denote the following operations.
=
Integrate the function for z between the limits 2 ^i, z^z^,
Then,
considering the remaining variables x and y constant.
whether the limits z^, z.^ are constants or functions of x and t/,
Next, considerthe result will be a function of x and // only.
ing X constant, integrate this function for y between the limits
which may either be constants or functions of x.
7/i and 7/2»
The result will now be a function of x only. Lastly, integrate
this function for x between the limits x^ and x^_.
Similarly for a function of any
number
of variables.
The clearest view of the nature of a multiple integral
afforded by the geometrical interpretation of a triple in-
1906
is
tegral.
* Tho integral
niivy
be read "
Sxiin a to b, j (x) fix''; | signifying
"sum."
lUTj
jXTnnriii'Tioy.
Taking
z
=
(,/',
<i>
//)
cylindrical
rectangular
coordinate
axes,
let
tlie
surface
by the
(A'ro7)i in the figure) be drawn, intersected
surface y —. xp {,r} (UMNiim), and by the
])lane
= a\
(LSmI). The' volume of the solid OLMNohan bounded
by these surfaces and the coordinate planes will be
x
»,
..
«.
Proof.
— Since
tlie
volume cnt
off"
c
parallel to OYZ^ and at a
must be smue J'nnct'ioti. of x.
change in its value due to a
I'Qqp X d.c, au element of the solid
varies continuously with r,
this volun:e, and let (/ T be the sniidl
distance x iVom
it,
Let Fbe
change dx in x. Then, in the limit,
shown by dotted lines in the figure.
^=r(2'ii'=J^
«.
(I
by any plane
JV=
it
Therefore
?>0^.'/)^///,
by (1002),
X being constant throughout the integration for y.
Making x then vary from to x,
function of a; only.
volume,
\Jl
t(..-..v).'y|.''
=
=
The result will be a
we have, for the whole
.'-].'."(./..
J,
!j,,
[J„
With the notation e.vplained in
z.
since f (x, //)
not required, and the integrals aro written as above.
(l'.iO:)j,
the brackets aro
(.»', y),
bounded by two surfaces z, =
?s = 0j («• !/)•
\//,(.r), i/^ = \p.i(x), and two planes x = x^, «=;•'».
//, =
the volume will then be arrived at by taking the diffV-rence of two similar
integrals at each integration, and will be expressed by the integral in (I'JUS).
if any limit is a constant, the corresponding boundary of the solid
becomes a plane.
1907
two
Tf the solid is
cylindrical surfaces
(/),
—
^
.
INTEGRAL CALCULUS.
316
METHODS OF INTEGRATTOX.
INTEGRATION BY SUBSTITUTION.
1908
( (^
Tlie formula is
(.r)
=
(Lv
(
cf>
^'
{.v)
some function
wlicre z is equal to/(.T),
tate the integration.
dz,
UZ
»y
«,
cliosen so as to facili-
Rule I. Put x in terms of z in the given function, and
multiply the function also by x^; then integrate for z.
a,
If the limits of the proposed integral are given by x
b, these mast be converted into limits ofz by the equation
X
=
=
^ = t-'W-
The following
substitution, and
Rule
1909
then
Ex.
^
<t>
1.— To
tbei-efore
II.
rule presents another view of the
useful in practice.
—//"^(x) can
(.r)
(Lv
=
=
1
-|
,_,"
—
—^^'^ dx
2.
J
Here
z
x
+ x'
be expressed in the
(z)
——•
Substitute
=
i?
^—
r
]
;
^^
= '°«^ =
l^i--±:^ dx
J
x''
+ x-'
= x-'-\- x-\ F (z) = - --,
•go
1—
-^^
d^
l+rx' y{i + ax'' + c'x*)
dJx-'^
r
—=
'"^ f"
x
--
+
+
—
^/(.r
+ a")
-,
„
^^''+'-'»-
= - logov(x-' -r
+ x-').
y
z,=
~
z =^
- (rxv-' + 2x--).
3'"'— c
dx
]x-' + cx ^(c-x^ + x-'' + a)
+ r.r)dx
1
1
XK/(a-2c)
,
V{a-2c) ^^
_
—
or
Bj
(1027) or (1026).
of
form F(z)Zj;
= f ^ (^) d^-
zA^'
=
;
\vw^> t " =
Ex.
V
I
integrate
---
method
is
Here
k
o
^/(l
+ ax' + rx*)
l+cx^
1
/
+
^
Vi2c-n)
Vi2c-a)
= x-'+cx
cos
—+ —
_,«N/(2r-rr)
'
;
7,
i
cx'
;
METHODS OF IXTECHATIoX.
31
In Examples (2) ami (3) the process is analytical, and leads to the diswifli respect to which tlie intejrrution is
covery of the particular fiuietioii
z be known, Ride 1. supplies the direct, thoujjh not always tho
effected.
;:,
U
simplest,
method of integrating the
function.
INTEGRATION
1910
T\v differentiating
general formula
The value
of the second
\
7(V
i(,r(Lv
of the
fii-st
PARTS.
\)Y
with respect to
=
iir
—
integral
\
is
./',
we
obtain
tlio
ur.d.v.
thus determined
if
that
known.
is
— Separate
the quantity to he integrated into two
Integrate one factor, and differentiate the other with
If the integral of tlie resulting quantity is
\'esprct to X.
known, or more readily ascertained than that of the original
one, the method by " Parts" is applicable.
Rule.
factor.^i.
1911
Note.
— In
subsequent examples, where integration by Parts ia
which is to be integrated will be indicated. Thus, in
"By Parts |6'"'(/.c" signifies that e" is to be integrated and
directed, the factor
example
sin
h.c
(10-">1),
differentiated afterwards in applying the foregoing rule.
more frequently integrated than any
The
factor 1
other, and this step will be denoted
is
by \dx.
INTEGRATION BY DIVISION.
1912
A
formula
is
^{a+hry d.v = a ({a + Lv"y-' + h
f.r" («
+
/>.<•")"-'
d.r
:
The expression to be integrated is thus divided into two terras,
the index p in each being diminished by unity, a step which
often facilitates integration.
Similarly,
=
1913
a (a + hx" + cx'"y-^
Ex.— To
By
Parts |
By
Divi.sion,
dx
j (a + bx" + ex'")''
+ hx" (a + i.c" + cj;'")''-' + ex'"
integrate
J.r,
|
+ cx''-y-\
\^(x' + n')dx.
(.r'
+ cr)
dx
=x ^
v/(.r'
+ a')
dx
=
y
(a + hx"
j
I
(.."
+ a')
^^^:^^^^
-|
+
-^f^^^y
j v/(xHa')-
Therefore, by addition,
=
;iv'(i'+(i')
+ J.<Mog{.c+
v'(r
+ a')}, by
(1900, Ei. 1).
INTEGUAL CALCULUS.
318
INTEGRATION BY RATIONALIZATION.
1914 In tlie following example, 6 is tlie least common denominator of the fractional indices. Hence, by substituting
Oz^, we have
z
x^, and therefore x^
=
=
'•-'-
z^-l dx
dz
1
\
dx.
Each term
of the result
is
directly integrable
by (1922) and
For other examples see (2110).
(1923).
INTEGRATION BY PARTIAL FRACTIONS.
Rational fractions can always be integrated by first
The theory of such
resolving them into partial fractions.
1915
resolutions will
1916
<}>{x)
now be
given.
If </>('^0 and F{x) are rational algebraic functions of x,
being of lowest dimensions, and if F {x) contains the
factor {x
— a)
once, so that
F(a>)-(.r-«)t/,Gi')
(1);
«.n|« = _l_ + |gana^ = iM
1917
Proof.
— Multiply equation (2) by
(,,
(I), thus
= A4^ {x) + (x-a)x (x).
Also, by differentiating (1), and
(a) = A4^ (a).
Therefore, putting x = a,
(a) -^ F'((i).
Therefore A =
(a).
putting x = a afterwards, F' (a) =
(x)
(j>
(p
<p
\p
1918
Again,
if
F{x) contains the factor
F{d^
that
Assume
=
(x
— a),
n times, so
(.{—«)"</» Or).
= 7{,v~a)"
^V^
T:r-i + ---H
hr, +
d-a rT7~TF{.c}
id—a)"
Mulliplij
(x — a)"; put
A,,.
determine A^, A^
7
xli[.r)
To
X^a
...
hi/
1919
and, dijD'crentiate, alternately.
If
^'(^0
=^
^^^ ^ single pair of imaginary
roots
—
.97'.
a±li5;
and the
tlu'H,
XD.
/«
I
31
D TSTIU! U.\ LS.
applying (1017),
lot
partial fractions corres[)on(ling to these roots will
A-ili
For
1
A-^ili
_'2\(,-a) + 2li§^
methods of resolving a fraction into
which occur, see (235-238).
practical
different cases
bo
partial fractions in the
INTEGRATION BY INFINITE SERIES.
"When other methods are not applicable, an integral may
sometimes be evahiated by expanding the function in a converging series and integrating the separate terms.
Ex.
j
- dx =
logx + ax
+
j-^^
+ ^-^, +
3-2-3-^,
+ &c.
(150)
STANDARD INTEGRALS.
1921 Some elementary integrals are obtained at once from
known derivatives of sim})le functions. Thus the deri-
tlie
vatives (1422-;)8) furnish corresponding integrals.
lowing are in constant use
The
fol-
:
1922
1926
j.vrf..=^.
f_^ =
}- =
''"'-'
lc„s-^or-lsi,.-.^.
[.SU..1
.
TNTEGRAL CALCULUS.
320
- log
{^v^V{^i'±a-)}.
1928
r
1929
(
1930
(-yf^ = ±^^{a'±.v-).
1931
^y
[
\/(cr^
1932
J
.
,/^:'\_
...
—
-^-4^^
=
-
v/(ci''-
f v/(«'-ci-)
/5'
Jf V{(r—cV-)
r
1935
d.v
.,,
T i«^ log
= ^a' siu-^ ^" +
}
±a'-)}
.
1937
f^!^,
= — cos X,
1
sin
1940
\
tana dr
= — log cos
1942
\
seCcvrfcr
= log tan (-4- -J.
'2),
[By
^^-;:^.
= ^log'i±i-.
1938
.i\
substitute cos.c.
in (1032)
(^36)
a
a
_^, = J- log
r
Partial fractions
Do.
[
\
cos
.v
dx
= sin
j
cot
a:
dd'
= log sin
1
a;.
<r.
coscCcvrAr^logtan
(1941,
•
V'(«--cr^).
[As in (1031)
ict'
- 1 cot"^ i-
or
«
1936
— (1940,
v/(a'^
[As
_^, = 1 tan-^ ±
a
-I' t?'*^
{.r+
= i«'- «m- ^a - i.r v/(«-.ro\
J .r--h«
Meiiiou.
(1434)
± a') ± i«' log U + V{.v^ ± «")
i.i-N/Gf'±«^)
-^
1934
- cos-' —
or
Parts, Division, and difference of results,
7^^ =
1933
—
and adding results (1913), we obtain
Parts, Division,
± a-) dx = U
By
siu-^
(1009, Ex. 1)
'3),
substitute sin
1944
I
«in"'
<^'
d.v
=
.V
sin-^
.v
+ v^(l —d")
1945
\
cos"^
.r
d.v
=
.r
cos"^
.v
— v/(l —
.r")
.
.c.
'-.
(7/7? (•
'L A
I
Ji
Fl 'XC TJONS.
321
1946
\iin\-'
a- ill
=
.r
-
]
lo-
1947
\voi-' Jill
=
.r1j»ir'.r+
I
l()-(l+.r-).
1948
(*soc-^
=
.r
1949
\
.r (/.r
1
according as a
is
>
+.»").
(1
- loir [r + v^^C' '- M
.r
to (1949), integrate
log
1950
soc"'
.v
= jeosec'\r + log
co.scc"^^^/.r
^Ieihod.— (104-i)
tnir'
.r
=
d.V
<
or
.r
by Parts,
}
•
[.v-^ ^'\d--—l)\
.
.?.c.
J
log X^.V.
[By
Parts,
J Jx
h.
[Siil).s.
tan
o.i;,
and integrate by (1035 or
'37)
VARIOUS INDEFINITE IXTF.GRALS.
GENERALIZED CIRCULAR FUNCTIONS.
1954
f sill"
(Lv.
Method.
COS" dr.
\
—When
n
is
»
i\
For
Phoof.
•which
is
1958
.r
—
cosec"
i^.e,
(772-4).
tan''-^r
see (2008).
=
tan"'-x sec* x
,
n
— tan""".
tlie first
c,
term of
;
—
i-^r^
— By
n;
=
substituting
,
tan
V-.-i
.
^^
=
in the place of cos-r, substitnte -if
1959
i
= ^7^r--^;^Tr+^;^r-^^^-
By Division; tan" .r
integrable and so on.
^Method.
pin
tan"-'.r
t^nl"-^r
/
n
W«-"-^'-
(Ia
.
integral, integrJite the expansions in
Otherwise by successive rednction, see (2uG0).
^nrw
1957
coscc"
\
«
«.
]
*^
d'l-
cos w.r
\
Jsin;/r
d
I
U"-''
^^" TT
\/
(
'-'"'->'" ''-
"XT )
'
Similarly with
— t.
.
dd\
\
J cos
n.f^
djr.
\
J
siii
/m
INTEGRAL CALOJJLUS.
322
Method.— By (809 & 812), when p and n arc integers, the first two
functions can be resolved into partial fractions as under, p being < n in the
The third and fourth integrals reduce to one
\ in the second.
first and < n
or other of the former by substituting ^tt x.
—
—
^^^^
q63
1
•*-^"*^
visin.u
«,(;
^.^^^ ^
with
S—'C-l)'-'^^"'^'^""^''"^
''"'
cos
fractions in (10G3) are integrated
.f
— cos rO
by (1952)
;
.
^
2u
cos.f-cos(2r-l)0
'"^
2^- = -^sin
(-1)-' sin(2r-l)Qcosn2r-1)0
2'-"
n
cos«.^
1964
*-*^
*
The
=-
= -^.
7i
those in (19G4) by (1990).
Fonmdce of Bednction.
1965
'^''^("-J)-^
{'^^^^civ=
J
2f
COS",l'
r'-:?^^ rf.r
1967
l'^i^'./.=
2f ^^l^^^-^^'" ^aH-f
J
J cos'\f
— In
'-"^^"7-^'''
rf.i-
J
COS''.!-
= -2 f'^'".^"-!^''- rf.r+ ('£^(£llll£
1966
Proof.
rf..-r
COS"-'.l-
.'
2
(1965).
cos
(n— 1)
'^"^"7^^'"
^/^i-
cos'\r
J
cos'^-\v
cos.i-
.t?
=
rf...
cos 72.^4-cos (?i— 2)
.r,
&c.
Similarly in (1966-8).
1969
^
\
siu^'
__ sin^.reos7i.r
1970
I
X
—
i
sin^'.r
si"".. sin«.r
p-\-n
1972
cos
_ _?>_
+«
i>
1
cos
(»-l)
.r r/.r.
cos'' cr sin »,r f/.r
'-
1971
dx
_j^ ^sm-^^
=—
^
sin n.v
cos'' .r
cos
I
eos^ \r sin (n
— i) x a.r.
7i<r (/.r
r
;,,,., ^, ^i^,
(„_i)
,.
,,,,
^'
?*.r
(Lv
»-'
cos"
^I-5llii!i^
p-\-7i
P
+ --ii+
/>
;/
iVos"J
.r
cos (;2-l)
.r (Lv.
—
"
j
CIRCULAR FUNCTIONS.
PiiOOF.
By
(lOtil*).
rnrts,
—
sin
^
)/.»•
In
(/./.
323
tlio
new
change
intcf^'ial
—
By sncci'ssivo reduction in
sin >/./• sin .r.
into cos (;<
1)
.Similarly in (llTU-'i).
tbis way tlio integral may be found.
Otlierwise, expand t-in" x or cos''.e in mnlliplo angles by (772-4), and
integrate the terms by the rollowing formulie.
cos
COS
>/a;
./•
.*•
1973-1975
r
am
1 /sin (/J
=
.
.
.
sin p.r
\
J
ua' (Ij
-2
'
-^
and so with similar forms, by
1976
I
cos n.v
r
i
/i)
.r\
.
/*
/
((jijG-i)).
GOspx—&m]Kv
cos
+
+
(/>
'-—^
lOLiud
trom
cos ud'
J
J
sin
;>
^'^'^
^111^^
\
J
— ;0,r
p—n
\
7
,
_
-"-'^"^""^ '^'^
o
f
)ui
+
1
J
.-•"
wlicn j) and u are integers, by equating real and imaginary
parts after integTating the right side by (2023).
—
=
=
Multiplying numedz.
z
therefore izdx
Proof. Put cos.r + i sin x
rator and denominator of the fraction below by cosH.i; + i sin /laj, we get
cos/ix
+
_
siny)j;
i'
"'"
therefore
^'^
f
sill
wr
way from
Pi;ooF.
cos
+
W.C
^
1
^^'^
•
,
^ cos
.7^
+
I
—
-
-•-.
Hence,
The
real part
'
.
i
+ ^'
=
—
'J
—-.
numerator and denominator of
•
,,,,1
r^iiiii^
=
.c
^^.^
g^^j
tan «x
i
(1-^/"),
J"i'l
therefore
«.»)
by
/,
we have
/cos.r-si ,Kr^^r
J
J>^^
2-;/'
coefficient of;' in the expansiou of the integral on
are the values required.
and the
'2),
^•"
1 -t z'"
sin?ix'.
C-'Ccos^a-)
J
n
r!!iy^:I^' are found in the same
sin ;hr
J
mulli:»lyiiig
r
the right by (2021,
sin
_
= - 2/ f '^^-^.
in (107<)), by multiplying
"^(cos
2-7/"
dx
?i.) a;
J
|-^:21^
Putting y
=
^'"
'— dx
I'.r
'—
by co3«A' — I
1980
ydx
^
^—r-'
— As
sin
.
+
and
f<:i^^P:!:^'
J
1
cos nx
J
1978
+ i sin ( p +
+ cos 2?iU! + i sin ^.nx
cos (p-\- » ) x
j,
"
cos nx
;
'
IXTEGBAL CALCULUS.
824
1982
"^^
o
j
a con' X +
}
a-^h tau X
[
19o3
,
am- X
b
=
7—
1934
f
J
—^
198d
l-a^cus^«
1987
cos
1988
sin.-B
a;
\o^( a cos X -Y
=
=
—-
3
,,T
a\'^(l
tan"' (a cos
li
[Subs, tan
a.
I
[Subs, tana;
&\nx)-\-ax\.
"
[Substitute cot.
[Substitute a cos a'
.
— —~.
^(1— a')
[Subs, sin
a;
a'
— a" sin"
sin »•/(!
J
^—
a;)
-
tan
— a-)
— a-&m^x)dx =
s^(\
\J a
\
.c
5-
Ty
:;
J
\/{i-ih)
tan- ^f^^+^^l
-TT^r-Tl
^/{a -\-ab)
cot
v/a
".''," dx
1985
/^).
-^/-IT^a^'' (tan,r
—c~~r-> {h
a--\-b'
=
a -\- b am- X
=
+ —- sin'' (a sin a;).
x')
[Substitute a sin.o
v/(l— o^sin-.r)
^-
1989
sin
.T
(1
= — |cos.? v/(l — a-sin^f)
(Z.y
— log
a cos x
j
+ V ( 1 — a"
sin'-'
x)\.
= — ^cosa;(l — frsiu^a;)^
sina; v/(l — a" sin-.r)
+ 1(1 —
— a'sin^a;)-
tZaj.
log
bill .<;C't
J
+
/;
sin
a;
t""
1991
(7
(a+ 6
•''
cos
(tr
a;)
=
^•''
[
cosec x-\-h cot
— b cos
— b^)
tan" |a;}
a;)^
^
sin
?>^
a;
sin
a;
1
- cos"'
cos
,7'
a;
— b"^) {a + i
(a'''
cus
.v)
— 0~)
x/f?)
-.
;
Vb
V'(fc-a )
v/('t+i tau^'t)
J
{ (<t
[Subs, acosa?
COS*)
1
By
[Subs. COS
-^992
Via + l^m'^x:) j^
[
sin
J
a cos «
cZ.r
a')
1^^^
[Subs."
^
^^
^^^_,
a'
^/(h
- «)
^hccx
\/{a + b)
a;
— ^/a log
{
\/a cot
a;
^/(a cosec'a^ + 6)}.
+
—By Division
(1912), making tbe numerator rational, and integrating the two fractions by substituting cot a; and cos a; respectively.
Method.
2993
wliere
1994
=
'li
I"
J
a+
m=
+ cco.s2.i;
^\\j^—1c {a—c)\.
/.'•
f
Method.
-//cos.<;
—
Sul)stituto
.
c
r«
(.
Then
J 2ccos.t;
+ 6^/»
'^-^
f
J
2c cos *
+ ^ i-
I
v/i
^
j
integrate by (1053).
^ =
= x — u,
_
d i'
f r
f
,
where tan
..^/f
a
=
.
,^
.
(i'«3)
—
EXrOXEXTIAL AND LOaAlUTlIMlC FUNCTIONS.
325
F(.\n.v..n.,),l.
^
J^ a cos ,r-\-b sin .v-\-c.
1995
F bring an integral algclmiic function of sin.r and cos
— Subslituto = x — n as in (1991), and the resulting
!MKTii(in.
—
A
—=
f /'(sinfl, t>nsfl)(/0
.
,,
,,
takes the form
--^
fi) iJd
<h
]
AcosO + B
,.
+ Ji
cos
J
(cos
[
f sin: 0»i/
+
,
^,-
rr—r,
r,
integral
Tens ^0
tt,
—Acosd n
+
:
J
since /contains only inteii:«"il j)o\vcrs of the sine and cosine, and may Ihercf .re be resoivod into the two terms as indicated.
To lind the lirst intet,n:il on tlie ri^'ht, divide by the denominator and
To tind the second integral, substitute tho
integnite eaeli term se|)anitely.
denominator.
Avlicre
F(co^,r)(rr
r
1996
J
(^^
F is
Mrthod.
+
an
_^
'>i
— Resolve into
(.r)
<f>
{a„-\-b,^ cosct')'
...
— Let
+ lls\n.r + V
a COS d-\-o
</»(.}•)=« cos
.1 Cvis x-t- JJ .sin
.J
-f-
Each
partial fractious.
C Ac„..v
J
Method.
COS.*)
((t2-\-t).,
int(^gral function of cos x.
1997
Assume
t-os.r)
C
=
and <}>'{x), and equate the
becomes
a;
+
/^
sill
sin
j'
integral
be of the
will
^^^,
d-\-c
+c
+
=
—a sin.r + Z> cos ar.
Substitute the values of
.'.
;
\<p(.r)+fx<ft'(j:)
<;»'(.>•)
t'.
coefiicients to zero to
determine
A,
/u,
r.
The
integral
and the
^'
C
I
v-
(•')?>
last integral is
°^^
^
(-'J 3
^^J
9{x)
found by (1991-).
EXPONENTIAL AND LOGAIITTILMIC FUNCTIONS.
1998
*
("^F(.r) (Lv
can be found at once wlien
expressed as the sum of
dcrivatii't'
tiro
f
(ructions,
F
(,r)
one of ivhirh
can
in
l)o
tlic
of the othrr, for
j"'-!<^(.'-)+f(.'))'/..='-<;t(.').
1999
*
f"-"
cos" La (Lv
——
^
a -\-no-
and
c^'^cos"
<'"''
|
bd.\-
sin" Lr(l,v are respectively
—
-^-
a-j-M
:.
6-
€'"
I
J
=
cos" -Oddu-.
INTEGRAL CALCULUS.
326
and
g
&.r-»& cos
sill
fta>
^K^^-l)^^
^,. ^j^„_, ^^,
f..^. ,i^.-2 j^^^.^.v^
—
In either case, integrate twice by Parts, j e"^dx.
Otherwise, these integrals may be found in terms of multiple angles by
expanding sin" x and cos" x by (772-4), and integrating each term by (iyol-2).
Proof.
2000
I
e-^siu'^ci'cos'^crdr
found by expressing
is
sin^d'
and
cos"aj in terms of multiple angles.
Ex.
:
J
e^ sin^a: cos"
2''e^ sin*
Then
a;
Put
e'^
(2 cos
;7;)'
x dx.
(2i sin
a;)"^
cos'*
—
e""
(sin 7x
= z in (7G8),
= (z-z-'y (z + z-y
—
'6
sin
o.f
+ sin
3.1'
+ 5 sin a-).
integrate by (1999).
2001 Theorem.
^Pdx = P^,
fP
J
^P,Q.dx
=
—Integrate successively by Parts,
Theokem.
2002
iP^Jx = F,,
,
^^
of x
and
;
'"'"
J
P dx,
...
P
±^/\+i.
of x
successively by Parts,
|
r/Vi^/^
/i-i J
— ^^^,„ —
g
^'"^^'^•
f.r--^(log.r)'W/.r
:^
^(/''-^/"-V^^i^^
Method.— By
(2001).
P,
=
and
Qjr-^
1
Examples.
2003
;
I\
(^^-1) {n-'l)
(>*-lj (/.//-^
i\
(/<-i)(«-2)(vi-;5)g/»>"-^
Proof.— Integrate
Then
&c.
— Let P, Q, as before, be functions
-
let
= P,,&c.
P,Q'-nP,Q-'-i-n {n-l) P,Q-'-
(pQ"(lv
Proof.
—Let P, Q be functions
•'""',
i',
=
/''-.. .+(-1)'''^)-'",
P,
=
""".
'•tc
ALGKVUAh' FrXCTfOXS.
327
2004
and
cacli
term
of
tliis
result can be integrated
2005
l)y (200:^).
fiP^.
'*'
\n-l^
{u
''''
— l)[n-'2)'^
{u-i){u--2){n—l.i)
+—
Method.—By
The
last
case, Avriting
(2002).
method
/
P.r
j-",
P,
=
wx-'"', P,
not applicable
is
=
when
vrx"-\
n
1
..
•
etc.
= 1.
In this
for log.v,
=—
METnOD:
=
liwr
,
1
Expaiul the numerator by
.
(1''0),
and
inte-
X log X
k)g X
grate.
See also (21G1-G) for similar developments of the exponential forms of the same functions.
PARTICULAR ALGEBRAIC FUNCTIONS.
n(v
+ ^)...
X
2(11-1) ,^_
\-x
« being even.
2008
f^^_,,^^,^^^
=
-^
[Subs.
2009
J
oniA
f
^T"^r^V(TT7) "
<?•'•
V'l
1
log
'"*=
,._
y(i-*'')
gy2
4-v/(»'-l)
(1018)
^('-^'^
INTEGRAL CALCULUS.
328
=
2011
^sm-\/(-H:^^r^'^^),
V{b<r-acc)
2012
2013
2014
2015
[
=
-r^
-^
V{c + ex^)
^(-^^'^n-^^a
^""^-'r^
l^q^- = ;^[.o.^^^H6*^--iai~
c]^
f
,
/%
^
n/(1
+ .(")=
(2^^-11*-^._ 1l^o,(^ff^
_ 1 tan"
.„
=
a;
f
J
s/2 in (2015
-f.)
1
,,
(2,r-lji
[Substitute
2018
1
[--r^.
a^:^Ai:..') =;^-'°-f^-
[Substitute
2017
^S,b3.
^
;^,--^^^£^-
i(a^5lfT^, =
f
log
°
aobc
(l-\-x'){^/{l
n
tau'
+ x')-x'^^
[Substitute
—-
a'
=
,-(2.f2-l)*
1
.
,.'-1
INTEGUATION OF -i-—
i-.r^ ^
—
2021
.r
,'
=
where 3
,
n
//
r
rnakiii*,^
be odd
?i
i log(.,_l) + (zi)'log(.r+l)
and Z denotes that the sum
'^-
n
obtained by
be taken.
If
1
329
=
2,
G
^i,
...
)i
—2
of
all
the terms
successively,
is
to
J
Ji^' = llogGr-l)
2022
n\
>-
n
'
ti
with
= 2, 4, 6
r
.
.
.
/i
—
8111
rp
successively.
1
n be even.
If
2023
^^r^'
+1
\
J
.{'"
= -LtvosrI/3\og(.v—2.rco.rfi+l)
n
<
S sm
-
.
•
-4
//0
4-
rlB tan
—
-1^ «*'—; eos>'/3
n
with
r
=
If n
1, 3,
be
5
...
»
—
^,
sill >'p
successively.
1
o(irf,
f£^=. til^logGr+l)
2024
- i ^ cos W)81ogGi--2.i^cos
ry8+ 1) +? S
^
n
» — 2 successively.
with r =
3, 5
sill
W/8taii-^ l^^:i^l^^
sill rytJ
/2
1
Proof.
of (1917).
...
,
— (2021-4).
We
Resolve
have -f-^^
1'
in)
=
-'^
into partial fractious
-
^
=
?/<«""'
—
,
since a"
by the method
= ^ 1.
Tho
diirorent
=
±
7i
±1 =
cos r/3
values of a are the roots of x"
0, and these are given by x
(See 4-80, 481 2r and '_*>•+
t sin r/5, with odd or even integral values of r.
The first two
of those articles being in each case here represented by r.)
the remaining
terms on the right in (il021) arise from the factors .r
1
terms from quadratic factors of the type
;
±
(y
These
— cos
last
rfi
—
i
sin
r/3) (a-
— cos r/3
-t-
(
sin r/3)
=
(.r
— cos
terms are integrated bv (10l^3) and (1935).
cases (202-2-4).
2u
;
rp)'
+ piir
r/3.
Similarly for the
—
INTEGRAL CALCULUS.
830
2025
in formulae (2021-4),
If,
added to the
+
—S
-
Q-tt
sin
7-/3)
W^
be
term for the constant of integration, the
term becomes
last
integral vanishes with x, and the last
=F
-
reading
+
^,
^
in (2023-4).
f4^=i(«)^f^,,/.
where
az"
202^
=
Then
Ihif.
,,T+1
J
where
71
and
in (2021-2),
2026
or
— ^ sin W/3 tan
= T ^ "" ""^ *"" -TmTT'
'
r
=
according as n
is
= tt-^v?,
)3
— 2,
integrate by (2023-4),
and
1, 3, 5,
...
successively up to n
—1
even or odd.
2028
.1
£
1
^^^,
^
-^ Scos?«ry8.1og(.r--2.rcosr^+l),
when n is
{x^l)^n.
with the same values of r; but
(--1)'" 2 log
additional term
Proof.— Follow the method
Similar forms are obtainable
2029
—
^
=
— .^2 sin (h —
where
But
m
•
as
<p
=
Q-\-
"^ and
the iiiteo'i-al
/oAorA
(2025).
if
—
of (2024, Proof).
when the denominator
is
S
/)
r
<^.log
= 0,
odd, supply the
is
x"
—
1.
008 (»—/)</). tan
(ci-
2, 4,
^
— —-X
;
— 2.r cos <^+l),
...
2
(»-l)
successively.
to vanish with x, write tan"^—^'^
-
1— .rC0S(/)
—
Proof. By the method of ("2024). The
given in (1^071, putting y equal to unit}-.
faf'tors
,
of the denominator are
.
TNTEGRA TION OF
2030
^^-
r^-^^^T^'
=
(a
^'
-
=^
x'"
J
+ 6^"
^"-^
)
831
«
(*''^+>-)
I
Xlog(a'-2.rcosr)8+l)--'si»(W)8+i//*7r)tair':^^^=^^
with the values of
and
/3
,
(2021-4).
/-in
—
Proof. Differentiate the equations (2021-4) lu times w ith respect to /,
by (1427) and (1461-2). If m be negative, integrate m times witb respect
to I, and tlie same formula is obtained by (21.!>5-0).
In a similar manner, from (2027-8) and (2029), the general
terms may be found for the integrals
2032
.•;.,,,.-.±(_lV,,.^.-.;(l.,^.,).^^^
J
INTEGRATION OF
— When
2035
Hulk
% sh6s/
i7
Expand
the binomial,
2036
But
I.
^
.r-"
— 2.t"cos/j^+l
{cLV-\-b.v"y^ (Lr.
.v'"
"^"'
.r'-(lopr..-)'V^
r
^^^^
.r'dtl
J
a positive integer, integrate
/s
n
1
;/ /
z
/^?.v
if
=
(a
=
2038
But,
the separate terms by (1922).
the positive integer be
n
(ax-'
+ b)".
1,
the integral
is
known
= v — \.
becomes
^
Expand and
.
II.— When I^+i
r.uLE
substitute z
Th us
"^
and integrate
at sight, since ra then
2037
+ bx")
+
J^-
is
a
negative
integer,
q
Thus
integrate as before.
if
the negative integer be -1,
it in the form
tlie
integral
found immediately by writing
/•
"P
p
r
V
i—-(..«- + ^0^^'= _—
nn{pJrq)
is
;
INTEGRAL CALCULUS.
332
Examples.
2039
To
find J.c*(l
+ «*)*cb.
Here
Xy
=
6?/"
m = \, « = i, p = 2, g = 3, '-^ = 3,
x^ )^
x = {tf — \f,
y =. {\
Therefore, substituting
(i/'— 1), and the integral becomes
a positive integer.
the value of which can be found immediately
the separate terms.
2040
'^-^
[
=
For '-^^^
(2036)
1
n
of lx\
2041
=
"^
'^^
Z'"'^'
cZ.r,
is,
or
[
Vjy
,
expanding and integrating
= ^1 (a + hx*)^.
m + l= n, and the factor x^
(« + fe-0*
that
;
-\-
^^^
X-'- (l
+ x^f^
dx.
Here
is
the derivative
m = -l,n = ^,p = 2,
I
J^
=_
Therefore, substitute
a negative integer.
2,
q
-6y^ (z/'-l)'"- Writing the integral in
(i/'-l)"', a-y
(a;-*
l)% X
y
the form below, and then substituting the values, we have
q
3,
'''^^tJ:
4-
n
=
=
=
+
Sx-'(x-i + l)^x,dy
which can be integrated
2042
f
t; M
,
= -6^y'(y'-l)chj,
at once.
=[-'(" + ^-")
-'
Here 'ii±l
dx.
+
^=-
1
therefore, by (2038), the integral
=
.«-"-'
I
(ax~"
+ b)-'
dx
= -~ log(a.c-"-f?0-
REDUCTION OF J .r'" (a+6cr")^ dr.
neither of the conditions in (2035, 2037) are fulfilled, the integral may be reduced by any of the six following
rules, so as to alter the indices m and p, those indices having
any algebraic values.
When
2043
I.
To change
m
and p
Integrate by Parts^
2044
n. To change
into
m-fn and
m
and p
into
m—n
Integrate hy Parts,
Jx""' (a + bx")'nlx.
2045 ni. To change
Addl to ^. Then
integrate hy V<irfs,
by Dii'isiov
^
p
— 1.
Jx'^dx.
ajid crjunfp
m
into
Ifir
and p + 1.
m + n.
yrsii/ls.
jx™dx; and
al^o
REDUCTION OF
2046
Add
IV.
'/'<'
to
1
rhini>j<^
p,
Parts, fx"'ilx;
2047
Add
integral
2048
y-
'/'-'
to
1
bij
////.'
+ Lc")" dx.
333
111-n.
and siihfnict n from ra. Thrri integrate by
and also hi/ ])lrif<ion, and equate the results.
rhaiKje p Into \)-\-\.
p.
rarfs,
7'Ar/i,
iufojrnf,-
(x''-^
(a+"bx")".
VI. To change ^ into
hg Division,
Integrate
J
111
j u:'" {a
Inj
Division,
and
I.
II.
the
new Integral
Mnemonic Table for the same Rules.
III.
IV.
V.
VI.
111
+
11,
new
p-1.
and
x"-^(a + bx")".
2049
the
hij
Parts,
INTEGRAL CALCULUS.
334
Examples.
2056
To
^'^p'^^
find
Apply Rule
dx.
I.
Formula
or
I.
;
thus
f
^x-'(a'-x-)hlx
2057
To
2058
—
find
I
^x'
= -lx-\a'-x'y- + l
=
Substituting sin
fcosec"'^fW.
or
II.
Formula
II.
thus
;
= a^a'-x'r'-S ^ x' (a'-x'r^ dx
(a'-x'yUx
j"
Apply Rule
-dx.
^x'' (ar-or)-^ dx (1927).
^
siu-'»0
dx
=
Apply Rule HI. thus, increasing
\x'"' dx, and again by Division
;
{ .«-'"
by
j^
(1934).
the integral becomes
a;,
(1 -x-)-^- dx.
1
and integrating,
first
by Parts
;
{
=
x-'-(l-^)idx
'''""
[ .c-"'
By
results,
—m
{x-"'(l-x')-Ux-
we
obtain
m
according as
is
is
Ja-m-z (i_a;2)^ dx.
To
have,
first,
(,r
[
+
Integrating the
made
is
to
|-=^-a-''')-'<i-
depend
finally
upon
[(1-xT^dx,
or
found in a similar manner by Rule IV.
is
The
integral
{l—x^)~' dx and the integral operated upon
Otherwise apply Formula IV. See also (1954).
^x"'
;
Apply Rule V.
'^
find
,.
.,
J
we
{x'-'"(l-x"-)-idx.
an odd or even integer (1927, "29).
'\sm'"ddd
to be evaluated
1,
J
j*-(i-«r'A. = fr^.(i--')'+ fES
{x-'(l-x'y^dx
2061
{x'-"(l-x-)-^dx,
9/i
repeating the process, the integral
2060
+ -^^
1—
=
il-x')hlx
Equating the
2059
}^ -'^^-
i
J
(y;--|-
.
p = — r, and
increasing
a')
by Division,
((-)'-'•
dx
=
new form by
{ ,r (..-
+ a')-" dx + a- j
Parts, j x
(x'^
(.r
+ «')
+ a")-'' dx, we
Substituting this value in the previous equation,
we
-''
dx.
next obtain
have, finally,
2062
C
d.r
^
.V
,
2r-a
i
d,r
is
p by
REDUCTION OF
335
cos" OdB.
(sin"*
Thus is
This e(|uation is i/wvu at oiiro ])y Foriimla \'.
clmncred into r-1, and by ropeatiii^r the process of ivduction,
the original integral is ultimately made to depend upon (1935)
for its value if r be an integer.
/•
Another formida
20^3
^Tq:^.
\
Prook.— Write
times for
The
-
To
find
last integral,
f
,r
j'
(a"
is
'•
1.2...
0-1) W^'\,
+ .r )''%/./.
(a-+/-)^'^-\lr
=
1
..
(«r
+ /-)''" -
</..•
a result given at once
=
-^
[
"
'i
'•'';;+f"'
1
Division,
we have
we
(a^^'^;*"-
J
obtain
+ ,-;:fi
j"(«H..=)'"- rf.'-.
by Formula VI.
be an odd integer,
reduction in this manner, at
The
-
by Parts, becomes
|(«=+..-^)'«
2066
By
Apply Rule V[.
Substituting this value in the previous equation,
If n
/•
in ('JJ.")5).
J
2065
/3
,
iS
and differentiate the eqniilion
(10;!")),
for a" in
ft
by the principle
/>
2064
Un- this integral
we
\
by successive
arrive, finally,
(1031).
{(i'^-\-,r)^ d^v
integral j'sin'"0 cos^Oc/0
is
reducible by the fore-
going Rules I. to VI., if, in applying them, n he ahrays put
equal to 2; if p be changed info p±2 iuiitead of p±l; nnd
if Division he always effected by separating the factor
l-sin^e.
cos-e
=
Proof.
n
=2
Jsin"
cosPOf/fl
=J
always, and the index i
Thus, Kule
I.
(jJ
x"'(l-a:')* "-'
— 1)
is
(/.r,
where
increased by
.>•
)u-\-
1
)n-\-\
»'
=
sin
fl.
by adding 2
gives the formida of reduction
2067
.'
1
Thus
to p.
—
INTEGRAL CALCULUS.
336
integral can be found by substitution in the fol-
But the
lowing cases
If
/•
:
be a positive integer,
2068
1
cos-''^^ct^siu''crf/cr
=
yl—zY^^dz, where
z
= sin X.
2069
I
siir''^\r cos^'ct'^Ai'
=—
{\—z-Y^^fi^, where
z
\
= cosx.
If
m
2070
\
-{-}:)
=
-2r,
=
sin'"cr cos'\i'^/cr
{l-\-z'y'''^"'(f^,
\
where z^tarix.
FUNCTIONS OF a + b.v±Cd'\
integrals are found either
The seven following
+ bx + ex' =
and substituting 2cx + 6
a + bx-cx" =
2072
and substituting 2cx — b.
2071
om'i
a
2
O^
according as
2074
r
-771
//
;
{2cx
>
or
^^-^^
<
+ by + Uc -W\^ 4^c,
or by writing
{A.ac
-
^-^'
(
{
by writing
+ b'-{2cx-bY] -r 4c,
^
-
Tn '^"
^
_i
2c.r+6-v/(6^-4ac)
2rcr+6
—771
7n>
4ac (2071, 1935-6).
-
1
log v/(6--+4..)
+ (2..r-6)
(2072, 1937)
2075
2076
^^r-J^
:r,
= 4-
''^i"-'
:7^rv
(2071-2, 1928-9)
—
FFXCTIONS OF
a
+ bx ± ex}.
=
.lr-5
j\/(<f+/>,r-r.r'-')
dv
=
^c"^
= 2cx + b.
integral^ are given at (1931-3).
f
2078
wliere y
X
The
^
2079
_
I'
\
O^.P-irn-^
[By
2080
(-2071), tlio
f
I
'In-
^/{lur-^/r-jr)
fh/,
'IlL
I'
— h')"
integral being reduced by (20G2-3).
/^+'"j%
{'2('A'+b)(Lr
i
~ 2c J
,/(//H 1^/^—^^)
J {ir-\-luc
J {(i-{-Li+i\r)"
{a+Lv+c.ry "^
C
(
bl\
V'''
2c/ J {a-\-Ou'-\-c.r)"'
<h'
The value of the second integral is {a-\-hx-\-rx"y-''
l, when the value is log [a -^ bx -\- cx^)
unless p
third, see (2070).
=
Method.
first
2rx
2081
I.
by
+b
I
J
If
that
—
is,
into two fractions,
hx
the derivative of a
+
—
^^'^
'^
r/,r
may be
/r
>
4>ur,
2082
If b'
<
4rtr,
put
«
and
put
""-
^
III. If h'
f
:
is
^
(1036)
y
S
^^(''')
^'
=
nr,
and tho
into
^^
^,
_
r
(^
-j^^7-
i*
is
^^^^^
resolved into
J .r-fi
fa-;>>0^r+</m
\
z±±jyi^z:^:!l^
for
jS
= n' and
may be decomposed
the value of Avhich
2083
numerator of the
+ cai\
integrated as follows
U .t--a
c{a-fi)
integral
tlic
For tho
(i-\-h.r--\-Cd'^
Partial Fractions, the integral
II.
making
(1— p),
-r-
.
— Decompose
;
,
83
'{n-^hr-VcA^) (iv
2077
7
^,,
^
found by (2080).
= 4ar,
~
=
—V-.+
\
.>n-|>JA).
(2062)
INTEGBAL CALCULUS.
338
2084
^^^^
2085
(—£JL}-—--—^
C
2a
y-(Lr
J ,+,,.-.+e..^
REDUCTION OF
Note.
iv"'(a
/h\
_,/
,
.v
2086
— 6mj9
is
J a?'" (a
+
?Ai'''
+ cx^'^' dx.
I
(7n-{-n,2)
2088
2c>j (;>
-(m-2« + l)
+ l)
j
=
f (m,i>)
f
i
2090
-{-2anp
{m,
j)
=
+ l) J (m,
— l) — cnp
^
j
^m + ;,y>+l)
=
/>)
(»j
I
f(m,
(m+1,
(m-»i,7?)...(3).
7>)
+ 2>j,;> — 1)
0», p)
{m,p — l)-\-bnp
(1).
(m-2>i + l,;) + l)
(m-2w,;)+l)-6/i(i)+l)J
(m + 2;i/> + l)
\
(m+2»,;> — 1)
I
= (m-» + l,7>+l)
(m,7>)
(m+//7;
2089
0^*+l,7>)
— l) — 2cnp
+ !)
h7i (;>
+ a«7)
for the sake of clearness,
denoted by (m,p), and the integral merely by j (w, p).
(m^-l)Jo>^.7>)
2087
=
{m-^l,p)
{m-\-n,
/>)
=
(4).
p — 1)
(»*-;*
(5).
+ !, />+!)
^
^
2092
)
—In the following Fnrmulce nf Iiedudiov,
+ hx" + cx'-"y
2091
1
(
-T-l7(2^)^"^ \-V2;i)-2;r+^-j
/»'(/>
+ !)
('>)
|'(m,/>)=-(m-;^ + l,/>+l)
^
(0-
—
-REDUCTION OF
2093
rN{p-\-\)
-\-an(i>-]-\)
\
\
f
{m,p)
.f'*
(a
=
+ ^f^-h wr')"
im-'2n^\,l>-\-\)
[)n-'2n,i>)-{in-^nj>-n
2094
an (/>+l)
2095
-V;/(/>+l)
(m, p)
\
((/>»,
^'^^
^^'*''-
+ \)
\i
in-'2n, p-\-l)
= -(m + 1, ;>+l)
;>)=-(//< + !,
/>
+ l)
^
2096
+ !)
«(»'
f
(>",/>)
=
(^>»
(!<')»
+ l,7>+l)
--b{n,-^)^p+)l-]-^)\(m-\-n,p)-c(m+'2np-^'2il^l)\{)n
+ '2n,p)
^
2097
c{in
+ '2up-{-l)
— b()u-\-)ip-)f^})
\
^{m,p)
=
(m-2n-^l,
— n,p)—(i{m — 2ii-\-l)
iin
(11).
p-\-l)
\
{)n
— 2n,p)
'
(1^).
^_
Proof.
— By diftereutiatiun, we have
2098
Formulw
indices
and (3) are obtained from this equation by altering the
so that each integral on the right, in turn, becomes j {))i,p).
(1), (2),
m and p,
Again, by division,
2099
J('«.i')
= a\l(in,p-l) + hj^(m + v,p-l)+cyni + 2n,p-l)...iA).
And, by changing
2100
JO"-",
i'
m
Formula! (1) to (12)
(4),
into
hi
—
+ l) = "3
)i,
and
j)
intnp +
may now bo found
((//I
(6),
by eliminating
(7),
by eliminating J
3
('"+". iO
p — 1) between
(1)
and (A);
(1)
and (A);
(m-H,
(ju
+ ^.
as follows:
+2m, p — 1) between
by eliminating j(j» + n,
(5), by eliminating
l,
(»i-«,iO + '\l"(»'.i')
+ »,
j)-hl) between (2)
j>)
and (B)
between (2) and 0');
;
(.^)-
;
;
INTEGRAL CALCULUS.
340
(8),
from
(4),
(9),
from
(4),
(10), from (5),
(11), from (6),
from
(12),
If
and
o
a-\-bx"
(G),
by clianging
by changing
/3
+ c\r" = 0,
into rn
— 2n,
and p intop +
1
?» into vi
—
n.
are real roots of the quadratic
then, by Partial Fractions,
-"::;-
2101
in
by changing p into p + 1
by clianging p into p + l;
by changing on into yn + n;
Jf «-f /At'"+ect-"
.„.
equation
i^K4^_f£^{,
^ e{a—/3)
— a J a'—p )
iJ
cV"
and the integrals are obtained by (2021-2).
But,
if
the roots are imaginary,
where
cos?i0
= — -—
-^^
—
and
r
z
=
2 Viae)
2103
f-
^^^^
is
,
f"'^^^'^
2105
f
2106
f
rrr
(.i^
+
/;)-i,
^^£-=^
^r=c,
=~J
B=
y(-44-%+Cy)'^"^'^^
h-2ch,
C
= a-hh+ch\
= --l-cos-'i±^=-i=cosh-I±$.
^^iL__=
Method.— Substitute
\a/
reduced to (2079-80) by (2097).
J (.,^+/Ov(«V^'cr+e.t^)
where y
l—Y''x.
(.i;
l_sin-'l±^.
+ /0"\
[By (2181).
as in (2101). Observe the cases in
which
A=l.
'^^"'
J (.r+/ov(«+^>''+^'0
~
^'
V u+7i//+(y)'
same values for A, B, C, and y as
integral is reduced by (2097).
with
tlie
91 OR
r
(
^'+»')
'^''
in (2104).
The
INTEGRATION BY RATIONALIZATION.
341
=
p tan (0 + y), and (lotcrrninc tlio
:Mktiiod.— Substitute 1) liy puttint,' .>
constant y by equating to x-cru the coenicicnt of sin 2« in the denominator.
„,,
..
,.
rcsu Iting mtcgi-al is of the (orm
.
The
,
f.
f
—
cos «
//
.
-f
iV sin
,
tT"''"-
,,
Separate this into two terms, and integrate by substituting sin
in tlie first
anil COS y in the second.
C__±u)jrr
2109
wlicrc
mid
«/)(,'')
F{,r)
the former being of
Method.
— Resolve
are either of the form
of F(x)
a-
—o
= 0,
in this,
and are
tlie
arc rational alo:('l)raic functions of
lowest dimensions.
'^
into partial fractions.
^,
(2107), or oIfs
of the type
resulting integrals
they arise from a pair of imaginary roots
.-V^'-'l
f
and the integral (21U8)
The
.t,
is
//
'^"l! 7
_l,
n
'
Substitute
obtained.
INTEGRATION BY RATIONALIZATION.
F
denotes a rational algebraic
In the following articles,
In each case, an integral involving an irrational
function.
function of x is, by substitution, made to take the form
]'
This latter integral can always be found by the
(z) d::.
F
method
2110
of Partial Fractions (1915).
f;.'|„..,(_^:f,(5+|i;)Uc.(
Substitute
v
=
'',
where
nominator of the fractional indices
h-fjz"
the powers of
2111
z
being
dz~
now
(b-fjz'y
'
</..-.
the least
/
is
;
thus,
common
\f+gxl
all integral.
j>'^1..->(5^)M5$^)-.^^-('^'-
Reduce
to the
form
of (2110) ])y substituting .r\
de-
INTEGRAL CALCULUS.
342
F{^(a + \/nLV-i-n)}
^
2112
2114
f
Writing
^{a-\-hv+c.v')} dx.
tlie
\
Ldx-\-\
J
J
F
for a-\-hx-\-cx^,
in
,
C-\-V\/ Q
functions of
rational
takes
Q
^, g = _ _^^.
± c.^ =
^ Iv,
^ + ^^^
form
tlie
z -\-c
y(^x
AncULereforc
M\/Qdx,
wHcli ^, 5,
x.
L-\-M
form
may always be reduced
\/Q.
D
of
are constants or
wliicli
two
r
.
.
—
2115
a
Otherwise. (i.)
by substituting
— cz'
dx
2cz-b'
_
2c
When
c
(hz-cz^-a)
(2cz-hy
dz
is
integrals
MQ
—~z
positive, tlie integral
+ j., + ,.,.)'
y(^
'
^
y,
"
'
dx,
is
wMch
^^
the form in (2075).
rational
to
Eationalizing tliis fraction, it
Thus the integral becomes
first
tlie
0,
rational, wliile the second is equivalent to
is of
x=^-^.
Substitute
y/{hx±c.v~)\(lv.
(V{.r,
J
2113
Subs, ^{a-^^/mx-\-n).
ilv.
may be made
+
{B^
V2cz-h
''
a, /3
let
be the roots of the equation
(ii.) When c is negative,
a+l>x — M^=0, which are necessarily real (a, b, and c being now all
—
—
a) (fi—x).
The integral is now made
cx'
c (x
positive), so that a + hx
rational by substituting
=
In each case the result
2116
f
^'
when
IS
is
.r"'F
is
of the form j F(z) dz.
lv\ Va-\-bA--{-c.r"} dv,
an integer,
is
reduced to the form (2114) by
substituting x^
2117
f
r
[.r,
y^{a^h,),
,
/(/+ -.r)}
r/.r.
Substitute
r}
= a + hx
TNTEOUALS EEDUCIBLE TO ELLIPTIC INTECHiALS. 343
and, therefore,
.r
The form
J
=
.,
'
.
7^ [", ^(f/.r-//)]
(/.v
—
—
V{"-\-'>-^)
,
77"'-'" /\»
obtained,
is
Avlilch
is
compreliended iu (2114).
^^^
when
an integer,
is
by substitnting
2119
\
d"'
(fi
;:
=
/u-"
is
rednced to
+ \/{a + h'\c-"),
+ b.r") F
''
(,r")
iLr is
form
tlic
2120
1
rationahzed by snbstituting
or
"-
+^
whether positive or negative.
d^''-^
F
[d''\
.r",
{(i-^1hv")'^\ (Lr,
positive or negative integer,
(rt
(v) (h:
and therefore
either (<i-\-Li:"Y or (ax-"-\-hY according as
is integral,
/•'
J
is
when
rationalized
—
is eitlicr
a
by substituting
+ ia,'")".
INTEGEALS REDUCIBLE TO ELLIPTIC INTEGRALS.
2121
f F{.r,
Writing
A
^{a-{-b.v-\-Cd''+(Lv'-\-r.r^)\ fLr.
fur the quartic, the rational
always be brought to the form
and
this again,
\.^
y
function /'
may
,
by rationalizing the denominator, to the form
INTEGRAL CALCULUS.
344
M-\-N\/X, where P,
of
^
X.
Mdx
(n^X(Lv =
By
P\ Q\ M,
or
f-^ civ
substituting^
°
•^
Q,
N are
all
rational functions
has already been considered (1915).
a^
= ?l±Vi/^
l
+
where
f^^\
B
B
rational.
and determining
j;
and
q'
so
;y
that the odd powers of // in the denominator
last integral is brought to the form
sum
is
may
vanish, the
being a rational function of ?/, may be expressed as the
an odd and an even function; thus the integral is
of
equivalent to the two
The
J
first
r_F^{!rUhi__
?/F.(.v^)rfy
(•
91 oq
^^'^'^
y(«+6r+<-/)
"^
J V(''+*/+<-/)'
integral can be found
The second, by substituting
depend upon three integrals
2124
f
,,
by substituting
/.,
f^
of the
foi
for
7/^
\/y.
can be made to
forms
fv^lES^^
^";
,,
.,
.
J {\-^nar)Vl—M'.\—lraj
By
substituting
2125
r
«^
(/>
'^^.
= sin
fx/l-A:^sm^^#,
,^,
,,
— ^- sm"9
Vl
•^
H
(•
J
above become
^x, the
(1 -{-n
siir^)
v^l— /r
sin-<^
These are the transcendental functions known as Elliptic
They are denoted respectively by
Integrals.
2126
y{.K<i>),
E{k,<t>),
li («,/.-,</>).
TNTEGT?ATS nEDUCIBLE TO ELLIPTIC INTEGIiALS.
APPHOXIMATIONS TO
AVIuMi
E{Ji,
2127
loss
is
/•
from the
tp),
=
/'(A%<^)
2128
/<:
(A-,
F{1-,
unity,
lltiiii
oriL;-in
AND
^>)
<s>
=
<>,
</>-| t.+
llio
in
^^^
.
.
.
_
Valiums
of
converging
/''(/.-,«/>)
mid
series, aro
« = </>+ J ^1^-2^. ^'+ iTi:^^ -^''-•••
2.1.0.../*
[vi
.^
sill
\6
sill
0(i
(I
sill
siiw/(/>
;/
siii(;/
.
IN SEKIES.
(/•, ^,)
^^.-^445^''+
+ (^^
^
A'
345
i sill 2(f)
n
n
,
If/)
,
.,
2^^
'*'"
'^^•'
being an even integer.
,
15sin2<i
— 2) j
—1
C (/?,
>
i^jl
2) sin (»
n
_^(;^;{)sin(»-G)<^
—
i)
</>
—h
_^(_^).„,^.(.^^^-,^,)^^
—
In each case expand by the Binomial Theorem; substitute from
Proof.
(773) for the powers of siu^, and integrate the separate terms.
The values
^
= 0,
^
=
^TT,
of F(l',cp)
and
E{Jc,(}>),
between the limits
are therefore
2129
2130
But
series Avhich
converge more rapidly aro
2131
2 Y
)
INTEGRAL CALCULUS.
346
2132
l-^/(l-A-)
0133
^
r
pressed in
tlie
substituting
(-'^)
^^'^^
^Yhen
_,
form (x
;)fi'^+-T)'
can be ex-
i^(.')
^^
integrated
by
x-^—.
If h is negative,
and
i^(r(,)
form U-\-
of tlie
^
)/ (^'— ^j
5
substitute x
x
^(•')"'-
2134
y(a+i»u'+cu-4-^At''+fu^'+6cr'+«a''')'"
<r+
Substitute
-^^
Hence
x/cf'
=
""
I
V2 V
—=
^
—
=t-
2
2
y ;H^
,
)
^Z^,
^
and the integral takes the form
J
^ {«(:s^-;k)+Z'
z,
"
2v/^--4
Writing
we see that the integral depends upon
where P, Q
cubic in
(;s'-2)+c;^+^/}
are rational functions of
f_il±_
and
z.
2135
Tlie integrals therefore fall
(
for the
f^|±^:,
the radicals in which contain no higher power of
fourth.
Z
z
than the
under (2121).
^^•^"
F
Expressing
{x) as the sum of an odd and an even function, as in (2i23), the integral is divided into two; and, by
substituting .r, the first of these is reduced to the form in
(2121), and the second to the form in (2134) with a
= 0.
iNTEcnALS ni'DrrrniE to eluptjc iXTEaRMs.
2136
Put
.v
r
iV
=
i/-\-a,
i.v)
fh'
lliiallv
of
under
falls
;:i/
//,»-
-f
nr + (/,(''=
0;
for the duuomiiuitor.
obtained will bo
+ r»>\/a + y8^V/-
f(//
wliicli
a+
oheino-ii root of tliooquation
niul, in tiie rcsulliiiu- iiitooral, siibstituto
The form
317
(i!!:^),
/'and Q
ration,.!
bein.^r
functions
^.
IMjLH
r
2137
.
tlie sum of an odd and an oven functwo integrals are obtained. By puttin.cr
the denominator eciual to z in the first, and equal to .cz in the
second, each is reducible to an integral of the form
Expressinir F(.r) as
tion, as in ('2123),
which
falls
2138
under (2121).
j
J_
^
,,^
=-^2^' (72'
^'''''""'
V-
''^'
^"^''
= i/.-(-L,.^
f_^,
2
\\
l+.f'
2139
:
cos'x
PhOok.— Substitute
in (21.38), iuul 'Jtau-'.r in
2140
i
ill-
i'
according as
Pkoof.
//
is
>
— Substitute
Pkoof.— Substitute
or
<
,,
,,
,./
rill
l-i.
,\
ri
2*/.
aceurdiu^jly, x
.r
(l!l:'.0).
=
'la siii^
= a-(a-6) 8in->,
^ or
* being
.v.
=
<
b siu'^.
a and >h.
'
''
,
k\
—
INTEOBAL CALCULUS.
343
SUCCESSIVE INTEGRATIOX
In conformity with the notation of (1487), let tlie
operation of integrating a function v, once, twice, ... n times
2148
for
X,
be denoted either by
\v,
«,
V, ...\
\
J 2.C
X
«.
V,
nx
or by
rf.^-,
d_.2x,
•••
(^-nx^
the notation d_^ indicating an operation which is the inverse
Similarly, since y^., y..^, y^.r, &c. denote successive
of 4.
derivatives of y, so y_^, 7/_2«., y^x^ &c. may be taken to represent the successive integrals of y wdth respect to x.
2149 Since a constant is added to the result of each integration, every integral of the n*^' order of a function of a
single variable 'x must be supplemented by the quantity
^
?kl!_=+...+«,._.r+«,.=
f 0,
—l + n—2
Jnx
\7i
where
a^,
flg?
(h
\
•••
'^n
are arbitrary constants.
Examples.
The
six following integrals are obtained
from (1922) and
(1923).
When p
is
any positive quantity,
When p is any positive quantity not an integer, or
positive integer gi^eater than oi,
2151
(-J)"
f J-
When 7)
is
f
I
a positive integer not greater than
lowing cases occur
2152
2153
f
f
any
— = ^"-"^T' log.r+f
0.
i = tiili:(.,iog.r-uO + f
J(y'+l).i-.*''
L/^~i
J(p+l)x
0,
n
,
the fol-
—
310
SUCCESSIVE INTEGHATION.
2154
-i=^^'{f
(•
lor the integral
(..i..,..)-^}+r
•••
witliiu the brackets, see (21 GO).
The following
foriuula
is
analogous to (llGl-2)
tt]"..=J.tl('"-^-)+j>
2155
SUCCESSIVE INTEGRATION OF A PRODUCT.
Leibnitz's Theorem (14(;0) and its analogue in the Integral
Calculus are briefly expressed by the two eipiations
2157
/>«x(«0
=
i>-«.r("'')
(^/r+8,-)"''is
=
(^/.,
+8,)-"
where D operates upon the product ur, d only upon
only upon v. Expanding the binomials, we get
2159
J)Jifr)=:i(„,i' +«W(«.i),r,
+"
^J~
ffr
and
u,
;
S
+&C.
^/ („-:) x^:.
—
The first equation is obtained in (14G0). The second follows
Proof.
from the first by the operative law (1488) or it may be proved by Induction,
;
independently, as follows
AVritiiig
f
ikc u
=
1
it
in the equivalent
(uv)
=
uv-n
f
form
f
».,+
'iI;^
.a,.-&c
(i.),
f
til
;
(,/()=
pry—
nv^+\
nv.^
— &c
(ii.),
may be obtained directly by integrating; the left member sucNow integi-ate equation (i.) once more for .r, integrating
cessively by Parts.
each term on the right as a product by formula (ii.), and eiiuatiou (i.) will
I) in the place of?;.
be reproduced with ('i
a result which
+
2161
f
e"''.v"'= c'''-(a-\-(i.)-\v'"-{-\
0.
Or, by expansion,
2162
a"
]nx
If
m
L
be au integer,
a
tiie
1.2
a*
series tcrminuleb with
)
l)*"
(—
?i ""
-i-
a".
]
nx
—
INTEOBAL CALCULUS.
350
Siniilarlj^,
by cliangiug the sign of m,
2163
f
e^
]„x
a;'»
Proof.
_
~
e^
_1_
(
nm
,
n(n + l) in(m + l)
— Putting
u
=
e""",
v
,
= x"'
7
^^
'
1.2 aV'-^
a" la;'»'^aa)""i'^
')
in (2158), tLe formula
,
f
q
'
J«x
becomes
J Ha:
written before ^^ within the brackets, because ^ does not
affects only the operaObserve, also, that the index
tive symbols d^ and ^j., but it therefore affects the results of those operations.
Thus, since d^e"^ produces ae"-^', the operation d^. is equivalent to aX, and is
retained within the brackets, while the subject e""", being only now connected
as a factor with each term in the expansion of (a + ^a,)"", may be placed on
Here e"^
operate upon
the
is
—n
e"^.
left.
=
2164 P-."V/.
Proof.— Make w =
2166
^ f,.»-^,...->+!il^) ...»-.c.(
l in (21G2)
and (21G3).
cr''(log.r)'"
Jfnx
=
fn.ve^^-*-"^•^^•'^
[Sabs. log...
«.
Hence the integral of tlie logaritlimic function may be
obtained from tliat of the equivalent exponential function
(2161).
For another method, see (2003-5).
HYPERBOLIC FUNCTIONS.
—
The hyperbolic cosine, sine, and tanDE^INlTI0^•s.
gent are written and defined as follows
2180
:
=
=
2181
cosh
2183
siub
2185
ianh.r^:
.V
.V
= cos
—
= -/ «iu
c-')
I {v'
}j
{e'--\-c-')
'—^
6''
4"
<-'
(/.r).
(7G8)
(/'.<).
=.-iii\n{h).
(770)
TTYrElUJOLIC FUNCTIONS.
"By
equations the
tlicsc
351
relations are readily
followinu^
obtained.
2187
2191
2192
2193
0()>li
0=1;
=
siiili
0;
cosli
eosh-.r — sin]r.r=
S'ii»l»
('^H-//)
cosh
(t -f//)
=
=
2194
tanh(,r4-//)-
2195
2196
2197
^iii^^ -•^'
x
= siiih
j:
=
r.
.
1.
siuh.i' eosh//4-cosli.r siiih//.
ooshii'
cosh ;/+siiihcf
siiih//.
fanh.rH-tnnh//
l-ftaiihct' taiihy*
cosh 2.1'
cosh
== cosh'.r+sinh'cr.
= 2cosh'-.r-l = l + 2sinh-.r
siiili
.i\
'^'
2199
2200
siiih
coshiU
= o siuh .1+4
= 1 cosh\t -3 cosh
2201
tanh2.r
=
2202
tanli.J,r=
ar
siiih'' .r.
2tanh.r
1+taulr.f
—
,
1
.
1
.,
,
•
'
.
,
4". J taiili.r
— = y/cosh.r — 1
.?'
2203
siiih
2205
taidi—
2208
cosh
.r.
ry
1
;
2
= Wvcosh.r+l
+—r =
.r
=
j
.r
cosh—
—
= \j/coshr+l
=
r-j
smli.r
-,
—
j-T.
cosh.r+l
2 tanh Iv
l+lanlr^r
^iuli
I— taiilr
l-lanlr.'„r
.\.t''
INVERSE RELATIOXS.
2210
2211
2212
= cosh.r,
r = sinh
7r = ianli,r,
T.et u
,r,
= cosl|-' =
.r= sinlr^ r =
= tanlr^c=
/. .r
/.
.'.
.r
t(
lo^r (,/_|_
h)g
^/„-i_l).
(r+ \/rM- I)l+»c^
.lloirf
_
)•
INTEGRAL CALCULUS.
352
GEOMETRICAL INTERPRETATION OF tanh
/S^.
2213 2V<(^ tangent of the angle which a radius from the centre
is
of a rectangular hyperbola snakes with the principal axis,
equal to the hypcrholic tangent of the included area.
Proof.
— Let 6 be the angle,
r the radius,
isec20cW
i>
e^« = I±li^
Therefore
— tan y
1
=
ilogtan(i7r + 0).
(1942)
= ^^^-^ = tanh 8.
tan
therefore
;
area, in the hyperbola
and S tbe
e
(2185)
-re
VALUE OF THE LOGARITHM OF AN IMAGINARY QUANTITY.
log {a+ib)
2214
a -\-%b
."t^",.,,
,
Proof.-
= ilog {a-+b')-^iimi-^^^.
log
= log
,
1+^a
/
J
^[ =
/
^
i
tan"^
A. By
(771).
DEFINITE INTEGRALS.
SUMJklATION OP SERIES
2230
BY DEFINITE INTEGRALS.
(/Gr)f/.t>= [/(.OV(«+^^^0+-.-+/(«+^*^/aO]^^^^^
where n increases and dx diminislics
qidx
= h—a
2231
Ex,
1.
—To find the sum, when n
+ _1_.
+ JL
n n
is infinite,
1 + _.l.
n+ 1
n
n+2
Ex.
wH 1'
2.— To
»i'
^?a>
l
^
+
+ 2*
.
find the
+
''
«"^
+ 3-
da:
>i
2a
sum, when n
+
+
thus,
J„
log2.
x
of the series
Put «
= ^,
.
^l + (>a/.0='
then
f^-«
'i'
^^'^^
,
+ (Jx)''^l + (2(?,ry^^
= -^;
r— =
is infinite,
+ —!L»-'
of the series
dx
+ -'- =
^--^
^
Put
+
i£ + _^£_+
+
a + 2dx
a + dx
a
2232
so tliat
indefinitely,
in the limit.
=
f
^''L
1^+'^'
=
5..
4
(1935)
OF INTEQRA TION.
R ESVECTING LIMITS
TnEOn TIMS
.'5
5 ;i
THEOREMS RKSPECTIXa THE LIMITS OF
INTEGRATION.
"^
2233
2234
or
= fV (^'-'^O
according as
::t'ro,
and
Ex.—
=
dx
= </)(—.r),
,/,(,r)
that is, if
values of x between
all
(""
J -a
<t>
(')
Ex.—
„
^/'^'
cos
.i'f/x
=
~
cos
.i;rfa!
0.
a.
('"^
= T „cos a;(7j.
"
that
is,
if
and
a.
= -( V W
be an odd function
</>(.r)
*'*(''') ^^'''
a^^^"^
<^'^'
i
•-
*
„ sin
X d.r
=—r
«iii
•«'
'?•*'
f^n^
"^
"^
*'
i
[<p{x)dx
may bo
^'^'-
'
.,
Given ^/<c<^, and that x =
value of
=
~2
between
,/'
*/>(•*) 'f''
j
J
'^
""
J -a
Ex.—
and
i
= _^(_a>),
<^(,,.)
for all values of
r
aj
be au even function
<|.(.r)
= JofVO*') '^' = T^J-a ^
J~2
2238
values of
'^^^
^^^'
Jo
(1401) for
If
= i '/'("—'*')
'/>('')
2 ['
*'o
2236
a-z.
a.
smxdx
[
[Substitute
'''»'•
= 2fV(.r)r/.
f>(,r)r/.r
between
If
(hv
(.,)
I
c
~
„
^'"
•''
-a
'^''
=
^^-
~ ^^
>
makes
*/>('•)
investigated by
infinite,
putting
/t
the
= 0,
after integrating, in the formula
2240
\
V
('*)
<^'''
If the function rp{r)
expression, when ^
the
j>ri'»c/j>a/
is
=r
"^^^ ^-''^
^^"''+
i
*^ *^''^
^'''''
elianges sio-n on b.'ccnung iniinito, tliis
an indelinitely small (puiutity, is called
value of the integral.
2 z
—
,
.
INTEGRAL CALCULUS.
354
If, however, n be
is the principal value.
co
sion takes the indeterminate form oo
which
—
Given a < c
2241
< 6,
value while n
finite in
— Let ^
the integral
is less
made
to vanish, the expres-
.
I
'
—3^'
will
always be
than unity.
{x) shall
(2240) be taken so near to c in value that
remain finite and of the same sign for all values of x coinpi-ised between
Then the part of the integral in which the fraction becomes infinite,
c
iu.
Ppoof.
in
\//
±
and which
is
omitted in (2240), will be equal to
a constant whose value
which occur between 4'
the last integral
2242
where
\"f{^)
6 lies
'^
.
„,
multiplied by
(x)
between the greatest and least values of
and 4' (c + /")• By integration it appears that
in value when « is < 1.
lies
{c
is finite
•.
'
;//
— f^)
d.v
=
{b-a)f{aJrO{h-a)},
a
and
between
1
in value.
The equation expresses the fact that the area in (Fig. 1901), bounded by
the curve y =f(x), the ordinates f (a), f{h), and the base h — a\s equal to
the rectangle under I — a and some ordinate lying in value between the
greatest and least which occur in passing from /(a) to f{h).
If
^ (,T.) does not change sign while x varies from x
2243
f/G^')
2244
If ^(^j
^
W
~)
d^^^
is
=f{n + e(h-a)} fV Gr)
PiJOOF.
=
—
2fV(,,.,i)l^.
— Separiitc the integral into two parts by the formula =
,,,.,,
and substitute
— in..,,.-.
the last integral.
1
to
(Iv.
^ sj^mmetrical function of x and
|->(,..,i)l^
=a
Jo
+
Jo
Ji
,
METHODS OF EVALUATING DEFINITE INTEGRALS.
2245
li/nii ts
Rule
I.
fiiih^titute
a new rariahle, and
acljust
the
a ceo rding ly
For examples,
250G, 2605, &c.
see
nvmhers 2201, 2808,
2;342,
231-5,
241G, 2425, 2457,
—————
METHODS OF EVALUATISC, DEFINITE INTEGRALS. 355
22-16 Rl-hilU.—liilr./ralr //// rarl,i (lUlO), so as
duce a kiLoivu dfjiniic inlfijiuil.
For
see
ej:ai,i2>les,
2608-13,
numbers
2J^:3,
21'J0,
21:30,
2-I-5:i,
li>
2-4Go,
intro-
2484-5,
2(32o, 2G25, &c.
D ijjc rent III tc or integrate ivith respect to
other than the carlable concerned; if a known
integral is thus obtained, evaluate it, and then reverse the
operation of differentiation or integration before performed
with respect to the secondari/ variable.
224:7
i<onie
Ivii.i:
111.
tiiiantltif
For examples, see nnmhers 2346-7, 2364, 2391, 2417, 2421-4, 242G, 2428,
2407-8, 2502-4, 2571, 2575-6, 2591, 2604, 2614, 2617-8, 2632, &c.
Sntjstitute iniaginary values for constants,
the expression into one capable of inte-
IvULE IV.
2248
and thus transform
gration.
For examples, see numbers 2430, 2404, 2577, 2504,
2615, 2641-2.
25J8, 2603, 2606,
2249 'Rule V. Expand the function, if possible, in a finite
or converging series, and integrate the separate terms.
For examples, see numb'^rs 2395-7, 2402-3, 2418-9, 2479, 2506, 2571,
2593, 2598, 2614, 2620, 2625, 2629, 2630-2, 2639.
—
2250 Rule YI. Decompose the integral into a numbrr of
partial integrals, and change all these by some substitution
Bij summing the
into integrals having the sa)ne limits.
resulting series^ a new integral is obtained which mag be a
known
one.
For examples,
2251
see
numbers 2341, 2356-61, 2572, 2638.
Separate the function to be iitf<-grard
Rule VII.
and replace one of them by its value in the
into two factors,
taken between constant limits with
The doutde integral so obtained
may frequently be evaluated by changing the order of integration as explained in (22G1).
form of a
definite integral
respect to some
For examples,
see
variable.
numbers 2507, 2510, 2573, 2619.
Rule VIII. Mnltiply a known definite integral u-hich
discontinuous between certain values of a constant which it
2252
is
new
INTEGRAL CALCULUS.
356
contains, hy some function of that constant, sucih that the
integral of the j^t'oduct with respect to the constant is hnoiun.
A new definite integral may thus he obtained.
For
exaviples, see ntnnhers 2518, 2522.
Particular artifices uot included in the foregoing rules are employed in
2293, 23U5, 2310, 2314-5, 2317, 2367-9, 2404-15, 2422, 2429, 2456, 2495,
2514, 2518, 2585, 2000, 2626, 2635, 2637.
Additional formulae for integration will be found at 2700,
et seq.
DIFFERENTIATION UNDER THE SIGN OF
INTEGRATION.
Let
11
=
f(x) dx, where
J
of each other
;
Proof.
— Let
Let u
u
of
u,
Phoof.—
=
225V
if
and
n„
=
—<j>'
(a)
= —f(a).
Then, when a and
c) dx.
f{x,
are inde-
h
a
c,
= f {/(.r, c)
^=
j
}
(Iv
and
\f{x,c + h) dx- ["/ (x,
'^— dx
--^
But
= (j>(b)—f{a).
rb
=
J
pendent
^ = -/{")
and
% = 0' (&)=/(&)
Therefore
2255
then
^=/W
2253
and f{x) are iudepeudent
a, b,
a
(since h
is
n„,
c)
= f {/(.r
dx
I
c)
} „.
d.v.
~h
constant relatively to x)
a and h also are functions of
,
=
^•'''
(
dx.
c,
^=j:irq£-)j<..+.m,.)f-/(«..)S.
—
Proof. The complete derivative of u with respect to c will now be
u,+ a,b^ + u^a^. But
=f{b, c) and u„ = —f (a, c), by (2253-4).
III,
ArvnoxuiATE ixTEauAriox.
INTKCl
357
RATION BY DTFFKRKNTIATTNd UNDKll THE SIGN OF
INTFCi RATION.
2258
by (Mr>
i:x.
I),
2259
The
1.-
a
and
.r
2.—
i:^-
\y'>^'-^ir
boiiif,'
Ex.
^(c'-),,,,.h
=
j.,.^.-dr (2250)
transposed.
{x"c"-^\uh.rd.t
last intet,M-al is
2260
=
=
il,Jc"ii{nh.rJx.
given in (1999), putting
n= 1.
3.-
INTEGRATION UNDER THE SIGN OF INTEGRATION.
When
2261
That
are constant,
the
liiuits
(""
^"\nv, y) dvdj,
=
the order of integration
is,
But an exception
of the integration,
intesrrals
above
an
T" r7(.r,
may
ij)
fh/flr.
be changed.
to this rule occurs wlien, at
infinite value is
will not
any stage
The double
produced.
then have the same value.
ArrrxOxiMATE integration.
BERNOULLI'S SERIES.
2262
J/('0^/.^^=
Proof.
— Integrate
/'(aj) into/(j;)
2263
in
^{/X^O-j^^/X^O + y-^^
successively by Parts, J
(1510).
-/.c,
l.c/r,
Ac.
Or
cliange
—
IXTEGRAL CALCULUS.
358
—
Proof. Put/(a) for V)'(tt) iu the expansion of
(19U2), by Taylor's theorem (1500) viz.,
tlic
right side of equation
;
\^f{x)dx
= ^(h)-<^{a) =
The following
Let
{h
a nearer approximation
is
— a) =
(6-a)^'(a)+fc|I%"(a)+&c.
where
nh,
« is
an integer
:
then
;
|7W civ = h {lfib)-\-if{a)^f{a+h)+,..Jrf(h-h)}
2264
-7j!rw-r(«)}+&c.
— Expand
(<?" "
— 1 ) -4- (e"-'' — 1) by ordinary division, and also
and opciate upon J\x) with each result; thus, after multiplying
we obtain, by (1520),
Proof.
by
by
-^
(15oi)),
h,
h{f(x)+f(x + h)+f(z + 2h) +
which expression, by changing
.-«
...+f(ix
and x + n]i, into
into a
=
d_-^{f(x + nh)—f{.c)}
the above, since
+ l^^h)}
/
b,
is
equivalent to
dx.
(,7.)
2265
Proof.
— Assume
in the expansion of
a;
=
—log
"
ce''^.
f
1
^
=
^+
—
2//C--
Then
—
].
,
1:2+
and a
upon
\
the coefficient of
to
1:273
,
+
4V,V
"^,
1:^:3:1.
.
•'
.t,
An- x.
A
more general
result, obtained iu the
i*a+nh
2266
\
—
Tliug
3-//-r'
- /Id
it
equal
for c in this equation, and operate
and therefore dj.e
f (x + h) dx, employing (1520). Finally, write f(x) for f (x).
Substitute d^ for
with
is
a;
/Or)
same
7,9.
(Lv
wa^-, is
= nhf{n-h)-^rn {n^-2)^f {a-2h)
—
350
Tin: L\Ti:i!i:.\Ls h{l,i,i) asij V(,i).
THE INTEGRALS B (/,
AND
m)
EULER'S FIltST INTEGRAL h
Till' tlirco ])fiiici|):il fornix;
2280
I.
2281
II.
2282
111.
B{l,nt)=
\r'-\\-.ry'-'(h'=Ti{m,l). [ny(2233)
.'!'''
lUl,m)=(
Jo (l+.r)
I
aud
m
/
[By substituting-^-
f'^'-
(/,
w)
in I.
•'
-L
[By substituting!—^ in I.
•*
arc positive, and
B
/
is
au iutcgcr,
= i|^.
be the integer, iutercliangc / and
integers, the forms are convertible.
rti
Proof.
If Ijotli
ui.
/
and
— Integrate (2280) by parts, thus,
f
.r'-'(l-.^0"'"''^-^'=
Jo
Repeat
"
lUl,m)=\ -4^1-1— fAl-.
2283
If
{!,,„).
arc
Jo (1+cl')
"Wlieu
m are
\
r(w).
—m
f
.^•'-'(l-a;)"•.
Jo
this step successively.
EULER'S SECOND INTEGRAL T{n).
n being a
2284
real
y{n)
and positive quant it v,
=
\\-\i--\Lv
=
\
'(logiy
The second form being obtained by substituting
2288
2290
=
r(l)
2286
I'(7'
r
(>*
+ l) =
+ l) =
Proof.— By^ Parts,
'
!
f
Jo
«
r(2)
l,
>'!'(")
,
c
=^n{u-\)
^vhen n
Cx^'-'dx
is
...
\Lv.
' in
=
the
first.
l.
(n-r) V{n-r).
an integer.
r^'x^lx.
= ^1 + —
f
n Jo
"e'Jo
The fraction becomes zero at each limit, a~s appears by (1580), dilTerenfiiiiiug
the numerator and denominator, each r times, and taking r>n.
.
INTEOBAL CALCULUS.
360
2291
=^ = f
Ce-'V-'d.,
Jo
^'- (logi)""'rf.r.
Jo
A,
\
tl /
—
Proof. Substitute l-x in tlie first integral, and so reduce it to the form
(2284). In the second integral, substitute — log.t;, reducing it to the former.
When n is an integer, C2291) may be obtained by differentiating n l
—
times for k the equation
AYhcn
in
is
an indefinitely great integer,
r{u)=
2293
Proof.
and
—
=—
e~'''^dx
I
1
log—
/^^
^
1
=
X
then substitute y
\;f /,
.
lim.
(1— x*^)
/u
(1583).
Give
it
this value in (2285),
1
=
.f^
;
thup, in the limit,
r(n) =/.«-' r(l-.rM)"-'cZx
+
= ^"
['>/-'
(l-yy-'dy.
Then, by (2283),
1 in the fraction.
changing
fi
2294
Let n he <1, ^ an indefinitely great integer, and
finally into
/i
logr(l+?2) IN A CONVERGING SERIES.
/Sf,
=
l+^ + J7+...-^,
then
log r
(!
+ »).
2295
={\ogii-S,)n+lS,n'-lS.y-\-iSy-iS,7i'-\-&c.
2296
= ilog^^^^ + (log/.-.SO n-iS,n^-iS-X-&c.
HIT
sill
+ |(l-S.) + |(l-S.) +
2298
log +i' +
=i\og-^^-i
1—n
^aiunir
'!tc.
1227813«
I
P.OOF.-By(2203),
1,
^ d + n)
when
/M
= CO
=
^^^^
.
^^;^;^^ ^^^ ^^^
,
Whence
+i
iogr(i+H)=«/^.-Ki+n)-/(i+ ;^)-z(i+
ii-tfi
;;)-...
-/(i+^j.
—
THE INTEGRALS
B(/,
AND
m)
<S>*
+ -i.V* +
iV + i'V +
&c.
=
301
l{ii).
Dovelopinf? the lopfarlthms by (155), the series (2295)
series is deduced from thi.s by substituting
is
.
obtained.
The next
hifr«7r-lo{,'sin«7r,
=
mc and expanding the logarithms
a result obtained from (815) by putting
by(15(;).
The series (2207) is deduced from the preceding by adding the expression
= -J- log J-"ti' ^n^—-^^\—n
=
B('.'")
2305
— Perform
Puoor.
tlic
from (157).
+«!^c.,
5
3
M^.
integrations in the double integral
by formula (2201), and then for//, by (22^1), and the result is
(i-\-m).
Again perform the integi-ation, first for //, by (2201), and
the result is r(/) T{m), by (2284).
first for
B
(/,
.r,
m) r
—
Note. The double integral may be written by the following rule
Write xy for x in r(0, 'i«t^ imdtiplij hij the factors of r(»i-|-l).
:
thus obtain
[
which
e-^^(.njy-'x-c-\rdxd>/,
\
jo J
I)
equivalent to the integral in question.
is
2306
m) B(/+m,
B(/,
= B(m, «) B(/** + «,
= B(>j,/)B(;i + /,»0
n)
/)
=%SW-
2307
2308
If
('\v^-\a-.v)""'(U'
Jo
2309
l+.t'
Jo rr^.
2310
I
V^
(f>
=
and
i'<'/,
if
m
= "^^
—
—
^r-^
-7 siii>/<7r
—
1-7^
'Iqiiuxmn
=
(i.) In (2023) put /
2p + ], » = 2-/, and take the value of tho
between the limits ±oo. The lirst term becomes log 1 = U llio
Proof.—
;
sec"ond gives
tlie
series
2q
q L
by (800).
(ii.)
=
Jo l—ar''
[Substitute ^.
"'
^^*
'
f^^<-->-
= a'^"'-'B{l, w).
and q are positive integers,
j;
integral
We
The
(2310)
is
deduced
Iq
2q
integral required
iu
is
)
qsmniir
one-half of this result, by (2237).
a similar manner from (2021).
3 A
.
iNTEnnAL calculus.
362
Jo
"wliere
l+.r
m lias
fi_,/,.=^L-,
= ^j^,
ii:,/,,.
2311
Jo 1—0,'
siiu/<7r
and
value between
ainj
Proof.— By substituting
in
;r'^
(2309-10).
taii/>i7r
1
^p
w = =^+—1
Also, since
by
,
takino" the integers p and q large enough, the fraction may, in the limit, be
made equal to any quantity whatever lying between and 1 in value.
r(m) r(l-»0
2313
'^
=
m
,
.
being <1.
siii/><7r
Proof.— Put
l+m =
r
thus,
two values of
in the
1
=
r (l-»0
{in)
f
Jo
2314
CoE.-
The following
r(i)
=
y
form the product of the last
d by the equations
= rs{nd\
^^,^^^
=
{r(i)}-
^^.^^
4l
\
Jo
<lr
s\n
by
two
(1
2
e-'-'dij
[
integrals,
009), cluch
e-'" *'''dydz
=
2
;
by (2311).
iiiTT
er'-dz.
|
and change the variables
= 'l['^'-)-drdd = rdrJe.
= 4[
\'
e-'\drdd
=
to
Hence
tt;
Jo Jo
=
i/
+
::\
tan0
=
^.
r(l)r(l)r(|)...r(^l) =
—
Proof. Multiplv the
apply (2313) thus
.
bui
-^,
and (2305)
= v/'r.
.'o
r^
2316
+
(228-2)
= —"—,
being obtained from
the limits for r and
2315
=
e-'\c-^cU
[
Now
r,
l
w)
(/,
an independent proof:
is
r(l)
'^x
B
TT
n
r(,,)
left side
.
sm
27r
-
n
by the same factors
.
(l!—
...sin
r(,,-+i)
n
V^
(27r)"-^
in reversed order,
-^^—,
and
by (814).
TT
11
...
vL+'^ = v^i^^^'.
THE INTEGRALS B (/,
Pljoor.
— Call
r is niiy
V(.c-\-r)
=
*f>{x)
a*.
=^
tlif
J-""
V{x)
(.(•-!-/•),
tp
cliaiifrt'
—
'//(.r).
Icfl
be.
3G3
T{n).
(!liaiij^o
(iaimiia fuiicti.'ii
result aftor rcilucrtion
The
great r may
'l'{x)
,
AND
x to
by
each
(2'JHS).
liowi'ver
But, wlu'U x =:
fore
cxincssiuii on the
mid
iiifc^'tT,
?//)
Tlii-relore
is
</>(.r)
is
./•
+ r,
ilic
wlicro
forinula
II<-iico
(/.(.r).
iinlrjimdvul of
There-
takes the value in ciut-stiuu by ("iolo).
(x) ulwa^-s lias that valui\
The formula may
also be obtained by
means of (220
NUMKiaCAL CALCULATION OF
2317
All values of
r(,i')
may he found
I).
r(.r).
terms of values
in
lying hetween r(0) and r(^).
in
"When X is > J, foi-inula (2280) reduces
which x is < 1 and wlien x lies between
;
reduces the function to
(2818)
between
and
Values of
made
to
r(.e)
^.
T(,r),
when
and 1, can also be
x lies between
in which x lies between J and J-,
depend upon values
by the formuh\3,
.
rw = 2'-=VTT.!5^.
2318
Proof.— To
obtain (2:U8),
make
n
r(,.')
=
2
—
=
I"
in (2:11
T.).
^-c
in
v
!(£)
f^-
To obtain (2310),
(2318), and then
(—7— J-
Mcfhods of employing the forniuhe
2320
,
^^^^
put i»=:|(l+,r) in (-:31o), and change x into
clinunatc
to the vahie
and ^^ formula
the value in which x lies
1
(i.)
When
—
x lies between | and
1,
reduce r{x) to
V{l-,v), by (2818).
2321 (ii.) When x lies between -J and I, reduce by (2:}K»),
the limits on the right of which will then be I, and -J.
(iii.) When x hes between (» and
rr.hicc by (2818)
r(^+.'') will then involve the limits .} and J, and will bo
reducible by case (ii.)
2322
'
,
;
If 2x is <i, reduce r(2./) by (2818), writing 2,r for ./•.
If
this gives 4,/'<^V, i-cduce again by the same formula, writing
4x for
Xf
and so
on.
INTEGRAL CALCULUS.
364
The
2323
figure exhibits
tlie
curve wliose equation in rectangular coordinates is ?/^r(,T),
Let the unit abscissa be OA
=
AB=1.
AD, BG
Then the ordinates
—
1 by
are also each
(2286-7).
The minimum value of T(x)
is approximatelyO'8556032, cor1 '461 6321.
responding to a?
The values of logr(a;) in
the table at page 30 correspond
to ordinates taken between
V{2).
T{\) and BG
AI)
=
=
=
INTEGRATION OF ALGEBRAIC FORMS.
f/Sw^^^'^' =
2341
^(^'^'^)-
Pr.oor.— Add together (2281) and (2282).
tegi'al into
2342
+
y
2343
2344
The
,
and substitute
.r'-^(l-.r)'"-^
^^'-^'-^y:
—
d.
rV-Vl-.r")"'-'^Ai'
Jo
Separate the resulting in-
in the last part.
=
Bilm)
4i\^
[Substitute ^-±i^
m\
= —B(L^
\n
n
[Substi tute x".
J
-Lz,(/,,0.tS"b.^;^3f^
f'^igi^^^,/,. = uo
Jo \a.v-\-b{l—d')\
integral
is
^
^
also equivalent to
(19,
and
siniilarlv in
other casi
(a sur
Jo (M.siii-^+6cos-^)"
(/.V
2345
[Substitute
'o
{H-\-OA-y"^
nab"
— ~.
—
,
'
.,
INTEORATIOX OF ATJlKBl^Air
2346
,
— Substitute =
.);
2348
where
l+.r'
m and
"When n
2350
\
cosec-.
Proof.
-I
x'" in
r=— COWOC—
')ii
Proof.
in ^2350),
— Se2)arate
>
and
l
= —m cot —n
—
+
'
.«
=— col—«
>'
7)i
and
in (2351).
1
«.o-*-ct
—
(2311-2)
^^d substitute
V
and
+l
for
smnin
hes between
+
<
?«.
'
*^
+cr
into
-^^
J"
\''!^::;±^.i.,=^^.
1
and
m
1
"
^<
Hes between
n^'t^
n
/*
in is
r
\
(2311-2) and change
— Make n = 2 and write
Jo
1—.*
Jo 1—.*'^
n
— Substitute —- —^
^
=
in
- cot —-,
z=
)
Jo
«
(2311-2), and then substitute x".
n
Jovl— U"
When m
—
and greater than unity,
— =—
— Substitute
Proof.
"^
positive
is
(2255)
for a.
a?i^ positive quantities,
-T-,
1
2352
I
;i
Jo l+.r"
Pkoof.
where
= - c'osec
n are
3G5
FDiniS.
in (2311), aiul tlicn difreirntiato n times for a.
aij
— Change m into
Proof.
2356
-
-j-
\
»',.
.
1/1.. = -^^ -^/
PuooF.— Differentiato (234d) hi— 1 times
Pkoof.
.
.
m
^
«^
in (231-8-9).
f'l!^
1—
d'
Jo
tan>/<7r
1.
each
into
two
integrals
by the formula
.f' in the last integral.
I
Otherwise, iu (2G01) substitute t'"^, aud change a Into
Tra
— jtt.
.
.^
INTEGRAL CALCULUS.
3G6
Pjjoof.— From (2354-5) by
2360
—In
2363
«
— cosec
—
n
= -cot _.
,/.,.
same way, from (2348-9).
/
!_;>
Proof.— In (2601)
tute e
=
n
_^^_^
tlie
J
(Lv
l+.r"
j
Proof.
method of (235G).
—
I
Jo
2361
tlie
substitute
2«
e
= ^.
and put a
In (2595) substi-
—
=
and put a
'^-|,^""l^-
71
/»00
(»-l)
T
-J
2364
j„
Proof.
„
f
x^
'
+ a^
— By
f-^
successive reduction by
= — w— 1
2365
-,
2a
f
to
times with respect
'
«
(2002),
an iBteger.
by
differentiating
or
(2255)
a^.
_ 7reosoe»;7r
^^^'-^1^^
l^eing
^gubs.
V^^
(2311)
^^{fL-^,. = lo,n.
2367
Proof.
= 5t^ 2;^.
— The value when a =
1 is
is a, is
Jo
which, by substituting
.<;"
i--«
The
\ogn.
dx — n
'
—
Jo
'-
difference,
—
when
the value
dx,
i--'-;'
in the second integral, is seen to be zero.
F(x) being any integral polynomial,
^'
2368
1
At^^^ = ^'^^
^vliere
J
is eciual
to tlie constant
J-:v/(l-.r-)
term in the product of F^i) and the expansion of
M
/
1\-^
r,
)
.
Loa.\nmniir axd expoxenttal
Pkoof.
1^07
fo7?vs'.
— By successive redaction by (2053), we know that
i^^ = *(')^<'-">-'[7(f^,
wliere
(r) is
<p
some
inti'Sjral
=
inti'cral in question
<•>•
polynomial and A is a constant. Therefore the
To detcrmino -1 write the last equation thus,
Avr.
f^(^-;;^)"'--'*«('-7)-'-'U.{'-'=)
the integrations and equide
of the two logarithmic terms in the result.
Expand each binomial; perform
F{.r) being
an integral polynomial of a degree
tlio
coenirients
less
tliaii
h,
f;^,.,. = |^"i-{.(,.)lo.^:}.
2369
J,i
But El£}
=
»-
(.'•—</
I
I
fix, c)
i--c
(421), and therefore
+ ^--^,
x—c
d,,,.^^
,
/
(.r,
=
c)
J
<.'^'
a
\X — Cl
f is of a dimension
where
0.
Hence the
)
lower than n — l
integral on the right
INTEGRATION OF LOaARITHMIC AND
EXPONENTIAL FORMS.
2391
f;,.''iog,../,,.=^,.
Pkoof.
integral
j
x"
(Ix
Jo
— These
are cases
diflerentiate,
=—
Y
/''-'•
2393
and
of (2202).
to obtain the
with respect to
;<
(2255 and
j"'.,"(log,r)",/.,
=
j;;^>iog(/<+i).
to obtain the first
Otherwise;
Other
second integrate, the equation
22<)1).
(-i)»Il|+ll.
—
Otherwise, when » is either a positive or negative
Sec (2202).
Proof.
integer, the value may bo obtained, as in (2:i01), by performing the diflerentiation or integration tliere described, n times successively, and employing
formula) (21GG), and (21G8) in the case of integration.
Jo
log.r
*!+*•
=
=
INTEGEAL CALCULUS.
368
<'"•-'''
2395
Jo
1— cr
Jo
1+.^'
Proof.
The
= -^«„y = -
2'^""
by
'is
..0
><
summed
l=Fa;,
first
;
^rf.r.
1
e'
and integrate by (2393)
The
in (1545).
equal to the
i—
—
c'
Jo
2/t
— Expand by dividing by
first series is
tiplied
rf.,.
difFerence of the
two
this gives the value of the
+l
;
thus
series
second
mul-
series.
2399
r^jiKii^
ri2££rf,.
Jo
l—t
Jo
-1-1
1
3-
2-
^v
1
Proof.— As in (2395-7), making n = l.
The series (2399) may also be summed by equating the
in (764)
2401
b
coefficients of 6
-l-&c.=-^.
= -l-l-i,
fM£rf.
«
1—
tii)^
.1^^
Jo
Proof.
2404
= -^.
and (815).
Proof.— The
terms.
&c.
4-
integral
—Expand
The
of those in (2399, 2400).
r ^""^^"'^'^ (Lv =
\
—x =
u(
i/;
\
"^^'^
integral
a being
<t>{(i),
<
1.
<(
Jo
Substitute
sum
the logarithms by (155) and (157) and integrate the
(2400-1) are reproduced.
series in
Let
The second
half the
is
therefore, writing
=
C
^''^
].-al-2/
=
r
Z
for log,
^^ "^y
-
l^-'J
C'"^1L^
I
1-'/
by (2399), and the third by Parts, make the right side
= _ .^l + Ja
I
(1
- a) -
f""
^-^—^
'?i/-
Therefore
LOQARlTUmO AND EXPONENTIAL FORMS.
2405
cf>(^a)-{-<i>[^i-a)
Again,
Put
for
-
,
2406
.<:
^ (.0
2407
•••
=-
?.
by (2
(-.»•)
(l+
[
+ (-x) = -
^=
—
x+1
1+ ^
!<?
03)
and by (2405)
tufi)
;
=
l(/y3r-;
(//3)'^-
+
+ |r
-^
+'^c.)
=
7^-?. (.c'j.
=
= 4{log(H-,r)}'tt
(/'O
'l-^
+
~t v^^
2
=
/'
say,
= ^ {^ (l+/3)r,
= l-/> and l + /3=~
[v
/5-^
=
2
(//3)^
Itt' and 9(1-/5)
jT'i^S^^
2408
(i.)
thus
05^)-'/' (/3)
(1 -ft)
=
...
+ |- +&C.)
2
|-9(l-/3)-9
(,5)
= iii—^
Hiog(i+.r)}"^.
"
-^ (•«'+ fr
and therefore x
a;",
by (2407),
<p
_a)- 'tt.
i^y -^-^3) 9 (r)
<^(;^j)+i'^(.'-0-^(.')
Let
or
'/•'•,
^ I
then
;
9(.r)
Eliraiuate
.-.
\oira lo^-
^(-•'•)+^HrT:r')
Also,
Henco
'-^^^^
= r
('•)
V
=
8G9
-
=
-j^
7r=,
(//5)=-
j^. tt',
that
is,
= (<«?^=-')' - W-
2409
2410
value
is
^ („)
Let a be
>
determinate.
=
1,
then 9 («) contains imaginary elements, but
have
We
ilk:^) ,i.+
1°
Ul=s) ,,. = -
I'
3 R
^ + [ " + '^-"
.
its
.
INTEGRAL CALCULUS.
370
—X =y
Substitute
]±
Hence, when a
If a
(«)+</»
= 2,
^
2412
2413
Let X
4/ (a;)
+
^
2415
2416
Proof.
4/
l^^^]
\l
+ x/
= T
Jf,
(l
^-^
\
1
W +^ ([=f
)
^:^^
,
-
—
+ -~]
1—
2x
l+.-B
—x
1
,
IX
»"/
—
-\-
/
'-,
and therefore x
=
X
T'l^gy^
4?-
-^^.„
dx,
therefore
= ilog.r- log l±f
vanishes, by (2403) and (2401), putting x
1
a
by employing (2405),
cZ,^
Z
=
\
^_^4 +,; ,og2.
rf,,
.V
f C^) = ^ (i^),
The constant
6
^(a.)=£^logi±|(Z*.
Therefore
2414
becomes
y
result becomes,
Let
therefore
last integral
= -^+^nog«+i(iog«)^
lo^(l-v)
f
Jo
The
(2214).
> 1,
is
(^)
tills
it
h
y
io
<^
and
in the last integral,
y
2411
= ^i by
l{-l)
the integration by 2300, and
v^2
—1
;
= 1.
then, by (2414),
= -* {logiy-'-i)}'-
f'M+iiOrf,_-,„g2.
Jo
— Substitute
o
l-|-cl'
^ = tan"'.r
;
then,
by (2233),
2417 l^^y (liffereiitiating or intooTating tlie equations (234.1)
to (2;]G3) with respect to the index w, the integrals of func-
=
LOGAlUTmilC AND I^Xl'oyhWTlAL
involving logc are produced
integrating for ))t between the limits
tioiis
2418
f' /i'I'Vr"'"
''•'
L'(WMS.
thus, from
;
!
and
in,
IWl
by
{'2-\o(')),
we have
= '<•& *"" ^-
(''«>
Otlierwisc, this vesult may be nrrived at by farming the expansion of the
fraction in powcM-s of.c, and integrating the terms by (2o'J2) ; the reduction
is
then efFected by 815-G.
^^^
In a similar manner, we obtain the more general formula
2419
'^ '''
f
'
'"S-
(l+ci' j log.f
Jo ^•fri";',".
2420
ii
(f'V
„,,
,,
]
Jo (l+.r") logci'
Proof.
=
m+*^p
in-\-p n-\-'2j)
-
log
tan —-.
"^
2n
— Integrate (23G0) for m from 2« to m.
''^''
2421
f'
Jo
""-''"Y^'"7^'"^''
(log.r)"
= (7.H-l)log(y>+l)-(i-+l)log(r+l) + (r-7>)|l + log(v+lj}
Proof.
— Integrate (2394)
for
between
j)
tlie
limits
and
r
(v-.-)..-+(>--/>) ..-+(;>-,)
2422
I"
Jo
.r-
j).
„.
(log.rj-
= log{(y,+ l)0^+')(7-.)(,^_^l)('/+i)('-i^)(,.4.1)(.+ n(/.-.)}.
Pkoof. — Write (2421) symmetrically
y and
for
the three equations, respectively, by
ry,
r, j>,
r,
;;
;
7,
/»,
and add,
r.
^Multiply
by
redu<.-ing the result
r '"^^^.^+"''%/.f=:^log(l+«i).
2423
—
Proof. Differentiate for o, and resolve into two fractions.
integration for x, and integrate finally with respect to n.
T^ffect
the
2424
Proof.
and
— In
(2423) put a
intei^rate for I
between
=
1,
and substitute
limits U
and
-,
-
=
ij
;
multijdy up by
and intho result substitute
b,
6y.
JNTEGBAL CALCULUS.
372
fV*-\/A=
2425
2426
f %-'-\i'^" chv
=
«..
Proof.
— Substitute
1^^
^•^••:if;^~^^
Z
for I:
2427
j
2428
f"'(^L^^^' _ (£=^)^')rf..
b
b
?i
=
and a
and c
1 in (2291),
'^ y^.
to obtain (2427)
e-"' ch
;
c
=
=
--.
Integ rate this for a
and integrate that equation
/
.r
for h
= „_i+HogA.
®
a
a in (2428),
— Integiating the
first
[£:!:n£:l']-
The indeterminate
= .-6+iog*;^.
to obtain (2428).
.r"
Proof.— Make
2430
[
r('-i:^:=ir_(^£^*)^),,,,
Jo \
Otherwise.
A^"
^^—d.i=loga-\ogb.
^
Proof.— Making
2429
lix^.
Otherwise, differentiate the preceding equation
Avr.
n times
between limits
between hmits
[Substitute
1 ^ k
Jo
fraction
is
term by Parts, the whole reduces to
+ ,,p-'-^-",,.
evaluated by (1580) and the integral by (2427).
|--j£!:z^_(«-^V-_(«-y>-"V^,^.
Jo
,v^
(
=
Proof.
— By
)
\ {(r-^'^lr-4.ah-\-2lr{\oga-\ogh)].
two successive
int(^grations by Parts,
'\x''^dx,
&c..
{'L^d. = '^'"'-[^-dr.
Also
Substitute these values, and
make
e
= 0.
by (1580), and the one resulting integral
In a similar
2.V
.r-
maimer
tlie
The vanishing
is
fractions are found
that in (2427).
value of the subjoined integral
may
be found.
.
TNTEOnATION OF CIRCULAR FORMS.
373
2431
i~~?
Jo
1.2.6-
.r'
1.2.;t,r'
S
•
INTEGRATION OF CIRCULAR FORMS.
Notation.
—Let
signify the continued product of n
a^"^
factors in arithmetical progression, the first of which
the common difference of which is h, so that
2451
a^;^
= a{a + h){a + 2b)
...
+ 0^-1)
{.,
is a,
and
/.}
Similarly, let
2452
a':l
= a{a-h){a-2h)... [a-in-l)
These may be read, respectively, " a to n
ence b"; "a to n factors, difference minus b."
2453
sin".r^Ar
Proof.— By (2048)
;
=
sin".i;tfx
=
I
.sin"-.r^Ar.
\
\vc have,
applying Rule VI.,
sin""^rtf«—
sin"
I
b].
fartors, differ-
by
ili\
cos"
"^i;
ision,
t/.r,
I
1
Ti
i
and by Parts,
1
f
•^
n-2
sin" ^x
7
coa-xdx
= sin""'.)'
—
71
J
i'l'
Tliereforc
h'ui"
'
.r
con' x
dx
The
If n
he an
n
iiite<^vr,
PnoOF.
H
first
—
l
—I
,
11
L,;„n vdx.
sni
}
xdx.
Jo
equation produces the formulu.
with the notation of (2I")1),
2"'>
siir"^\rr/r=
-^
m
1*"'
i-i-
and
— By repeated application of formula
Wall is' s Formula.— Ji
2456
1
.
1
'
ri'
2454
.V
= If*".sm
Jo
substitution of this value in the
cos
sin^ .rf/.r
"
= -^
7r
^
(•21:'>3).
be any positive integer, we liavc
INTEGRAL CALCULUS.
374
And siucc the ratio of these limits to each otlier constantly
increases, the value of either of them
approaclies unity as
m
when
Ex.
m is
infinite is
— With m =
4,
-Jtt.
^v
lies iu
2^4^^^8
Proof.
— Put
2m =
2-A-.C,K7
,
and
then
nrr
fin
sin"'^ xdx,
Jo
n,
magnitude boUveen
I
Jo
sm"xdx,
and
„_1
fl^r
sin""';
»^
dx
Jo
are in descending order of magnitude ; the first and second because sinx
< 1 tlie second and third by (•24-33) ; then substitute the factorial values
(24.54-5).
;
2457
i.s
by
^
TXTEanATTOX OF CIECULAB
If
both the indices
2465
ni-c
^i"'-''^''
Proof.
— Ileduce hy
the value of which
37'-
cvon, then
t'^*-"^-'''
Parts as
given at
is
FORiTS.
•'•'/•'•=
(2i:.i)
Tn;rr7^
Tlic liinl intiLrnil
Ix'forc.
is
siii-""'''a;'/j;,
("2 -too).
2466 Sliould eitlier of the indicos bo a nco^ative integer, the
vaUie of the integral is infinite, as the foregoing reduction
shows, for the factor zero vn\\ then occur somewhere in the
denominator.
2467
*
sin^j.r sin/).rr/<r
=
••0
when
n and
]
cos iLV cos jhtdv
= 0,
*
p
are unequal integers.
2469
\
^\nnr cos jhv (I, V
»'o
= n—p-^—
-,
or zero
according as the difference of the integers n and j) is odd or
[By (1973-5).
cnm.
2470
when u
is
Proof.
2472
\
shi\vd,i
=
\
The following four
2474
2475
}i
and
})
cos2iix',
^
if 7^
=
be even, and
2"-'hr-ir
=
Itt,
and then
\
integrate.
{l—,v')
~
(h\
integrals (2473-9) all vanish for inexcepting in the cases here specified.
(-l)"-'\;(^/>,'-^)-;^,
and n are both odd, and
But
cos-na(Lv
cos'\v(f.v=
fJsin^r sin n.vd.v
})
I
Jo
— E.xpress in term.s of
tegral vahies of
wlien
=
sin-/Mv/r
1
Jo
an integer.
u
n is
not greater than p.
odd, the value
is
7,-(p--2r'^ fr-{p—ly
.:.i-^y'^^l
—
i
1
IKTEORAL CAWUIjUS.
376
)
2476
.r
Csi»"
COS n,vd.v
= (
when j;
and n are hoth ecen, and
2477
But
1
i^"-^'
,
*
be
if j)
(
otZri
and n
2478
Jcos''
ecoi, the value
C (/),!)(?>-:;)
;>
.1'
cos
«.i J.i^
=
i
when
2^
and
greater than
2479
is
C(,., 2) (/,-<.)
(;)— 1)--«-
C (;>, ^') ^,
are either both odd or hoth even, and
7i
^i
not
is
j)-
smnxdx, when
rcos^is
n\
.
"^
(/)-2)--«-
2"-H/)'-H-
'
2^) ^,,
not greater tlianj;.
is
'?i
C (,,,
"
1
j;
~
C(/),2)
C(y>,l)
1
odd, takes the vakie
is
?i
^.,
?
the last tonn Avithin the brackets being
^v~2~/
—
/r—
w^^®^
5
P
71
Proof.— (For 2474
is ^^^^^
V
^^^^
^'
even, or
'"2") ^^^^^^
i'
is
^^"^^^
and n odd.
s
^'"
to
2470.)— Expand by (772-4), aud
11
being any integer,
apjily
(2467-2470)
to the separate terms.
CoROLLAEiES.
2480
i
"t*os"
'^^
cos
nddv
2482
= ^,
--
.'o
\
cos"
\
.f
cos udd.v
=
••
sin-"ciCOs2/?crf/.r=
(
-^^.
^
— 1)";^,
JO
m-"^\rsin(2w + l).rr/.r=(-l)".-^.
Jo""
2484
I
Jo
cos^.f cos
>ia7/ci'
—J= ^—^
jr-tr
\
Jo
cos"--.f cos nd(Li\
?
JXTKCIiWrioS OF I'lUCFLAR FOHMS.
377
2485
i*i'
=
cos'\rsiiUKr(/.r
\
Jo
Proof.
x,
by
eos''~-cr(M)s»,rf/,r
— ^lake p = u
When
k
— 1)
i'
p—trr-
in
e<)S.j-(/.r
j cos''
l\irts,
= 0.
"'a;
sin
fi
siunrd.r—-, -,.
p —n
;
the
an.l
new
iiiti-gral
a- (/j;.
cos" ^r sin
j
—
,
•
.
cos'^-.r
)
Jo
either fonuuhi) I5y Piirts, j
Proof.— (For
of highest dimensions in cos
2486)
i> r />
'—
/J.r^/.r
= -—j.
(2184-5).
a positive integer,
is
Mir
=
co^n.vdv
2488
)
2489
|%«os--^^r COS »u(Lv ==
cos'-'-^r
0,
^^^±^
^::^.
= —
n 2, n — 4-, ... ?; — 2/.: successively in
Proof.— The first, by putting ;)
u + 2, 7i + 4,
(2484) and employing (248t)). The second, by putting ^i
... n + 2k successively and employing (2481).
When
2490
k
is
not an integer,
"co.s'-'^r co.s )U(Lr
(
=
=
2' -""^^ sin
/.-tt
/i (/i
-2/,
+
1
,
A).
(270G) t;.kc /('0!= """'^ a"*^ transfurm by (7t'.t^). ^The covanishes by (22'jO), and the limits are changed by (22o7).
P f;noF.— In
efficient
mt of
2491
\
I
2+^+^ + + -
cos".rsin>M</,r=,^
Proof.— By
...
successive reduction by
making
(1070),
vi
= n,
.
and tho
integral definite.
2492 Wlien
;)
and
C^^
,
Proof.- Reduce
is
rW
1
Jo
(+,
(_lV("+"*'M<'''
I"— y').y
J„
J)
one odd and the other even,
n iire integers,
odd or even,
,
cos.rcosji.rcAc
successively by (2 IS
1).
<^
The
with n odd,
""' with^) e\en.
final integral,
will be
= cos'.J^Tr- =
1-n'
(-l)*"
-
i-"
|'4*
or
I
Jo
3 C
cosv.r
,
fix
according as
—
= Rin?.)nr = (_l)i(-')
'*
»
.
TNTEilRAL CALCULUS.
378
2493
)
cos^\icos)hvcLv
= ^-g-^\
I2
•.0
•-
sm^\r
cosP-".r(/cr,
"
p—n
where n and p are any integers whatever such that
is
>
0.
—"When p — n
Proof.
When
jjj-n
cos
I
is
odd, each integral vanishes,
is
even, let
bj (2478) and (2459).
then, by (2488),
(n + 2k)ii
•dx
«(
=2k;
it
V"
= (ji±^^
_
'T
(n
+ 2/,-)_,
'''''
2"
1,
I2
IT
IJ"
(by 2405).
[^\h,^>'xcos''xdx,
Bat n + 2l-=]), and by (2234) the Umit may be doubled. Hence the result.
2494
=
^cos'^'^a^ mn)Li(Lv
\
-—,
2"{n
Jo
—
— =^.
l)
=
Tn (2707), put /.•= 1 and /(.')
Proof.
Give e'"^ its value from
a;""-.
(76G). The iinaginai-j term in the result vanishes, and the limits are changed,
by (2237). Finally, write x instead of 0.
2495
= l.ii
/"(eoscr) »'m~'\rdd
\
...
(2>?
Jo
Proof.
— Let =
;:
cos.-c.
By
(1471),
— 1) f {cos
Jo
\
.v)
cos
rid (Lv.
we have
f7,^_,^,(l_,Y-^=(-l)""'l-3...-(2n-l)^'-
(i.)
n
Also, by integrating n titnes by Parts,
f f"{z)(\-r)"-Kh
Then
substitute z
=
=
(-1)"!' n,)d„,(l-zy'-idz
=
-l.S...
\-2.
-
i
)
['
^/(.)
.7,
(^)
dz,
by
(i.)
cos.i".
= J„ + A^z + AS- + &c. = ^A^z",
= ^p(l'-l) (r-n + l)A,z^-",
Othcrw!s''.—LL't f(z)
•••
f"i^)
...
/(cos.r) cos7ixdx
I
=
Jo
=
2i96
C
.((
—
-•
^Ap
—
cos''.);
Jo
cos
nxdx
.
[7"(cos.^) sin'"J,r, by (2403).
}1.0... (-/I — 1) Jo
=
'\'',
..
<r COS' .v-\-b- >^\n-
,v
^
lab
(i'->''-)
INTEGRATION OF CIUCULAR FORMS.
p
2497
.'n
*;""'/;^'.
'
2498
=
.
.
i'Os-,r-\-fr Sin
{(I-
.
= ~,.
-,
.
I)ilT,n.n.iM(..rJ.l.:H;)fora.
:
la h
I)-
^'"''^7^('^'.
r
-r^.
-^70
[
DifftTontiato (2VM\)
f..r h.
[A.id iuLTctlicr d'-i-lir-S)
^^^^
Jo
t'Os^r+/r
(«^
.siir
~
.r/
iJl^i^t
«+^/
/>^
+ ah' "^ A'/'
(2o00) and (2501) are obtaincil l)y rcpeidinp^ upon ('il'.*!*) the operations
by which that integral was it>clf obtained from (2490).
Jo
Proof.
.r
v'
(1— •")
— Denote the
-
intej^ral
by
u.
[by (2008)
</"
fii;il^,/,,
2503
Jo .r(l+.r-)
Proof.— DinVrentiate
then integrate for
2504
2
Jo(l+.«V)v/l-..^
for a.
=
v/(l+<r)'
^lo,«o+").
2
Integrate for x by partial fractions, and
a.
J>-f-'T^ = ^'"''[('+^)\' +
Proof.
— From
(2.j";{)
Integrate for a between
a'
("'
7:)'l-
ol)t;iiii
tiin-'.r
i'
Jo
we
/
_
+ •'•'/'"
liniit.'^
j
and x
'T
loL^(l
+'0
'-^
,
and
in the le.sult sub.stituto hx.
INTEQBAL CALCULUS.
380
2505
tnn- «--tni.-'/M-
1
Pkoof.
— Applying (2700),
J
since Ix
X
h_
2
Hence the required value
integral.
J
V
2506
A «
=—
evenj element of the
in
is
dx
[''
we have
Also, by Parts,
(0) liere vanishes.
and therefore tan"' (hx)
infinite
is
<p
a
^^
^,^.
TT
a
1
rT4^^''.'=T
h
+ COS-.l'
Jo
l
—
(i.) Substitute
Proof.
thus shown to be
z=
TT
—
f^
sin?/
i—
2 ]^l
+ cus,-y
tt
—
a*,
=
,
dii
-^
and the integral
1"
/.
(tan
2
_,
'
is
1-1
cos tt— tan
reproduced,
and
is
=—
^4
A\
cosO)
^
''^^
-,
denominator, and integrate
(ii.) Otherwise, expand
xpand by dividing by the denomniator,
Employing (247H) we obtain the series
It by Parts.
each term of the result
i
6
f cos.r
ncnrt
the
method
("^ filler
/ tt
,
V 2
^d-
Ju
Proof.— By
ij
Jo
,
v^^'
of (22.j1), putting
1
e-"""
,
^x
v/'tt
dii,
Jo
the integral becomes
—
The second
2509
e-''^'cosa;(7.r(7^=
[
[
integral
\
is
COS
— Substitute
f
\
c-''^
co^xdridx
obtained in a similar manner.
Jo
PitOOF.
—
>/,
?/-
(lu
=
^^-^
Z
Z^
=
\
siiw/Vv.
Jo
and (2507-8) arc produced.
(2201)
'
INTEGRATION OF CIRCULAR FORMS.
"Wlien n
2510
and
881
are intcg^ers,
j>
r^rf,.
r\\'-\--sm'.,.h,h:
=-i^^-i
integration for x in tlie double integral is given in
(2608-9), and the original integral is thu3 reduced to tho
integfral of a rational fraction.
The
Proof.
— Bj the method of (2251), putting
[By (2201).
2511
nc-in
—-T-
)
r^«^in\r
^^^
=-
^ f
,
-rXi
=^
[By (2510).
2510)
[Bv (25
tt
f/:::
081).
^—
2513
PijooF.
I
(r + 'l)
— By (2700).
=
'^
^ (I'
—
'l)
!—(Li=hv^J-.
Transforming the numerator by (G73), and putting
^'"s becomes
^''
=
f!iliI^!-2illA.%/,..=
2514
p
2515
liog^J.
oo.,.r-eos;M ,,,.^:.(^,_^)
p,;ooF.— Integrate (2572) for r between the
If u
and
p and
q.
are po.>itive quantities,
/y
l" ^"'"'''-"^^''''
."^iiw/.r eosA.r
r
Jo
2516
according as a
is
>
Pkoof.— Change by
2518
liniit.s
or
<
_
=^
""
,/
,
rf.r
.<
f>.
(GOO), and employ (2572).
r shwusinA,,- ^,,^^^
Jo
according as a or b
orO,
'*'
is
•^
„^
^<_
**
the least of the two numbers.
;
INTEGRAL CALCULUS.
382
—
Troof. From (2515), exactly as in (2513).
Otherwise, as an illustration of the method in (2252), as follows.
noting the integral in (2516) by w, we have, (i.) when 6 is > a,
that
^2 = r r ^^""^'""^^-^
dhdx =
X
is,
f ^^R^pRJ^dx.
x'
JJo
(ii.)
When
t is
<
{"ndh
o,
(22G1)
Jo
=C^db='^.
'^
Jo
Jo
De-
If a is a positive quantity,
2520
r!i^IJ2^\iv = ^{2-a)
according as a
is
Proof.
and the
—
or
sin^a; cos
is
ax
not less tlian
=
result then follows
\ sin
4
{sin (1
and
4
J^*2ilL™rrf,,
2522
according as a
>
is
or
2.
+ a) + sin (1— a) a;}
a;
from (2518), the value of the integral being in
-^—^ =
the two cases
a;
<
— ^ (a — 1) =
J^
4
4
=
\%,da= r
is
<
(2-
da +
;
then,
|
\
Jo
Jo
when
a
is
>
2,
the
To
a)
J
—
rnda
nda = {"^(2-a)da='^^ (2 — o) da = ^
2
2,
a).
|
—
a
—4 (2 —
2.
Proof. Denote the integral in (2520) by u
present integral is equal to
And, when
or 0,
4
^
INTEGRATION OF CIRCULAR LOGARITHMIC
AND EXPONENTIAL FORMS.
r£::!lf!nzi',A,
2571
Pkook. —
Jo
.f
—
tan-^.
tl
and integrate by (2584).
Expaul since by (704), and integrate the terms by (2201).
Difroroiitiato for
Oihcni'iae.
=
Gregory's series (7'JIJ
is
r,
the result.
=
CTBCULAn LOCAUlTllMir ASD
—
By making a = in ("2571).
By the method of (22oU).
imlepeudent of r, which may be proved
Proof.
is
Jx
Then
^J^d,.
r sin
.r
X
term
= r
f-
,
»
C
.
=
^
by substituting.
= (2.-1).-,;
1,1
1
Ctt-j/
7r+//
= I {^myUn^dy
1
Stt-i/
(2910)
oTT
-—
=2
+
.
i
t
.
+y
+ y,
f.,.
j„
"i
)
~.
"^
e-^^-^ s^'ydy
[
7r
ott—ij
-^
Jo
(22;j), putting
(2,i-l)
= \\m-^dy =
-
Jo
Proof.— (i.) By
either
is
by substituting x
sin.,/,
Jo
iU + &c.
X
i'n
,
--^"'.'/'^.V
in-
= 1.
X
]„
integer, the general
r---!il--J.= r
Jo
obsorvinp that the
by substituting rx, let r
First,
dx+l
ilv+\
\
J„
Now, n being an
r-'"
=
a;
Jo
'
383
(i.)
Olhencise.
(ii.)
^^
mirMS.
I.
('^•,/.,=^.
2572
tegral
l'(i\ h'STlA
i:.\
(2291),
•'
J
I)
the integral takes the form
2
[
I
cos
/.f
e"
"
-^ "'
'
'
y dx dy
=
(ii.)
Olherxvise.
„
By
,
,
tlio
= -[
e"
j
""'y
e ' "' -^
cos
7-.r
dx
c'"''
^^^\^
*y'
„
dy (2G14) ='";^''(2G04).
.
mutliod of (22o2), putting
?t
=
r* sinaj*cns/Ar
Jo
it
J(/
,
dx,
*
follows from (201 G) that
^ uc-^da = C Oc-" da + r
Therefore
2575
= ^ t-*.
c"dadx=\
«"''
•^
I c-^da
Jo Jo
«
Jo i
j,
+ **
dx,
by (2583).
r'T£^'''^=T^"
2576
r,7ff^"'=T(l-'-')-
—
Proof. For (257.'>) dillcrentiate, and
(2573) with respect to r.
for
(257G)
integrate
equation
INTEGRAL CALCULUS.
384
Proof.— By
(2-201),
=a+
and a
Put
/c
ih,
re-'-^x^-'dx
= r cos 9,
=
r sin
(A^)
r~v-' "'"
^
COS ^
Proof.— Make
= ^^
b'
'/..•
Sill /,
N
,^
Tim)
2
^
^
n
—. 1
'"'(
cos\ ^).
2 /
""
^
^.
— Put
real
N
A^^V- ./.r=—
Proof.
and equate
in (2577).
fl,=
2581
thus
;
«^-isin»)^)^^
(cos
Jo
'
^
side tor e"''-^ from (767),
as in (2259).
by (757). Substitute on the left
and imaginary parts. Otherwise,
*^*^
^
2579
=
i
rg-(a.ii)J-_^n-l^^_p
= ^^.
—
»?
in (2579),
sill/ »/ir
-^
cos\ 2
and employ
Vnr{l-n)
SUU^TT
SlU?i»r
2583
e-''^ sill «».r(/cr
1
=
-TTT-TJ.
=1
Proof.— Make n
Othcnoise.
in
c""" cos fearer
\
a-\-b'
Jo
Jo
(2577-8).
— Directly from (1999), putting n =
\v1ktc n
is a
Proof.
— In
positive integer
>
(2577), nut tan"'—
a
=
>r"\c^-'^mb.c<hv
=
[
Multiply this equation by
h
in the integration for
=
h"'^<lh
to oo
will be
,
=
1,
and
—a
for a.
1.
i),
thus, uriting
^-0'^^
""
Jo
and integrate from
= -:r^,
«-+6-
cos"
6^
sin
p
for
»i,
?.f,'.
a" tau"-'(i fccedd.
by (2579).
and }^w.
Then the corresponding
limits
CTUCULAR LonMurinnc Axn i^xpoxenttal forms. 385
2587
r'-.-«:>^)''''=;:r,:(f),7^-
Proof.
— Put n=p —
f .-'..- f^
2589
Proof.— In
.-.
I
{.V
^
(2700), let
Um 6)
(.»•)
,lv
= cos
-^*
= r(») cos" e
tan d)
(.r
_
^*~
ian-^
A^=l,
by (7G5),
in (2585).
l
= ~Y^'
&c.
;
172^3.4'
Ai, Jj,
&c.
Therefore
vanishing.
e-'a;"-' cos (x fan 0)
j
1
„e.
;
tan*
.
"^^
_ <L(^}±y) taa> + "
1.2
;;
tan*
- "4' tan" +
...
=
p
^^
i
1
j
dx
•
e-'x'^-'^dx
=
i (1 + t tan 0)-" + i (1 -'"^f^" S)"". which by the
series on the left
Then change a into n.
values (770) and (708) reduces to cos a^ cos"^ 0.
Similarly, with sine in the place of cosine.
The
^"^^~^"'%
m6T(Ar = tan"^^ -tau"^-^.
r
Jo
2591
-Jc
—Integrate (2583)
Proof.
2592
where
\
for
e"^"^
a between a=:-a and a
—
(i.
cosa.r sm"crr/cr,
Jo
n is
any pcsitivc integer.
See (2717-20) for the values of this integral.
2593
a
\
cO
L-ZJ__sin»M'f/.j
—
yrx
f'
—c
.=
irx
4
c"'-{-'lv()S(i-\-e
'"
being <7r.
Proof.
— The function expanded by
divi.sion
becomes
(^" + e-")sinm.r (e-" + e-'" + c"" + &c.)
The
Multiply in an<l integrate by (2583).
this series is also
(2i:t53).
Hence
produced by
is
|(2»-1)
\{2n-\)ir-aY-\-m^
But
result
7r
+ a}' + m»
dilTerciitiating the
tiie result.
;i
1)
logarithm of pqu.ition
INTEGRAL CALCULUS.
386
2594
Proof.
= —-r-;
COS ))hV(Lv
\
— Change m into
p
iO in
(2503), thus
(,,-»x^,-<xx)(,gx_^-ex)
^^^^
^
sing
e7rx_e-7ra!
COS a
j^^
Now
change a into hm and write a instead of
^^^^
2^'
e--6>--
Jo
Proof.— Make
2597
?u
=
—
— Expand sin'm« on the
= — |i tan {ihn)
2S00
\
f
"
=
in (2593).
=—
J-.
= ±-^B.,.
left side of
by (770).
Expand
(2596) by (7G4).
The
right
by (2917), and equate the
this
2(c'"
^r- sin m.:p(Za?
-T
Jo
^--,,— -4e^'«+e-^'
same powers of m.
coefficients of the
2599
'1±
—c
e
*.'o
Proof,
6.
in (2593).
^
2598
_
+ COS y
Jo
sm 7n.v (Iv
Proof.— Make ri=7r
side is
and a
in (-iSOi),
\
:
\
e-'"+2cos2«+6'-"'
ei-'-+c-^^"-'^
4^^±^
COS „M.d.
— C""') sillrt
=
,2^";+'-;) cos«
—
a—W
successively for a in
Proof. To obtain (2599), put a + ^ir and
(2G00) is obtained
equation (2593), and take the ditference of the rcsufts.
in the same way from (2594).
2601
1
Proof.— Make
2602
f ''sin
m=
4^1±^.fAi-=secw/.
in (2600).
(ri)-' f/.v
=
f
^M).M
{crY (h
= :/^y
CIRCULAIt LOOARITnMIG
PnooK.—•By
Bv (2125)
a=
Put
-c.
AND EXl'ONENTIAL
\\-"'
c'"''
|
Substitute on tho
a»7
FOliMS.
= ^^
d.c
from (700), and equate
luft
and
real
imaginary parts.
2604
Jo
Proof.
Bubstitute
fore u
=
i!
— Denote
— the resulting integral
the integral by u.
to prove that
in
Ce'-"^.
AVhen a
= 0,
}j^/n (2-125).
=
;A^c>--,
a,
and
and there-
-v/Zt
Jo
integrate by (2G0-i).
Pkoof.— Substitute x ^n:, and
J>-^''^'-^-":-K.'=+|)-^]'^r
=
+
cos
Pi;ooF.— In (2005) put
equate real and imaginai-y parts.
/.•
r"
=
e-'^^ur"^\v(Lv
/
2
siii\
2
2608
equation for
= — 2«,
=
C
.-.
re-("^-)%/.r
2605
(lie
--
we get
re-\lv=C,
2606
Diircrcntiate
sine
;
substitute from (700),
and
2
f2;j + l)
'"T
>—+rr^
r)
+ 'V-)...('' +2«
+ l)(fr
i\
1
."'^
-
.
Jo
(^r
].2.l\...'2n
n
Jo
(^r
+ 2-)(</- + l-)
{(r-{-2n
...
)
2610
1'"
(i(2n-\-\)2n
a
+
//--U- (2/i
'ii
<r
'.,
(""
r
+ l)-"^(/r
+
-4-
I
.T.,
I
ON
+ l)27i(2»-l)(2>t-2)
o;^Y^a' + 2n-V){a--{-2n-'S')
a(2»
+
/
2//
+ l'j(^r + 2;*-l')
(7
_^
2)( 4-
1
_^
(,r+-'»
+
l-)
.
(u'+l)
)
INTEGRAL CALCULUS.
38S
r* ^"""^
2611
„
,
cos-'*
\
xdx
a
=
.
.,
.
,
V'
^
(i2n{2n
+
r
..
^
.
/
— \)
o-A
7>
•>
,
^'-
aOn(2n-\)j2n~2)(2n-
+ 27i') (tr + 2m - :^) (a- + 2n-V}
(a=
-X
'
,
_"J
,
{a'
'
'
+ 2h)
•
,
.
.
.
(a-
+
2'-)
Proof of (2G08-11). —Reduce successively by (1999). The integral
part after each reduction disappears between tbe limits in the cases (2608-9),
See also (2721).
but not in tbe cases (2t)10-l).
2612
{a--\-l){a-+&)...((r-\-2n-[-l
J^in
2613
a i(r-\-2^){crJt4^)...{cr^2n)
J_^.
Proof.
— By successive reduction by
^"''"'
2614
i
(1999).
cos 2hi(Lv
^
=
Proof.
'-h
— Denote the integral by
=-
db
[^-«^x'^ 2.. sin
2hx dx
n,
e~''\
2a
Jo
then
=-
f^
J,
J„
^ e-^^' cos
2h.c
= log C- ~
;
and
b
=
gives
G
= ~^~
a
Othenvise.
=
(-1)"
Thus the general term
(
—
)'"
2a!
dx.
Therefore
Tf-"
.
the terms of tho
is
which gives the required result by
(150).
i^p-^^
•''
cosli 2hj
,'o
Pkoof.— Change
,
-<(>
-'^^-'^
2615
-^
(2-125).
— Expand the cosine by (765), and integrate
product by (2426).
=
a'
""•'"
the second integration being effected by parts, j c"
log u
dx
a'
b into ih in (_2G14;.
=
^
2(1
<'('')
.
(2181)
CIIiCULAR
LOOARITmrW AND EXPONENTIAL FORMS. 8S9
f ^•-'^r
2617
a=l
Proof.— To
obtain (2617), put
obtain (2018), dilVurentiate, in all, n
Jo
PitOOF.
— By
putting
order of integration, the integral
(2014), and
times for b.
~
.-
+
'^
!/
= iog„.
c'^
Jy
and changing the
(22'Jl),
r (e'^^cosx-e-^"'"'') dy dx
[
<*"^
•'+"')
=
^^'^^
Proof.
equal to, or less than, unity
is greater than unity.
is
^^'
but
;
is
equal to
when a
log a,
—
By
a=l.
(i.)
log 2 (1
(ii.)
<
a >
a
(iii.)
(2035), since
— cos x) = log 4 + 2 log sin
1.
By
integrating (2922) from
1.
As
in (2920), integrating
.r.
-J
to
if.
from U
to
tt.
y\og{l-nco^a)iI.v.
2622
?i
is less
on those
than unity, the values of
See (2933).
tliis
integral depenil
of (2620).
according as a
is less
PnuOF.— Integrate
or greater than unity.
log
(1— 2a
cos
./r
+ a"')
J^ by Parts,
\dx, and apply
f
Jo
(202U).
2625
/v.
Jo
when a
When
(lUreiviitiiiie for
+ ^''
H"?^ (!--"
2620
('>'•''')•
becomes
r r (cos x-e-"'') e-'" dy dx =
\1
^^
J'
(2261)
Jo
''\
v
in
+1
r£2££^£r,/,,
2619
27r
=
l^'-'.r"^^ sin (2/Ar+ Inn) ,Lr
2618
To
'^
=
sin 'ILrdr
\
cosr.i
according as a
log(l
is
— 2acoy.i+^/')
(/.'
less or greater than unity.
r
'
r
INTEGRAL CALCULUS.
390
—
Substitute the value of the logarithm obtained in (2922).
Proof.
integral of every term of the resulting expansion, excepting the one in
vanishes
by (24G7).
u r,
T
which
=
T" siu
^r«/%M
2627
according as a
Proof.
sill r.r
^?
mi''''^
r/,t'
irfr'-'"^^^
i-2„cos,,^ = -2-'
J^
°'
or greater tlian unity.
is less
—Integrate (2625) by Parts, J cos rx
f
2629
Proof.
— The
cos
n
TTCf
VcVcIj?
fraction
Proof.
= cos
— Expand
l-2a
1
2
+
1
+
cosCcT+tt'
""
.
« l^emg
«--.5
2a^ cos 2^3
ra? (1
2a. cos a;
result follows as in (2G25).
by (2919), and the
Jo 1+.:^'
dx.
—— 2acosa7+aT— = 1—
1
\
Jo 1
-f-(l-ft2),
—T-'
+ 2a' cos
3a;
.
<
-,
1.
+ ...)
2(l-a-) l-ae"^'
the second factor by (2919), and integrate the terms
by (2573).
r
2631
log'
(l-2« cose.^+a^)
Jo
Proof.
1
r/.r
^ ^ ^^^. (i-^.e-Q.
-|~ t^'
— Expand the numerator
by (2922), and integrate the terms by
(2573).
^
(l^x%l-'la
Jo
Proof.— By
Othenvise.
2633
Proof.
and
^TT,
cos c.r+«0
differentiating (2G31) for
— Expand
—^—
«
=1
fi^
= ]— — (cos
1
('"''
/
-k
_i^
N9
c)-
and take the integral between tlio limits
and c; the result is
between limits
in (1951),
then integrate for
Jo
c.
by (2921), and integrate the terms by (2574).
r^'^loiifri -l-rcos,r) ,
^ (Iv
\
— Put
~ 2 (<"-«)'
b
lo^d+ccos.)
t^^'«^''
and the integral on the right
is
j^
^
2
r
-1
Jo
v/i-6-
tan-
Jl^
V 1+6
found by substituting cua~'
b.
db,
CinCVLAU LOGABITHMIO AND EXPONENTIAL FORMS. 391
r log(l + 0COS£)
2634
— As
Proof.
—
Proof.
stitute
^-d.,-
^ ^ ,;„-, ^
tt
for the limits of x.
= ^ log i = f;75^
siniCtZ.tf=
I
and
by taking
in (2633),
J""log sin
2635
^,,.
COScl?
Jo
Add
(2233).
cosa;cZ.r
2637
log siu
^^
ci f/.i^
= ^ log i.
-^
Jo
—
=
(tt— .«)Mogsina;
Proof.
i?' log sin a; c?^;
Jo
Jo
difference of these integrals to zero.
^v
t
log siu^ xdcc
xl sin^
xdx
I
result
Proof.
xl sin' x
dx +
being an integer.
(tt
+ t/)
Z
sin"
sin'^
is
ydij
=
fZ?/+
?/
\
.
.
+
xl sin' x dx
...
{(n — l)Tr + y\lshrydi/.
+
'0
by (2635) and (2037)
2
.
J{)!-llir
Jo
integral reduces
+ y)
(ir
+ y)
I
sin y dy
;
for example,
= ^tt
i
ls'mydy + 2\ ylsinydy
= - 27r' log 2 - tt' log 2 = - 37r2 log 2.
-{1 + 3 + 5 + ... + (2?i-l)} rMog 2 = -«Vnog2.
— Develope sin mx by (764)
;
integrate the tei-ms by (2396),
the scries by (1539).
2640
7i
Jt
a'Zsin'a;t?a;+
Jo
sum
+
xl sin^ x dx
Jo
=
The
= — ;iV log 2,
of (2250),
=
Jo
(ir
by (2233). Equate the
(it;,
Jo
Proof.— Method
Each
these integrals and sub-
applying (2234) to the result.
2,1',
2638
<?.'•
r™rf,,=
+
Jo
e^
1
1
2)11
and
—r
INTEGRAL CALCULUS.
392
sin mx by (764)
integi-ate the terms by (2398).
The
IT
1
^ 2"-^°^^° "'^'^' ^J (2918), wliicli is equivalent to tbe
o
Proof.— Develope
resulting series
=
is
;
above by (769).
^ _
C cos(mlog.r)-cos(nlogcy)
logcf
2641
^
^,
'
Jo
(m
r^siu
o^yio
2642
loo'cr)
—
—
^—log-
^^
1
Jo
= im
Proof. Put p
See (2214).
and
—
(w loader)
sill
^^
,
d.v
^
j^^ l
° l
+ >>r
+
zi'-^'
= tan -1 m— taii -1 n.
±
i
^
^
ci'
= in
q
in (2394),
and
equate corresponding
parts.
2643
log.v
Jo
w=
Proof.— Put
in (2641)
''l
logci^
Jo
+ n-
and (2042).
MISCELLANEOUS THEOREMS.
FRULLANI'S FORMULA.
r J>(«-^-)-<^(M
2700
/i
Jo
being
Proof.
=
,/^,
'V
00
J
^'
find the last
,
^ ^(0) log^+ rilM ^.r,
—
— In the integral
term generally
rAi2
—zA-J-dz
'^'
= 0.
substitute
z
= arc
and
z
Jo
and equate the
I
results thus,
^
Jo
^
[^lOL^0:r:lI?i^:)^,_
J„
Then make h
X
inliiiite.
Jo
p 01^)
J A
a
a!
,7,
=
Jo
^
("
J»_
9lO) d,
a;
'^
= ^ (0)
'
^
log^.
" a
6
For applications see (2513) and (2505).
=
h,
}
.
MISCELLANEOUS THEOREMS.
393
+<^(0)(iog|:-«+*)-(«-6)^+jt«<^.
= CO
with h
Pkoop.-
.
rf„
{
r
^
=
,;.
I
«
£lHl
cfa
- »fi
I
(2i57)=r'S^<ix-f(0)Za-l('l>,
Jo
by making
=
6
['
aand^tbus
t^l^ d. .
Jo
and the
Integrate for a between limits
in the proof of (2700).
1
*-"
fi
tSM d.
^
Jo
= j' tiill^ilM^.- J"J*i|Hlfe
left is
POISSON'S FORMULA.
c
being
<
1
—
Proof. By Taylor's theorem (1500),
to the product of the two expansions
2
[/ («) +/'(«)
and
{
divided by (1
+ p^ /" (a) cos 2x + ^-^ /'" (a) cos 3x +
1 + 2c cos x + 2c^ cos 2a; + 26^ cos 3a; +
.
.
.
.
|
.
.
when
Hence the
(2471).
fraction is equal
cos X
— c^). By
vanishes, except
and by (2919), the
it
(2468) the integral of every term of the product
is
of the
form 2
cos^ nx,
I
and
this is
^
tt,
by
^^
result.
2703
r/(^^+e")+/(a+e-'-)
Jo
1— 2c
(i_, eos .r)
d.v
cos.i^+c-
=
rr
{/(«
+ e)4-/(«)}
•
2704
Jo
1
— 2ccoScr+c^
^*
Peoof.—As in (2702), adding unity to each side of (2919), and employing
(2921, 2467, 2470).
3 E
INTEGRAL CALCULUS.
394
ABEL'S FORMULA.
Given that
then
"
2705
f
Jo
can be expanded in powers of e"",
F{;v-[-a)
P(-+MyVF (.-«,)
1+
^ ^p( ,.^„),
^j,
''"'
= A+A^e-"'-irA^e-'''' + A.,e-^'' + &c.,
F (x + iaf) + F (x — iat) = 2A + 2AiCos at + 2A^cos2at + &c,
Pkoof.— Assume ^(« + a)
:.
Substitute and integrate by (1935) and (2573).
Ex.— Let F (.r)
=—
then
,
f
f,
.
^,
,
,.
=^
/_^
•
,
KUMMER'S FORMULA.
(\l-zY-'f{.vz) dz.
2706 J-i^f{^^ Qos0e'') e^'^'dO = sin kn Jo
then
+ h = 2.r cos Oe'^ by (766). Substitute these
Proof.— If h =
{^^
.re-'(>,
,r
values in the expansion of f (x + h) by (15U0)
tegrate thus, after reducing by (760),
;
multiply by
e-'''^
and
in-
;
f'j(2.cos e»», ^'-<W
Again, putting h
we have
tegrating,
= sin A, |.^ - fif +
= — xf
I
(pf''~^f(x
^ig^ -
(1500), multiplying by
in
— x<f)d<p=^
f'^'^
Wlien
2707
f^
is
I'
and changing ^ into
f\2.vQ^o^ee')e''''ede
— Divide equation (2706) by
by (1580),
in-
1
— 2,
the formula
= 'L^^^{\\-zf-'f{a^z)dz.
^l
Proof.
I
an integer,
J-l^
fraction
and
the foregoing series within the
Jo
Equating the two values
brackets.
is obtained.
For an application see (2490).
df,
fe.
sin
A-tt,
.'o
and evaluate the indeterminate
differentiating with respect to k.
For applications see (2490), (2494).
2708
If
^ ^e a function
rV(*,
and
if
of x so chosen that
Z:) ./.'
=
G,
f Xr(.r,
0)
(i.),
.7..
the series
Af{-r,0)+AjX.,>,
l)4-^1./(.>', 2)
+ &c.
...
= .K-r)
... (ii.),
MISCELLANEOUS TBEOBEMS.
wliere ^
is
a
known
function, then
^Co+J,C,+.4,a+&c.
=-4^-
...
\
Proof.
2709
—Multiply
(ii.)
sum
If tlie
395
(iii.)
Xf{,r,0)(lv
by X, aud integrate from a to
h,
employing
(i.)
of tlie series
be known, then
— Jo
-
Parts,
it
we have
follows that
j
C/,
'a
X = e-^.i!^-i and f {x, k) = a;\ Then since, by
e-^a;"*'^-'tic = a (a + 1)
+ h—1) e-''x"-'^dx,
—
= a (a + 1) (a + 1). Hence, conditions (i.) and
Proof.— In (2708)
let
...
...
(ci
(ii.)
Z;
being fulfilled, result (iii.) is established.
For an application see (2589).
Theokem.— Let
+
/(<«
= P + iQ
/?/)
2710 Then£f^rf,«?^=
(i.)
("•)
rrs"^''"
{"^''^dady^-^'^'^dyda:
2711
JaJa
Proof.
J Ja
(ly
— Differentiating
a.
independently for x and
(i.)
y,
=
=
••
(iii.)
ll<-l
Py+iQy,
P., + iQ,, ifix^iy)
f'(x + ii/)
-P^.
Qy-iPy, .-. P^
-Pa-f iQ^
Qy and Q^
=
=
=
Hence by (2261) the
equalities
(ii.)
and
(iii.)
are obtained.
Ex.— Let f(x + iy) = e~'^-'>^' = e'^V (cos 2xy-isin 2xy).
P = e-^V cos 2xy, Q = — e'^V sin 2xy, therefore, by (iii.),
Here
[\-^' (e^'cos
Put a =
(
Jn
a
= 0,
e-^' (e^'
2/3 a; -e'''
6
= co;
cos
2/3a;
cos 2a x) dx
=
(^
e^ (e-^\in 2by
—
e-""'
sm2ay)
dy.
therefore
- 1)
dx
= 0,
.-.
e^' [
Jo
e"^' cos 2p.r
dx
=
e"^^
•'o
f7.-».
INTEGRAL CALCULUS.
396
CAUCHY'S FOFtMULA.
Let
2712
x~''F{.v^)
\
Proof.— In
dx
the integral
f
anditbecomes
(.«
- J-V"
=
P
n being an integer,
F {z")
dz =: 2A.,,„
(,.+ -1) i^
is
proved by substituting -
(,^
-
substitute
-1)'
|
^^
=
C^,,,
in the first integral.
2i;8in^
in the expansion of cos (2u
= ;i;— —
,
where x
=
e'^
= x——^
(i.)
2«,„
then
Therefore by addition
I^--^0-[(^-^)'lf-^"
Now,
z
[
'
Let the integral sought be flenotcd by
This
tlien
1
z-''
'
Jo
A.^,,,
+ l)
(776), put 2 cos
-f^
and
= ^+ —
by (768-9), and multiply the equation by
X
_p
j
(ii.),
/a3—
—
in?
]
'
'-^,
[
and integrate from x
the required result
2713
is
=
to
a;
= oo
.
Then, by
(i.)
and
obtained
= e-"^ then
r"e-"\L-. = ^-^
'^^^^V^
^-^-'^-^V
A..= \^^^.^-1.
Ex.— Let F(x)
7r
and
A
A=^.
^"^
Therefore
FINITE VARIATION OF A PARAMETER.
2714
to the case of a
value of a quantity under the sign of
Theorem (2255) may be extended
finite cluingc in tlic
integration.
.
MISCELLANEOUS THEOREMS.
397
Let a be independent of a and ?>, and let A be the difference caused by an increase of unity in the value of a, then
CA(J>{.v, a) ilv
2715
re-'^dx=—,
Ex.1.
=A
.-.
ihatis
—
«(a^l)
I?
Also,
flv.
^e—^cJx= A~-,
[
(e-a-_l) Jx
VCr, a)
\
by repeating the operation,
A"e-'^dx
I
=
A"—,
that
is
Jo
r V-(e- -1)^^
2716
d.V
=
Jo
2717
Ex.
2.— In (2583-4) put
h for a and
A sin (2a- m)xdx
-''''
e-^'""
f
i7/^^^^_^
V
a(a+l)
... (a+zj)
A cos
(2a
=A
— m)xdx =
A
In
(ii.) let
m=
then
for h,
""~'"'
,.,
,
(i.),
r,
(ii-)-
,
^-r,
A-
J^
z^
+ (2a-»0-
an even integer, then
2jj,
= cos (2a + 22j) a;-2p cos
A'^cos (2a-2^) X
(2a-m)
(2a
+ 2jj-2) .r+
-f cos (2a — 2^)
...
...
= cos 2ax
[cos
2^x-2p
COS (2j)-2)
.c
+ (7
(2p, 2) cos
.
— sin 2ax [sin
2j.).«
— 2jj sin (2^j — 2
) o-
.
...
a;-...
+ cos 2j^.t]
.
+
X
(2p-4)
.
.
—sin 2px^,
in which equidistant terms are equal, xh.
while the coefficient of sin 2a« vanishes becausQ
(773)
Therefore
other.
each
the equidistant terms destroy
(-1)^ 2"^^ cos 2a.7; sin=^^.
A=^ cos (2a -'2p) a;
—
The
coefficient of cos 2ax,
(— l)^2-J'sin-^a;
;
=
Hence
(ii.)
becomes
2718 £e-^2719
COS 2a.r ^\n'^adx
Again, in
A-^^i sin
+ C(2p + 1,
=
sin
\.-^
+ (2a-27>)^
[sin
^
'[,
2)sin(2a + 2i)-3)a?-...-
sin
(2a-2p-l)
a;
s- (2p + l) cos (2j)-l)a!+ ...-cos (2^ + 1) a;]
+ sin {2j} + l) x].
(2j; + 1) a3-(2i) + 1) sin (2p-l)a;+
2ax [cos (2p + l)
+ cos2aa;
The
^''
m = 2p + an odd integer, then
= sin (2a + 2p + l) a7-(2jj + l) sin (2a + 2jJ-l) x
(i.) let
(2a-2p-l) X
= ^-^
coefficient of sin
...
2ax vanishes as before, while that of cos 2ax
=
(-l)''2-''^'sin2^*^r
(774).
is
'
INTEGBAL CALCULUS.
398
Therefore equation
becomes
(i.)
2720
To compute
tlie
right
member
2-^^-*-^
2p
Jc'
A;-+(2a-2;>-l)^
we have
of equation (2718),
=
^
A^^
2a-2;>-l
= i^^^ A'^^'
Ce-'^ cos 2a.r ^'m'^^Krdv
Jo
7c
^
r
C(2p,2)
,
+ (2a +2p- 2f
/r
1
+ {2a + 2^-4)-
A=
-]
+ (2a - 2pf]
Let a = 0, then the equidistant terms are equal, and we obtain in this case
07P1
(-l)n.2...2/).2^^
_
fe
A^.
Thus formula (2600) is proved.
Similarly, by making a =
in (2720) after expansion, formula (2G08) is
obtained.
Let p be any integer, and let q and
in (2722), and <2j9 + l in (2723).
a
be arbitrary, but
q<2p
2722
r
cos2a^-sin^^..-
z^^^
_ (-1)^ r ^,,
~ 'I'n^ (r/+l) Jo z'+ i2a-2py ^,
,
x^^'
Jo
2723
''
^^
Jo
(-^y
r^...: (2a-2;)-l)^^
= 2^^+^r(ry + l)
where A
Proof.
lias tlie signification in
z'-\-{2a-2zy
Jo
^'
(2714).
— Employing the method of (2510), replace
q being integral or fractional
c_os^a.. sin^^«
r
^^
^'"
^
;
therefore
1
rr
2^,^ si^..,.,-.r^., J, j^^
in2 + i)]oJo
Jo,
,
by changing the order of
the integral containing
in(k'])endeut of a.
x,
integration.
Substitute the value in (2718) for
z'' under the operator A, since it
Avriting the factor
is
Similarly, with
2j7
+l
in the place ofp,
we
substitute from (2720).
MISCELLANEOUS THEOBmiB:
It
may be shown
whenever
tliat,
p, formula (2722)
>
a
399
reduces to
2724
For a complete iuvestigation, see Caucliy's "
technique," tome xvii.
2725
Ex.-Let a = 2,p = l,q
r cos4aW^^^_^ =
^
h
Memoire de
I'Ecole Poly-
= i,
1
A= (_2a-2)K
srmsin^
6
A2(2a-2)*=
and
+ -2)*-2
(2a
(2a)*
+ (2a -2)^
=
r>-2.4* + 2i
FOURIEE'S FORMULA.
r~<^(.)./.. =
2726
Jo
when (X=
and
CO
-|<^(0),
sill cl
li
£i
not greater than
is
-^tt.
—
Pkoof.
(i.) Let <^{x) be a continuoas, finite, positive
creasing in value as x increases from zero to li.
quantity,
de-
(i-),
a
—
—
a
The terms
are alter-
nately positive and negative, as appears from the sign of sin ax.
lowing investigation shows that the terms decrease in value.
consecutive terms
The folTake two
being tbe greatest multiple of
a
contained in
h.
d'
"
r
Inn
sin
ft
Sm
»
.B
^ /
X
^
^
^
r
7
J (n
/
^
,
a
a
Substituting
sin aa;
"
+ l)7r Sin X
x— —
-1f
-
in the second integral, it
a
—
Sin
ax
becomes
T
\
,
/
r<ph+~]dx,
,
sin (x-\
a
\
and since f decreases as x
increases,
the corresponding element of the
an element of this integral
first integral.
is less
than
—
—
INTEGRAL CALCULUS.
400
we have
N'ow, by substituting ax-=y,
-.
—
f Or)
=
(ii^
(p[-]iii/
=
--—
<piO)\
fZ//
.
.
.
(11.),
a
n
when
is infinite,
fl
because then
-]
\a
{
(p
'
Hence the sum
=
^(0) and a
sin
of n terms of
(i.)
may
=
^
a
I
y.
be replaced by 0(0)
'
^^^cZv,
Jo
y
takes the value </)(0) ^tt by (2572) while the sum
of the remaining terms vanishes, because (the signs alternating) that sum is
less than the ?t-|-l"' term, which itself vanishes when n is infinite.
when n
which,
is infinite,
;
(ii.) If ^(a;), while always decreasing, becomes negative, let (7 be a constant such tihat (7+0 (») remains always positive while x varies from
to h.
The theorem is true for G-\-(p (x), and also for a function constant and equal
whatever its sign.
to (7, and it is therefore true for the decreasing function
If (j){x) is a function always increasing in value, —(f>{x) is a decreasing
function. The theorem applies to the last function, and therefore also to
{x).
2727
two
CoE.
—Hence
limits lying
tlie same integral taken between any
between zero and -^tt, vanislies when a is
infinite.
C^^^^[a:)(lv
sm <r
2728
Jo
=
7r{i^(0)+^W+<^(27r)+...+<^r«-l)^ + ^(«7r)},
when
a is an indefinitely great odd integer, and mr is the
But when a is an indefigreatest multiple of ir less than h.
nitely great even integer, the second and alternate terms of
the series have the minus sign.
sin ax
P —.
Peoof.
I
Jo
sin
/
^
(x)
j
dx
—
/""'^sin a.r ^
^
—
Jo„
a;
sin x
f
/
\
(.))
[''
dx+\
7
sin
,
I
J
ax
^
-0
„ sin
"f
.
,
(.r)
j
dx
,.
.
(1.),
.1;
decompose the second integral into 2n others with the limits to ^tt, ^tt to tt,
and in these integrnls put successively x = ?/,
to |7r, ... (2;) — 1) gTT to UTT
IT— TT + y, 2v — y, 2w-\-y, ... mr—y. The new limits will be
to \ir, \tc to
alternately, with the even terms negative, so that, by changing the signs of
to ^tt.
the even terms, the limits for each will be
Also, if a is an odd inTT
;
_;/,
teger,
—
;
'-
sm*
is
changed into —
;
- by each substitution, so that
(i.)
&n\y
+ 0(— + 0(-+:/)+... + 0("— 2/)}./.'/
r"^^{0(y)
^^^y
becomes
!/)
Jo
c"
+
,
But,
when a
is
even, the substitution of
sm ax
--.
J,,^
m =F
;/
/
•.
/•
J
?'(.^)(Z.c
for
\
(ill.)
sm.c
smy
MISCELLANEOUS THEOREMS.
whenever
—
{<!>
In the
The
r is odd.
(0')
+
limit of the first part of
2<p(w)+2cp
last part of
(iii.)
(27r)
put
sin
(iii.) is
+ ...+2^ (n-l)7r + (nir)}, by (2726).
x = nw + y, and the integral becomes
,p
^JRm^(n^ + y)dy=^cf,(n7.),
r
Jo
if
/,-«,r
is
>
'^
1/
between
If h—7iir lies
the one with limits
with limits ^tt to h
and
-Jtr
401
tt,
-^,
^
by (2725).
decompose the integral into two others
;
towards ^tti^ («t), while the other
— mr becomes, by putting y = Tr — z,
to ^tt will converge
the limit by (2727). Hence the last term of
tuting these values, (2728) is obtained.
is
(iii.)
iir(b (nir).
2729 Ex.— By (2614), [ e-^'^' cos 2hx dx = 2a
n successively, and add, after multiplying
Put & = 0, 1, 2
Substi-
'
...
by
tion
i,
the
first
equa-
thus
+ cos 2ar + cos 4^ +
(i
-a2ar>
.
.
.
+ cos 2n«)
c7a;
f,
The
=
left side
J
[%-.'»' ^'°
and,
w = 00
if
,
(^."
+ ^) " <;.,
by (801),
sma;
Jo
becomes
JL
jl
+ e-Tr2a2_^e-4^2aS_^e-9;r2a=^_,..},
Put
Tra
2730
=a
and
—=
a
/J
;
by (2728)
_1
/
_i
;
-i
therefore
v'a{4+<'~°'+«"*'''+<'~'°'+-}
= y/3{i+e-«'+e-«'+e-'^'+...},
with
tlie
condition
a/3
Jo
when a
is
Proof.
an
=
tt.
<r
-<i
infinite integer.
—The integral may be put
f^
sin ax
'^•'^
^^°
-
(.r)
c^,.,
in the
where
J„ sin.T
3 F
form
^(x)=^<t> (.r).
INTEGRAL CALCULUS.
402
therefore,
is
by (272G), when
h is
Itt* (0), since in (272«)
Hence the theorem
and
Ja
.V
J -a
Jo
.L
—
2734
1
I
;<;
AYhen
=
cp
(0).
= f^O)^
'
^
cV
1,
bj (2729).
(0),
ux du dcV
= -r-(l>{0),
when
a
= oo
.
Substitute (his in (2731).
oos«.i;c?«.
Jo
and
a
f"
Jo
cos
{.1}
—
X
the value
(Oj
second integral.
in the
<j)
^IIL^
PiiOOF.
>.^7r,
But *
=1° +j[=|*(0)+f*(0),
(ii.)|'
by substituting
is
are both positive,
/3
f= f-
PROOF.-(i.)
and by (2728), if /t
Cln), Ac. all vanish.
Jtt,
a>
proved.
is
o
Wlien
>
(ti),
<i>
2735
are positive, the limit
/3
is infinite
of
cos in COS UcvdudiV,
<l>Lv)
I
I
when a
Jo Ja
or of
is
^7r0 (/),
if
/
<^(,r) sill
I
1
.0
».'a
lios
between
for any other vahie of
Pr')OF.
— When a =
cc)
and
/3,
:{;?(/)
(/) if
f
= a,
and
xH'o
t.
we
f rr,)oo,nxco.Mrclu =
JaJfl
a
/« sin //,rf/wr/cr,
have, by (668), and integrating with respect to «,
:L
f/^
sin
Jo
a (.-/)
^~^
^ ^^.-^^^_^
, [^
J«
lillWllti^
'•'"^^
^
= xr'^i^c.+o,.+ir'^^^^Q:-,),,
(,,) cZ..
(i.),
by substituting z = x — t and 2 = .r + / in the two integrals respectively.
"When a is infinite, the limit of eacli integral is known.
When a and /3 are positive and Hies bt'i\vc(Mi tlieiu in value, the
limit of
(i.) is
When
value
is
V0(O,
a ami
zero,
/3
by (2782 8)
are positive and
by (2732)
(ii.)
/
does not
lie
between them, the
(iii.)
MISCELLANEOUS THEOREMS.
403
=
=
^ir<p (i) by (2731), and
Hu (i.), the first integrnl becomes
If a
the second vanishes as before so that the value, in this case, is ^tt cp (t) ... (iv.)
;
The same demonstration applies in the case of (2736), transforming by
(6G9) instead of (GG8).
Hence, by
/100
2737
be always positive,
(ii.), if t
/-tec
\
]
<t>
cos fu cos
('t^)
=—
iLV (In d.v
=
/»0D
/loo
1
\
<l){t)
(l){.v) iiiu til
sin u.vdiidcV.
Jo Jo
Ex.— Let
2739
<p
(x)
= e-'^^
= — e~"^
e~"^ cos tu cos ^lxdudx
Jo Jo
which
is
f^^^?^ du = ^
by (2584),
Therefore-,
^
equivalent to (2574), with
The expressions
t
=
e""*,
1.
in (2737-8) being even functions of
ii,
we
have, supposing
to be always positive,
t
I
J -00
^
(x) cos
cos uxdu dx
til
= 7r(j)(t) =
Jo
\
J
Replacing
<p
and
by <p{~x),
(x)
<p
-X
(x) sin tu sin
ux du dx
...
(i.)
J
afterwards
substituting
—x, these
equations become
(p
J
_x
From
cos tu cos ux
Whence, by
(ii.),
<j>
i
2741
du dx
=—
and
(i )
2740
2742
(a-)
J -x>
^
f*
= tt^ ( — ^)
f
(()
(x) sin tu sin
uxdu dx
by addition and subtraction, we get
(x) COS tu COS
(•'^)
sin tu sin
^lxdudx
=
ux du dx
=
irlfp (t)+(l> (
tt
[^ (J)—<f>
— t)],
(~0]'
addition,
\
f"
4> (ciO
cos u {t-x)
the original formula of Fourier's.
dudx
=
27r<l> (/),
(ii.)
INTEGRAL CALCULUS.
404
THE FUNCTION
The function
2743
V,(.r)
wlien
/t
is
wlien
= ]og^-
is
denominated
(p{x).
1
differentiating the logarithm of (2293).
= iog(>.-l-i.-i-...-^^,
Cor. ^(l)
/I
111
dj,\ogT(x)
an indefinitely great integer.
Peoof.— By
2744
t/i(a7).
= CO
,
= -0-o77215,664901,532860,G0
...
(EuJcr).
All other values of ^p{x), when a? is a commensurable
quantity, may be made to depend upon the value of ;//(!).
When
2745
X
is less
Proof.
r
2746
(1
1,
(x)
-.^0
r (l-.r)
TTcr.
V' i-^)
=
sin™
TT -f-
= nxjji^nd^^nlogn.
—Differentiate the logarithm of equation (2316).
'-To
compute the value of
^(—)
Find ^
when
(
^
is
a ^proper
^
^
fraction.
2748
(2313).
.^(..•)+'A(.'-+i)+'^(^-+|)+...+V'(..-+^^)
Proof.
2747
than
= ^ cot
—Differentiate the logarithm of the equation
^
- ) from the two equations
t/»(l—^)-tA(^)
'^
^
= 7rcotii7r,
(2745)
^
2749
V'(l-i^)+t/,(-^)
=
2]t/,(l)-log</4-cos?^log(2vers??)
+ cH>8^1og(2vors4^)+cos^log(2vors55) + ,^c.|.
}
,
THE FUNCTION
The
405
y\i{x).
term within the brackets, when
last
,
q is odd, is
^
'-^— log ( ^
cos (ild)j^lo,c.^2Yers(i::i5>);
\
9.
and when
is
term
q is even, the last
+
is
log 2 according as
_p
even or odd.
Proof.
—Equation (2743) may be written
.
being an indefinitely great integer.
1
2
-D
by
x successively u
Keplace
9.
<1
teger; thus
+1
fi
•
1
—
1
^
^(i
q/
2
+1
g
+2
-±+l,.
V
2
>>
/
;P(1)
Now,
if
"
— —
,
3
+
2^ + i
'
+7a_
1
1
...
;
where
is
q
any
'*-3^^+'*32+:
+ZA
?
22
+2
__1
32 + 2
(i-)
2-1
Fl^'"'' 22-1"^'^ 32-1
+12+ 11-
=-1
6 be any one of the angles
^=^,
— —
,
...
ilinllzr, we
= cos 220 = cos ^q<p = &c
1 = cos
= cos (2 + 1)0 = cos (22 + 1)0 = cos (32 + 1)0
by
cos0 + cos20 + cos30 + ...+cos (2 — 1)0 + 1 =
shall
have
(ii-),
q(p
COS0
By means
of the relations
cos0;p(— ]
cos 20 1//
in-
1
=-
(ii.)
COS0
2
and
+
(iii.),
equations
(iii.)>
(803)
(iv.)
be written
cos0Zf
—
2_cos(2 + 2)0 + cos20Zf-,
cos
I
cos(2-l)0^
may
^cos(2 + l)0+
cos 0^2
^\=-^2 20 +cos20Z2
q
^^
X3 cos 30 + COS30Z2
Hi)
(i.)
&c
q
+^
q
+o
2_ cos(2+3)0 + cos30Zf-,
^
(^— ) =--5jCos(2-l)0 + cos(2-l)0Z2
^cos(22-l)0 + cos(2-l)0Z|-,
= -!
-
+
-.
Zf
+ 12
h
the coefficient of each logarithm vanishes, by (iv.)
remaining terms on the right form a continuous series, and we have
4/(1)
Upon adding the equations,
The
cos0;/^(— )+cos 20 ;//(-)+...+ cos (2-1)0^ ('^^-)+^(l)
= — 2 cos + 1 cos 20 + i cos 30 + in inf.
= 12 log (2 -2 cos 0) by (2928)
{
(v.)
INTEGRAL CALCULUS.
406
—=
Let
3w
2w,
ties
—
\p (
Then, by giving to
w.
(q
...
— l)<*),
],
yp [
-we obtain q
— •
)
'^
to their
COS 2p(o
sum equation
(k
—
cos jJw cos
+
Ji(o
cos
2j:)<i»
To
unknown
solve these equations for
quanti\//
[~j
multiply them respectively by
q,
C0SJ3W,
and join
linear equations in the
)•
(
being an integer less than
p
in equation (v.) its different values w,
(p
—1
.
. .
cos {q
— \)pio,
x= —
(2746), after putting
and n
= q.
^
\
j
in the result,
cos 2/iw -|-
.
.
.
/.;
being any integer less than
q, is
+ cos (q — l)p(i) cos (g — l)ku)-rl.
each tei'm by (6C8), we see by (iv.) that this coefficient
vanishes excepting for the values k
q—p and k p, in each of which
Hence, dividing by ^q, we obtain
cases it becomes
^q.
By expanding
=
=
=
yp(9-:P\+^ (£.\
= 2xP (I) - 2lq + cos pio
+ cos2^wZ(2-2cos2w) +
+cos
(/?
I
(2
-2 COS co)
— l)27wZ {2-2 cos(g-l) w}.
the third term; the last but one
term = cospwZ(2 — 2 cos w)
cos 2^wZ(2 — 2cos 2a^ = the second term, and so on, forming pairs of
equal terms. But, if q be even, there is the odd term
The
=
last
=
cos iqpoj log (2
according as
2^
Examples.
— 2 cos ^quj) =
—By (2748-9) we obtain
2750
^(f)
= ^(l)-31og2+^,
4^
2752
^(l)
= ^(i)-Hog3+^^,
^(i)
(I)
= ^ (I) -Slog 2- j-,
= v/^(i)-fiog3-^,
;/.(i)=;/.(l)-21og(2).
2754
DEVELOPMEN'TS OF
Wlien
2755
dh 2 log 2,
even or odd.
is
,1'
is
-/,(«+,.•)
Proof.
r
= ^(«) + J- + -l,+-L.,+ ...+
— By (2289), putting n = a + x —
(a
t/»(rf+.r).
any integer,
+ x) =
(a
+ x-1)
(a + x-2)
l
...
and
r
^
— x—1,
(a + 2)(a + l) a r(a).
Differentiate the logarithm of this equation with respect to a.
2756
<l,(a+.v)
= ^ia)+^-pf:zll + £ip^kl^
a
2a{a-\-l)
.ia (« + !)(« 42)
,r{,v-l)(,r-2)(,r-i\)
,
THE FUNCTION
If
407
\fj{a^.
be a positive integer, the number of terms in tliis
and the value of ^p{a-\-x) can be found from
33
series is finite,
that of xp{a).
Hence, by this or the preceding formula, in conjunction
with (2747), the value of ^{N), when JV is any commensurable
quantity, may be found in terms of ^(1).
Proof.— Let (a + x) = A + Prx+Cx (x-l)+Dx (x-l) (x-2)
Change a; into x + 1; then,
\|/
A4^(a
Ax
=
+ x) =
Aaj
1,
^lJ{a
+ x + l)-^l^(a+x)
(x-l)
—
— 2x,
Ax
=:f?^{logr (a + x +
+ &c.
l)-logV(a + x)}
= ch\og (a + x) (2-288)=-^^,
{x-Y) {x - 2) = ox — 1), &c. Therefore
(.«
^-B^20x^'iJ)xix-V)^^Ex{x-V){x-2)\,
a-\-x
A
—=
2(7+2.3jDj; + 3.4E«(a;-l) +
,
a-\-x
A^^— =
2.3D+2.3.4.i;a- +
,
a-\-x
- = 2.3. 4^ +
A*—
a-\-x
Put = in each equation to determine
thus
A = 4^(a), B = -, 2G=a'^
.
a;
a'
a
= A^ — =
1
2.3D
—1
A
(a
2 3
1
2
a
a(rt+l)(a + 2)
4^ = A'— = A
the coefficients A, B, G, D, &c.;
^
a+
1
^
^
a
a(a + l)
-
2
a{a+l){a + 2y
+ 1)
—
—^3
=
—, and
-
a (a
+ l)(a + 2)
(fl
+ 3)'
SUMMATION OF SERIES BY THE FUNCTION
t/»(cv).
^ + ^^ + ^^ + + ^^
2757 Fonnulal.
Proof.
so on.
...
— Let S„ denote the
«.
terras of the series to be
summed.
We have
S„.,-8„ = f-^(^^+n+\)=^[^[-^ + n+2)-^[^ + n + l)']{2288)
or
>S«,i-
—
c
Hence the
4/
(-^ + n
\ c
+ 2] = S„I
difference is independent of n,
^
c
4/
(^ + n + l).
\
c
and therefore
I
1
INTEGRAL OALOALUS.
408
Ex.
2758
1 + 1 + - + 2;;^ = ^-^"^ (^^+-^^ '^"+^)-
1+
2759 Formula IL
^
7>
+ 3o
= .lFT^-|:{*(4^')-*('^>HHf^-)-^(^-)}-
— The
Proof.
series is equivalent to
+ r-^+,-V+...+ ,-A-- ^7^ +
and the result follows by Formula
Formula
2760
1
b
1
III.
.,
.
I.
—
1
•
+
"',
,
ininf.
i+c^i+2c
i !*(¥)-*(*-!?)!
Proof.
2761
—Make n=
Ex.
1.—In
00
in
Formula
(2760)
h
let
= c=\,
l-i + i-l + &c. = i + i
For ^
(2)
2762
= 1+4/ (1),
Ex.
by (2755)
;
The
II.
r|,
last
then
\^ (2)-4/
(f)
two terms become equal.
= 2+
4^
a)}
= log 2.
(l)-21og
2,
by (2754-5).
= 1, c = 2, then
l-\+\-\+&c. = 1 + 14^ (i)-!-]' (I) =
f-
2.— In (2760)
2763
V'(H-«)
let &
= Jor"-^rf.r+V(i)—1
.1'
Proof.
4-(l
+ a) -^(l) + ^-^|xF^ + "%rl^.2!^^^"'^'^a(a-l)
1_(1_9;)«
v|,
(1
+ a) = f
3
^^^ ^-^ ^^)
„
1.2.0
1.2
a;
therefore
.
a(a-l)(a-2)
^"(l--'^')"
dx + v|/ (1).
Jo
Substitute 1 —
2764
a;
in the integral.
i^(l+.0-V'(l+M
= r^!——4-%Ar.
,'0
.r
[By (2768)
THE F UNCTION
Ex.
— Put
= —a;
h
40 9
.
then
+ a) _;/.(!_ a) = - +^(a)-4^(l-a)
;/.(!
(.v)
yj/
(2756)
=- -TrcotTra
2765
--_i_-
Therefore
tZ»
= - -tt cot
Tra.
AS A DEFINITE INTEGRAL INDEPENDENT OF
i/j(a?)
2766
^(,.)
1,1.,
x+
X
—-with^ =
^-...
- + -^ + ...+
But
i/r(l).
= -j;(j_L.+i^jrf«.
v|/A^)=W«-—
Proof.
(2745).
0/
ex
1
^^
1
^
X+fl — l
,
l-z
J^
(2743).
dz,
,
,
oo
^X + fl.-l
-1
by actual division and integration.
iog^
Also
Put
2
=
i/'^
of
(i.),
by
i/
(2367).
in the first integi'al, therefore
1-'/
Jo
Replace
= p(^^-g)cZ.
1-2/'^
Jo
and suppress the term common with the second integral
z,
and we get
»//(«.)
=
|
'-j^^ - fj-^
\
dz.
J
Put
g'^
=
and
u,
= oo
But when ^
and
w'^
=
this
1.
becomes
the product /i(l
Hence the
— m^^)
has
— logtt
result.
1
2767
Proof,
for its limit (1584);
?
da
^(^*^)=ri'
r
(«)
=
e-^z'^-'^dz;
j
d^Y
(x)
But, by (2427),
=
e-'z'^-Hogzdz.
j
Jo
Jo
log2=[
^~"~^~°%
^«,
3 G
TNTEGUAL CALCULUS.
410
.•.d^r(x)
=
C' C^
/<
- "
/)
- at
dzda
e-':f-'-
i
J n J
=
r[c-.[%----'&-rc-"+->-r'-'&]
'^'
-wLich establishes the formula since
d,v{x)
2768
logro.)
-
= .z.iog r
r(.o
0^)
=^
(x).
^£[G.-i)c-^^-^-^!=£f]^'
=:fT!jl£l!_,,.+ill?£.
2769
^(„.)=j;[^-j^,/f
2770
—
Proof. Integrate (2767) for x between the limits 1 and
that logr(l)
0; thus
Jo
log(l
'
Subtract from this the equation obtained from
multiplying the result by x—\. We thus obtain
log r (.)
x,
observing
=
[0.-1,
+ «)
>
by making
it
(1+,.)- ^^""^";"""^1
= i;
Substitute ^ = log (! + «), and
stitute z = (l+a)-\
Lastly,
(2768)
(2770)
is
is
a
;c
=
2,
and
s^-
To obtain (2769), subthe result.
the result of ditferentiating (2768)
for X.
NUMKRICAL CALCULATION OF
\ogr{.v).
2771 The second member of (2768) can be divided into two parts, one of
which appears under a finite I'onn, and the other vanishes with x. If we pnt
r = (.-i-.1- 1-,)\
Iogr(,«)=^
then
If
'-i
=r
-,.w«,,_= ,-.£._,^,
Q
.
iP+Qe-i>-)d^
bo developed in ascending powers of
negative indices are --
+
^,
=
li
.--a^-.
I
,
l,
(i.)
the terms which contain
—
NUMEEIGAL CALCULATION OF
Put F(x)
=
411
log Fur).
{P + Re-^^jdi
{
=i:[((-^)-i^)--(M)Hf
= (Q-B)e-^-^-cU - (j^" j" 1) -^' f - ("^^
^^^•)'
and
[^
.. (..)
Then, by
(i.),
F (x)
now
can
j^^
log
T
=i^(.0 + w(a;)
(x)
be calculated in a
finite
form, and
ra-
(iv.)
(.f)
will liave zero for its
limit as x increases.
First, to
sbow that
and, by substituting
Again,
2;)utting'
«
F (^)
w
and
be exactl}^ calculated.
(|) cfin
-g-^,
=1
in
we have
(iii.),
-«=i:(i^4-i)-^f
.
and, by substituting
^^'-'^
A^,
(vii.)
"0)=j;(^,-,\-i)-«f
The
diffex-ence of (vi.)
and
(vii.)
gives
-r(r^-^-^)-f
Jo
since
1
l
Subtract
(viii.)
^
from
(1)
=
^
Also, by (iv.), -F(i)
i-'O^)
By
(ii.),
may now
^-•>'
^
— e^i
(v.),
—
=
1
-.
— e--5
=
,-
1-e
1^>-«
thus
r(^:^^ _ ^) ^^ = I
+ w(^)
=-
/r(i)
= i?7r,
.-.
be found by calculating Fi.r)
F(.)-Fii)
=
_ l£^
Fil)
— F{^)
(2429).
ilog (27r)-i...(ix.)
=:
as follows:
|^[(.-i)e-f+ (1 + |) (e-^-e-i^J
^'
^ Jo
Jo
= i-a'+(^-i)loga: (2427-8),
F(x) = ^\og{27r) + (x-l)\ogx-u:, by(ix.);
(x),
logrOi;) = 1 log (2;r) 4-(a;-i) logx-x +
r(*) = e--^-a;-r-iv/(27r)e-('-)
|
^
.-.
.-.
by
(iv.)
2772
xjr
••
(x.)
;
(xi.)
412
INTEGRAL CALCULUS.
When
W
X is very large, e^
differs but little from unity.
For vt(x) diminishes
without limit as x increases, by the value (iii.)
Replacing xsr (x) in (x.) by its value (iii.)) and observing that
log
we
log
get
r
(.c
+ l)
r (a; -I-]) =loga; + logr
=-i-log
(Stt)
-|-
(aj
+
|) log x
(a:),
—x
("^-^
+J„(n^^-T--2)^'^^
Now, by (1539),
\l-e-f
where
ti
is
21
i
<
1
.
1.2
1.2.3.4
^"'^'
Jo
So that equation
2773
k
1
...
1
2;i
...
2« + 2'
Also
(xii.)
log r (.r+ 1)
,
'^
Jo
produces
= i^^^ + (.r + i) log .v^a.^
^2
^i_
I
^J^in + 2
zr
This series is divergent, the terms increasing indefinitely. The complementary term, which increases with n and is very great when n is very great,
For instance, when
is, however, very small for considerahle values of n.
a
10, the values obtained for log r (11), by taking 3, 4, 5, or G terms of
the series, are respectively,
=
10-090820096, 16104415343,
16^104412565,
16-1041 12563.
CHANGE OF THE VARIABLES IN A DEFINITE
MULTIPLE INTEGRAL.
2774
Let
,7',
?/,
^
be connected with
^, v, I
by three equations
Then, when tlie limits of the integral containing the
variables can be assigned independently, we have
new
TRANSFORMATION OF A MULTIPLE INTEGRAL.
where
of
^,
what F becomes when the vakies of x, y, z, in terms
obtained by solving the equations u, v, w, are sub-
is
<l>
7),
413
I,
stituted.
F (x, y,
Proof.
cluhj dz
z)
=
f
1^
(^^ n,
III
To
ti,
w
for
4',
entiate for
We
r/.
J
^^^
^^
<^^-
aud ^ constant, and differentiate the three equations
To find y^, consider and x constant, and differTo find Z/-, consider x and y constant, and differentiate for ^.
find Xt, consider
V,
ff
rj
as in (1723).
i^
thus obtain
d (ttvw) d (uvw)
d inzj) d (46O
d (uvw) d {uvw) d (uviv)
d {xyz) d {yzS) d {zkn)
d
(itviv)
dxdych^_ d {kyz)
dE,
dt]
dZ
observing' that two interchanges of
value or sign (559).
columns
^_
d (uvtv)
d (tnQ
d.
(uvtv)
'
d {xyz)
in a determinant
do not
alter
its
Similarly in the case of any
number
of independent variables.
When, however, the limits in the transformed integral
have to be discovered from the given equations, the process
is not so simple.
In the first place, we shall show how to change the order
of integration merely.
2*7*75
Taking a double integral
cb
r4>
in its
most general form, we
rp r*
(.1-)
shall
have
t
F(x,y)dydx
F(x,y)dxdy=^\
(i.)
right member will generally consist of more than one integral, and S
denotes their sum. The limits of the integration for x may be, one or both,
^ is the inverse of the function \p,
constants, or, one or both, functions of y.
i//(.x'), so that x
^(^). Simiand is obtained by solving the equation y
larly with regard to (p and 4>.
The
=
=
examination of the solid figure described in (1907), whose volume
make the matter clearer. The integration, the
order of which has to be changed, extends over an area which is the projection of the solid upon the plane of xy, and which is bounded by the two
(.^')>
straight lines x = a, x = h, and the two curves y
V = 'p(x)The summation of the elements PQ.qp extends from a to h, and includes
in the one integral on the left of equation (i.) the whole of the solid in
An
this integral represents, will
=
'4^
question.
But, on the right, the different integrals represent the summation of
a, y
elements like PQ^^rj), but all pai^allel to OX, between planes y
fi, &c.
drawn through points where the limits of x change their character on account
cp (x)
of the boundaries y
not being straight lines parallel
\p{x), y
=
=
to
OX.
=
=
;
INTEGRAL OALCULUS.
414
—
2776
Example. Lot the figure represent the projected area on the xy plane, boanded by the curves
b.
a, x
(b (x), and the straiglit lines x
y
\p (x), y
c.
cp (x)
have a ruaxinium value when x
Let y
The values of y at this point will be ^(c), and at the
points where the straight lines meet the curves the
=
=
=
=
=
=
values will be (p(a), <P\h),
(a),
i//
v/'(fc).
According to the drawing, the right member of
equation (i.) will now stand as follows, U being written
for F{x,y),
C't'
{a)
rb
[<!>
(b)
Cb
(a)
J*
Vdydx+\
Udyd,,
Uia)]a
Udydx
J*
(2/)
The four integrals represent the four areas into which the
by the dotted lines drawn parallel to the X axis. In the last
and ^iiy) are the two values of .r corresponding to one of y
the curve y = f (x) which is cut twice by any x coordinate.
To change
2777
from
z,
part of
the order of integration in a triple integral,
we shall have an equation of the form
y, x to y, x, z,
F {x, y, z)
dx dy dz
ixii^i{x)}4>i(x,y)
is
divided
=
F (x, y, z) dz dx dy
I.\
\
J
2^1
J
^1 U)
J
*i
(c, a-)
(ii.)
Here the most general form for the integrals whose sum is indicated b}^ 2
that in which thf limits of y are functions of z and x, the limits of a; func-
Referring to the figure in (lOOG),
tions of z, and the limits of z constant.
Every element
the total value of the integral is equivalent to the following.
multiplied by F(xyz), a function
in
is
dxdydz of the solid described
(1907)
of the coordinates of the element, and the sum of the products is taken.
This process is indicated by one triple integral on the left of equation (ii )
the limits of the integration for z being two unrestricted curved surfaces
the limits for ?/, two cylindrical surfaces y
4^i{x),
(x, y), z
2
02 (x, y)
<^i
x
x.^.
a'j,
^^d the limits for x, two planes x
y
\p2 (^)
=
=
=
=
;
=
;
=
But, with the changed order of integration, several integrals may be
The most general form which any of them can take is that shown
required.
Solving the equation z
on the right of equation (ii.)
0, (.'', y), let
<t., (z. x)
be two resulting values of y then the integra<l>j (z, x),
?/,,
?/,
tion for y may be efiected between these limits over all parts of the solid
(x, y) is cut twice by the same y coordinate.
where tlie surface z
(pi
The next integration is with respect to a;, and is limited by the cylindrical
fi{x, y).
surface, whose generating lines, parallel to OY, touch the surface z
At the ])oints of contact, x will have a maximum or minimum value for each
=
=
=
;
=
=
= 0. Eliminating y between this equation
(z) for the limits of
x = 4', (z), x =
value of z therefore J^^j (a;, ;/)
.t.
"ir^
and that of the surface, we get
Lastly, th.e result of the previous summations is integrated for z between
z,, z
Zj, drawn so as to include all that portion of
two parallel planes «
the solid over which the limits for x and y, already dotermiued, remain the
Bame.
;
=
=
415
TBANSFOBMATION OF A MULTTPLE INTEGRAL.
=
z^ and similar
The reiuaiiiiiig integrations will take place between z
successive parallel planes and, according to tbe portion of the solid which
any two of these planes intercept, the limits of x for that integral will be one
or
or other of the bounding surfaces, curved or plane, the limits of y, one
other of the curved surfaces.
;
to change the variables in a multiple
and determine the limits from the given equations,
The general problem
integral,'
may now be
solved.
First, in the case of a double integral,
2778
/•.r2
/-MS
(j)
F{x,y)(Udy
(m).,
to change from q^, y to S, v, having given the equations u
v
0, involving the four variables.
= 0,
=
To change y
and dy
= /^ {x,
for
?/)
F(x,
eliminate ^ between these equations thus y =f(-v, v)
Substituting these values, we shall have
rj,
;
dt].
= F{xJ(x, 7) }/„(»•, v) dn =
=
=
y) dy
F, (x, n) dn.
corresponds to i/^, the equations y^
^^ (f) and
Similarly V2
^2 C^)4'i (^')?7i
Hence the integral (iii.) may now be written
Also,
give
if
T)^
=
F, (x, v) dx fZ^
=
2
F,
(X,
y^
r,) dr,
=/(«,
Vi)
dx
will
(iv.),
JliJ^iC,,)
J.riJ>/'i(.r)
the form on the right being obtained by changing tJie order of integration, as
explained in (2775).
0, v
Next, to change x for £, eliminate y between the equations «
Substituting as before, we shall have
g{t,, v) and dx —
thus, x
g^ (^, >?) dt,.
=
=
;
=
=
F,(x,rj)dx
Also,
^1
duce
^1
cori'esponding to
= m^
(»?),
and
F^{t,-n)dl
= "^^
= m.^ (n)-
the equations x^
x^,
in the
same way
F{x,y)dxdy
=
^3
(jj)
and «i
Hence,
F,{^, ^)
:z\
dii
=
f7
(^d ^)
W°'
finally,
di
(v.)
In the last transformation from x to ^, the most general form of tho
integrals which may be included under S has been chosen. When any of the
limits of* are constants, the process is simplified.
2779
Again, to change the variables from x,y,z to
^, r?,^,
in the triple integral,
F{x,y,z)dxclych
(vi.),
rJ>^iU-)Jxi(.r,2/)
having given the equations u
six variables x, y,
z, ^,
»?,
t-
= 0, = 0,
'y
iv
=
between the
INTEGBAL CALCULUS.
416
First, to cliange from z to C, eliminate ^ and n between
equations, and let the resulting equation be z =/(*, y, 0dz
f^(x, y, 4) dZ] therefore
three
the
From
this
=
F(x,
»i
y, z)
dz
=
F{x, y,f(x,
y,
if
V,
d'C
=
F,{x,
^1
=
=
F,(..,y,0fZ.^%c?^
J
0}/^ (^.
corresponds to the limit z^, the equations
Similarly C^
^3(2?,
=/('^ 2/' ^1) give Ci
<^i (.e, 2/)The integral (vi.) may therefore be written
Also,
.vi J
=
M£) Ui(^-,y)
z^
y,
dl
= Xi (">
!/)
^^*^
?/).
F,(x,y,OdCdxdy
2
J^i J*i(^) J*i(^,j-)
(vii.),
the last form being the result of changing the order of integration, as exwe therefore
have now to change from y to
plained in (2777).
eliminate z and ^ from the equations u, v, w, obtaining an equation of the
form y =/(f, x, j;), and proceed exactly as before. The result, as respects
the general foi'm of integral in (vii.), will be
We
>)
;
(vm.)
F,(x,v,Od^dxdr,
The order of x and t) has now to be changed by (2775). Since C is a
constant with respect to integrations for x and ?j, '^x(0' ^".(O "will also be
constants, while \^(ii,x), XjC'T, «) will be functions of the single variable x.
A^{i;,r]).
Similarly, x
A^{!i,ri) maybe
Suppose r]
\{ii,x) gives x
the other limit.
At the point where x '^i(i^) and ri
\(^,x), we shall obtain by
A,, (
:c)
eliminating «, say, v
Similarly, from x='^i(Q and »?
l^i(0suppose, we get r)
fA2(i^) for the next limit; then a general form for the
=
transformed
=
=
=
=
integi'al will
=
=
=
(T,
be
r^2 f/^2(^) rA2(^,r,)
F,(x,r,,Odi:dr,dx
(ix.)
J^Jmi(6")JAi(^,7,)
Eliminating y and z
It now remains to change from the variable x to ^.
between the equations u, v, iv, we have a result of the form x-=f {l, v, 0Substituting for x and dx as before, we arrive finally at the form
F,{^,V,0<-Udr,d^
=
(x.)
=
Ao(C, v), in (i^-), are
It should be noticed that the limits x
A^ii', »;), x
not necessarily different curves. They may, in some of tlie partial integrals,
be different portions of the same curve. This was exemplified in the last
integral of (2770).
MULTIPLE INTEGRALS.
The following theorems, (2825) to (2830), which are
given for three variables only, liold good for any number.
Let X, y, z be quantities which can take any positive values
MULTIPLE INTEGRALS.
pfi--ri-^-^
«o«r
I
"^^"^"^
JoJo
Here x+ij + z
=
n
n
7
r(/+m+« + l)
then for y by (2308)
function by (2305).
for z
gamma
T(l)T{m)r{n)
J
^
the hmiting equation.
is
I
(
—
)
-}-(^)
finally for x
;
;
+( —
Proof.— Substitute x
not greater tlian
is
j
=r
= l—Y, y =
^"V
^f^
the hmiting equation.
is
1
by (2280),
^^^
fflV>y-4--V/^/,r/^^^^g^^
2826
when
to the
1
-^
Proof.— Integrate
and change
m
1
Jo
sum
tliat their
subject to tlie condition
unity; then
417
(^-jY,
2827 When the hmiting equation
the value of the last integral becomes
z=
is
(—)'', and apply (2825).
simply
^
+ + ^ = A,
r)
r{l)T(m)T{n) \
.z^.n^n
r(/+m+n + l)'
2828
The value
limits h
and h-\-dh
of the
of the
same
sum
.^..«.n-i r(/)
Proof.— Let u be the value
value required
is
which reduces
to the above,
in
integral, taken
between the
of the variables, is
T(m) T(n) „
(2827)
;
then,
by Taylor's theorem, the
by (2288).
2829 j]p-^r-'^'^'y('^'+^+-)
-
T{l
3 H
f^'^'^^f^-
+ m-\-n)
X^
^
'
INTEGRAL CALCULUS.
418
=
to c.
In otlier words, the
h and 11 varies from
variables must take all positive values allowed by tlio condition that their sum is not greater than c.
if x-\-}/-\-z
Proof.— For each value
by (2828),
fih)
—
of 7t the integration with respect to x,
//
»
y, z
gives,
r(l)r(rn)r(n)
the variations of .r, y, z not affecting h.
tegrated as a function of h from
to c.
This expression has then to be in-
fij>->v»-'f-/[(l)''+(iy+(-^)''( didridC
2830
'dh,
rl
1"P'
itli
'
the hmiting equation
Proof.— From (2820) by
2831
If X, y, z
(
—
)
"
I
=
J«
\
+(-^)
substituting x
+() =^-
f-^~
Y, &c.
he n variables, taking
subject to the restriction
CCC
'"
1
cX'-
+ + r+
?/-
...
_
(lv(h/(h.&c.
all
>1
;
positive values
then
7r'("+^)
But, if negative values of the variables are permitted, omit
the factor 2" in the denominator.
Proof.— In (2830) put l=im = &c. = l;
f(^'^
^^~7r\
)T'
''
The integral is =
Hence the result.
But
— I;
a
^'^^ t^iG expression
17 (in,
\) (2280)
= = &c.=l;
j3
];i
= q = &c. = 2;
on the right becomes
r(i»)r(i)
_ t\
=
jy ' U;
Iil(«+i)}
(2305
5).
.
negative values of the variables are allowed under the same restriction, .i"+ij^ + ~-+ ... :^ 1, each element of the integral -will occur 2" times
for once under the first hypdtliesis.
Theivt'ore the former result must bo
multiplied by 2".
if
.
MULTIPLE INTEGRALS.
2832
If
condition
<f)
n positive variables,
a;'^-|-?/"-^
(
+ 2;'^-f-&c.> 1,
,r,
y,
z,
&c,, are limited
then
ax -\-hy-\-cz-\-kQ.) dx dy d.%
PI
_i(w-l)
i(w-i)
2'-^r{i(«+i)}Jo^'
F=
wliere
a^
+ K^ + c' + &c.
—
Proof. Change the variables to £, rj, ^ by the
oi'thogonal transformation (171*9), so that
a^
+ h^ + c^ + S^c. = k\
The
intearral
cp
and ax + hy + cz + &o.
then takes the form
(JcO di
ch]
cU ,.. with
k^
=
H.
'
+ >f + T" + &c. >
1
&c., considering ^ con
Now, integrate for
by adapting formula (282G). The limiting
>/,
stant,
equation
is
<r,
419
by tbe
INTEGRAL CALCULUS.
420
Here
I
=
f (h)
= m = &c. = 1 p = q = &c.= 2;
_ c = 1 and the reductions are
a
;
:
;
= =
ft
&c.
=
7(1-^)
;
similar to those in (2832).
2834 If in (2832-3) negative values of tlie variables are
admitted (since the limiting equation is satisfied by such),
each element of the integral with respect to »?, t, &c. will then
occur 2''"^ times, and therefore the result in each case must
be multiplied by 2''"^, and the limits of the integration for ^
and 1.
will be —1 and 1 instead of
EXPANSIONS OF FUNCTIONS IN CONVERGING
SERIES.
The expansion of a function by Maclaurin's theorem (1507)
can be at once effected if the n}^^ derivative of the function is
known, or if merely the value of the same, when the independent variable vanishes, is known. Some ?z*^^ derivatives of
different functions, in addition to those given at (1461-71),
When the general value would
are therefore here collected.
be too complicated, the value for the origin zero alone
is given.
DERIVATIVES OF THE
/i*'^
ORDER.
a general formula for calculating the n^^
derivative of a function of a function.
The following
If
be a function of
7/
w^iere r
term
is
=
1, 2, 3,
of the
...
7?.
z,
and
z
a function of x,
=zm each
successively, and a is put
after differentiation.
expanded binomial,
Proof.— Assume
?/„^
=
A,y:+ ~2/2z+
rf
2/3--
+
•••
+
ryf
2/'«-
form r equations from this by making
nmltiply these r equations respectively by
l)'*^'?;-'", and add the results.
All the
yz-\ —C(r, 2) z-\ G(r, 3) z-\ ... (
This is shown by differentiating the
coefficients excepting A^ disappear.
To determine any
y
=
z,
2^
equation
z^,
...
z''
coefficient A,.,
in succession
:
—
(1 -x)'"
= 1-rx + G (r,
2)
z'-C (r,
3) ;»»+... d=
x*-
421
EXPANSIONS OF FUNCTIONS-
successively for
with a put
=
z,
x,
and making x zero
after
after each diflfereutiation.
expanding and differentiating the
Thus,
finally,
binonaial.
—
Examples. The formula may be applied to verify
equations (1416-19).
Jacobi's formula (1471) may also be obtained by it.
2853
(hrv+Dx siu-' .V
2854
__
""
n_y
U
1.3... (2/1-1)
2''{l-.if(l-j:y^
l.3C{n,2) ^,
2/1-1. 2/i-3
'^n-1
I
Z = i=^.
l-3-5C(^^,3)
Z-^+..,±Z"L wliere
V
2/1-1. 2/i-3. 2/1-5
^ ^
(sin-^ cb)(„,i), =
Proof.
Expand
the right
2855
</„;r
member by
tan-^t-.
The following
is
\{l-x)-^- {l
+ x)-^n.
l-\-x
(1434).
(1460).
This derivative is obtained in (1468).
another method, which also includes the
result in (1469).
tan"
•••
by (1425)
=
d„^
ic
—
=
tan- .
1
,
o
,
—
t
(
c,
I
1
1
l«-l(-l)«f
1
1
)
^^^^ - ^—y„ j
(cos
± sin 0), which
•
1^
I
x±i=
z
y(H-a;'')
cote, therefore
a;
substituted in (1) convert the equation, by (767), into
Put
(tan-^ x}„^
2856
=
(-1)"-'
rf„;,{e^'°'^cos(c«'siua)}
\n-l
sin"
.(1).
values
sin nO.
= e^«°^«cos(.^sina+n«).
Proof. — By Induction.
LAGRANGE'S METHOD.
2857
Lemma.— The
?^"'
derivative of a function
u=f(x)
% times
the coefficient of h"" in the expansion of fix-\-h) in powers of
h by any known method.
will,
by Taylor's theorem (1500), be equal to 1.2
...
2
INTEGRAL CALCULUS.
422
=
=
and therefore ?/^
i + 2c,c;
tlien
equal to either of the following series,
with the notation of (2451-2).
Let u
a-\-hx-\-Ciir,
d^_^{a-\-hx-\-cx^Y is
i(H,,n-.,,.[nS
U^
U
1
(r)
I 1
2858
^005
2859
or,
Pkoop.
putting
this trinomial.
{(ic
.
•
(2858)
=
|
lr
/i
in
?t", it
becomes
,^o,,
].
+ •••]'
(f,
(7t
+ «^7i + c7(.')".
r times the coefficient of h^ in tlie
Tlien,
expansion of
it may be obtained by expanding
and collecting the coefficients of A'' from the
is found by taking
the result, and
is
as a binomial,
subsequent expansions.
expanding,
l(r,)l[r-Z,)
7?i
will be
+ ttj.h) + ch^}"
••
j^2
— Cbangitig x into x +
by (2857), drxU"
n^:r'c^n-
n^^r'cu
+ l(r-l) + +
2?i =
and Aac — =
The value (2869)
and multiplying by
collecting coefficients of h^,
1.2...
r,
as
before.
Ex.— To
2860
u
=
a^
+ x^,
(a'
ti^
find d„^(a'
= 2x, q=
+ x'r =
2a, r
(2n)'.";
I
L
+ x-)".
=
X-+ f^;'~^], aV-^
2)1 (2ii — l)
1
2861
with
Applying fornuila (2859), we have
Therefore
n.
.
2 .2?i
rf«.t-'= e-'[a«(2.r)«+...
7-
=
...
{2n~
3)
)
+ :^«-'-(2.r)''-^''+&<).|,
&c. in succession.
1, 2, 3,
=
Proof.— By the method of (2857). Putting e<^(^*A)'
e"^ e'"^* e"''\ expnnd
the factors containing h by (150), and Irom the product of the two series
collect the coefficients of h".
ciocn
1
siu
<,
/o
\„sin/
.,
•••+1(Tt(^^^)
,
mr\
,
eo^V
+
—
j"^
EXPANSIONS OF FUNCTIONS.
Proof.— i„^
putting
i"-'"
=
(cos
^^.iin-r^l^
and then equate
2864
A'-
=
=
+i sill ic")
'"-''''",
e'
by
since,
and imaginary
real
fh.r-^^
=
Expand the
(7„,,e''-^-'.
=
(760), i^""
(iiZ^I +
\s^+
eos
423
sin
*
[a'H
^sin
right
~=
i.
by (2861),
Also put
^^^^" |
,
parts.
i-^r [^"'+ {n+l-2'^}
c^"-'^^
+ {(7(^^ + 1, 2)-2«(« + l) + 3"[e^»-''^
+ \C(n + l, 3)-2«(7(H + l,2) + 3''(» + l)-4"}e'"-^>^ + &c.]
Proof.
— Let u be the function.
To determine
the A's being constants.
+ l)"'^S
and
also (e^
(e-
+ l)«-i =
From
e("'^
By
differentiating
u
it is
seen that
expand u
their values,
=
(e^
+ 1)"*,
by the Binomial theorem; thus
•)-+(«.+
1) e'-
+ a(n + l,2)
e^'^'^'^+C (n
+ 1,
3)e(«-^)^+&c.
the product of the two expansions the coefficients A^, A^-x, &c.
may
be selected.
'-1
2865
dr,,,iim-\v
according as n
is
=
(-1)
Proof.— By Rale IV. (ISo-i)- The
(see Example 1535) are, in this case,
(l
+ x')y,^ + 2xy,
=
(i.)
or zero,
first
last differential equations
and
+ ^ («+ 1) 2/»xo =
(ii.)
;
=
=
^.osin-^t^
according as n
Proof.
\n-l
2/(». 2)0.0
;
0with y^
1 and f/2xo
Otherwise.— B J (1468), putting x
2867
-^
odd or even.
is ocZtZ
= 0.
=
1.3^o\.. {ii-2y or
or
eveii.
^ero,
—By differentiating (1528).
Othenvise.—As in (2865) where equations
case
(l-a!^)i/2^
2869
^4.0
according as
Proof.
in (2867).
— As
=
{sm-\vy
^i-
is
(i.)
a:?/^
=
(i.)
and
2/(„.2)xo
2.2\4?.6'
...
=
(ii.)
will
become
in this
^^^2/»^o ... ..-(ii-)
{n-2r
or zero,
even or odd.
in (2865)
;
equations
(i.)
and
(ii.)
being identical with those
INTEGRAL CALCULUS.
424
d„,,i-o^(mHm-\i')
2871
=
or zero
(-l)f m2(wr-2-)(m---4-)
according as n
;
2873
is
sin
f^?„ ,0
(r>i
;
according as n
is
odd or
f4^o eos
2875
2876
and
>
if
even.
siu \r)
= (_1)"-T m(m2-l)(m2-3^)
or zero
[/>i--(7i-2)-],
...
even or odd
...
ei-e?^
rm COS
[m--(n-2)-],
and
>1
if
odd.
\v)
or
mir
according as n
2877
—
and >
is 0(7(?
«/„,o
1,
or even and
siu (»i cos-^r)
(_l)fm2(m'— 2-)(?n^— 4-)
2879
or
(
n
is
Observe that,
cos~^
...
[m^—in—2f]
siu JmTr,
...
[m--{n — 2f]
cos-imTr,
^^^
= — l)^'m(m2-l)(m2-3-)
according as
= ^i
>0.
odd and > 1.
in (2871-3), sin-^0 = 0, and in (2S75-9),
ei'^'^^
and >
0, or
are the only values admitted.
As
Pp^oof.— For (2871-9).
in each case
in (2865)
;
equations
(i.)
and
Cii.)
now
bo-
coming
(i-)
(!-»') 2/2x-a32/x+»'V =
Otherwise.— By the method of (1533).
2/(«.2);ro= («'-«i')2/r,xo
(ii.)
= x cot then
wcos^i^ = ?/oSiu — ir+...+//,^o^(^',>*)sin—2" '^+
2880
Let y
,r,
7r
...+//(n-l):rO>iSill^,
with integral values of
r,
from
to
n—
l
inclusive.
—
EXPANSIONS OF FUNCTIONS.
425
y„-,Q shortly by ?/„, we find, by making
&G. successively in the formula,
Thus, denoting
2881
n= 1, 2, 3,
_
_
2
_
_8_
_
32
128
—
=
Proof. Take the nth derivative of the equation a; cos a;
y sinx by
0.
(14G0), reducing the coefficients by (1461-2), and putting x finally
2882
The
derivatives of an odd order
independently, as follows
Let y
^ (x), then <p (x)
all
vanish.
=
may be shown
This
:
=
is
an even function of x (1401)
9"-'(x)
...
=
^2„U(0)=:_^^-'(0);
...
d„,,{{l+.v-y siu(mtau-^.r)}
2883
according as n
odd or
is
(l,,,{{\-\-a)')''
according as n
Proof.— As
(2885), are
Formula
(ii.)
;
2887
Equations
2/2^-2
2/(«t2)a'0
in (2888)
=
0.
=
(-1)"'^ m^^l or zero,
=
(-1)
n-2
cos(mtan-i.:i?)}
(2865).
now (l+«')
and
f-^^(0)
'
m^^^l
(i.)
and
(ii.),
both for (2883) and
m (m-1) y =
{m — n — 1) y^^
xy^ +
(m-1)
= — {m — n)
gives the factors in succession, starting with y^
1, y^=0 in (2885).
y,
(i.),
(ii.)
=
=
and with
f/.xo{(l+'?")~^cos(mtan-^a7)}
according as n
or zero,
even or odd.
is
in
thei-efore
eve7i.
ni
2885
;
-f"^'(~x);
is eveii
= — l)'m^'^^
(
0,
y^
=m
or zero,
or odd.
— Change the sign of m in (2885).
— In formulaa (2883-7) zero the only admitted value of tan"'
Pkoof.
Note.
is
2889
according as
2891
7?
is
When |>
— Put
y
(-1)'
"'^'^
^^ '''^'
even or odd; by (1539).
is
a positive integer,
or zero, according as n
Peoof.
=
'^"-(^l)
is
= e"^ cos hx
>
and
< j;.
= x^ in (1460),
or
z
3
I
employing (1465).
0.
2
INTEGRAL CALCULUS.
426
MISCELLANEOUS EXPANSIONS.
Tlic following series are placed here for the sake of
reference, many of tliem being of use in evaluating definite
Otlier series and methods of
integrals by Rule V. (2249).
expansion will be found in Articles (125-129), (149-159),
(248-295),
(756-817),
(1471-1472),
(1460),
(1500-1573).
tests of convergency, see (239-247).
For
Numerous expansions may be obtained by differentiating
These and
or integrating known series or their logarithms.
other methods are exemplified below.
2911
cot.i
= l--i— x + -l---^-iUtt —
TT+ci"
,v
IT
a:
+ _i
^ 27r+.r
Proof.
2912
^—+-L
^
'dir^.v
— By differentiating the logarithm of equation
^cot7r.t=i--h--i-:+
0?
cV
^
—i
— By changing x into
2913
taD.r
= ^-i
irx in
(815).
^
a'-\-l
.r—
1
Proof.
&c.
aTT+A-
,
1
+
1
+&e.
(2911).
1__|_.^
&c.
iTT+.r-
Proof.
— By changing x into ^t — x in
2914
cosec
.r
=—
V
'
Proof.
into
53!.
— By
fTT— .r
Itt+cI.^
(2011).
'
TT— .r
27r+.t'
77+
.J'
t\7r—.r
27r— .r
tin-^-.v
47r— .r
adding together equations (2911, 2913), and changing x
EXPANSIONS OF FUNCTIONS IN SERIES.
=lm + -i— m -^
+
-.J!—
smmiT
2915
wi
1
1
1,1
3— m
2—m
—By putting x =
Proof.
cot.
2916
m-n-
2-^m
1
3+m
^
in (2914),
= i_?^_^-^'-&c.
[o_
cV
For proof
427
L4_
|_2_
see (1545).
The
reference in that article (first edition) should
be to (1541) not (1540).
2917
2918
= i--f^i^pi^A.r
coseCci^
Peoof.
IA
\Jl
\JL
— By (2916) and the relations
tan x=. cot »
— 2 cot 2x,
cosec
a;
= cot 4a; — cot «.
2919
-kz^
1— 2acoStr4-a
-,
=
l+2acos.r+2a'-cos2^^+2a'cos3<i^+&c.
cos cr
^^^^
—
= 1— a-
i-2acos.tM-a^
i-_i-^(coScr+acos2cr4-a^cos3cr+«^cos4i'+&c...)
1—a^^
2921
—_sinf
1— 2a cosji+a
Proof.
^
_
sin^-j-a sm2.v-^a' sin3ci
—By (784-6) making a=l3 = x and
When
a
is less tlian
c
+ a^sm4i + &c.
= a.
unity and either positive or negative,
.
INTEGRAL CALCULUS.
428
2922
1
--^log(l
— cosSci— &c.
— cos2.:r+ a^
+ 2acos.iH-«^) = ttcoscr— a^
—=
tan-i_l+«coSci'
2923
1
Proof. — Putting
log (1
+ 2) =
— sin2a7+--T-sin3ci'— &c.
asino?
2
z = a (cos x +
sin
i
ti
we have
a;),
+ a cos 4- m sin a;)
= |log(l + 2acosa^ + a'') + aan-'
log (1
a;
''"'"•''
1
and
log(l
also
+ 2) =
Z^
z
o
"^"
gS
jgl
"q
t-
+ a cos
(2214),
.«
+ &c.
Substitute the value of z and equate real and imaginary parts.
2924
Otherwise.— Ho obtain (2922),
log (1
Expanding by
2925
Otherwise.
changing a and
2926
+ 2a cos X + a?) =
log
(1
+ ae'') + log
(154), the series is at once obtained
/3
When
(1
+ ae"*^)
by (768).
—Integrate the equation in (786) with respect to
into x,
a
is
and
c
into
a, after
— a.
greater than unity, put
= loga^-fiog (l+2«~^ coscr+w"^),
term can be expanded in a converging series by
log(l-l-2acos.r+«^)
and the
last
(2922).
2927
=
log 2 cosher
cosc^-— ^cos2cr+icos3^'— Jcos4r-|-&c,
2928
log 2 sin ^x
2929
2930 i
2'*'
(it— .r)
= —cos .r— ^cos 2jg—^ cos 3.r— ^ cos4r— &c.
= sin cv—\ sin 2x-\-\ sin 3.i'— J sin 4i -f &c.
= sin .v-^l sin 2x-\-\ sin 3cr+i sin 4r+&c.
Proof.— (2927-30) Make a =^
2931
4^
=
77=
2932
sin
cV+i
±1
sin
in (2922-3).
3<r+i
sin
2y2(l+i"i-|4-i+-A— &c.).
Proof.— Add together (2029-30), and put
"When
2933
71
OcV+&c.
is less
than unity, and
log(l-f w coSct)
=
(/,
,r
=
.|t.
= 1 + \/(l +
^r'),
Iog(lH-2acosjr+a-)— log(l-h«'),
EXPANSIONS OF FUNCTIONS IN SERIES.
and
equal to twice the series in (2922), minus
if a be greater than unity, expand, as in
is tlierefore
log (!-]-«-).
429
But
(2926), by
2934
log(l + ;icos.iO
= log (l+2a~^ cos.^+a"-)+loga^— log (1+a^).
2935
(l+2«
= ^ + ^1 coScr+^2 cos2.v-\-A^ cos3.v-\-&g.,
coscr)"
where
A = l + C(n,2)2a'+... + C(n, 2p) C (2p, p) 0"^+ ...,
= 2a\n+C(n,S)Sa'-{-... + C(n, 2p + l) C (2p + l, p)
A,
-
A
and
a-P+
...},
^r-^(n-r-]-l)a-rA,
(n+r+l)«
be a positive integer, the series terminates with the
term, and the values of A and ^j are also finite.
If n
n-\-V^'
Proof.
— Differentiate
and equate
coefficients
the logarithm of the first equation
of sin ra; after transforming by {GG6)
;
;
multiply up
thus ^^^1 is
obtained.
To
find
A
and
the powers of cos
A^,
a;
expand
(l
+ 2a cos re)"
by the Binomial Theorem, and
afterwards by (772).
LEGENDRE'S FUNCTION" X^.
2936
(l-2«.r+«^)-^=
x„
with
l+X,«+AVr+...+XX+
= -J— 4^ {^' -
)^
1
—
Proof. Expand by the Binomial Theorem, and in the numerical part
of each coefficient of a" express 1.3.5 ... 2/i — 1 as 2/< -^ 2" /i
j
.
|
Consecutive functions are connected by the relation
2937
d^x,^^
=
(2;i+i) x,+^,Z„_,.
—
Proof. Difierentiate the factor once under the sign of difierentiation in
the values of X„^.i and X„_i given by the formula for X„ in (2936).
A
2938
differential equation for
X„
is
(1-^-) d^Jin-'^xd^X^^n (w + 1) X,
= 0.
1
INTEGRAL CALOFLUS.
430
2939
When jp
is
any positive integer,
1^+2^+3"+... + (/i-l)P
5ji^
_
~'^? + l
Bm^-^
n^
2 p
|/9
concluding, according as n
_
(_1)
=
is
or
1
^^
B,n^-'
B,n^-^
+|2 \p-l
4 \p-3'^\ii |7i-5
even or odd, with
(1)
i„_i|2^
Expcand the left side by division, and
1
X
e^
each term subsequently by (150). Again, expand the first factor of the right
side by (150), and the second by (1539), and equate the coefficients of x" in
the two results.
See (276) for the values of the series when p is 1, 2, 3, or 4. But the
general forraula there is incorrectly printed.
Pkoop.
.
e-^—
—
.
Let the series (2940-4) be denoted by S^,,,
under, n being any positive integer ; then
S,,=
2941
S^^^l-±+l--±+&c.
s,„
Proof.
—
=
(i.)
(ii.)
(iii.)
l
...
=
^^
S^n
is
-'-"B,...
obtained in (1545)
8,,-s:,,
8,,-s:,
s,„
= 22l^^^
(_4 + J^ +^"-) =
^^^-
^^''
'^'^^ ^•"••
= ^(s,„-]-s:„)
^..^l+.p + y;: + ^+&c-==-
2944
4...1-5i.+^-,-i.+&c.=
—
sL+i5 as
+ ± + l, + l:+&o....= ^^^'-"B,,.
2943
iT
So,,,
l+±+^, + ^,+&c.... = ^^"B^-
2940
2942
S'.,,,
'
i;,.|2„-l
lf^
ird.,,,,.
cot
TTj;
Proof. By differentiating equation (2912) successively, and putting
in the result.
To compute d„x cot ttx, see (1625).
=J
'
EXPANSIONS OF FUNCTIONS IN SERIES.
The following values
2945
431
been calculated by formulaa
liave
(2940-4).
77"
TT
o(
^' ""12'
'
~
'
720'
'
~
945'
'
9450'
^30240'
'
1209600
tt'
TT
"^"900'
-4-j^^,
g,
TT
3946
'
_
161280'
'
57r^
"n"
GItt'
'
T^^-(^-5)[^-(^][^-(^^
(47r-a)-J L
L
0Q4.7
^y^<
tt'
TT
o.
'~"90'
0'
cosa;
T
o;^!
l
L
(TT-aj^J L
+ cosa
+ cosa
^
(7r
(37r
L
(47r
£!_"!
H
+ a)^J
L
+ a)'dL
+ aj-J
^]
(3T-a)^J
(57r-aj=J
^osa^-^o^^ = sin |(a + g)sini(a-a;)
Expand the sines by
Proof,
sin^ |-a
1— cosa
The two n + V^ factors of the numerator divided by the corresponding
(815)
ones of the denominator reduce to
.
/
2ax
+ cc^
\
/
2ax—x''
\
\
2nn-a/
2u.n
\
(2n7r-a)V
\
+ al
(2»7r
+ a)2
Similarly with (2947) employing (816).
2948
cos. + tan|siu.=
(1+-I-J (l--|-J (l+jl^J
x(l-.^)(l + ^)(l-r^)...&.,
2949 c„s.-c„t|sin.= (1-1)
(1
+
^^J (1-^J
Wl+ 2^)(l_ 2^)(l+^
Proof,
and reduce.
—
=
SSlil^L-^. Expand the cosines by (816),
sin x
a; + tan
cos ^a
2
Similarly with (2949), employing 815.
cos
INTEGRAL CALCULUS.
432
2950
^=4^-('^)'][^+(^:r][^+(3^)'j-
2951
^'=
Proof.— Change
[i+(;01[i+(if)1[i+(s)]6 into ix in (815)
and
(81 G).
e>-2e.s.+ .
2952
fe^
Oft CO
2953
2 (1
— cos a)
+2
cnsa + e"'^
2(1+ COS
a)
Mi^S] M^S] Ms7r:n
=
Proof.— Change
M^n •-
x into ix in (29'lG-7).
FORMULJE FOR THE EXPANSION OF FUNCTIONS
IN TRIGONOMETRICAL SERIES.
When
2955
X has any value between
I
and —I,
*W = ij>W'^''+|C:J>('')eos'iI^rf«...(i.),
where n must have
from 1 upwards
But,
Proof.
x=l
if
all
or —I, the left side becomes Z^
—Bj (2919) we have, when h
l—h^
l
Put
—
Z
l
— 'J.koosd + lv
d
positive integral values in succession
=
''^^
to I; then
——
make
n
J-^
;
h
<
is
^
which
lie
/(/>(
—
/).
1,
The left
(v),
and integrate
for v
side becomes, by substitutiing z
jl-lr) f (z + z) dz
_[•'-"
0-h')<t,(v)dv
l-2^cos^^''~''^
<p
+ /r
from
= v — z,
^
J-^-a-(l-/0H4/isiu^g
=1
When
+
+ 2/iCO3 + 2;i-cos29 + 2;i'cos3^ + &c....
multiply each side by
= 1.
(/)
each element of the integral vanishes, excepting for values of v
near to x. Therefore the only appreciable value of the integral
arises from such elements, and in these z will have values near to zero, both
Let these
1.
positive and negative, since x has a fixed value between I and
/3 to a.
Then between these small limits we shall
values of z range from
—
—
,
have
sin' irz
____
ttV
= _,
,
and
/
ip{a:
N
+ z)
,
^
= <p{x),
,
X
433
EXPANSIONS OF FUNCTIONS IN SFPdES.
and the integral takes the form
when
is
7;
made equal
to vmity,
which establishes the formula.
In the case, however, in which x =
is,
when
z =
from -/5 to
and when
and
0,
also
2
I,
siu"
= — 2Z. We
vanishes at both limits, that
^^-
to integrate for a
have therefore
from -21 to —2l + n,
and
a
/3
being any
sra^all
=
The
U.
The first integration gives Icp (I) as above, putting a
quantities.
2l, produces a sin\ilar form with
z
second integration, by substituting y
(x) giving lij> (
1) when
to a, and with f (ji;-2l) in the place of
limits
Thus the total value of the integral is Jf (l) + i<!' ( 0- The result
l.
x
= +
—
(j>
—
=
=—
l.
the same when x
That the right side of equation (i.) forms a converging series appears by
integrating the general terms by Parts; thus
is
^
(.) eos
—j—^ dv = --
)
^ (.) s:n
_^—
^ _^
(v)
which vanishes when n
is infinite,
provided f (v)
is
sm
^
^-
dv,
not infinite.
Hence the multiplication of such terms by h" when )b is infinite produces
no finite result when h is made = 1, although 1" is a factor of indeterminate
value.
function of the form ({>(:x) cosnx, with % infinitely
''
a fluctuating function,'' for the reason
that between any two finite limits of the variable x, the function changes sign infinitely often, oscillating between the
2955«
A
great, has been called
The preceding demonstration shows
values (^(03) and —<l>(x).
that the sum of all these values, as x varies continuously
between the assis^ned limits, is zero.
By similar reasoning, the two
obtained.
2956 If 3> ^^las any value between
.^(,,.,)=
1
= 0,
-^ (0) on
if
If X
has any value between
2957
=
1
and
write
the left; and
and
are
Z,
j'V(.)rf.+isrfV(,Oeos!^^:ipr^rf.
But
a;
following equations
if
...
x=l,
(2).
write
?,
fV(„)rf.+ l2r[V(.)cos^!:^i^)rf«...(3).
3
K
INTEGRAL CALCULUS.
434
But
im
2958
if
,i'=0, write
= 4 fV
^ (.<)
This formuk
both inclusive.
But if X be
left,
where %nl
('•)
rf"+
and
left;
if
x
=
l,
write
1 sr cosi^ (".OS ^>(/-) rfr
true for any vahie of x between
is
>
on the
^(}>(0)
and
/,
write (^ (,/' ~ 2ml) instead of <^{x) on the
that even multiple of I which is nearest to x
/,
is
in value.
be changed on the right, the left side of
If the sign of
the equation remains unaltered in every case,
,/'
2959
H^v)
= |2rsm^fsm^>(r)<fo
(5).
This formula holds for any value of x between
and
I
exclusive of those values.
If X be > I, write ±(\>{x~ 2ml) instead of (p{x)
on the left,
or < 2nil, the even multiple of I
which is nearest to x in value.
But if X be or /, or any multiple of /, the left side of this
equation vanishes.
If the sign of x be changed on the right, the left side is
numerically the same in every case, but of opposite sign.
+
or
—
according as x
>
is
—
Proof.
For (2958-9). To obtain (4) take the sum, and to obtain (5)
take the difference, of cquatJDns (2) and (o). To determine the values of
2ml =b '/, so that x' is < Z.
the series when x is > /, put ,e
=
Examples.
For
ftrt/>/\
2960
all
values of
'^'=
PiiOOF.
4
TT
^
from
x,
^
7
f^
inclusive,
tt
cos elr
=
(x)
<j)
rvsinJiv
L
y
=x
,
h
n
,
and
on/^i
2961
lQI
values of
4
{
x,
.
from
—
-^tt
siiiJlr
=
I
co?,nv~\'"
•
to
Jo
Pkoof.— Change
,*
into Itt—x in (29G1).
o
,
then
2
"T
n
°^
is
reproduced.
^tt
inclusive.
siiio.r
*•= —Unuv-—^j:r--}-—^,
,
tt,
=~
'
^r-
W
according as n is odd or even.
Similarly, by foi-mula (5), equation (2929)
For
cos 5.r
^ ?
cos.r+—;^^ + -^^:^+&c.^.
— In formula (4) put
V cos nv dv
to
,
^
^j
)
^C.K
n
EXPANSIONS OF FUNCTIONS IN SEBIES.
435
^':l^^
= ?^-l^
+ ^l^-&c.
—
rr
ir
ir-\-z^
2962
2
Pkoof.
c
e'^
"^
— In formula
a
+
-\-l
(5) put 6
(.c)
= ^<'-v_(,-ax
j^,^(|
i
—
„
-^
then
=
be not a continuous function between x
to x
a, and
the function be <p {x) from x
to c^=/; then, in formulse (4) and (5), we
from
;//(<iO
shall have (pio') or
(.c) respectively on the left side, according
and a, or betvveen a and I.
to the situation of x betAveen
But, if x
a, we must write i (</>(«) +^(«)} foi' the left
2963
and
r/'
If
=
/,
i>(^')
=
let
c'K
=
=
rt.
i//
=
member.
—
Proof. In ascertaining tlic valae of the integral in the demonstration
of (2955), we are only cjiicerned witli the form of the function cl ise to the
Hence the result is not affected by the di.^continuity
value of aj in question.
with (p (x)
/3 to
unless x=a. In this case the integration for z is from
for the function, and from
to a with 4' G^) for the function, producing
^<p(a)
l^ (a).
—
+
2964
Hence an expression
involving
x
series of sines of consecutive multiples of --^
in
an
may
infinite
be found,
such that, when x lies between any of the assigned limits
(0 and a, a and h, b and c, ... h and I), the serie3 sh dl be equal
respectively to the corresponding assigned functions
provided that the integrals
\\m^^)M.) ds,
can
all
j'sm'^,/;
W
dr.,
...
j'.i.i(«f )/„(.) d.
be determined.
The same is true, reading cosim for slii' throughout,
with the additional proviso [as appears from formuLi (4)] that
the integrals
2965
[fMdx,
Jo
^f.^dx,... U\„{,e)dx
J
a
Jk
can also be determined.
2966
—
Ex. 1.
To hud in the form of a series of cosines of multiples of
X a function of « which shall be equal to the constants a, (i, or y, according
as X lies between
and a, a and b, or h and vr.
3
,
436
INTEGRAL CALCULUS.
Formula
2
2
H
=
(4) produces, putting
— cos nx
Tr,
x 1
» =
« cos
<
».i; (/./!
+
/3
cos
ax
vi.f
+
y cos
Jx
/(.c
>
+ Z.(/3-y) + -y}
{« (a-/3)
1
=
l
—2 S"^^^-y
jz
+
=
X
1
cos?;x-
{(ct
— ft)
shi7ia+(fl—y)iimnh + y
sin
jitt},
—
2967 Ex. 2. To find a function of x having the value c, when x
between
and a, and the value zero when x lies between a and L
By formula (4), we shall have
f
'"r*^ J
-—
cos
\
z'
7
ax
(v)
["
=
c
Jo
since
<p
(v)
= c from
P
rpi
—-
^"^^'
cos
= — sin
'^^
7
ctv
/
,
Ihereiore ^
N
(.f)
to a,
ft
=—
+
and zero from a
to
—
+
2c
,
-
TTX
ira
C
.
j
sin
-
•
^'""
Jo
cos
lies
—
nir(Z
^
Z.
1
-sm
.
27r(i
'2.TZX
cos -
-
oira
1
o~X
7
+ -3-sm-pcos-^+&e.J.
.
=
When
may
n
=
x
a, the vaUie is |^c, by the rule in (2963).
[0 (a) + 0]
be verified by putting a
] in (2923).
=—
2968
X
,
Ex.
3.
between
By formula
— To
and
lies
</)
find a function of
|/,
and equal
—r-
tZy
=
^
Jo
This reduces to
^/'
f2
TT^II-
\
is,
liv
or
is
f
(v)
cos
=_
(Zi-'+
^
—2 -cos
not, of the
dv
=
cos
'"'
— i')
{l
=-
4?/i
+
i'^'^\
^
—- + —
IF
txiJ.
or 0,
Also
(l-v) du
=
Ji/'
r COS
''^^
-
TT-ltr
2.
rl
-^
cos
^
/
dv + k
Jo
-]1-2-^
A'
Ji/
viTr- 1)
form
nil
{' V
Jc
Jo
(,()
cos
Jo
rl
(p
which becomes equal to Jcx when
when x lies between II and /.
— x)
(4),
{v) cos
according as n
.r
to k (l
This
l.n
'4-
^
;
-— +
— '"^
cos -—
^h &
"T- ^^^T" "^ Fu^
2
APPROXIMATE INTEGRATION.
2991
Let
/(.?')
^?.c
be
tlie
integral,
and
let
tlie
curve
I
By sunirainp^ the areas of tlie trapezoids,
7/ =_/'(.') be drawn.
eipiidistant ordiuates
Avliose parallel sides are the
)i-{-l
.
.
AVl'BOXniATE INTEGRATION.
2/oj
?/i5
Vm we
••
find, for a first
437
approximation,
f/W ilv = tzIL (,y,^+2y,+2^,+ ...+2//,._,+//»)
(i.)
SIMPSON'S METHOD.
ordinate intermediate between
and y,=f[h), then, approximately,
2992
If V\
be
tlie
— f {a)
= *r;^0A+4//,+;y,)
f/(.r)rf,c
(ii.)
^
* a
—
y^,
=
write y.^ for y^, and suppose two
?j
in formula (i.)
Proof. Take n
The area thus obtained is equal to
intermediate ordinates each equal to //j.
what it would be if the bounding curve were a parabola having for ordinates
Otherwise by Cotes's formula (2995).
axis.
//]' Vi parallel to its
;
?/o'
A
closer approximation, in terms of
distant ordinates, is given by Simpson's formula,
2993
Proof.
2/^
+1
equi-
— We have
\
f(x)dx=
\y(,v)clc+\'\f(x)dx+...
^ —
+
Jo
Jo
Apply formula
(ii.)
to eacli integral
value of y corresponding to x
and add
f(x)Jx.
{
J
'-^
denoting by
tlie results,
y,.
the
^—
2994 When the limits are a and b, the integral can be
and 1, by subchanged into another having the limits
stituting X
= a-\-{h — a)
n
equidistant
Let
abscissge,
II
COTES'S METHOD.
ordinates, and
Vo, Vi, y-i--. Vn-i,
2995
A formula for
]f{d^) cLv
wli(
=
1
and
Vn
0,
-,
corresponding
the
be
—2
n—1
.
.
.
-^,
1.
approximation mil then be
A,y,+A,y,-\-...
+ A,i/,,+ + A,?j,
...
(iv.),
r' (<«•)'_",'
-1>
.= (_i)»"
\^^!^^,U:
—
(-2460)
r
I
n
—r
Jo
nj;
r
INTECmAL CALCULUS.
438
Proof.
— The
method
consists
in
substituting
for
the
/ (.'^)
integral
function
=
Wlieu x
to « inclusive.
r taking all integral values from
\P(^r)=yr; so that \l (x) has n+l values in conimou with
approximate value of the integral
tis
is
therefore
(x) dx,
and
we have
The
/(a;).
nui}^
be written
•''*
in (iv.)
By
\p
r,
1—x,
substituting
it
appears that
(«)
joOiX
—r
]^)HX
and therefore A^ = A„_,.. Consequently
half the number of coefficients in (iv.)
2996
The
line 11.
11
= 1:
number has been
;
is
only necessary to calculate
corresponding to the values of 7/ from 1 to 10 ai'e
carefully verified, and two mis]>rints in
namely, 2089 for 2'J89 in line 8, and 89500 tor 89600 in
coefficients
Every
as follows.
Berti-and corrected
it
— {)l — r)
APPBOXIMATE INTEGnATION.
2851.
^^
8DGUU-
_
.
in
_
.
^ _
_
,
10067
_
4
^
12i">9
A
A
= ,,,= 1^,
^
.
.
5r>75
b'JGUO
_
439
A,^
= A,=
,
,
^
^,
224U'
2S89
26575
,
1H175
_
TTSO?
.i_^__4825
GAUSS'S METHOD.
When
f{x) is an integral algebraic function of degree
2n, or lower, Gauss's formula of approximation is
2997
=
Cfi.v)
wliere
^'o ...
c^?^
J,/(.r„)
A^= r
Jo
and
are the
... cr,^
rPi.v)
+ ...+^./W + ...+^«/GrJ
(v.),
roots of the equation
7?-f-l
= d,,,,,.^U-+^{.r-ir-'}=0
(vi.),
0.-.r).. G.-.v_OGr-.v,O...G.-.g
(.tv— ^t'o)
.
.
.
(ctv— cr,._i) (.r,— civ+i)
.
.
.
^^^^,
i'-Vr—^r,,)
_
_
_
^^->^
evidently applicable to a function of any
form which can be expanded in a converging algebraic series
not having a fractional index in the first 2)i terms. The result
will be the approximate value of those terms.
The formula
Proof.
— Let
and let
where /(«)
is
is
'•P
=
('-')
(.i'
— .-Co)(x—a\)
=
f (x)
Q-sly(x)
...
(.»
—
a-,,),
+B
(viii.),
—
B
of the n"\ since \p{x)
of the -211^^ degree, Q of the n V'\ and
of the 71+ I"' degree.
Then the method consists in choosing a function v^ (;)!) of the ?i + 1"' degree,
is
Q 4^
so that
(x)
dx shall vanish
shall coincide with/(.7;)
(i.)
writing
when
To ensure that
N for
i//
{x),
x
and a function
;
is
B
of the n^^ degree, which
any one of the n + 1 roots of
Qi/{x)dx
=
().
and with the notation
We
i//
{x)
= 0.
have, by Parts, successively,
of (2148),
\xm=x^^\N-,l(^x^-^\N)
= x^
=
=
X''
Now
[
N-pxP-'
[
N-px''-'
Q\P (x) is made
to n
1 inclusive.
—
[
N+p (p-l){
[
l^x^'-' f
-v)
&c.
&c.
N+p
(p-1)
X''-' {
N-...zk\ji\
^^^N
(ix.)
of terms like x^ 4> (x) with integral values of jj from
Hence, if the value (vi.) be assumed for ^(x), wo
up
TXTEGnAL CALCULUS.
440
see,
by
X
=
(ix.),
—1
that
(2\p
(x)
dx
will
vanish at both limits, because the factors
appear in every term.
Then, when
(ii.) Let It be the function on the right of ef|nation (v.)
1, and that the other coefficients all vanish.
Xr, "we see, by (vii.)* that Ay
X and
.X
will
=
Hence
R
becomes
/(.r)
whenever x
is
a root of
xp
Oc)
=0.
The values of the constants corresponding to the first six values of n,
according to Bertrand, are as follows. The abscissas values, only, have been
recalculated by the author.
0:
1:
X,
as
w
= 2:
a-„
x^
X.,
71
= 3:
a;.^
X,
=4
= •2113-2487,
= -7880751;].
Jc
Ai
= '5,
= •11270167,
= '5
= •88729833,
Ao
= A, =
A,
= f,
= •06943184,
= -33000948,
^o
A,
=
—
-^?^,
log
•6989700
log
= 9-44:36975
log
=
;
;
X,
=
ar,
= -93056816.
X,
=-04691008,
•66999052
;
^»
A.,
=
=
9-6478175.
'1739274,
log
^3260726,
log
= 9-2403G81
= 9^5133143
;
CALCULUS OF VARIATIONS.
FUNCTIONS OF ONE INDEPENDENT VAEIABLE.
3028
Let
?/
and
rr /(.?]),
and a certain number
F be
let
a
known
of tlie derivatives
t/.^,
chief object of the Calculus of Variations is
of the function /(a;)
which
will
function of x, ij,
&c.
The
to find the form
7/2^, 7/3^,
make
Vd^
(i.)
maximum
See (3084).
or minimum.
Denote
?/o.r, ^3^,., &c. by j;, q, r, &c.
For a maximum or minimum value of U, ^U must vanish.
To find S?7, let Bij be the change in y caused by a change in
the form of the function y =f(x), and let dj^, dq^ &c. be the
a
//.,.,
consequent changes in
&c-
j^, q,
Now,
=
2^
Therefore the
value of p,
the form of the function y, is
therefore
Sp
Similarly,
Sq
=
=
(S//).,
y,^.
when
new
that
;
a change takes place in
is,
g
{jA = '^^.
{^p)ri
^r={Sq)^, &c
Now
m=rWdx
(1488).
(ii.)
Expand by
Taylor's theorem,
.ro
rejecting the squares of ^y,
Bj),
Sq, &c.,
and we
find
m = r iVJy-^VJp+V,^^...)
or,
denoting F„, F,„
F„
...
by N, P, Q,
...,
BU= r {my + PSpi-Q^q-^ ...)
J .To
3 h
dx,
dx.
(iii.)
CALCULUS OF VABIATIOXS.
442
Integrate eacli term after the
(ii.)
\^2^dx
=
Nhjdx
^F^pdx
3029
by Parts, observing that by
Thus
integrals involve hjdx.
j"
first
&c., and repeat the process until the final
Sij,
Q^pdx
is
unaltered,
= ny-^PJydx,
= Q^p-QJy + \Q,Jydx,
Hence, collecting the
coefficients of
BU= ^'\N-P,+ Q,,-Rs^„+
^p, S^, &c.,
8//,
...) ^i/clx
+ Sg,(il-~8, + T,,-...X-Sr/o(il-^^. + T,.-...)o+&c.
(iv.)
must have x
by the suffixes 1 and
x^ and x^ respectively after differentiation.
Observe that P^., Q^, &c. are here complete derivatives;
Vi P» ^'j ^'5 &c., which they involve, being fvmctions of x.
Equation (iv.) is written in the abbreviated form,
The terms
made equal to
affected
3030
ZU ={Khij8.v-\-Ih-H,
The condition
mum
value of U,
for the vanishing of S?7, that
iV-P.+Q,.~P3.+ &c.
and
Proof,
—For,
is,
for mini-
is
K=
3031
3032
(v.)
if not,
=
K),
(vii.)
//i-//o=
we must have
r
KSydx=:E,-n,
that is, the integral of an arbitrary function (since 1/ is arbitrary in form) can
be expressed in terms of the limits of ^ and its derivatives which is imposfor, if the integral could vanish
Tlierefore II,— H^
Also
bible.
0.
;
=
K
K=
witliout
vanishing, the/or>Ji of the fuaction
inadmissible.
;
^1/
would be
restricted,
which is
FUNCTIONS OF ONE INDEPENDENT VARIABLE.
443
K
is twice that of the highest derivative contained in V. Let
The order of
n be the order of K, then there will be 2n constants in the solution of equaFor
tion (vi.) and the same niimber of equations for determining them.
If any of
there are 2)i terms in equation (vii.) involving o?/i, Bij^, dp^, &c.
these quantities are arbitrary, their coefficients must vanish in order that
equation (vii.) may hold; and if any are not arbitrary, they will be fixed in
their values by given equations which, together with the equations furnished
coefficients which have to be equated to zero, will make up, in all, 2?i
by the
equations.
PARTICULAR CASES.
3033
I-
—When
V
does not involve x explicitly, a first
can always be found. Thus,
integral of the equation
if, for example,
a
first
K=
integral will be
+ QP.-Q.P
The order
by one than that
of this equation is less
Proof.— We have
of (vi.)
V^=Np-\-Pq+ Qr + Bs.
Substitute the value of iV from (vi.), and it will be found that each pair of
terms involving P, Q, E, &c. is an exact differential.
—
3034 II- When V does not involve y, a first integral can
be found at once, for then
0, and
0, and therefore
= 0;
—
P—Q^.-\-Roj. &g. = A.
we have
-P«— Q2.c+^3a:~~<-^c.
and therefore
3035
K=
N=
III-
—When V involves only y and p,
V=Fp-i-A,
3036
IV.
—When V involves only
i?
by Case
and
q,
V = Qq+Ap+B.
Proof.
K = —P^+Q2x =
Also
Integrating again,
0, giving,
I.
See also (3046).
by integration,
P=
Q^ + A.
= Pq+Qr = Aq+ Q,q + Qr.
find
V = Qq + Ap + B,
V,
we
a reduction from the fourth to the second order of
differential equations.
—
CALCULUS OF VABTATIONS.
444
3037
Ex.
—To
find the bracliistoclirono, or curve of quickest descent,
taken as origin to a point
from a point
wards.*
Velocity at a depth
=
y
—-=i-
I
V = J^-±^ =
Here
By
measuring the axis of y down-
v^gij.
=
Therefore time of descent
x^ij^,
dx.
-^(-~- +A,
by Case HI.
= —^ = 2a, an arbitrary constant.
= tan 6, y = 2a cos^ 9, the defining property
reduction,
y (l+j?^)
That is, since p
having its vertex downwards and a cusp at the origin
H,-n,
reduces to
-~^_^{(ph\- (p^yX)
of a cycloid
= 0.
If the extreme points are fixed, Sy^ and hj^ both vanish.
The values
Suppose
its
x^,
«!,
t/i,
at the lower point, determine a.
but not
— ^
=
l^y(l+p^)^i
C
)
lower point
3038
(i.)
(ii.)
is
y^,
Then
fixed.
is
(P— Qa!+&c.)i must
coefficient in (3)
therefore
0,
horizontal,
«,
=
0,
and the curve
arbitrary ; therefore
that is, (Vp)^
0, or
h/^
is
=
vanish;
which means that the tangent at the
is therefoi'e
a complete half cycloid.
Tn the example of the brachistochrone, it is useful to notice that
extreme points are fixed, ^//„, cy^ both vanish.
If the tangents at the extreme points have fixed directions, fj\„ ^Pi
If the
hoth vanish.
If the curvature at each extremity
(iii.)
fixed in value, ^p^, cq^, ^pi, cq^
is
all vanish.
If the abscissfB x^, x-^ only have fixed values, hjf„ hji are then
and their coefficients in II^ I1^, must vanish.
(iv.)
—
arbitrary,
3039
Win
Wlien the
limits x^, x^ are variable,
V^dx,-VJx,.
(3029)
PkoOF.
add to the value of
— The partial increment of U, due to changes in
dx, =
^dx,+ 4^
dx,
V,dx-Y,<U,.
a\
and
a-^,
By
is
(2253).
dx-^
3040
Wlicn
equations,
EuLE.
r/^1
and
y^
7/i,
=^
and
o\
{,,\)
,
t/o
i/o
J^^c
=x
connected by given
{''o)-
—Put
2.^1= W(^i'i)—Pi}
fJr,
and
8//,=
IxM-^Po]
^^-^'o'
* The Calculus of Variations originated with this problem, proposed by John Bcrn.ulli
in 16i)6.
FU^X'TIONS OF OXE INDEPENDENT VARIABLE.
and afterwards equate to zero the
coefficients of dxi
445
and
dx^^,
because the values of the hitter are arbitrary.
Proof.
—
iji
+ %,
being a function of
d., iVi + hi)
(2/1 + hi) +
therefore cij^+j^dx^
Ex.— In
=
\p'
d^-^h
(x^) dx^,
=
^
x^,
(•*.
+
^•^•1)
=
^
(«i)
+ "P' (-^i)
'^^i
5
neglecting Sjulx^.
the brachistochrone problem (3037), the result thus arrived at
each of the given curves at its
signifies that the cycloid is at right angles to
extremities.
3041
If
V
involves the limits
x^, x^, y^, jh,
Ih, Ih^ &c.,
the
terms to be added to 8?7 in (3029), on account of the variation of any of these quantities, are
dx^\
[V..+ VyjH+V,/2,+ ...\dx
JXo
+
r {^oSyo+T;,g//x+T;>o+T;,gpi+&c.}
dx.
In the last integral, g//o, ^j/i, ^Jh, &c. may be placed outside
the symbol of integration since, they are not functions of x.
Hence, when F involves the limits x^, x^, y^, y^, i?q, Px, &c.,
and those limits are variable, the complete expression for
gifjis
3042
8t7= p{iV-^.+ Q2.-^3a+&c.]
J Xo
%J
Xq
hyclv
-
.
CALCULUS OF VARIATIONS.
446
3043
Also,
if
= xp{x,)
y,
and
= x{^\)
y,
be equations re-
stricting the limits, put
^y,
=
{i.'{x,)-2h] dx^
and
=
8//0
(3040)
dx,,
[x'(«'o)-B}
The relation 7^= is unaltered, and, by means of it, the
additional integrals which appear in the value of H^—B^
become definite functions of x.
—
Ex. To find the curve of quickest descent of a particle from some
^ {x^).
point on the curve ?/o
xC^'o) ^o the curve y^
3044
p, and
T/o-
=
=
Equation (3042) now reduces to
^U = r(N-P^) ^ydx + V,dx,-{V,-\Vy,podx}
dx,
+ P,cy,-{Po-\
E=0
Now
gives
N-F^ =
/"i^'
f
therefore
.1
\
Vl/-?/o
-
7^'
wTTT^ +
A
=
^^
F=l±/^;
Hence
=
=
A
becomes
2a
(2).
= ^k)
= -7, = -^ = -P.
^JyA^ = ^""^^ =
therefore
this
,
y(.-.!ici+/;i
F,„
Substituting the values (2), (3),
(4), in (1),
(by
By^
and
By^ the values in
the condition
(3040)
;
and
0),
H.-Uo
=
produces
0.
thus the equation becomes
being arbitrary, their coefficients must vanish
p^^P'(x,)=-l
i:=
^^^•
{l+P,^'(^i)]d.r-{l+p,xi'^-o)}dx,
dx^, dx^
(^>-
^^
(l+2^\)dx,-{l+Po2h)dxo+PiCy^-p,cyo
Next, put for
(3035)
^^
(2/-2/o)(l+r)
^=
(I).
v/(!/-2/o)(i+i')
Clearing of fractions, and putting
^^^°
V = Pp + A
0; therefore
-77
Vy,dx]cy,
i'lX'C'^o)
;
=
(5);
therefore
=-!•
the tangents of the given curves \p and x ^^ ^^e points x^yo and x^i/i
point a-,?/,.
are both perpendicular to the tangent of the brachistochrone at the
Equation (2) shews that the brachistochrone is a cycloid with a cusp at
00
t/q, and thorclbro jjj
the startiug-poiut, since there y
That
is,
=
=
FUNCTIONS OF ONE INDEPENDENT VARIABLE.
OTHER EXCEPTIONAL
447
CASES.
(Continued from 303G.)
3045
V.--Denoting
and
T;, F,_, T;
//,
...
y,, yo.
T;^
••.
Vn. "by y,Pi,2h --.Pn;
by N, P„ P,
...
P,,;
let the first
m
Then ir=
fZ,„.P,„-fV+i).P,„+i+... (-ir-"cZ„,P.
of the quantities y,lh,p-i, &c. be wanting in the
function F; so that
which equation, being integrated
m times,
=
0,
becomes
p„-4P,«+i+^z2.p.+2-...(-ir-'"^?(.-.).-p.
= + Ci.i'4-...+c,„_iaj"'-^
(i.),
Co
a differential equation of the order 2n
3046
VI.
—Let X
also
— m.
be wanting in F, so that
y = f{Vm,Pm^l -Pn);
K=
the same as before, and produces the same
From that equation take the value
differential equation (i.)
of P,„, and substitute it in
then
is
V^
= P,nPm+l + Pm+lPm+2+--- + Pn Pn+1'
Each
pair of terms, such as Pm+2P>m+3~ <^2gPm+2Pm+ij
exact differential ; and we thus find
is
an
F=C + P„, + iJ^,„4.1+(P«. + 2B« + 2-4^», + 2iWl)+...
+ (PnPn-d.PnPn-l + d,,P,p,.,) + ^ {co-\-CiX+ +c,^_-,x'''-^) p)„,+idx.
. . .
(" l)^^-'"-^^(.-.-l) .PnP>n^l
...
The
m+
resulting equation will be of the order
1 degrees lower than the original equation.
2n—m — l,
—
or
3047 ^n. If V. be a linear function of p^, that being the
highest derivative it contains, P,, will not then contain p^.
Therefore d^^P^ will be, at most, of the order 2n l. Incannot be of an order
deed, in this case, the equation
(Jelletf, p. 44.)
higher than 2?i— 2.
—
^=
CALCULUS OF VARTAT TONS.
448
—
3048 VIII. Let 2^,„ be the lowest derivative whicli V involves; tlien, if P,^ =f[x), and if only the limiting values of
X and of derivatives higher than the 7?^*'' be given, the problem
cannot generally be solved.
(Jelletf, p. 49.)
3049
—
N=
IX.
Let
0, and
then the equation
;
be given
let
the limiting values of x alone
K=
becomes
or, by integration, P—Qj, + B.,,,—&c. = c,
and the two conditions furnished by equating to zero the coefficients of
Sv/i,
%o)
^iz-,
{P-Q, + &G.\
=
(P-Q. + &c.)o
0,
= 0,
=
are equivalent to the single equation c
0, and therefore
supplies but 2n
l equations instead of 2n, and
Hi Hq
the problem is indeterminate.
—
=
—
3050
Let
F= F{x,
U=rVdx-\-V', where
7/, j9,
q
...)
and
V =f{x,,
x,,
y„ '!h,p„Pu &c.)
U
arising
for a maximum or minimum value of
and the terms to
from a variation in y, is, as before,
;
be added to H^ Hq are
The condition
K=
—
r;r/.To+i';;s.'/o+F;„^i^o+
If the order of
V be
n,
...
+F;d^+F;%i+&c.
and the number
of increments
cIxq, S//o,
&c. be greater than n-{-l, the number of independent increments will exceed the number of arbitrary constants in K, and
no maximum or minimum can be found.
Generally, U does not in this case admit of a maximum
contains either of the limiting
or minimum if either V or
values of a derivative of an order
or > than that of the
highest derivative found in V.
(JcUetf, p. 72.)
V
=
FUNCTIONS OF TWO DEPENDENT VARIABLES.
3051
and
Let
F
be a function of two dc})eudent variables
their derivatives with I'cspcct to
,r;
that
V=f{A,y,iy,q,..z,p\q
ij,
z,
is, let
,..)
(1),
FUNCTIONS OF TWO DEPENDENT VARIABLES.
449
where p, q, ... as before, are tlie successive derivatives of ^,
and p', q, ... those of z.
Then, if the forms of the functions y, z vary, the condition
^
for a
maximum
W=
or
minimum
r{KSf/-\-K'B:^)
value of
?7
or
P'
.
Vdx
is
dj+H,^H,-^H;-H^=0
...
(2).
z, }'>'> ?'» •••5 precisely as K, jff involve
the values of the latter being given in (3029).
Here K', H' involve
y, p, q,
...
;
3052 First, if y and ^z are independent, equation (2) necessitates the following conditions
:
^ = 0,
H,-H,+H,'-H,'=0
^'=0,
K = 0,
(3).
=
K'
give y and z in terms of a;,
and the constants which appear in the solution must be determined by equating to zero the coefficients of the arbitrary
The equations
quantities
^y^,
which are found
%i,
^p^, Sp^
Szq, S^i, ^j)^, Sjh't ...
...
»
in the equation
H,-H, + H,'-H,'=
3053
The number
number
of constants to be determined.
Proof.— Let
of equations so obtained is equal to the
V = f{x, y, y„ y^
K
is
of order
2/t in y,
K'
is
of order
2w in
z,
and
and
(4).
... i/„^,
z, z^,
z^
(p (a;, ?/,
y^...
1/2,,^, «, Zj, ...
.-.
of form
.'.
of form ^
(a*,
y,yt
...
2^^),
y{m*H)xi
m + n times,
Differentiating (i.) 2m times, and (ii.)
obtained, between which, if we eliminate
equation in ?/, of order 2{m-^n),
2(m + n) arbitrary constants. The
2 {w,-\-7i) in number, viz., 2rt in H^
...
^t
«(«+„)x)
^x
•••
^jibx)
•••
•••
(i-)i
(ii-)
3wH-n+2
equations are
z, jJj. ... 2(3,„+„)a., we get a resulting
whose solution will therefore contain
equations for finding these are also
— Hf^
and
'Im in
H[ — H'q.
—
3054
Note. The numberof equations for determining the constants is not
generally affected by any auxiliary equations introduced by restricting the
limits.
For every such equation either removes a terra from (4) by annulling some variation (cy, ^p, &c.), or it makes two terms into one in each
case diminishing by one the number of equations, and adding one equation,
;
namely
3055
itself.
Secondly,
let
y and
be connected by
z
3
II
egjiie
equation
—
CALCULUS OF VABLiTTOXS.
450
(j,
(^xy:c)
ously
= 0.
tlie
(j)
y and
equations
{a\
Proof —
(p
forms of y and
by
(2).
3056
y,z)
are
z
=
(x, y, z)
tlieii
found by solving simultane-
K
and
= 0,
and therefore
vary. Therefore
Hence the proportion.
z
Thirdly, let
tlie
(p^^y
<}>
:
(x,
+ cpJz =
<l>y
y
=
K'
+ ^y, z +
:
<^^.
Sz)
(1514), Also
equation connecting
7j
= 0, wten the
K^y + K'h = 0,
and
z
be of
tlie
more general form
4>{^^>y>p>q
By
differentiation,
we
...
^,p',q'
...)
=
(5).
obtain
If (wliich rarely happens) this equation can be integrated so
as to furnish a value of ^z in terms of %, then dj)', dq\ &c.
may be obtained, by simple differentiation, in terms of Sz/, ^_p.
Generally, we proceed as follows
:
^V=my+P^p-hQh + ...+N'h + F^P+Q'^q' +
Multiply (6) by
X,
,„
and add
it
(7).
to (7), thus
+ {N'-^\<t>,)^z + {P'+Xi>,)^jy +
(8).
for SZ7 will therefore be the same as in (2),
by P X<^^, &c., thus
replace iV by iV+Xe^^,
The expression
we
3057
P
if
+
BU=:^\{{N+\<l>,)-(P+\ct>X-\-...}Si/
+ [{N'-\-\<l>,)-{P'^\<t>,)^+...}
+
{P+X(^,-(Q+X(^,).+ ...},8^,
+
{g+X(^,-(i?4-X<^,.).,+
Bz](Lv
-{P+X(^,-(g+X<^J,+ ...}«8i/o
...},87>,
-;g+X(^,-(/{+X(/>,),+
+
3058
To render
similar terms in
W
...}o87>o
&c.
&c.
P,
Q
...
p\
q
...
&c.
...(0).
independent of the variation h, we must
FUNCTIONS OF TWO DEPENDENT VARIABLES.
tlien equate to zero
integration; tlius
under the sign of
of Sz
tlie coefficient
451
N'+\<l>,-iP'']-H,).+ {Q'+H,)2.-&G.
=
(10),
the equation for determining X.
3059
F= Fix,
Ex. (i.)— Given
z='\vdx
Tlie equation f is
0i/
<p,
now z—^vdx
=
= 0,
9p
'"j/.
=
7j,p,
v
=
or
...
—1,
where
2),
v—Zj.=
'Pa=
^3'=
'"p^
(j>p,=
q
= F(x,y,p, q ...).
and
:^f^
0,
^1' ^^-^
0,
the rest vanishing.
we obtain
Substituting these values in (9),
SU= r[{N+Xv,-{P + Xv,),+ (Q + Xv,),^- ...} ^y +
{N'+X,\8:]dx
J J-u
+ \P+\v^-(Q + Xv,)^ + ...\Jij,-{P + \v^-(Q + \v,), + ...],Si/,
Tor the complete variation DU add V^dx^—VQdxQ. To reduce the above so
Let
j N'dx.
as to remove Sz, we must put N' +
0, and therefore \
\ = M be the solution, u being a function of a;, y,p,q ... »• Substituting this
=—
K=
expression for
Ex.
v—Zjjj,
(ii.)
Here
the value of
— Similarly,
= 0;
3061
A,
and, to
if z in
cU becomes
Q'=0; and
P,
;
N' =
the equations
= 0,
P;
= 0,
Solving these equations,
y^
= ?n
;
vanish,
x/l
f
we must put
A
^z.
(2148),
(p
becomes
= —y^^N'.
+ yl + zldx
—=^==^
P=
;
= j^^'"
the last example be
make N'+\^
Ex. (iii.)— Let Z7=
N=
independent of
K—0, K'—O
;
(1).
P'
=
Q=0;
J---==;
become
or
we
z-^
get
= n;
or
y
=
mx-\-A
;
z
= nx + B.
3082
First, if x„ y„ z^, x^, y^, z^ be given, there are four equations to
determine m, n, A, and B.
This solves the problem, to find a line of minimum length on a given
curved surface between two fixed points on the surface.
3063
Secondly, if the limits
{P\
= 0,
are only equivalent to the two
undetermined.
.Tj,
x^ only are given,
then the equations
= 0, (P')o=0,
equations m = 0, n — 0, and A
(P)o=0,
(P')i
and
B
remain
—
CALCULUS OF VABL-ITIOXS.
452
3064:
"We
Thirdly,
shall
have
Substitute
<}>^^
let
+ ^^, jh +
(^^.^
=
the limits be connected by the equatlous
m^cpj.^,
(1
<Pz, 2'i) ''•^'i
((>2^=: 7i^<p^^,
+
<Pi^,
^V
i
+
2h=^'>^h
'•I'^-i
=
p'\
+ m???i + MHj) (ZXi + Wj ^i/i +
7(i
^'-i
'>i
^Zj
=
^•
thus
',
=
0.
Eliminate dx^ by this equation from
and equate
to zero the coeflBcients of
m^V^
Replacing
we
find
Fj,
=
(P)i(l
+
??z?i2j
and
^i/i
+ 7mi)
Sz^
=
?ijT^i
;
;
then
(P')i(l+mm-i + nn^).
Pj by their values, and solving these equations for
m = m^,
n
= Vi-
=
we
m = m^,
Similarly from the equation
\p (.?•„, t/^,
Eliminating
between these equationp, and
y^
x^, y^, z^,
= mx^+A;
a^^, i/^,
z^
=
(--^i, 2/i,
z^
vx^
«i)
+ B',
=
;
four equations remain for determining
On determining
3065
Denoting
and for
tlie
j;,
q,r
...
y^
y\>
=
mxQ-\-A;
O^o,
vi, n,
derive
2/o,
2o)
=
and
n
n,
= n^.
z^=nx^-]rB;
;
A, and B.
the constants in the solution o/ (8056).
by l^uP^jPs
•••
j
"^6 liave
limiting equation,
<l>{^^\I/,Pl,Pl,
."Pn,^,P'l,P2, ".p'm)
V is
of the order
n
in y
^
of the order
oi
in y
is
z^)
m
m
and 7?i in
and m' in
=0.
z.
^.
m', and n r?7/it'r > or < n, the
order of the final differential equation will he the greater of the
two quantities 2(mH-n'), 2(m'
n); and there will be a
sufficient number of subordinate equations to determine the
arbitrary constants.
3066
Rule
I.
If
Z^c
>
+
—
3067 Rule II. 7/'in be < m', and n < n', the order of the
final equation ic ill generally be 2(m'+n'); and its solution
may contain any number of constants not greater than the least
of the two quantities 2(m'
For the investigation, see
— m),
Jelleff, pp.
2 (n'
118
— n).
— 127.
3068 If V docs not involve x cxplicitl}^, a single integral of
order 2{nii-n) — l maybe found. The value of V is that
given in (303:^), with corresponding terms derived from z.
FUNCTIONS OF TWO DEPENDENT VARIABLES.
dV =
Proof.—
Ndij + P,dp,
+
...
453
+P„d2}„ + N'dz + P\dp\+ ...+P'^dp^.
^
K=
Substitute for
and iV' from the equations
and integrate for V,
0,
K' =
as in (3033),
0,
RELATIVE MAXIMA AND MINIMA.
3069
In
tliis
maximum
class of problems, a
value of an integral, Ui
=
Vich,
condition that another integral,
U.^
=
V.^^dx,
J
same
or
minimum
required, subject to the
is
involving the
Xq
variables, has a constant value.
—
Find the maximum or minimum value of the func+ aV.;,, and afterwards
Ui + alJg; that is, take
determine the constant a hy equating Ug to its given value.
For examples, see (3074), (3082).
Rule.
V^Yi
tion
GEOMETRICAL APPLICATIONS.
3070
Proposition
F [x, y) ds
I.
— To
maximum
a
find a curve s
minimum,
or
which
will
make
F being a given function
of the coordinates x, y.
The equation
(5), in (3056),
here becomes
=
= Xg, p' y^, x and y being the dependent variables, and
dependent variable.
In (3057), we have now, writing u for F(x, y),
where p
N = u^,
N' =
the rest zero.
The equations
Multiplying by
Xg,
^
-M^— cZs(X.Cs)
My,
fp
= 2p,
fp,
=
s
the in-
2p';
of condition are therefore
=
and
yg respectively, adding
\
Uy
— dg{\yj) =
(1).
and integrating, the
result is
= u,
the constant being zero.*
Substituting this value in equations (1), differentiating
putting Wj = u^Xg + u^yg, we get
Vsiu^Vs-UyXg)
Xg(u^Xg—U:,yg)
= uxog
= uyog
* Bee Todhunter's "History," p. 406.
tta;^
and
uyg,
and
(2),
(3).
—
CALCULUS OF VABLATIONS.
454
Multiplying (2) by y
,
and (3) by x„ and subtracting, we obtain
«
— «»
(2/» a'2»
=
2/2«)
«x 2/»
finally
or
^h ^s,
«_^dud^_dud.
3071
*^"*
dy
ds
(Lv
p
(4)^
ds
p being the radius of curvature.
To integrate this equation, the form of u must bo known,
and, by assigning different forms, various geometrical theorems
are obtained.
3072
II.— To
Proposition
which
find the curve
make
will
\F{x,y)ds^-\f{x,y)dx
a maximum or minimum, the functions
form.
F{x,
Let
Equation
=
7/)
Tberefore
=
v.
now bave V=u + vp; and for (p, p'^+p^ = 1,
= 2p-,
P = Vp = v;
N = u^ +pu^
as in (3070).
</)j,
;
N' = u,^+pv^
of given
\{u-\-vx^ ds.
(1) is equivalent to
In (3057) we
F and / being
and /(x, y)
n
(1)
=
(pp'
;
2i>
the rest zero.
;
Therefore, equating to zero the coefficients of ^x and cy, the result
is
the two
^x + P^x — (v + ^p)s = 0,
n^+pvy—{\p)s = 0;
equations
d, (Xx,)
or
+v,
= Ux +
^>>'^x,
=n^ + x^v^.
dsl^y,)
we
Multiplying by a;,, y, respectively, adding, and integrating,
«, and ultimately,
(3070), X
obtain, as in
=
1
1_
-~
Qn»7Q
dU/d
/du_(ly_du. d^
dy
w \d.v ds
p
ds
^
dv\
dyl'
— To
find a curve s of given length, such that the volume of the
which it generates about a given line may be a maximum.
Here \{ifx,—a})ds must be a maximum, by (3069), a? being the arbitrary
The problem is a case of (3072),
constant.
3074
Ex.
solid of revolution
u
= a\
u^
= 0,
v;
Hence equation (3073) becomes
Givinrr D its value,
-
^^
"^^^"-'^
PP,,
—
='
'
:
=0,
1
—=
f where
V
p
=
V
-2ij
v^
y\
=
2y.
•
"i
=
from which
-f^],
dxy
x
and integrating, the
= —
y--
\
„
—^—
.
result
—
FUNCTIONS OF TWO INDEPENDENT VARIABLES.
455
FUNCTIONS OF TWO INDEPENDENT YAEIABLES.
V = f(:x,y,z,p,q,r,s,t),
Let
3075
in whicli x, y are tlie independent variables, and p, q, r, 5, t
stand for z^, Zy, z.2^, z^y, z-^y respectively (1815), z being an indeterminate function of x and y.
U=\
Let
\
Vdojdy,
and let tlie equation connecting x and y at the limits be
^ {x. y) = 0. The complete variation of Z7, arising solely from
an infinitesimal change in the form of the function z, is as
follows
:
Let F„ Fp, &c. be denoted by Z, P, Q, B, S,
T.
^ = {P-B,-\8y) ^z-^\8^+mp,
^ = (Q-Ty-^S,)^z+^S^p + nq,
Let
X={Z-Pr-Qy + R2. + S,y + T,y) h.
The
3076
variation in question
81/
Proof.—
is tlien
=£(v.,=, -V',=..+<^,=,.g -^'-'S)
Ydxdy=\
^
J Xo
hVdxdy
}xo]ya
J yo
J Xo J
2/0
as appears by differentiating the values of ^ and
Jj,„
d.^
by (2257), and
Hence
''"
-^
cZa;
^'
Jj,^
T"!^
'^^
^
"'
»//.
But
^^
dx
-^
dx
'/'iz-^'i-^y-V
the result.
3077
U are,
The conditions for a maximum or minimum value of
by similar reasoning to that employed in (3032),
</»
= 0,
^
= 0,
X
= ^-
CALCULUS OF VAmATTONS.
456
GEOMETRICAL APPLICATIONS.
3078
I.— To
Proposition
make
[I
F{x., y, z)
dS
maximum
a
function of the coordinates
Here, putting
F (x,
y, z)
\i^.
and V„
Vg,
Vt are
dx
i^
will
being a given
[Jelhtt, p. 276.
z.
= «, V=u^l +/ + 2';
"^
Q=-y^J^
F •
(du
V
\
^
^ dzl
(du
du\
q
v/l+/ + 2'
dy
du
^
^i^f^(^^\dx
dQ^
(1
x, y,
minimum,
or
which
S,
all zero.
dP_
The equation
find the surface,
x=
or
dz
d'j
^
Z-P^-Q, =
+ r/^) r-2pqs+ (1 +r)
(l+/ + 22)i
a + q') r-pq.^
^
'
(i-|-^/
^^
+ 22)S
(l-\-p')t-pqs
(H-/ + 2^)t
I
gives
1
t
^^
(
du^ du_du\Q^
dy
u^/i+p' + q:'^ dx
dzl
W
be the principal radii of curvature, and Z, m, n
If E,
the direction cosines of the normal, this equation may be
written
3079
i + i.+l(.|^ + .J + n3 =
0,
function
u
and according to the nature
geometrical theorems
3080
of the
different
may be deduced.
Proposition II.— To find the surface
S which
will
make
\\F{x,y,^S)dS+\\f{x,y,z)dxdy
a maximum or
the coordinates
Let
(3078),
F
(a;,
y, z)
we have
minimum
a3,
=
F and / being
;
given functions of
y^ z.
lo
and / (x,
y, z)
=
Proceeding throughout as in
v.
V = u^/Y^^fT^+v,
Z=
s/l+p^ + q*U, + V„
and the remaining equations the same
reBultiiig ditloroutial equation the
term
as in that article if
—
V,
*
u
on the
left
wo add
to the
;
APPENDIX.
may then be put
This equation
R
where
I,
R
u\
457
form
in the
dy
cLv
dz
dz
m, n are the direction cosines
of the
/
normal to the
surface.
—
3082
by
Ex. To find a surface of given area such that the volume contained
be a maximum.
it shall
By
(z-a-yT+pT^) dxdy
(3069), the integral
|
|
must take a maximum or minimum
We
u-=—a,
have
and the
v
=
m^.
z,
value.
= 0,
14^
differential equation of the surface
3083
The problem
= 0,
t/^
is
= 0,
a case of (3080).
«^
=1
(3081) reduces to
\^^,-\.
or
APPENDIX.
ON THE GENERAL OBJECT OF THE CALCULUS OF
VARIATIONS.
3084
Definitions.
called determinate^
—A
function whose form is invariable is
and one whose form is variable, indeter-
minate.
Let du be the increment of a function u due to a change
magnitude of the independent variable, hi that due to a
change in the form of the function, Du the total increment
from both causes then
in the
;
Dv
= du-^hL.
Thus, in (3042), the terms iavolving dx^ and dx^ constitute du,
the whole variation being Du.
8//
and the remaining terms
hi
3085
is
;
called the variation of the function u.
A primitive
indeterminate function, u, of any number
of variables is a function
constant form
;
in other
whose variation
words, ^hi
3 N
= 0.
is
of arbitrary
but
CALCULUS OF VARIATIONS.
458
—
3086 Let V =^ F.u be a derived function, tliat is, a funcdenoting
tion derived by some process from the function u ;
a relation between the forms, but not between the magnitudes,
of u and v.
The general object of the Calculus of Variations is to
determine the change in a derived function v, caused by a
change in the form of its primitive ?l
The particular derived functions considered are those
whose symbols are d and
entiation
and
,
F
denoting operations of
differ-
integi^ation respectively.
SUCCESSIVE VARIATION.
Let the variation of the variation, or second variation
due to a change in the form of the involved function,
y =f(x), be denoted by S (§F) or ^'F; the third variation by
B^V, and so on.
By definition (3085), y being a primitive indeterminate
function, and ^?/ its variation, ^-y =
(1).
3087
of
V
3088
zero,
•
The second
i.e., ^"^p, ^^q,
Proof.-
P(y„,)
variation of any derivative of y
&c. all vanish.
= B (c^,,,) =
=
3089 If V f{x, y, 2^,
indeterminate function of
{S (hj)},^
&c.
then
q, r,
x,
=
...),
(chj)„,
=
where
by
?/
is
is
also
(1).
a primitive
where, in the formal expansion by the multinomial theorem,
hj, ^j), &c. follow the law of involution, but the indices of r/^,
dp, &c. indicate repetition of the operation dy, d^, &c. upon F.
Proof.—First,
^V=(^yd^ + dp dp + ^d^+...)
In finding PV, each product, such as hj dyV,
is
V.
differentiated again as a
function of i/, f, q, &c.
but, since the variations of ^?/, cp, &c. vanish by (2),
it is the same in effect as tlioiigh ey, ^j), &c. were not operated upon at all.
They accordingly rank as algebraic quantities merely, and therefore
;
PV=
(^yd, + Spdp + ^qd,+ ...yV.
Similarly for a third differentiation
;
and so
on.
IMMEDIATE INTEGRABILITY OP THE FUNCTION F.
Def.— When the function F (3028) is intcgrable
3090
without assigning the value of y in terms of
x,
and therefore
APPENDIX.
459
integrable whatever tlie form of the function y may be,
said to be immediately integrahle, or integrable per se.
3091
The
integrable
Peoof.
—
is
requisite
Vclx
V to be immediately
be identically true.
condition for
K= ^
that
it is
shall
must be expressible
in tlie
form
where (p is independent of the form of y. Hence, a change in the form
which leaves the values at the limits unaltered, will leave
'Vdx
=
;
that
is,
i
^K^y
of y,
= 0.
I
But the last equation necessitates 7v = 0, since cy is arbitrary. And
must be identically true, otherwise it would determine y as a function
K=0
of x.
DIFFERENTIAL EQUATIONS.
GENERATION OF DIFFERENTIAL EQUATIONS.
3050 By differentiating ordinary algebraic equations, and
eliminating constants or functions, differential equations are
produced.
Some methods are illustrated in tlie following
examples.
3051 From an equation between two variables
arbitrary constants, to eliminate the constants.
n
and
—
+
Rule. Differentiate r times (r<n), and from the r 1
equations any r constants may he eliminated, and thus C (n, r)
different equations of the r*^ order (3060) obtained, involving
—
Only
&c.
-3-^, -r
^,
dx'" dx''"^
r-|-l, however, of these equations will he
By
independent.
differentiating n times and eliminating the
constants, a single final differential equation of the n*^ order
free from constants may he obtained.
3052
Ex.
— To eliminate the constants a and
y
Differentiating,
we
-
find
Eliminating a and b in turn,
a.
we
4^ +hx
differentiating
(iii.)
from the equation
(i.)
(ii.)
get
= 2y,
a:^ =
ax- + y
ax
and eliminating h produces the
dx
Now,
"^
h
= ax^ + hx
= 2ax + h
(iii.,
final
iv.)
equation of
the second order,
^^
The same equation
To
3053
where
z?
is
(v.)
obtained by differentiating (iv.) and eliminating a.
eliminate tbe function
a function of x and //.
^v
Therefore
is
|| -2x^ +2y =
r
V
/
'^. i
<p
from tbe equation
have
We
z
=
(p
(r),
GENERATION OF DIFFERENTIAL EQUATIONS.
3054
To
eliminate
tions of X, y,
from
</>
'ii
=
(p
{v),
where
ti
and
461
v are func-
z.
Consider x and y the independent variables, and
differ-
entiate for each separately, thus
^*x
Uy
and, by division,
3055
To
a^, a^, ...
Rule.
^^z
y,
{I'x
^^:c
,
<}>'{v)
(v) is eliminated.
eliminate
F {x,
where
<{>'
= 1>'{v) + ^z ^x)
+
{Vy+V,Z^),
+ U,Zy =
(p^,
<p2,
...
z, «^i(ai),
a^ are
<pn
from the equation
faW,
known
...
i>nM]
functions of
= 0,
x, y, z.
—Differentiate for x and y as independent
variahles,
—
1)*^
of each order up to the (2n
Fj^,;
Fg^.,
F.^^,
F^,
F;
Sfc.
that
is,
;
possible
ivay
in every
¥y;
There will be 2n^ unhnoivn functions, consisting of (pi, (p^, ... ^«
and their derivatives, and 2n'^ + n equations for eliminating
forming the derivatives of
¥
^
them.
3056
To
eliminate
^,
<^i(g,
(p.,{^),
...
<p,,{l)
between the
equations
F [x,y,z,l,
.p,{^),
U^)
...
<PM]
=
^^
f{x,y,z,K,i>,(^),<p,i^.)...<p^{'.)\=0.
—
Consider z a,nd ^ functions of the independent
Rule.
variables x, y, and form the derivatives of F and f up to the
2n 1*^ order in the manner described, in (3055). There ivill
2n equations for eliminating
be 4n^+n functions, and 4n^
—
+
them.
3057
To ehminate
^ from the equation
F{x,y,z,w,<p{a,^)}
where
a,
Rule.
j3
are
known
— Consider
functions of
x, y, z
=
x, y, z,
0,
u\
Difthe independent variables.
between the four
<p, (p„_, (p^
ferentiate for each, and eliminate
equations.
DIFFERENTIAL EQUATIONS.
462
DEFINITIONS AND RULES.
3058
Ordinary differential equations involve the derivatives
independent variable.
of a single
3059
Partial differential equations involve partial deriva-
tives,
and therefore two or more independent variables are
concerned.
3060 The order of a differential equation
highest derivative which it contains.
3061 The degree of a differential equation
which the highest derivative is raised.
3062
A
Linear differential equation
derivatives are
all
involved in the
is
the order of the
is
is
the power to
one in which the
degree.
first
3063 The complete primitive of a differential equation is
that equation between the primitive variables from which the
differential equation may be obtained by the process of differentiation.
3064
The general solution is the name given to the complete
when it has been obtained by solving the given
primitive
differential equation.
Thus, reverting to the example in (3051), equation (i.) is the complete
which is obtained from (i.) by diilerentiation and elimination.
The differential equation (v.) being given, the process is reversed.
Equations (iii.) and (iv.) are called the first integrals of (v.), and equation
priviifive of (v.)
(i.)
the final integral or general sohifion.
3065
^
particular solution,
differential equation is obtained
or particular integral^
of
a
by giving particular values to
the arbitrary constants in the general solution.
For the definition of a singular solution, see (3068).
3036
To
oi'dor
have a
—
when two differential
common primitive.
find
equations of the
first
Rule.
ViffrrentiatG each rguatioii, and eliminate its
The tivo results will agree if there is a
arbitrary constant.
common primitive jivhichy in thai case, will be found by eliminating y, between the given equations.
Ex,
—Apply the rule to equations
(iii.)
and
(iv.) in
(3052).
—
.
SINGULAR SOLUTIONS.
To
3067
find
each, involving
when two
463
solutions of a differential equation,
an arbitrary constant, are equivalent.
—
Rule. Eliminate 07ie of the variables. The other will also
disappear, and a relation betiveen the arbitrary constants ivill
remain.
=
Fand v
c he the two solutions
Otherwise, if V=G, v
being functions of the variables, and G and c constants ; then
:
dV dv__dV
is
dy
dy
cLv
du_
d.v
the required condition.
Proof.
and Fj,=
— V must be a function of
—
Ex. tan"'
tions of 2xyy^
equation
then eliminate
'pv'^'y't
is
h
v.
Let
+ 7/) +tan"' {x — y) = a and
Eliminating y,
x^ + y^ + ^'
(aj
=
tan
a.
V= (p(v);
therefore Fj
= ^^v^
<p„.
=
x^-\-2hxare both solu2/^ + 1
x disappears, and the resulting
=1
SINGULAR SOLUTIONS.
— "A
singular solution of a differential
3068 Definition.
equation is a relation between x and y which satisfies the
equation by means of the values which it gives to the differential coefficients t/.^, y.^j., &c., but is not included in the comSee examples (3132-3).
plete primitive."
3069
To
^{x,y,c)
find a singular solution
from the complete primitive
= 0.
Rule I. From the complete primitive determine c as a
0, or else by solving
function ofx, by solving the equation j^
The
0, and substitute this value of c in the ptrimitive.
Xc
result is a singular solution, unless it can also be obtained by
giving to c a constant value in the primitive.
=
=
3070
If the singular solution hivolves j only,
the equation
from
Xc
tions Xc
jc=0
= only.
= 0, yc =
only,
If
it
and
involves
give the
it results from
x only, it results
both x and y, the two equa-
if it involves
rame
result,
OF TKB
TTNI^ERSITT
DIFFERENTIAL EQUATIONS.
464
integral function,
ye
=
is a rational
WJien the primitive equation (p (xyc)
or
'^(^1/ ^^ used instead o/ Xg
^c
3071
= 0.
Proof.
— Let
Then,
if c
<p
(x, y, c)
=
=
be expressed in the form
y
and,
if c
When
=
= fChc)
(1).
yx=fx
be constant,
(2);
yx=fx+fcCx
varies,
(3)-
constant, the differential equation of which (1) is the primitive is
But it will also be satisfied b}' the same
satisfied by the vahie of ?/^ in (2).
co and in
or /^
value of y^ when c is variable, provided that either fc
either case a solution is obtained which is not the result of giving to c a
constant value in the complete primitive; that is, it is a singular solution.
(Xi , and therefore
oo makes y^
is equivalent to y^
But
0, and /j.
c is
=
=
X
=
f^
= constant.
=
=
,
=
GEOMETRICAL MEANING OP A SINGULAR SOLUTION.
3072 Since tlie process in Rule I. is identical with that
employed in finding the envelope of the series of curves
obtained by varying the parameter c in the equation
the singular solution so obtained is the equa(x, y, c) -—
;
tion of the envelope itself.
An exception occurs when the envelope coincides with one
of the curves of the system.
(j)
3073
w
Ex.
— Let the complete primitive be
= ca;+ v^l—
c",
therefore yc
=x
;
yc=^^ gives
c
= —P
=
vl + x"^, a singular solution.
Substituting this in the primitive gives y
the equation of the envelope of all the lines that are obtained by varying
the parameter c in the primitive; for it is the equation of a circle, and the
See also
pinmitive, by varying c, represents all lines which touch the circle.
(3132-3).
is
3074
"The determination of c as a function of aj by the solution of the
equation y^
0, is equivalent to determining what particular primitive has
contact with the envelop at that point of the latter which cori-esponds to a
given value of x.
"The elimination of c between a primitive y=f(x,c) and the derived
equation y^
0, docs not necessarily lead to a singular solution in the sense
above explained.
" Por it is possible that the derived equation ?/p
may neither, on the
one hand, enable us to determine c as a function of x, so leading to a singular
solution
nor, on the other hand, as an absolute constant, so leading to a
=
=
=
;
particular primitive.
—
—
SINGULAR SOLUTIONS.
"
Thus the
= 0,
particular pi-imitive y
=
465
e" being given, the condition yc=
— oo if « be positive.
+co if a; be negative, and
The resulting solution
a dependent constant.
gives e"^
whence
c is
=
does not then
It is
J/
represent an envelope of the curves of particular primitives, nor strictly one
It repi^esents a curve formed of branches from two of them.
of those curves.
It is most fitly chai^acterised as a particular primitive marked by a singularity
in the mode of its derivation from the complete primitive."
Differential Eqiiations,'^ Supplement, p. 13.
l^Booles
'''
DETERMINATION OF A SINGULAR SOLUTION FROM THE
DIFFERENTIAL EQUATION.
3075
Rule
Any
II,
relation
a singular solution luhich,
is
ivhile it satisfies the differential equation, either involves
makes Py
iifinite, or involves
x
3076
aiid
makes
(
—
\
j and
infinite.
" One negative feature marks all the cases in which a solution
ao
It is, that the solution, while
involving y satisfies the condition p^
expressed by a single equation, is not connected with the complete primitive
by a single and absolutely constant value of c.
" The relation which makes p,, infinite satisfies the differential equation
0, and this implies a connexion
only because it satisfies the condition ?/c
between c and x, which is the ground of a real, though it may be unimportant,
=
.
=
singularity in the solution itself.
" In the first, or, as it might be termed, the envelope species of singular
solutions, c receives an infinite number of different values connected with the
value of a; by a law. In the second, it receives a finite nnmber of values also
connected with the values of x by a law. In the third species, it receives a
finite number of values, determinate, but not connected with the values of x."
Hence the general
inclusive definition
"^
singular solution of a differential equation of the
3077
first order is a solution the connexion of which tuith the complete primitive does not consist in giving to c a single constant
value absolutely independent of the value of x."
[Boole's " Differ ential Equations," p. 163,
and Supplement,
p. 19.
RULES FOR DISCRIMINATING A SINGULAR SOLUTION OF
THE ENVELOPE SPECIES.
3078
Rule
equating
^'
may
3079
III.
to zero
—
]Y]icn
py or
(
—
is
j
made
a factor having a negative index,
infinite
hy
the solution
be considered to belong to the envelope species''
"In
other cases, the solution
3
is
deducible from the
—
DIFFERENTIAL EQUATIONS.
466
as a constant of multiple
complete primitive by regarding
value,
its particular values being eitlier, 1st, dependent in
some way on the value of x, or, 2ndly, independent of x, but
still sucli as to render the property a singular one."
t;
—
\_Boole's " Bifferential
A
Equations," p. 164.
ivliile it makes py infinite
equation of the first order, does not
satisfy all the higher differential equations obtained from it, is
a singular solution of the envelope species.
Rule IV.
3080
and
solution which,
satisfies the differential
Ex.
:
i/x
= my
m-\
'"
lias
the singular solution y
=
when
m is
>1.
The
solution
m -r
Now
and,
2/ra;
when
thei-efore,
?•
>
is
by the rule of the envelope
3081
Rule V.
u
let
= 0,
and X
= "i O'i— 1)
=
•••
m, the value y
O't— r + 1) y '"
makes y^-x infinite.
,
—" The proposed solution heing represented hy
the differential equation, transformed hy
= 0.
the variables, be u^-l-f (x, u)
f^
dn
as a function of
-^-
tcgral
Jo
U
is,
species.
x and
making u
Determine the in-
u, in ivhlch
U
is
either
equal to f (x, u) or to f (x, u) deprived of any factor ivhlch
0.
neither vanishes nor becomes infinite when u
If that
integral tends to zero ivlth u, the solution is singular" and of
the envelope species.
[Boole, Sttjyplement, p. 30.
=
3082
Ex.
— To
determine whether ^
Here w
= ^y,
and
I
J
As
=
y^ =z y (log
ticular integral of
—
2/
•'
^.,
,
(log
a singular solution or par-
is
?/)^.
=—
yy
,
.
log'^
this tends to zero with y, the solution is singular.
—The
Verification.
c will
assigned to
complete primitive is
produce the result y =: 0.
?/
=
c"--',
and no constant value
3083 Professor De Morgan has shown that any relation
oo
involving both x and y, which satisfies the conditions j>y
p^=: OD , will satisfy the differential equation when it does not
make 7/2.r> as derived from it, infinite that it may satisfy it
even if it makes y^.^ infinite and that, if it does not satisfy the
differential equation, the curve it represents is a locus of
points of infinite curvature, usually cusps, in the curves of
=
,
;
;
complete primitives.
[Boole, Sxpiilcmmf, p. 35.
—
FIEST ORDER LINEAR EQUATIONS.
4G7
FIRST ORDER LINEAR EQUATIONS.
M+N-^ = 0,
3084
or
Md.v-\-Ndi/
=
0,
M and N being either functions of x and. y or constants.
SOLUTION BY SEPARATION OF THE VARIABLES.
This method of solution, when practicable,
and is frequently involved in other methods.
3085
is
the
simplest,
xy(l + x')dy=z(l + ,/)
Ex.
therefore
/,
l
and each member can be
'
.,
+ y^
dx,
= x(l
—rr-+—x^)
jr,
at once integrated.
HOMOGENEOUS EQUATIONS.
Here
if
and
N, in (3084), are homogeneous functions
3086
of X and y, and the solution is affected as follows
Rule. Put y = vx, and therefore dy = vdx + xdv, and
For an example see (3108).
then separate the variables.
:
—
,
EXACT DIFFERENTIAL EQUATIONS.
3087
Mdx-\-Ndy
=
is
an exact
M,
and the solution
is
differential
then obtained by the formula
SMd.v-\-^{N-d,{^Mda^)]
Proof.
— F=0
If
therefore V^y
= My =
with respect to
3088
Here
C.
=
x.
N = V^ = dyj Mdx +
Ex.
=
ij)
<}>
3Iy
dij
=
M, Vy
N;
be the primitive, we must have Vj.
Also V ==\ Mdx + (y), (p (y) being a constant
JV^,.
Therefore
therefore
when
= N,,
(t/)
(p'
(y),
= ^ {N-dyJMdx} dy+G.
(x'-Sx'y)dx + Of-x')dy
= —Sx^ = N^.
=
Therefore the solution
C=.^-xhj + ^[f-o:^-dy
(I
0.
is
-a)
dy
I
—
DIFFERENTIAL EQUATIONS.
468
3089 Observe that, if M(Jx-{-Ndij can be separated into two
parts, so that one of them is an exact differential, the other
part must also be an exact differential in order that the whole
may be such.
3090 Also, if a function of x and y can be expressed as the
product of two factors, one of which is a function of the
integral of the other, the original function is an exact differential.
ijnyi
JiiX.
—
—
cos
Here
—
— dx
,-
cos
r
y
y
= cos —
- dy
y
the integral of the second factor.
is
.
y
;
—
r
^
= 0.
Hence the solution
is
y
sin -^
=
G.
y
INTEGRATINa FACTOR FOR Mdx-\-Ndij
When
which
= 0.
is not an exact differential, a factor
such can be found in the following cases.
this equation
will
make
it
—
When one only of the functions Mx Ny or
vanishes identically, the reciprocal of the other is an
integrating factor.
3092
I.
3093
n.
Mx — Ny
is at the
+
If,
ivhen
Mx + Ny =
identically, the equation
same time homogeneous, then
x~^°"^^^
is
also
an
in-
tegrating factor.
3094
III.— If neither
cally, then,
when
Mx + Ny
nor
Mx-Ny vanishes identic
— 1^ is an
the equation is homogeneous,
integrating factor
;
and ivhen
form <f>{xy)xdy-\-x{xj)jdK
the equation
= 0,
-
can be put in the
^*«
^.^^_-^
an integrating
factor.
Proof.— I. and IH.
M dx + Ndy =
— From the identity
i f
(Mx + Nij) d
log
x>j
+ (Mx - Ny)
d log
-
|
,
—
FIRST OBDEB LINEAB EQUATIONS.
469
assuming the integrating factor in each case, and deducing the required
forms for 31 and N, employing (3090),
11,
—Put
3095
M=
v= -^,
N=
a?"^ (v),
x'"\p (v),
and
cly
= zdv + vdx
in
and 3Ix + Ny.
Mdx + N(hj
The
form
general
Mdx+Ndy =
an
for
integrating
factor
of
is
wliere v is some chosen function of x and y ; and the condition
for the existence of an integrating factor under that hypothesis
is
that
M—
—N
3096
XT-^
TT—
Nv^—MVy
a function of v.
be
'iniist
—
=
Proof. The condition for an exact differential of M/jdx + NfJtdy
is
Assume f^
(v), and differentiate out; we thus
(Mii)y-= (Nn)
(3087),
=
J.
2—
obtam
The following
=
<!>
—-f-
—-^
,
are cases of importance.
—
3097 !• If an integrating factor is required which is a
that is, v
x; and the
function of x only, we put
<p {x),
necessary condition becomes
i^i
—^^—
—
M —N
-
must
=
=
he a function of
x
only.
IIIf the integrating factor is to be a function of xy,
v,
the condition becomes, by putting xy
3098
=
-
Ny—Mx
3099
III-
—
miist be a fmiction of
wy
If the integrating factor is to
only.
be a function of
-^, the condition is
x
—i^——-^
Mx+Ny
If
must
he a -^function of -^.
-^
Mx-\-Ny vanishes, (3092) must be resorted
X
to.
DIFFERENTIAL EQUATIONS.
470
In this and similar cases, the expression found will be a
function of v
= -^
takes the form
if it
F (v)
when y
is re-
X
placed by
3100
vx.
of the n^^
— Theorem. — The
condition that the equation
a homogeneous function of x and y
degree for an integrating factor, is
IV.
Mdx-\-Ndy
=
may have
^^K,.-M,)+nKv^j,^^^
3101
= x^xpiu)
tlie differentiations,
from
Nvj.—Mv,j'
and, by reduction,
we
get
Mx + Ny
>pIu)
right member
by integration.
The
= i^.
in (3097), thus
V
Perform
t.
factor will then be obtained
The integrating
Proof.— Put fx—v
where
must be a function
of
u
in order that
4>
(u)
may
—
3102 Ex. To ascertain whether an integrating factor, which
geneous function of x and y, exists for the equation
0.
(y'^ + axy^) dy — ay^dx + (x + y)(x(ly—ydx)
is
be found
a homo-
=
M = -(ay^ + xy +
Here
N=
y-),
Substituting in the formula of (3100),
the fraction reduces to
^^V-^^^
,
(y^'
we
+ ax/' + xy + x-).
fiud that,
and, by putting y
by choosing n
= ux, it becomes
= -3,
-"":
a function of
u.
fi
= x-'e
^^
-^=y-'e
the integrating factor required. It is homogeneous, and of the degree
in X and y, as is seen by expanding the second factor by (150).
3103
If
^
^'
y
by means
=
of the integrating factor
V=G
^it
-3
the equation
for its complete
found to have
primitive, tl'ien the form for all other integrating factors will
be /"/(^)j wlicre/ is any arbitrary function.
fLMd.e^l^Ndy
is
471
FIRST ORDER LINEAR EQUATIONS.
Proof.
— The equation becomes
fiMf (V)
Applying the
dx+fxNf(V) chj =
we have
0.
test of integrability (3087),
{i^^ifiy)},
Differentiate out,
and the equality
=
{i^Nf{v)u.
remembering that
is
established.
—
Ascertain by the determmcdion of an
integrating factor that an equation is solvaUe, and then seeh to
effect the solution in some more direct waij.
General Rule.
3104
SOME PARTICULAR EQUATIONS.
(acr+%+c) d.v-]-{a\v+b'y+c') dij =
3105
This equation
I.
—
may be
X
Substitute
0.
solved in three ways.
= ^ — a,
y
= — ^,
r]
and determine a and j3 so that the constant terms in the new
equation in E and n may vanish.
11.
—Or substitute
ax-\-hy-{-c
=
^,
a'x
+ b'y -{-c =v.
=h:b\
the methods I. and II. faiL The
written as a function of ax-\-hy.
adx, and afterdz
ax -{-by, and substitute hdy
Put z
wards separate the variables x and z.
3106
But
equation
3107
if
a:a'
may then be
III.
—
—A third method consists in assuming
(Ayi
and equating
+ C) d^-h{A'^ + G') ch =
coefficients
0,
with the original equation after sub-
^=zx-\-7n-^y,
stituting
m^^,
=
=
V
= ^-\- m.2y.
?% are the roots of the quadratic
anv'
The
+ ih + a)m + h'=0.
solution then takes the form
{(ami— a')(.r+m,?/) + cw*i—c' }"'"'""'
{
(am2 — a'){.v-\-m.2i/)-\-cnh—c' } «'«^-«'
—
.
DIFFERENTIAL EQUATIONS.
472
3108
(1,
2/
= —
>?
thus
(3,
(Sj,-?^) c^+(7v-3^i,) dn
with equations for a and
therefore
(i.)
7a —3/3 + 7
/3,
=
=
(i.),
3a — 7(3
;
+3 =
(ii.)
/3
r]
Putting
I,
= x—\
and n={y
y,
by
as in (2080), with
log
(ii.)
(»/
— ^) =
;
iV—7
c,
The second member is integrated,
reduction, we find
5 log (»j + ^) + 2
3109
0;
= — 1,
=
= v^, and therefore drj = vd^ + ^do (3086)
(7v'-7)Uii-\-(7v-3)^ulv = 0, or ^+pfldv = 0.
a
being homogeneous, put
.-.
= 0.
(Sij-7x + 7)dx + (7ij-Sx + S)chj
Ex.
= ^—
X
PiTfc
&
= 0,
and, after
G.
the complete solution
is
+ x-Yf{y-x + \y=G.
When P and Q
are functions of x only, the solution
of the equation
^+Py =
by merely separating the
3110
y=Ce-^'^
is
(i.)
variables.
Secondly, the solution of
%^-Py =
This result
imrameters.
y
is
(i
= c-^"''
{
c+Jeci"%/..}
obtained by the method of variation of
is
EuLE. Assume equation (i.) to he the form of the solution^
considering the parameter C a function of x. Differentiate (i.)
on this hypothesis, and put the value of j^ so obtained in the
proposed equation to determine C.
Thus, differentiating
therefore
Then
Q
^=
CxG
(i.),
we
get
J''"",
?/j.
=
therefore
substitute this expression for
C
J
(7j.e
C=
in equation
Otherwise, writing the equation in the form
integrating factor
3111
is
J
'^
may
'
—Fy,
[
qJ'^"" dx + C.
(i.).
(ry—Q)dx + dy
=
0,
the
be found by (3097).
y^J^Py=Qf
reduced to the last case by dividing by
^
—y
•
//"
and substitutiug
FIEST OEBEB LINEAR EQUATIONS.
P,(Lv+P,d^+Q
*3212
{^itIi/-^/(lv)
Pj, P3 being liomogeneoiis fimctions of
degree, and Q homogeneous and of the
473
= 0.
and
,*;
q^^"-
of the
1/
degree,
is
|/''
solved
by assuming
=
Put y
ox,
and change the variables to
may be reduced
which
is
,»j
and
v.
The
result
to
form with (3211), and may be solved
identical in
accordingly.
3213
To
+ Brr + G,y)(xdy-ydx)-(A, + B,x + G,y)dy
-(iA, + B,:c^G,y)dx
equation, put x = + a,
y = v+
(A,
solve this
t,
and determine a and /3
and the form of (3212)
=
0.
ft,
so that the coefficients
will be obtained.
may become homogeneous,
RICCATI'S EQUATION.
3214
U,
+ bu' =
(A).
C.V^'^
=
Substitute //
ilv,
and this equation is reduced to the
It is
1.
Jii
2 and a
form of the following one, with n
or a positive
—4t, t being
solvable whenever
(2^+1)
= +
=
m
=
integer.
3215
I.
y
d'i/,-ai/-]-bt/
— This equation
= vjf,
dividing by
is
solvable,
x-"",
thus obtain
= c.e
(B).
when n = 2a, by
and separating the
substituting
variables.
We
r— = af~'^dx.
— ov
:,
c
Integrating by (1937) or (1935), according as h and c in
equation (B) have the same or different signs, and eliminating
1; by
vsf, we obtain the solution
y
=
3216
,
= ^|,,.C£^^
(1).
* The preceding articles of this section are wrongly numhered.
Each number and
reference to it, up to this point, should be increased by 100. The sheets were printed off
before the error was discovered.
3 p
—
.
DIFFERENTIAL EQUATIONS.
474
3217
01-
3218
II-
n
—— 2a
=r
Rule.
= ^[- -^),..«tan C- -iVtzM I
,/
— Equation
•-•
,
(2),
I
•
may
(B)
be solved whenever
also
.
a positive integer.
f
— Write
m the second
z
term,
for j in equation (B), and nt+a for a
and transpose b and c if t he odd.
Thus, we shall have
xz^—
xz^^
{nt
-{-
-{-hz'
a) z
—{nt-Y a)
z
+
cz^
= (when
= haf (when
c-x"
t
is
even)
t
is
odd)
(4)
Either of these equations can be solved as in case
11
=
2 {nt-\-a), that
is,
when
—-
'
--
t.
.^'
(3),
(I.),
when
having been de-
termined by such a solution, the complete primitive of (B)
will be the continued fraction
y
=
--^ + —^— +...+
where h stands for
b or c
— Equation
3219 III71
2a
—= a
—
according as
(B)
+—
j:
t
...ioj,
odd or even.
is
can also be solved whenever
-\~
t
271
positive inte2:er.
°
^
The method and
result
mil be
the same as in Case II., if the sign of sl he changed throughout
and the first fraction omitted from the value ofj. Thus
=
y
—
2n — a
—— a h—
r— + -+^(t—l)n a + -—
^v
71
,
,
,
,
T
Proof.
— Case
— In
equation (B), substitute y
a
the absolute term to zero.
This gives ^
or 0.
,
II.
= A+ —
-
first
value, the transformed equation
and equate
^'
=
Taking the
,
(6).
becomes
x^-(n
+ a)y, + cy\=lx''.
dx
Next, put
equation
)/,
= n+a
(3) or (4)
is
r
^—,
and so
obtained with
,;;
on.
In this
written for the
way the
^"'
/"'
transformed
substituted variable
yt-
FIB ST OEBEB NON-LINE AB EQUATIONS.
475
—
Taking the second value, ^ = 0, the first transformed equafrom the above only in the sign of a and consequently the same
The
series of subsequent transformations arises, with —a in the place of a.
successive substitutions produce (5) and (6) in the respective cases for the
Case
III.
tion differs
values of
3220
;
y.
u, +
Ex.
Putting
^
-ii
= -^,
a =1,
-r-—
dx
X
and the equation
5
= 1,
7^
= cx-^.
i<.'
is
= 1,
and ^-^±^=2, Case
changing the sign of a
Solving as in Case
x^
xy^—y^if
reduced to
for
I.,
Case
III.,
we put
z
(3214)
>
2
= cxi
of the
III.
By
form (B).
Here
the rule in (3218),
equation (3) becomes
= vx^,
&c.
;
employing formula (1)
or,
directly,
,
=
ye.*
^^^^
;
and then, by
(6),
y
-^
=
3c
-
,
+
the final solution.
FIRST ORDER NON-LINEAR EQUATIONS.
3221
Type
where the
and y.
coefficients
P^Pzi
...
Pn i^ay be functions
of x
SOLUTION BY FACTORS.
3222
and
if
If (1)
can be resolved into n equations,
the complete primitives of these are
V,
= c„
V,
=
c,,
...
n=
c„
(3),
then the complete primitive of the original equation will be
(V-c)iV-e)
...
(F,-c)
=
(4).
——
.
DIFFERENTIAL EQUATIONS.
476
Proof.
eliminate
— Taking
The
c.
n
=
assume
3,
Differentiate
equation.
last
tlie
and
result is
(r,-r,y-
(v-v,y {v-v,rdv,dr,dv, =
(5).
=
Sub&c., where ft, is an integrating factor.
(2), dV^
f<i {ij^—i\)dx,
stitute these values in (5), rejecting the factors which do not contain differ-
By
ential coefficients,
which
is
and the result
is
the differential equation (1).
3223
Ex.— Given
+
yl
The component equations
are
^/,
+ 2 = 0.
+ l = and
^>j,,
7/,
+2 =
0,
giving for the
complete primitive
(y
3224
+ ,r-c)(y+2x-c)=0.
SOLUTION WITHOUT RESOLVING INTO FACTORS.
(.r, jy) = 0.
Class I.—Type
<l>
When
only is involved with jj, and it is easier to solve
the equation for x than fov ji, proceed as follows.
aj
=
Differentiate and eliminate dx
f (p).
Obtain x
liuLE.
pdx. Integrate and eliminate p hy means
hy means of dy
of the original equation.
Similarly, when y =f(p), eliminate dy, &c.
=
3225
x =
= adp-\- 2hp dp,
Ex.— Given
dx
ay.,
!/
jfj
between
=
this equation
(ax + 6hy - hcf
3226
Class
i.e.,
dy
therefore
therefore
Eliminating
+ hy'i,
=
o
^
—^
and
(6a7/
= ap + hf
= p dx = apdp + 2hp-dp,
x
r ^
(1),
•
(l),the result
- 4i'" - ac)
is
(a'
the complete primitive
+ -ihx)
11.-%;.
<<^
(;>)+.# 0^)
= x(/>)-
EuLE. Biff'erentiate and eliminate y if necessary.
grate and eliminate p by means of the original equation.
If the equation be
simphfied into
3227
first
divided by ^{p),
Inte-
the form
is
//-.*<^(/>)+x(/>)-
and an equation is obtained of the form
»r^,+ P.r = Q, wliere P and Q are functions of p.
This may be solved by (3210), and j) afterwards eliminated.
Differentiate,
FIEST ORDER NON-LINEAR EQUATIONS.
477
Otherwise, a differential equation may be formed
jj>, instead of between (c and |7.
3228
between y and
Or, more generally, a differential equation
3229
may be
formed between x or y and t, any proposed function of ^, after
which t must be eliminated to obtain the complete primitive.
3230
Glairaufs equation, which belongs to this class,
the form
Rule.
—
and two equations are obtained
Differentiate,
Rr = 0,
(1)
and
is
of
=pc€+f{p).
i/
.-.
p
=
e;
s^+f(p)
(2)
—
= 0.
Eliminate -p from the original equation hy means of (I), and
again hy means o/(2). The first elimination gives y
cx-f-f (c),
the complete primitive.
The second gives a singular solution.
=
—For,
Proof.
if
we have x+f{c)
Rule
= 0;
I.
=
(3169) be applied to the primitive y
ex +f (c),
c between these equations is the
and to eliminate
elimination directed above, c being merely written for
3231
This
is
in the
two equations.
y=px + x^/T+f.
Ex.1.
of the form
entiating,
p
=
?/
i>'(p
(p),
and therefore
xdp + dx\^l+p^-\
we obtain
'''
^'^
falls
^
^
=
under (3227).
Differ-
0,
\/(1+F)
since dy
^=^
pO.x; thus
(70^) + !^?)'''"+^"='''
in
which the variables are separated.
Integrating by (1928), and eliminating p^ we
primitive x^+y'^
3232
This
is
Ex.
=
find
for the
complete
Cx.
2.
y
=px+V¥^'p\
Clairant's form (3230).
^
I..
dx
The complete primitive
is
L
y
we have
DiflTerentiating,
^—
\/{lr
1
=
0.
d-p-)
= cx+ \/(h^— ctV)
;
and the elimination of p by the other equation gives for the singular solution
a^y' — b'x" = a-F', an hyperbola and the envelope of the lines obtained by
varying c in the complete primitive, which is the equation of a tangent.
3233
—
Ex. 3. To find a curve having the tangent intercepted between
the coordinate axes of constant length.
—
.
DIFFERENTIAL EQUATIONS.
478
The
differential
equation which expresses this property
is
= 42/=i-+-7=
(!)•
«./•
-/dx
crives
Differentiating "
^
p between
Eliminating
1st,
(1)
the primitive
;»;
j
H
L
7 [
=0
(2).
(l+/)t3
and (2) gives,
ij
=
—V
(3);
cx-\
2nd, the singular solution x^
+ y^
1
+c
= a?
(4)
the equation of a straight line (4) is the envelope of the lines
obtained by varying the parameter c in equation (3).
(3)
is
;
3234
Homogeneous
Class III.
Type
in
x and
j.
a^^<l>(^^,p^=.Q.
—
=
Solve for p, and
vx, and divide by x".
Rule. Put j
vx ; or solve for v, and
eliminate p by differentiating y
=
eliminate v by putting y
= ^;
and in
either case separate the
variables.
3235
y=px + xyi+p\
Ex.
and therefore p = v + xv^. This gives u = p + -/_! +i^*.
Eliminate p between the last two equations, and then separate the variables.
Substitute y
,
,
The
result
= vx,
2vdv
(IX
.
is
X
from which
a;
The same equation
C^'H^)
is
=
h T~r~^^
v
1
+
(7
or
_
—
a;-
r.
'^'
+ r = Cc
solved in (3131) in another way.
SOLUTION BY DIFFERENTIATION.
3236
To
—
solve an equation of the form
respectively to arbiRule. Equate the functions (p and
Differentiate each equation, and
trary constants a andh.
elimindte the CDiislanfs. If the results aqrce, there is a common
1//
EIGEEB OBDEB LINEAB EQUATIONS.
479
primitive (8166), which may he found by eliminating j^ hetiueen
b, and siibser[iiently eliminating one
a, ^
the equations ^
0.
of the constants hy means of the relation F (a, b)
=
=
=
^-yy.+f{f-fyl)
Ex.
=
Here the two equations x—yjj^.
on applying the test, are found
eHminating y^, we obtain
Hence the
solution
to
a+ h
by the given equation,
Also,
a,
=
^>
have a common primitive.
Therefore,
= 0.
f {y^ — {x + hf} =
is
= ^'
f (y^—y^yl;)
h.
HIGHER ORDER LINEAR EQUATIONS.
3237
Type
where Pi
...
P„ and Q are
—
Lemma.
which
of X,
case
^\-ill
be
g+p/J3 + ...+P„..,,.-^+/',,; = Q,
If y^,
?/2,
...
either functions of x or constants.
y,,
satisfy (3237),
=
t/
— Substitute
C^i/,
be n different values of y in terms
when Q
+
C.2ij.,-{-
= 0,
+
. . .
the solution in that
C„?/,,.
Multiply
in turn in the given equation.
the resulting equations by arbitrary constants, G^, C^, ... C« respectively;
original
in
the
add, and equate coefficients of F^, P^, ... Pn ""-ith those
equation.
Pkoof.
y^, y^,
... ?/„
LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS.
3238
3239
The
;A.v+«i?/(«-i):f+...+«(«-i)«/..+««3/=
Casel.—mien
(i)-
roots of the auxiliary equation
m*^+aim"-i+...+«.-im+«.
being ruj, hlj, ... m,„ the complete
equation will be
3240
Q
Q = 0.
p
=
(2)
primitive of the differential
Cie-^-^-+C,e--+...
If the auxiliary equation (2)
=
+ C\e-«-^-
(3).
has a pair of imaginary roots
.
DIFFERENTIAL EQUATIONS.
480
(a^ih),
terms
tliere
will
3241
be in
tlie
value of y the coiTespondiug
Ae"'' cos kv-\-B(f''
If any real root m' of equation
corresponding part of the value of
3242
A
(2) is
//
(4)
repeated
if
and
/•
times, the
will be
{A,+A,.v+A,a;'-^...+A,_,.v^-')
And
for
s'mkv
e^>'\
a pair of imaginary roots occurs r times, substitute
in (3241) similar polynomials of the r
V^^ degree
—
B
in X.
—
=
Pkoof.
(i.) Substituting y
Ce'"^' iu (1) as a particular solution, and
dividing by Ge'"^, the auxiliary equation is produced, the roots of whicli
furnish n particular solutions, y
Cje"'"^', y
C'oe'"^^, &c., and therefore, by
the preceding lemma, the general solution will be equation (2).
=
=
±
(ii.) The imaginary roots a
ib give rise to the terms
which, by the Exp. values (766), reduce to
(G+C) e"^ cos hx + i{G- C) e"^ sin
(iii.)
two equal roots m<^ = m^, put at
ae""'""'-^- become e'"-^' (C^i + G/'-O-
If there are
two terms (7ie"''^+
and put G^-\-G.j = A, CJi
By
=
Ce"^''''^-'''+ C'e"^''''^,
fe.c.
first
???.,
=
Expand
=
=
1n^-\-^l.
e*^
The
by (150),
=
B in the limit when k 0, G-^ cc, G^ —oo.
repeating this process, in the case of r equal roots, we arrive at the form
(A, + A,x
and similarly
3243
+ A,x' + ...+A,.,x'-')e"'^^;
in the case of repeated pairs of imaginary roots.
Case
Il.—When Q
First method.
Putting
in (3238)
—By variation
Q = 0,
as in Case
is
a function of x.
of parameters.
I., let
the complete primitive be
y = Aa-]-B(3 -h Cy + &c.
to n terms
(6),
The values of
y being functions of x of the form c"'-\
the parameters A, B, G, .,., when Q has its proper value
assigned, are determined by the
equations
a, j3,
;/-
3244
=
+B,.(3
+to n terms
yl,«,
+J9",i3,,.
4-
„
-0,
^%v
+y|,i3,,,
+
„
=
A^a
0,
0,
B,., &c. being found from these equations, their integrals
must be substituted in (6) to form the complete primitive.
A^,
EIGEEB OBBEB LINEAR EQUATIONS.
Proof.— Differentiate
tions of X
;
481
Gy&a. are func-
(6) on the hypothesis that A, B,
thus
;y,= (^a, + i?/3,+
...)
+ (^.a + B,/3+...).
—
1 relations may be assumed between
in addition to equation (1), n
Equate then the last term in brackets to zero,
the n arbitrary parameters.
and differentiate y, in all, n 1 times, equating to zero the second part of
Now,
—
each differentiation
;
we
thus
y^
=Aa^
+BI3^
y,^
=Aa,^
+B(3,^
obtain
= 0,
= 0,
+&C.
+&C.
and
Aj,a
+B^l3
-j-&c.
and
A^a^
+B,ft^
+&c.
y{n-l)x= ^«(«-ljx + -BA«-I)x+&C.
and
^x"(»-2)x
+ -5x/3(H-2)a;+&C. =
0.
—
1 equations on
&c. are now determined by the n
For, differentiating the value of ^(,j.i)xj we have
the right and equation (1).
The n
quantities
y,^
and
if
=
A^., B^.,
{Aa„^ + B(3„^ + &C.}
these values of ^^,
y>x, ... y,ix
^x«(»-l)x
for the other part vanishes
+
{^^a(,_i)x
+ -B,.A»-lia-+&C. =
...
y^,
it
reduces to
Q,
by the hypothetical equation
ynx+(hyin-i)x+---+ci>iy
since the values of
in this equation.
+ 5^/3(„_i).. + &c.},
be substituted in (1),
and the
y(H-i)x,
first
= o,
part of y^x: are the true values
—
Second ifef/iot?.— Differentiate and eliminate
11.
resulting equation can be solved as in Case I. Being
of a higher order, there will be additional constants which
may be ehminated by substituting the result in the given
3245
Case
The
Q.
equation.
3246
y,,-7y. + V2y
Ex.— Given
1st Method.— Pxxiimg x
therefore
m=3
and
4.
= 0,
Hence
y
is
The corrected values
of
A
and
=:^
x
(1).
the auxiliary equation is m^—7m + 12
the complete primitive o{tju—7y^+12y
=
B
= 0;
=
Ae'''+Be"'
(2).
for the primitive of equation (1) are
found
from
A,e"+ 5,6*^=0")
SA,e''+4iB,e''
.-.
A,^-xe-'^-
and
A=
B,=
and
B
= xy
Substituting these values of ^ and
y
tive
3247
2nd Method.
B
xe-'^
in (2),
we
=-^^
e-'^'+h.
find for its complete primi-
= ae\+le + -J^y,,-7y.+ 12y = x
S Q
^i^g-^^+a.
(1).
;
DIFFERENTIAL EQUATIONS.
482
Differentiating to eliminate the term on the
The
we
get
m = 4,
=
therefore
= Ae'^' + Be'^+Cx + D
= 4>Ae"' + dBe''+G', y,. = 16^e^ + 95e^
7
1
values in (1) thus C = —; D = y—
aux. equation
Therefore
I'iglit,
m'^—7m^ + l2m^
is
;
3, 0, 0,
(2)
y
7/,
Substitute these
;
r
;
therefore, substituting in (2), y
= Ae'^ + Be^+ ^ + ^44^^ before.
When
may
a particular integral of the linear equation
in the form ij =f{x), the complete primitive
be obtained by adding to y that value which it would
take
if
3248
(3238)
known
is
Q were
zero.
Thus, in Ex. (3247),
the complementary part
=
Ae^'^
+
?i^
iZ
+
7
TT7
144;
Be^'' is
^^
a particular integral of (1)
when
the value of y
the dexter
;
and
is zero.
of the linear equation (3238) may always
if a particular integral of the same
The order
3249
2/
be depressed by unity
equation,
Thus,
and
ii
y
of (1).
when Q
= 0,
=z
be known.
Q
y-i.+Piy-2.+P2y..+P3y=
if
be a particular solution when
Q
=
;
(!)>
let
y
= vz be the
Therefore, substituting in (1),
(z^,
+ P^z,, + P,z, + P,z)v + &c. =
Q,
the unwritten terms containing i\, v^x, and v^,-.
The coefficient of v vanishes, by hypothesis therefore,
we have an equation of the second order for determining w.
;
V
3250
solution
The
if
we put
it
v^
= u,
being found,
= yudx-\-G.
linear equation
where A,B^ ... L are constants, and Q is a function of x, is
solved by substituting a-\-hx = e\ changing the variable by
formula (1770), and in the complete primitive putting
t
= log {a + hv)
Otherwise,
(lA-bd-
= X^,
.
reduce to the form in
and solve as
in that article.
(3446)
by putting
HIGEEB OBBEB NON-LINEAR EQUATIONS.
488
HIGHEE ORDER NON-LINEAR EQUATIONS.
Ty,e
3251
K-....g.^.-g)-0SPECIAL FORMS.
3252
F{.v,
?/,,.,
?/(,.+a).^.
... ?/„,.)
=
0.
When tlie dependent variable y is absent, and 7/,.,. is the
derivative of lowest order present, the equation may be deIf the equation
z.
pressed to the order n—r by putting 7/,,,.
in z can be solved, the complete primitive will then be
=
y
3253
=
F{i/,
2
f
+
(2149).
f
?/,,, 7/(,^i), ... I/,,)
= 0.
be absent instead of y, change the independent variable from Q3 to y, and proceed as before.
Otherwise, change the independent variable to y, and
make j^ (= y^) the dependent variable.
If X
For example,
let
the equation be of the form
3254
(i.)
F(y,
This
may
7/„ y,^, 2/3.)
=
(1).
be changed into the form
F(y,
av, x,„ x,,)
=
by (1761,
and the order may then be depressed to the 2nd by (3252).
'63,
and
The
'66)
solution
will thus give x in terms of y.
3255
(ii.)
may be changed
Otherwise, equation (1)
F (y, p, py, p,^) = 0,
the form
at once into one of
by (1764 and
'61),
the order being here depressed from the 3rd to the 2nd. If the solution of
get, for the comthis equation be ^
(?/, c^, c.^), then, since dy —pdx, we
=
x=\
plete primitive of (1),
J
^-^
{y,
+C3.
Ci, ^2)
fc=i^W.
3256
Integrate n times successively, thus
jnx
—
DIFFERENTIAL EQUATIONS.
484
?hv=F{yy
3257
Multiply by
KchJ
2ij^^
and integrate,
^^^'
^-^^
J
3258
?/«x.
tlius
^^{2 F{y)dy + c,]
= i^{!/(«-i)x},
an equation between two successive derivatives.
Put y(n-i)x = ^j tben z^. = F{z), from which
«•
Hm+_^
If, after
that
z
=
integrating, this equation can be solved for
we have
(l>{x, c),
V(«_i)a-
=^
(a?,
^)5
which
falls
z
so
under
(3256).
But
3259
follows
cannot be expressed in terms of
if z
x^
proceed as
:
_
2/(n-3).-j^
dz
^
„
Fmally,
T^.
j,
=
f
J
r
dz
j^^;
<i;2
f
^^
j
f
_...
(^2;
dz
_.;
J
the number of integrations and arbitrary constants introduced
being n—1.
3260
?/«.
^(«_2)^ = ^;
= ^{y(«-2)4=
i^(/), which is (3257), the
then 2^2^.
Put
solution giving x in terms of z and two constants. If z can be
found from this in terms of x and the two constants, we get
Z
or
2/(„_2).^.
==<^(^',^l, ^2)5
which may be solved by (3256).
3261
But
if z
cannot be expressed in terms of
in (3259).
DEPRESSION OF ORDER BY UNITY.
3262
is
When
F
{.V, ;/, 7/,, ;/,,,
)
. .
.
=
rendered homogeneous by considering
a\ y, ?/,., ?/2,., ?/3,,., &c.
x,
proceed as
—
,
;
HIGHER OBBEB NON-LINEAB EQUATIONS.
to be of the respective dimensions 1, 1, 0,
= e\
,v
y
= ze\
—1, —2,
485
&c.
;
The transformed equation
will contain the same power of
every term, and will reduce to the form
F{z,ze,
the order of which
is
z.,,,
put
&c.
e
in
= 0,
...)
depressed by unity by putting
= u.
Zq
3263 When the original equation is of the 2nd order, the
transformed equation in u and z may be obtained at once by
into 1, z, if+^j u-\-uUg, respectively.
changing a\ y,
y.,^
The solution is then completed, as in example (3264).
?/.,.,
Peoof.
—We have
y^=
e,
=
e*
;
+ z,)
;
a?
+ z; y^^^
e"" (zo,
y =.
ys^
ze^
= e"-" (^39— «»)
;
and so
en.
The dimensions of x, ij, y^, &c. with respect to e^ are 1, 1, 0, —1, —2, &c.
Therefore the same power of e' will occur in every term of the liomogeneous
equation.
3264
x'y,,= {y-xy,)\
Ex.:
Making
the above substitutions for
^20
Put
Zg
=:
71^
ii;
—uuz,
t?ar^u=a—z
therefore dd
1/2^.,
the equation becomes
thus
+ u = — «« =
therefore
and
x, y, y^,
+ ^9 = — ^e-
=
cot
(a— z)
But
therefore
6
therefore
= log X
and
—),
or
^
=
—dz,
= « = tan (a— 2),
= — log & sin (a — 2) (1941).
z = -^,
therefore
(1935),
dz,
+ 1 = — «r,
tt^
z^
X
therefore
by
=
cosec la
= sec
bx
f
c
+—
|
altering the arbitrary constant.
3265
is
bx
When
F
{.V,
y,
?/,, ij,,,
)
. .
.
made homogeneous by considering
the respective dimensions
x
=
1,
?i,
e^,
When
y, y^,
n— 1, w— 2,
2/
and depress the order by putting
3266
=
cV,
=
Zq
y^^i,
&c.
;
&c. to be of
put
ze'"'^
= u,
the original equation
is of
as in (3262).
the 2nd order, the
DIFFERENTIAL EQUATIONS.
486
final
equation between u and
may be
z
obtained at once by
changing
.r, ;/,?/.,,
?/2.r
iiito
u+nz, m«^+(2n— 1) u+n{n — l)zy
1, z,
respectively.
3267
Ex.:
y2,y.
=
(l)-
^y
With the view of applying (3265), the assumed dimensions
member of this equation, being equated, give
n—2 + n—l =
Thus
x=:e';
Substituting in
tion
is
y
=
(1), e
y^
ze''',
disappears
e'' {z,
+ 4^z);
y,,=
and by putting
;
n
therefore
l-\-n,
=
= 4.
e'' {z,,
= u,
z,
of each
+
z.,
'?z,
+ l'lz).
= uu„
the equa-
reduced to
0tH4«z)cZtt+(7ttH40H2 + 482--2) dz
which is linear and of the 1st order. This equation is
by the rule in (3266).
3268
When
F(cl^
?y,
y,:,,
?/.,,
=
0,
also obtained at once
=
...)
\n(lx
(tec, put y — e'
homogeneous with respect to ?/, y,^,
and remove e as before by division. The equation between u
and X will have its order less by unity than the order of F.
•
1
j_
is
y.^,-,,
3269 Ex.:
P and Q being
Here
2/2.+P2/x+Q2/
functions of
?/
=
=
(1),
x.
eJ
;
y^
Substituting, the equation becomes
If the solution gives
1st order.
,
= uy;
?t^
u
— {u^ +
+ «^ + P«+Q =
=
y.^^
(p(x,c),
then
ti
) y.
0,
|
^
an equation of the
(re,
c) J.«
= log
?/
is
the complete primitive of (1).
EXACT DIFFERENTIAL EQUATIONS.
3270
Let
dU=<l>
{x, y,
y„
y,,,
. .
.
?/,,,)
dx
=
be an exact differential equation of the n*'' order.
derivative involved will be of the 1st degree.
—
The highest
Rule for the Solution {Sarrus). Integrate the term
y,,^ nnt]b resjyect to J(n-i)x only, and call the result Ui.
Find dUi, considering both x and y as variables. dU — dUi
I*''
order.
ivill be an exact differential of the n —
3271
Inculcing
MISCELLANEOUS METHODS.
487
Integrate this with respect to J{n-2)x only, calling the
U2,
The first integral of the proposed equation
3272
Ex.
:
Here
.'.
TJ.i
= —xy„
the
ivill
dU =
he
= 0.
{y' + (2x7j-l) yl + xy2. + ^y3x}
= x%^, dU^= (2xyi^ + x'ys^ dx
dU-dU^ = {7/+ (2xy — l) y^-xyo^) dx = 0;
dU2 = —{y;^+xy2x)dx, dU—dUi—dU2—(y^ + 2xyy^T.)dx',
= xy^, and JJ = xSjix — xy^ + xr/ = C
Let
d.v
U^
tlierefore
is
resiilt
so on.
a?i(i
;
JJs
first integral.
Denoting equation (3270) by clJJ — Vdx, the series of
and amount to the single condi-
3273
steps in Rule (3171) involve
tion that the equation
N-P,-{-Q,,-R,,-\-&G.=
0,
with the notation in (3028), shall be identically true. This
then is the condition that V shall be an exact 1st differential.
3274
V shall
Similarly, the condition that
be an exact 2nd
differential is
P-2Q,+3i?,,-483,+&c.
The condition that
3275
and so
be an exact 3rd differential
^ s^-^-j^
g-3i?,+
is
V shall
on.
= 0.
= 0,
T3.+&C.
[Euler,
Comm.
Petrop., Vol.
viii.
MISCELLANEOUS METHODS.
3276
where
ih.-\-Pi/.+
P
The
Proof.
and Q are functions
solution
— Put
'^y:r+Qzijl^O.
e^
Put
y
"'
=
zy,
R
(1),
= ^e'^'''-' {2^Qe~'^'^''\b^-hIx.
z,
and multiply (1) by
.-.im^-Qz--, .-.
= u-\
=
3277
where Q and
is
Qtjl=0
of x only.
z/2.+ Qz/i+^
are functions of y only.
2;
71
then, since
=
z^
= Pz,
V(2 ^ Qz-dx),
&c.
(1),
DIFFEBENTIAL EQUATIONS.
488
The
solution
Proof.
—Put
= lJ'"(2\Be'^'""dy)-hl:,.
x
is
=
"
e^
and multiply (1) by
2,
3278
ih^'+Pi/..-+Ql/'^
= ^>
where P, Q involve x only.
= z, and the form (3211)
Put
is
y^.,
3279
z.
y-2.+P!/l-{-Qi/:
arrived
at.
= 0-
This reduces to the last case by changing the variable from x
to
7/
by (1763).
3280
For a few cases in which the equation
y.+Py' + Qy+B
De Morgan's "
can be integrated, see
=
Differential
and Integral
Calculus," p. 690.
3281
2/2.
Put ax + hy
=
t
=
t,
= ci.v+hij.
Eesult U,
(1762-3).
=
Solve by (3239)
bt.
or (3257).
(1 -^^') ih-^^'y.+q'y
and obtain y-zt + c/y =
3282
Put
sin~^rtf
Solution,
y
Put
(l
,''
I
+
—^=
«ct-'0
t,
i^{l+ax')
3284
0.
= A cos {q sin~^ x) + B sin (q sin~^ x).
3283
'*'
= 0.
= 0.
y.H±fv = ^
yz,-\-axy,±Ti)
and obtain
LlouviUo's equation,
^^ above.
yi.v-\-f{x)y^-\-F(y)yl
=
{).
Suppress the last term. Obtain a first integral by (3209), and
The complete primitive is
vary the parameter.
3285
Jacohrs thcuron.
equation
?/..,
=/0r,
y)
—
If
is
one of the
?/.,
=
first
<^ (.r,
integrals of the
y, c)
(i., ii.).
—
—
APPROXIMATE SOLUTION.
tlie
489
complete primitive will be
— Differentiating
obtain <p:,-\-<p<p,j =/(«, !/), and differenBut, by (3i87), this is the condition
shall be an exact differential therefore fc
(i>,<pdx
^^
an integrating factor for equation (ii.), l/z—f (j^, V, c)
Proof.
tiating this for c,
for ensuring that
is
f:,c
+
(t>c<py
(p,d}j —
(ii.),
+
(p<p,,c
=
we
=
0.
;
=
Equations involving the arc
3286
3287
-\-yl =
a-^-hy^,..
Find
3288
y^^
=
ci'2,
Change from
or
or
1+7/2
^"
to
s
<«
= \/l-{-yl.
s^
= cLv-\-btj.
s
Here \/l
having given
s,
= (M + chf
ds'
by (1763)
= —i
a.
/.
;
.
from the quadratic equation.
^.
—
^r
^
2ax-\-c'
s'^s.^^.
= a,
.'.
1
/(
s~^
= 2ax-\-Ci
l)clx+i
]\\2ax-\-c
)
APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS
BY TAYLOR'S THEOREM.
3289
The following example
Given
GeneraUj,
y,,
will illustrate the
= xy,-]ry,
= A,,y^,+Bjj
.'.
let y,,.
y-s.,
;
method
:
= {x'-]-2) y^,+xy.
= A,^,y^,-\rB,,+,y.
//(„+i).,
But, by differentiation,
+ {A,,+B:,) y,
= A,, + B'^.
A,= x' + 2, B, = x;
B, = x' + S, &c.
y(H+i).«= {A,,x-{-A',,+B,,) y^
.'
But
.
^,,+1
= A,,x-\- yi; + B,,
A,=
x,
B,=
l,
= x'-{-bx,
=
x
a, let y = b
A,
Now, when
.'.
and ^„+i
and
y^
=
ir,
then,
by Taylor's
theorem (1500),
y
= a-}-p {x-a) + (A,2Ji-BJj) (l^ + iA,p+B,h)^-^^^
+ &C.,
which converges when
x—a
is
small.
3 R
[De Morgan,
p. 692.
;
DIFFERENTIAL EQUATIONS.
SINGULAR SOLUTIONS OF HIGHER ORDER
EQUATIONS.
DERIVATION FROM THE COMPLETE PRIMITIVE.
Let
3301
v/,,,
=
•
<i>{'>',y,y.;y2.
be the differential equation, and
^7=/(»''«j^'^
0)
^(^-dJ
complete primitive be
let its
(^)'
^)
•••
containing n arbitrary constants.
3302
Rule.
eliminate abc
y
=U
— Tu find
...
the general singular
between the equations
s
jx
=
and
fx>
is
fax
•
•
•
y(n-i)
J
tax
Ia2x
•••
t;i(n-l)x
tlj
Ibx
tb2x
•••
lb
Is
The result
=
y-ix
til
ls2x
Isx
•••
(11-1
(11-1) X
-
suliitloii
of (1),
= f (n-l)x
VV
=
Is(n-l):
a differential equation of the n
the integral of it, containing
sing^Uar solution.
(4).
n—1
— V^'
—
Proof. Differentiate (2), cousidering the parameters
thus y^
fx+fa 't.r + •• +/,Sx. Therefore, as in (3171),
=
and
a, b
...
s
variable,
•
2/2X
= />.
fax a. +fi,x b.v +...J\xS.
if
and so on up to y„^ = f„x- Eliminating
on the right, the detei-minaut equation
columns interchanged.
3303
order,
arhitrarij constants, is the
Ex.
a^, h^,
(4) is
= 0,
6\
y-^y^ + ^^y^^-yl- (y^-Xy,^'
:
The complete primitive
is
y
=
From which
-j- +h,c
+
c(r
+
h'^
y^.=:ax + h
and the determinant equation
pn2a,
X +2Z/,
as well
between the n equations
produced with the rows and
...
=
.(2).
.(3),
is
x\
1
I
^
or
+ 2Z^.v
(4).
;
SINGULAR SOLUTIONS OF HIGEEB OUDBB EQUATIONS. 491
Eliminating a and
from
b
and
(2), (3),
M!, + (2x + f)cU
the integral of
-whicli,
(4),
we
get the differential equation
^ ^
,
and the singular solution
^/(iey + 4^x' + x')
=
a;v/(l
^
of (1), is
+ rc*)+log{^+ y(l + x')} + G.
[Boole, Sup., p. 49.
3304
Either of the two * first integrals' (3064) of a second
order differential equation leads to the same singular solution
of that equation.
The complete primitive of a singular first integral of
a differential equation of the second order is itself a singular
solution of that equation but a singular solution of a singular
first integral is not generally a solution of the original equa-
3305
;
tion.
Thus the singular
is
integral (5) of equation (1) in the
first
example has the singular solution
last
not a solution of equation
16y-\-4<x^-{-x'^
= 0i which
(1).
DERIVATION OF THE SINGULAR SOLUTION FROM THE
DIFFERENTIAL EQUATION.
—
3306 Rule. Assuming the same form (3173), a singular
solution of the first order of a differential ecptation of the n^^
LYnx;
order will make
d
second order
and
infinite
;
a singular solution of the
(y(n-l)x)
mahe
ivill
^-^"^^
^
,
^•^"'^-^
, ,
^
,
,
d(y ,,_,),)' d(y („.,),)
so on.
3307
Ex.
infinite
-^
[Boole, Sup., p. 51.
— Taking the
differential equation
(3303) again,
y-^y^+i^%z-yl- (y^-xy.^y
and
hoth
differentiating for y^
{^x' + 2x
The condition
and
?/2j,
=
(y.-xy.J-2y.J d(y,;)-{x + 2(y^-xy^)} d(y,)
,;
/
=^
^x'
(1),
only,
= 0.
requires
+ 2x {y..-xy,,)-2y,, =
0.
Substituting the value of yo^^ obtained from this in equation (1), and rejecting
the factor (;^- + l), the same singular integral as before is produced (3303,
equation 5).
DIFFERENTIAL EQUATIONS.
493
EQUATIONS WITH MORE THAN TWO VARIABLES.
Pd.v+Qdii-^Rdz
3320
=
(1).
{S
P, Q, B being here functions of x, y, 2, the condition that this
equation may be an exact differential of a single complete
primitive
is
P (Q-R.) + Q {R-P^R (P-Q.) = 0.
3321
—
=
Then
0.
Proof. Let f^ be an integrating factor of Pdx + Qcly + Bdz
—fiPcIx
ijQdi/ + fiIlc1z, and, by (3187), for an exact differential, we must
Write this symmetrically for P, Q, and B, differentiate
have (i^Q)~
(At-R)v
out, and add the three equations after multiplying them respectively by P,
Q, and B.
=
To
3322
=
find the single complete primitive of equation (1),
Rule.
— Consider one
of the variables z constant, and
Integrate, and add
(z) for the constant of
= 0.
therefore dz
integration.
(j>
Then
and
differentiate for x, y,
z,
and compare
with the given equation (1). If a primitive exists, ^(z) will be
determined in terms of z only by means of preceding equations.
The complete
system
primitive so obtained
of surfaces, all of the
same
is
the equation of a
species, varying in position
according to the value assigned to the arbitrary constant.
3323
is satisfied
;
M,,
now
for
.t,
(x-Sy- z)
Hence the
dx + (2
= — 3 = iV„
^x"'
Difierentiating
//
-
and
3.1')
(z)
(l>
with
{
f (,v) - x
f'(z)=z,
(1),
single complete primitive
x'
= 0.
z,
dx +(2y- Sx) dy +
coefficients
dy
(1).
we have
and integration gives
- Sxy—zx + y- +
y,
= 0,
= 0.
therefore, putting dz
(x -Sy-z)
Applying (3187),
Equating
=
(x-Sy-z)dx + (2y-dx)dy + (z-x)dz
Ex.:
Condition (3321)
}
dz
= 0.
therefore
(f>(z)
=
^z-+C.
is
+ 2y^ + z- — 6xy — 2zx =
C,
the equation of a system of surfaces obtained by varying the constant G.
3324
When
the equation Pdx-\-Qdy-\-Rdz
geneous, put x
=
of dz vanishes,
is
3325
=
vz.
y
of the form
2iz,
The
Mdu-\-Ndv
result,
=
i),
=
when the
is
homo-
coefficient
:
EQUATIONS WITH MORE THAN TWO VARIABLES.
solvable
by (3184).
Otherwise
it is
493
form
of the
~ = Mdu-{-Ndv,
3336
z
and the right
be an exact differential
will
if
a complete primi-
tive exists.
3327
Ex.:
y%ilx^-zxdij-^',mjih=^^
Condition (3321)
X
=
is satisfied.
y
%{jZ,
= vz,
(1) becomes
z
and the solution
When
= tidz-\--4 du,
+ f^ + f = 0,
dx
^Zf
=
= vdz + z dv,
dy
6v
oil
log {zu'v^)
is
(1).
Patting
G
or
xyz
=
G.
the equation
Pdxi-Qdy + B(h
=
(1)
has no single primitive
3328
and
Rule.
—Asswne
differentiate
;
(p{x,
y,z)
=
(2)
thus
(^,dxH-<^ydy
+ .^,dz==:0
(3).
The form of <p being given, eliminate z and dz from
The result, being of the form
(2) and (3).
Mdx + Ndy =
(1) hy
0,
can be integrated, and the solution taken lolth (2) co7istitutes a
solution of equation (1), and represents a system of lines {by
varying the constant of integration) drawn on the surface
^ (x, y, z)
3329
= 0.
Ex.:
{l
+ 2m)xdx+(l-x)ydy-\-zdz =
0.
=
condition (3321) not being satisfied, assume «- + t/^ + z^
r^ as the
function <p, therefore xdx + ydy + zdz
0; and by eliminating z and dz,
4imx =. G, a cylindrical
^mdx ydy
0, the integration of which gives y^
surface intersecting the spherical surface in a system of curves (by varying
G), whose projections on the plane of xy are parabolas.
Tiie
=
—
=
—
The condition that
3330
where X, Y, Z,
Xd.v-{-
T
Ydi/+Zdz+rdt
are functions of x^ y,
= 0,
z,
t,
may
be an exact
,
niFFJEIiENTIAL EQUATIONS.
494
differential, may
be sliewn, in a manner similar to
to be expressed
by any three
tliat of
(3321)
of the equations
Y{Z,-T,)+Z{T,-Y,) + T{Y,-Z^)=0,
3331
= o,
T {X,- r,) +x(r, - r, ) + y{t,, -x,)
X(Y,-Z,) + Y{Z,,,-X,)+Z{X^-Y,.)
= 0,
the fourth being always deducible from the other equations.
If this condition is fulfilled, the solution of equation (3330)
is analogous to (3322).
if z and t were constant, and therefore dz and
adding for the constant of integration ^ {z, t).
hy
Differentiate next for all the variables, and determine
comparison with the original equation.
Integrate as
(It
zero,
(jt
3332 If a single primitive does not exist, the solution must
be expressed by simultaneous equations in a manner similar
to that of (3328).
SIMULTANEOUS EQUATIONS WITH ONE
INDEPENDENT YARIABLE.
GENERAL THEORY.
3340
Let the
first of
n equations between n + 1 variables be
Pdd+P.du+P.ih-]- ...+P,,dw =
Pi ... P« may bo functions of all the variables.
(1),
where P,
Let X be the independent variable. The solution depends
upon a single differential equation of the n^^^ order between
two variables.
Solving the n equations for the ratios dv dy dz &c., let
:
:
dx
.
"
dy
dx
dz
d.ii
_Q^
Q
dz_
'
dx
_
~
(/?/'
Q2.
Q'
^_
Differentiate the first of these equations ii
from the others the values of r;,. ...
tutino:
:
dx
—
Oil
Q'
\
times, substi-
ii\,,
and the result
SIMULTANEOUS EQUATIONS.
is
u
33,
?/,
equatioDS in
2 ...
,f/^,,
t/^.i-
•••
495
and the primitive variables
1J,lv>
W.
Eliminate
all
the variables but x and y, and let the differ-
ential equation obtained
be
F{x,y,y_,..,y,,) == 0.
of this, each of the form
substitute in them the values of
•••
ill terms of x, y, z ... w, found by solving the n
2/.1-) ?/2.rj
i/».i-5
Thus a system of n primitives
equations last mentioned.
G.
is obtained, each of the form F{x,, y,z ... lu)
the n
Find
F{x,
y,
?/,.
...
first
y(n-i)x)
integrals
= G, and
=
The same in the case of three variables.
Here n = 2. Let the given equations be
3341
P.,(Lv-\-Q,di/+R,dz
Therefore
3342
From
these let
Therefore
q^^/^^q^^^
?/^,
=
(|)
t/o.,
= B,pf-P.,B, = P^-F^^^
{x, y, z),
=
«^.,
+
= 0.
z,,
y,,
<l>,
+
=
1>z
^p
{x,
y,c).
-.v
Substitute the value of ^.,, and eliminate - by means of
An equation of the 2nd order in x, //.„
y^,
<p (x, y, z).
is
the result.
Let the complete primitive of this be
=
=
=
Then we also have (p (x, y, ^)
y
X ('«j (h ^)These two equations form the complete solution.
FIRST
//o,,.
_?/,
(/.,x
{>''>
«?
^^)-
ORDER LINEAR SIMULTANEOUS EQUATIONS WITH
CONSTANT COEFFICIENTS.
3343 In equations of this class, the coefficients of the
dependent variables are constants, but any function of the
independent variable may exist in a separate term.
Such equations may be solved by the method of (o34U),
but more practically by indeterminate multipliers.
3344
Ex. (1):
I +7.-2/ =
Multiply the second equation by
m
JL+2x+5y =
0,
and add.
;^>a.) +(„„ + ,)
(..+
The
result
0.
may be
|1=-1,|=0
written
(.).
.
DIFFERENTIAL EQUATIONS.
496
To mako
5*y;-l
the whole expression an exact diftercntial, put :^-—-=
^,
gives
and the solution
(^
"^
now becomes
(1)
= ce-
is
This
^n'==J^
(2);
=
+ ^»?/) + (2m + 7)ix + my)
x + my
= ^»-
-'"^'''^
0,
=
x + vi'y
and
c'e--'"'*"'-.
Solving these equations, and substituting the values (2),
e-'''\{c-c')cost-i(c + c) sin^j,
iy =zce-^'''^^-c'e'^'''^^
^.
=
= .--|(^ + ^^^)cos^+f^-^^)sinfl,
1
or
7/
"
2
\
3345
Ex. 2
a-,
:
+
5.7;
+
/
2
'
rre-^'CC^cosi-C^'sinf),
=
2/
Multiply the second equation by
e*,
vi,
cl(x±my)_^^._^^^^
"*"
Put
2^
+
5
(5
—m
_ „i)
]vfoTE.
of
2
(
^y
^
+ Sy -
a;
cost +
_^_^
l±3m
(0-0')
sin t}.
= e".
and add to the
|
first
^
^.^^^^^.._
= thus determining two values of vi, and
2 =
+ «ie-^ This is of the form (3210)
iji,
put x-\-m.y
=z;
thus
e'
— The equations of
example, written in the symmetrical form
this
_
dx
(3342), would be
e^
3346
V
= le-'mC+C)
•'«
— hx — y
dy
e-*
+ x — ^y
_ dt.
General solution hy indeterminate
dx
Let
__
dy_
_
rnultijjliers.
dz
be given with
=
^2
Assume a
third variable
dt
T
The last
wc have
r/, /
t
a.,x-^h.y-^c.^z^d.2.
and indeterminate multipliers
Idx + mdy+ndz _
_
""
ZPi
fraction
is
+ (/.,(;;+((..(
/t
+ mP.^ + rtPs
an exact
=
A/,
+ n = Xin,
+
+ m + Cj = X»,
Ci
dil + ditn + d^n = A?-,
b^
I
b.^
I
c'a
III.
h-^
Jt
dift'ereiitial,
:
J,
m,
)i
suc'h that
+ ndz
~\{lv-\- my + nz + r)
Jdx-\-m dy
(/j
—A
^1
and, ia determine
a.
a^
b,-K
b.
(1).
\,
/,
)/?,
«, r,
.
.
SIMULTANEOUS EQUATIONS.
497
The detei'minant is the eliminant of the first three equations in I, m, n.
The roots of this cubic in \ furnish three sets of values of I, in, n, r, which,
being substituted in the integral of (1), give rise to three equations involving
three arbitrary constants thus,
;
c^t
Eliminating
t,
we
arbitrary constants.
similar solution
A
variables.
3347
To
Assume
ji
{l^x
+ m^y -f n^z + r^) ^s.
may be
when
obtained
more than three
there are
^^^
-\-
by
+ c,
=
P^
= ai,-\-h] + cZ,
^j
aj^
=
Pi
the solution of which is
these equations become
= a^c + &iy +
=
—=
and take
P-2
known by
Substitute
^ ydi: + (dy ^
cU_
P2
P'
i,
=
xi^,
n
= y^,
and
—=—
=
p.-yp
Dividing numerators and denominators by
solution
&c.,
(2),
Pi
p,-xjp
&c
Ci ,
Ci<^,
xd^ + ^dx
and therefore
its
+ tj?? +
= &e. ...(1),
—P
(3346).
'—
duced, and therefore
of (2) into x^ and y^.
two equations involving two
=
solve
P == ax
where
=
find for the solution
]J
first equation in (1) is proobtained by changing ^, rj in the solution
is
C,
the
Certain simultaneous equations in which the coefficients
are not constants may be solved by the method of multiphers.
Thus,
3348
Ex.
Xt-^P{ax + ly)
(1):
P, Q, B being functions of
as in (3344),
determine
m
^ + my
t.
=
Q,
y^
+ P (ax + h'y) =
E,
Multiply the second equation by
The solution is obtained from
= e-'""""'^''|a+f
e^'^""''''-(^''(Q
in,
+ .;iE) dt\,
add,
and
(3210)
with two values of m.
334:9
Ex.(2):
x,+ ^{x-y)
=
y,+
1,
]
(x
+ 5y) =
t
are equations solvable in a similar mannei*, and the results
ai-e
[Boole, p. 307.
3
s
———
—
'
DIFFERENTIAL EQUATIONS.
498
REDUCTION OF ORDER IN SIMULTANEOUS EQUATIONS.
3350
TiiEOKEM.
n simultaneous equations of any orders
between n dependent variables and 1 independent variable are
reducible to a system of equations of the first order by sub-
new
stituting a
highest.
variable
every derivative except the
for
3351 The number of equations and dependent variables in
the transformed system will be equal to the sum of the indices
of order of the highest derivatives.
This will, therefore, in
general be the number of constants introduced in integrating
those equations.
If, after integrating, all the new variables
be eliminated, there will remain n equations in the original
variables and the above-named constants.
These equations
form the complete solution.
In practice, such reduction is unnecessary. The following
are methods frequently adopted.
:
Rule I. Differentiate until by elimination of a variand its derivatives an equation of a higher order in one
3352
able
dependent variable only
3353
Rule
3354
Ex. (1):
By Rule
I.,
is
obtained.
Employ indeterminate
II.
Xit
=
ax + by,
differentiating twice for
^'4i:~
(^ +
&') *2!;+
which may be solved by (3239).
Otherwise by Rule II., exactly as
y2t= a'x + b'y.
t and eliminating y and
{ah' — ah) x = 0,
in (3344),
am" + (a — h')m—h
and
=
we
y.,t,
we
obtain
find
0,
for the exact differential
= (a + ma) (x-\-viy),
(_x-\-iny).^t
the solution of which, by (3239),
X
in duplicate with the
3355
Ex. (2):
Diilerentiate,
+ my
=
is
Cye
two values
^'^^
solve by (3239).
y, y^, y^t
+ C^e
"
^^^
^ '"«'' *
(2a:-
y2t
+ 2a.Vt + hy =
0.
thus
',
+ h)x2t+b-x =
0,
Otherwise assume
;<;=:£ cos at
+ V sin at,
and the given equations reduce to
^,, = -{a' + h)l,
which
»'«'' *
of m.
x.i-2ayt-\-hx=i0,
and eliminate
Xu + 2
and
multipliers.
are solved in (3257).
y '=
i)
(^^s at
— l sin at,
^„,= _(aH?')r;,
[Boole, p. 311.
PARTIAL DIFFERENTIAL EQUATIONS.
—
3356
=
499
=
be three equations in x, y, z, t,
Ex. (3). Let u
0, V =z 0, iv
involving derivatives of t up to x^t, ijco ^uTo obtain an equation between x and t. Differentiate each equation.
6+7 13 times, producing 3 + 13 X 3 42 equations involving derivatives
Between these 42 equations eliminate ?/, ?/<, ... 3/19^,
of t up to aJie;, 2/i9i, Zos,f
order in x and t
2!, 2!^, ... 020^5 in all 41 quantities, and an equation of the 16th
[De Morgan.
is the result.
=
=
3357
If a
&c.,
x.if,
may be
iff,
number
y5t,
i/sf,
of equations involve the quantities x, X2t,
&c., all in the first degree, these quantities
eliminated by assuming
X
= L ^mpt,
y
=
M
cos, pt.
If there be n linear homogeneous equations in n variables x,y,z, ... and their derivatives of the 2nd order only,
3358
the equations
cV
3359
may be
= L sin
Ex.
Putting
:
solved by putting
y
j)t,
=
Msiii^^,
z
= N smpt,
= ax + hy,
y.^ — gx +fy.
= L sin
y = Msinpt,
(a+p') L + hM=01
.1 a+p\ b
_
9L+(f+p')M = 0y --Ig, fi-f\
Xot
X
2yt,
I
p and
the ratios
L
:
Suppose
X
then
and
Jc
and
t
&c.
q
'
M are thus found.
L = —Ich and M = l-(p^-\-a),
= —Jcb sin pt,
y = ]c(p^ + a)
sinj^iJ,
are arbitrary constants.
PARTIAL DIFFERENTIAL EQUATIONS.
3380 An equation is termed a general primitive or a complete primitive of a partial differential equation, according as
the latter is obtained from it by eliminating arbitrary functions
or arbitrary constants, as illustrated in (3150-7).
LINEAR FIRST ORDER
3381
u-=
<\>
To form
(?'),
the
P.
P. D.
EQUATIONS.
D. equation from the primitive
where u and v are functions of
x, y, z.
;
DIFFER EN TIAL EQUATIONS.
500
—
Rule. Differentiate for x and j in turn^ and eliminate
See (3054).
b; thus
a, v
Otherwise.
Differentiate the equations u
^'(v).
—
=
-^
Therefore
= -^ = ^,
=
= 0,
= 0.
Uidx-hUydy+Ugdz
Vj^dx+Vydy+v^dz
P
where
=
^x.
^|'^Jj^,
Tlien the P. D. equation loill he
Proof.
— Since
= kQ, dz =
dij
3382
Ex.
/
point (a,
.
—The general
V
7
,
0,
2 is a function of x and
kB, therefore IPz^ + lQzy
+ Zydy = dz.
z^dx
y,
= hB.
But dx = hP,
equation of a conical surface drawn through the
II
.
'^
c) IS
x
—h
—a =
^
(t>
— c\
/ z
,
\x—al
the form of ^ being arbitrary.
Considering z as a function of two independent variables x and y, differentiate for X and y in turn, and eliminate <p' as in (3154).
The result is the
partial differential equation
{x—a)
To
3383
B
+ (y—b)Zj,-]-z—c =
Pzj.-\-Qzy
— Solve
is,
to solve
= E,
being either functions of
Rule.
0.
obtain the complete primitive; that
the P. D. equation,
P, Q,
z^
x,
//,
z
or constants.
the equations
dx
"P
_ dy _ dz
~ Q ~U'
u
Let the two integrals obtained he
u
then
=z
<l>
= a,
v
=b
(i^)
ivill he the comiilete jprimitive.
Propositions (3381) and (3383) extended to any
number
of variables.
To form
3384
the partial diiferential equation from the
primitive
where
ables
?6,
a.',
^
v, ...
i/i
...
z
w
are
(«,
i?,
...
iv)
=
n given functions
and one dependent
t.
(1),
of n independent vari-
—
PARTIAL VIFFEBENTIAL EQUATIONS.
501
Differentiate for all the variables thus,
KuLE.
<^,du
+ (/),dv+
...
=
+<p,A^
is arbitrary, du,dv...dw
Therefore, since
vanish, giving rise to the n equations
(2).
must separately
(f)
du
dv
= u^dx + Uy dy 4= Vx dx + Vy dy +
dw = Wjdx + Wydy +
.
. .
.
.
.
. . .
+ Ut dt = 0,
+ Vt dt = 0,
+ Wtdt =
Solving these for the ratios, by (583),
^_4L-
0.
tve get
il^ii
(3),
Q ... R, S being functions of the variables or else constants.
Noiv, t being a function of all the rest,
P,
t^dx
therefore,
required
by
(3)
+ tydy+...+tzdz = dt
and
(4), the partial
(4),
differential
equation
is
Pt,-^Q%-^...^-Rt,
3385
=
S,
3386 If ^^ V ...whQ n functions of n variables, x, y ... t, the
condition of interdependence of the functions or existence of
some relation expressed by equation (1) is J{u, v ... w)
(see 1606) ; that is, the eliminant of equations (2) must vanish.
=
Conversely, to integrate the partial differential equa-
3387
p^,+g^,+ ...+iiL= s
tion
Rule.
— Solve
(1).
the system of ordinary equations
| = |=.c. = J = |
and
then
let
=
m,
=
w=k
b, ...
v
a,
the integrals obtained &e u
tvill be the complete primitive.
... w)
^ (u, V,
;
=
the
I/" P, Q ... R, S are linear functions of the variables,
integrals of equations (2) can always be found by the method of
(3346).
:
,
DIFFERENTIAL EQUATIONS.
502
—
Suppose, in equation (1), tliat any coefficients P, Q
Note.
0, dij
0, and therefore the corvanish; then, by (2), dx
The complete primih.
a, y
responding integrals are x
tive thus becomes
0.
(\>{x, y, u, V ... w)
=
=
=
=
=
3389 When only one independent variable occurs in the
derivatives of the partial differential equation, the equation
may be integrated as though the others were constant, adding
functions of the remaining variables for the constants of
integration.
~~ —
^
3390
(^^- 1)
stant, the
complete primitive
•
dx
^y^ — x^
2
Integrating for x as though y were con-
is
= ?/sin-^— +0(2/).
y
Some
equations are reducible to the above class by a transformation.
Thus
3391
Ex. (2):
therefore
ii„
K,,
= x^ +
z
or
j
?4
x^'y
(p
(^)
>
\P
= c-z.
(:x-a)z^+(y-h)z_„
Ex. (3):
-^
= "^ =
x-a y—h
Solving by (3283),
^
^
^'
•'
The
z, = «,
= ^^ = + i^/* +
Put
therefore
= ^x'^y + ^xy'^ + (x) dx + (y)
z = \ (a;V + xy"") +x(^) + 'P(y)'
therefore
3392
= x' + y\
if,
——
z
c
integrals are
log(2/-Z>)-log(a;-a)
\og(z —
therefore
= loga
)
lLl±
^^,
x—a
c)—\og(x—a)=logC'j
^~ =
x—a
^~'^
o
(
)
\x—a/
For the converse process
is
in respect of the
x(j)
(x,
^^^=C',
x—a
same equation, see (3382).
Ex. (4). To find the surface which cuts
spheres whose equations (varying a) are
Let
'
the complete primitive.
—
3393
= c,
+ y' + ^'-2ax =
y,z) =() be the surface.
all
the
(1).
Then
(.f-a)0^ + //f„+~0,-
by the condition of normals at right angles.
(1), and divide by 0-; thus,
{x^-y''-z'')z,
orthogonally
=O
Substitute the value of a from
+ 1xyz^ =
2zx.
—
PARTIAL BWFEEENTIAL EQUATIONS.
_
dx
By
(3383),
^ = —X
^=c
z
=
gives
y
Substituting y
_
dy
603
dz_
for one integral.
cz,
we then have
_
dx
dz
x'-{c' + l)z'~2zx
which, being a homogeneous equation in x and
= vx
z
plete primitive
3394:
Ex.
is
(5).
^ "^^
— To
~'"~'
=^ ^
dx+(x7f-2x') dy
=
(1).
z for
(xf-2x')z,+ (2f-xhj)z,
of ordinary equations (3283)
dx
xy^—2x^
The
first
_
~~
dy
2y^—xhj
of these equations
is
_
~
-y
-f
ydx + xdy
Al=o
^^'°
xy'-2xSj + 2xy'-xSj
which reduces
to
^
1
x
-f
=
(Nz)^ (3087) pro-
= 9(x^-y-^)z
(2).
is
identical
—^
Its integral is
always occurs).
Hence the com-
J
that factor, the condition (Mz),
duces the P. D. equation
The system
G.
an integrating factor of the equation
(a;hj-2y*)
Assuming
be solved by putting
=
and the equation of the surface sought.
f
find
may
z,
'
'^
Tlie resulting integral is
(3186).
dz
9 {x^—y^) z
with
=
^
—=
(1)
(and such an agreement
c.
dz
9 (a^-y')
z'
;
z
y
and thus the second integral is x^y^z = c.
Hence the complete primitive and integrating
factor is
Any linear P. D. equation may be written as a homogeneous
equation with one additional variable; thus, equation (3387)
may be written
l\. + Quy +
3395
.
•
•
+ R^^'. =
^^h-
SIMULTANEOUS LINEAR FIRST ORDER
3396
to
Pkoi'. I.
P.
13.
EQUATIONS.
The solution of sneU equations may
he
made
depend upon a sijstem of ordinary 1st order differential
—
;
DIFFERENTIAL EQUATIONS.
S04
equations having a nnviber of variahles exceediiir/ hij niore than
one the number of equations.
Let there be n equations reduced to the homogeneous form
(3395) involving one dependent variable P and n-\-ni independent. Select n of the latter, x,y ... ^, and let the remaining
From the n equations find P,,, Py ... P^ in
??t be
^,r]...l.
terms of P^, P,, ... P^, and arrange the results as under
:
P..
+ ci,P, + h,P^...+hP, =
0'
Py + a2P^ + h,P^...-j-h,P^
= 0[
P. + c''nF,-\-h.P^...+KP^
= o,
Multiply these equations by
A^, X25
... X,^
(1).
respectively,
and add
thus,
A1P..
+ X2P,
+X.P. + 2
...
(Xa) P,
+ S {U) P„
...
+2 (XZ-) P, =
(2).
From
this, as in
(3387),
we have
the auxiliary system
dx___ dy_
_ dl
_ dz_ _ d^ _ dn
Y^~ \ "'~ K ~2(Xa)~2(X6) "'~^{U)
and, by eliminating
X^, X2
...
/on
^^^'
X,„
— a^dx — a-idy ...—andz =
— Jj^dz = 01
dn — h-^dx — h^dy
d^
r,^\
...
Then,
if
u
dl
- 1\ dx - h,dy
=
a,
will be values of
...
-
/.•„
=
&c. be the integrals of (4), they
v
I),
satisfying the equivalent system (1), and
P
the integral of that system will be
3397 Prop. II.
P. D, equations.
ad^^-\-hdy
AP =
represents a
so that
tion of the 1st order.
Rule.
F(u,
v,
...)
= 0.
To integrate a system of linear 1st order
A=
Let
= 0.
d^.
— ^'Reduce
...
-\-kd,,
homogeneous
the equations to the
linear P. D. equa-
homogeneous form (1);
express the result symholically hy
AiP
= 0,
A.P
= 0,
...A,P
= 0,
PARTIAL DIFFERENTIAL EQUATIONS.
505
and examine ichether
the condition
is identically satisfied
for every pair of equations of the system.
the ec[iiations of the auxiliary system {Prop. I.) will
he reducible to the form of exact differential efiuations, and
c, ..., the complete
b,
a, v
their integrals being ii
If
it he so,
=
w=
=
of P will he F (u, v, w, ...)j l^^^ farni of F being
arbitrary.
" If the condition be not identically satisfied, its ap2)lication
will give rise to one or more new partial differential equations.
Gonibine any one of these with the previous reduced system, and
again reduce in the same toay.
" With the neio reduced system proceed as before, and continue
this method of reduction and derivation until either a system
of P. D. equations arises, hetiveen every two of lohich the above
condition is identically satisfied, or, wliicli is the only possible
value
=
=
In the
0, ... appears.
0, Py
case, the system of ordinary equations corresponding to
the final system of P. D. equations, will admit of reduction to
alternative, the system P^
former
the exact form, and the general value of P ivill emerge from
In the latter case, the given system
their integrals as above.
can only be satisfied by supposing P a constant.''
3398
"Ex.:
P,+ (/+.-v/ + a-r)P,+ + .— 3,OP, =
+ -.r//) P, + (./ -y) Ft = 0.
P,+
(7/
(a^„-;
0,
//
0, A.P = 0, it will be found that
Representing these in the form A^P
0,
becomes, after rejecting an algebraic factor, xP^ + Pt
(AiAj — zi,.,Ai)P
and the three equations prepared in the manner explained in the Rule will
be found to be
=
=
=
P, + (3.r + OP.
No
one
= 0,
P, + yP,
= 0,
P, + .7;P„,
We
other equations are derivable from these.
= 0.
conclude that there
is
but
final integral.
"
To
obtain
it,
eliminate P^, P^, P^
fi'oixi
the above system combined with
Pjx + P,jdy + P,dz + Ptdt=0,
and equate to zero the
coeflBcient of
P, in the
result.
dz—{t + 3x-) dx -ydy-xdt
the integral of which
"
An
is
%
— xt — x^ — lif =
arbitrary function of the first
value of P."
For Jacobi's researches
in the
member
We
find
= 0,
c.
of this equation is the general
[Boole, Sup., Ch. xxv.
same subject,
3 T
see Crelles Jonrnal, Vol. Ix.
—
, ,
DIFFERENTIAL EQUATIONS.
606
NON-LINEAR FIRST ORDER PARTIAL DIFFERENTIAL
EQUATIONS.
3399
F{x,j,z,z.,y^,)
Tupe
Chaepits's
assume
ilic
Solutioj^.
— Writing
=
^
(!)•
p, q instead of z^ and
A^^dy = -Al- = -4Bthese by integration,
Find a value of p from
ponding value of q from
the given equation,
the equation
dz
and integrate
Proof.
= q^.
z,.,
ecjnations
(2).
and the corresand substitute in
= p(U-\-qdi/
by (3322) to obtain
tJic
(
3)
final integral.
— Since dz=pclx + qdy, we have, by the coudition of integrabihty,
of
x, y
Express p„ and q^^ on the hypothesis that z is a function
a function of x, y,z q a function of a;, y, z,p considering x constant when
finding j5,/, and y as constant when finding q^. Equating the values of p^ and
the result is the equation
(/,, so obtained,
j/y
jj
Ap, + Bpy + Cp,
in
;
;
;
= I),
which A, B, C. D stand for —q^,, 1, q—pqi„ q^+pq-.
Hence, to solve this equation, we have, by (3387), the system of ordinary
equations
3400
(2).
—
More than one value of p obtained from equations (2)
give rise to more than one complete primitive.
The first two of equations (2) taken together involve equation (3).
Note.
may
DERIVATION OF THE GENERAL PRIMITIVE AND SINGULAR
SOLUTION FROM THE COMPLETE PRIMITIVE.
EuLE.
Let the complete primitive of a F. D. equation of
the \st order be
z
3401
^i^^t-G
the
general prunitlre
z
bet'lveen
= f {x,
form of (p being
3402
^-I'lie
h between
= f(x,y,a,b)
y, a,
«^
is
(a) }
(1).
obtained
and
f.,
hy
=
clluiliiating
;i
(2)
specified at pleasure.
singular solution
is
the coinplete 'primitive
t-0,
obtained by eUtninating a and
and the equations
f,.
=
(3).
PABTIAL DIFFEBENTIAL EQUATIONS.
—By varying a and
Proof.
p
h in (1),
= f^+faa^+f,h,
Therefore, reasoning as in (3171),
nating
fa,
= 0,
/,,
= 0,
= f,+faa,+f,Ar
and
f,,a,,+f,h,,
=
(3),
leading to the singular solution
«..
ft,,
q
we must have
faa^+fiK^O
therefore either /„
507
^y
;
or,
elimi-
- a,J h, = 0,
and therefore, by (3167), h = (p(a).
spectively, and add, thus fada+fi,db
in (1), and the equations "(2) are the
Multiply equations (3) by dx, dy re(a) in this and
Substitute l
= 0.
=
(j>
result.
SINGULAR SOLUTION DERIVED FROM THE DIFFERENTIAL
EQUATION.
Rule.
3403
—Eliminate p and q from
means of
tion by
the differential equa-
the equations
Zp
= 0,
z,j
=
0.
the D. E. be z =/(«, y,p, q), and the C. P. 2 = Fix, y,
and q being implicit functions of a and h, we have, from the
Proof.— Let
Now p
equation,
Hence the conditions
z,
= 0,
= z^pa + z^ qa,
=
z„ = 0,
z^
z,,
z,,
= z^po + ^a
a, h).
first
?Zi.
in (3) involve,
and are equivalent
to,
= 0.
z,
3404 All possible solutions of a P. D. equation of tlie 1st
order are represented by the complete primitive, tbe general
and
primitive,
tlie
To connect any given
3405
[Boole, p. 343.
singular solution.
solution witli
tlie
complete
primitive.
Let
^ r=
^
(^x,
z
= F{x,
y)
y, «, h)
some other
be the complete primitive,
Determine the values of a and
equations
F=
and
solution.
F^.
(}>,
=
(^^.,
h
which
F^
=
satisfy the three
(p^.
If these values are constant, the solution is a particular
case of the complete primitive; if they are variable so that
a function of the other, the solution is a particular case
if they are variable and unconnected,
the solution is a singular solution.
one
is
of the general primitive
3406
CoE.
—Any
;
two solutions springing from
complete primitives are equivalent.
different
;
DIFFEBENTIAL EQUATIONS.
508
3407
Ex.:
By
(3299),
z=px + qy+pq
^=
1
— ^—
px
= z—
andT we have
q
He.ce
i^
.-.
dp
becomes
(2)
=
0,
p
=a
;
.'.
q=:
(3322),
making
z
+
—
xy
z
7-^-
n?
(p+yy
= ., = ,-MiL, = f
,.
Substituting in dz
+y
;
= pdx + qdy,
= (idx+^'^ dy
(3).
a+y
-^ + -^ =
constant,
— log (2 — oaj)+log (a +
therefore
(2).
.
'-.
a
(1).
= |. = .|
A = —q„ =
;
p+y
dz
By
.7.^
^z)
0,
=
(^)-
(^)
Differentiate for x,y,z, and equate with (3), thus (p'{z)={), therefore
ax by al), the C. P.
constant (say —log 6); therefore, by (4), z
<p {z)
=
=
+
+
of (1).
3408
To
between
z^^
find a singular solution by (3402), we
and y
that is, x + h
2;,,
;
=
= 0,
=
therefore
is the singular sokition.
To
z
h
0,
= —xy—xy + xy = —xy
general primitive by (3401), eliminate a between the two
find the
equations
z
must eliminate a and
+a =
=
ax + (y + a)
<p
(a)
and x + {y + a)
^' (a)
+
(a)
= 0.
P. D. EQUATIONS WITH MORE
THAN TWO INDEPENDENT VARIABLES.
NON-LINEAR FIRST ORDER
3409
Pkoi'.
—To
find the complete primitive of tlie differ-
ential equation
F{d\,a^,
3410
Rule.
''•
—Form
I Vdx,
"^
^^-
...
the linear P. D. equation in
dz ) dpr
the summation extending
auxiliary system (3387) n
^l
= «l,
=
a\„z,j)„pi ...p,)
dpr Vdx,
from
—1
^2 =
r
=l
...
$
denoted by
^ ^'
to
r
= n.
integrals
rt2,
(1),
^„-i
= an-i
From
the
SECOND OBDEB
EQUATIONS.
P. D.
509
onay he obtained. From these equations, together with {l),find
^''^
terms of Xi, x,, ... x,,, substitute the values in
Pi P2 ••• Pn
J
= pidxi + padxg
dz
+PnClXn,
...
the integral of this last equation will furnish the solution
re([uired in the form.
and
f (xi, X2
.
x„, z, ai, aa
. .
.
= 0.
a^)
.
.
[Boole, Biff. Eq., Ch. xiv., and Snp., Ch. xxvii.
SECOND OEDEE
3420
Type
P. D.
EQUATIONS.
= 0.
F{.v,i/,z,z,,.,^„z,,,z,,„z.,,)
The
are briefly denoted by
derivatives 2.,.,
respectively.
z being a function of tlie two independent variables x and
y, the following values are of freqnent use
z^, z^^, z^y,
z.,y
_p, q, r, s, t
dz
3421
If
= pcU-\-qdy
dp
;
= rdx-\-sdij
;
= sdd?-\-tdi/.
dq
u be any function of x, y, and z, the complete derivau are indicated by brackets, thus
tives of
M=
3422
A linear
(Uy) := Uy
2nd order P. D. equation
is
+ qu,.
of the type
Rr-^Ss+Tt= V
3423
in
u, +pu,,
which
_B,
Fare
S, T,
functions of
x,
(1),
//,
z,p,
q.
—
Any P. D. equation of the 2nd order which
Pkoposition.
first integral of the form u =f{v), where u and v involve
y, 2, p, q, is of the form
has a
'C,
Rr+Ss + Tt+U{rt-s') = V
3424
where B,
S, T, U,
Fare
x,
ij,
z,p,
(2),
q,
and
U=u,,v,'-u,Vp
3425
Proof.
functions of
— Differentiate u = /
p, q all involved in n
with the values
and
v,
(v) for x and y separately, considering x,
and eliminate f'{v). The result is equation
(3).
y, z,
(2),
niFFEBENTIAL EQUATIONS.
510
3426
= n^ {Vy) - (Uy) Vp,
f^l^
^iT
= v^ (u,) - {v,)
u,^,
with the notation (3422), ^ being an undetermined constant.
3427 Cor.
equation (1)
— The
condition to be fulfilled in order that
a first integral of the form n =f{r) is
may have
SOLUTION BY MONGE'S METHOD OF
Rr-^Ss+Tt=V.
3428
Rule. — Wri'tr the
tiro
equations
Rihf-S(iidij^-r(ii^
=o
=
(1),
Rdp(Ii/-Vclvdi/-\-Tdqclv
Besolve (1) into
its
(2).
factors, producing the tnv equations
dy — m^ dx
=
dy — mgdx
and
=
,
= m^dx and equation (2) combined, if necessary,
= pdx + qdy, find two 1st integrals u = a, v = b;
then u = f (v) will he one 1st integral of the given equation.
Similarly from dy = modx fnd another 1st integral.
From dy
with dz
3429
The final 2nd integral may he found from one of the
method (3383).
1st integrals hy Lagrange's
3430 Otherwise, determine p and q in terms of x, y, z from
the two 1st integrals; substitute in dz =pdx-\-qdy, and then
integrate by (3322) to obtain the final integral.
If equation (1) is a perfect square, there will be only
one 1st integral, and Lagrange's method only is applicable.
3431
= mvp and
Proof. — By (3427) we may put n^ =
= pdx + qdy (3321) in the complete derivatives
(du) = u,,dx + u,idy + %(,ulz-\-Updp + u^dq = 0, (dr) = &:c. = 0;
dx +
dy + u, (dp + m dq) = |
.-.by (3422)
(v,)dx+(v,)dy + v,(dp + mdq) = 0^
mi(,^„
v,^
;
also
dz
(71,)
(?/.„)
Solving these equations for the ratios dx
dx
_
dif
ir~
with the values of
J?,
S, T,
+ mdx _
s
V in
:
dy
:
_
~ T ~
mdij
(342G).
dp
+ nuhj, we
dj> -f hi
V
dq
,3.
obtain at once
/,>.
^^'
SECOND OEDEB
EQUATIONS.
P. D.
511
Equations (1) and (2) are the result of eliminating »i from (4). These
two equations with dz = pdx + qdy suffice to determine a first integral of
(3428) when it exists in the form « =/(i),
3432
q{l+q)r-{p + q-^2pq)s^p{l-]-p)t
Ex.(i.):
2nlx
+ qdy =
dz
First,
Monge's equation (2)
= —qdii,
which, by pdx
Hence a
first
2 (l
is
integral
+ (l + q)
(1+2^) dx
or
0,
= x>dx-\-qdy =
+ s) dpdy+jp (l+p)
= --^
—
is
'-
=
p
=
qD
now
,
dx
= — T-.
^
dy
^
(x
+ y + z)
dz
= -TT'
0'
•"•
=
-
p
is
=
—
For the
.-.
3433
Ex.
——L—B.
x + y + z = C.
= pdq, and by integration,
= qyp {x+y + z).
z^~\pz,, = 0, by (3383)
dx —-^
+ dy + dz
—
and dx =
+ z)
.-.
0;
i.e.,
All
A,
;
.
l-xP{x + y
d(x + y + z)
Hence the second
0,
reduces to qdj)
therefore the other first integral
final integral integrate j;
^^'
;
=
(6).
=
(5), equation (2)
dqdx
(p{z)
=
by
(5).
i)
A.
and, integrating,
;
Next, taking the second equation of (5) with
dx + dy + dz
0,
dz
p dx + q dy,
Also,
=
dij
=
z
.•.
0,
gives -^£-
= 0.
we hud
Solving the quadratic equation (1),
^j,(,,^^,)
C
^-^^ ^ +
^
]l-^(x
+ y + z)
integral is ;«— /(;^' + + ~) =^ F {:).
j,>.
2/
(ii.)
= 0.
^,,.-a'^,,
:
E=
= 0, T=-a\ F=0; therefore (1)
dpdy — ahJqdx = 0.
dij^—a^dx- = 0,
and (2) become
and converting (2) into
From (1) dy + adx = 0, giving y + ax =
therefore a first integral is
dp + adq = 0, which gives p+aq = c
(3).
p + aq = f(y + ax)
gives rise to another first integral
Similarly, from (1), dy — adx =
— aq = yp(y — ax)
(4).
find
and q by means of (3) and (4) from dz = pdx + qdy,
Eliminating
dz = (2a)-'
(y + ax)(dy + adx)-xl^(y-ax)(dy-adx)},
—
therefore, by integrating, z = ^ (y + ax) +^
(i.)
Here, in (3428),
1,
S
c,
;
2)
j>»
{(/.
(i/
«'*-')•
For the symbolic solution of the same equation, see
SOLUTION OF THE
3435
EQUATION.
Ri'^Ss-^rt+U{rf-s')= V
3434
Let
P. D.
(3-5G6).
iii^,
iiiz
be the roots of the
qLiaclratic
7rr-Sm-\-RT-\-UV=0
(1).
equation
(2).
;
.
:
BIFFEBENTIAL EQUATIONS.
512
Let
»!
=
a,
L\
=
= a',
and
two systems
b,
the solutions of the
equations.
Vdp
Udq
3436
th
Then the
^i.,
= riuthj- Tih ^
= mJx-Bdi/
= dx + q
=
be respectively
b'
ordinary differential
ITdp
Udq
(3),
dij )
jj
d:i
=
=
—
m,dij-Tdx
m,dx
j;
dx
^
-Edy
[
+
)
q dy
(4).
integrals of (1) will be
first
To obtain
r.y
of
a second integral
—When
m^, m.^^ are unequal, assign any particular
forms to /i and /,, then substitute the values of p and q, found
from these equations in terms of x and y, in dz = pdx-\-qdy,
which integrate. Otherwise, assign the form of one only of
the functions f\, f.i, involving an arbitrary constant C, solve
for jj and q, and integrate dz =pdx-\-qdy, adding an arbitrary
3437
1st.
function of
3438
C
2nclly.
for the constant of integration.
— When
')%,
S-'
Equations
reduce to
(o)
m.2,
are equal, and therefore, by
= 4.{BT^-JJV)
and
(4)
coincide,
and,
since
m = ^S,
mp=iSdy-Tdx
3439
Udq
dz
Here py =
(2),
(5).
(6),
= iSdx-Bdy
= p dx-\- qdy
(7),
(8)
and therefore the last equation is integrable
if the values oi p and q, obtained by integrating (6) and (7),
be substituted in it. Let n = a, v = b be the integrals of (6)
and
(7)
;
and
7.,.,
let
z
=
(}>
{x,
//,
a, b, r)
(0)
be the integral obtained from (8).
The general integral is found by making the parameters
F (a)
b
f{a), c
(/, b, c vary subject to tAVO conditions
that is, by differentiating
z
<l>{x,y,a,f{<i), F(n)}
=
=
=
for
(i,
3440
and oHmhiatino'
ti.
The general integral therefore represents the envelope
whose equation is (l>).
uf the surface
SECOND ORDER
— (Boole,
P. D.
EQUATIONS.
513
Assuming a Isfc integral of the form
from equations (o426) by multiplying (i.) by
(H,J?f^, (ii.) by («„)«,„ (iv.) by (».r)0'v/)> (/Ob'^^'vz, andadding. Again,
eliminate /x and v by multiplying (i.) by («.,,)"> (ii-) by («v)^ (iii.) by (Ux)(Sh)i
The two resulting equations are
(v.) by (^iij.) 7ip + (Uy) Uq, and adding.
Proof.
u =f(v), eliminate
Sup., p. 147.)
/j,
and
v
=
u, - U (u,) («,,) + Vu,,
E,(n,y+soQM + TOhy+v{(n,r) »,+k)«j =
B (u,) H, + T («,)
n,^
I
o)
Multiply the 2nd of these by m, divide by V, and add to the 1st equation
the result is expressible in two factors either as (11) or (12),
{BOtd + ^^hM + V<h,}{'>^hM+TOh,) + Vu,^}
{B(u.)+vi,(u,)
+ J%}{m,M +
TM +
ru,}
=
=
;
(11),
(12),
being the roots of the quadratic (2). By equating to zero one factor
of (11) and one of (12), we have four systems of two linear 1st order P. D.
equations.
Taking each system in turn with the equations
m,,
vi.,
(ti^)+stip
+ tUq =
0,
B
and eliminating (uj), (u^), u^, u,^, we have the determinant annexed for the case in which the 1st
factor of (11) and the 2nd of (12) are equated to
In this case, and also when the 2nd factor
zero.
of (11) and the 1st of (12) are chosen, transposing mj, Wj in the determinant, the eliminant is equivalent
V{Br+Ss + Tt^U{rt-s')-V}
=
0,
to
;
DIFFEltENTIAL EQUATIONS.
514
POISSON'S EQUATION.
P = {rf-srQ,
3441
P
a function of p, q, r, s, f, and homogeneous in r, k,
t ; Qis a function of a3, v/, 2;, and derivatives of ?:, which does
not become infinite when rt s^ vanishes, and n is positive.
where
is
—
Rule. — Assume q = ^
and r;
rt
thuSi
tion 0/ p, q,
—
and
s"^
and express s and t in terms ofq^
and the left side becomes a func-
(p)
vanishes,
qp.
Solve for a Isf integral in terms
o/p and
and integrate
q,
again for the final solution.
Proof.—
therefore
rt
s
= 5, = q^p^ =
— s- =
Also
0.
P
(/,,
r
;
f
=
of the
is
Hence the equation takes the form
=
q,
q.rpy
form
(1, g^, ql)'"'
=
qlr\
=
(r, s, f)'"
=
r"'
(1,
q^,, cj],)'".
0.
LAPLACE'S REDUCTION OF THE EQUATION.
Rr+Ss+Tt+Pp + Qq+Zz= U
3442
(1),
where B, S, T, P, Q, Z, U are functions of x and // only.
Let two integrals of Monge's equation (3428)
Edy'-Sdxdy + Tdir
be
(p
{x, u)
Assume
^
=
3443 To change
we have
r
t
=
=
;:,,.
Z,^
^
=
a,
{>', [/),
^
(;>j,
v =
//)
4^ {^v,
2;v^,^,7,.,
is
of the
;
form
ilf, N, V are functions of ^ and
written in the form
{d,
-r,V,y
(2),
>;.
This equation
+ M){d^ + L), + {N-LM-L^::=V
N-LM-L^==i)
If
with
(J„
+ L).v =
(3).
(4),
have
(d.-^M )::'=¥
j?,
(1701)
r,.„,,
Nx=V
where L,
shall
y).
= %e +
+ ^,^r,H;:^L, +
= Z2^ti + 2^^^t,^V, + ^2A+-A!j +
,,^^Lz, + ]\P, +
we
h.
the variables in equation (1) to i and
The transformed equation
may be
=
=
c',
LA W OF BEGIPBOGITY.
and the solution by a double application
from
By symmetry,
equation
515
of (3210)
is
obtained
(1) is also solvable, if
y-LM-M^ =
But
in
if
neither of these conditions
terms of
z'
from
(3).
.
= ^4 + 7?.' +
may be
similar to (4) or (5),
3444
CoK.
(1),
if
of the equation
3445
LAW
wbeu
viii., p.
^,p, q)
cV
and
-^
(*,
ij)
z,
see
[Booh, ch. xy.
be
=
(1).
]),
^(p, q,px-\-qy-z,x,y) ==
=
contain also
differential equation of the 1st order
Let the result of interchanging
changing z into 2hi-\-qy—z, be
z
V
L, M,
159.
OF RECIPROCITY.
<l>{^v,ij,
ii
the
fulfilled,
+ hz^ + ahz = V
For the solution of equation (2),
Prof. Tanner, Proc. Lond. Math. Soc, Vol.
then,
not
is satisfied.
— The solution
Let a
for this equa-
if fulfilled
and,
in
z
repeated imtil one of the equations,
z^,-\-az^
3446
z
^',
Substitute this for
of the form
v.
of integrability,
tion, will lead to a solution of
transformation
found to hold, find
form
is
It will be of the
where A, B, G contain ^ and
(c/, + i) z = z', and the result is
The same conditions
(5).
y and
0...
^,
and of
(2);
be the solution of either (1) or (2), the
DIFFERENTIAL EQUATIONS.
516
solution of the other will be obtained
by eliminating
and
^
»?
between the equations
a;
=
3447
d^xfi {^, f}),
Ex.— Let
ij
=
(l^xlf
zz=pq
z= iv-^rji/-^ {$,
{^, rj),
(1),
2^^^
+ r2y-z = xy
rj).
(2),
be tbe two reciprocal equations.
The
i,,
T)
integral of (2) is
z
=
xy+xf {^^\
.-.»//
= bi + t>i ( y )
(bj)
•
have now to be eliminated between
»^=v-|/'(|)+/(|),
!/
=
Each form assigned to/ gives a particular integral
the equations (3) become x=^rj-\-l,
and the elimination produces z
(x
=
y
'-f"
£+/'(!).
If
of (1).
= ^ + a,
— a).
z
=
/ ^
(
)
(3).
= y+
'^
Z^,
tn,
— l)')(y
3448
In an equation of the 2nd order, the reciprocal equaformed by making the changes in (3446), and, in
addition, changing
tion
is
r into
—
—
rt—s
-,
s into
^„
t
into
rf—s-
rt—s"
then, if the 2nd integral of either equation he
of the other will be found by the same rule.
z
=
\p (.r,
//),
3449 The above transformation makes any equation
form
<^ Q), q) r+^ {p, q) .v+x ilh q)t =
dependent for solution upon one of the form
X (^% «/) »'-^ G*'^ 1/)^+^ (^^ y) t = 0.
3450
is
And,
in the
same way, an equation
of the
dependent for solution upon one of the form
Rr^-Ss+Tt=
V.
See Be Morgan, Camb. Phil. Trans., Vol. VIII.
SYMBOLIC METHODS.
FUNDAMENTAL FORMULA.
Q denoting
3470
a function of
{do-m)-'Q
0,
=
e"''j6'-'"^
Qd0.
form
that
of the
8 YMBOLIG METHODS.
517
—
—
—
= Q by
= Q (see
= dx.
Hence
The right is the value of 1/ in fcbe solution of d,y my
Proof.
(3210). But this equation is expressed symbolically by (dg ra) y
(cZ^— 7?i.)'^Q.
1492), therefore y
=
Let X
(3470)
=
may
therefore
e\
d^
= xd^
and xdQ
be written
3471
{.v(h-m)-^
3472
Cor.—
3473
or
Q
=
cr"^
{de-m)-^0
Ja-»^-i Q(Lv.
=
Ce"'\
(.vd.,-m)-'0= av'\
Let F(m) denote a rational integral function of m; then,
since
dee"^'
= one"'\
= mV"^ &c., the operation
is
mX Hence, in all cases,
F{de)e"'' = e'"'F(m).
F(de) e''"Q = e'>''F(de+m) Q.
f^o, e'"'
f/,
always replaced by the operation
3474
3475
Formula (2161)
is
3476
e^''F(de)
.
a particular case of this theorem.
Q
= F {de-m)
e'»^
Q.
e'«^
Q.
Also, by (3474-6),
3477
3478
3479
To
= e-'«^F(fye'"^
F (f/,+m) Q = e-'F {d,)
F (de) Q = e-"''F{de-m)
F(m)
&'''
Q.
the last six formulce correspond
= x"^F{iii).
F{a:d,) .v"'Q = .v"'F{M,-\-m) Q.
Q= F - m) a^'^Q.
F ixd^
3480
.r"^
3481
3482
^v'^'F (ctY/,)
3483
F (m) = .v-"'F (erf/,)
3484
F{a^d^-]-m) Q = .v-"'F{a'd,) .v"'Q.
F i^d,) Q = x-"'F {.vd,-m) .r'"Q.
3485
If
3486
{.vd,,
.v'".
U=a-{-hx-\-cx^ + &c., then, by (3480),
F{.vd,)
U= F(0) a+F(l) bjv-{-F{2) c.r+&c.
DIFFERENTIAL EQUATIONS.
518
U= F-'{0) a-\-F-'{l) Kv-\-F-\2) cx^^kc.
3487
F-'(.vd,)
3488
F(.v(]_,,7/(Iy,
3489
or,
...
Otherwise
Proof.
— As
in
(1770).
and remembering that
Note.
D for
{D-n-\-l)n
a-^
or
either
.r'y«^..
U,
D'"in
— l)
(2452).
(.?y/,-/<
...
+ 1) /^
Otherwise, by Induction, differentiating again,
= x.
—In the symbolic solution
we may
{(h-tl+l)
...
...)
dg,
=: .vd^{.rd,,
,r"«,„,.
= F{m., n,p
<r'"ifz\..
succinctly, writing
D{n-l)
3490
...)
= (h{cU-\) {do-2)
^v^'Un,.
more
z(L,
of differential equations,
employ the operator
operator d^ after substituting e^ for x.
(3474-9) will be required accordingly.
3491
xd,,
directly,
or
the
Formulae (3480-5) or
{(^(/))e'-^j"g
= ct>{D) ^{D-r) (t>{D-2r) <l>{D-(n-l) r\
— By repeated
of (3475) or (3476).
...
Proof.
e"'-'Q.
apjilication
For ready reference, formulas (1520,
'21)
are reprinted
here.
3492
/(.*'
3493
+ /0 =
fi'V+h, 7/+
Let
cIq
+ «! X
A-)
=
e^'"^-f{.r).
e'"'..--H./(.r, ;y).
CLuCr ...-{- cr„
-\-
r/;"
=
/' {,r)
,
then, denoting d^ by D,
=
3494 /( D)
in-
where
means that
/ (D)
uf{D)
v^uof (D) r+
D
is
^/" (i>) r + &c.,
to be written for x after differ-
entiating f{x).
—
Proof. Expand uv, D.nv, I)- .uv ... B'^.uv by Leibnitz's theorem (1400);
multiply the equations respectively by a^, a,, a^ ... a„, and add the results.
3495
uf(D)
r
^f{D) uv-f" (D) u,v+ ^f" D.u,^ i—&c.
.
.
SYMBOLIC METHODS.
Proof.
— Expand
uv^, uv-i^i uv-m
•••
519
by theorem (1472), and proceed as
''f''^'-.!)
in the last.
^^"
Fid,)
3496
m.v
A
more general theorem
3497
F (7r^)(7.,„+ »,_„„)
= F(-m') ^"^
m.v.
is
=z/^ (->H^)(»,,^^+ «_,,),
where ii and tt have the meanings assigned beloAV (3499),
and i = \/ — 1
Theorem.
and
//,
it
—
3498
^
3499
and
If ^
denote any algebraic functions of x
(3474) and (3475), that
x^
may be shown, by
Let
{(!.+!/)
more
u, or,
H^r)
=
<!>
definitely,
a homogeneous function of the
and
ables,
{(h^^v)
n,^
n^^^
xl^Qf).
=
(x, y, z, ...)", represent
degree in severable vari-
let
= ^4+K+-^.~ + &c.
^
3500
Then, by (3480),
3501
TTW
= nu,
3502
= 7ru,
TT^H
F {'n)u =
Hence
REDUCTION OF
irhi
=
nhi,
&c.
F{n)u.
F{Tr,)
TO /{it).
Let u be any implicit function of the variables, and
TT.,, where tt^ operates only upon x as contained in
TTi
u, and TTg only upon x as contained in ttil, &c. after repetitions
of the operation tt.
Then
3503
let
TT
= +
3504
=
'"'i"
<« =
3506
{rr
Proof. —
since
tt^
— r-\-l)
= (^r —
tt^u
=^ (tt— l)7r?/,
...
tt,,)
(tt— 2)(7r— 1) nn.
u
=
ttk,
has here no subject to operate upon.
TTj ((
for, Ttu
'^[ft
'^ti,
=
(tt
next step,
Cor.
tt^
and 2
—
n-.J
TTH
=
(tt— 1)
7r«,
and 1 are equivalent
are equivalent, and so on.
being of the 1st degree,
—"When
u
is
tTo
as operators.
a homogeneous function,
we
In the
have, by
,
DIFFEEENTIAL EQUATIONS.
520
= n^'ii, therefore ir and n are equivalent operators
Hence (3506) may be written
(3501),
upon
TT^'a
u.
3507
=
7T[i(
(»->• + !)
=
{n-2){n-l)nu
...
n^:lu,
which is Euler's theorem of homogeneous functions (1G25),
since in that theorem the operator is confined to v.
3508
As an
then
=
Trjit
+ ydy) u =
= {tt^ +
Tr^mi = (xd^ + ydy) (xd^ + ydy) u,
ttu
illustration, let
+ 2xy cl^y + y'doy)
(.t;^ J-j.^
Here
=
(xd^
ir^u
u,
ttjM,
ttu
tt.,)
=
tTj
u
+
ir,
ttu.
the operation being confined to x and y in the second factor (.3503), and therefore producing {xd^-\-ydy)i(, merely.
Hence
li
tt^m
=
{x'd.,:r-^'lxyd^,i-\-y'^do,j-\-xd^-\-ydy) v,
which proves (3505).
U = Uo-{-Ui-{-U2-\- ...,
of dimensions 0, 1, 2,
...,
F {n)U= F (0)
3509
a series of homogeneous functions
then, by (3502),
u,-\-F{l)
u,+F(2)
u,-^...,
3510
F-'(7r)U= F-'{0) u,+F-'{l)u,-^F-\2)u,+ ...
3511
3512
Ex. 1
a-''U
Ex. 2
:
= + + ahi^ +
= u^ + a-^u,-{-a~hL2-\-...
a" Z7
:
^i^.ti
^(o
. .
.
u have the meaning in (3499),
Ji
3513
and simikrly for the inverse operation F~^
Proof.
— By
__
3514
where
j)-\-q-\-r-\-
Proof.
—
(tt).
(8502) applied to the expansion of the subject by (150).
Equate
(l
^
....
_______
= m,
coefficients of
pi
and
a"'
in the
=
1.2
u,^,
...
i).
expansion of
+ rt)'^f7=(l + a)'''"-^(Ua)^"''(l + a)--"'...
tr,
reducing by (3490).
3515
F{de)a
The
=
Q
general
symbolic
solution
of
the
equation
is
u
=
F-'
((/,)
Q-^F-'
((/J 0,
by (1488-90).
-
SYMBOLIC METHODS.
3516
The
solution of the equation
521
(:^)238), viz.,
is a function of x, is most readily obtained by the
symbolic method. Thus 7%, 7iu, ... m,, being the roots of the
the numerators
auxiliary equation in (3239), and A,B,G .,.
of the partial fractions into which (7;i"-|-fti^)i"~^+ ••• +««)~^ can
be resolved, the complete primitive will be
where Q
N
3517
=
(4-''0~'Q
where
(3470)
e""-je-'"'^-QcZa-,
and the whole operation upon zero produces, by (3472), for
the complementary term,
3518
Proof.
— Equation
(1)
may
be written
(4-w^i) (d^—ini)
or
.-.
by (3515),
y
=
is
If r of the roots m^, m,,
{A +
— By
{dx-mn) y
are each =.of the form
...
term
54 + Cch.
•
•
•
-h
Rd,,)
(A + Bd,+
or
Ex. (1)
e'"' I J
e-™^e»'^
6-"^'^'
[
Q.
7n^—vi"
{m-3)(m + lf
y
L-"" Qdx
— om- 3 =
^
1
.
-^R'(d,-m)-'\ Q,
Q= (d^-ni)-'e'"^^e-"'^ Qd^
| dx
y^^-y.^-5y^-3
:
Here
therefore
e-'-
those roots give
ni,
Cd,,... +Bd,.,?)id.-m)-'-Q.
(d^-vi)-'
But, by (3470),
=
3521
Q,
(1918), the r roots equal to in will produce
\A\d,-my + B'{d,-vi)-'-\..
3520
=
id,-m,)\-' (Q+0),
...
converted into the formula above.
rise in (3517) to a single
3519
...
\(d,-ind(d,-m,)
whicb, by partial fractions,
Proof.
+C,e'v.
(7,e'""'-+a,e'"^''-...
=
= e'"4
e"'"^
Qdx,
and so
on.
Q.
{m — '3)(m + iy,
1
1
16(Hi-3)
16(w + l)
L_^
4(m + l)-^'
= yV (d.~S)-'Q~^\ (4 + 1)"' Q-i O^. + l)"' Q
= i^e'^ J Qf?"-— tV'^^ J eie-'j^e" QdxK
e-^'-
^
3 X
Qr/,.'
(3520)
—
DIFFERENTIAL EQUATIONS.
59.2
3522
Ex. (2):
=
/'
Here
+ cr)-^ =
(vi^
=
u
therefore
=
u,,,.Jrfrf(
thei-efore
('/..,
(2m)
(2ia)'^ {(dj.
-'
+ «-)
Q,
'(^.
-/a) -'-(m + wt)"'},
{()»
— ia)'^ Q — (Jj.4-m)~' Q\
= (2ia)-' {e'"^ Je-'"^ Qdx-e-'"=' je'"^ Qdx}
(3470)
= a" sin cw j cos o,f Q — a cos j siu ux Q dx,
"
(?.«
'
a.e
'
by the exponential values (766-7).
3523
CoE.
1.— The
u
3524
Cor.
Change a
2.
solution of
=A
iu,-^a^u
cos ax-\-B sin
— The solution
of
jf^,.— rt^jt
into ia in the fifth line of (3522),
=
is
=
is
ac«.
and put
Q = 0.
3525 When Q is a function whose derivatives of the 7i^^ and
higher orders vanish, proceed as in the following example.
+ a'^u = (1+x)-,
+ xy + (d.,, + a^) "
(a-'-a-%, + a-%,-&c.)(l + 2x + x') + (d,,-\-aY^0
a'^ (l + xy—2a'^ + A coaax + B sina.i",
Ex. (3):
Uo,:
n
therefore
=
=
=
(d.,
+ a") "'
(1
'
the last two terms by (3523).
Exceptional Cat<e of the Inverse Process.
+ «"« = cos JW,
1 (d,, + a^)''
+
= I (e'"^ + e-''")(-w- + (r)'' + ^cosaa; + 2?sinaa;
= coswa; (a''— «^)"^ + &c.
3526 Ex. (4)
= (d,, + aY'
.-.
'II
Now,
if
as follows
Put
n
= a.
n
M2.r
:
= a,
(cosux + 0)
the
=
(e'"-'-
term becomes
first
infinite.
e-'"-''
+ 0)
by (3474) and (3523)
In such cases proceed
:
A = A'-(a'-n')-\ and find the value pf ^"^ ".r-cosa;?; ^^^^
By (1580) it is = iL^Mf Thus the solution is
^
.
u
=
—a
X sin ax
,
+ 1)ti sina.t;.
•
,,
\-A cos ax
,
2
The same
result is obtained by making
For another example, see (355'.').
Q =
cos
aa;
in the solution of (3522),
!
SYMBOLIC METHODS.
3527
Ex. (5)
therefore
+ {(ih-4)((l-5)}-'
= e''-{((l-l'){cl.-2)}-'x' + Ae'^- + Be'\ (3475,
y
{(d,-4^)(cL-5)}-'
x'e'"^
C^^ 3,.-mHence the
solution
y
3528
Ex. (6)
3529
e'"-
=
^""
i+
{
{
f
4 + H:^ + &c-
Ex. (7):
(d,
y
=
('
^'
+ ^6*' + ^^'
= e--\
{d^-aYu
e^'^'
}
+^^''+-^^''-
Y +T+T1
:
there fore
^/3m^y^^^
becomes
=
= {d^-ay'e"- =
therefore n
3517, 3472)
= \ (l-^J^)''
0,^^-3m + 2)-
Now
= .^V%
y,,-9>u + 20y
:
=
523
= e^^^^ +
(d,)-^ (3476)
+ ayy =
K + «)
"
"
sin
sin
»ia)
o).
f
(2149)
mx,
+ (fZ^ + a) '
^
[by (3478) with ^ = 0]
= (4-a)' ('k-«')"' sin ma; + e-"^'
= (-vr-a^)-- (d,—ay sinonx + e-'"' (Ax + B) (by 3496)
(Ax-\-B).
= (ui' + a")"- ( — VI' smmx — 2amcos7nx + a^ sin ma;) +
(c/,.)''
e'"''^
REDUCTION OF AN INTEGRAL OP THE
3530
^ Q=
;^3Y] {
^"''
J Q^^^^'- («
+ C {n, 2) .r"-«
where n — l\
=
1.2
...
Proof.— Bj^ (3489) d,M
d.,M
therefore
'^-1-
=
=
j
- 1)
^^"~'
V''^/^''
ORDER.
w"'
.
. .
J Q^^'^'
±
\Qd''-' cLv,
n.
= e-'"(d-n + l)(d,-n + 2)
{{d~n+l){d,-n + 2)
...
...d,Q
(1),
d,]-'e^''Q
+C'(«,2)(t7,-« + 3)-^-&c.}e"^(3
(3517)
-J_{e(»-i)VJ,)-ie'-(7i-l)e("--'^(cZ,)-ie=" + &c.}Q.
w— 1
Then
replace
e'
by
a*.
The equation
3531
«<y..+6ct^'^//..,+ &c.
=
^+J5.v+ at'^+&c.
=
q
.
DIFFERENTIAL EQUATIONS.
524
may, by (3480), be transformed into
U' W,-)'';'
The
solution
+
!j
=
3532
y
=
F-'
Q + ^-'
(.''4)
ij
=
Q.
0.
(j^'l,)
of the 1st part is given in (3487).
=
F{m)
3533 If a, i3, y, &c. are the roots of
part gives rise to the arbitrary terms
3534
F{xd,)
or
(2
then obtained from
is
The vahie
+ &c.}
(.v/...)^';
^>
0,
the second
the corresponding
If a root a is repeated r times,
terms are
.1-
t\ (log aY-'-\- C, (log .r)'-^+
{
—
.
.
.
+ C,}
can be resolved, as
Proof. The partial fractions into which F'^{xd:,)
Bnt
(jc)
being a root of
0.
ony^ 0,
in (3517), are of the type G (a-d^
(xd^.—m)"^
Gx"^ (3473), G being an arbitrary constant.
For a root in repeated r times, the typical fraction is C (xd^.—m)''^, p
—
F
m
=
=
being less than.
Now
r.
(xAl-my Gx>"
(log.r)^'-'
--=
{de-my Ge'"' iP''
=
(xd,.-m)-P
therefore
=
Gx'" (log
e""
(d,y CT^"' (3475)
=
0,
xy-\
The equation
3535
ai/,,e-{-bi/ne+&G.
=f{e%
siu 6, cos 6)
reducible to the form of (3531) by x
from (768), it may be written
is
Fide)y
=
and the solution
will take the
3536
=
^
(1
)
;
or, substituting
:^{A„,e'"%
%A,,,e-'F-'{m)-^F-^{(h) 0,
term of which the forms
substituted with .v changed to e\
Ex.
e^
form
for the last
3537
=
a;V
:
in (3533-4) are to be
= ax'" + hx"
= ax"'+hx",
xd^ (xd^-l)(xd^-2) y
.-.
u
=
=
{xd, (xd,-l)(xd,.-2)}-'
^
m {m-l)
(7U-2)
by (3180) and (3533).
A
(ax"'
+ h.r") + {xd,
^
+
71
(xd,,-l)(xd,-2)}-'0
+ .1 + Ih + Gx;
(7i- l)(ji-2)
result evident by direct integration.
:
,
(tjni--
SYMBOLIC METHOb'S^c:.
By (3490)
x%, + 3xy,, + y= {l-x)-\
+ 3xd^ + 1 = (-^4 + 1 )' y = 1 + 2.^ + 3a;2 + &c.
2/=GiY7, + l)-^(l + 2,« + 3cuH) = (0+l)-^ + 2(l + l)-^a) + 3(2 + l)-V +
3538
Ex. (2)
{
.-.
525
,
xd^ (xd^ - 1 )
} !/
=l+|.+^+&c.+ ^-^^^ + -^
(3480)
o
<i
3539
Ex. (3)
Let
=
TT
cZ,
=
The
3540
a;
,y,^+(4.7J-l) 2/^+(4aj'^-2.i;4-2)
:
+ 2a".
Then
=
7r(7r-l)^
Let (7r-l)-^0
=-llog(l-..) + &c.
.'('
a'
0,
7/
= 0.
be Avxntten
= {7r(7r-])}-'0= (7r-l)->0-tr->0.
— 1)?< = 0, ov u,+ {2x — l)ii —
« = i4e^'-^
.-.
(tt
.-.
?s
may
the equation
7/
(},
.-.
solution of a P. D. equation of the type
7/1, u.,, &c. are homogeneous functions of the 1st, 2nd
xd^,-\-yd,j (3503), is analogous to
degrees, &c. in x, y, and Tr^
(3531), and is obtained from that solution by substituting
Gx"-, an
7ti, U2, &c. for Bx, Gx?, &c. ; and, for such terms as
arbitrary homogeneous function of x and y of the same degree,
where
=
3541
Solution of
where
F(7r)
F{tt)u
=
7r"
Q=-
and
=
Q,
+ ^i7r"~^H-yl.27r""-+^„,
u^^-{-Ui-\-Uo-\-kc.,
a series of homogeneous functions of
tive dimensions 0, 1, 2, &c.
u
Here
general value
F
{711)
of the respec-
= F-'{7r) Q + F-\7r)0.
of the 1st
of
=m
=
;
C{w-m)-PO=C{a{\og.vy-'+v{\og.vy-\..-^w},
3543
where
...
term is given in (3510). For the
the last term (see Proof of 3533), let
then
have r roots
The value
3542
x, y, z,
u, v,
...
w
are arbitrary functions of the variables
all
of
the degree m.
3544
that
is,
degree
(7r-m)-i
CoE.—
a sing^le
m
=
(cr, 1/, ... )'«,
homogeneous function
(1620).
of the variables of the
5
3545
Ex.
x-zox
:
+ 2xyzj.,, + y'eo,,—a {xz^ + tjz^^ + az =
u„ being homogeneous functions of the m^^ and
may be
or,
.
DIFFERENTIAL EQUATIONS.
526
u,„,
,
written
(tt-
by (3505),
therefore
z
— OTTj-f a) z =
(7r-rt)(7r-l)
=
=
{(7r-a)(7r-l)}-'
(u,,
!!i»
{m — a)(m—l)
\
(n
z
/t"'
n.,„
+ «„,
degrees.
The equation
";« +
= «,„ + «,„
'?'«
+ n„) + {(7r-a)(7r-l)}-'
^*»_
— a)(n — l) + cr„-i-[7i.
The first two terms by formula (3502) the last two terms are arbitrary
functions of the degrees a and 1 I'espectively, and result from formula (3543)
1 and
a and 1.
by taking ^
;
=
m=
To reduce a P. D. equation, wlien possible, to the
symbolic form
3546
{WJ^A,n"-^+A,W^-\..~\-A,)n= Q
n = Md^ + Nd^ + &c
where
(1),
.
and Q, M, N, &c. are any functions of the independent
variables.
Consider the case of two independent variables,
=
{Md,-\-Nd,Y u
]\Pu,,
+ 2MNu,y + N'u,,
+ {MM, + NM,) 71, + {MN, + Ny^
n
n^,
.
.
.
(2)
obtainable from the right by considering the terms involving the highest derivatives only, for
these terms are algebraically equivalent to (Md^-^Ndy)'^.
The reduction being effected, and the equation being
brought to the form of (1) then, if the auxiliary equation
Here the form
of
is
;
3547
have
w''+A,m''-'-^A,in''--...-{-A,
its
of the
roots a,
h,
...
all
=
(3)
unequal, the solution of (1) will be
form
3548
u
= {n-ay Q,+ {u-h)-U}+&c
(4).
The terms on
first
the right involve the solution of a series of linear
order P. D. equations, the first of which is
Mn, + Nil,, J^ ...-au
3549
and the rest involve
Q,
&c.
imaginary roots occur in the auxiliary equation,
proceed as in the following example.
If equal or
we may
h, c,
=
SYMBOLIC METHODS.
3550
(l
527
Ex.:
+ xyz,^-^xy (l+x') z,.,,-\-4.x'i/%, + 2x (1 +x') z^ + 2y
Here
IT
=
(1
+ x")
Let the variables
since
n
(4)
cl^.
n
=
(4)
(l
+ .r)
to
so that 11
t],
^,
=
ia.-2,r^£,
= -^ =
-^
1+a-2xy
Therefore, by (3383),
(x'-l)
and the equation becomes
now changed
y be
x,
= 1,
— 2xijd„,
+ a'z =
+ a') z =
z,
(11^
= d^.
0.
0.
Therefore,
1.
J£,
from which, by separating the variables and integrating, we obtain
and, by (1430),
Also, since 11
(?/)
=
7/^
=
+ x"')
(l
0,
-^, =
Therefore
l
the solution of which
.y-
= tan~^ X
^
solution,
+
by (3523),
z
(2).
^^
=
-2xy
=
0.
-^,
0'
Thus
and
now
is
(1),
t]^—2xyn,,
equation (1).
is
The transformed equation
and the
x'y + y = A
= tan-^x + B
^
(_d2^
rj
=
x'-y
+ a-) z =
+ y.
0,
is
= <p(r]) cos ak +
sin a£,
\p (ji)
arbitrary functions of the variable, which is not explicitly involved, being
Therefore finally,
substituted for the constants (3389).
z
=
(x-y
<l>
+ y)
cos (a ta,n' x)
^
+ \p
(x-y
+ y)
sin (a tan' ^
a').
MISCELLANEOUS EXAMPLES.
3551
Put
«2.v
d2y
+
d.2z
is
= a'.
ii
Thus
=
<p
+ W2.+ M2.-=0.
tt-.^
(y, z)
+ a^u = 0, the solution
cosax + (y, z) sina.i;,
of which,
by (3523)}
tp
arbitrary functions of y and a being put for the constants A and B, Expand
the sine and cosine by (764-5) replace a^ by its operative equivalent, and,
in the expansion of sin ax, put a\p (y, z)
x (y, ") thus
;
=
u
= f (y, z)- 1^ 0?,, + fU
o
cp
(y, z)
;
+ 1^
(cZ,,
+ c?,,)
o
;
^
(y,
z)-&c.
;
[See (3626) for another solution.
3552
u,
Here
a
=
{d.,-d_,,
=
+ + =
ti^
?/_,
ciT/^-.
+ d, + d,)~' (xyz -f 0)
{d, + d,) + d.,, id, + d,y-...]{xyz + 0).
(rZ,
Operating upon xyz, we get
u
= ix'^yz—^x^ + + Ta^*,
(2
2/)
DIFFERENTIAL EQUATIONS.
528
For symmetry, take
the rest vanishing.
expressions thus
of the
^rcl
sum
of three such
;
Operating upon zero, we have, in the
instead of a constant, therefore
The
{1-x
result
(d„
d_j.Q
first place,
= xf (yz),
d.^.^O
=
<p
(yz)
&c.
is
+ d,)+hy (d. + d.y- ...}<!> (yz) =
e-^^"^^"--'cp
= 6 (y-x,
(yz)
z-x)
(3493) the complementary term.
3553
= 5), we have, by (3478),
(d, + ^)-'xyz = e-^-^d.,e^-^xyz = e--*-3^cZ.,{.. (y + x) (z + x)],
= e-^^{Wyz + :Lx'(y + z)+}x^}
Otherwise, putting
=^ ^x'
cZ^
Substitute x=.
which
is
a^,
tZ^
(y-x)(z-x)+ix'
which agrees with the former
3554
+
« Ux
+ bUy
=
z
y
hri,
-\-
cu^
= c(,
= cvyz.
and the equation becomes
The same methods fm^nish the
ttu,.
3556
.^'5!.,+/p,
-\-bUy'\- cu.
Put
TT^
= a cos f
.
.'.
TTi",
solution of
=
x'"y''z^\
= 2ajij\/a^-z\
z z^
.-.
TTV
a sin
=
y,
.".
2,13?/,
z-=.as\n(xy-{-c).
aa;u^-{-hyUy-\- czu.^nu
3557
Put
X
.-.
^H^
+
The
y]i(^
=
i,",
+ ^u^—nu =
7/
0,
=
.-.
T]'',
z
=C
= 0.
;
by (3544) u
=
(x"
,
y'',
z''
)".
solution of any P. D. equation of the type
F{.vd,,
is,
+ z-2x)+^x\
solved in (3552).
3555
3558
(y
solution.
ijdy,
zd„
...)«
= XA.v"y^''
•••
by (3488) and (3557),
W
= ^ 177
"^
T
+ 777—1
7
-,
7
<'•
(3493)
SYMBOLIC METHODS.
3559
Ex.
xu^^-^yUy
:
— au =
529
Q,,^,
where Q,,= {x,yy^ (1620).
Here
ti
terminate.
=
=
m,
(ni—a)'^Q„^+ TJa- When a
In that case, as in (3526), assume
m—a
Differentiate for a,
=
u
becomes inde-
m—a
by (1580), putting Q„
thus
this solution
iQ,„ (log X
first in
the form
+ log y) + F,„.
Similarly, the solution of
3560
xu^.-\-yUy-\-xu-—mu
n
is
=
^Q,„ (log X
3561
The
ct
solution,
=
2
by putting
(3476),
ot^j,
z
(d^-ad^)-'0
=
c.
or
+ Fo-
{d,—adyyz
(fZ.,-a(^^)-'
(y) e"^"y
= 0.
(3472)
and ^ (y) for C. The second operation produces, by
e"''"v{x<p(y) + ^P(y)}=x<p(y + ax) + ^P(y + ax).
(3492)
=
^v%^, —if^-iy
+
'J'%-
—y^u
= 0.
{;xd^ — yd,J) 3 =
m = in (3544),
—
^ = («, i/)"+
(xd^ + yd^)
This reduces to
tt
=
=
Q,^
for m,
3563
Here
w,+i/i*^+5:w^
by (3560), is
u = ic (log x + \ogy + log 2)
^2x—^a^xy+tt%y
3562
—
+ log y + log ^) + V,„.
= xd^^+yd^,
therefore
and
( i^',
j
0.
,
=
y', and so converting
the second term being obtained by substituting y'^
The above may also be written
the second factor into (xd^ + y'd^,).
F and/ being integral algebi'aic
3564
Putting y
functions.
x,,,-a%y-\-2aM,-{-2a%!:^y
=
at],
= 0.
this equation is equivalent to
(d,-d.,
+ -2ah)
(d^
3 Y
+ dr,) z =
;
DIFFERENTIAL EQUATIONS.
530
putting X
= log
and
.7/
= loy-
?;
by
this gives,
;;',
=
(354-i),
e-'^"'-F(,j
+ ax)+f{y-ax),
the functions being algebraic and integral.
3565
.-.
u
=
U2^—ahi.2y
=
=
{cl,,-a'ch,)-'
{2ad,Y'
{
I
j
y)
e"^"v fe""^";'
e"^";/
Here
3566
If
*2
=
= 0,
H =
xj/^
For the solution
^; (a^,
(',
J0
(3470)
]
(3492).
9/-«,r)
(Z.t;
+
+ «.')
fZ.i'
+ x (!/)•
2/
dx
c7//,
j-
(a;, 1/
./.
(x, y)
v/.
(?/),
the solution therefore becomes
0^', //)
«/>
^., (,r,
=|
y)
(3470)
(x, y)
}
y — ax)
=
+ ax)
(a-, ?/)
*, (x, y)
(',
y).
(x, y) dx-e-"-'"'.' (e"^"«'
(^
*i {x, y + ax) —
since
(.V,
<!>
(3515)
(,l-ad„)-'-{(l + ad,)-'
[
= (2ad^)-^
= (2a)"'
{x,
[-BooZe, ch. 16.
0/4-«^?')H~Xi (Z/~"'0
by Monge's method, see
in this case
(3433).
%^^.—aZy
3567
z= (d^— ady) -^e"'^ cos ny —
=
e"^".'/
[
cos
e""'
e
'
'"'"i'
e'"'
tii/.
cos ny dx
(3470)
=
[e'«^cosn (y-ax) dx (3492), and this by Parts, or by (1999), is
_ f^axiiy^mx
pQg
— aa?) — awsinn (y — ax) (Hr4-«V)"' +
(y)
—
=
2
m cosny an sinvy (m- + a'ii')'^ + (y + ax), by (3492).
e«^''i/
^^ (?/
j^^
e"^''.v0
]
I
.-.
e'"-'^
\
(]>
z-a::,,=
3568
,
=
(J,_ar/,,)
-^
=
e'""^'-^</>
i).
(0),
^ («) taking the place of the constant G.
Therefore
s
(a!) + ai^2.-l-iaV^4^ + &c.
=
Otherwise, to obtain
z in
powers of
a;,
j;o.-62»i
.-.
z
=
{(d,
+ ldh(d.+ hd])}-'0 =
we
=
by (3472),
(3492)
have, putting
e'''-"'(t>
(t)
+ e-'"-">x}. (0
then expand by (150).
3569
«2a -f^2//
=
Ir
= a~\
0,
^os 1UV COS mi/.
(3518)
;
531
SYMBOLIC METHODS.
=
z
Treating
(h,
and cosmy
+ c?2;,)~^ COS nx cos raij.
(t?>^.
as constants,
(diy — n-)
we
have,
by (3526), putting
cos my + A cos ax + B sin a,^,
= cos nx
= cosnx cosmy ( — m--n^)-' + (ij) cos (xd,,) +
s
\p
(p
A
and
becoming ^ (y) and
J5
3570
where
=
tt
(tt
r7,
+
f?^.
+ ia) (tt -
for a^
d.,y
by (3496),
(y) sin (*f7^),
vp (//)•
=
^,^+2zh-\---24'+(''-
Therefore
or
''
=
ia) z
i
COS
{e'C'^-'^^'
(w4+w^).
+ e-f'"**"^'},
Therefore, by (3510), with x
Ca^— (j^i + w)
— (m +
a^
?i)
=
e*,
(/
= e^
-^
a^—^m + ny
Prop.
3571
of the
I.— To transform
a linear differential equation
form
into the symbolical
form
MD) u^-f,{D)eUi+MD)e^'u^kc. = T
where Q
a function of
is
.v,
T
Multiply the equation by Qf
becomes, by (3489),
(,,
+ le' +
cr'-'
;
a function of
6,
x
(2),
=
e^
then the 1st term on the
+...)D{D-1) ...{D-n-i-l)
and
left
u.
This reduces, by the repeated appUcation of formula (3476)
with the notation of (2451), to
3572
aD^"Ui-\-h
{D-iy^^ eUi^-c {D-^Y"^
The other terms admit
e-'u-^ka.
of similar reductions.
Conversely, to bring back an equation from the sym(2) to the ordinary form (1), employ formula (3475)
so as to transfer e"" to the left of the operative symbol.
3573
bohc form
3574
Ex.
:
x' {xhi,,-\-
7xu^.+ hu)
=
=
=
g''{I)
(i»-l)
+ 6 J> + 5) « =
(X»-l)(D + 3)e"tt
e-'
(D-
e-^
+ 7D + 5} «
(D + 1) (D + 5) u
(3476).
DIFFERENTIAL EQUATIONS.
632
For
converse reduction, the steps must be retraced, employing (3475).
tlie
See also example (3578).
3575
Prop. II.
—To solve
u+a^<l> (D) e'u+a,<f>
U is
wliere
By
a function of
tlie
(D) ^
equation
(D-l)
3576
p"it
0.
for this, the equation
qi,
q.2
...
and
q„ are
A,.
becomes
the roots of the equation
=
q'l.-^
•••
iq,'-qi){q>--q-^
U
?/^
...
4-^nW„,
u,-q4iD)e'u,.=
3578
U.
Ex.:
+ 5x' + G.v') «o. + (4x + 25x' + S6x')
=
Putting X
(l
A^Ui-\-A.2Ui
given by the solution of the equation
is
3577
+ Se' + Ge^O
The
=
(qr-qn)'
by
solution will then be expressed
(x'
...
Therefore
where
where
u
(3491)
Putting
The
e''
first
e*,
u,
i>(-D-l) H + (4 + 25e'' + 3Gc-')
term
+ (2 + 20a; + 3G.r'0
D((
= D (D-1) « + 5 (D-l)(D-2)
by applying (347G).
or
therefore
if
(1
=
this
The other terms
similarly
it
;
thus, after rearrangement,
=
20e^
with {(D + l)(D + 2)}-', we get
if
+ .V + Cp^)« = e^
p = (i)+l)(D + 2)->e^
« = {3 (l + 3p)-'-2 (1 +2,))'}
= 3;/ -22,
= (1 + 3p)
= (1 + 2p) e^*.
and
<>''
-
2/
20.v».
+ (2 + 20e'' + 36e-^) = 20e^
e'^ + G (D-2)(D-3) e'"«
(D + l)(i' + 2)« + 5(D + l)-e'« + GD(i) + l)e'-'«
Operating upon
u
and transforming by (3489),
'
e="'
.-
"'
-
;
;
SYMBOLIC METHODS.
Hence
(l
+ 3p)y
=
y+S
or
e''
533
(D+l)(D + 2)-' e'y
=
e''
(D + 2) y + 3 (D + 1) e'y = e"> (S-\-2), by (3474),
by (3475)
(-D + 2) y + Se' (D + 2) y = 5e^
(x + Sx')y, + 2(l + Sx) y = 5x'.
therefore
or
;
that
is,
(x
Similarly
+ 2x')
2,
+2
(1
+ 2.v)
=
Solve these by (3210), and substitute in m
3579
— To transform
Prop. III.
=U
u =
put
Proof.— By
3580
(8474), because
Prop. IV.
?f+(^(i>)
tlie
(D)
equation
U—
and
e'^^v
= 5x\
— 2z.
v-\-<f>{D^n) e'v
into
u+(f>{D) e''u
z
3y
e'^^'-^'tJ
=
=
V,
e^^ V.
e»> (D + n)
e'^v.
— To transform the equation
e"-'
u
=
t/
v+V'(i>)
into
e^^'v
=
F,
put
P
where
3581
HD) ^JD-r)
HI>)
<l>{B-2r)
...
—
Proof. Pnt u =f(D)v in the 1st equation, and e'''f{B)v =f{D—r)e'''v
(3476). After operating with f'^ (D) it becomes
v
+ <l>(D)f(D-r)f-'(D)e'-^v=f''iD)U,
therefore
and so
(p
in inf.
Also
(D)
f(D-r)
U =^ f (D)
/"' (D)
=
4^
(D)
by hypothesis
V.
3582 To make any elementary factor x(^) ^^ 'Pi^) ^®'
come, in the transformed equation, x (-^ i ^^'0' where r is an
integer; take ^ (D)
xi^ + nr) Xi(-^)- See example (3589).
=
3583
To make any
factor of
(p
{D) of the form
—^y /^
=
disappear in the transformed equation, take '^(-D)
xi{D),
where Xi(-^)' ^^ ®^^^ case, denotes the remaining factors of
<^(X>).
See example (3591).
.
DIFFERENTIAL EQUATIONS.
534
3584 III tlie application of Proposition IV., differentiation or integration will be tlie last operation according as
^j-
/^7^
lias
(3581)
its
factors,
after
reduction,
in
tlie
numerator or denominator, and therefore according as \p (D)
is formed by algebraically diminishing or increasing tlie several
factors of (p (D). However, by first employing Proposition III.,
the given equation may frequently be so prepared that the
final operation with Prop. IV. shall l^e differentiation only.
See example
For further
(1).
investigation, see Boole's Dif. Eq., Ch. 17,
and Supplement,
p. 187.
To reduce an equation of the homogeneous class
(3531) to a binomial equation of the same order of the form
3585
The general theory of such solutions
the given equation be
it+q {(i>+«i)(i> + «2)
%,
w
rtg,
...
ctn
=r e'^'^v,
'V
...
is
{D+a,)}-\^'Ui
=U
being in descending order of magnitude.
by Prop.
Let
as follows.
...
(1),
Putting
III.,
+ q{D{D-7i:;^,)
...
(Z>-^^7=^J}-^^«^•
= e"^'U...
(2).
To transform
these factors, regarded as ^ (D), by Prop. IV.
into ^{D)=D{D
l)...{D-n l), we convert
into
+ rn
(3582), r being an integer.
Hence for the j/^^ factor we must have
—
and therefore
D
— ai-{-aj, = D— + 1,
= rn -\-p — 1
a^ —
D-\-rn
3586
D
+
j;
(3)
iip
If this relation holds for each of the constants
equation (1)
is
which, by (3489), is equivalent to !/„.,. + </'/
being found in terms of x from the
1/
V being
3588
while
=
„
P, /
=
... rt„,
reducible to the form
u^-q{D{D-l)...(D-n-\-l)]-\"'i,=^
3587
a^
^
(^
ij
=
1,,.,.
Y
(4),
= X.
last equation,
and,
(3580), the solution will result from
(/;-i)(_^>-^)y(/^-"+i)
!-...»P,.
U and Y are
,;
connected by the same relation as u and
y.
>
SYMBOLIC METHODS.
3589
I^x- 1
Putting X
Given
:
= e"
./•>«3.,
635
+ 1 8a;-«,, + 84to. + 96a + Say'u =
{D (D-l)(D-2) + 18D (D-l) + 84D + 96} u + 3e'ho
(D + 8) (D + 4) (D + 3) « + Se'' tc = 0,
or
0.
and employing (3489), this becomes
therefore
« + 3 {(D + 8}(D + 4)(D + 3)}-^e^^^«
Employing Prop.
III.,
therefore (3476)
To transform
=
(1).
= e~^"v,
+ 3 {D (D-4>)(D-5)}-' e''v
v
this
u
put
= 0,
=
(2).
=
(3),
by Prop. IV. into
y + S{D(D-l)(D-2)}-'e'^y
we have
P
D(P-l)(J-2)(D-3)(J-4)(J}-5)
tm = Z)(D-4)(D-3)(I»-3)(D-7)(Z>-8)
...
'4,{D)
i;= (D-l)(D-2)
.-.
2/
ii
.„_! wr,_9>)
"^'
'^
^
= e-''(D-l)iD-2)7j
(4),
obtained by differentiation only, performed on the value
as obtained by the solution of (3), that equation being equivalent to
and the solution
of
.-.
;/,
^
...
is
D(D-l)(D-2)y + -Be''y =
0,
or,
by (3489),
y,,
+ Sy =
0.
however. Prop. IV. were used to pass directly from (1) to (3), we
should have
<p(D)
I)(D-l)(D-2)(D-S)(D-4)(D-b) ...
If,
^
'4^(D)
(i)
+ 8)(D + 4)(X» + 3)(D + 5)(D + I)D...
(i'
+ 8)(D + 6)(D + 4)(D + 8)(D + 2)(D + l)'
1
and equation
(4)
would involve integrations of y as high as
—
3590
D'^^y.
Note. By the literal application of Rule IV., the right side of
2)}"^
but no such term is
equation (3) ought to be F
{(!> -1)(Z)
required when the original and transformed equations are of the same order,
for in such cases the arbitrary constants introduced by the operation upon
zero disappear with the terms containing them in the final differentiation
The result is the same as if the operation upon zero had not been performed.
In the following example, V has to be retained.
3591
—
=
(x-x')u,,+ (2-l2x')u,-'30xu
Ex.2:
;
=
Multiply by x, transform by (3489), and remove
function of B by (3476), thus
(1).
e"'
to the right of each
M =
u— (J + 4)(J + 3)„,,,_
D^D + l)
Transform
this
(2).
by Prop. IV. into
v-^±^e-v = V
We have
?i
== P,
^^
u == (-D
+ 4) (D + 2)
(8).
n,
7= {(D+4)(D + 2)}-iO = Ae--' + Be-''
The operation upon
zero
is
(3518).
required in this example (see 3590), because
(8)
DIFFERENTIAL EQUATIONS.
536
but only one term of the result need be retained,
is of a lower order tliun (2)
because only one additional constant is wanted. Hence (3) becomes
;
(D + l)v-(D + S) e-%
Changing again
to x, this equation
=
(D + 1)
(x'-x') v,-4:xh- + A
=
0.
obtained from this by (3210) will contain two arbitrary consolution of (1) will then be given by
The value of
The
stants.
v
u=
3592
= -Ae-'\
Ae-'-'
becomes
Ex. 3
W2«-«
:
(D + 4)(Z> + 2)r.
(h + 1)
= 0,
x-'u-qho
[Boole, p. 424.
n being an integer.
Multiplying by
x"
and employing (3489),
this
becomes
u-cf {(D + n)(D-n-l)}-^ e-'u
This
is
changed by Prop.
v-q'{D(I)-2n-l)}-'e''v
and
=
0.
III. into
=
0,
with m
= e'-'V,
by Prop. IV., into
this,
or
y-q'{D(D-l)}-'e''y =
t/,.-5V
we then have
^-'^
=
(3489).
y being found from this by (3524),
= e-'-P. — — y = e-'-(D-V)':,y = x-"(xd^-my.
Zn 1
JJ
F (xd,-m) = x"'F (xd,) x-'\
(3484),
x'"-' (xd^^) x'^'^'y,
w = X-" X (xd,) x-\x' (a;4) x-'
u
But, by
.
...
.'.
u-=x-^"''^(x'd,yx-"'*'y
or
This
Vol.
may
= X-'"' (x'd^y x--"-' (Ae'^ + Be-"') (3525).
(See Educ. Times
z = x'-.
be evaluated by substituting
3593
Ex. 4
n,,-a'u,,-n (n + 1) x'^ u
:
=
0.
solution is derived from that of Example (2), by putting q
arbitrary functions of y after the exponentials instead of A and B
The
and
Reprint,
XVII., p. 77.)
to
= X-'"' (xM^y X-""' {e"^"." (y) + e
= X-"-' (x'd^y X--" " {0 + ax) +
"'"
./)
(2/
>//(?/
-4 (y)
=
;
ad^^,
thus
}
+ ax) },
by (3492).
[Boole, p. 425.
(l+ttd-) u.,,'\-axu,±nUi
3594
To
= 0.
solve this equation or its symbolical equivalent obtained
by (3489),
viz.,
u^.^liJ^=g^e-u = il
3595
Substitute
/
= [-tt^—^
v/(l + aa;^)
J
i" the soluticm of
u.t^n-n
= 0,
by (3523-4).
—
SYMBOLIC METHOBS.
3596
or, the
Similarly, to solve the equation
same
Substitute
(3596)
3598
is
in its symbolical form,
t
=
•
r
in the solution of w^,
±
?t*i(
=
0.
the symbolical form becomes
0,
= 2,
w be not
b(D-n)(D-n-l) + e(D-n) +
aD{D-l) + cD+f
substitute
Oj, aj
nd,
'
and therefore
2cfe
"-^
^^^'
= nd^..
(2).
are the roots of the equation
h
/3i, /3j
20'=
c,
„+^i|5^1(|5|le-. =
Thus
3599
and
—7—^dx
Pfaff's equation,
"^
where
[
Jx^(x^ + a)
obtainable from (3593) by changing 6 into —0.
When Q =
If
537
(^na-v)(^na-n-l) +e (^na-n) -{-g
=
(3),
are the roots of
a |n/3 (|7i/3-l)+c i"/3+/
=
0.
Four cases occur
3600
I.
—
Oi— Qj and /3i— /Bj
If
Prop. IV. (3581) to the form v +
^
are
odd
integers, (2) can be reduced
^^~"'^ ,^^~"^~^^
and then resolved into two equations of the
first
e'^'v
=
by
0,
order.
—
II.
If any one of the four quantities Oi— /Jj, aj— /Pj, «2 — /3i, fj—Z^a
is an even integer, (2) can be reduced by Prop. IV. to an equation of the
first order.
3601
3602
III.
—
If Pi—Ij-j
Props. III. and IV., (2)
3603
IV.
—
If
aj-oj and
reducible in like
manner
— The
integers
Note.
may
— —
aj + uj
/jj are both odd integers, then, by
/3j
reducible to (3595).
and
is
Oi
+ Qj— /3i—
to (3597).
may be
/Sj
are both odd integers, (2) is
[Boole, p. 428.
either positive or negative,
be zero.
3 z
and when even
DIFFERENTIAL EQUATIONS.
538
SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
BY
3604
/o
in
ao
Case
I.
— Solution
(D) u-f, (D)
e^'
u
SERIES.
of the linear differential equation
=
or
u-f, {.vcQ
f, {.vtQ
.v^
u
= 0,
which fo{D), fi{D) are polynomial expressions of the form
and f,{D) = {D-a){D-h){D-c) ....
+ chD-\-cuD\.:-\-a,D'^
Let </>(!))=/, (D)-f-/o(D), and
3605
3606
^
(«)
= 1+
3607
Proof.
27-)
<!>
-{-<p (rt
Then the
let
+ r) X' + {a + <p{a + r) x'^
+ 3r) {a + 2r) {a + r) x^''-{-&c.
<^ (rt
cp
<p
solution will be
u = A.v'<P{a)+B.v'^{b) + av'^ic) + &c.
— Operating with f~^ (D) and writing for ^ (D)
p
e''",
= f-' (D) = Ae"' + Be''' + &c. (3518)
by (3515), « = (l-p)"' (Ae''' + Be'"+ ..,)
= (l+P + p'+...)AG"' + (l+p-^p'+...)Be'' + &c.
u -pu
Therefore,
Now
in each terra substitute
formula (3474).
Case
II.
for p" the value in (3491),
where
A(D)
=
...
+/,(D)
{D-ct){D-h){D-c)
c^^'u
(a) := 1
where the
determined in succession by the formula
3609
/o (m) v,„+f, (m) r„,_:
The
3610
=
^.r«*(a)
Hero
and
.
.
+/. (m)
i'„,_,
=
and
v,
=1
+ /^.f'*(«')-f CV*(c) + &c.
— From (1)
«=
where
.
&c. are
solution will then bo expressed by
u
Proof.
=
...
+ F, (a + l) x-\-F^ {a + 2) x' + Slc,
coefficients F^{a-{-l) or v^, i^2(« + 2) or v^,
^
Let
by
— Solution of
MD) u+MD) e'u+MD) e~'u
3608
D
and remove
{l
+ 0.(/J)e"...+0,(D)e'-}-7o-'(^)-O
(3),
= fr{D) ---fo{D).
...}-'
=
= Ae"" + Be^' +
(3518);
{(D-a)(D-h)
fo-'(r))0
{l + ^,(D)e"..,+./.„(D)e-j-' = l + F,(D)e« + i^,(X»)e^^+...
l'r(D)
.
.
.
:
SOLUTION OP LINEAR DIFFERENTIAL EQUATIONS. 539
To determine F^, F^, &c., operate upon each side with {l+0 (D) e' + ctc.},
and equate coefficients of powers of e thus formula. (2) is obtained.
(3)
now becomes
(4).
{l + F,(D)e' + F,(D)e''+...}(Ae'"^+Be"^+...)
;
u=
Multiply out; apply (3-i74), and put x for
3611
or,
xht2x—(a + h — l) xio^+ahio — qxhi
Ex.
by (3489),
Here
=
f,(D)
{B-a){B-h) ti-qe^'u
=
A (D) =
0,
(D-a)(D-b),
(m—a)(m—b)
v,„
=
therefore F^, F^, &c. vanish, and ^o («)
=
1»
Therefore (2) becomes
F,(a + 2)
=
F,^a + ^)
=
qFg
we
Similarly
A a
^-^
The
--^M^
(a+4-a)(a + 4-6)
4.2 (a + 4-&)(a + 2-Z>)
^^'^°''
^?-^"''
I
I
is
—
= 20^^^'
series
±
<^(a
*(''
+ 4)
*=
*) =
= 4-(^;^^'
+
by the value of *
(a).
=
Similarly with
$ (6),
B—a,
the corres/o(-D) lias r factors each =:
of the value of u in equation (4) will produce
^o+"4,log.^^+^2(log.^f
coefficients A^,
But
We
(D-aXB-hy
When
where the
'"
arrived at more quickly by formula (3607).
{!>)
ponding part
3615
F^{h + 2), &c., and thence ^(Z^); and, sub-
I
solution
producing the same
then Gr
4.2(a-6 + 4)(a-6 + 2)
^2(6-a + 2)^4.2(6-a + 4)(6-a + 2)
(»+2)
3614
-.2,„4
qr
+ 6-2)
+2(a4-6-2)"^4.2(a-6 + 4)(a-i + 2)
have
3613
q
we have
^
3612
f,(D)=-q.
2(a + 2-5)'
find F^(h-\-2),
stituting in (3610),
_
u-
0,
qVin-2,
_
(g)
=
0.
(a + 2-a)(a + 2-Z>)
^ (a) = 1 +
2(a
Therefore
e".
if
A-^,
...
...
+^._x(logcr)-S
are each of the form
any one of the quantities F, (a + r)
also.
=
(3608),
DIFFERENTIAL EQUATIONS.
540
Proof.—/"' (D) now contains a term of the form
=
e''"(co+Ci0...+c,e'-O
The corresponding part
of
ii
{l
=
[e'''
+
e^''*'^'F,(D
t^'v, say.
in (4) is
+ ^i(D)e''+...}
e'"'v
+ a + l)+c"''"'F,{D + a + 2)
+ ...]v
by (3475).
F
function
by Taylor's theorem in powers of D, operate upon
and arrange the result according to powers of 0.
Expand each
v,
In practice, proceed as in the following example.
3616
Ex.
a;u2;,-{-u^+cf^u
:
= 0.
Multiplying by z and changing by (3489), this becomes
DV + s'e'" u = 0.
I>-u
=
gives u
= A + Bd.
B as variables,
Substitute this value and operate with I), considering A and
thus
and equate to zero the coefficients of the powers of
;
B'A + q'e^'A +
Then change
D into m,
and
WB =
B'B + q'e'^B
0,
e'^J. into a„_r, to
m^a„,-^q^a^.i+2'mb^
=
;
= 0.
obtain the relations
m''b„+ q^b^.^
=
0,
which determine the constants successively in terms of a^ and
l^
(which are
arbitrary) in the equation
u
=
ao
+ aiX^ + atX*+ ...+\ogx (bQ + b2x'' + h^x*+ ...),
which thus becomes the solution sought.
[J5ooZe, Biff.
Eq., p. 439.
SOLUTION BY DEFINITE INTEGRALS.
3617
La2)lace^s method.
— The solution of the equation
^(^((/,)w+^(c/,)i.
=
(1)
u=C^{e'''^S'^'\cf>t)-'}dt
is
(2),
the limits being determined by
/'*i«^'=0
Proof.— Assume
= ^(0e^'
t*
=e
e'^Tdf,
(3).
and substitute
in (1), putting
c''
(3474), thus
{xe^'f (0
Tdt+
[
c"xP
(0 Tdt
= 0.
• This mothod of solution is meruly indicated here, and the reader
Ch. xviii., for a comploto investigation.
jD//. Hq.,
f (cZ,)
is
referred to Boole's
—
SOLUTION OF DEFINITE INTEGRALS.
Integrating the
e'^cp
first
(0
term by
parts, this
541
becomes
r- je-' [cZ, [<|>(t)T}-^p (t) T^t^O
(4),
an equation which is satisfied by equating each term to zero. The second
term thus produces a value of T by integration by (3209), and this value
substituted in the first term, and in the value of u, g-ives the results (3)
and (2).
3618
a^u,:,+au^'-q\vu=0*
Ex. (1):
= c^a,— j^
Here
(4)
and (8) become
<}>
(2)
u
=
;//
-
(d^)
ad^,
<p
G{e''(f-q')^''dt;
a being positive, the limits are
t
= Joq,
(t)
=
e''' (t'
(4).
f^-q",
;//
- q')^ =
and, putting
t
(0
=
Hence
at.
0;
= qcos
we
d,
find
u=z C V"'^sin"-'etZ0
(5).
I
3619
The solution in series by (3608)
(2) of that article are in this case
D(D + a— 1)m— gV^M =
Thus, a in (3608)
"
=0, and
I
and
= 1-a.
is
as follows.
Equations (1) and
m (m + a — l)vn—q^Vn-2 = 0.
Therefore (3610) becomes
= ^{^+2^) + 2.4(a + li(a + 3)
+^^-
^^-^-1^+2|g^ + 2.4(3-ai(5-.)
Both
convergent by (239
series are
The
results
3620
(5)
-^^-}
^'^'
ii.).
deduced by Boole are these
is
equivalent to the particular integral represented by the
first
series of (6).
3621
be,
when
A second particular integral, by assuming u =
2 — a is positive,
u = C^x'-" e'^^'^sin'-^flcZfl
e'^'^'^t;,
is
found to
(7).
I
Jo
3622
When
a
tt=
*
The method by
important equation.
lies
between
and
2,
the complete integral
is
(7ijV^«"'«sin"-^0(Z9+C2a;^"'[V*'='"«sin^-"^(^6l
definite integrals is elucidated
by Boole
(8).
chiefly in the solution of this
DIFFERENTIAL EQUATIONS.
542
3623
But,
a
if
= l,
the solution becomes
u=
fV*'°««
3624 If a does not lie
= a—2n, and replace the
a
(7,
(xd,
{A + Blog(xsm''d)]
and
between
term
of (8)
fix'st
+ a-l)(ixd, + a-S)
...
(xd,
2,
dO
(9).
a be negative, put
tlieu, if
by
+ a-27i + l)
[V"^^sin<^'-ic^0
...
(10),
the transformation being eflFected by (3580).
And
3625
verts a into
one.
if
=
=
This conx'^'''v.
e^'^-''''''v
a be positive and > 2, put u
a negative quantity, and the case is reduced to the last
2— a,
3626
Ex.
wlien r
=
(2).
— To solve by
metliod
tlie
P. D. equation
+ U2z=
ihx+i(2y
\/{x'^
tliis
(1)
+ i/)'
Eliminating x and y, (1) becomes
rU2r
Now
+
l('r
+
the solution of this equation
change x into
r,
q into id^,
A
and
and
'>'^''2z
is
B
=
(2).
(9) of Example (1),
into arbitrary functions of z.
number
if w^e
We
thus obtain
n=
or,
[V<=°^^^'^={./)(2)
+ ;//(2)log(rsiu=e)}
dd
(3),
by (3492),
M=
^j
j^ + tVcoseJ fZ9+
;p(2 4-tVcos0) log(rBin-^)
c?^
(4).
See (3551) for another solution.
3627
and
if
If
it,
« be the potential of an attracting mass at an external point,
then, since log r = oo
when r =
^ (z) must vanish
= F {z)
;
therefore
F(c)
=
Hence
«
=
—
(4) reduces to
,
<\,
I
JT"
(s)
[
dd
;
= tt^ (z).
-.fiV cos Q
\
dO.
ParsevaVs Theorem.
3628
and
If, for all
values of n,
A'-\-B'ii-^^G'u-^^
...
= ^p{u)
(1),
DIFFERENTIAL RESOLVENTS.
543
then
AA'+ BB +
Proof.
and add the
+
<^
(e'^ 4
results.
dO.
(e"''^)]
=
(1), and in it put u
Multiply by dd, integrate from
e'"
and
to
e''"
and
tt,
EQUATIONS WITH MORE THAN TWO
INDEPENDENT VARIABLES.
By means
3629
(e^-^) tA (e'-0
27r.
D.
P.
L^^^[<l>
—Form the product of equations
separately,
divide by
=
...
of Fourier's
theorem (2742), the sohition
of the equation
may be deduced by
ti
=
(1
+
f/,)
a general method in the form
(((((( e'^^'-'^^^'^
xjj
{a, b, c)
dadbdcdXdiidp,
—
oo to go
and the functhe limits of each integration being
tion ^ being arbitrary and different in the two terms arising
from the operator {l-\-df).
Boole,
matique,
Ch.
Tom.
xviii.,
and more
fully in
,
Cauchy's Exercice di'Anahjse MathS'
pp. 53 et 178.
I.,
Poisson's solution of the same equation in the form of
a double integral is
3630
7i=(l + dt)\
I
^sin.^4'
O'^
+ ^^'^siulsinT?,
?/
+ /iLsin^cos
with the same latitude in the interpretation of
/?,
z
+ ht coa ^) d^dr/
xp.
[Gregory's Examjyles, p. 504.
DIFFERENTIAL RESOLVENTS OF ALGEBRAIC
EQUATIONS.
3631
Theoeem
I.
(Boole).
—
''If y^, y^ .-.Vn are the
n roots
of the equation
y^-air'-^l
and
if
=
()
(1),
the m^^ power of any one of these roots be represented
—
DIFFERENTIAL EQUATIONS.
644
by
and
u,
if
= g\
a
then u as a function of
B satisfies the
differential equation
„-(!izix)+i!i-iy"-'Y^_»_i)rz)<»)-i-.e-„
n
\
n
\n
/
and the complete integral
" Cor.
3632
I.— If
n
of the
m=-l
/
same
and
if
will
n be
J
•-
= o,
be
> 2,
the differential
equation
/)(»-.)
„_i (!i^ i>- ± _iy""'e.»„ =
n \
has for
y iV
•••
its
n
n
I
general integral
Vn-i being
any n — \ roots
of (1).
" If be changed into —0, and therefore
above results are modified as follows
D into — D,
the
:
3633
" CoE. II.— The differential equation
has for
its
complete integral
u
Vii
2/2 ...
=
C,y'^-^ C^yl
+ C^yZ,
Vn being the roots of the equation
ay'-?/"-^
3634
. . .
+a =
(2).
" CoK. Ill,— The differential equation
„-„(/>-2).»-..[(!i^i>+±)'"-"]-V«„
supposing
7i
2/i>?/2 ••• 2/n-i
>2
has for
being any
its
com})lete integral
n—\
roots of (2),
= 0,
BIFFEUENTIAL RESOLVENTS.
" Theorem 11. {Ilarley).
3635
^
—
" The differential equation
-"^
\
is satisfied
n
by the
n
??i^''
power
of
n
\n
I
any root
if—xif-'^-a
u
545
I
of the equation
= 0,
beino; considered as a function of x.
3636
"
Cor.— The
is satisfied
by the
differential equation
Wi^""^
power
?/'*— ////"-'
of
any root of the equation
+ — !) X = 0."
(«
[Boole, Biff. Eq., Sup.
191—199.
See also Boole, Phil. Trans., 1864; Harley, Froc. of the Lit. and
Phil. Soc. of Manchester, Vol.11.; Bawson, Proc. of the Bond. Math. Sac,
3637
Vol.
9.
4 A
CALCULUS OF FINITE
DIFFERENCES.
INTRODUCTION.
III this branch of pure mathematics a fuuction
(,')
denoted by %., and (j){x-\-h) consequently by u_^+,,. The
If Ax denotes the increment
increment h is commonly unity
h, and A^^^. the consequent increase in the value of ?;^.5 we have
3701
(/>
is
.
3702
3703
A«.,
When Ax
= «,+^,,-w,,.
diminishes without limit, the value of
Ai/,
—
-
3708
AND
FOBMULJE FOB
FIB-ST
Hence the
difference of a rational integral funcconstant.
71th.
tion of tlie nth deo'ree
is
7;*"
DIFFFBENCES.
547
= 1.2.3...7i.
3709
So also
3710
Notation.— Factorial terms are denoted as follows
A".r"
^
3711
3712
x{x-l)
Thus
=u\r-l
{x-m + l) = x^'-\
...
1
3713
Hence
\m, ml,
and
m^'"^
-...i-m)
are equivalent symbols.
3714
According to (2452), x''"' would here be denoted by a;'.'"'.
suffix, however, being omitted, it may be understood that the common
ence of the factors is always —1.
and,
if
3718
m<Cn,
= 0, since
= -ma.^^-"-'\
A^i-^'"^
A.v^-'")
3720
AmI;"^
3722
:
Ac
=
if
A\v^->"^
=
c
=
The
differ-
constant.
(-m)^"^^^-'"-"^
= {u..i-u.-,n^^) "i'"-^^
Ex.:
A(ax + by'"'>
=
am(ax + by"'-^\
^(ax + hy"">
Alogw,=:logh + ^^(,
3724
3726
Art-^-
owoo
3728
**^^
= {a — 1)
AM^iu/
7\
A"
(«ct"+^)
^
cos^
I
'
Pr.oOF.
—
A
sin
(art;
+
«'',
A'Vr"-''
?>)
•
2/
= logiiH±2_.
Alog<-i)
=
= /o2sm—«\"siii
cos
V
= -am (ax + hy-'"-'^\
{a"'
^
^
— iya"'-'\
,
/
,
rtci'+64'
—^^-r^—^f.
n(a-\-'rr))
2
I
)
= sin (ux + a + b) — sin (ax + b)
= 2 sm ~ sm \ax-\-b^ —
.
j
That
is,
A
is
equivalent to adding
———
2
sine
by
^ 2
sin -.
2
to the angle
and multiplying the
CALCULUS OF FINITE DIFFERENCES.
548
Conversely, tlie same formula holds
be changed throughout.
3729
3730
then
3731
good
x
=
sign of n
GO
=
<j>
+
(0)
A(/) (0)
.r+
^^^
.v^'^
+ ^^^
= 0,
x
A"'/'(0)
^'^)
+ &c.
AcV=/^ instead of unity, the same expansion holds
If
if
tlie
EXPANSION BY FACTORIALS.
denote the value of AX-v) when
If
(^
if
for A"<^(0)
we
write (A"<^(<(')-r/^")^^o; that
is,
making
after reduction.
—
Proof. Assume
Compute A0 (a;), A^^
(a?)
(a')?
= aQ + a^x+a^x^-^ + a^x'^'-^&c.
&c.,
and put x=:0
to determine
fto,
ai, aj,
&c.
GENERATING FUNCTIONS.
3732
then
If
(i)(t)
he the general term in the expansion of
called the generating function of n^. or (p{t)
=
is
0—t)-'=G(x + l),
Ex.:
expansion.
=
3733
Gu.,
3734
GA»,,
Proof.
for
a!
+
= (l-l)<^W,
—
-E*
l
is
G».,«=M),
<i>{t),
G^u^=
tbe coefficient of
Gh,.,„
...
GA"»,,
...
Eu^,
27,
A,
AND
=
H.,+1
=
jf.,+Ait.,
E^ + A =
1
—
Eu^
(1520),
letter
d
= M.
d^.
=
(1
+A)
E'
3736
* The
the
= (l-l)V(0.
and A both follow the laws
commutation, and re;petition (1488-90).
By
Gi',^.
in
denotes the operation of increasing k by unity,
The symbols
Proof.
t''
Gu^^^—Qu^, &c.
THE OPERATIONS
3735
(p{t),
'?f.r^'"
is
A.r
=
a^.x
c-^-
or
=
u^ + (lv^ix+h^2x^''^+
=
(l
t/..,.
of distribution,
e^.*
:^-^<-^ix^'x
2.3
+ '7,,+H.,+ ^jh. + &o.)
+ &'Cu,
= e"'u,.
being unity.
reserved as a symbol of differentiation only, and the suffix attached to
Sec (1187).
it indicates the independent variable.
FOBMULzE FOB FIRST AND
= 6'^^!
A
Hence
3737
3739
u"'
DIFFERENGBS.
549
D = \ogE.
and
Consistently with (3735) E"^ denotes the diminisliing
X by unity
For
thus
;
= u^,
Eti^.i
Ux+n
"2^'^^
3740
=
E~'^n,.
u^^^i.
= E'^tCj;.
w^.i
.".
terms o/u^ and successive differences.
«^.+«
= u,-\-n^u, + C
—
Proof.
(i.) By induction, or
the symbolic law
A2?f,+C
2)
(».,
A%,+&c.
(w, 3)
by generating functions, or
(ii.)
(iii.)
by
:
=
Gu,,,
(ii.)
Expand by
(i)'V (0
= [ 1+ j -i)
(
I
'V(o-
the Binomial theorem, and ajiply (3734).
= E««^=(1 + A)«tv
«,,„
(iii.)
Apply the laws in (3735) by expanding the binomial and
operation upon %.
Conversely to express
3741
A"Jf^.
=
/y'u^.
in terms of
tf,^^^^--nff,,+„_!+
AX- =
Proof.—
Expand, and apply (3735) as
C (71, 2)
7?.,+i, v^^.+c,,
u^,,,
w.,+«_2
. . .
(-1)"
&c.
u,.
(^-l)""a; (3736).
Putting
before.
AX = ^K-nu„-,+ C,,,u,_,
3742
distributing the
.»;
= 0,
...
we
also have
(-1)" ii,.
3743
A"cv"^
=
3744
A'^O'"
= M™-n (7i-l)"^+C(w, 2)(w-2)'«
(,t'+?j)'"-« (.v+n-l)'"
-\-C (71,2) {.v-\-n-2y"-&c.
-C{n,3){n-3y"-^&G.
3745
Ex.
:
By
(3717) A"0"
=n
Hence a proof
!
theorem (285)
of
obtained.
A'%,v,,=
3746
where
E operates
Proof.
3747
Expand
A?(,_,,v ^
Ex. (1):
=
{EE'^iyu,v„
only upon
Uj, +
i
i\,
,
i
—
u^^
i\
if^
and E' only upon
= Euj.
.
E'v^
— n^r^ =
i\^,.
(EE' — 1 )
ApiMcations of (3746).
{~iy{\-EE'Yu,v,.
^''u,v,=
the binomial, and operate upon
tlie
subjects
u-^,
v^
;
thus
u^ v^..
is
—
,
CALCULUS OF FINITE BIFFERENGE8.
550
374:9
Kx. (2)
A"
a""
sin
To expand cfsmx by
:
=
a)
|
(1
_E?
+ A') — 1
1
"
successive differences of sin
sin
a""
a;
=
|
^ + ^^'
}
""^
a-.
sin x
= A" + «A"-'J?A'+(70^2) A"--E-A'H&c. a^sina;
= A" a\ sin + /iA"-^ a^ A sin + C 2) A''--a^"- A^ sina; + &c.
= n"'! (a — l)''sina' + ?i(a — l)"''«Asina)+6'(ii, 2)(a — l)"'-o^A^sina! + &c.
}
{
'^
.c
by (3727),
(ii,
a?
known from
wliile A'' sin « is
3750 Ex. (3) To expand A"Mj,?;j, in
E = l + ^, E'=l-^^' in (3746), thus
:
wliicli
|
(3728).
differences of u^
and
v^ alone
:
put
A«M^'y^= (A + A' + AA')"m^.v^,
must be expanded.
A"Ux in
differential coefficients of \\.
3751
A«w.,
= C%+^i^/r'+^2C'?r,-+&c.
Proof. —
A"»^.
Expand by (150) and (125)
=
(e"-"
as if
— 1)"?^,.
cZ^
(3737).
were a quantitative symbol.
See also
(3701).
-r-Yi
i^'>^
successive differences of u.
g={log(l+A)}"«,
3752
The expansion by (155) and (125) will present a series of
ascending differences of u.
Pkoof.—
3753
If
3754
e""=:l+A,
Ex.:if%
G be
=
i,
^=
.-.
A^^
J,
=
log(l
+ A).
2^'Y~~¥'^^'''
a constant,
i>{n)C=cl>{A) €=<!>({))€ and
<PiE)C
= <t>{l)C,
Since every term of (1>(D), or of
(A) C, operating iipon C, produces 0;
and every term of (E) operating upon
produces G.
(j)
IIERSCHEL'S THEOREM.
3757
3758
<p{c')
= <l>{E)c^^-^
=ci>0)+ct>(i£)i)j-j-ci>{E)iy^.^+&c.
,
tNTEEFOLATION.
Proof.— Let
=
=
^ (e^
A,e'-'
A^^-A^e*
+ A,Ee'-'
= <^(iJ) e°-^ =
= 0(l) by
and 0(i;)l
...
...
+^„e"*
+AnE"e'-^
(E)
1
{
551
+ O.f +
=
(A, + A,E
j*^ +&C.
...
+A„E„)
}
(3756).
A THEOREM CONJUGATE TO MACLAURIN'S
3759
<l>{f)
e"-'
(1507).
= <l>{D)e'-^
=^{0)+<l>{(QO.t+<j>{(h)0\^-^&G.
3760
= ^ (log i') e°'
<p{t)=<l> (log eO
PiiooF.—
= </.(D)e°-'
(3738)
=^
(-D)
,/;(D)l:=</)(0)
and
n being a
(3757)
(1 + O./+ +&c.|,
(3754).
positive integer,
3761
"
~
"^
^At'"
1 2
.
Proof.— By
.
.
.
(yi+ 1)
(3758), putting
f/a'"+^
(e')
"^ 1.2... («-}-2)
"^
^/ci""^-^
= (e'-l)",
(e'-l)"=(i;-l)"0./+(:EJ-l)"0-.-^'-+&c. =:A"0./ + A"0^^
Put
t
— d^,
+ &c.
and employ (3736) and (3737).
INTERPOLATION.
Aiyproximate value of u^ in terms of n particidar equi'
distant values.
3762
n—1,
an integral algebraic function of the degree
0, and
vanishes, and therefore by making x
If u^ is
^''u,.
=
writing x for n in (3740),
3763
"...
=^ u,-]-.rAu,-\-C,^,A'u,
is formula (265).
&c., corresponding to «,
This
3764
...
+C,,,_iA'-^w,.
The given values
h, c,
are
For an application of the formula to the problem
example x = 1-54 and u^ = log 72-54.
see (267), in whicli
u^,,
Aii^, A^ii^,
...
of interpolation,
CALCULUS OF FINITE DIFFERENCES.
652
3765 When tlie term to be interpolated is one of a set of
equidistant terms, employ (3741). A"?(^;
0, as in (37G2) ;
=
tliercforc
3766
'^-""«-i+^.,2*^.-2-C\3'^.-3
3767
Ex.:
From
siuO, sin
sin 45°,
SO'',
...
and
+ (-1)"'^ = 0.
sin 60', to
deduce the value
of sine \h\
The formula
sinO— 4 sin 15° + 6 sin 30°— 4 sin 45" + sin GO^ =
-4sinl5°+3-2y2 + iN/3 = 0,
sin 15" = 1(6-4/2 + 73) = '2594.
gives
or
from which
The true value
is
"2588
0,
the error "0006.
;
LAGRANGE'S INTERPOLATION FORxMULA.
3768
wliicli
Let
ft,
b, c,
be
...h
the values of
are
?/.^,
r^
x, not equidistant, for
then generally
values of
known
;
3769
"-
^^'-(a-b){a-c)
...
{a-k)'^^^'lb-a)(b-c)
[b-k)
...
'{k-a){k-b){k-c)..:
Proof.
— Assume
i(,„.,
+ B(x--a){x-c)
and determine A, B,
...
Ji)
(x-k) + C(x-a)(x-b)(x-il)
C, &c.
If the values
= A (x — h)(x — c)... (x —
by making x
of a,b,c,
...
Jc
=
a, h,
c,
...
(x-Jc)
+ &c.,
&c., in turn.
are 0,1,2,
...
n—l,
(3769)
reduces to
3770
_
u
K^
-
u
*/«-!
^-Gr-l)...(.r~/i+2)
1.2.8...
Oi~l)
,rOr-l)...(.r->i+a)Gr-n+l)
1.1.2.3... (;i-2)
+.,,"«-^
3771
a>(,r-l)
(.r-»+4)(.t-» + 2)(.r-;^ +l)
2.1.1.2.3...(>,^
,
'^^•'
""'
.
!
MEGBANICAL QUADBATUBE.
553
MECHANICAL QUADRATURE.
=
The area of a curve whose equation is y
in terms of
z/.,.
n-\-l equidistant ordinates u,Uy, ... n,^, is approximately
3772
nu+!^..+{l^-f)^+il^-.,+,.^^
4
\6
+iy
Proof.
— The
-—
area
is
=
+1^'^
—
i-
o
+-3- -^^N iry-
Take the vakic
u^-dx.
I
/
,3
of
n.^-
in
terms of
Jo
i6(„ 16, ...
3773
from (376o) and integrate.
?«„_!
When n = 2,
{\ijLv
Jo
3775
«
= 4,
f »,fe =
=
^^-^^[h+ik
t>
l4(«+»,)+64.(».+-0+2t»,_
4o
Jo
3776
n=
6,
)
f^,,^/cr
Jo
In the
= ^ {u-\- + "i + '^u+') ('^1+
iU
J^2
''5)
+6/rjj
which is due to Mr. Weddle, the cotaken as -^-o instead of 1-4-0, its true value.
These results are obtained from (3772) by substituting for
each A its value from (3742).
efficient of
last formula,
A^u
is
COTES'S
3777
AND GAUSS'S FORMULA.
•
These give the area of the curve directly
in
terms of
fixed abscissae.
They are obtained by integrating Lagrange's value of
(3769-71), and arc fully discussed in articles (2995-7).
4 B
//.,.
—
CALCULUS OF FINITE DIFFERENCES.
55-4
LAPLACE'S FORMULA.
=
^"ujLv
3778
l^^-t-
wi+wo
•••
+ 1^
the coefficients being those in the expansion of
{log (1
^
Hence, putting
?f j
•^1^2
= A^y^.,
f'
and so on
,
24
12
Mn
+ 0}"-
+ «i
A- Wo
72U
5
"^
o^
Adrift
,
then add together the n equations.
;
3779 Formula (3778) contains A?f,„ ^\„ &c., which cannot
be found from 7(o5 ''i ••• ^hr
The following formula does not involve differences higher
than A?/.„_i.
^\Ar = '-^^-ui+u,
3780
+1^
...
- ^ (Af/,_i- A^^,)- ^ (A-?/„_i-A-/^,)-&c.
Proof.— In
and put E'hv_,
the proof of (3778), change
SUMMATION OF
3781
Definition:
3782
Theorem
where G
is
,.,
,
= iv^.i
:
into
E
—
--
t^jttk'
log(l — Aii
log(l + A)
(3739) after expansion, and proceed as before.
,.
SERIES.
= //„4-Mv,+i+'^.+2- +
2//, = A-^m,+ C,
^m.,
constant for
all
••
"..•-!•
the assigned values of
x.
)
SUMMATION OF SERIES.
— Let
555
=
=
^i^,
then (p (x)
A'^u^,
he such that Af (x)
<l>(x)
Write thus, and add together the values of
(a + l)—(j) (a).
S?*^,
Therefore,
by
?> (")>
(3781),
(«)— <^ («)
^"'"o^*a) «a+i> ••• "x-iand ^ (o) is constant with respect to x.
Proof.
therefore «„
=
=
Taken between the
limits x
=
= a, x = h — l,
—
we have
notation,
3783
^iifl
or
tl-\,.
Functions integrable in
3784
3785
Class
I.
= tih-tu^, = ^-'ih-^-'u„.
finite
terms
:
^.r(-)=lL_. + C\
m+1
the
—
.
CALCULUS OF FINITE DIFFERENCES.
556
ir t;]iis function bo presented as «,., and ^l'^ u_, be required, we
then the differences A?*,,, A^«u, ... A*«^
71^= 1, ?t,= 5, «.,= 15, &c.
4, 1, and then, by (3780), the required sum, as in the example re-
3791
first find
1, 4, G,
=
;
ferred to.
3792
For another example,
Here
=
^x'
3x'
AO^
therefore
= 1 + 2^.. +n^ be required.
= 6x + 6, AV = 6,
^'0' = 6,
A'O' = 6
let S".7;'
+ 5x + l,
=
1,
A\c^
expressed in factorials, and the summation
First, by (3730),
effected by (3784).
x^
may now be
x^zzz
therefore,
3
_ n(n + l)
3793
SlV
= 2 (n + lf
=
(n-l)(n-2)
4
3
=
u„
x^,
(3783),
(n + l) n
S(n + l)n(n-l)
Otherwise, by (3789), taking a
7(^
(1),
then bo
x + Sx(x~l)+x(x-l)(x-2);
by (3784),
'^~2'^
n
may
A?<„
0,
=
A-?t,|
1,
= 0, we have
= 6, A'«u =
6,
-4"
_ n\n + lf
as above.
Therefore
y,-i
'''
"'
6n(n- l) (n-2)
^ n(n-l)
therefore,
6n
(n-l)(n-2)(n-S)
changing n into
?i
+ l,
_ n'jn-lf
1.2.3.4
1.2.3
"•"
2
S"'?fjr
4
4
—
3794 Class IV. When the general term of a series
rational fraction of the form
—
'
!—
'
= — ——
,
where
and the degree of the numerator
resolve the numerator into
is
7/
^.
is
a
= ax-\-b,
not higher than cc+m
—2
;
The fraction then separates into a series of fractions with constant numerators which can be summed by
by (3730).
(3787).
3795 If the factors u.^.... ?/.,.+,„ are not consecutive, introduce
the missing ones in the denominator and numerator, and then
resolve the fraction as in the foregoing rule.
3796
Fx.
:
To sum the
series
--—
1.4
+ -— +
2.5
-
-
3.0
+
to
n terms.
,
SUMMATION OF SERIES.
Tbe ..^ncrna
—^— =
is
0^+^)0^ + ^)
_
n (n + l)+2n + 2
557
1
«(n + l)(% + 2)(n + 3)
0i + 2)(« + 3)
^
(ri
+ l)(u + 2)(« + 3)
2
The sum
/!_
\3
of
w terms
form in (3787)
by making n
U^f-^
3 \2.3
^
(w + 2)(« + 3)/
is
11
^7t?
(ri
+ l)(^j+2)(7i + 3)
+ 12n + ll
3Oi + l)(n + 2)0i + 3)"
18
used, the total constant part
which gives G
f{E)
Theorem.
3797
f
= 0,
by the rule (271),
^
^
~
If the
therefore,
is,
J_Ulf^
2 \2 3
« + 3y
G
determined
is
finally
=—.
a^j> {x)
= a^f{aE)
<i>
{x),
being an algebraic function.
then the left
Proof. — Let a =
= f{E)e'''^<l>{x)=f{e")e'^^^<p{_x) (3736) = e'»V(e'' "")«/> C^O
e'",
(3475)
^a^f{aE)<p{i,).
Class V.
—
If
<p
be a rational integral function,
(x)
3798
The upper limit is understood to be x—1, and a constant
be added, (3781-2).
Proof.— Sa^^
(3797)
(a-)
= a-
{
=
=
^-\i^f(x)
a (1
(i;-l)-'a^0 (x)
=
3799
}
(x).
w.ere
in successive derivatives of ^(x).
2«-^-<^GiO
.„
=
^
[
i+^if Oi')+
^
<^"Gr)+&c.
(
= (S^iyV = (l + ^yV.
Proof.— By (3757), \P (e^) = (E) e"-^; therefore (see last proof)
a^{aE-l)-u\x) (putting ^ = e^) = ft^ (aE-l)"' e"-^*^ (a;)
xj^
to
a^ (aE-l)-'<p (x)
+ A)-l "V (x) = j£^l^l+^^y'cp
Then expand the binomial.
2a'''^(x)
is
CALCULUS OF FINITE DIFFERENCES.
558
3800
Ex.:
AVo require
To sum
the series
2.1+4.8 + 8.27 + 10.04+
2^-.f*
If ^~"''.r be known for
be rational and integral,
all
(O.f
3.«
(3a;'^
{
3801
^\.
to n terms.
2V + 22'.«* = 2^ + 2^ (a;*-2A.«' + 2-A-.i;''-2'AV)
=
+ 2^ x^ - 2
+ 0) - 8
+ + 1) + 4
= 2^-{2.i!»-0.r + 18.T-20}.
integral A^aliies of n,
Proof. 2«^.t;^. = {EE' -lyUi^v,^ = (A^' + A')-^
Expand tbe binomial operator, observing (3738).
.
}
and
if
(3740) and (3730).
«,,v^
3802
^IX-^x = i^ + ^^y^^v^,
= A-" v^u^-n (A-"-^E) v^^u^+
Pkoof.
(7„,.>
as in (3801),
(A-«--J5;-) v^Ahi,,-&c.,
producing tbe above by (3735) and (3782).
Observe that, in (3801) and (3802), two foi'ms ai'e obtainable in each case
by expanding the binomial operator from either end of the series.
3803
Ex.
The sum
know
:
is
To sum
the series sin a
+ 2"
= x' sin ax + '^x' sin ax.
A~"sinfta3, by (3729)
ax
2a;^ sin
— (2 sin|n)~-sin
,
(cc
=
;
sin2rt
Taking
+ 3^ sin
u_^
3a +
= sin ax
to
terms.
a-
and
v^
= x^,
we
therefore (3801) gives
(2 sin ^a)~' sin
— (a + 7r)
(2.i!
j
I
ax — I (a + tt)
— 3) + (2
(x
— 1)"
j
sin |a)'^sin
|
a.«
— f (a + 7r)
|
2.
APPEOXIMATE SUMMATION.
3820
The most
useful formula
1
Pkoof.—
of
a;.
2vA^
=
((-"-l)-'»^.
is
the following
(Pif,.
E.xpand by
,
(l.'')3;t)
1
wiMi
f/'-'j/'"
D
in
the place
AFPBOXIMATE SUMMATION.
Ex. 1
3821
:
The value of
Ex. 2
:
%v^' at
To sum the
(2939)
is
1+
series
559
given at once by the formula.
— + —...
approximately,
H
- -i-, + -l^-&c.
- +2- = - +C + loga;-f
2x
r2x'
X
X
Put X
= 10
from which
x
120a;*
to determine the constant
C
=
sum
is
^-^-|^ + ^-^,-&c.
i+^ + i^ + i^+&c.,
Ex.3:
'
2
x^
The convergent part
4
2
2x'
12
12
2x^
20
2x"''
12'
of this series, consisting of the first five terms, is
approximation to the sum of
A much
2x'
2x^
2x'
x^
3823
thus
"577215, and the required
.577215 + log.+
3822
;
all
an
the terms.
nearer approximation
is
obtained in this and analogous
at a more advanced term.
summation formula
cases by starting with the
1+i + i + i+^^^i^
^2035 J^ _L
3^_M.+&e
^2.5'^ ^2.5^ ^2.5*
2.5«^
E.,.:
111
1728
^
2035
1728
The converging part now
and the convergence at
50
250
2500
1
187500
"^
'''
consists of a far greater number of terms than before,
much more rapid.
first is
3824
Ex. 4: The series for logr(ie + l) at (2773) can be obtained by
the above formula when x is an integer.
For, in that case,
logT
(x
+ 1) =
logl
+ log2 + log3
and (3820) gives the expansion
by making x infinite.
3825
...
+log.<;
=
log.u
in question, the constant
Formula (3820) may
also be used to find
process of summation, and
Laplace's formula (3778).
thus
+ S logs;,
being determined
[
uj.c by the
answers the purpose of
—
CALCULUS OF FINITE DIFFERENCES.
560
2"Uj in a scries of derivatives of
u^.
Lriwma.—
3826
...W + l){6''-l]-\
Proof.— Put
Vn.i
S%j.
may now
3827
= 0,
=
Then
(e/-l)-".
=—(dt + n)
=
i\
{dt
+ n){dt + n — l)
i'„.i.
be developed.
Ex.— To
with jLy
n-l\
v„ for
{Booh,
develope 2^*.,
p.
97)
and
- (2r + 2) = I - I + I
(-l)"A,..i
^..1
!
+ 2r{(r + 2)(r+l)J,.2 + 3
Therefore, changing
t
into
f?^.,
we
('•
+(2.1 + 3.1,-1)
+ !) ^..1 + 2^1, }r.
get
2V = |||«.^/.«-|||«.^/.^+j«.c7..-|«.+ ^^'^-&c.
3828
m
2"Uj
Let
33"
a series of derivatives of Uj_n.
cosec'*
= 1—
3;
C^'k^ -H O^rc*
— &c.
,
then
iLoole, p. 98.
3829
2
^(0)-^(l)+,^(2)_&c.
By
= iJl-| + ^-&e.(^(0).
tins formula, a scries of the given type
may
often be trans-
formed into one much more convergent.
p.ooK.-Tbo
loft
=
the expansion of which
3830
-'^-^'
^^
s^^^^
^m
is
'^
=
= i rTp:*^.
2+^*'(0)
the series on the right.
~"
"*"
"o^
"^
~T
"^^^^
•^"Q^i^iiug ^'^c
fi^'st
six terms,
+ &c. Takiug (0) = (0 + 7)-\
+ I7 - ^
o
2
2.3
^o
^
1,„
1(1,1,
y- Q +&C. = - I y + ^-7-3 + ^--^-^g-^ + 8.7.8.9.10+'^''
it
becomes ^^
00
1
after six terms converges rapidly by this formula, and more rapidly
the formula had been applied to the scries from its commencement.
The sum
than
if
PLANE COORDINATE GEOMETRY.
SYSTEMS OF COORDINATES.
CARTESIAN COORDINATES.
In tliis system (Fig. 1)* the position of a point P in a
determined by its distances from two fixed straight
These distances
lines OX, OY, called axes of coordinates.
4001
plane
is
PM
They are the abscissa
are measnred parallel to the axes.
denoted by //. The
denoted by x, and the ordinate
or
is
The abscissa
axes may be rectangular or oblique.
reckoned positive or negative according to the position of
to the right or left of the y axis, and the ordinate // is positive
lies above or below the x axis
or negative according as-
PN
ON
,*'
P
P
conformably to the rules (607,
'8).
These coordinates are called recMngular or ohlique
according as the axes of reference are or are not at right
4003
POLAR COORDINATES.
4003
The polar coordinates
of
P
(Fig. 1) are
and 9, the inchnation of r to OX, the
measured as in Plane Trigonometry (609).
vector,
4004
To change rectangular
the equations
4005
,r
To change
r
r,
the radius
initial line,
into polar coordinates,
employ
y = r sin 6.
= r cos 6,
polar into rectangular coordinates, employ
= \/c^+/,
*
e
= taii-1 nty
See the end of the volume.
4
PLANE COORDINATE GEOMETRY.
662
TRILINEAR COORDINATES.
P (Fig. 2) arc
perpendicular distances from three fixed lines which
form the triangle of refeiryiice, ABC, hereafter called the trigon.
These coordinates are always connected by the relation
4006
o,
|3,
The
trilinear coordinates of a point
y, its
aa+by8+cy=2,
4007
4008
or
a
where
a, b,
c
sill
A +y8 sin B-\-y sin C
= constant,
are the sides of the trigon, and
%
is
twice
its
area.
4009
If
y are the Cartesian coordinates of the point
the equations connecting them with the trilinear
coordinates are, by (4094),
<'«j
afty,
=
=
y =
a
fi
oV
cos a-f // sin a—p^,
,1'
cos )8+//
OS
cos y-\-y sin y—pz-
sill
^—pi,
Here « has two significations. On the left, it is the
length of the perpendicular from the point in question upon
On the right, it is the inclination
the side AB of the trigon.
of tliat perpendicular to the x axis of Cartesian coordinates.
Similarly /3 and y.
4010
4011
The angles
a,
[3,
y are
connected with the angles
Af B, C by the equations
y— 13=
a—y =
IT— A,
7r^B,
a—fi
=
'rr-\-C,
only two of which are independent.
upon
7^1, P-i, l>:i are the perpendiculars from the origin
the sides of the triangle ABC.
4012
ARKAL COORDINATES.
A, B, C (Fig. 2) be the trigon as before, the areal coordinates a', /3', y' of the point F are
If
4013
a-±-!U!!.'
B-^-!l£A
y=x=£ii^.
—
SYSTEMS OF GOOBBINATES.
Tlie equation connecting the coordinates
563
is
now
a+/S-fy'-l.
4014
To convert any homogeneous trilinear equation into
the corresponding areal equation.
4015
4016
aa
Substitute
= 2a',
=
byS
Cy
2y8',
= 2y'.
Also any relation between the coefficients /, m, n in the
equation of a right line in trihnears will be adapted to areals
Similarly for a, b, c,
for /, m, n.
by substituting /a, rii{\
a conic (4656), substitute
f, g, h, in the general equation of
aa\ bh^, cc^ /be, gca, hah.
M
In either the trilinear or areal systems, a point is deterif the ratios only of the coordinates are known.
P Q B, then, with trilinear coThus, if a (^ y
mined
=
:
:
:
:
ordinates,
4017
a
J—
—/>2
=
—p:
^;
aP-^hQ+cR'
and, with areal, a
-^
'
'
P
P+Q-^R
TANGENTIAL COORDINATES.
4019
111
this
of a straight line is deter-
system the position
of a point by an
be the trilinear equation of a
then, making a, (5, y constant,
m, n variable, the equation becomes the tangential
mined by coordinates, and the position
If
equation.
straight line
and
I,
+ )»/3 + y/y z=
EDF (Fig. 3);
/o
whilst /, m,n are the co(a, /3, y)
equation of the point
ordinates of some right line passing through that point.
Let X, ^, V (Fig. 3) be the perpendiculars from A, B, G
upon EDF, and let pi, p^, ih be the perpendiculars from A, B,
G upon the opposite sides of the trigon then, by (4624), we
;
;
have
R\ =
4020
where
l2h,
RiM
= nip,,
Rv
= np,,
B = ^{I^-\-'m^-\-n^ — 27nn cos A — 2nl cos B — 2lm cos C).
Hence the equation
becomes
of the point
4021
V
X
a
/8
.,
Y
-\-aJ—-\-v^-=0
,
where
,
Jh
Pi
p,
=
OA,
X
or
Ih
6,
X
sin 6^
= ABOG,
.
i+/x
Pi
&g.,
sm
0.
=+v
Pi
and 2AJUJG
.
sin 6.
^'
^
:= 0,
P-
= p,p,^me.
FLANE COORDINATE GEOMETRY.
564
Eormiila (4021 ) sliows tliat, when tlie perpendiculars X, ^, v
are taken for tlie coordinates of tlie line, the coefficients become the areal coordinates of the point referred to the same
trigon.
Any homogeneous
4023
coordinates
/,
m, n,
equation in I, m, n as tangential
expressed in terms of X, m, v by substituting for
is
— — —
,
,
Ih
Ih
By
respectively.
(4020).
Ih
An equation in X,^t, v of a degree higher than the first
represents a curve such that X, ^tt, v are always the perpenThe curve must therefore be the
diculars upon the tangent.
envelope of the line (X, ^t, v).
4024
TWO-POINT INTERCEPT COORDINATES.
Let X
two
= AD,
^i
fixed points A,
= BE
B
(Fig. 4) be variable distances from
fixed parallel lines,
measured along two
then
4025
a\-^rhii-^c
=
{)
through which the line DE
the equation of a fixed point
always passes. This may easily be proved directly, but we
shall show that it is a particular case of the system of three-
is
point tangential coordinates.
Let one of the vertices (0) of the trigon in that system be at
Then equation (4022) becomes
infinity
(Fig. 3).
X sin
//
1 -j. ti
0,
,
sin
Pi
= sin COE always.
For
I'
&c.,
and the equation becomes
:
p3
0.,
1
nni:^
a = A
sin
COE
0.
•
.
,
-I-
sin
6*3
P'2
COE
Divide by sin
5iEii^D+ 515^^1^ + sin ^3
The only
variables are
AD
and AE.
n\
written
the form taken by
^/X
+
?)^/
-^ sin
COE =
AD,
= 0.
P2
Pi
may be
then X
+
c'j'
Calling these X and ^, the equation
+ h^-\-c =
=
when
Q,
v
=
00
and
c'
vanishes.
ONE-POINT INTERCEPT COORDINATES.
4026
(Fig. 5)
Tjct
;
(I,
and
h
let
be the Cartesian coordinates of the point
the reciprocals of the intercepts on the axes
^^YSTEMS OF COORDINATES.
of
any
line
DOE
he ^
passing tlirougli
565
= -j^,,
v
= -j^.
Then, by (4053),
f^^+h
4027
=
1
the equation of the point 0, the variables being
is
This
is
two of the
t„
rj.
a case of the system of three-point tangential coordinates in which
Equation (4022)
vertices (B, G) of the trigon are at infinity.
now becomes
^l?]B-^
+ sin BOB sin d, + sin COE sin 63 =
0,
Pi
—
-p^^-AD^-lE-^'
a^-^-hr] = \.
sin
which
is of
the form
0,
,
sin
0.,
sin Q^
,
r\
TANGENTIAL RECTANGULAR COORDINATES.
name has been given to the system last described
when the two fixed lines are at right angles (Fig. 6).
which are defined as the reciprocals
The coordinates S,
of the intercepts of the line they determine, have now also the
4028
This
r/,
following values.
4029
Let X, y be the rectangular coordinates of the pole of
the line in question with respect to a circle whose centre is
the origin and whose radius is h ; then
f=|
since
x
x.OM=y.ON=
= 0.
and
¥;
for
The equation of a point
a, OS
coordinates are OB
4030
=
a^^br,
=
= l,
this equation being satisfied
,
= i,
M,
N
are the poles of y
P
on
NM
h,
is
= 0,
whose rectangular
by (4053),
by the coordinates
of all lines
passing through that point.
In all these systems an equation of a higher degree
represents a curve the coordinates of whose tangents
satisfy the equation.
4031
in
K, V
.
ANALYTICAL CONICS
IN
CARTESIAN COORDINATES.
LENGTHS AND AREAS.
Coordinates of the point dividing in tlie ratio n
right line which joins the two points ;/'//, x ij'
f^'-i^^^i^.
4032
r,
=
n-\-n
Proof.
— (Fig.
7.)
=
I
= x + AG = x-\
f='^,
Similavly for
'—, (.«'—.<;).
?/.
= ^-
If«
4034
Lengtli of the line joining the points
v
the
n
^UL±2LJL.
n-\-n
4033
»',
:
xi/,
x'y'
= y(,r-y)^+(//-//7.
Tlie
4035
same with
V(^v-^v'y-{-{t/-t/Y-\-2(.v-.v')[f/-i/)i'OS<o.
Proof.— By
Area
J.
(Fig. 7), Euc.
A
I.
=
and (702).
47,
of a triangle in
angular points
4036
oblicjuo axes
terms of the coordinates of
I {.*'i//2-.t\.//i
+ .*"2;/3-'^V/2 + '^'3.'A—
—
PudOF.
(Fig. H.) By considering the
and the sides of the triangle, we have
i
+://,)(.r,-.'',) + i (!/o +
A
=
(//,
its
xYi/i, x.^y.,, r^y/g.
f/3 )
tliree trapezoids
Of, -''•:)
-^
(.'/»
'^'i.^/^}-
formed by
i/,,
+ '/,)0''3--«i).
y^, y^
+
'
LENGTHS AND AREAS.
Area
of the triangle contained
1/
= miX-{-Ci,
A = J^^ill^
4037
Proof.
— (Fig.
Cob.
—Area
= —
Ci
Co.
=|
by the \ axis and the
y =
+
m.2,x
(4052)
Co,
= .Jjlfllf'j^y
(.'^'i
— ^dl^y
The sign
a.nd
p
is
lines
(«5G)
found from
of the area is not regarded.
by the
of the triangle contained
lines
= i|(£i=£5)!+(£j=^' + fe=^
4
4038
Area
9.)
pm^—pm^
567
(.
4039
_
4040
^"*"
=
—
~
ACiA'i
^"^^
Pkoof.— (Fig.
_____^
{ Ci
}}ii—m
(ma
— mg)
c-i
(^%
2 (mi — m.y) {m.2
iBi^--B,C,Y
2B,B, {A,B,-A,B,)
— m-^
— +
— nis) — nii)
^^^1)
^'
i
(>><
]
1
{ni-^
^^
Square of Determinant (^4iR,ty
2{A,B,-A,B,){A,B,-A,B.^{A,B-A,B,y
ABC = AEF+CDE-BED. Employ (4087).
10.)
E
Area of Polygon of n
sides.
First in terms of the coordinates of the angidar points
4042
2A
=
Secondly,
(criy.,---.r.,^?/0
when
+ (<^'2y3-^%?A') + + G*^.yi-^^'i?/J
--.
the equations to the sides are given, as in
(4037).
4043
4044
Proof.
Also three values similar to (4039, '40,
'41).
— By (4367), adding the component triangles.
Each expression for the area of a triangle or polygon
be adapted to oblique axes by multiplying by sin w.
4047
will
2A=i^l^^^i^^^^+...+i^^^^.
CARTESIAN ANALYTICAL GONICS.
568
TEANSFORMATION OF COORDINATES.
To transform
4048
Put
.i>
tlic
origin to
tlie
point
Id-
y = ij'-^k.
= .i'+A,
to rectangular axes inclined at an angle B
To transform
to the original axes.
Put
4049
= Qo^e-y' siii^, =
as
as cos0 and
Pkoof. — Consider
.!>'
.r
i/
=
+ ^)
sm(./.
//'
//
t/'
a;'
cos^+.i'
sin^^.
Then
siii^.
.<;
(Fig. 11.)
= cos (^
-|-
9)
and
(627, '9).
Generally (Fig. 12), let w be the angle between tlie original
axes and let the new axes of x and // make angles a and /3
respectively ^vith the old axis of x.
;
4050
Put
and
.r
siu
o)
= x' sin (w — a) +//' sin (w — ^)
y
sin
w
=
.v'
sin a-\-y' sin ^.
— (Fig. !-•)
The coordinates of F referred to the old axes being
OG = X, PC = y, and referred to the new axes, OM = x, I'M = y', we have,
by projecting OCP and 02IP at light angles first to CP and then to OC,
Proof.
CD =
which are equivalent
MF- 31E,
FN = ML + PK,
to the above equations.
To change Rectangular coordinates into Polar, hk being
the pole 0, a the inclination of the initial line to tlie x axis
(Fig. lo), and xy the point P.
4051
Put
.V
= A + r cos (^+a),
THE RIGHT
//
= k+r sin (^+a).
LINE.
EQUATIONS OF THE RIGHT
LINE.
4052
yy=->/M+c
(1),
4053
- + f =1
(2),
(I
.
THE BIGHT
4054
A'
4055
Peoof.
— (Fig.
=
14.)
=
Aa'-\-Btj-\-C =
Let
COS
a-\-i/ sin
AB
PN =
be the
a
lino.
coordinates OiV
Tlien, in (1),'
y.
the inclination to the
axis therefoi^e mx
OB. In (2), a, h are the intercepts OA, OB.
pendicular from
upon the line a
Z AOS.
.7;,
X
=
;
569
LINE.
(3).
j)
(4).
Take any point P upon it,
m = tanO, where 6 = BAX,
— OG, and c is the intercept
In
(3),
=
p = OB + LP = X cos a + y sin a
;
(4) is the general equation.
4056
p
=
OS, the per-
,
CARTESIAN ANALYTICAL CONIC S.
670
To
oblique axes
tau
4072
— As in the
Proof.
Equation of a
4073
4074
4075
4>
:
=
last,
,
,
7v^
j
through
line passing
»^
{^^
x'y':
-^^'1>
^^'s- 8)
y — tn.v = y' — wicv'
or
or
+% =
.4.2
— From
Figure (13),
m
A.v'
being
=
+ By'.
tan
0.
Condition of parallelism of two lines
m=
4076
by a constant.
differ
Condition of perpendicularity
mm=~l
The same
1
4081
or
Af\ork
4082
or
A
+
:
AA^BB =
or
to oblique axes
4080
:
AB=AB.
or
7)i',
Hence the equations
4078
7
employing (40GG).
y-y =
Proof.
—
/
,
=
l
'
+'—«<eoscu
+ COSW
(4072)
0.
{AB'-^rA'B) cos
m =
(4070)
:
{m-\-m') cos 6>4-mm'
AA -^BB'=
{i.
co,
.
Wl
line passing
through the points
iaii
4083
Proof.— (Fig.
4084
Or
4085
or
—
IG.)
By
,y
=
Ilii:]li
x^y^, avyo
=
'
,n.
the simihir right-angled triangles
=
PCA, ABB.
,,M4-'M2ZJMj,
(^v-^r,)[ii~!r;)
=
{.v-.v,){y-n,).
Proof.
This equation represents a straight line because it is of the first
degree; and the coordinates of each of the given points satisfy the equation.
:
THE BIGHT
A
line passing tlirougli x'y
line (m)
:
571
LINE.
and perpendicular
y-y = - ^i^-^)'
m
4087
(4073, '78)
Bx—Ay= Bx — Ay
or
The two
(=
to a given
:
4086
j3
:
lines passing
through xy' and making an angle
tan'M^^o) with a given line {m^)
?rz]L
4088
x—X
A
joins
line
Xii/i
=
p:zl!h.
1
and
+ m^m.2
I!h±2!h..
1
— m^tn.2
(4073,70)
passing through hh and dividing the line which
??2
^2//2 ii^ the ratio 1^
and
^
4089
:
= ^jrfiZkl^-)
(4073, '32)
Coordinates of the point of intersection of two lines
Jl=^ =
ff^:f|.
.^.=
4090
4092
y
"^
Length
= ^j!n2=££L.
= _ 4#=4|A^B^—A.B^
m.2—m^
perpendicular from a point
of the
("i*^)
x'y'
upon a
given line
=
4094
—
^^'
cos
a-\-i/'
sin
a— p.
Proof. Let AB (Fig. 14) be the line, and Q the point x'l/'. Then, by
upon a parallel line
OT, the perpendicular from
(4054), a;' cos a + 2/' sin u
through Q, and
=
j)
=
08.
Otherwise, the same perpendicular
4095
=
Aa,'-]-Bi/ -f
A/yl'
g
+ jy^
The same with oblique axes
(Ax -\-Bi/-\-C) sin (o
{Ax'-\-By'
_
Qfi
4nqfi
^^
~ ^^-i_^B'-2ABcos(oy
^/{A-^+B-^-
obtained in a similar way from (4065-69).
^^QgQ^ ,gj^ ,94^
:
CARTESIAN ANALYTICAL CONIGS.
672
Condition of three lines intersecting in one point
4097
CiWJo— rjomi+Ca/Wg— CaWia+Csmi
The area
4098
make
in
— CiWig = 0.
1009 must vanish.
—
Otherwise.
If certain values of the constants
the expression
/,
??i,
n
vanish identically, the three lines indicated intersect in one
point.
—By (4099),
Proof.
vanish also
A
make
line
hj
I,
and
A'x-{-B'i/-\-G'
=
is
or l{A.v-\-By-\-C)-m{A',v-]-B'i/-\-C')
= 0,
m being any constants.
and
4101
=
A.v-\-B?/^C= k {A\v-\-B'i/+C'),
4099
4100
and y which make (1) and (2)
passing through the point of intersection of the
Ax + Biji-C
lines
for then values of x
(3) vanish.
k
—
If the equation of a right line contains a third
in the first degree, the line altvays ])asses through a
Rule.
variable
iixed looint.
—
Proof.
For the values of x and y, which satisfy simultaneously the given
equations, also satisfy (4099), whatever k may be. See (4604).
=
0, the coequation of a line Ax-\-By-^G
A, B, G involve x, y\ the coordinates of a point
which moves along a fixed right line, then the first hne passes
4102
If in the
efficients
through some fixed point.
—
Proof. By means of the equation of the fi.xed line, y' may be eliminated,
and X then remains a third variable in the first degree (4101).
4103
To
find the point in Avhich the line
sects the line joining the points xy,
A,v-\-Bij-\-C iovn,
—
xy;
Ax + Bij + G
inter-
substitute
and y4.j'+%' + C for
// in
(4032).
Proof.
By (4095), since the segments intercepted are in the ratio of the
perpendiculars from xy^ xy upon the lino -4a; + J5y +
C
THE BIGHT LINE.
'/
573
—
CARTESIAN ANALYTICAL CONICS.
574
If
be the angle between the
(j)
yntn
4112
JL
tan (^=
4-
\/(lr
^
— ab)
^
^
lines,
—
2
or
siiio)
v/(/r
— «^)
\
^.,\
,
,
according as the axes are rectangular or oblique.
—Assume
Proof.
{ij
— m.^a'^{y—m^x) =0,
and apply (4088).
Equation of the bisectors of the angle
ka:^-^a-h)mj-hif
4113
— Let y =
Proof.
-^^^^
1
fj.^
The
roots of
tliis
ixx
be a bisector
(/i
= ^^h±J!h_
= -2^,
a —b
1—
= tan
<p
:
= ^.
i//)
by (4111)
;
;
then, since 2J/
and ^
?;Ji«2.2
equation are always
=
(^,
+
l
=I
X
real.
GENERAL METHODS.
APPLICABLE TO ALL EQUATIONS OF PLANE CURVES.
4114
Let F(.^^;/)
=
(i.)
^nd
f{.v,ij)
=
(ii.)
be the equations of two curves of any degree.
4115
To
find the intercepts
Put y =
in
Similarly, 'put
4116
To
x
(i.),
=
on the x and y axes.
then x becomes the intercept on the
for the intercept on the j axis.
find the points of intersection of
(i.)
and
x
axis.
(ii.).
Solve as simultaneous equations.
Each pair of values of x
and J so obtained gives a point of intersection. Imaginanj
values give an hnaginary point.
To determine equation (i.) so that the
through certain fixed points, x^y^, x^y^ &c.
4117
line
may
pass
Substitute x^j^, Xayg, ^'c. for xj successively, so forming
From these equations
as many equations as there are points.
the constants iii (i.) must be determined in terms of Xi,ji, x.,,y..,
The number of arbitrary points cannot exceed
number of constants in the equation.
4118
the
GENERAL METHODS.
4119
Condition
tliat (i.)
and
(ii.)
may
575
touch.
At a point of contact tiuo or more points of intersection
must coincide, and therefore the equation for x or y, obtained as
in (4116), must have tiuo or more equal roots for each point of
contact.
The contact is said to he of the second order when
there are three coincident points; of the third order when
there are four, and so on.
To find the equation of the tangent at a point x'y' on
the curve f(x,y)
0.
Form the equation to the secant through two adjacent points
4120
=
Xiyi, x,j2
(4083),
and determine
the limiting value of
-^^
'^'^
Xi
ivhen the points coincide by
means of
the equations
f (x^, yi)
X2
= 0,
f(x3,y2)-o.
m=^, % (5101).
4121
Otherwise
4122
For the equation
tanp^ent into
4123
m
of the normal,
m
m
of the
(4086).
To express the equation
terms of
change
and the constants
of the tangent, or normal, in
of the curve.
From
the equation of the tangent or normal, the equation to
the equation furnished by the value ofm, eliminate
x' y', the coordinates of the point of contact of the tangent.
the curve,
and
THEORY OF POLES AND POLARS.
=
represent the equation to the
Let F(x, y, x, y)
tangent of a curve at the point x'y'.
Then F{x, y, x, y')
0, the equation obtained by interchanging the constants x',y' with tlie variables ;«, 7/, represents
the polar of any fixed point x'y' not on the curve.
4124
=
B
on the curve, and let the tangents
Let Xyiji, x^y^ (Fig. 19) be points A,
Consider the equations
at those points intersect in x'y'.
Fix,.y,,x,y)=Q. ..(}), F(x„y„ x,y) = ...
Here (1), (2) are the tangents, and (3)
according to the dimensions of x and y. Also
(2),
is
F(x,7j, x',y')
=
...
(3).
some straight line or curve
(3) passes
through the points
CARTESIAN ANALYTICAL CONICS.
576
of contact .^'j//,, r/'o'/o, and raay therefore be called the cicrve of contact-; or, if a
right line, the chi)nl of contact of tangents drawn from x\ y\ i.e., the polar.
Heuco the coordinates of the points of contact of
tangents from an external point ,i"'v/' will be determined b}^
solving (3) and the equation of the curve simultaneously.
4125
4126
Again, let x'y (Figs. 20 and 21) be any point
Then, from the equations
P
not on the curve.
F(x,y\x,y)=0...i4), F (x,y,x.„y,) = ...(o), F (x,y, .r,,y,) = ... (G),
we see that (4) is some straight line that, if x.j,y^ and x^y^ are any two points
upon it, (5) and (6) are the curves of contact of tangents from those points
;
;
and that these curves of contact pass through the point
x'y'.
4127
If the points x.^j/^, x^y^ are taken at A, B, where (4) intersects the
curve, (5) and (6) then become curves touching the given curve at
and B,
may call these lines the curve tangents
and passing through x'y'.
A
We
from
x'y'.
4128
Lastly, let xy' in (o) be a point within the given curve (Fig. 22),
then the equations
F(x,y,x',y')=0...(7),
F (x„y,,x,y) =
...
(8),
F (x,,y„x,y) =
...
(9)
show that
(7) is the locus of a. point, the curve tangents from which have
their chord of contact always passing through a fixed point.
When x'y' is
without the curve, as in Fig. (19), the same definition applies to every part
of the hicus (3) from which tangents can be drawn.
4129
If the given curve be of a degree higher than the
second, the line of contact of the tangents from a point is a
curve, and the line of contact of the curve tangents from a
point is a straight line (Figs. 19 and 20).
A similar converse
relation is exhibited in Figures (21) and (22).
If the curve be of the second degree, equations (o)
become
(4)
identical.
The
line
and
of contact or the polar is
always in this case a straight line, and so is the locus (7).
Figures (19) and (20) now become identical, as also (21)
and
(22).
The polar of the point of intersection of two right
lines with regard to a conic passes through their poles.
4130
Proof.
—As
their poles,
and
Let (1) and (2) be the two lines,
point of intersection.
in (4124).
x'y' their
(,t,.Vi),
(^VJi)
.
GENERAL METHODS.
577
which the line joining two given
0.
cut by the curve /(x, y)
Substitute for x and y, the supposed coordinates of the
point of intersection the values
To
4131
points xy,
find the ratio in
=
x'y' is
i
n-\-n
n-\-n
and determine the ratio n n' from the resulting equation.
The real roots of this equation correspond to the real points of
:
intersection.
To form
4132
drawn
the equation of all the tangents that can be
from a point x'y'
to the curve
Express the condition for equal roots of the equation in
(4131), and consider xy a variable point.
To form the equation of the lines drawn from
the points of intersection of two curves.
4133
all
Substitute 7ix-\-n'x, ny'
and eliminate
Proof.
—Take
The
intersection.
fore
the ratio
may be
n
:
-\-n'y
n
x'y'
to
for x and y in both curves,
.
any other point xy on the line through xij and a point of
ratio n n (4131) is the same for each curve, and there:
eliminated.
To find the length, r = AP or AP' (Fig. 23), of the
segment intercepted between the point A or x'y' and the curve
/ {x, y) = on a straight line drawn from A at an inclination
9 to the
That is, to form the polar equation with
axis.
x'y' for the pole and the initial line parallel to the x axis.
Substitute for X. and y, the assumed coordinates of the point
of intersection, the values x = ON or ON', y = PN or P'N',
4134
X
that
is,
X
= .r'+r cos^,
j =: y'-\-r ^m.6,
That is, put
the resulting equation.
in (4051).
The real values of r are the distances of the points of intersection from x'y'.
and determine r from
a
=
When an equation has been obtained for determining
X the length of a line, important results may frequently be
arrived at by applying theorem (406) respecting the sum and
product of the roots.
4135
4 E
CARTESIAN ANALYTICAL CONIC S.
578
THE CIRCLE.
Equation with the centre for origin.
a?^+i/^=r\
4136
(Fig. 24.)
Equations of the tangent at the point
or
x'y'.
=-jr (.r-ciO.
^v^v'+m' = r\
4137
(4120)
y-l/
4138
Also, by (4124), the polar of
4139
?/
-V
cos
any point not on the curve.
x'y',
= ma^-^r Vl-^ni'
4140
a
P
m = — ^.
;
a-\-i/ siu
a
(4123)
= r,
being the inclination to the x axis of the radius to the point
'<^'y'-
Equation of the
circle
with
the coordinates of the
a, h for
centre Q.
(Fig. 24.)
(.,,_«)'^4.(,^_6y^=,.^
4141
Tangent at
x'y',
or Polar,
= r\
= r,
Op -«)(y -«)+(// -6)0/ -ft)
4142
4143
or
(cP
— a)
cos a+ (//— b)
sin a
a being the inclination of the radius to the point
General equation of the circle
4144
4145
Centre
{-^g,
Equation of the
4146
4147
circle
= 0.
Eadius
-/).
— By equating coefficients
y{g^-\-f^-c).
with (4141).
with oblique axes
.(••^+ 2d'i/
cos
0)
+//"
(Fig. 25.)
:
{.v-aY + 0/-0y-\-'2 {a'-^a){,/-b) cos
or
x'y'.
:
.^--|-/+2^>^^•+2^/+c
Proof.
(4138)
<o
= r~,
— 2((i-\-b cos
— 2{b-\-a cos6>)//
co) .v
-|-
cr
+ 2ab cos (D-\-b- =
r~.
(7u2)
—
:
f
THE GIBGLE.
579
General Equation.
4148
The coordinates
of the centre are
fcOSCU
4149
4150
=
Ptadius
= 0.
cos (o-{-7f-\-2g\v+2fi/-\-e
ci'^+2.r//
—
o-
—
\i>'-
2fi>'
COS
,
2-
cos
(o-\-f-
CO
—
— c siir
<o
SlUOt)
— By equating
Proof.
coefficients
with (4147).
Polar Equation.
r'+P-2rl
4151
4152
or
r^
— By
Proof.
r''-^2gr cos
tan a
4154
Proof.
4156
points
'*'l
— By equating
P
.Co//.,,
^+2/r
= ^.
being
/
sin
where
d.
:
d+c
= 0.
= V^+f-
circle
passing through the three
x^y.^.
— Eliminate g,f, and
c
w y
1
<^i III
1
from (4144) by (4117).
Equation of the chord joining
circle x^
4158
and
r
Ih
Proof.
4157
(Fig. 2G)
coefficients witli (4152).
Equation of the
rt'i?/i,
c\
— 2l cos a r cos 6—21 siu a r sin 6-\-l^—r — 0.
(702), the coordinates of
General form of the polar equation
4153
=
(^-a)
cos
x^ij-^^
xjj.,
two points on the
+ y' = v"
^r
= ^r^2+Ih!h+r', (4083, 4136)
= rcosi(^,-^.,),
&c.
7'sin0i = y^,
Grx+^r,) +// (y.+y^)
or .rcosi(6'i+6',)
rcos^i
=
+
r^i,
7/sini(6'i+6',)
CAETESIAN ANALTTIOAL G0NIC8.
580
—
=
''^^ may
Note. The coordinates x, y of a point on the circle x- -\- yoften be expressed advantageously in this way in terms of 0, a single variable.
4159
4160
=
S
Let
{.v-ay-\-{?j-hy-'r'
=
Then, if xij be a point P outside the
If xy be
circle, S becomes tbe square of the tangent from P.
a point P' within the circle, S becomes minus the square of
the ordinate drawn through P' at right angles to the radius
be any
circle (Fig. 27).
throu2:h P'.
CO-AXAL CIRCLES.
(See also 984 and 1021.)
4161
be two
If
= .v'+ir+2g.v +2/// +c =
-S
circles, the
If x
=
0,
equation to the radical axis
is
be taken for the radical axis, the equation to any
system of coaxal circles (1021) is
circle (radius r) of the
4162
± 82 =
.v'-{-f-2K\v
and
+ in Figure (1), — in Figure
and h=. 10 a variable.
4164
The polar
of xy' for
(2).
any
= ± S\
constant,
d = IB
k'-r^
Here
ii
circle of the
system passes
through the intersection of
.vx
-\-i/i/
Proof.— Its equation
± S^ =
is
xx'
and
.r
+ = 0.
.r'
+ yTj'-h (x + x')±P
=
(4121).
Then by
(4099).
4165
When
Jc
=
^,
then h
= ID = ID'.
D
and D' are
Poncelet's limiting points.
The polar of D with respect to any of the circles
passes through D\ and vice versa, by (41 G4).
4166
4167
Tangents from any point on the radical axis
system are equal (41(30, '01).
circles of the
to all
THE GIBGLE.
581
The radical axes of three circles, S^,
point called their radical centre.
4168
S^,
S.^,
meet
in a
D
The reciprocals with respect to the origin
or D' of
the system of co-axal circles are all confocal conies (4558).
4169
The equation
of the circle, centre Q, cutting the
= h,
.r'+/— 2%— = 0.
system of
circles orthogonally is, putting IQ,
4170
8'^
This circle passes through
The common tangents
(x-aY + (y-hy
D
and
to the
= r'
—Assume (4143) in
Pboof.
a, h, r, a,
Then tana = tana'; therefore
two equations.
The chords
a'
and
=a
also in
or
tt
a', b',
+ u.
0.
r,
—
a as coinciding
Take the
difference
of contact are
{a-a'){ai-a')-\-{b-b'){i/^b')+r(r^r')
for exterior tangents,
—For these are straight
Pkoof.
r'\
is
ia-a'){^-a) + {b-b){i/-b)-\-r{rzfr')
4172
4173
with
circles
(x-ay+{y-hy =
{a— a) cosa+(6— 6') sina+r=Fj''=
4171
of the
two
and
(See also 1037.)
The equation for a in (4143)
lines.
(1230, 1236)
D'.
+
= 0,
= 0,
for transverse.
and they pass through the points of
contact of each pair of tangents respectively, by (4171).
lines,
The centres of similitude 0, Q are the intersections of the
external and transverse tangents respectively.
4174
4175
Coordinates of 0,
Coordinates of Q,
a'r—ar'
r—r
'il±^,
r-\-r
b'r—br'
7
—
r
Vr+br
r-\-r
— By
equating coeflBcients in (4172) and (4142), the polar of
The
six centres of similitude of three circles lie
Proof.
or Q.
4176
on
CARTESIAN ANALYTICAL CONICS.
582
Sec the figure
four straight lines called axes of similitude.
of (1046).
Proof. The coordinates of the three centres of the forms (4174, '75)
—
will in each case satisfy equation (4083).
The equation of the external axis of similitude
determinant notation (554),
4177
is,
in
= 0.
(IrA) a3-(l?v/3) y + {nh(h)
— By forming
the equation of the right line passing through two
of the centres of similitude whose coordinates are as in (4174).
Peoof.
The remaining three axes are found by changing in
turn the signs of i\, r^ ; r^, r^ rg, r^.
4178
;
one of the circles touches the other two, one axis of
similitude passes through the points of contact.
4179
If
4180
The angle
6,
at
which the
r (Fig. 29), intersects the
radius
U is
circle
Proof
:
y)
whose centre
=
radius
0,
is
lih,
and
given by the equation
iJ2-2iJrcos^
6
F [x,
circle
=
B'- 2.Br cos 6
OQP,
= jP(/i,
+ r- =
A').
FT' +
/•-
^F
(Ji, 1-)
+ r\
(702) and (4160).
4181
x^
Cor.
1.
—
If the circles are given
+ + 2g'oj-{-2f'y-{-c' =
y''
the equation for cos
2Rr
4182
CoE.
orthogonally
4183
CoR.
2.
becomes, since
cos 6
— The
0,
;,^-{-y'
=
2g'g'-^
by the equations
+ 2i/x + 2j'y + e =
h=—g, k= —/,
2ff'-c-c.
0,
(4145)
condition that the two circles
may
solving three such
we can
cut
is
3.
—By
equations,
find the circle cutting three given circles orthogonally (4186).
4184
circles
— The
condition that four
orthogonal
the determinant equation
CoR.
4.
may have
circle is
a
common
^3
g^ Jl 1
TEE FABABOLA.
583
=
+ +
cuts
4185 Cor. 5.— If the circle x- if iax-^2Fy^G
three other circles at the same angle Q, we have, by (4081),
The resulting deterthree equations to determine G, F, G.
minant equation may be written
^v'+if -.V
-w
-7J 1
-{-2R cosp
4186
The
first
determinant, put
=
-11
1
CARTESIAN ANALYTICAL CONICS.
584
is
in a constant ratio (a) to its distance
directrki).
XM (the
When
e
from a fixed
= unity, the curve is a parabola.
riglit line
(See also p. 248,
et seq.)
Equation of the Parabola with origin of coordinates at the
vertex A.
4201
= 4«ci.
a = AS,
y = FN.
!r
Here
x
= AN,
Proof.— Geometrically, at (1229).
Analytically, from
PS''
= if + ix- af = PM" = (x + af.
The equations with the f^
S and X respectively
r
origin at
are
4202
= 4^a{a^-\-a),
y = 4«(cV-«).
t/
Equations
(4048)
^
:
:
:
THE PARABOLA.
585
Equation of the parabola
with a diameter and tangent for
axes of coordinates.
f = ^acc,
4211
where
4212
'r-
= acoBeG^e = SP;
x = PV;
y = QV.
a
Proof.— Geometrically,
at (1239).
=
-VQ'hc equal
Otherwise, let VQ
V being the point xy,
roots of opposite signs of the quadratic (4221),
therefore
y'^
since
?/^
= 4a x
or
r^
=
(y'"'—4^ax') cosec^
=
4a cosec"
d.x,
abscissa of P.
Equations (4204-10) hold good for these axes, with a
written for a in each.
For the polar equation of the parabola, see (4336).
4213
4214
Quadratic for n^
which the
line joining
the parabola
i/—4ax
:
n.2,
the ratio of the segments into
x^yi, a'o^a is divided by
two given points
= 0,
(4131)
Equation of a pair of tangents from any point
4215
{y"-4^a.v'){t/-^tt^')
The condition
x'y'
= W-2« (^4-.^0}' =
0.
for equal roots in (4214).
Quadratics for the coordinates of the points of contact of
tangents from x'y'
4216
aj^^-(y'^-2ax')
x-aw"
/-2?/y+4a.r'
4217
= 0.
= 0.
—
Proof. Solve simultaneously the equations of the curve and the polar
(4205) and (4125).
Coordinates of the point of intersection of tangents at
and
x^y.^
4218
y=y^.
2
a'=m
4a
4
F
Xj^yi
:
:
GAETE8IAN ANALYTICAL CONIGS.
686
Quadratic for
m
of the tangent
m\v
4220
from xy'
— my + « = 0.
(4206)
General polar equation of the parabola, or quadratic
the segment intercepted between a point, x]j\ and the
to
curve on a line drawn from that point at an inclination
the X axis (4134),
4221
for
r,
r'
sin2^+2r
{ij sin
^-2a
cos ^) +?/''— 4ay
= 0.
Quadratics for the coordinates of the points of intersection
4<ax
(4116)
Ax + By + G and the parabola y^
=
A\v^-2 {2B'a^AC) .v+C = 0.
Ai/'-^4<Bai/-\-4^Ca = 0.
of the line
4222
4223
Length
:
of intercepted chord,
W{{Bhi'-ACa){A'-^B')}-^A\
4224
Equation of the secant through
x^i/i, x^y^,
(4oa4)
two points on
the parabola
4225
= ^jiVz-^^ax,
y{m^-\-m2) = 2a-\-2m-^mo^oc.
!/
{yi+y2)
4226
or
4227
4228
The subtangent
The subnormal
Pkoof.— Put y =
in
NT = 2x.
NG = 2a.
(4083)
Fig. of (4201)
(4205) and (4208).
4229
The tangent PT'
4230
The normal
= 4^.v{a-^.v).
PG' =
ki (a^.v).
The perpendicular p from the focus upon the tangent at xy
4231
p
= ^/aia^+a) = Vu^-
:
(4212), (4095)
w
THE ELLIPSE AND EYPEBBOLA.
The part
of the
4234
normal intercepted by the curve
4«(l +
m^
4233
4«
^
^ cos d
= 6rt\/o
and
9?i
=
v/2.
= -^ = 4a'.
Coordinates of
rtrt
4237
*
=-
Coordinates of
4239
its
extremities, with the focus for origin
2« cos 6
^-^j^^i,
its
(4212)
centre
'2a
«/
= - ^^^em-
:
y
^=^-|S^'
(See also
:
sin 6
= 2a.oie.
THE ELLIPSE AND HYPEEBOLA.
4250
equal to
(4221), (4135)
_
siu-
is
of a chord thi^ough the focus
4235
-
m')t
The minimum normal
Length
587
p.
'
233, et seq.)
Referring to the definition (4200) when e is less than
is an eUii^se ; when greater than unity, an
;
unity, the conic
hyperhola.
Equation of the
and
ellipse
with the origin of coordinates at
X
SX=p.
4251
Proof.
y^-^{.v^py
— Bj the definition in (4200).
Abscissae of vertices
4252
= e\v\
:
XA = ^,
(Supply A'
XA'
in the following figure.)
= j^.
(4115)
588
OABTESIAN ANALYTICAL 00NIG8.
m'
)
THE ELLIPSE AND HYPERBOLA.
^3
4273
Proof.— By
7/+
(4200),
Def.
Ki
+ GSf =
e^(a;
QN
6
PiV:
4274
4275
{x
.2
^
«2
::
:
+ OX)^
X and y in terms of the eccentric angle
4278
(4271)
(^,
y
<^,
of the point P.
:
= h siu
y = b tan
= acos<t>,
X = a sec
a^
&c.
a.
— QGNis the eccentric angle,
4276
689
<ji.
<^.
(Ell.)
(Hyp.)
Five forms of the equation of the tangent or polar of the
point xy'
:
4280
4281
y—ti
=
oox_,inL—
„2
"T 12
(4120)
7-;(<l?— 0?).
1
*•
(4123)
4282
4283
4284
4285
•y
aft
a7C0S(^
?/sin(^
^
^sec(^
ytaii(^
_-^
X
COS y-\-y siu
y
being the inclination of ^.
=
1
(Ell
(4276)
(Hyp.)
(4278)
\/ d^ cos'^y+ft'^ siu^ y,
(4054)
&
(4372)
:
.
:
:
CARTESIAN ANALYTICAL CONICS.
690
Five forms of the equation of the normal at
x'y'
:
= g; (---')•
4286
2/-y
4287
^-^=a'-h\
y
^''''^
ha^-h/
or
= a'-b\
oc
where h and h are the intercepts
»=m.r-i!Lig!=^
4289
sec j>—hy cosec
4290
«.^'
4291
.*\^-^x2/
where
a^iyi is
=
<^
of the tangent.
(«23)
.
= a^—fy'.
(4276)
(^^^2)
{^x^'-Viy').
the extremity of the conjugate diameter.
Intercepts of the tangent or polar on the axes
-^
4292
—
and
(4115), (4281)
y
00
Intercepts of the normal
4294
On
the X axis,
4296
On
the
7/
axis,
Focal distances
r,
4298
4299
Proof,
(4287)
_
—5— x
,,2
—"
1^2
or
~
or
y
r of a point
<r?/
{a±cx)
in Ell.
(ca7±a)
in
— From = {ae±x)--\-if,
r'
e^x.
— y37.//-
on the curve
:
Hyp.
and (4272).
Perpendiculars from the foci upon the tangent
4300
p
= byj^,
p'
= byj'-.
(4095,4282)
THE ELLIPSE AND HYPEBBOLA.
4302
591
CARTESIAN ANALYTICAL C0NIG8.
592
Quadratic for abscissae of points of contact of the tangent
from xy'
4312
:
cr^
{bV-^aY^)-2aWxiv'-^a' ih^-y^)
m
Quadratic for
4313
of the tangent
from
xij
= 0.
(4282, 4125)
:
= 0.
m' {x'--a^)-2mxy-\-y'-h''
(4282)
General polar equation of the ellipse, or quadratic for r,
the segment intercepted between the point q/i/ and the curve
to
on the right line drawn from that point at an inclination
the major axis and x axis of coordinates.
(«' sin^^+i^ cos-^) r^
4314
+2r (ay sin ^+6V cos^) + {aY+bV-a'b')
4315
Length of
4316
Distance
4317
intercej)ted chord
to
=
= 0.
difference of roots.
middle point of chord
= half sum
of roots.
Rectangle under segmc7its ^products of roots.
—
If two chords be drawn to a conic at two constant
inclinations to the major axis, the ratio of the rectangles under
their segments is invariable.
CoE.
For,
if xij
be their point of intersection, the ratio in ques-
becomes a^sin^0-j-Z>^ cos-0
constant if B and B' are constant.
tion
:
Locus
a^sin-0'-f-^'cos''O^',
of centres of parallel chords
4318
a^y sin e-^-h'x cos ^
which
is
:
=
(4314)
0.
Quadratic for abscissae of points of intersection of the line
Ax-\-By-\-G
and the ellipse Wx^-\-a"y^—crlr'
0.
(411G)
=
4319
>iQOA
=
{A^a'-\-B-}r) x''-{-2ACd\v-\-Chi"-B-a-¥ = 0.
_
-ACa''±Bahx/ A'a'-i-n-fr-C'
A-a-\-B-b^
4321
For the ordinates transpose A,
B
and
a, h.
THE ELLIPSE AND HYPERBOLA.
Length
of intercepted chord
693
:
Hence the condition that the hne may touch the
A'a'-{-B%'
4323
=
The chord through two points
*^'^*
0\
Xiij^, x^y.i, is
~
P
"^
^?
a"
"^
^
b'
denoting the points by their eccentric angles
chord joining a/3 is
or,
The coordinates
tangents at
V
(or
a/3
— ^^y^'^^^-^y^ ^
V
^
=
^\+^2
QQo
"^^^
as above).
^'
iVi—y^^
oi\yx-x^y^i
""^y^'^'^^-'y^
The following
>1
the
a, /3,
chord or intersection of
of the pole of the
x-^y^, x.^y^
Vx-Vy-i
4329
'
±cos^ + |-sin5+^ = cos^.
4325
4326
ellipse is
=
^'(-^^^"-^-J
(«+^)
-a ^^^^
cosi(a-^)"
=
b
^i^i(^+^)
.
cosi(a-/8)
^^^^2-^*^1
relations also subsist
—
«^ sin
g sin
^
_
6^
__ b (sina+sin/S) __ a
-
-
cos a cos
^
(cosa+cosff)
2.V
2y
" which are of use in finding the locus of
connected by some fixed equation."
{x, y)
when
a, /3
(Wolstenholme's Problems,
4 G
p.
are
116.)
CARTESIAN ANALYTICAL OONIGS.
694
If a, i3, 7, S are tlie eccentric angles of tlie feet of the
four normals drawn to an ellipse from a point xy, tlien
4334
a+)8+y+8 = 37r or
— Equation (4290) gives the following biquadratic in
5ir.
Proof.
by/ + 2(ax + a'-h')
Let
a, h,
(406).
^
c,
d be the
+2
+ —r;
+ ca= — +
ab
ca
-=^
= —1
= tan|a,
ahcd
from which, since a
+ y) + sin(y + a) + sin(a + /3) =0;
l — (ab + ac + &c.)+ahcd = 0,
and, since
a + + y + 2 = Stt
tani(a+/3-f y + o) = 00,
we
}^(j),
be
^
&c.,
= tan
z
= 0.
= and
(ax-a' + h') z-by
Eliminate d from ah + ac + &c.
roots.
Thus ab + hc
z'
get
sin(/3
.-.
/3
or
Stt.
The points on the curve where it is met by the
normals drawn from a fixed point xy' are determined by the
intersections of the curve and the hyperbola
4335
a^'cc'y-y'yx
=
(4287)
c'xij.
POLAR EQUATIONS OF THE CONIC.
The focus S being the pole
any conic
(Fig. of 4201), the equation of
is
r(l+ecos^)
4336
=
/,
being measured from A, the nearest vertex.
1.
For the parabola, put e
=
Proof. —
r=SP;
d
= A8F;
l
=
SL;
r
= e{8X + SN)
(4200)
= + er cob
l
6.
The secant through two points, P, F, on the curve, whose
angular coordinates are a+/3 and a— /3 (Fig. 28), is
r {ecos^4-seci8cos(a-^)}
4337
=
/.
a, I'SQ = FSQ = ft.
Take (4109) for the equation of PP'. Eliminate r, and r,
by (433G), and substitute 2a for e, + d^ and 2/3 for 0,-0,.
Geometrically. Let PP' cut the directrix in Z then QSZ is a right angle,
by (IIGG). Take C any point in PP' SC = r ASC = 0. Draw CD, CE,
CF, CG parallel to SL, SP, SQ, SX, and Dll parallel to XL. Then
Proof.— Let
ASQ =
Analytically.
;
;
;
l=SL= SH+HL.
BL
^P _ SL _
^'^
_
SL
arr
^^=^^^
= "''°^^- GG=PM-^-lJX-GG'
or,
.-.
IlLz=
= r cos (a — 0) sec
r sin CSF sec
= er cos/(3 + >'sec/3 cos(u— 0).
CE =
.•.
IlL
n
Z
/?
/3,
THE ELLIPSE AND HYPERBOLA.
The equation
595
of tlie tangent at the point a is, conse-
quently,
4338
r {e cos
A
=
I.
Focal Chord.
_
"~
Length
4339
^+cos (a-^)}
2/
(4336)
1— e^cos^^'
Coordinates of the extremities, the centre
origin
4340
4342
a^
The
meet in the
_
_ a(e±cos6)
l±ecos6
the
/
sin 6
^"Idzecos^*
'
lines joining the extremities of
two
focal chords
[By (4337)
directrix.
Polar equation with vertex for pole
4343
r^ (1
-e^
cos^ 6)
:
= 21 cos d.
Polar equation with the centre for pole
r^ («' sin^
or
Proof.—By
(4200)
:
= a'b\
r ^/(l-e' cos^ 6) = b.
4344
4345
G being
:
(4273).
e+b'
Otlierwise,
cos^- 6)
by (4314), with
x'
= y' = 0.
CONJUGATE DIAMETERS.
dr
T
A
A'
Equation of the
coordinate axes
4346
:
ellipse referred to
conjugate diameters for
CARTESIAN ANALYTICAL C0NIC8.
596
where
4347
Here a
a"
=
=
»'^'
h"
a-sm'a+6'cos'a'
CD,
h'
=
GP,
a is the
=
a- sin'
angle
—+
DCB, and
.
6 cos^
/8
'
/3
the angle
jS
FOB.
Proof.
tan a tan
—Apply
/3
=
When
4349
(4050)
^,
a
=
to the
equation (4273),
putting
^
=
-^
and
-^
by (4351)
h',
= and equation (4346)
^-^ir = « = i {(I'+b').
a+/3
tt,
becomes
'
Let the coordinates of D be x, y, and those of
equation of the diameter GP conjugate to GD is
P
x,y; the
:
TEE ELLIPSE AND HYPERBOLA.
xy
597
The perpendicular from the centre upon the tangent
given by
at
is
7 = -^+^-
4366
The area
of the parallelogram
PCDL
where p
= PF,
(Fig. of 4307)
is
= ab = a'h' sin
= CP, = A PCD.
a' = GD,
pa'
4367
(4281,4064)
<u,
i^
b'
Proof.— From (4366), and (4352), and a'=^x' + y'\
Other values of p^
4369
P'^-i^-^W^r
4371
p^
Proof.
— From (4844,
=
a" sin^ e-\- b' cos^ 0.
'67),
p^
4372
=
putting r
a^ cos'
y being the inclination of
p^
4373
= a.
y +6'
(4371)
sin' y,
j9.
= a" (1 -e' sin' y).
Equations to the tangents at
x, y
D being
(^362)
P
(4372, 4260)
and P\ the coordinates
of
:
ocy'-yx
4374
=±ab
(4073)
m = -^.
DETERMINATION OP VARIOUS ANGLES.
j)Cd=-^.
4375
tan
4377
where
c
=
^{a'-h')
PCD = - ^,
= G8.
Fig. p. 595.
(4356)
(4070, 4352-3)
,
CARTESIAN ANALYTICAL CONICS.
698
tan
4378
i//
be
tlie
c^
^p
= + /SPr.
tan
^P^'
figure
i.no'
y
»',
(4280)
= tP^.
2¥
= —^,
tan
CPG =
(652, 4378)
^.
to an ellipse,
TOF' =
and construction of
(1180),
OS'+OS"-S^
2(70H20/S^-4a^
ITOS'.
_
Therefore
„
are the coordinates of 0,
tanPOP'
4386
x axis,
Then by (631) and (4379).
OP^ OP' are tangents
Proof.—By
tlie
= £+^.
= -^
sm ^
ai/
tan SPS'
4382
4383
^^^^^^ ^^SG, 4336)
^
e
inclination of the tangent to
ianxlf
^
Proof:
If
— = 1+gcosg
sm
[See figure on page 588.
4380
If
=
= P8T.
where
If
(SPT)^
= 2v£(*!£!+£2:!z^.
—
Proof. By (4311), taking terms of the second degree for the two parallel
through the origin and tan^ from (4112).
lines
It is
may
worthy
of
also be usefully
conjugate diameters
signification of ^
remark that the substitutions (4276-8)
employed when the axes of reference are
:
though, in that case, the geometrical
no longer
exists.
I
aN TV- f'sittI
TEE HYPERBOLA.
599
THE HYPERBOLA
REFERRED TO THE ASYMPTOTES.
.ri/=JK+6^).
4387
Proof.— By (4273) and
Here x
(4050).
= CK,
Equations of the tangent at P, {x,
4388
ciy+y?/
4389
y
y
= PK.
?/).
= i(«'+^')-
(4120)
= m.v-{- v/m(«^+6'0.
(4123)
4390
4391
Intercepts on the axes
4392
Here a
CI
=
2a?' ,
CL =
2?/.
THE RECTANGULAR HYPERBOLA.
ordinary axes
h,
e
= \/2
;
^'-1/
4393
4394
=
and the equation with the
is
Tangent
= «'•
xx'-yy'^a''.
(4273)
(4281)
OABTESIAN ANALYTICAL GONIGS.
600
Equation with
tlie
asymptotes for axes
:
= a\
a:y-\-xij = a\
2ay
4395
Tangent
4396
(4387)
(4388)
THE GENERAL EQUATION.
The general equation
4400
of the second degree is
+ ^hxij + hf -H 2gx + 2/7/ + c = 0,
+ hif + cz^+2fyz + 2gzx-{-2hxy = 0,
ci9^
4401
or ax^
The equation
will
be denoted hj u ov
with
<\>{x,y)
=
z
l.
= 0.
THE ELLIPSE AND HYPERBOLA.
When the general equation (4400), taken to rectangular
axes of coordinates, represents a central conic, the coordinates
of the centre, 0' (Fig. 30), are
440^
^1
-
^^_j^
-
y
c'
ab-h'
-
C
—
Proof. By changing the origin to the point xt/ and equating the
g and /each to zero (4048).
/t^, see (4430).
For the case in which ab
new
=
4404
The transformed equation
4405
where
c'
is ax^-\-2hjcij-\-hy^-\-c'
= aa^''-\-2Lv)/-\'hir+2g.v-^2fi/'-\-c.
4406
=g*<^'+/y+^.
^407
- «ft^+2fyA-^(f-^fA''^-^/^'^ _ A.
The
X axis
4408
= 0,
(4466)
inclination B of the principal axis of the conic to the
is ffiven
by
tan 2^
=
^.
Proof.— (Fig. 30.) By turning the axes
(4049) and equating the new h to zero.
in
(4404) through the angle 6
,
TEE GENERAL EQUATION.
now becomes
Tlie transformed equation
«V+6y+c' =
4409
4410
iu
which
4411
a and
b'
=^{a-\-b+ \/m^+(«-6)^ }
b'
=i
a-\-b'
The semi-axes
4414
V--^,
Note.
(4408),
{a-\-b-v'4sh'-\-{a-by}
—
we have
two equations
= (i-^b,
ab'
ab
— h\
[See (4418).
e=^'(^l-|^).
^^'"^
V-T'
If B
(4273) (4261)
of the foci, see (5008).
be the acute angle determined by equation
to choose
between
tion in question, since tan 20
Rule.* — For
=
aiid excentricity are
For the coordinates
4416
0,
ti
are found from the
4412
601
is
9
and 0-\-^ for the
also equal to tan (20
the ellipse, the inclination of the
+
7r).
major axis
+ ^7r,
X
axis of coordinates will be the acute angle 6 or
according as h and c' have the same or different signs.
hyperbola, read " dfferent or the same.''
to the
inclina-
—
For
the
Proof. Let the transformed equation (4409) be written in terms of the
semi-axes p, 2 thus cfx'^+p'y'^
Now turn
jfcf, representing an ellipse.
the axes back again through the angle —d, and we get
^
;
{q^ cos^ 6
+f sin^ d) X'- (p'^-c/) sin 2$ xy + {(f
Comparing
this
we have
ip'^
— 'i)
sin 29
sin20
=—
c'
p-q^
2/i,
t/^
= p\\
= —g\
= — c';
= ^.^^„.
p'
c
when h and
sin^ d-\-p^ cos' 0)
with the identical equation (4404), ax^ + 21ixy-\-hi/
Hence
— q"
have the same sign, p being >q.
applies to the hyperbola by changing the sign of
* This rule and the demonstration of
it
are due to
Hammersmith.
4 H
A
^
is
<
-J
2
similar investigation
g'.
Mr. George Heppel, M.A.,
of
'
,
CARTESIAN ANALYTICAL C0NIG8.
602
INVARIANTS OF THE CONIC.
Transformation of the origin of coordinates alone
does not alter the values of a, h, or h, whether the axes, be
This is seen in (4404).
rectangular or oblique.
4417
When the axes are rectangular, turning each through an
angle 9 does not affect the values of
ab-h',
4418
When
in
«+6,
^^-4-/^
or
c.
the axes are oblique (inclination w), transformation
affect the values of the expressions
any manner does not
4422
.
—^
and
,
—^^
These theoi'ems may be proved by actual transformation by the formulae
For other methods and additional invariants of the conic, see
in (4048-50)
.
(4951).
4424
If the axes of coordinates are oblique, equation (4400)
transformed to the centre in the same way, and equations
(4402-6) still hold good. If the final equation referred to axes
coinciding with those of the conic be
is
4425
and
shall
a\v'-^h'ii'-\-c'
the inchnation of the
c unaltered,
new
axis of
,i'
we
to the old one,
have
4426
tan
20= 2/,sm^-„sm2<«
2/t
coso)—-a cos2a*
4427
,
= 0,
_ CT+fe— 2Acos6j+yQ
2 sin-
,
'
J,
—
/>
_ (i-\-h—2hQO^<o — ^Q
2
0}
sin-
w
= a^-\-'U^-\-2ah cos 2w + 4A {a-\-h) cos a)-{-4^^.
Proof. — (4404)
now transformed by the substitutions
^
'
where Q
putting
a'
and
the
//
new
is
in (4050),
+ 9O°, and equating the new h to zero to determine tan 20.
are most readily found from the invariants in (4422). Thus, putting
/t
and the new w
dO\
/3
=
=
=
a
7/
+0
.
=
—
——
a + 6 —~
2/i cosw
-
sm
equations which determine a and
u>
?/.
n
/w
and ab
h^
= ah-r-5——
sin
u)
THE GENERAL EQUATION.
The
eccentricity of the general conic (4400)
603
is
given by
the equation
__£!_
4429
_
(a
+ b-2li cos coy _^
— By (4415), and the invariants in (4422).
Proof.
THE PARABOLA.
When
—
=
Jr
ab
0, the general equation (4400) represents a parabola.
For X, y' in (4402) then become infinite and the cui^ve has
no centre, or the centre may be considered to recede to
4430
infinity.
Tui^n the* axes of coordinates at once through an angle %
(4049), and in the transformed equation let the new coeffiEquate li to zero; this gives
cients be a, 111, h\ 2c/, 2/ c'.
(4408) again, tan 26
=
—
,
2/?
-.
If 6
be the acute angle deter-
this equation, we can decide whether 9 or 9-\-^Tr is
the angle between the x axis and the axis of the parabola by
the following rule.
mined by
—
The inclination of the axis of the imvahola to
KuLE.
the X axis of coordinates will be tJie acute angle 9 fh. has the
opposite sign to that o/a or b, and 9-\-\Tr if it has the same
4431
sign.
—
Let that sign
Since ah — h^
Proof.
0, a and h have the same sign.
be positive, changing signs throughout if it is not. Then, for a point at
infinity on the curve, x and y will take the same sign when the inclination is
But, since
the acute angle 9, and opposite signs when it is O + ^tt.
co, the terms of the first degree
+00 we must have 2hxij
ax^^lif
vanishing in comparison. Hence the sign of h determines the angle as stated
=
=
=—
,
in the rule.
4432
-"<'=V46'
Proof.— From the value
and from
4434
For
(4431).
/t^
=
of tan 20 above, d
a
=
and
h'
= ab — lr = 0, and we ensure
a+b' = a+b (4412).
Also
= V^-
being the acute angle obtained,
ab.
Also
a'b'
•^°«''
= a+
b.
that a' and not
6'
vanishes by
:
CARTESIAN ANALYTICAL CONICS.
604
= g COS e+f sin e =
4436
ff
4438
r = .cos^-^si„<»=^-:^i=^*.
4440
But
if
/i
same sign
lias tlie
-
"^^ff^*
as a
and
b,
change
6 into
(4431)
O+Itt.
Pkoof.— By
(4418, 4432-3).
The coordinates
4441
of the vertex are
y=--y-
''^-W^'
Obtained by changing the origin to the point x'lj and equating to zero
The coefficient of x then gives
the coefficient of y and the absolute term.
the latus rectum of the parabola viz.
;
L- -
4443
^- -
2 ^J>±/V^.
(4437)
METHOD WITHOUT TRANSFORMATION OF THE AXES.
4445
Let the general equation (4400) be solved as a quadThe result may be exhibited in either of the forms
ratic in y.
4446
//
4447
y
4448
1/
4449
4454
,.
-
>>
—j,
fi
= -j,
(^
y and
Hero
v/
=
(„b c+2fkh-ar-hf,--''lr)
{(ib—lr)-
4456
4458
= arJrfi±Vf^ {(^^-pY+iq-f)}^
= a.'+,8± v> (.r-y)(.r-S),
a=z
where
,
= aa^+/3± \/ft {x'-1px^rq)^
= aa'4-/3
S
^."
_
.
bA
C'-
= ± V{f—q).
;>
is
the equation to the diameter
DD
THE GENERAL EQUATION.
605
D
and D\ its extremities,
(Fig. 31), 7 and g are the absciss93 of
the tangents at those points being parallel to the y axis. The
OM. The axes may be rectwhen x
surd
angular or oblique.
—
= PN = FN
When
ah
— ]r = 0,
4459
1/
=
=
4460
where
4462
In this case,
p
equation (4446) becomes
aa +/3 d=
J ^q'-2p^v,
bg—hf,
q=p—bc.
^,
is
the abscissa of the extremity of
the diameter whose equation
infinite branches.
is
= ax-}-^
y
and the curve has
RULES FOR THE ANALYSIS OF THE GENERAL EQUATION.
First examine the value of ab— h^ and, if this is not zero,
calculate the numerical value of c (4407), and j^roceed as in
(4400) et seq. If ab— h^ is zero, find the values ofip' and q'
The following are the cases that arise.
(4459).
4464
ab—h-
positive
—Locus an
ellipse.
Particular Gases.
4465
^=
See (4402).
4466
By
4467
By
4468
6A
— Locus the point
positive
—No
(4447-54), since q—p^
/i
=
(4144).
xy'.
For, by (4404), the conjugate axes vanish.
and a
=
locus.
is
b
then positive.
— Locus a
circle.
In other cases proceed as in (4400-14),
ab—h^ negative
—Locus an hyperbola.
Particular Gases.
4469
A
=
— Locus
two right
lines
intersecting in
the
point xy'.
By
(4447), since
g—p^ then
vanishes.
In this case solve as in (4447).
2
CARTESIAN ANyiLYTICAL CONICS.
606
bA negative
4470
4471
By
a-\-b
—Locus the conjugate hyperbola.
—Locus the rectangular hyperbola.
=
(4414), since
a'=
= =
—
—h'.
Locus an hyperbola, with its asymptotes
a
b
parallel to the coordinate axes. The coordinates of the centre
4472
—^
now
are
— -^,
and
Transfer the origin to
by (4402).
the centre, and the equation becomes
''^=T-
4473
In other cases proceed as in (4400-14).
4474
ah
—Locus a parabola.
— h^ =
Particular Gases.
4475
4476
By
4477
^
By
By (4459).
= — Locus two parallel right lines.
= q = — Locus two coinciding right lines.
|/
J)'
(4459).
=
Ji'
(4459).
and
(/
negative
—No
In other cases proceed as in (4430-43).
of a,
li,
h,
(j,
f, c
are respectively
G
here.
locus
The
= 0.
2x'^-2xy+y' + 3x-y-l
Ex. 1.:
Here the values
The
locus.
ah
2,
—1,
3
1,
--,
— h-
1
——
,
—1,
4
therefore an ellipse, none of the exceptions (44G5-7) occurring
coordinates of the centre, by (4402), are
is
'
_
kf-bg
ah
— h'
_
_i
Hence the equation transformed
„'
_ f/^'-"/ _ _
ab — k
to the centre is
2x'-2xy + y'-^
=
0.
Turning the axes of coordinates through an angle
(4408), we find the new a and h from
a'
+ 6' =
3,
1
'1
ah'
=
\;
(4412)
so that
tan29=—
;
:
THE GENERAL EQUATION.
therefore
and the
The
= i (3-^/5),
a'
final
equation becomes
2
b'
(3—
=
v/5)
607
^
(3+
a;^
+ 2 (3+ ^/5)
v/5),
?/*
= 9.
major axis to the original x axis of coordinates
the acute angle ^tan"^ ( — 2), by the rule in (4416).
inclination of the
12x' + 60xy
Ex. (2):
The
values of
a6-7i^
= 0;
+ 75y'-12x-8y-6 =
p'
= bg-hf=
jj'
ta.29
By
= ^^ = -|;
g=-gsme+/cosg=
fore
y
and
sm« =
we must take O+^tt
the rule (4431),
6'
cos e
=
(4432)
-l^.
for the angle, instead of 6.
There-
= -^.
%43
=729-
87 (4435).
Consequently the transformed equation
Q^7
2
^^^
The coordinates
x'
of a, h,
I
b,
+"729"+
^
729^-^
-^
=
^
^-
computed by (4441), and the
44
87x/29*'
+ 6xy+9y' + bx + 15y + 6 =
g,f,
ab
is
,64
44
of the vertex are
Ex. (3)
The values
^,
(4460)
Proceeding,
a parabola.
is
^^^J^^
= -/sine-, cos 0=
= a+6 =
—4, —6,
q'=f-bc.
-330;
does not vanish (4475-7), the locus
therefore, by (4430-43), we have
Since
0.
are respectively 12, 30, 75, —6,
a, h, b, g,f, c
is
c
—W =
and
p
0.
— —
= bg — kf = 0,
are respectively
1, 3, 9,
final
,
,
6,
by (4475-7), if there is a locus at all, it consists of two parallel or
coinciding lines.
Solving the equation therefore as a quadratic in y, we
therefore,
obtain
it
in the form
(x
+ oy + 2) (x + Sy + S) =
0,
the equation of two parallel right lines.
The equation
4478
of the tangent or polar of x'l/ is
u,,'<.v-\-Uy,i/-\-u,,z
=
or
u,.a:'-\-Uyt/-]-u.z
=0
(4401, 1405) obtained by (4120) in the form
= 0,
{ax+hy-]-g) a;'+ {Kv^hy+f) y'J^^cc^fy^c = 0,
4479
(«.!-'+%'+§•) .r+ (Ay +%'+/) y-\-g^'-^fy+c
4480
or
CARTESIAN ANALYTICAL CONICS.
608
4481
or
AVlien the curve passes through the origin,
the origin is
4482
ga)-\-fy
And
tlie
=
4483
4484
0.
(4479)
normal at the same point
nv-g\i/
tangent at
tlie
is
= 0.
Intercepts of the curve on the axes,
— — — -/.
,
a
Length
4486
of
b
normal intercepted between the origin and
the chord
= ^^ V
a-\-h
(4483-4)
Rigid Line and Gunic with the general Equation.
n\ the ratio in which the line joining
cut by the curve.
Let the equation of the curve (4400) be denoted by
^ (aj, ?/) =0, and the equation of the tangent (4479) by
(a3, y,
^')
then the quadratic required will be found,
;
by the method of (4131), to be
Quadratic for n
4487
xy,
:
is
x'lj
t//
=
c-^^',
n^^ {x\ y) -i-2mi\f/
The equation
4488
<t>
Proof.
CoR.
4489
{'V\
y, od\
of the tangents
y)
i>
Gr,
//)
=
y) + n^^
from
{^
— The equation
of
{x, y)
= 0.
is
y')}\
iu (4487).
two tangents
B.v^-'llliy^r Cy-
aul
.r'//'
(.r, //, a>',
— By the couditiou for equal roots
Tlie equation of the
4490
(.r,
asymptotes of
=
u
tlirougli the origin is
(46G5)
i).
(4400)
+ 2/mj(, + bill = 0.
is
:
:
THE GENERAL EQUATION.
The
equation
=
ax^-\-21ixy-\-hif
the
of
eqiii
— When the conic
Proof.
conjugates
conic
the
of
1 is
(a+&)(«-r+2/u7/-f 6//-)
4491
-
609
(a + h) (ax' + hif)
ax^ + bif =
is
=
=
2ah
(x'
2 ((ib-1i^(^-\-if).
1,
+ ^f)
the similar equation
(ax'-hf)
or
is
= 0,
given by the intersections of the conic and a circle. Transformation of the
axes then produces the above by tbe invariants in (4418).
4492
When
the coordinate axes are oblique, the equation
becomes
{a-h){ax'-hif) + 2x (hx + hy) (h-a cos w)
+ 2ij
(ax
+ hj)(]i-h cos w) =
0.
Greneral polar equation
4493
(« cos^ 0-\-2h siu ^ cos e-\-b siir 6) r'
+ 2 fe cos ^+/siii 6) r-\-c =
Polar equation with
{x,
ij)
for the pole
0.
(4134)
:
{a cos' e^~ 2/i siu 6 cos e-\-h sin 6) r4494
+2 {{ax^hy+g) cos^+(%+Act'+/) siu^} r^F{wy)
'
= 0.
Equation of the hue through xif parallel to the conjugate
diameter
4495
Proof.
{^—^v){a.v'-^luj^g)^{y-y'){luv'^hy'^f)
— By the condition for equal roots
=
0.
of opposite signs (44'. »4).
Equation of the conic with the origin at the extremity of
L being the latus rectum.
the major axis,
4496
if
=
L.v-{l-e')
Equation when the point ah
Ax -\-Iju-\-G
4497
=
^{(•'-")'+0/-'')'5
is
(4269, '59)
.v\
the focus and
the directrix
:
=''7n§Jf-
(^20"'«»^)
:
:
CARTESIAN ANALYTICAL CONICS.
610
INTERCEPT EQUATION OF A CONIC.
Tlie equation of a conic passing through four points whose
intercepts on oblique axes of coordinates are s, «' and t, t\ is
^+2/..,+ |:-.(± + J.)-,(l + |)+l =0.
4498
Equation of a conic touching oblique axes in the points
wliose intercepts are s and t
:
^+2/u^//+^-^-^+l = 0,
4499
4500
(^ + |-^iy+nr// =
or
0.
Comparing with the general equation (4400), we have
4501
'S'
=
—
,
= --p^
*
f
g
Perpendicular p from
chord of contact
wi/,
_
AKOP.
*^"^
=
= 2h-—
St
2
^•^
(4096, 4500)
s'+t'-26'tQOS<^'
of the tangent at xy'
:
4507
2(^+^-l)(i^ + |--l)+.-(.<.'/'+.'» =
4508
The equation
(l
+
lstc)(x^
+
y'
4509
Proof.
/i
(•l-'300),
(x
+ y cosw) — k (y + x cos uj) + hk cos w =
same coordinate axes
the
(^ + f-l) =
— From
0-
of the director- circle is
+ 2xy cos w)—
The parabola with
.
c
any point on the curve, to the
vs'f^l/f^co
W2 -~
^'
Equation
v
^
putting
o..
h= — ~
st
0.
as in (4109)
V7 + Vf=l(4174), and tliorefbro v
=
-.
St
:
:
:
TEE GENERAL EQUATION.
Equation of the tangent at
4511
or
,;
611
x', y'
» = -Vi?-
= ,„.r+_-j-^,
(4123)
Equation of the normal at xy'
4512
.'/
= »-+(^;^7+;)F.
Normal through the
4513
The equations
of
S
Proof.— Diameter
t
tbrotigh
Of,
-^^
Coordinates of the focus
6
451
4010
cV
=
are,
=-
^
the figure of (4211).
with any axes,
by tte property
OB =
i?Q, iu
'
:
"^^
y
^
s'-\-f-\-2stGOSto'
Equation of the directrix
=
—
.
(5009)
s'+f+2stcos<o
:
cV{s-^t GOS(o)-\-1/{t-\-S COS
4518
(4122)
n/-^-
^-f
St = -l-
and
l
=
xVs = yVt.
origin
two diameters
iL_X =
4514
'«
6))
= St COSG).
—
Proof. Expand (4509), and form the equation of the polar of the focus
by (4479) and (4516).
When
the axes are also rectangular, the latus rectum
i>= ^fi-^.
4519
(4095,4516-8)
Locus of the centre of the conic which touches the
axes at the points sO, 0^
4520
t,v
4521
tute
xy
To make
=
Si/.
(4500, 4402)
the conic pass through a point xy'
in (4500),
and determine
v.
;
substi-
CARTESIAN ANALYTICAL GONICS.
612
SIMILAR CONICS.
—
If two radii, drawn from two fixed
Definition.
points, maintain a constant ratio and a constant mutual
inclination, tliey will describe similar curves.
4522
If the proportional radii be always parallel, the curves
are also similarly situated.
4523
If tliere be two conies
form (4400), then—
The condition
situated is
of
The
(2),
being
with equations of the
and
similar
similarly
b
— By (4404), changing to polar coordinates,
condition of similarity only
r
:
r'
= constant.
is
M!=(^:;
— ab
h^—ab
4525
or,
their
and
^a = -^h = t
4524
Pi^OOF.
(1)
(4418-9)
Ir
with oblique axes,
^^gg
(^^
+ 6-2Aeosa))- ^
h^
{u'^b'--2h'Q0f^0iy
— ab
li-
— ab'
(4422_3)
CIRCLE OF CURVATURE.
CONTACT OF CONICS.
—
When two points
common tangent,
of intersection of two curves
the curves have a contact of
coincide on a
ihQ first order; when three such points coincide, a contact of
To osculate, is to have a conthe second order ; and so on.
tact higher than the first.
4527
Def.
4528
The two
conies (Fig, ^2) whose equations are
=
+ />/r +
a'x' + 21i'xii-\-h'tf + 2g'x =
ax^ +2//r*'//
2j/<i!
(1),
(2),
GIBGLE OF GUBVATUBE.
613
touch the y axis at the origin, 0, by (4482). EUminate the
0, the line
third terms from (1) and (2), and we obtain x
through two coincident points, and
=
4529
{ah'-a'h) a'+2 {hh'-h'h) 1/-2
fe -?>»
=
0,
the equation to LM, the line passing through the two remain(4099)
ing points of intersection of (1) and (2).
Again, eliminate the last terms from
(1)
and
and we
(2),
obtain
4530
«-
« g")
^'+2
{hg'-h'g') .vy+ (bg'-b'g) 2/
the equation of the two lines OL,
4531
If the points L,
coincide, the conies have contact
for this is that (4530) must
The condition
of the first order.
have equal roots
M
OM.
therefore
;
{ag' - ag) (bg' - b'g) =
4533
= 0,
[By (4111) and (4099)
- h'g)\
{lig
M
must
If the conies (1) and (2) are to osculate,
Vg.
Therefore, in (4529), hg'
coincide with 0.
h'g, the conies have a contact of the
If in (4532) hg'
4533
=
=
third order.
CIRCLE OF CURVATURE.
(See also 1254
The radius
et seq.)
of curvatui^e at the origin for tlie conic
(Lv^-^2JLVt/-\-by^-\-2gx
= 0,
the axes of coordinates including an angle w,
4534
Proof.
P
—The
circle
= - h siu
is
(si
touching the curve at the origin
x^-\-2xy co?,b)-\-if—2rx sin
w
is
= 0,
by (4148), and the geometry of the figure, 2rsinw being the intercept on
the X axis. The condition of osculating (4533) gives the value of p.
p is positive
when
the convexity of the curve
is
towards the y
axis.
Eadius of curvature for a central conic at the extremity
P
of a semi-diameter a, the conjugate being
h'.
:
CARTESIAN ANALYTICAL CONICS.
614
4535
= ^^=*l=if!*: =
p
Proof.
a sm<y
— Take the equation and
,
xijf
=
CF).
Transform
= (*V:My)!.
(5138), or from (4538) and the value of h at (4365).
The coordinates
point
(a
in terms of x, y, the coordinates of the point P.
4539
Pkoof.— By
figure of (434G)
(4367)
no
Then by (4534).
to parallel axes through P.
The same
-*;.
jr
j)
for P, the
of the centre of curvature
are
i=
4540
P,..00P.-(.ig.
with the values of
-~,
..o.
33.)
= — ~^^
V
5| = t^ = i,
PG, and PG'
p,
where 0^= a^—b\
at (4535)
ana
lj
=
^^^=f^.
and (4309).
Radius of curvature for the parabola.
Taking the diameter and tangent through the point for
axes,
4542
By
p
'
^=J^= ^^.
Si
= sin u
sm^
(Fig. of 4201)
ff
(4534), and equation (4211).
Coordinates of the centre of curvature at
xij
(rectangular
axes)
4545
^
Proof.
—From
= 2a cot 6.
and y
i/
The evolute
— i}:p = y:PG
where
^
and
p
= -|7.=
2a cosec" 0,
of a central conic (Fig. 33)
4547
4548
= 3^+2«,
:
(aa^^-^{bf/)^={fr-Ir)K
or
c"
{a\v^-^by-c'Y-\-27(rhh^\2nf
= (r —
lr.
= 0,
PG'
=
2acosec0
CONFOCAL CONIC 8.
615
—
Proof. Substitute for x, y in the equation of the conic (4273) their
values in terms of ^, t} from (4540).
Otherwise as in (4958), or by the
method of (5157).
The curve has cnsps at L, H, M, and K.
The evolute
Proof.
of the
parabola
:
—As in (4548), from the equations (4201) and (4545).
CONFOCAL CONICS.
^ + |l=:l
4550
are confocal conies,
or the sign of
4551
if
a^-a" = ¥-h'\
may be changed.
h'''^
For the confocal
4 + 1^=1
and
of the general conic, see (5007).
Confocal conies intersect,
—If
if
at
all,
at right angles.
are the two conies in (4550), changing the sign of h"" to
make the second conic an hyperbola, « «'=0 will be satisfied at their point
this proves the tangents at that
If' -\-h'')
and {hj a' a"^
of intersection
Proof.
«, «'
—
—
;
=
point to be at right angles (4078, 4280).
Otherwise geometrically by (1168).
conie
P on one conic to a confocal
angles with the tangent at P.
[Proof at (1291)
Tangents from a point
4552
make equal
4553 The locus of the pole of the line Ax+By G with
h" r= A, is
respect to a series of confocal conies in which d^
the right line perpendicular to the given one,
—
BCA-ACy-\-AB\
Proof.
and I?6^=
is
{S.
— The pole of the line for any of the conies being xy
—Cy
(4292)
—
4554
locus
=
;
also d-
— =
lr'
A.
+
Eliminate
a"
;
Aa?
= — Cx
and W.
CoR.
If the given line touch one of the conies, the
the normal at the point of contact.
CARTESIAN ANALYTICAL CONIC S.
616
—
The two tangents drawn to an
Grai-cs' Theorem.
from a point on a confocal ellipse together exceed the
intercepted arc by a constant quantity.
4555
ellipse
—
Let P, P' be consecutive points on the confocal from
(Fig. 132.)
Proof.
which the tangents are drawn. Let fall the perpendiculars FN, P'N'. From
The
(1291), it follows that /.FP'N = F'FN', and therefore P'jV = P^".
increment in the sum of the tangents in passing from P to P' is
But
this is also the
BB' - QQ' + P'N- FN' = BB' - Q Q'.
increment in the arc QB, which proves
the theorem.
M
the tangents are drawn from a confocal hyperbola,
is
as in (Fig. 133), the difference of the tangents PQ,
equal to the difference of the arcs QT, BT.
4556
•The proof
FB
is
quite similar to the foregoing.
^t the intersection of two confocal conies, the centre
of curvature of eitlier is the pole of its tangent with respect
4557
to the other.
Pkoof.
— Take -^ + ^ =
a-
At
the point of intersection,
loy
a'
(i.)
— a"- =
arc
see,
x"-
and -^
a'
= ^^
— ^, = 1
(ii.)
for confocal conies.
'
and y"-=
— (where = (r—h'-);
c"
+ h"\ The coordinates of the centre of curvature of x'y' in
The polar of this point with
x" = ^\ y" = -^(^ (4540-1).
respect to
we
1 (i.)
h'
h-
(ii.)
will be
by the values of
^^ + -Kr =
x', y',
1-
that this
Substitute the values of x", y"; and
is also
the tangent of
(i.)
at P.
A
system of coaxal circles (4161), reciprocated witli
respect to one of the limiting points 1) or D\ becomes a
system of confocal conies.
4558
—
B
is one common focus of the reciprocal conies, by
The origin
Pkoof.
(4844). The polar of P with respect to any of the circles is the same line,
by (41GG). P and its polar (both fixed) reciprocate (4858) into the line at
The centre and one
infinity and its polar, which is the centre of the conic.
focus bcinc: the same for all, the conies are confocal.
ANALYTICAL CONICS
IN
TRILINEAR COORDINATES.
THE RIGHT LINE.
For a description
The square
system of coordinates, see (4006).
between two points aj3y, a'/3'y is,
of this
of the distance
with the notation of (4008),
4601
4602
^
=
Proof.
it is
[a
cos^ (a-a
— Let P, Q
easily seen,
pg^
=
)-hl)
cos^
be the points.
(i8-/8')^+r
By drawiag
cosC(y-y)1.
the coordinates
/?y, fi'y',
by (702), that
[(/3-/3T+{y-yT + 2(/3-/5')(r-y')cos^]cosec^4
a (a- a') + b(/5-/3') + 0(7-7') =0,
(I).
Now, by (4007),
from which b(f3-i3'y = -a(a-a')(/3-/3')-c(/3-/3')(7-y'),
and a similar expression for 0(7 — 7'/. Substitute these values of the square
terms in (1), reducing by (702).
Coordinates of the point which divides the straight line
joining the points afty, a^^'y in the ratio I
:
4603
'^±^\
'-^j^,
l-\-m
l-\-7ri
m
:
'-^^.
By
(4032).
l-{-m
=
ABC
=
being the triangle of reference, and a
0,
0, |3
the equations of its sides, the equation of a line passing
is
through the intersection of the lines
0, (5
7
=
4604
=
=
/a-myS
=
or
4 K
a-A:.5
= 0.
TRILINEAR ANALYTICAL CONIC S.
618
—
Proof. For this is the locus of a point whose coordinates a, ft are in the
constant ratio vi I or k (4099).
When I and m have tlie same sign, the line divides the external angle C
of the triangle ABC; when of opposite sign, the internal angle C.
:
The general equation
/a+my8+«7
4605
and
it
of a straight
may be
Hne
is
= 0,
referred to as the Une
(/,
m,
??).
is aritj line through the point C, and (la + mft)-^ny
Proof: la + mft =
is any line through the intersection of the former line and the line 7=0
(4604), and therefore any line whatever according to the values of the arbitrary constants /, m, n.
=
The same
4606
+
(/
Proof.
Or,
/8+ n
sin y) ?/— (/;?!+ >m;>2+ ups)
—By substituting the values of
if
o,
ft,
^•
y at (4000).
=
{lA,-{-mA,-]-nA,) .v-i-{lB,+mB,-\-nB,)i/
-\rlC,-\-mC,-\-nC,=
Proof.
=
the equations of the sides of ABC are given in the
0, &c., the hne becomes
form Arr-\-B^y-\-Gi
4607
in Cartesian coordinates is
a+ m cos ^+?i cosy) cV
cos
sin a-\-m siu
(/
hne
straight
— By
the constants
/,
(4095), the denominators like
0.
\/(AI +
II\)
being included in
»j, v.
=
=
=
^ are the general equations of
If ''
^i '"
^K '
is, like
the hnos a, /3, y, then it is obvious that lu-\-mv
(4604), a line passing through the intersection of u and Vy and
represents any straight line whatever.
Jn-\-mn^nw
4608
=
=
To make an equation such
geneous
in a,
/3,
as a
=p
(a constant) liomo-
y; multiply by the equation
(4007), thus
(a;i^-S)n-fb;v3
wliicli is of
the
same
foi'iti
as
(
+ Cjiy =
llJOo).
0,
S
=
aa -f^/S
+ Cy
.
THE BIGHT
The point
4610
/a
619
LINE.
of intersection of tlie lines
+ m/3 + ?iy =
and
ra-\-m(5-\-ny
=
determined by the ratios
is
y
/3
— mn
mn
The values
VI
of a,
(5,
— n'l
I
t
{nV
— n'l)
D = n (mn — m'n) +1)
+
1)/3
+
[nl'
—I'm)
D
'
— n'l) -f t {Im' — I'm)
or by solving the three equations
= S,
cy
(Ini
2>
Proof.— By (4017),
aa
t
'
,
j^
where
and (4017).
— m
Im
y are therefore
— m'?i)
^ (mn
ei i
4611
nl'
^a
+ m/3 + ny =
0,
I'a
+ m'ft + n'y =
0.
The equation
4612
aa+l)i8+Cy
=
a sin ^+i8sin
or
J5+y
sin
C
=
represents a straight line at infinity.
Proof.
— The
Za+WjS + ny
4613
=
coordinates of its intersection
are infinite by (4611).
Note:
=
ar«
+ B/3 + cy =
a quantity not
2,
with
any other
line
The equation
zero.
therefore in itself impossible, and so is a line infinitely
The two conceptions are, however, together consistent; the one
distant.
0, the ratios
involves the other. And if, in the equation Za + m/3 + ?iy
n approach the values a fc c, the line it represents recedes to an
I
(^a-\-hft-{-(.y
is
=
:
m
:
:
:
unlimited distance from the trigou.
4614
to (4612) in Cartesian coordinates is
Cartesian
the intercepts on the axes being both infinite.
coordinates may therefore be regarded as trilinear with the x and y axes for
two sides of the trigon and the other side at an infinite distance.
Oa;
The equation corresponding
+ 0?/ + 6^ =
4615
0,
The condition
that three points
lie on the same
the determinant equation,
aii3i7i, a.,l^zy2, «3/3373
straight line
is
may
tti
/3i
TRIUNE AB AXALYTICAL
620
CONK'S.
—
Tlie above is also tlie equation of a straight line
Cor.
passing tlirough two of the fixed points if the third point be
considered variable.
4616
Similarly, the condition that the three following
straight lines may pass through the same point, is the determinant equation on the right,
4617
=
The condition
4618
is
Proof.
4619
two straight
of parallehsm of the
the determinant equation
— By taking tbeliue at
0.
I
)n
n
I'
m'
n
a
1)
r
lines
= 0,
infiuity (4(jl2j for the third
Hue
Otherwise the equations of two parallel Hues
in (4617).
differ
by
Thus
a constant (4076).
/a+m)8+«y+A- (aa+byS+ry)
=
(4007)
(/+A-a)a+(m+A'l))/8+(;i + AT)y=
or
represents any line
the value of k.
The condition
to
i)arallel
/a
+ zyz/S-h^/y =
by varying
of
perpendicularitv oP the two lines
nn'
— {uni' m h) i'osA—(nl'-\-n'l) cos B
in
(•1618) is
4620
W -\-mm'
4621
or V
-\-
(l—m vosC—ii
-\-
—
-\-
n
{n
—
I
cos B) -\-7h
cos
B—m
cos
{m—n
^)
cos
A—
I
cos C)
= 0.
Proof. Transform tlie two ccjnations into Cartesians, by (4G06), and
apply the teat AA' + BJi'
(4078), remembering that
=
voaift-y)
=
-eoKJ,
&c. (kill).
THE
When
the second line
4622
is
AB
= m cos
"
621
lUailT LINE.
or 7
.4
+/
=
is
cos B.
by (4676), that (4620)
It also appears,
the condition
0,
is
the condition that
to the conic
two lines may be conjugate with respect
whose tangential equation is
tlie
/^4-m'+/i' — 2m/j cos
4623
The length
A —2nl vosB — 2lm
of the perpendicular
line la-{-'}}ij3-\-uy
=
a/
}
C=
0.
a(i'y' to
the
:
+ m ^' + ny
la
4624
from a point
cos
i:--{-)n~-\-n-—'2mn cos
__^__^^__
A — 2nl cos ii— 2/m cos C]
is equal to the form in (4006), with
by the square root of sum of squares of
The numerator ~ la' + miy + ny'. The denominator
Proof.— By (4095) the perpendicular
x',
y' in the place of x, y, divided
coefficients of x and y.
reduces by cos (/3 y)
—
= —cos J.,
Equation of
4625
a
a
^
)8'
y
y
tlie
&c.
same perpendicular
:
QOsC — n cos B
m — /icos^— cos C =0.
n — cos B — m cos A
l—m
/
I
—
This is the eliminant of the three conditional
Proof.
0, and equation (4621).
0, La->rMi^'-\-Ny'
ia + Mj3 + A>
4626
line
(/,
Equation of a
m, n)
a
line
drawn through
a'j3'/ parallel to
the
:
a
Xm—hn
=
y
Proof.
—
la-\-mii->rny
It
=
0,
is
la'
y
I)/
eliminant
the
+ mjj' + tiy'
4627
The tangent
and
m\ n)
{V,
equations
=
=
= 0,
0.
— am
of
the
three
conditional equations
at (4618).
and the equation
of the angle
between the
lines
(Z,
m, n)
is
{mn—mn)
sin
A-\-{nJ!—nl) sini?+ {Im'
— l'm) sin (7
— (/m'+Z m)cosC
ll'j^r^nwi-\-nn— {mn-\-m'n) GOsA—{nl'-\-nl) cosjB
Proof.— By (4071) applied to the transformed equations of the lines,
(4606), observing (4007).
—
:
TlilLINEAR ANALYTICAL CONIC S.
622
EQUATIONS OF PARTICULAR LINES and COORDINATE RATIOS
OF PARTICULAR POINTS IN THE TRIGON.
C
Bisectors of the angles A, B,
4628
/B-y =
4629
y-a = 0,
{),
a-y8
= 0.
Centre of inscribed circle (or in-centre)*
The coordinates
1:1:1.
are obtaiaed from their mutual ratios by the formula
(4017).
4630
Bisectors of the angles
y^,
y+a =
/3-y=0,
ir
— B,
tt
— C:
a+y8
0,
= 0.
Centre of the escribed circle which touches the side a (or
a
—1:1:1.
ex-circle)
4631
drawn through opposite
Bisectors of sides
P^inB = y siu C,
smC = asluA,
y
cosec^
—
—
coseci^
:
:
=
asinA
Point of intersection (or mass-centre)
4632
vertices
fi
:
siu B.
:
cosecC.
=
0, by (4G04), as the form of the equation of
Proof. Assume inp ny
a line through A, and determine the ratio ih n from the value of y /3
0.
when a
The coordinates of the point of intersection may be found by (4G10), or
thus
cosec A
cosec B,
sin B
sin A
o
ft
:
:
=
:
:
ft
a
therefore
4633
vertices
)8cosB
4634
y
:
:
/3
=
=
:
y
=
:
sin
=
Perpendiculars
C
:
sin
cosec
A
B=
:
cosec
sides
to
:
cosec
B
B
:
:
cosec G,
cosec C.
drawn
through
opposite
:
= y cosC,
y cos
Orthocentre:
C= a cosyl,
secyl
:
secB
:
acosA
= ficosB.
secC.
suggested by Professor Hudson, who proposes the following
... mid-circle for inscribed circle, circumscribed circle,
circle escribed to the side a, and nine-point circle; also in-centre, circum- centre, a exand in-radiufi, circum-radiits, a excentre, ... viid-ceritre, for the centres of these circles
radius, ... mid-radius, for their radii central line, for the line on which the circum-centre,
mid-centre, ortho-centre, and mass-oentre lie and central length for the distance between
the circum-centre and the urtho-ccntro."
* This nomenclature
"
In-circle, circum-circle,
:
is
a
ex-circle
;
;
;
:
THE BIGHT
623
LINE.
If the Cartesian coordinates of A, B, G be x^y^, x^iji,
the coordinates of the centre of the inscribed circle are
^
4635
x^y-i,
m+bm+e>h
= '±!v±*fi+^3_
—
D
where the bisector of
Proof. By (4032). Find the coordinates of
the angle A cuts BC in the I'atio 6 c (VI. 3), and then the coordinates of
where the bisector of i? cuts
in the ratio b + c
a.
E
:
AD
:
4636
For the coordinates of the centre of the a
change the sign of a in the above values of x and y.
The coordinates
4637
ex-circle,
of the mass-centre are
The coordinates of the orthocentre are obtained from
the equations of the perpendiculars from X2y2i ^Hlzi viz.,
4638
=
{x^—x.^x-\-{y^—y.^y
x.,{w^—A\^-\-y^{y^—y.^.
Perpendicular bisector of the side
a^mA-^^\nB-\-y^m{A—B)
4639
4640
=
acos^-y8cos5--^sin(v4-jB)
or
4641
0,
= 0,
or
/
,
rtsiiiBsinCX
2
\
4642
sm A
/«
.
/
ft
,
sin
C sin ^\
2sinB
\
^
cos^
cos^
sin^ — /3 sinB + ?(y
n
:
:
cosC.
through the intersection of y and a sin ^1-/3 sin 5 (4G31)
0, and, by (4622),
line
of the form a
:
^
J
Centre of circumscribed circle (or cir cum- centre)
Proof. — A
is
AB
=
—?,\nB cos x4-)-
=
sill ^1
cos
5=
sin
(A — B).
Otherwise, by (4633) and (4619),
a cos J.
is
any
line perpendicular to
the value which
it
AB
;
—
/?
cosB + k
=
and the constant k
has at the centre of
A
l>.
is
found by giving a
:
(3
:
624
T.R
4643
L IXl<!A
I
Centre of
nine-point circle (or mid-cpntre)
tlie
(B-C)
cos
A SA L YTK 'AL COXTrS.
h'
(G-A)
cos
:
cos
:
(A-B).
— By (955)
the coordinates are the arithmetic means of the corresponding coordinates of the orthocentre and circum-ceutre. Therefore, by
(4(334, '42) and (4017),
Proof.
^
a
cos A
7
+ sin5secJ5-|-sin(7 6ecO sinJ. cos^4 + sini?cosi?-|-sin(7cosC )
reduces to cos (B — C)x constant.
sec
s'mA
which
A
,
sec.l
—
Ex. 1. In any triangle ABC (Fig. of 955), the mass-centre E, the
orthocentre 0, and the circum-centre Q lie on the same straight line ;* for the
coordinates of these points given at (4632, '34, '42), substituted in (4615),
give for the value of the determinant
4644
cosec
A
(sec
B
cos (7— cos
B
sec G)
+ &c.,
which vanishes.
Similarly, by the coordinates in (4643),
centre iV lies on the same line.
Equation of the central
—
line
may
it
be shown that the mid-
:
drawn through the orthocentre and mass-
Ex.
To find the line
The coordinates of these points are given at (4632, '34).
centre of ^-BC.
Substituting in the determinant (4616) and reducing, the equation becomes
2.
a sin
2^
(B-C)
sin
25
+13 sin
sin
(C-yl)
-|-y sin
—
Similarly, from (4629, '42), the line
Ex. 3.
of the inscribed and circumscribed circles is
a (cos
Ex.
4.
2C
(A-B) =0.
sin
drawn through the centres
B— cos 6')+/3 (cos C—co{iA) + y (cos J. — cos B) =
0.
— A parallel to ^B drawn through C:
a sin
A +/3
sin
B=
0.
a line through a/3, by (4604), and the equation difiVrs only by a
constant from y
0, for it may be written
For
this
is
=
(a sin.l
Ex.
5.
—A
+
/^
sinL'
poi })endicular to
+y
BG
For
a peipeiidicnlar is
line
through C
constant Ic (a sin
Thus
nate y.
/J
(3
sin
/3
is
A + ji
cos
7?
-h
/3
— y cos (7
B + y sin
= 0.
=
+
=
0.
is
(4633)
=
(1),
Hence, by (4619), the
C) must be added to (I) so as to elimi-
of the form /d
sin
("
sin
drawn through C
« cos
and a
C)—y
sin
C cos B + a sin A cos (7
{B+C) + « sin .1 cos C =
sin
Tlif Luutriil \\uv.
»i/j
-|-
/3
sin
or
0.
=
5
cos
C
/3
+o
cos
iSce uut«f tu ;4G29).
0,
C
=
0.
—
ANHAEMONIC
625
RATIO.
ANHARMONIC RATIO.
For the
definition, see (1052).
ratios of that article are the vakies of the
in the three following pencils of four lines respec-
The three
4648
ratio h
k'
:
tively
a
= 0,
a-kfi
a=0,
a
= 0,
= 0...
= 0...
a^k'l3 = 0,
a-^y8==0,
= 0, a+A-/3 = 0, a+k'/S = 0...(iu:}
= 0,
= 0,
/8
a+///3
(i.)
(Fig. 34),
/3
(ii.)
(Fig. 35),
/3
(Fig. 36).
becomes harmonic when
form a harmonic pencil
with the hnes a, j3, the first dividing the external and the
second the internal angle between a and /3 (Fig. 37).
4649
h
=
h'.
4650
The anharmonic
Hence the lines
Similarly, the
ratio
(i.)
a-\-h^, a
— l'P
anharmonic
ratio of four lines
whose
equations are
is
(a^i— M2) i^h—fh)
the fraction
— Let OL be the Hue a = 0, and
= 0.
fx^—/d^ = difference of perpendiculars from
Proof.
OB,
A
13
and
B
upon OL, divided by
p.
These differences
&c.
i^i,
are proportional to the segments AB, CD,
Similarly, ^3
AD, BC, and jj
—
is
a
common
divisor.
Homographic pencils of hnes are those which have the
same anharmonic ratio. Thus the two pencils
4651
a
and
—
/iiii3,
a'— ^ii/3',
a —^^2/3,
o'
—
^i./B',
a
—
^tg/S,
a—f.L.^(i',
are homographic pencils.
4 L
a
—
^ii/3,
o'— ^ii/3',
TBILINEAE ANALYTICAL CONIGS.
626
THE COMPLETE QUADRILATERAL.
4652
Def.
—Any
called diagonals,
foiii'
riglit lines
together with the three,
of intersection, make a
which join the points
figure called a complete quadrilateral.
be any point in the plane of the trigon ABC.
4653 Let
Draw AOa, BOh, GOc, and complete the figure. The equations of the different lines may be written as under, with the
aid of ijroposition (4604), the ratios I
and dependent upon the position of 0.
= 0,
= 0,
Cc,
la —)n^ = 0,
)n^-\-iiy — la =
ny -\-la —ni^ = 0,
la -\-nifi — ny = 0,
In;
nb.
m
:
n being arbitrary
Aa,
))(/B-~iiy
AP,
mP-\-ny
=
0,
J3b,
ny -la
BQ,
ny +/a
=
i),
CR,
la
-\-m^=
i);
OP,
m/i-\-ny
{),
ca,
:
PQR,
Proof.
OQ,
ny
OR,
la
la-\-nil3-\-ny
=
=
+/a -2/;/^ =
-^m^-'Iny =
-2la
i),
0,
i),
0.
— Aa, Bb,
Cc are concurrent by addition, he is concurrent with
and with Cc and y, by (4G04). AP and OP are each concurrent
with be and a. PQB is concurrent with each pair of lines be and a, ca and
Similarly for the rest.
ft, ah and y.
lib
and
ft,
4654
ing
Every pencil
AP, BQ,
Proof.
sum and
Git)
—By the
is
of four lines in the above figure (supplya liarmonic pencil.
test in (4G4'J), the alternate pairs of equations
being
tlie
other two in every case.
Otherwise by projection. Let PQRS be the quadrilateral, with diagonals
UP, QS meeting in C. (Supply the lines AC, hC in the figure.) Taking the
plane of projection parallel to OA 11, the figure projects into the parallelogram
2jqrs; the points yl,7j pass to infinity, and therefore the lines J L-, ]>C become
difl'erence of the
:
THE GENERAL EQUATION.
lines
e>27
harmonically divided by the sides of the parallelogram, the centre,
and the points
at infinity.
Theorem (974) may bo proved by taking o, /', y for the lines BC,
CA, AB, and Va^-mj^ + ny, la^-m'fl ^ny, Ja + mft + n'y for he, ca, ah, the last
form being deduced from the preceding by the concurrence of Aa, Bh, and Cc.
4655
THE GENERAL EQUATION OF A CONIC.
The general equation
4656
of tlie second degree is
+ cf-\-2fl3y-{-2fiya-\-2hal3 = 0.
or
(o,
will be denoted by
7) =
aa:'+b/3'
This equation
/3,
<^
n
= 0.
Equation of the tangent or polar
4657
i(a'Ci-\-u^l3-\-Uyy
=
or
u^^a+u^P' -\-u^y
=
0,
the two forms being equivalent and the notation being that
(1405).
4659
The
first
equation written in
full is
of
TBILINEAB ANALYTICAL CONICS.
628
—
Proof.
By the methods in (4120). Otherwise by (4678); let a/3y
Next let the point where the line cuts
0.
be on the curve then <p (a, /3, y)
Then the line becomes a tangent and the
the curve move up to afty.
ratio n n' vanishes; the condition for this gives equation (4G58).
=
;
:
Cor.
— The
polars of the vertices of the triangle of refer-
ence are
= 0,
4660 aa-]-hft+oy
4661
/,a+^,;8+/y=:0,
o-a4-/;8+fy
= 0.
The condition that u may break up into two linear
two right lines is, by (4469), A = 0,
factors representing
where
4662
A
=
abc-\-2fgh-af'-bg—cJr.
a
(4454)
h
g
THE GENERAL EQUATION.
629
determinant form annexed corresponding to (4GG3), or in
full
{BC-F') a'^(CA-G') /B'-]-{AB-H')f
4667
+ 2 (GH^AF) /8y+2 (HF-BG) ya+2 (FG-CH) ayS =
4668 or A {aa'-^bfi'+cf+2fl3y-^2oya-\-2ha/B) = 0.
The result
and the given
from U
Proof. — Eliminate
=z
)'
+ 2EAjLt + JIju'^ =
the form
(4792).
I/\^
4669
The condition
line.
is
and therefore the envelope
0,
LM — B^,
is
0.
of
by
The coefficients are the first
This produces equation (46G7).
minors of the reciprocal determinant of A (1G43), and therefore, by (585),
are equal to aA, 6A, &c.
is,
as in (4661),
4670
U may consist
of
two Hnear factors
where
D = ABC+2FGH-AF'-BG'-CH' =^\
(1643)
becomes the equation of two points, since the
Aa+^ij3-|-vy must pass through one or other of two fixed
In this case
line
that
D = 0,
See (4913).
points.
4671
?7
The coordinates
A\+H,.i-\-Gv
4672
of the pole of Xa+/i/3
H\-\-Bfi+Fv
:
U^:U,:
or
:
AiVJQ
^"''^
A\ + H/ii + Gy
—
'^^^.io,T!^'Z
"'
:
_^
—
V
/^
II\ + Bfi
are as
U,.
—
Proof. By (4659) we have the equations in the
margin, the solution of which gives the ratios of a (3 /.
«
+ vy
GX-^F^i+Cv,
:
+ Fp
G\ + Ffi + Gy
j
^
_
_|_*^
I'
^.^^^
ii
A'
+
+
v'y,
of the pole of X'a
//3
vy may pass through the
the condition that Xa ^ijS
pole
or, in other words, that the two lines may be mutually
conjugate, is
Hence the tangential equation
+
i.e.,
+
;
4674
\U^'+^iU,'-\-vU.'
=
or
XX^;,+itt'C/^
+ v'f7.= 0,
the two forms being equivalent, and each
=
4676
^xv+i^^i^/+Cvv'
+ F(^v'+/i'v) + G(vV+v'X) + i/(X^'+XV).
The coordinates
4677
where
of the centre
oq, /3o, yo
An+m + Gt m+Bh + Ft
:
a,
tj,
C
:
are the sides of the trigon.
are in the ratios
Gn-\-Fh
+ Ct,
TRIUNEAB ANALYTICAL
630
—By
Proof.
aa + bfl + cy
since the centre
(4-G71),
=
CONIGS.
the pole of the line at infinity
is
(4612).
n' of the segments into
Tlie quadratic for the ratio n
which the Une joining two given points a/3-y, a'/3'y' is divided
by the conic is, with the notation of (4656-7),
:
4678
y) n'-\-'2 ((^„a +(/»,/8'+(^y)
Proof.— By the method of (4131).
<!>
{a', p',
nn'-{-ct> {a,
0, y) n"
=
0.
The equation of the pair of tangents at the points where 7
meets the general conic u (4656), is
aitl-{-2hujip-\-bui= 0.
4679
Pjjoof.
— The
point
a'/3',
where y meets the curve, is found from
The tangent at such a point is
+ 2ha'iy + hiy'- = [y = in (4G56)].
ny + u^jy' = (4658). Eliminate a',
au"-
/3'.
The equation
4680
By
(aW)
^
Proof.
of a pair of tangents
^ i^Py)
=
from
al3'y is
{<t>.ci'+<t>,li'-\-cl>yy7.
—By the condition for equal roots of (4678).
actual expansion the equation becomes
(^]>y2+Glo'-2Flh) cP + iCa' -{-Ay'-2Gya) /3'H(^l/3' + iV -2/fa/5)
+ 2(-Al3y + nya + Gui3-Fa')tyy'
+2(Hl3y-Bya + Fal3-Giy)y'a'
+ 2 (Gi3y + Fya-Cafi-Ey') u'lY
4681
In which either
a', /i',
y' or o,
/?,
y
may be
=
y'-
0.
the variables, for the forms are
convertible.
Otlierwise the equation of the two tangents
4682
ya-ya,
4>(/3y-)8'7,
—
a/3'-a/3)
—
^
is
0.
(4665)
Proof. By substituting py' /)'y) ^^- f^*'' ^^ P^ "^ (4664), the condition
that the line joining "'/j'y' to any point a/Sy on either tangent (see 4616)
should touch the conic is fulfilled. The expansion produces the preceding
''
equation (4681).
The equation
4683
wluM'c
of tlie
asymptotes
(^(a,A7)
(/„,
/^,„
y,, :ii't'
tiie
=
is
<^(ao,A.7o)-A-S
coordinates of the centre.
(1),
THE GENERAL EQUATION.
Otherwise
equation,
in
a
(aa,+lj;8o+rro) ^(^,^,7)
=
tlie
631
homogeneous
form
in
a, /3, y, is
4684
k{na-\-hl3+CyY
(2),
where n, I), t are the sides of the trigon.
And, finally, if the tangential equation (4664) be denoted
by O (A,^i, v) = 0, the equation of the asymptotes may be
presented in the form
4685
a>(a,Ij,r)(^(a,A7)
Proof.
—
the centre
;
(i.)
The asymptotes
therefore, put
(">
<t>
since the polar
a^, jj^,
becomes the
(aa+l)y8+r7rA
(3).
are identical with a pair of tangeuts from
thus
y^ for o', /3', y' in (4680)
;
y) f ("o' l\, ra)
fi,
=
=
^•'
('^«
+ fV' + cry = k'T
(4),
line at infinity.
N'ow, multiplying the three equations in (4672) by a, /3, y respectively,
and therefore
Jc (Xa + nft + yy),
(a, ft, y)
and adding, we obtain
(j)
0K,/3o,yo)
=
= A-(ar. +
[v3
+ cy) =
/.-2
(5),
the pole of the centre.
From (4) and (5), by eliminating A-, equation (1) is produced
dividing (4) by (5), we get equation (2).
since the line at infinity (4612)
Again, taking the values of
^-
HKj^^
+ ^^^ + ^y =
^
k
By
the last equation, (2)
CoK.
— Since
is
is
a,
ft,
and by
y from (4673), we have
^,a therefore
^^o+J^/'o
k
converted into (3).
the centre
;
(oo, 1%, y^)
+ ^/o = *1^^M).
A
See also (40GG).
is
on the asymptotes,
we have
4686
4687
<^
K, A,
7o)
=
The semi-axes
values of
/
^'^
- * i^> ^^
0-
of the general conic (4656) are
obtained from the quadratic
(«+*^>
h,
tlie
TRILINEAB ANALYTICAL CONICS.
632
Proof.
— The centre being
a„/3„y„,
and
a/3y being a point on the conic,
put a — a,^
/•
/3—l% = y, y — yo = s.
from the centre, we
= x,
the radius to
it
have, by (4G02),
=
r-
Also (4656),
f (a,
^(x\\
13,
y)
Expand and write m, n for
The terms in x, y, z become
/,
U-\-r,:y^nz=
Z
COS
A + 7fb cos B + zh
cos C)
+ x, /3„ + y, y, + z) = 0.
aa^ + hl3^ + (jY,„ lia^,+ hi%+j\,
=
<p
(n
(1).
(a,
-a^) +&c.
= 2-2 =
f/"o
+//^o +
(4007)
(2),
y,z)=-<p (a„, /3„, y,) = :s=A-f-<I» (a, i\ c) (4686)
The maximum and minimum values of r" and therefore of
and we obtain
(3).
(.<,
3racos^ + 2/"^cosI?-|-s^ccos6'
are required, subject to the equations (2) and (3).
tei'mined
ained multipliers (1862), the quadratic above
Coord., Ch. 4, Art. 18
The area
4688
7rri?-2.
The
the absolute term
coefficient of
is
—4'
The conic
4689
method of undefound.— i^erre/'^'s Tril.
the
^
TrSalJCA
=
±rr\ rfcr.;'-, the area
reduces by trigonometry to — SV, and
Hence the product of the roots is found.
the roots of the quadratic (4687) are
Proof.— If
will be
of the conic
(4)
By
is
<'7u-
according as
^^
r"''
(a, t, c).
will
(n,i),C)
be an ellipse, hyperbola, or parabola,
(4664) is positive, negative, or zero.
—
Proof. The squares of the semi-axes have opposite signs in the hypei*Therefore the product of the roots of the quadratic (4687) must for
an hypei'bola be negative, and therefore <1» negative in (4688).
4> (a, h, c)
makes the curve touch the line at infinity (4664), a property which distinguishes the parabola.
bola.
=
The condition that the general conic (4656)
rectangular hyperbola is
4690
r«
Proof.
+ ^*+c=
may
be a
2/cos^4-2i,'- cosi^+2/i cosC.
— Let the asymptotes be
/a
+ mft + ny =
I'a
0,
+ m'(3 + n'y =
0.
Forming the product, equating coefficients with (4685), and denoting
we get the proportions
(a, b, c) by
<?>,
(j)
W
a^j
_
— a-A ~
_
— b-:l ~
))im'
h(p
u.ii'
c^
— rA
_
~~
mn' + m'n
2(/^ — bcA)
id'
2
(g<p
+
11' I
— ca A)
hii'+l'vt
2 (^hf
— ab A)
TEE GENERAL EQUATION.
633
We
may therefore substitute these denominators in (4620) for the condition
The result reduces to the equation
of perpendicularity of the asymptotes.
above, by (837).
For another method, see (5002).
The general conic (4656) will become a circle
the following relation exists between the coefficients
4691
when
:
sm'B-2fsmB sinC
c sinM+« sin^C— 2^smC siii^
a shr B+b siu^^ — 2/i sin A shiB.
b
=
=
PuooF.
— Equate
sin^C+c
(465G) with
coefficients of the equation of the conic
those of the circle in (4751).
The equation
4692
of the pair of lines
drawn from a point
and the line
a(5'y to the points of intersection of the conic
|ii/3'
is, writing L' for Aa'
\a-\- 1^(3 ^vy
=
L=
+
(/>
+ vy
,
with the
notation of (4656-7),
V'cl> (a,
fi,
y)
Proof.— By
The
4693
- 2LL'((^y +
the
<^,/3'
+ <^,y + ^'* («
)
^'^
V)
=^
method of (4133).
Director-Gircle of the conic, that
of intersection of tangents at right angles,
is,
the locus
in Cartesians,
is,
C {.v'^-if)-2G.v-2Fij^A^B
Proof.
'
= 0.
— Let the equation of a tangent through xy be
mi,—n + (y — inx) =0.
=
vi,
Therefore in the tangential equation (4665) put \
and apply the condition, Product of roots of quadratic in
The equation
4694
or, in
of the
same
= — 1,
m = —I
yu
v=:y — mx,
(4078).
circle in trilinears is
iB+G + 2FcosA)a'-\-{C-{-A + 2GcosB)fi' + iA-i-B + 2HcoiiC)y"'
+ 2 (A cos A- H cosB- G cos G-F)(3y
+ 2(-HcosA + B cos B-F cos G-G)7a
+ 2 (-G cos A-F cos B+CcosG-H) aft = 0;
the form of (4751),
4695
(aa
+
fe/3
+ cy)(^+^+|^^^^a + &c.) = ^l^Ail (a^y + 6y« + c«/3).
4 M
:
:
TBILINEAB ANALYTICAL CONIGS.
634
—
Proof.
The equation of a pair of tangents (4681) through a point o^y
in trilinears, when the tangents are at right angles, represents the limiting
Therefore the ecination referred to must
case of a rectangular hyperljola.
have the coefficients of a', /3'", &c. connected by the relation in (4690),
which thus becomes the equation of the locus of the point a^y i.e., the
;
director-circle.
When
=
in (4G93)
is a parabola, C
(4G95), by (4430) and (4689), and these
equations then represent the directrix.
4696
and ^
{a, h, c)
the general conic
= m
PARTICULAR CONICS.
4697
A
tions of
whose
conic circumscribing the quadrilateral, the equasides are a
(Fig. 38)
0, S
0,
0, i3r=0, 7
=
— This
Proof.
is
A-/38.
a curve of the second degree, and
points whei'e a meets
/3
and
h,
and
also
The circumscribing
4698
=
=
ay =
where y meets
it
passes through the
/3
and
d.
= d=y8S; +
—
as
or
the origin of coordinates lies without or within the quadricircle is a-y
,
lateral.
—
Transform (46'>7) into Cartesians (4009)
Proof.
X and y and put the coefficients of xy equal to zero.
4699
A
;
equate coefficients of
conic having a and y for tangents and
j3
chord of contact
=
ay
Proof.
4700
—Make
^
coincide with
ft
A
conic having two
given conic ^S^
S
A conic having a
given conic 8
4701
k^K
in (4698).
common
common chord
Coil.
—
If
UPQ
o
and
/3
with a
(Fig. 40)
of contact a with a
(Fig. 41)
S
line,
chords
= ka^.
:
4702
for the
(Fig. 39)
=
Aa'.
be drawn always parallel to a given
rN'ozRP.FQ, by
(4317).
:
.
FABTIGULAB CONICS.
635
4703 A conic Laving a common tangent T
and a common chord witli the conic S
4704
A
conic osculating
^S'
at the point xij
common chord
at one extremity of the
/
(x
at a point x'y'
(Fig. 42)
where
T touches
— iG)-\-m (y — ?/)
S=T {Lv + 7mj - Iv - my')
A
4705
conic having
points Avith the conic 8
common
common
tangents T, T' at
(Fig. 44)
:
= kTT.
S
4706 A conic having four coincident
S at the point where T touches
points with the conic
(Fig. 45)
:
= kTK
S
S+L^
The conies
4707
:
(Fig. 43)
=
S+M' = 0,
0,
S-\-N'
=
0,
N
(Fig 40) having respectively L, M,
for common chords of
contact with the conic S, will have the six chords of intersection
M±N=0,
L±M=:0,
N±L=0,
passing three and three through the same points.
(S + iP) - (S + N')
Proof.— From
By supposing one
may be obtained.
or
more of the conies
= (M+N){M-N),
to
become right
The diagonals of the inscribed and
quadrilaterals of a conic all pass through the
Proof.— (Fig.
theorems
circumscribed
4709
form a harmonic
&c.
lines, various
same point and
pencil.
47.)
By
(4707), or by taking
by (4784).
LM=E?
and
L'W =
E''
for the equations of the conic
If three conies have a chord common to all, the other
three chords common to pairs pass through the same point.
4710
Take 8, S
48.)
to all
then M, N,
Proof.— (Fig.
chord
common
;
+ L3I, S + LN for
the conies,
L
being the
M—N are the other common chords.
TEILINEAB ANALYTICAL CONIGS.
636
The hyperbola
4711
.vy
=
(Ocr+0//+j>)-
is of the form (4699), and has for a chord of contact at infinity
Oxi-Oy+2^ = 0, (i; y being the tangents from the centre.
The parabola
4712
y^
=
(0^-\-Oy+p)
has the tangent at infinity Ox-{-Oy-{-p
=
.v
0.
So the general equation of a parabola
Thus
the form of (4699).
4713
may be put
(a.r+/3^)-^+(2^-.v+2/i/+c)(0^+0^/+l)
in
= 0.
Here ax + fty is the chord of contact, that is, a diameter; 2gx + 2fy + c is
the finite tangent at its extremity, and Ox + Oy + 1 the tangent at the other
extremity, supposed at infinity.
The general conic may be written
4714
{cLv'+2kvy-{-bt/)-\-(2g.v-\-2fy-hc){0x-j-0y-^l)
For
this is of the
4715
The
form ay
conies
S
being at
The parabolas
infinity,
infinity.
and are
similar.
8 and S-k''
have a contact of the third order at
Proof.
0.
8 and S-k{0.v-\-Oij+iy
have double contact at
4716
+ kl3o,
=
infinity.
—For S and S—(0x +
of contact
;
0y-\-Jiy have the line at infinity for a chord
and, by (4712), this chord of contact is also a tangent to both
4717 All circles are said to pass through the same two
imaginary points at infinity (see 4918) and through two real
or imaginary finite points.
Proof.
— The general equation
(x
of the circle (4144)
may
be written
+ iy)(x-iy) + (2gx + 2fy + c)(0x + 0y + l) =
0;
and this is of the form (4G97). Here the lines x±iy intersect Ox + Oy + 1
in two imaginary points which have been called the circular poi)ds at {njinity,
and 2(jx + 2fy + c in two finite points F, Q and these points are all situated
on the locus x^ + y'^ + 2gx + 2fy + c — 0.
;
4718
infinity.
Concentric circles touch in four imaginary points at
PARTICULAR CONIC S.
637
— The
centre being the origin, equation (4136) maybe written
iy
{Ox
Oi/ + ry\ which, by (4699), shows that the lines a;
have each double contact with the {supplementanj) curve at infinity, and the
Compare (4711).
variation of r does not affect this result.
Proof.
{x
+ iij)(x—{y) =
The equation
4719
±
+
of
any conic may be put
^v'+y'
=
in the
form
= ey.
=
are two sides of the trigon intersecting at
Here x
0, y
right angles in the focus ; y
0, the third side, is the directrix,
and e is the eccentricity.
=
The conic becomes a
ey
= r,
circle
when
Two imaginary
4720
=
and y
= oo
,
so that
tangents drawn through the focus
by (4699),
are,
e
the radius, (4718).
{^v+ii/){a:—ii/)
= 0.
These tangents are identical with the
lines
drawn through
Hence, if two
circular points at infinity (see 4717).
tangents be drawn to the conic from each of the circular
points at infinity, they will intersect in two imaginary points,
and also in two real points which are the foci of the conic.
two
the
All confocal conies, therefore, have four imaginary comtangents, and two opposite vertices of the quadrilateral
formed by the tangents are the foci of the conies.
mon
4721
If the axes are oblique,
equation of the conic becomes
this universal
form
of the
The two imaginary tangents through the focus must now
be written
{cV-\-y{cosQ}-\-i siuco)}
4722
lines
{^v+y (cosw— i siuw)}
=
0.
Any two lines including an angle 9 form, with the
drawn from the two circular points at infinity to their
point of intersection, a pencil of which the anharmonic ratio
is
e^('^-2«).
—
Proof. Take the two lines for sides /3, y of the trigon. The equation
of the other pair of lines to the circular points will be obtained by elimin-
TBILINEAB ANALYTICAL CONICS.
638
between the equations of the
ating- a
a«
viz.,
The
+ tv3 + cy =
result is
(/3
The anharmonic
y,ft-\-c-''y
is,
and
«
+
+ (''"y
y-
(4738)
0.
y
) (/3
;
formed by the four
lines
ft,
/S
+ e'^y,
i.),
—
—
n
cii'cum-circle,
+ =
+ e-'V) = 0.
ratio of the pencil
by (4G48,
and the
^ + -^ + ^ =
2/>y cos ^
/3^
or, in factors,
line at infinity
e'«
p-'«
:
= — e"« =
'"--"'.
e'"
=
^Tr, the lines are at right angles, and the
Cob.
If 9
four lines form a harmonic pencil. [Ferrers' TrU. Coords., Ch. VIII.
4723
THE CIRCUMSCRIBING CONIC OF THE TRIGON.
The equation
4724
of this conic (Fig. 49) is
=
Wy-^mya+nap
1 + !± + ^ =
or
0.
—
Proof. The equation is of the second degree, and it is satisfied by
simultaneously. It therefore passes through the point aft.
a
0, ft
Similarly through fty and ya.
=
=
The tangents
4726
^^
Proof,
A, B, and
at
G
are
^+^ =
+ -;=0,
— By writing (4724)
in the
=
is seen,
by (4G97),
now
of a and y, with the curve,
the two coincident points.
4729
The tangent,
(my'
4730
+
*j/3')
to
a+ {na' + ly')
The tangents
at A, D,
R
line
of the tangents at
"-
—
A
^
(now
^-
G
ly
;
for the intersection's
+ na)
a'(5'y
passes through
is, b}^
^8+ (Ifi'-^ma') y
(4G59),
= 0.
(Fig. 49) meet the opposite
line
on the right
^ Aj_>:==0.
m^n
,
I
The
and
or polar, of the point
a
4731
= 0,
be the tangent at ay
coincide,
sides respectively in P, Q,
0.
form
mya + ft(Iy + na)
ly-\-na
1+^ =
0.
By(4G04).
passes through (D), the intersection
and B.
639
PARTIGULAB GONICS.
The diameter
4732
A
gents at
and
B
tlirougli the
intersection
of
the tan-
is
nna—nhl^+{ltl-mh) y
= 0.
—
The coordinates of the point of intersection are I vi —n, by
Proof.
The
a
0.
(4726-7), and the coordinates of the centre of AB are b
diameter passes through these points, and its equation is given by (4616).
:
:
:
The coordinates
4733
/(-/a + >>*t) + /ir)
Proof.
on the
The secant through
and the tangent
—The
= cu,
comes when
4735
n{l^ + mh-nt).
the point being the intersection of two diameters
Otherwise, by (4677).
conic,
Proof.
by a
also
:
—By (4610),
like (4732).
4734
of the centre of the conic are as
m(/a — mlj + wr)
:
n'2
{a^ftai), {c^-Ay-^, ^^ny
two points
at the first point are respectively,
a right line, and it is satisfied by a = Oi, &c., and
by (4725). The second equation is what the first beFor the tangential equation, see (4893).
&c.
first is
&c.,
=
"i'
The conic
is
a parabola
when
f^'-^m'h^-{-nH'-2mnht-2nUil-2lmnh
4736
:
y(/a)
or
= 0,
+ v/0Hl))+y(»a) = O.
= 0,
—
Proof. Substitute the coordinates of the centre (4733) in aa + 6/3 + Cy
the equation of the line at infinity (4612).
Otherwise, the conic must touch the line at infinity therefore put a,
for A, ^i, r in (4893).
;
The conic
4737
/
and
in this
is
a rectangular hyperbola
cos J + m cos B-\-ii
case
it
cos
C=
when
0,
passes through the orthocentre of the
triangle.
Proof.— By (4690), and the coordinates
of the orthocentre (4634).
THE CIRCUMSCRIBING CIRCLE OF THE TRICON.
4738
b, C
Py siu4 + 7a sinB+afi siuC
siu^
,
siiiB
.
sin (7
..
=
0,
'
TBILINEAR ANALYTICAL CONICS.
640
—
Proof. The values of the ratios I vi n, in (4724),
metrically from the equations of the tangents (4726-8).
For the coordinates of the centre, see (4642).
:
:
may
be found geo-
THE INSCRIBED CONIC OF THE TRIGON.
4739
ra'-]-m'P'-\-ny-2m7i/3y-2nlya-2lmalB
4740
or
^{la)
— (Fig.
= 0.
+ y (»*« + ^(ny) = 0.
The first equation may be written
ny (ny-2la-2mi3) + (la-r>i(3y = 0.
By (4699) this represents a conic of which the lines y and ny — 2la — mft are
the tangents at F and /, and la — m^ the chord of contact. Similarly, it may
be written so as to shew that a and /5 touch the conic.
Proof.
4741
50.)
The three
pairs of tangents at i^,/, &C.5 are
2my8+2«y-/a
)
and a
J
and they have
the right line
4742
2ny-\-2la-mfi
and 13
)
2la-\-2m^-ny
?
and y
J
->
their three points of intersection P, Q,
/a
+ mjS + ^iy.
The coordinates
By
ZC+wcl
:
—By putting
:
?/ia+/I).
=
a and ft
zero alternately in (4739),
the coordinates of the points of contact,
at D,
/3
=
on
of the centre of the conic are as
nh-\-mC
Proof.
B
(4604).
-i^^',
+ mi
'
nh
and
at
E, a
'
therefore the equation of the diameter through
a
Similarly the diameter bisecting
^
C
=
we
find,
for
^"^'^
m\ + h
bisecting
'
DE
is,
by (4603),
ft
DF is
la
+ lb
7ib
+ mc
Therefore the point of intersection, or centre, is defined by the ratios given
above.
Otherwise, by (4677), and the values in (4665), writing for a, h, c,f, g, h
the coefficients in (4739).
4743
The secant through
ai/3iyi,
nj^.^y,
any two points on
the curve.
^y^/n (\/^o+v/^x)
= 0.
:
641
PARTICULAR CONICS.
Proof.
— Put
a^fi^yi for afiy,
and shew that the expression vanishes by
(4740).
The tangent
4744
Proof.— Put a^=
a^,
at
tlie
point
ai/3iyi
and divide by ^^(a^fi^yi).
&c., in (4743),
The equation of the polar must be obtained from
(4739) by means of (4659).
4745
The conic
4746
is
a parabola
a
Proof.— Similar
*
when
b
c
to that of (4736).
THE INSCRIBED CIRCLE OF THE TRIGON.
4747
a' cos*
A +^2 cos*|-f f
cos*
^
-2l3y cos^:|cos2-| -2ya cos^^cos^i _2a/8 cos^^-cos^^.
or
4749
The
rt-escribed circle
(4629)
:
cos— v'^^+siu— y^+siu—
—
Proof. At the point of contact where y
metrically, r being the radius of the circle,
Z
+
:
i(5
:
a
4751
have, in (4740), geo-
:
tt
at a(5'y\
eosA -^ +COS
The
we
:
The tangent
4750
= 0,
\/y.
= rcot-^sin^ root— sin5 = ±cos^|J. cos^-B;
— B instead of B.
inscribed; — for the escribed circle and
m=
for the
= 0.
cos^ ya+cos^ V^/8+cos-^ ^y
4748
polar
is
by (4744),
4^
+COS
is
I -f =
,
0.
obtained as in (4745).
GENERAL EQUATION OF THE CIRCLE.
(/a+?«)8-f «y) (a siu ^+^8 sin 7?+y siu C)
\-}i [j^y siu
^+ya
siu /^
4 N
+ a/3 siu C) — 0.
TinLINEAB ANALYTICAL CONIC S.
642
— The
second term is the circumscribing circle (4738), and the
by (4G()9) therefore the whole represents a circle. By varying
k, a system of circles is obtained whose radical axis (4161) is the line
la + mft + ny, the circumscribing circle being one of the system.
Proof.
first is linear
4752
;
If l'a-^'}n'(5-\-n'y be tlie radical axis of a second
of circles represented
of
any two
will
circles of
system
by a similar equation, the radical axis
tlie two systems defined by Ic and Jc
be
^•'(/a^-my8
Proof.
+ ?iy)-A•(ra+m'^+»^V) =
^
—By eliminating the term
/3y sin
A + ya
sin i?
+ a/3
To find the coefficient of
only the trihnear equation is given.
4753
Rule.
— Mahe
a,
/3,
X'
sin C.
+ if
in the circle
when
y the coordwates of a j^oint from wliich
the length of the tangent Is hioivn, and divide by the square of
that length; or, if the point he within the circle, substitute
^' half the shortest chord through the poinV^ for ^Hhe tangent."
—
=
If S
be the equation of the circle,
then, for a point not on the curve, S -^
serai-chord, by (4160).
Proof.
efficient
and
m=
;
m
the required cosquare of tangent or
THE NINE-POINT CIRCLE.
4754
a' sin
2A
+)8-' sill
'2n-\-y sin
— 2 (/8y
2^
sin/1-fya
sill />
+ a/3siiir) =
il
;
PARTICULAR CONICS.
Proof.
— The
643
equation represents a circle because
it
may
be expressed iu
the form
(a cos J. 4-/5 cosi)
See
+ y cos
C)(a
sin J.
+
/3
sin i?
+y
sin G)]
— 2 (/3y sin A + ya sin B + afi sin C) =
Proof of (4751). Now, when a = 0, the equation becomes
(/3 sinS — y sin(7)(/3 cos5-y cos G) = 0,
BC
whicli shews, by (4631, '3), that the circle bisects
the foot of the perpendicular from A.
0.
and passes through D,
The equation of tlie nine-point circle in Cartesian
coordinates, with the side BC and perpendicular on it from A
for X and y axes respectively, is
4754«
x^
where
+ y--B, sin {B- G) x-E cos (B-G) y =
B is
0,
the radius of the circum-circle.
THE TRIPLICATE-RATIO CIRCLE.
47546
*Let the point S
be chosen, so that
(Fig. 165)
its
trilinear coordinates are proportional to the sides of the trigon.
Draw
lines through S parallel to the sides, then the circle in
question passes through the six points of intersection, and the
intercepted chords are in the triplicate-ratio of the sides.
[The following abbreviations are used,
a, h,
this article written for the sides of the trigon
c,
and not
a, i\ c,
being in
ABG-I
A = ABC
K=a- + b' + c-; \= ^Qrc" + e'er + d-J)-)
=
= A BFD = DE'F', &c.
6
DFD' = DE'D',
;
f^
By
= ~;
A
u>
;
i?-
hypothesis,
= A = X = __1A__ (4007)=^
BF
therefore
If
BF
.
AS, BS,
BF'
CS
&c.
=
BD.BD',
c
y
^
^
BD'
therefore F, F', D, D' are concyclic.
produced meet the opposite sides in
B)i
An
_
g sin
(1),
BC n _
6sin^C'«
aa
__ a?
Z>/3
6"
,
I,
^n
m, n,
,^.
* The theorems of (1 to 36) are for the most part due to Mr. R. Tucker, M.A.
original articles will be found in The Quarterly Journal of Pure and Applied Mathematics,
Vol. XIX., No. 76, and Vol. xx., Nos. 77 and 78.
Other and similar investigations have been made by MM. Lemoine and Taylor and
Prof. Neuberg, Mathesis, 1881, 1882, 1884.
The
TBILINEAB ANALYTICAL CONICS.
644
SF'
= BD = -;^ =
smB
v^
= ^.
A
(1)
KsmB
Similarly
BF' =
&c.
^,
J^
...
= 7)P^=^.-^ = $,&c
Ac A
sin
7)D'
(3).
(4).
(7
^D'
= ^D + DI>' = ^^^^^^,
FD = y(BD' + BF' -2BD.BF cos B) = ^,
Hence
DBF and
&c
by
..(5).
(2)
and (5)
(G).
B'B'F' are triangles similar to ABC, and tliey are equal
FSF = B'SF = E'SF', &c. (Euc. I. 37.)
to each other because
= y(BB'- + BF"-2BB.BF' cosB)
BF'
BF'
Hence
B'F
=
=^
(7).
= EB'.
FB'
= ^BB'=^-^\+^
A
a
^^^^^BF^±FB^-mi^cr
+ <f+]^
^^^"^
2BF.FB
&c
&
(5
2X
^
(8).
6)
=
^
-^
(9).
^
^
2\
(^°)-
sin
-"=V^-£)=ir('««)
A
COS0
AFE' + BBF' + CEB'
=
,
,
cos (/I
= a- cos ^ + 6c
,
J,
&c.
^^"^^'^"^^'^
=
Or, geometrically, by Euclid
+
a6c(cr +
(h'
+ c')
a'
=
&c.
-1
s
(.iij.
/x^A
= BEF,
by (G)
...
(12).
37.
I.
= ^B,
by (G) (B
=
(13).
/3Hr)=^(«" + ^/^ + ^y)' + «'PV + ^V + c'«/3
(1-i),
circle,
p
trilinear equation of the T. R. circle
The
^-1
A
circum-radius)
Radius of T. R.
or
— w),
+ (r + tr) /r + (a^ + Ir) y' =
+ ((L= + r)(Z.= -ffr) + cV} ^^ +
is
{{a- +
{(c'
h') (cr
+ r) +
hV'
]
^
+ cr)(,y + K-) + a'h'} ^...
(15).
Obtained by substituting the trilinear coordinates of B, E, F, through
which points the circle passes, in (4751), to determine the ratios I 7n n
and k. The coordinates of B are
:
^
g (a^ +
^'
Similiuly those of
B
and F.
IJ^)
K
sin
C
ac' sin J?
'
'
K
•
:
.
PARTICULAR CONIC S.
G45
THE SEVEN-POINT CIRCLE *
A, B, G (Fig. 1G5)
DEF, B'E'F', as in the
figure, intersecting each other in P, P\ L, M, N.
Let Q be
the circam-centre then the seven points P, P', L, M, N, Q, S
4754c
Let
be drawn
lines
tlirougli
parallel to the sides of the triangles
;
all
lie
on the circumference
of a circle concentric with the
T. R. circle.
(IG)
The proof depends on Euclid
III. 21,
and the similar triangles
DEF,
2PP^
C(17),
B'E'F'.
The radius
p'
of the seven-point circle is
P
'
obtained from
Expand and
sm TDD
3p
= COS
(11),
^
sin 2a;
= p' + 8D"^—2pSD cos (B- TDD')
TDD'
substitute cos
+ p'= R\
The
p'-
Qx^N
/(V^~
cosi?
= —— = -r-j^,
= —!—
by
s]n_B
,
2ca
^
by (17) and (13)
=—
and
(5),
cosJ.
,
= —-~
^^
*^^^^
^"^
"^^^
-
1 (20).'
triliuear equation of the seven-point circle is
= a'/8y + iV« + c'a/3
= ^ (5ca-fca/3 + a5y) (aa +
a6c(aHiff' + y')
or
o/3y
+ Jya-t-ca/3
If the coordinates of
P
are
a^, l\, y^,
"i"!
The equation of
STQ
a sin
And
.
2bo
etc
^ = V'cS^'
;
(3)
the equation of
is,
/3i/5i
and those of P'
=
a
The point S has been
* This circle
a{,
ft'i,
y[
(22).
;
then
(23).
yiyi
sin
(C-yl)
+ y sin (A-B)
(24).
=
(25).
is
JL(a'-hV-) + 4-(h'-chi')
triangle.
-fey)
by (4615),
(B-C)+f3
PP'
=
(21),
Z)/3
+
^(c'-a'b')
c
called the
It has also this property.
Symraedian point of the
The line joining the viid-
was discovered by M. H. Brocard, and has hoen called " The Brocard Circle,"
the points r, I" being called the Brocard points.
—
;
TFILINEAB ANALYTICAL CONICS.
646
point of any side to the mid-point of the perpendicular on that
through S.
Proof.— Let X, Y, Z (Fig. 166) be the feet of the perpendiculars
the mid-points of the same, and X', Y', Z' the mid-points of the sides.
the trilinear coordinates of X', /S', and x in order are proportional to
;
0,
c,
h
a,
h^
c
1,
since
CX-
cos
(7,
Now
This determinant vanishes
.
cos I?
therefore the three points are on
the same right line, by (4015).
lines X'x, Y'y, Z'z are concurrent appears at once
That the three
x, y, z
by (970),
2Y'x, &c.
The Symmeclian point may also be defined as the intersection of the three lines drawn from ^, i?, C to the corresponding
vertices of the triangle formed by tangents to the circumcircle at A, B, G.
respectively.
CX, AY,
Let Ba, (7/3, Ay be taken
=
BZ
a point 2, by (976), and this point
by similarity of figure is the Symmedian point of the triangle
formed by lines through A, B, G parallel to the sides BG,
Then Aa, B^, Gy meet
in
GA, AB.
If the sides of
that
X'Y'Z' be bisected, similar reasoning shews
Symmedian point
the
a,
the triangle X'Y'Z', lies
of
on S^.
It can also be shewn that,
if A'B'G' be any triangle having
those of xiBG and its vertices on SA, SB,
8G, the sides of the two triangles intersect in six points on a
circle whose centre lies midway between the circum-centres
When A'B'G' shrinks to the point Sy
of the same triangles.
the circle becomes the T. R. circle.
its sides parallel to
A
is
more general theorem respecting the triangle and circle
the following
Take
ABG
which any
any
From BD.BD'
which are
let
DD'EE'FF'
CE =
BD = pc,
Let
be the points in order, in
=
qa,
AF -
rb
|
^^O)
CD' = p'b, AE'=qc, BF'=r'a)
BF. BF', &c., Euclid III. 35, we can write three equations
satisfied
p
by the values
= r' = tac,
and from these equations
DF=ffc; B'F'
BO that
and
triangle,
circle cuts the sides.
it
q=p'=iah,
r
=
q'
= the
(27),
appears that
= aa,
BEF and B'E'F' are
&c.,
where
both similar to
ff --=
/(/-.V-/A'-M)
ABC.
(28),
<
SELF-CONJUGATE TRIANGLE.
Also DF'=tabc,
-^^-^
^
From &mBFD
=
.
therefore
tac sin
DF'
B we
=^
= ED'
FE'
i,
64
(29).
.
can obtain
a
cot
BED =
cot
(l>
=
=F^-^^=^
(30).
=
(tR
The radius of the circle
and the coordinates of its centre are
t(Kcr-a'-h'-c*)
a
=E
j cos
^+
of the circle
al3y
1
_
zbc
(.
The equation
(31),
gij^ii^^j.]y ;3
^nd y
(32).
)
is
+ hya + cai3 =
t
(aa
+ b[3 + cy)
{ahc(l — ta-)+&c.}
(33),
tabc[a\l-ta')+l3'(l-th')+y'{l-tc')}
or
When
When
= a(3y{(l-tb'){l-tc') + ebx~\ +&c
= 1 and the circle is the circum-circle
tK=l, = tX=— and the circle is the T. R. circle
t
= 0,
<t
(T
(34).
(35).
(36).
CONIC AND SELF-CONJUGATE TRIANGLE.
When the sides of the trigon are the polars of the opposite
vertices, the general equation of the conic takes the form
/V+m2)82-?iy
4755
Proof. — (Fig.
three ways,
=
/V
51.)
= 0.
The equation may be written
m^^^
{ny-\-m(3){ny—m(3),
n^y^
=
Hence, by (4699), a or
{la
BG
=
in
any one of the
(ny + la)(ny—la),
+ im^){la—imf3).
is
the chord of contact of the tangents
ny:hml3(AQ, AS) drawn from A, and ft is the chord of contact of the
tangents ny ± la (BR, BP) drawn from B. Hence a, ft are the polars of A, B
respectively
and therefore y or AB is the polar of G (4130). Also y may
be considered to be the chord of contact of the imaginary tangents Za±im/3
;
drawn from
G.
and j3 with the conic
be joined, the equations of the sides of the quadrilateral so
formed are
4756
If the points of intersection of a
QR,
PQ,
= 0,
=
-^la-\-tnl3-{-ny
0,
Ia-\-mfi+ny
Hence QB,
and
B'.
SP
and PQ,
BS
= 0,
la—m3-{-ny = 0.
SF, la-\-mfi-ny
RS,
intersect on the line y in A'
TEILINEAB ANALYTICAL CONICS.
648
4757
Eacli pencil of four lines in the diagi^am
is
a liarmonic
pencil, bjtlie test in (4649).
The
4758
triangle
A'B'C
also self- conjugate with regard
is
to the conic.
Proof.
— The equations of
lu-mft
Denote these by a,
=
its
0,
GB\
sides
Za
CA', A'B' are
+ m/3 =
and put a, /8
A'B G thus becomes
/3\ y,
equation referred to
same form as (4755).
y
0,
=
in (4755) in
0.
terms of
a'^-|-)8'^— 2ft'y^
= 0,
The
a', /S'.
which
is
of the
It is clear that the triangles AQS and BPB, formed
pair of tangents and the chord of contact in each case,
4759
by a
are also self-conjugate.
Taking A'B'G for the
4760
by
trigon,
and denoting the sides
the equations of the sides R8, PQ, QB,
quadrilateral become respectively
a, /3, y,
ny :^la=
mfi
0,
—
Ex. As an example of (4611), we
the equations
+ b/3 + Cy = S
+ m(3 —ny = ^
— lu+ +ny = 0)
aa
To
-s
may
of the
i ny = 0.
find the coordinates of
P
from
~ ^lm^(amn + hil + dm)
= '^mn-i-(amn+bnl + clm)
^7= ^)il^(amn + hnl + clm).
C
from which
SB
"
< /S
obtain the coordinates of Q, B, and S, change the signs of m,
n,
and
I
respectively.
ON LINES PASSING THROUGH IMAGINARY
POINTS.
—
Lemma I. The right line passing through two conjugate imaginary points is real, and is identical with the line
passing through the points obtained by substituting unity for
4761
^/
—\
in the given coordinates.
— Let
{a + ia\ b + ib') be one of tlie imaginary points, and therefore
the conjugate point. The equation of the line passing through
them is, by (4083) and reducing, b'x a'y + a'b ab'
0, which is real.
But this is also the line obtained by taking for the coordinates of the
points (a + a'j b + b') and (a— a', b
b').
Proof.
(a
— ia, b — ib')
—
—
=
—
—
Lemma II. If P, S and Q, B are two pairs of conjugate
imaginary points, the lines BS and QR are real, as has just
been sliown, and, tliorefore, also their point of intersection is
——
SELF-CONJUGATE TRIANGLE.
The
real.
otlier pairs
of lines
But the points
imaginary.
BS
PQ,
649
and PR,
QS
are
of intersection of each pair are
and are identical with the points which are obtained by
1 in the given coordinates, and
substituting unity for \/
drawing the six lines accordingly.
real,
—
Proof.
— Let the coordinates of the four points be as under
P
a + ia,
h
S
a — ia,
h
The equations
of
L
PB
=
(h'
,i'
— (a — a)
y
a + ia,
jj
u—ia,
(o—ift'.
+ a — ah
jj
L—iM, where
+a'iy — a'b',
+ i3')x—(a—a)^j + afi-ab — aiy
=
Now the
i ± ill =
+ ifl':
Q
R
and QS, by (4083), are L + iM and
(b—p)
M=
+ ih',
— ih',
+cib'.
intersect in the same real point as the lines
Unas L :iz iM
satisfy both equations
values L
0,
0, because the
Hence, to determine this point, we have only to take i as
simultaneously.
unity in the given coordinates.
=
M=
—
Lemma III. If P, S are real points, and Q, B sl pair of
conjugate imaginary points, the lines PS and QB are both
real, by Lemma I., and consequently their point of intersec-
BS
and PB, QS
The remaining pairs of hues PQ,
But the
their points of intersection are all imaginary.
line joining these two imaginary points of intersection is real,
tion
is real.
and
and
is jdentical
for \/
—l
with the line obtained by substituting unity
in the given coordinates
and drawing the
six lines
accordingly.
Proof.
— Let the coordinates of the four points be as under
P
x,ij„
Q
a
+ ia,
i8+//3',
S
x.2y,,
B
a
— ia,
ft—ifi'.
Since the coordinates of B are obtained from those of Q by merely changing
the sign of i, the equations of the four imaginary lines will take the forms
PQ
PB
SQ
SB
A-iB,
A + iB,
C-iD,
C+iD.
PQ
and SB be
point of intersection of
L'
iM' be the coordinates of the intersecgot
from those
lines
are
pair
of
this
tion of
and SQ, for the equations of
of
and SB by merely changing the sign of i. The points of intersection
are therefore conjugate imaginary points, and the line joining them is real,
by Lemma I. Also, since that line is obtained by writing 1 for i in the coordinates of those points, it will also be obtained by writing 1 for i in the
original coordinates of Q and B and constructing the figure as before.
Now let the coordinates of the
L+iM, L' + iM', then will L —iM,
—
PB
PQ
4
TBILINEAE ANALYTICAL CONICS.
650
To
4762
find a
common
pole and polar of two given conies
:
four real points P, Q, R, S,
Then A'B'G
construct the complete quadrilateral (4652).
(Fig. 51) is a self-conjugate triangle for each conic, by
(4758), and therefore each vertex and the opposite side form
a common pole and polar to the conies.
(i.)
If the conies intersect in
(ii.) If the conies do not intersect at all in real points, the
triangle A'B'G is still real, by Lemma II. (4761), and can be
constructed in the manner shown.
(iii.) If two of the points (P, S) are real, and two
{Q, B)
imaginary, then, by Lemma III., the vertex A' and the side
B'G are real, and may be constructed, and they form a common
pole and polar of the given conies.
Returning to the triangle of reference ABG,
m(i= ny sm<p; then the chord
ny cos
4763 Let la
joining two points <^i, <^2 is
=
la
(j>,
cosi (<^i+(^2)+»^/3 sini
and therefore the tangent
4764
4765
loi
Putting V
at the point
cos ^'-\-mfi sin
= L,
m'
= M,
Hence the pole
''
4768
(^o)^
the conic (4755)
(1).
of a(5'y' is
=
of \a-\-iJ,p-\-vy
—
(2).
i)
(^^-
H'-M'i)'
The
—
ny.
=
Laa+Ml3fi'-\-Nyy'
4767
(<^i
is
n''=-N,
La:'JrMP'-\-Ny'
The tangent or polar
<(>'
=
<f>
becomes
4766
= ny cosj
(<^i+<^2)
tangential equation
is
i+i+i='
and this is the condition that the conic
by the four lines
\a±fil3±vy = 0.
(^)'
(1)
may be touched
SELF-CONJUGATE TRIANGLE.
In like manner,
4769
La'^+M)8'^+iVyis tlie
651
condition
may
tliat (1)
(5)
pass through the four points
±^\
(a,
=
±y').
Xa+^^+vy
The locus of the pole of the line
respect to such conies is
4770
a
—
Proof. By (3), if (a,
the equation of condition.
^
/3,
"^
/8
with
r
y) be the pole, "
=y
&c.,
.-.
L=
—,
in (5),
The locus of the pole of the line la-\-mB-\-ny, with
/ti^
vy
respect to the conies which touch the four hues Xa
4771
±
±
Proof.— By
(3),
if (a,
ft,
n
tn
I
=y
y) be the pole, «
&c.,
.*.
i=
-, &c., in
equation of condition.
(4), the
The locus of the centre of the conic
case (4770, '1) by taking the line at infinity
4772
5+7
asin J.+j3 sin
sin
is
given in each
(7
for the fixed line, since its pole is the centre.
Thus the locus
4773
of the centre
through the four points (a±/8'±y')
a ^ sin
~^
A
^"^ sin
Proof.
sin
C
_q
y
of the centre of the conic
M^
La
__ = _^
,
given by
y'~
+^i8~~+
The coordinates
4774
B
the conic passing
of
is
:=
are
Ny
—Z-
— Let the conic cut the side a in the points (0^i7i),
A
(1)
(O/S.^y.^.
The
bisecting the chord will pass through the centre of the
Now + is the
conic, and its equation will he ft
y
(3^ + ^.2
yi + y.2sum of the roots of the quadratic in y8 obtained by eliminating y and a from
Similarly
and
aa
2.
a
the equations La' + Mft^ + Ny^
0,
0,
hft + Cy
The equation of the diameter thi'ough A being
for yi + y2 eliminate a and /3.
right line from
:
=
found, those through
B
=
=
A A
:
=
+
and G are symmetrical with
it.
TEILINEAB ANALYTICAL CONICS.
652
The condition that the conic
4775
Proof.
— This
by
is,
(4), the condition of
i\a
The condition that
and in
L-\-M-\-N
i),
4776
is
+ hft-\-Cy =
=
(1)
(1)
may be
a parabola
is
touching the line at infinity
0.
may be
a rectangular hyperbola
this case the curve passes
through
the centres of the inscribed and escribed circles of the trigon.
—
Proof. By (4G90), (a, b, c are now L, M, N).
a := ±/3
±y, the four centres in question.
(1)
is
now
satisfied
by
=
4777
Circle referred to a self-conjugate triangle
a' sin 2A-\-fi' sin
Proof.
— The
—
-,
N ^
cos
A
to the centre is
sin
2C
= 0.
~- = —~
(4774).
the condition of perpendicularity to a by (4622).
Therefore
Similarly
ccosO
iicosB
c
line joining
2B+y'
:
L
G
therefore (1) takes the form above.
a cos
A^
IMPORTANT THEOREMS.
CARNOT'S THEOREM.
if
If J, B, G (Fig. 52) are the angles of a triangle, and
the opposite sides intersect a conic in the pairs of points
a,
a
4778
;
h, h'
c, c
-,
;
then
Ac.Ac'.B(i.Ba'.Cb.Ch'= Ah.Ah'.Bc.Bc'.Ca.Ca'.
— Let
Proof.
a,
/3,
by (4317), Ah Ah'
.
:
y be the semi-diameters parallel to BC, GA, AB then,
/3^
y^.
Compound this with two simihir
Ac
;
Ac
:
=
:
ratios.
4779
Coil
—
If the conic
touches the sides in
Ac\Ba\ Cb'
=
Ah\ Bc\ Ca\
n, h, r,
then
THEOREMS.
653
reciprocal of Carnot's theorem is
If A, B, G
are the sides of a triangle, and if pairs of tangents
from the opposite angles are a, a' ; h, h' ; c, c' ; then
The
4780
:
(Fig. 52)
sin (Ac) sin
=
(Ac)
sin
(Ba) sin (Ba) sin (Cb) sin {Cb')
sin (Ab) sin (Ab') sin (Be) sin
where (Ac)
signifies the angle
(Be)
sin (Cm) sin (Ca),
between the
lines
A
and
c.
— Reciprocating
the former figure with respect to any origin 0,
polars of the vertices A, B, C.
Then,
AB, AG; and b, h', the polars of the
points h, V, will intersect in
and touch the reciprocal conic. Similarly, c, c'
will intersect in Q.
A, h' are perpendicular to OA, Oh', and therefore
/.Ah'= Z AOh', and so of the rest.
let
Proof.
A, B, G
(i.e.,
by (4130), Q,
B
BQ, QP, PB) be the
will be the poles of
B
PASCAL'S THEOREM.
The opposite
4781
meet
in three points
sides of a hexagon inscribed to a conic
on the same right line.
—
Proof.
(Fig. 53.)
Let a, ft, y, y', ft', a be the consecutive sides of the
hexagon, and let u be the diagonal joining the points au and yy'. The
equation of the conic is either ay — Jcftu
or a'y'—k'ft'u
0, and, since
these expressions vanish for all points on the curve, we must have ay — l-ftu
Therefore oy — a'y'
ay' — k'ft'to for miy values of the coordinates.
= li (hft k'ft'). Therefore the lines a, a' and also y, y' meet on tlie line
kft — k'ft'; and ft, ft' evidently meet on that line.
Otherwise, by projecting a hexagon inscribed in a circle with its opposite
The line at
sides parallel upon any plane not parallel to that of the circle.
=
=
=
—
which the pairs of parallel sides meet, becomes a line in which
the corresponding sides of a hexagon inscribed in a conic meet at a finite
distance (1075 et seq.).
infinity, in
With the same vertices there are sixty different
hexagons inscribable in any conic, and therefore sixty different Pascal lines corresponding to any six points on a conic.
4782
—
Proof. Half the number of ways of taking in order five vertices B, G,
D, E, F after A is the number of different hexagons that can be drawn, and
the demonstration in (4781) applies equally to all.
BRIANCHON'S THEOREM.
The three diagonals of a hexagon circumscribed to a
conic pass through the same point (Fig. 54).
4783
TBILINEAR ANALYTICAL GONIOS.
664
Proof.— Let the three conies /S + L\ S + M\ S + N\ in (4707), become
three pairs of right lines, then the three lines
become
the diagonals of a circumscribing hexagon.
Pascal's and Brianchon's theorems may be obtained, the one from the
other, by reciprocation (48-40).
L—M, M—N, N—L
THE CONIC REFERRED TO TWO TANGENTS AND
THE CHORD OF CONTACT.
Let
trigon
4784
4785
;
L
L,
=
M=0, B =
0,
(Fig. 55) be the sides of the
M being tangents and B the chord of contact.
The equation
The
lines
IlL
of the conic is
AP, BP, and GP
= R,
LM = R\
(4G99)
are respectively
iiR= M,
irL
= M.
[By (4604).
P
on the curve is determined by the value
Since the point
of ^5 it is convenient to call it the point /n.
4788
ju^Jv
The points /i and
drawn through
=M
4789
—
ju
(P and Q) are both on the
The secant through the points
/m/x
Proof. — Write
L-
line
G.
(ft+fi)
^,
/x'
(P, P'}
is
R + M = 0.
fi(fi'L — ]i) — (fi'Ii—M),
and, by (4604), it passes
it
^'.
Otherwise, determine the coSimilarly through /j.
and
and of {.I'L
and
^L
of
f.ili~M,
the
intei-section
ordinates of
fx'B—M by (4610), and the equation of the secant by (4616).
through the point
— The
4790
Cor.
are therefore
4791
—U
—B
tangents at the points
These tangents intersect on B.
/n
and
—
^i
(P, Q)
[Proof by subtraction.
—
4792 Theorem. If the equation of a right line contains an
indeterminate ^ in the second degree, it may be written as
R^.
above, and the line must therefore touch the conic
LM=
4793
The
])olar of
the point (//,
M\
B') is
LM'-2RR-\-L'M =
{).
TEE come LM = Rl
—
655
Proof.
For /i + ju' and ju^', in (4789), put the values of the
product of the roots of /li'^L' -2fiB' + 31'
(4790).
4794
=
sum and
Similarly the polar of the point of intersection of
aL — B and bB —
M
is
abL-2aR-^M=0.
The
4795
line
two tangents
of
GE
at
joining the vertex G to the intersection
/.i, or at —/n and —jn', is
and
^t
iHLL—M= 0.
Otherwise, if two tangents meet on any line
through G, the product of their ^u's is equal to
— M,
ciL
drawn
a.
— Eliminate R from the equations of the two tangents (4790).
The chords PQ, FQ and the line GE all intersect in
Proof.
4796
the same point on B.
Proof.— The equations
of
FQ', P'Q
l^fi'L± d-i-i-i')
and, by addition and subtraction,
we
are,
by (4789),
E-M =
obtain
fxfx'L
M
0,
—M =
(4795), or
i?-
= 0.
The lines i^fx'L +
(GD) and B intersect on the chord
PP' which joins the points jn, ^t/; or The extremities of any
chord passing through the intersection of aL-\-M and B have
4797
—
the product of their ^'s equal to
a.
The chord joining the points ^i tan ^, /i cot ^ touches a
and chord of contact B.
conic having the same tangents L,
4798
M
Proof.
— The equation of the chord
fi^L— fiE (tan
and
this touches the conic
4799
The tangents
on the conic
Proof.
eliminate
by (4789),
+ coi<p) + M=
= jB^
at the points
LM = B^ sin^
—Write
(l)
JyMsin^2^
is,
0,
at the point
;ii
tan
<}),
yu,
/i
by (4792).
cot
(p
intersect
2(j>.
the equations of the two tangents, by (4790), and then
)u.
—
4800
Ex. 1.
To find the locus of the vertex of a triangle circumscinbing
a fixed conic and having its other vertices on two fixed right lines.
Take
for the lines CD,
B"- for the conic (Fig. 56), aL + M, hL +
GE. Let one tangent, DE, touch at the point /x then, by (4796), the others,
M
LM =
;
TEILINEAU ANALYTICAL CONICS.
656
PD, PE,
— —
will toiicli at the points
,
/.«,
P
and the locus of
therefore,
by (4790), their
^L--B + M.
l!^L-~li + M,
Eliminate
and
,
r-
r-
equations will be
found to be (^a-\-h)-LM = A^abp-.
[Salmon, Art. 272.
is
—
4801
Ex. 2.
To find the envelope of the base of a triangle inscribed in
a conic, and whose sides pass through fixed points P, Q.
P- for the conic ciL—M,
(Fig. 57.) Take the line through P, Q for E
for the lines joining P and Q to the vertex G. Let the sides through
hL —
and Q meet in the point fx on the conic then, by (4797), the other extremi;
M
LM-
;
P
;
ties
will be at the points
——
and
+ ?0
LM =
4802
Ex.
3.
— To inscribe a
and therefore, by (4789), the
,
equation of the base will be a&L + (a
line always touches the conic ^ah
(a
A'-R
+ /-3'^ =
By
0.
(4792), this
+ by B^.
\_Ibid.
ti-iangle in a conic so that its sides
may
pass
through three fixed points. (See also 4823.)
We have to make the base ahL+ (a + h) ixB + in^M (4801) pass through
a third fixed point. Let this point be given by cL = P, dB = M. Elimi-
=
nating L, M, B, we get ab + (a + b) ixc + fx'^cd
0, and since, at the point /u,
B, fj}L
M, that point must be on the line abL + (a-\-b) cB + cdM.
The intersections of this line with the conic give two solutions by two posi-
=
nL =
[^Ibid.
tions of the vertex.
RELATED CONICS.
4803
A
conic having double contact with the conies
S and
S' (Fig. 58) is
where E, F are common chords
S-S' = EF.
PiioOF.
— The equation may be written
(fiE
+Fy =
4^N
or
S and
of
in either of the
(fxE
- P)-
=
8',
so
that
ways
4^. S',
There arc three
=h F are the chords of contact ^IP, CL.
such systems, since there are three pairs of common chords.
showing that fxE
4804
C'oR.
1.^
— A conic touching four given hues
the diagonals being
J^,
F
ii:'E'-2ii
Here S
= AG
and
(Fig. 59)
{AC-\-IJD)-\-F
S'
=
A,B,
C, D,
:
BB, two
=
i\.
pairs of right lines.
ANEABMONtO PENCILS.
Otherwise,
4805
ii'L'-iM
For
(L' +
L, M, Nhe
becomes
if
sides, the conic
tlie
657
diagonals and
L±M±N the
{U+3P-N')+M' = 0.
always touches
this
M'-Ny-WM'
or
(L + 3f+N)(M+N-L){N + L-M)iL-\-M-N).
[Sahnon, Art. 287.]
4806
Cor.
circles G,
2.
—A
The chords
4807
conic having double contact with
of contact
^+C-C' =
The equation may
4808
which
two
C is
signifies
that the
become
and
^-C+(7'
=
0.
also be written
sum
or difference of the tangents
to the circles is constant.
drawn from any point on the conic
ANHARMONIC PENCILS OF
CONICS.
The anharmonic ratio of the pencil drawn from any
point on a conic through four fixed points upon it is constant.
4809
—
Proof. Let the vertices of the quadrilateral in Fig. (38) be denoted by
Multiplying the equation of the
A, B, C, I), and let P be the fifth point.
conic (4697) by the constants AB, CD, BC, DA, we have
,
AB.CD _ ABa CDy ^ PA PB sin APB PC PD sin CPD
PB.PC sin BPG. PD PA sin DPA
BC.DA BCfi DAS
sin APB. sin CPD
sin BPC. sin DPA
.
.
.
.
.
.
Compare (1056).
If the fifth point be taken for origin in the system
(4784, Fig. 55), and if the four lines through it be
4810
L—iJLj^R,
L — ix,R,
4
L—fJiJi,
p
L—jM^R,
TBILINEAU ANALYTICAL CONICS.
658
the anliarmonic ratio of
tlie
pencil
is,
by (4G50),
—
CoE. 1.
If four lines through any point, taken for
the vertex LM, meet the conic in the points ^tj, /n.,, fx-s, /t^, the
anharmonic ratio of these points, with any fifth point on the
conic, is equal to that of the points —/Hi,
i^
a'o,
/"s, —l^a
which the same lines again meet the conic.
4811
—
—
—
—
4812 CoE. 2. The reciprocal theorem is If from four
points upon any right line four tangents be drawn to a conic,
the anharmonic ratio of the points of section with any fifth
tangent is equal to the corresponding ratio for the other four
tangents from the same points.
4813
The anharmonic
made by four
to a conic
segments of any tangent
ratio of the
fixed tangents is constant.
—
Proof. Let fx, n^, fj^, yug, fx^ (Fig. 60)
anliarmonic ratio of the segments is the
lines from
to the points of section;
HH-Jj—M, fifi^L- M, a pencil homographic
LM
be the points of contact.
same
The
as that of the pencil of four
M,
that is, of nn^L—M, iiii.Jj
(4651) with that in (481U).
—
4814
If P, P' are the polars of a point with respect to the
conies 8, S\ then P-\-hP' will be the polar of the same point
with respect to the conic S-\-h8'.
Hence the polar of a given point with regard to a
conic passing through four given points (the intersections of
8 and 8') always passes through a fixed point, by (4101).
If Q, Q' are the polars of another point with respect to
the same conies, Q-\-hQ is the polar with respect to 8-\-h8'.
4815
Hence the polars of two points with regard to a
system of conies through four points form two homographic
4816
pencils (4651).
The locus of intersections of corresponding lines of
two homographic pencils ha^ang fixed vertices (Fig. 61) is a
4817
conic passing through the vertices and, conversely,
conic be given, the pencils will be homographic.
;
Proof.— For
climinatiiio-
A-
fnmi
P + AT'=
0,
if
the
Q + IQV, we get !'(/=
r'Q-
CONSTRUCTION OF CONICS.
659
—
The locus of the pole of the line joining the
Cor.
points in (4816) is a conic.
4818
PROOr.
—For the pole
The
4819
(Fig. 62) of
is
the intersection of
P + hP'
two
and Q + hQ'.
right lines joining corresponding points AA', &c.
two homographic systems of points lying on two
right lines, envelope a conic.
— This
Proof.
is
the reciprocal theorem to (4817)
;
or
follows from
it
(4813).
If two conies have double contact (Fig. 63), the anharmonic ratio of the points of contact A, B, G, D of any
four tangents to the inner conic is the same as that of each
set of four points (a, b, c, d) or {a, h\ c', <V) in which the
tangents meet the other conic.
4820
— By (4798).
The ^'s for the points on the latter conic will be
of the points of contact multiplied by tan <p for one set, and
cot^) for the other, and therefore the ratio (4810) will be unaltered.
Proof.
equal to the
by
4821
/.t's
Conversely,
if
three chords of a conic aa
=
,
hb', cc
be
then
dd' envelopes a conic having double contact with the given
fixed,
and a fourth
rZc?'
moves
so that \ahcd\
[ctb'c'd'l,
one.
For theorems on a right
line cut in involution
by a
conic, see (4824-8).
CONSTRUCTION OF CONICS.
THEOREMS AND PROBLEMS.
If a polygon inscribed to a conic (Fig. 64) has all its
sides but one passing through fixed points A, B, ... Y, the
remaining side az will envelope a conic having double contact
4822
with the given one.
Proof.
— Let
a,h, ...z be the vertices of the polygon,
Then, by (4811),
a.
four successive positions of
a,
I
a, a",
a"
}
=
{ ^j
^" ^"'
^'
>
>
|
= &c. =
|
2;,
Therefore, by (4821), the side az envelopes a conic, &c.
and
a, a', a", a'"
TBILINEAB ANALYTICAL CONIGS.
660
4823
Poncelet's construction for inscribing in a conic a
its n sides passing through n given points.
polygon having
+
Inscribe three polygons, each 0/ n
1 sides, so that n of
may pass through the fixed points, and let the remaining
sides he a'z', af'z", ?\1"t:!" , denoted in figure (65) hy AD, CF, EB.
Jjet MLN, tlie line joining the intersections of opposite sides of
the hexagon
(4781), meet the conic in K; then
will be a vertex of the required polygon.
each
K
ABCDEF
Proof.
—
iD.KAGIj]
= iA.KDFB\,
K, P, N, L; therefore the anharmonic
vertex on the conic, by (4809)
;
i.e.,
I
ratio
KAGE = KDFB
]
Kaa'a" =
\
|
be the remaining side of a fourth polygon inscribed
by (4811), as
where a and z
in (4822),
aa'a'a"
\
=
zz'z'z"
i
|
\
Kz'z"z"'
\
like the others,
.
Hence
K
for
\
But,
.
is
any
if
az
we have
the point
coincide.
Lemma.
4824
|
passing through
each pencil
|
—A
fixed points meets
involution (1066).
system of conies passing through four
any transversal in a system of points in
—
Let u, u be two conies passing through the four points then
in
Take the transversal for x axis, and put ?/
will be any other.
and
each conic, and let their equations thus become ax- 2qx c
These determine the points where the transversal
0.
2(7'a)+c'
two points given by
Jcu' in
It will then meet u
meets u and u
c')
2g'x
ax^
2gx c k (a
0, and these points arc in involution with
Proof.
;
=
u-\-'kitj'
+
=
aV +
.
+
+ +
V+
+
+ =
+
=
the former, by (1065).
Geometrically (Fig. QQ),
[a.AdhA']
therefore
{ AGBA'
]
=
=
AB'G'A'
{
}
[c.AdbA'] (4809),
=
A'G'B'A
{
,
}
therefore by (1069).
—
One of the conies of the system resolves
Cor. 1.
Hence the points B, B\
the two diagonals ac, bd.
G, G' are in involution with D, D' where the transversal cuts
the diagonals.
4825
itself into
,
—
A transversal meets a conic and two tanCoR. 2.
gents in four points in involution, so as to meet the chord of
contact in one of the foci of the system.
For, in (Fig. 66), if h coincides with c, and a with d, the
transversal meets the tangents in G, G', while B, B\ D, B', all
(Fig. Ql), one of the foci on the chord of
coincide in
4826
F
contact.
CONSTRUCTION OF CONIC S.
661
—
The reciprocal theorem to (4824) is Pairs of tangents
from any point to a system of conies touching four fixed hnes,
form a system in involution (4850).
4827
4828
tion
\x-\-iiiij-\-v:: may be cut in involuthe vanishing of the determinant
The condition that
by three conies
is
a.
^1
B,
As
B.
TEILINEAB ANALYTICAL CONICS.
662
Otherwise, let a, fl, y be the sides of ABC; la + mft + ny,
/x/3
the moving base ah.
and a
the fixed Hues Oa, Oh
Then the equations of the sides will be
(Ifi
Eliminate
/i;
then
I'u
=
;
+ m) l3 + ny — 0, (I'fi + m) a + n'fiy = 0.
Imajj— (ml3 + ny)(l'a+n'y), the conic in
+ ra'ft + n'y
question, by
(4697).
Given
4831
points on
tlie
five points, to find geometrically any number of
circumscribing conic, and to find the centre.
Let A, B, 0, D, E {Fig. 70) be the five points. Draw any
meeting CD in P. Draw PQ through the intersection of AB and DE meeting BO in Q; then QE ivill meet
PA in F, a sixth point on the curve, as is evident from PascaVs
theorem (4781).
To find the centre, choose AP in the above construction
parallel to CD, and find two diameters, as in (1252).
A
line through
To
4832
find tbe points of contact of a conic with five right
lines.
ABCDE {Fig.
AG and
Let
intersection of
point of contact of
Proof.
—By
71)
BE.
AB, and
be
Join D to the
the pentagon.
This line unll pass through the
so on.
(4783), supposing two sides of the hexagon to become one
straight line.
To
4833
describe a conic, given four points
upon
it
and a
tangent.
Let
b, h'
a, a',
Then,
points.
{exterior letters
ifAB
a tangent, c,
the ratio A.&
is
theorem (4778) gives
Since there are tivo values of
conies
may
4834
To
be
drawn
this
in
Fig. 52)
be
the
four
and GarnoVs
Then by (4831).
Bc'l
Be), two
ratio, + (Ac
c'
coincide,
:
:
as required,
describe a conic, given four tangents and a point.
a
b, V {interior letters in Fig. 52) be the four
Then, ifQ be the given point on the curve, the lines
tangents.
c, c' must coincide in direction, and (4780) gives the ratio
-sin2(Ac)
sin^ (Be), by ivhlch the direction of a fifth tangent
Then by (4832). Tlie two values
through Q is- determined.
Let
a,
,
:
+
Ac sin Be) furnish ttvo solutions.
Otherwise by (4804), d.elcrmiiiivg ^i by
(sin
:
the given point.
tlie
coordinates of
.
CONSTBUCTION OF CONIGS.
4835
To
describe a conic,
663
given three points and two
tangents.
Let A, A', A'' he the points {Fig. 67, supplying obvious
Let the two tangents meet AA' in the points C, C.
in involution (1066)
Find F, F', the foci of the system AA\
Similarly, find Gr, G', the
determining the centre by (985).
Then, by (4826), the chord
foci of a system on the line KK".
of contact of the tangents may be any of the lines FG, FG',
There are accordingly four solutions, and the
F'G, F'G'.
construction of (4831) determines the conic.
letters).
CC
To
4836
conic, given
a
describe
two points and three
tangents.
Let
points.
AB, BO, CA {Fig. 167) be the tangents, and
Draw a transversal through PP' meeting
P, P' the
the
three
Find F, a focus of the system PP', QQ'
tangents in Q, Q', Q".
in involution (1066, 985) ; G a focus for PF, qq',and H/or
Cunstruct a triangle luith its sides passing through
PP', Q'Q''.
N
on BC, CA, AB,
F, G, H, ami ivith its vertices L, M,
by the method of (4823), lohich is equally applicable to a rectiwill be the points of
L, M,
lineal figure as to a conic.
The reason for the construction is contained in
contact.
N
(4826).
There will, in general, be four solutions.
If the conic be a parabola, the foregoing constructions
can be adapted by considering one tangent at infinity always
to be given.
4837
To draw a parabola through four given
points a, a,
b, b'
This is problem 4833 with the tangent at infinity.
In figure (52), suppose cc to coincide and AB to remove to infinity so as
to become the tangent at c, the opposite vertex at infinity of a parabola, and
Cc then becomes a diameter of
therefore to be perpendicular to the axis.
the parabola, and Caruot's theorem (4778) shows that
^4^Ba^^
Ca.Ga
Gb.Cb'
AV''
Bo''
sin'
sin'
AGc
BGc
since the points C, a, a, h, V are all on the axis of the parabola relatively to
This result, however, is at once obtained from
the infinite distance oi AcB.
Cb Gb' being the ratio of the products of the roots
equation (4221), Ga.Ca
Thus a diameter of the parabola can be drawn
of two similar quadratics.
through C by the known ratio of the sines of AGc and BCc.
Next, describe a circle round three of the given points a, a', b. By the
property (1263) and the known direction of the axis, the other point in
:
.
circle cuts the parabola can be found.
Five points being known, we can, by Pascal's theorem, as in (4831),
which the
TEILIKEAR ANALYTICAL CONICS.
664
obtain two parallel cliovds, and tben find P, the extremity of tlieir diameter,
by the proportion, square of ordinate cc abscissa (1239).
Lastly, draw the diameter and tangent at P, and then, by equality of
angles (1224), draw a line from P which passes through the focus. By
obtaining in the same way another pair of parallel chords, a second line
through the focus is found, thus determining its position.
To draw
4838
a parabola
when
four tangents are given.
This is effected by the construction of (4832, Fig. 71). Let AB, BC, AE,
be the four tangents, and CD the tangent at infinity. Then any line
drawn to C will be parallel to BC, and any line to D will be parallel to ED.
ED
To draw
4839
a
given
parabola,
tliree
points
and one
tangent.
This is effected by the construction of (4835, Fig. 67). Let hC be the
must be at C, so that
then the centre of involution
tangent at oo
CF-, determining F. F', another point on
CA.CA'
CC.CC 0. CO
the chord of contact, being found by joining AA" or A' A", FF' will be the
diameter through a, since the other point of contact h is at infinity.
;
=
=
=
To draw
4840
given one point
a parabola,
and
tliree
tangents.
the case of (4834), in which one of the given tangents h' is at
and the tangent h,
must therefore be at infinity, and QB,
The ratio found
since they all join it to finite points, must be parallel.
determines another tangent, and the case is reduced to that of (4838).
This
is
FB
infinity.
B
4841
To draw a parabola, given two
points
two
and
tangents.
This is problem (4836).
tangent at infinity. F, G,
The chords
proportionals.
AC in that construction to be the
be determined as in (4830) by mean
Suppose
H
will
NM
M
is at
will become parallel, since
LM,
to
and we have to draw LA'' and the parallel lines from L and
may
pass through F, G, H in their new positions, so that the vertices L,
lie on BC and AB.
Otherwise by (4509), the intercepts s and i can readily be found from the
two equations furnished by the given points.
N
N
infinity;
4842
To
describe a conic touching three right lines and
touching a given conic twice.
Let
AD, CF, EB
the given conic.
{Fig. 65) he the three lines as they cut
Join
AB, AF, BC, BE, and
determine
K
hi/
K
MLN.
will he one point of contact of the
two conies, hy (481^2) aiid the proof in (4823), since AD, CF,
the
Pascal
EB, and
side " in
line
the tangent at
K
are four iJositions of the " remaining
The prohlem is thus reduced to
that proposition.
665
BECIPEOGAL P0LAE8.
(4834), since four tangents and
them are
To
4843
K
the i?oint of contact of one of
noio Jcnoivn.
describe a conic toucliing eacli of
two given conies
and passing through a given point or touching a given
twice,
line.
Proceed by (4803), determining ^
hij
the last condition.
To describe a conic touching the conies S-^-L^, S-\-M'\
ISahmn, Art. 387.
S-\-N'^ (4707) and touching S twice.
THE METHOD OF RECIPROCAL POLARS.
—
The polar reciprocal of a curve is the envelope of
Def.
the polars of all the points on the curve, or it is the locus of
the poles of all tangents to the curve, taken in each case
with respect to an arbitrary fixed origin and circle of reciprocation.
Thus, in figure (72), to the points P, Q, B on one
curve correspond the tangents qr, rp, and chord of contact j^q
on the reciprocal curve and to the points p, q, r correspond
the tangents QB, BP, and chord PQ.
The angle between the tangents at P and Q is evidently
equal to the angle j^Oq, since Oj), Oq, Or are respectively perpendicular to QB, BP, PQ.
4844
;
Theorem.
4845
from
— The distance of
Proof.— (Fig.
73.)
Take
for origin
CP
the polar of c, pt the polar of C,
Then
dicular on polar of G.
r-
and
that
a point
from a
line is to its
the origin as the distance of the pole of the line
the polar of the point is to its distance from the origin.
from
distance
is,
-
00. Ot
=
Oc.OT
)
:
FT
of auxiliary circle,
c, cp perpen-
Therefore, by subtraction,
0G.0iH=0c.0M)
OP 00
— By making CP
and centre
perpendicular on polar of
or
::
cp
:
cO.
OO.mt=Oc.MT,
00 .cp =Oc.GP;
Q. E. D.
constant, Ave see that the reciprocal
a conic having its focus at the origin and its
directrix the polar of the circle's centre.
4 Q
CoE.
of a circle
is
TBILINEAB ANALYTICAL CONICS.
GENERAL RULES FOR RECIPROCATING.
4846
^ point
hecoiues the polar of the point,
and a
rigid line
hecoines the pole of the line*
4847
^
^^'^e
through a fixed point becomes a point on a fixed
line.
The intersection of two
4848
lines becomes
the
line ivhich
joins their poles.
4849
Lines passing through a fixed point become the same
line, the polar of the point.
number of points on a fixed
4850
^
tangents
right line intersecting a curve in n points becomes n
reciprocal curve passing through a fixed point.
to the
Tivo lines intersecting on a curve become
wliose joining line touches the reciprocal curve.
4851
tivo
points
4852 Tivo tangents and the chord of contact become tioo
points on the reciprocal curve and the intersection of the tangents at those points.
4853 -4 pole and polar of any curve become respectively a
polar and pole of the reciprocal curve; and a point of contact and
tangent become respectively a tangent and its point of contact.
4854
4855
4856
The locus of a point becomes
the envelope of a line.
-4n inscribed figure becomes a circumscribed figure.
Four points connected by six lines or a quadrangle
become four lines intersecting in six points or a quadrilateral.
The angle between tivo lines is equal to the angle subtended at the origin by the corresponding points.
(4844)
4857
4858
The origin becomes a
line at infinity,
the
polar of the
origin.
4859
Tivo
through the origin become two points at
polar of the origin.
lines
injiniiy on the
4860 Tivo tangents through the origin
points at infinity on the reciprocal curve.
to
a curve become two
4861
The points of contact of such tangents become asymptotes
of the reciprocal curve.
4862
The angle between the same tangents
angle betiveen the asymptotes.
is
equal
to
the
(4857)
* Tliat is, with respect to the circle of reciprociitiou, and so throughout with the exception of (4853).
BEGIPBOGAL POLABS.
4863
667
According as the tangents from the origin to a conic are
the reciprocal curve is an hyperbola or
reat or imaginary,
ellipse.
if l^i'& origin be tahen on the conic, the reciprocal curve
a parabola.
4864
is
For, by (4860,
4865
A
'1),
the asymptotes
ai'e
parallel
and
at infinity.
trilinear equation is converted by Teciprocation into
a tangential equation.
Thas ay
= hfth
a conic passing through four of the intersections of
Reciprocating, we get a tangential equation of the same
hBB, and this is a conic touching four of the lines which join
form AC
0.
the points whose tangential equations are A=- Q, JB
0, D
0, (7
See (4907).
the lines
o,
/3,
=
y,
is
<).
=
4866
= aW
The equation
same
witli the
=
=
of tlie reciprocal of tlie conic ahf-\-l>^ii?
and axes
origin
is
where h is the radius of the auxiliary circle whose centre
the centre of the conic.
—
Proof. Let
then fcV-2=/
4867
be the perpendicular on the tangent,
p
= a2cos^0 + 6-sin2
its
is
inclination
;
(4732).
6l
The same when the
origin of reciprocation
is
the
point xy\
{aw' -\-yii' -\-W)'^
A;V~'=p=
Proof:
4868
The
auxiliary
^
+
b'^
=
a\v'^-{-b\i/\
sm-d—{x' cosd + y'
sinO).
reciprocal curve of the general conic (4656), the
in tribeing x^ y^
or iG^-\-y^
z^
=¥
+
circle
linears, will
replacing
-/a^ cos^d
+ =
be symmetrically
by
— A'^
—
Proof. Let i/j be a point on the reciprocal curve, then the polar of i>/,
Therefore, by
namely, a'^ + i/'/ —
0, must touch the conic, by (4853).
for A, /,/, u in tlie tangential equation
Jc'
(4(365), we must substitute 4, ?/,
^\- + &c.
0.
A'''
=
—
=
From the reciprocal of a curve with respect to the
origin of coordinates, to deduce the reciprocal with respect to
an origin xy', substitute in the given reciprocal equation
4869
'*'
for
a?.z--\-iji/'-{-k''
.V
and
—
,
'^'\i.i ^^^ ^•
d\v-\-yij-\-k'
;
ANALYTICAL GONIGS.
668
Proof.
^.
PB =
—
Let, 7'
he the perpuiidicular from the origin on the tangent and
The perpendicular iVom
— = —-—3!
.•.
h'
Jr
p
It
,
cos
a
—y
'
x'y' is
a
smU,
•
F — x'cosd — y'sind,
= ccx +yy' +
.. ^^'
.
--^
Ih
^
;
Icp cos
Rcos
•.
Jc^
p
aV +
2/2/'
+
^''
TANGENTIAL COORDINATES.
By employing these coordinates, theorems which are
merely the reciprocals of those already deduced in trilinears
may he proved independently. See (4019) for a description
4870
of this system.
The following proposition serves to transform by reciprocation the whole system of trilinear coordinates of points and
equations of right lines and curves, into tangential coordinates
of right lines and equations of points and curves.
THEOREM OF TRANSFORMATION.
4871
Griven the trilinear equation of a conic (4656), the
tangential equation of the reciprocal conic in terms of X, fx, v,
C
upon the
the perpendiculars from three fixed points A\ B'
being the origin of
tangent (Fig. 74) will be as follows,
,
reciprocation and
4872
0A\
OC =p,
OB',
^/,
r:
—
^^X^
hiL'
cv"
2ffiv
2^-v\
2h\fi
jr
ff
r^
qr
rp
pq
—
=
=
_
^^
=
Prook. Let a
be the sides of the original trigon ABC.
0, /3
0, y
The poles of these lines will be A', B', C, the vertices of the trigon for the
reciprocal curve.
Let BS be the polar of a point P on the given conic
a, /3, y the perpendiculars from
i.e., the trilinear coupon BC, GA,
upon BS;
ordinates of P.
Let X, n, V be the perpendiculars from A', B',
Then,
i.e., the tangential coordinates of the p6lar of
referred to A', B', C.
AB
P
;
C
P
by
of
(484.5),
a,
^ = ^^„ A=^,, -r = ^.
y in (4656) and divide by
ft,
The angular
4873
A'B'C
Substitute
those values
0P\
relation between the trigons
ABG
is
JiVCr
= TT-A,
evil'
= iT-B,
A'OB'
=
TT-C,
and
—
TANGENTIAL G00BDINATE8.
4874
If
^BG
reciprocation,
4875
Now
it
be self-conjugate with regard to
will coincide with A'B'C.
it
tlie circle
of
be the circum-centre (4629) of A'B'G'
ABC, and, by (4873),
let
(Fig. 74), then
669
will be the in-centre of
2B'=n-B, 2C'=7r-C.
2A'='n-A,
=
= q = r in
0, so
Also p
(4872), which becomes ^ (X, fi, v)
that the conic and its reciprocal are represented by the satiie
Consequently any relation in trilinear coordinates
equation.
have
has its interpretation in tangential coordinates.
then the following rule
We
:
—
Rule.
To convert any expression in trllinears into
tangentials, consider the origin of the former as the in-centre of
the trigon, change a, |3, y into X, ^, v, and interpret the result
by the rules for reciprocating (4846-65).
If the angles of the
4876
original trigon are involved, change these by (4875) into the
angles of the reciprocal trigon, of luhich the origin ivill now be
the circum-centre.
4877
ABC,
Referring trihnears and tangentials to the same trigon
the equation of a point, as shown in (4021), becomes
AX+l/x+X.^0;
Ih
Ih
4878
01^5
by multiplying by
Ih
-|-2,
BOC\^COAiL-^AOBv =
The equation
of a point can generally
from the figure by means
(Fig. 3)
{).
be obtained directly
of this formula.
EQUATIONS IN TANGENTIAL COORDINATES.
direct demonstrations of the following theorems, the reader
consult Ferrers' Trilinear Coordinates, Chap. vii.
For
AB
i\ that
in the ratio a
The point dividing
intersection with the internal or external bisector of C,
4879
aXdbV =
The point
4881
0-
in (4878)
Mass-centre,
is
:
Centre of
now on
the side
X+/li+i/
=
AB X+^ =
is,
may
the
is
0.
AB.
0.
[For
BOC=:COA^AOB.
7
ANALYTICAL CONICS.
670
4882
In-centre,
4883
«-
4884
nX+V+Cv^O.
(4878), for
a
t^
Circum-centre X sin 2 A +/x siu 2B-\-v sin
a sin
cation (4876),
to the trigon
ABC
;
=
J.
+ ^ sin
\ sin 2^'
=
Z?
^+
/Li
sin J5
+
»'
sin
+ ^ sin 21/' +
of perpendicular
J'
=
(7
sin 26'',
A'B'C
that
;
is,
by (4875).
from C npon AB,
Xtan J[+/u,tan^
=
0.
4886
Orthocentre X tan
A +fi tan B-^v tan C =
4887
Inscribed conic of
ABG,
4888
Point of contact with AB,
MX+Lfi =
4889
0.
2.4,
therefore
X sin
Foot
2C =
0//ie?7t!ise.— By recipro&c. in (4878).
is the line at infinity referred
+ y sin (1
^B' sin
the equation of the pole of that line referred to
4885
By
L
— aX+b/x+Cv = 0. L—
ex-centre,
Proof.— For JJOC
is
r
0.
[Proof below.
0.
In-circle (4629),
= o.
4890 Point of contact witli AB, (d-l)) X + (5-a) = 0.
ex-circle, (6—1)) X/A+(d — r) vX—^fxv = 0.
4891
the equaProof. — Since the coordinates of AB of the trigon are
(^-a) iiv-\-{^-h) vX+(s;~r) x^
/m
ii
0, 0, v,
must be satisfied when any two of the coordinates
Otherwise by
X,
vanish, therefore it must be of the form (4887).
reciprocating (4724).
(Fig. 'S), X
S-ci t^-h
in
n
If the circle touches
(Fig. of 700), which proves (4890).
tion of the inscribed conic
/i,
V
AB
D
:
- = AB BD =
:
:
(4888) is the equation of the point of contact, because the line (0,
passes through it and also touches the conic (4887).
(4889) is the in-circle by (4887) and (4890) and what precedes.
0, r)
[By (487G) applied to (4739, '40).
4892 Circumscribed conic,
L-X-+M>'+iVV-2MA>v-2iV2:vX-2LJ7X/i, = 0, (4740)
:
:
TANGENTIAL GOOEPlNATES.
4893
or
^{L\)-\-y/Mii-^^Ny
4894
Tangent
at A,
4895
Cir cum- circle
4896
or
Mfx,
= Nv.
to (4747, '8),
4897
Eolation between
4898
Coordinates of
tlie
— The
= =
/3
The point
= 0.
Proof. — By (4876),
= =
/x
y,
(4876).
m/LL -{-
uv
for the line
la
IX
=
sin^' (4875)
coordinates of any right line
trilinear cooi'dinates
ciprocal conic are o
and by cos—
tlie line at infinity
X
4899
= 0.
a^/x+IJv//*+ryv=0.
Proof.— By (4876) applied
Proof.
671
-{-
v.
of the origin
and centre of the
re-
It is also self-evident.
=
will be at infinity
when
l-^m-i-n
origin a
= /3=:y
4900
the
l
+ mft + ny
=
will pass
through the
+ m + n:=0.
curve will be touched by the line at infinity when
A
sum
when
of the coefficients vanishes.
Proof.— By (4876),
a
= =y
/3
for this is the condition that the origin in trilinears,
shall be on. the curve.
4901
The equation
4902
or
of the centre of the conic
(p
(X,
(a+h-\-g')\+{h+b+f)iM-\-{g^f+c)v
ABC
(4876) are
Proof.— The coordinates of the in-centre of
therefore the polar of this point with regard to the conic f
This point and polar reciprocate into
(4658).
0^ + ^ _1_,^
point, of which the former, being the reciprocal of the in-centre,
the line at infinity, and therefore the latter is the centre of <p (X,
=0
its
equation
is
as stated.
^t,
v)
is
= 0.
a=fi'=y\
is
ft, y)
a polar and
(o,
or origin,
is
u, y), while
ANALYTICAL CONICS.
672
4903
(X',
/Li,
The equation of the two points in which the line
v) cuts the conic is
<l>
(V, im\
y')
<f>
(X,
The coordinates
4904
v)
II,
=
of the
((^^V+<^,,x'+</,,v7-.
(4680)
asymptotes are found from the
equations
(l>
{\,
fjL,
=
v)
and
<^a
+ <^m+<^''= 0.
—
Proof. These are the conditions that the line
curve and also pass through the centre (4901).
The equation
4905
of the
two
(A,
/x,
j-)
should touch the
circular points at infinity is
a^(X-^)(x-v)+lj^(^-v)(^~x)+cHv-x)(v-ft)
—
Proof.
Put X'= ii'= v' in (4903) to
the conic take the in-circle (4889).
The general equation
4906
a^
make
=
the line at infinity, and for
of a circle is
(x-^)(x-v)+I)'^ {i,-v){i.-\)^e {v-\){v-ix)
= {lX-\-mix-^nvy
where l\-\-m^i-^nv
Proof.
put
0.
in the
=
is
(1),
the equation of the centre.
— The general equation of a conic
in trilinears
may, by (4601), be
form
+ K7-yo)(«-«o)+c(«-"o)(/5-/5o) = {la + mr^ + ny)\
+ m/3 + Ky = is the directrix, and ao/3oyo the focus. Now let the
a(/3-/3o)(y-yo)
where
la
focus be the in-centre of the trigon, and therefore a„
By this relation and aa 6^ + cy 2, the equation
=
+
a(S-a)(a-/S)(«-y) + &c.
(a— /3)(a— y)
or
cos"
|^'1
+ &C.
=
=
= /3o=yo= |SS~^
is
(709).
expressed as
(Z'a+w'/3 + 7t'y)-,
(ra + m'/i +«'y)''
Reciprocating by (4876), this becomes
(\—fx)(X — v)
ain-
A' -\-
&.C.
=
(Z\+
"'A'
+«»')',
the constant factor introduced on the right being involved in I, m, n; and
sin7i'
cos-j^, by (4875).
And we know that this is a cii'cle by (4845
Cor.), and that the directrix of the conic reciprocates into the centre of the
=
circle.
Otherwise.
—The
left
side of (1)
represents the two circular points at
infinity (4905), and, if for the right we take the equation of a point, the
whole represents a conic, as in (49(»9), of the form
In this case,
B'-.
AC =
A, C, the points of contact of tangents from B, being the circular points, the
conic must be a circle with i' =
for its centre.
TANGENTIAL COORDINATES.
673
Abridged Notation.
Let ^
= 0, B = 0,
D=
=
(Fig. 75) be the
G 0,
tangential equations of the four points of a quadrangle, where
A = a^X-\-hiiii-\-Ci^v, B = a.2\-\-biiii-\-C2v, and so on. Then
hBD.
the equation of the inscribed conic will be AG
4907
=
— The
equation is of the second degree in \, ju, v; therefore the
line (\, /J., J') touches a conic.
The coordinates of one line that touches this
That is, the line
0.
conic are determined by the equations A
0, B
joining the two points A, B touches the conic, and so of the rest.
Proof.
=
4908
If the points B,
AG =
becomes
contact of
4909
and
;
coincide (Fig. 76), the equation
are the points of
=
= 0.
ABG
(Fig. 78), and
k^B^ for its equation, let a tangent ef be drawn,
eB
h m.
The equations of the points f and
Referring the conic to the trigon
taking
AG =
Ae
let
/ will
D
and A = 0, G
tangents from the point B
JiB-
=
:
=
:
be
Proof.
eliminate
mA + hB =
— The
A
first
from
mhB + 6'
0,
= 0.
equation corresponds to (4879).
mA + kB =
and
AG —
For the equation of/,
Jc^B^-.
AB
whose equa4910 Let e, h (Fig. 77) be two points on
0.
The equation of the
tions are mA-\-hB
0, 7iiA-\-kB
point jj, in which tangents from e and h intersect, is
=
=
mmA-lr{m-\-m) kB-^C
Proof.
=
0.
— The equation may be put in the form
(mA + JcB)(mA + l-B)
= 0,
because k'^B'^=AG if the line touches the conic. The equation being of the first
degree in A, B, G, must represent some point. That is, the relation between
A, /.I, V involved in it makes the stivaight line \a-\-i.ip-\-vy pass through a
But the equation is satisfied when mA + lcB
0, a relation
certain point.
which makes the straight line pass through e. Hence a tangent through e
Similarly, by '^n'A + A-Jj
0, another
passes through a certain fixed point.
tangent passes through li and the same fixed point.
=
=
—
m =
m, then the equation of the point of
CoE.
Let
liB and ml'B-\-G
contact of the tangent joining the points ma
(4909) (e and/, Fig. 78) will be
4911
+
m'A-\-2mkB-\rC
4912
If
ill
=
().
Fig- (78) the trilinear coordinates of the points
4 R
ANALYTICAL OONtCS.
674
A, B, G are ii\, y^, z„ x.„ 7/2, ^2, ^h. y.s, ^3, the coordinates of
point of contact p of the tangent defined by m will be
'in\
+ 2mJcx.2 + ^3,
and the tangent at
ratios h
:
m
m^T/i
j^
and mh
:
+
2?ri/i;?/2
1,
mh^ + 2 » Jj^a +
+ Va,
divides the
two
tlie
z-^,
fixed tangents in the
by (4909).
=
(4G65) expresses the
When U is about
to break up into two factors, the minor axis of the conic diminishes (Fig. 79).
Every tangent that can now be drawn to the conic passes very nearly
through one end or other of the major axis. Ultimately, when the minor
axis vanishes, the condition of the line touching the conic becomes the conIn this
dition of its passing through one or other of two fixed points A, B.
case, Z7 consists of two factors, which, put equal to zero, are the equations
The conic has become a straight line, and this line is
of those points.
touched at every point by a single tangent.
4913
Note.— The
conditiou that Xa+fift
4914
If
TJ
and
equation
+ i^y
V
coordinates, hJJ-[-TJ'
f7
or
$
(A, ^, v)
shall touch a certain conic.
(Fig. 80) be
is
two conies
in tangential
then a conic having for a tangent
every tangent common to Z7 and TJ' and kU-\-AB is a conic
having in common with U the two pairs of tangents drawn
from the points A, B.
The conic U' in this case merges into the line AB, or,
more strictly, the two points A, B, as explained in (4913).
;
hU+AB
breaks np into two
hJJ^JJ' or
represents two points which are the opposite vertices of the quadrilateral formed by the four tangents.
4915
If either
factors,
it
ON THE INTERSECTION OF TWO CONICS.
INTRODUCTORY THEOREM.
Geometrical meaning' of
^{ — 1).*
4916
In a system of rectangular or oblique i)lanc coordinates, let the
operator \/— I prefixed to an ordinate ij denote the turning of the ordinate
about its foot as a centre through a right angle in a plane perpendicular to
The repetition of this operation will turn the ordinate
the plane of xy.
* [The fiction of imaginary lines and points is not ineradicable from Geometry. The
-\, and, as it appears
theory of Quaternions removes all imaginarincss from the symbol
that a partial application of that theory presents the subject of Projection in a much clearer
light, 1 have here introduced the notion of the multiplication of vectors at right angles to
each other.]
V
ON THE INTERSECTION OF TWO
675
GONICS.
through another right angle in the same plane so as to bring it again into
the plane o£ xy. The double opei'ation has c onv ert ed y into —y. Bu t the two
1
—y,
-s/
1 .y or (V —lYy
operations are indicated algebraically by v
which justifies the definition.
It may be remarked, in passing, that any operation which, being performed twice in succession upon a quan tity, changes its sign, offers a con-
—
sistent interpretation of the multiplier v/
.
—
=
— 1.
4917
With this additional operator, borrowed from the Theory of
Quaternions, equations of plane curves may be made to represent more
Con sider the equation a^ + y^= dr. For values
extended loci than formerly
For values of
of » < a, we have y
zk Va'—x^, and a circle is traced out
.
=
.
=
= —
—
±i \/x^ a^, where i v/ 1. The ordinate
y
\/x' — c^ is turned through a right angle by the vector i, and this part of the
locus is consequently an equilateral hyperbola having a common axis with
the circle and a common parameter, but having its plane at right angles to
Since the foot of each ordinate remains unaltered in posithat of the circle.
tion, we may, for convenience, leave the operation indicated by i unperformed
and draw the hyperbola in the oi-iginal plane. In such a case, the circle may
be called the principal, and the hyperbola the supplementary, curve, after
When the coordinate axes are rectangular, the supplementary
Poncelet.
curve is not altered in any other respect than in that of position by the
transformation of all its ordinates through a right angle but, if the coordinate axes are obliqiie, there is likewise a change of figure precisely the same
as that which would be produced by setting each ordinate at right angles to
its abscissa in the xy plane.
In the diagrams, the supplementary curve will be shown by a dotted line,
and the unperfoi'med operation indicated by i must always be borne in mind.
For, on account of it, there can be no geometrical relations between the
principal and supplementary curves excepting those which arise from the
This law is in agreement
possession of one common axis of coordinates.
with the algebraic one which applies to the real and imaginary parts of the
a in both curves.
al When y vanishes, x
equation x^ — (iyy
d-bd-b- or the hyperbola b-x'' — a-y'
If either the ellipse b^x^ + a-y'^
be taken for the principal curve, the other will be the supplementaiy curve.
It is evident that, by taking diflTerent conjugate diameters for coordinate
axes, the same conic will have corresponding diflFerent supplementary curves.
The phrase, "supplementary conic on the diameter DD," for example, will
refer to that diameter which forms the common axis of the principal and
supplementary conic in question.
x>a, we may
write
;
=
=
4918
=
=
=
=
h.
d^ and the right line x
circle x~ + ythe line intersects the su pplementary right hyperbola in two
By increasing b without limit, we
o.^.
points whose ordinates are ±^' v &'
These lie on the
get a pair of, so-called, imagiyiary points at infinity.
asymptotes of the hyperbola, and the equation of the asymptotes is
When
Let us now take the
& is
>
a,
—
(^x
+ iy)(x-iy) =
0.
We can now give
The two
lines
m
_
_
a geometrical interpretation to the statements
drawn from the focus of the conic
b'-x'
+ a^y- =
d'b''
(4/'20).
to the
" circular points at infinity " make angles of 45° with the, major axis,
they touch the conic in its supplementary hyperbola b'^x^—d' {iy)An independent proof of this is as follows.
=
and
d'lt\
—
;
ANALYTICAL GONICS.
Q1&
Draw
let
a tangent from S (Fig. 81) to the supplementary liypei'bola, and
Tiicii
y be the coordinates of the point of contact P.
a;,
^
= ^,
(115^0)
by the value
Therefore y
= _-f^;
of x.
and
>j
s/(^^-a^)
-
^'
Also
SN = x-CS=
,^'f
=
SP makes
SN,
=A
therefore
,..
- y^F^'
an angle of
45'^
with CN.
The following results are required in the theory of projecand are illustrated in figures (82) to (86). Two ellipses
are taken in each case for principal curves, and the supplementary hyperbolas are shown by dotted lines. As the planes
tion,
of the principal and supplementarj^ curves are really at right
angles, the intersections of the solid lines with the dotted are
only apparent.
The intersections of the solid lines are real
points, while the intersections of the dotted lines represent
the imaginary points.
Two
4919
in
(i.)
conies
in ttvo real
(ii.)
(iii.)
may
intersect
four real imints (Fig. 82);
and two imaginary
])oints (Fig. 83)
in four imaginary i^oints (Fig. 84).
[When
the two hyperbolas in figures (83) and (84) are similar and
two of their points of intersection recede to infinity (Figs.
85 and 8G). Hence, and by taking the dotted lines for principal, and the
solid for supplementary, curves, we also have the cases]
similarly situated,
(iv.)
infinity
in
ttvo
real finite 2^0 ints
and two imaginary points at
;
(v.) in tivo imaginary finite points
points at infinity ;
(vi.)
infinity
in two
and two imaginary
imaginary finite points and two real points at
;
(vii.)
in
ttvo
real finite points
and
two real points at
infinity.
4920
Given two conies not intersecting, or intersecting in
but two points, to draw the two supplementary curves which
have a common chord of intersection conjugate to the
—
TEE METHOD OF PBOJEGTION.
677
diameters upon whicli tliey are described, or in other words,
to find tlie imaginary common chord of the conies.
p.
Poncelet has shewn by geometrical reasoning (Proprietes des Projectlves,
The following is a method of deter31) that such a chord must exist.
mining
its
position
Let
(abcfgh'^xi/iy
=
and
(a'b'c'fg'li''^xyiy
=
(i.)
C
(Fig. 89), the coordinate axes being
be the equations of the conies G,
Then the
Suppose PQ to be the common chord sought.
rectangular.
diameters AB, A'B' conjugate to PQ bisect it in B, and the supplementary
curves on those diameters intersect in the points P, Q. Now, let the coordinate axes be turned through an angle B, so that the y axis may become
This is
parallel to PQ, and therefore also to the tangents at A,B, A', B'
accomplished by substituting for x and y, in equations (i.), the values
.
XG0s9 — y &\nd
and
7/
cos
+
a;
sin 0.
and
Let the transformed equations be denoted by {ABGFGH'^xyiy =
{A'B'G'F'G'H''^xyiy = 0, in which the coefficients are all functions of 0,
quadequations
as
a
Solving each of these
excepting c, which is unaltered.
ratic in y, the solutions take the forms
y
=
a,-(;
+ /3±\//*
{x^
— 2]px + q),
y
=
a'x
+ fV ±
\^fJ.'
(x-
— 2p'x + q) ...(n.),
p, q given in (4449-53), if for small letters we
and the
Thus, a, /3, fi, p, q are obtained in terms of
substitute capitals.
original coefficients a, h, h,f, g, Ji.
aE l^
DN, we have //
ON, n
being 4
Now, the coordinates of
with the values of
a,
13, /j.,
D
and
jy
= a'^ +
/3',
therefore
a^ +
ft
=
= a^ +
=
=
+
(iii-)-
fi'
represents the ordinate of the conic conjugate
and
For values of x in the diagram >
<0B, the factor \/—l appears in this surd, indicating an ordinate of the
Hence, equating the values of the
or A'B'.
supplementary curve on
common ordinate PD, we have
The surd
in equations
to the diameter
AB
(ii.)
OM
or A'B'.
AB
I,
(e-2p-Uq)
=
1^'
(e-2p'Uq)
(iv.).
Eliminating between equations (iii.) and (iv.), we obtain an equation for
determining 0; which angle being found, we can at once draw the diameters
t,
AB,
A'B'.
THE METHOD OF PROJECTION.
—
Problem. Given any conic and a right hne in^ its
plane and any plane of projection, to find a vertex of projection such that the line may pass to infinity while the conic is
projected into a hyperbola or ellipse according as the right
and at the
line does or does not intersect the given conic
same time to give any assigned proportion and direction to
the axes of the projected conic.
4921
;
—
ANALYTICAL CONIGS.
678
—
HCKD
BB
the right line, in
be the given conic, and
Let
Analysis.
Draw
Fig. (87) not intersecting, and in Fig. (88) intersecting the conic.
to be the
Suppose
the diameter of the conic conjugate to BB.
parallel to OBB,
Draw any plane
required vertex of projection.
in F, and draw the
intersecting the given conic in CB and the line
in
and
cutting the former plane in E, F, G and the line
plane
on the plane parallel
be the conical projection of
let the curve
UK
EGGD
UK
BB
OEK
to
A
;
HCKD
EGGD
OBB.
By
similar triangles,
FF^
^FG_^aA
_ OA
HF~HA^^ FK
Let
a, ft
.
AK'
EF.FG ^
"EF.FK
be the semi-diameters of the given conic
^^''
EF.FK ~
^"^
+1,
EF.FG~
^^''
(^
•-
a''
OA^
.^.
HA.AK
pai'allel to
P^'-SA.A
UK
K
^
and
^'
GB
;
.^.
^^"
a'.OA'
Now, since parallel sections of the cone are similar, if the plane of ECKD
moves parallel to itself, the ratio on the right remains constant therefore, by
;
Fig. (88).
parallel to
EGGK
an ellipse in Fig. (87) and an hyperbola in
Let a, h be the semi-diameters of this ellipse or hyperbola
and GD, that is, to OA and BB then, by (2),
(1193), the section
is
EG
;
^ = £ Ej^,
a^
ct
OA-
...
OA'
= $^HA..AK
(3).
b-a-
^EA.AK=
AB-, where AB in Fig. (88) is the ordinate at A of the
a
given conic, but in Fig. (87) the ordinate of the conic supplementary to the
given one on the diameter conjugate to BB. Therefore
But
AO'=^AB'
(4).
Hence AO, AB are parallel and propoi'tional to a and b. And, since AB
is given in magnitude and direction, we have two constants at our disposal,
namely, the ratio of the semi-conjugate diameters a and b and the angle
between them, or, which is the same thing, the eccentricity and the direction
of the axes of the ellipse or hyperbola on the plane of projection.
4922
The construction
will be as follows
:
Determine the point A as the intersection of BB ivith the
conjugate to it.
Choose any j^Iane of inojection,
diameter
and in a plane through BB, parallel to it, measure AO of the
length given by equation (3) or (4), maJiing the angle BAO
will he the
equal to the required angle hetioeen a and b.
vertex of projection^ and any plane IJM.'^ parallel to OBB will
serve for the plane of pirojection.
HK
4923
=
AO AB, the projected curve in Fig. (88)
be a right hyperbola.
OoR. T.—'If
will in every case
TBI]
—
METHOD OF PROJECTION.
679
BAO
is a right angle, tlie axes of tlie proIf
CoE. 2.
jected ellipse or hyperbola are parallel and proportional to
AO and AB. Hence, in this case, the eccentricity of the
4924
hyperbola will be
e
—
= OB
:
OA.
=
AO = AB
and BAO
a right angle,
the ellipse becomes a circle and the right hyperbola in Cor. 1
has its axes parallel to AO and AB.
4925
Cor.
3.
If
4926 To project a conic so
may become the centre of the
that a given point in
projected curve.
plane
its
BB
the polar of the given jjoiiit, and conTahe for the line
its
For, ifV be the given point, and
struct as in (4922).
polar {Fig. 87 or 88), p the projection ofP ivill have its polar
at infinity, and ivill therefore be the centre of the projected
ellipse or hyperbola, according as P is within or luithout the
BB
original conic.
To project two intersecting conies into two similar
similarly situated hyperbolas of given eccentricity.
4927
and
BB
{Fig.
Take the common chord of the conies for the line
88), and project each conic as in (4922), employing the same
and
Then, since the point
vertex and plane of projection.
are the same for each projection, corresand
the lines
ponding conjugate diameters of the hyperbolas are parallel and
A
AO
AB
proportional
to
AO
and
AB
;
therefore,
Sfc.
To project two non-intersecting conics into similar
and similarly situated ellipses of given eccentricity.
4928
Tahe the common chord of a certain two of the supplementary curves of the conics (4920), in other words, the
imaginary common chord of the conics, for tlie line BB, and
proceed as in (4927).
To project two conics having a common chord of
contact into two concentric, similar and similarly situated
hyperbolas.
Tahe the common chord for the line BB, and construct as in
4929
The common pole of the conics projects into a common
(4922).
centre and the common tangents into common asymptotes.
TBILINEAB ANALYTICAL CONICS.
680
To
4930
project any two conies into concentric conies.
common pole and polar of the given conies by
(4762), and take the common polar for the line BB in tlte
construction of (4922).
The common pole projects into a
Find
common
the
centre.
—
Ex. 1.
Given two conies having double contact with each other,
*xuOX.
any chord of one which touches the other is cut harmonically at the point of
contact and where it meets the common choi'd of contact of the conies.
\_Salmons Conic Sections, Art. 354.
AB
PQ
be the common chord of contact,
the other chord touching
the inner conic at G and meeting AB produced in D.
By (4929), project
AB, and therefore the point D, to infinity. The conies become similar and
similarly situated hyperbolas, and C becomes the middle point of
(1189).
The theorem is therefore true in this case. Hence, by a convei'se projection,
the more general theorem is inferred.
Let
PQ
—
49oA
Ex. 2.
Given four points on a conic, the locus of the pole of any
fixed line is a conic passing tlirough the fourth harmonic to the point in
which this line meets each side of the given quadrilateral.
[Ibid., Art. 354.
Let the fixed line meet a side AB of the quadrilateral in D, and let
be in harmonic ratio. Project the fixed line, and therefore the point
D, to infinity.
C becomes the middle point of ^ J5 (1055), and the pole of
the fixed line becomes the centre of the projected conic.
Now, it is known
that the locus of the centre is a conic passing through the middle points of
the sides of the quadrilateral.
Hence, projecting back again, the more
genei'al theorem is inferred.
AGBD
—
4i70o
Ex. 3.
If a variable ellipse be described touching two given
while the supplementary hyperbolas of all three have a common
chord AB conjugate to the diameters upon which they are described the
locus of the pole of AB with respect to the variable ellipse is an hyperbola
whos^e sup})lementary ellipse touches the four lines CA, CB, C'A, C'B, where
are the poles of AB with respect to the fixed ellipses.
C,
(Salmon, Art. 355.)
ellipses,
;
C
Pkoof.
—Project AB to
infinity and the three ellipses into circles.
The
the centres 2', c, c of the circles.
The locus of p is a
hyperbola whose foci are c, c
But the lines Ac, Be now touch the supplementary ellipse of this hyperbola (4918). Therefore, projecting back again,
we get AC,
touching the supplementary ellipse of the conic which is the
locus of P.
Similarly, AC,
touch the same ellipse.
poles P, C,
C become
.
BC
BC
Any two lines at riglit angles project into lines wliicli
cut liannonically tlie line joining the two fixed points which
are the projections of the circnlar points at infinity.
4934
Proof.— This
follows from (4723).
—
INVARIANTS AND COVAEIANTS.
081
The couverse of tlie above proposition (4931), wbicli is the theorem
356 of Salmon, is not nniversallj true in any real sense. If the lines
drawn through a given point to the two circular points at infinity form a
harmonic pencil with two other lines through that point, the latter two are
not necessarily at right angles, as the theorem assumes.
The following example from the same article is an illustration of this
Ex.— Any chord BB (Fig. 88) of a conic HCKD is cut harmonically by
through P, the pole of the chord, and the tangent at K.
any line
The ellipse BKB here projects into a right hyperbola B, B project to
The harmonic pencil formed by PK and the tangent at K, KB and
infinity.
KB projects into a harmonic pencil formed by 2^^ fii^d the tangent at h, kh
and l-b, where 6, h are the circular points at infinity but j>fc is notat right
angles to the tangent at J: of the right hyperbola. The harmonic ratio of the
latter pencil can, however, be independently demonstrated, and that of the
(Note that h is G in figure 88.)
former can then be inferred.
If we may suppose the ellipse to project into an imaginary circle havmg
points at infinity, the imaginary radius of that circle may be supposed to bo
The right hyperbola, however, is
at right angles to the imaginary tangent.
the real projection which takes the place of the circle in this and all similar
and it is only in the case of principal axes that the radius is at
instances
4935
in Art.
PKAH
;
:
_
;
I'ight
angles to the tangent.
INVARIANTS AND COVARIANTS.
4936
Let u ={al>cfr/]tjxyzy,
u'
=
{a'h'c'fg'h'Xxyzf
notation of (1620).
represents two
n'
for which ku
The three values of
right lines, are the roots of the cubic equation
be two conies as in (4401)
witli the
+ =
/.-,
AA:-^+0A--+0'A'+A'
4937
4938
where
=
0,
A = uhc-\-'lfgh—af- — hg--ch\
= Aa'^Bh'-^Cc'^2Ff'-^2Gg-^'lHh',
4939
A = bc-f\
and
For the values
of A'
F=gh-af,
&c.
(46(;5)
and 9' interchange a with a,
h
with h\
&c.
—
Proof. The discriminant of ku + u, which
must vanish (4661), is evidently the determinant here written, and it is equivalent to the
ka+a',
kh + h',
kg +
kh +
kh+h\
kf+f
+ g', ¥+f'^
^"'+^'
•
h',
g'
cubic in question.
4940
A, e,
e',
and a' are invariants
4 s
of the conic
ku-\-i('.
,
ANALYTICAL CONICS.
682
That is, if the axes of coordinates be transformed in any
manner, the ratios of the four coefficients in (4937) are
unaltered.
—
Proof. The transfoi-mation is effected by a linear substitution, as in
Then hu + u' becomes Icv + v', and
(1704). Let «, «' thus become v, v'.
represents two right lines, it will
k is unaltered. If the equation ku + u'
continue to do so after transformation but the condition for this is the
vanishing of the cubic in h aud k being constant, the ratios of the coefficients must be unalterable.
=
;
;
4941 The equation of the six lines which join the four
points of intersection of the conies u and u is
Proof.— Eliminate k from (4937) by
4942
The condition that the
A-h
+ «' =
conies
0.
and
ir
may touch
vf
is
(&&-9SAy = 4 (©•^-3A0')(0'--3A'0),
4943
or
4A0 3+4A'0^+27A2A'^-18AA'00-0-0^
— Two of the four points
(3941) must coincide. Hence two out
The cubic (4937) must therefore
have two equal roots. Let a, a, ft be the roots then the condition is the
result of eliminating a and ft from the equations
PuoOF.
must
of the three pairs of lines
in
coincide.
;
A
('2a +/-5)
=
A
-e,
(o-
+ 2a/3) =
G',
Aa'/S
= -A'
(400).
The expression (4943) is the last term of the equation
whose roots are the squares of the differences of the roots of
and when it is positive, the cubic in h has two
the cubic in
imaginary roots when it is negative, three real roots aud
when it vanishes, two equal roots.
4944
/.-,
;
;
=
Proof.—By (543) or (579). The last term of /(a;)
27F(a) F(ft), a, ft being the roots of 3Ax' + 2ex + e'
in
(543)
is
now
= 0. When this
therefore F (x) has
term is positive, f(x) has a real negative root (409), and
then two imaginary roots for, if (a &)'•'
—c, a &
ic, and a and b are
both imaginai'y.
When the last term of /(;c) is negative, all the roots of
—
;
/ (x)
— =
=
are positive, aud therefore the roots of
F (.^)
are all real.
INVARIANTS OF PARTICULAR CONICS.
4945
A=
^y\wn
afjc,
u
=
=
aiv^-\-
by-
bc-^-ca
+ cz^
+
ab,
and
0'
=
n
=
a^
-{- 1/- -\-
a-{-b-{-c,
:r
A '=
1.
,
INVARIANTS AND GOVABIANTS.
4946
u=
When
{abr.fghXxijzy and u'=x'+i/^ + z\
e = A + B-\-C,
=
0'
A'=l.
rt+6+e,
= x^ + if - r^ and n = {x - a)- +
A = -r\ A' = -^s\
4947
When
4948
The cubic
u
u
(//
-(^f- s\
for k reduces to
(k-^l) {s'k'+{r'+s'-a'-fi') k-{-r'}
4949
683
=
0.
When
= h\e' + aY-a%'' and v' = (->'-«)' + (^-/3)2-r%
A = - a'b\
= (rb' a' +/3- - a' - W - r^
0' = u'^'-\-h'a^-a'¥-r' («'+6^),
A' = -rl
{
4950
When
A = —^m\
4951
A,
}
= if-4^mx and u' = {x-a)-^{y-^Y-r\
= —4m (a + m), 0' = P^—4ima—7'\ A'= —
u
r'.
When
?(,= {(^^(^fg^K^^^y^y
and
?f'
= ^^ +
2rt37/
cos w
+ ?/,
0=c(..+6)-f-g-+2(/g— c/0cos6>,
A'=0,
0'
=
(•
sin^co.
Hence the following are invariants of the general conic,
the inclination of the coordinate axes being w.
abc-\-2f{j:h-af'-bcr^-ch'
4952
c
4953
c
shr
(a-\-b)-f--^'-\-2 ifg-ch) cos 6)
c siu"
4954
^ A
"^^
snroi
(3).
For these are what
(1)
^
.2),
0'
0)
"+b--^l"^-o^^
and
snvQ)
and
p.
0'
0)
(2)
transformed so as to remove /and
(4).
become when the axes are
g.
ANALYTICAL CONIC S.
684
If the origin be unaltered, c is invariable, and transformation of the axes will then leave invariable
^^ffA-qf-y-
4956
as appears
4958
=
.^j^j
.
f+ff--^fffcos6)
^
siu"
crlr.
CO
siu"-
by subtracting
(3)
from
(1)
and
(t)
(2)
from
(4).
Ex. (i.)— To find the evolute of the conic h'X'-{-ah/
See also (4547).
—
Denote tlie conic by n, and by u' the hyperbola c"xy -\-lh/o: — ci?x'y
which intersects « in the feet of the normals drawn from x'y'. Two of
these normals mnst always coincide if x'y' is to be on the evolute.
and n'
must therefore touch. We have
Pkoof.
(4335),
ti,
A
= -a'h\
e
=
0,
e'
=
-ft-i"
(a-x''
4959
Ex. (ii.)
obtained from
+ bY-cy-h27a'h'o'xhf =
— Similarly
A'=
(aV + ty-c*),
Substitute in (4942), and the equation of the evolute
the
is
-2a^h-c\vy.
found to be
0.
evolute of the parabola
is
u = y- — 4<mx, u' =
+ 2 {2m— x") y — imy',
= —4<m\ e = 0, e' = -4 (2m— x), A' z= 4my,
the equation 27ruy- = 4 (x — 2my.
See also (4540).
2a;?/
A
producing
4960
Ex.
(iii.)
— The locus of the centre of a circle of radius
E, touching the conic h\'-\-(rif—a-h\
the conic.
+ R'
is
called a
par all d
Ii;V-27^V {r (a'Hi'O + id'-2K-) x' + {2a'-V-)
f]
+ 4a-6- + h') - 2c- (a" - n^lr + 3&*) x- + 2c- {^a'-a-V + V) i/
+ (a* - C)^'^ + G60 + {Cm' - Ga'b' + h') y' + {(Sa' - lOcr't^ + (j?>0
2ar}rc' (rr + ^0 + 2c- (3rt'- a-h' +
x- - 2r {a'-aV + 3h')
{
-{(ja'-10aV- + Gb'){b\c' + dY) + {W^-6a*b--6a;'b' + 4b'') xSf
+ 2 {a--2b') 6V + 2 (&--2a') aV-2 {a'-a'b-^Zb') xSf
-2(3a*-a-/r + 6^).^y|
[ c' (a"
;o'
+ E-
to
Its equation is
Z/')
+ (/rV + aV--a'Z>-j-{(-«-c)' + r}
PROor.— Tf
the curve
and A'
in
{(:i!
+ c)- + r} =
xhf }
i/'
0.
the curves in (4940) be made to touch, o/5 will be a point on
u at a distance r. Therefore put the values of A, O, 9',
equation (4042).
Itiahnon, p. 325.
pai-allel to
—
INVARIANTS AND COVABIANTS.
When
4961
u
685
(4936) represents two right lines. A'
of
vanishes, and
=
0'
4962
two
the condition that the
is
intersect on u
should
lines
;
=
9
is the condition that the two lines should be
conjugate with regard to u.
4963
Proof.
— Transform
=
e
= 2(/^-c70,
origin xy
makes « pass through the
For
is
= 0,
A'
(4937),
c
=
=
into 2xy
n'
0, so that the axes x, y are the
We now have, by
This will not affect the invariants (4940).
right lines.
in (4671),
given by
=
fg-ch
Xx + fxy +
if
H
B
:
But
F.
:
= 0,
=a=
becomes y
v
a;
e'
fg
;
= -c.
= ch makes
x and y conjugate.
then \
v
0, and the pole
at the pole, therefore II =.
= =
0.
The condition that
4964
touch u
is,
either of the lines in u' should
by (4943),
0^
= 4A0'
O and
with the above values of
AB =
or
0,
9'.
equation of the two tangents to u, when Xa3 +^t^
the chord of contact, is, with the notation of (4665),
4965 The
is
u^ (X,
fjL,
v)
=
{\a;-{-iii/-{-vzf
—
A.
Proof. The conic of double contact with u, ku + (\x + fiy
and Q'
must now become two right lines. In (4937) A'
A-A
+6 =
=
<& (\,
i-i,
v).
Hence eliminate
Cor.— Taking the Hne at infinity
the equation of the asymptotes (4685).
4966
The invariant 9
4967
(i-)
of the conic
Whenever an
l-u
+ u'
<ix
+ i'y-
(4G99),
therefore
= 0,
=
But
0.
+v
Jc.
+
bjj
+ C^,
we obtain
vanishes
inscribed triangle of u
is
self-con-
jugate to u.
4968
(ii-)
conjugate to
li
II
in (i.)
and
e
is self-
in
(4937)
u
(ii.)
Proof.— (i.) u becomes
fore
a circumscribed triangle of u
.
0' vanishes under similar conditions, transposing
4969
and
Whenever
vanishes
f'yz+g'zx+h'xy (4724).
axif
+ hy- +
cz"-
(47G5),
a'=l'=c
=
();
and/= g =
i.e.,
if
u'
li
is
— 0.
There-
of the form
ANALYTICAL CONIGS.
686
(ii.) In this case, /'= 9'=
the line x
touches u, &c.
h'=
and
vanishes
hr
if
=
4970
If
u'
?')
be two conies, and
if
6'^
= /",
= 4A9',
&c.,
i.e.,
if
any triangle
inscribed in u' will circumscribe w, and conversely.
—
=
A=-4,
Q
Proof, Let u
x^ + 1/ + z^—2yz — 2zx—2icy and u
both referred to the same triangle, (4739) and (4724).
therefore 6^
= 4.(f+g +
= 4A6',
]i),
= 2fyz + 2gzx + 2hzi/,
A'=2fg1i',
a relation independent of the axes of reference (4940).
—
4971
Then
Q'^-(f+g + hy,
Ex. (i.)
The locus of the centre of a circle of I'adius r, circuma'b-, is
scribing a triangle which circumscribes the conic h-x-+a-i/
(x""
from 9-
=
=
+ - a- - + r-)2 + 4 [ h'x- + ahf - a-lr - r
2/-
Z>-
4A9' and the values
(ft-
+ b')} =
0,
in (4949).
—
4972
The distance between the centres of the inscribed and
Ex. (ii.)
circumscribed circles of a triangle is thus found, by employing the values of
^/(/-±2jt'), as in (936).
e, e', and A in (4947), to be
D=
4973
The
tion of the
tangential equation of the four points of intersec-
= 0,
two conies u
u
=
is
with the meanings
U={ABCFGHJ\iivy;
4974
(4GG4)
U'= {A'B'C'F'G'H'JXiivf.
V = {A"B"C"F"G"H"J\iivY.
4976
A = bc-f\
4977
&c.;
= h'c'-f\
A'
&G.,
as in (4665), and
4978
4979
4980
A" = bc'+b'c-2fr,
B"^ca'+c'a-2si^\
C"
= ab'-\-a'b-2hh',
—
F" = gh'-\-<r'h-af-a%
G"^hf^h'f-hi>:' -h'^,
H"=f<r'-\-fo'-cf/-c'h.
Proof. The tangential equation is the condition that Xa + fip + yy may
pass through one of the four points of intersection of ?t and «'.
The tangential ecjuation of the conic u + Jiit' is obtained by putting a + ha' for a, &c.
in U (4GG5), and is U+kV + JrU' =: 0.
The tangential equation of the
envelope of the system is V"
4UU' (4911). This is the condition that the
/i,
line (\,
v) may pass through the consecutive intersections of the conies
obtained by varying A-.
Put these conies always intersect in the same four
points.
The above is thcrefoi'e the tangential equation of the four points.
=
—
INVARIANTS AND COVABIANTS.
The equation
4981
conies a, u
of the four
commou
681
tangents of two
is
F = (a"h"c"f"^"h" X ^^y)'->
where
and
a''
= BC'-\-B'C-2FF\
&c.,
f" = GH-\-G'H-AF'-A'F,
&c.,
as in (4978-81).
—
Proof. This is the reciprocal of the last theorem.
ZJ+^ZJ' is a conic
touching the four common tangents of the conies U and TJ'. The trilinear
equation formed from this will, by (4007), be u^-^-hT + k'u'iX'
0.
The
envelope of this system of conies is the equation above, which must therefore
=
represent the four common tangents.
The curve F passes through the points of contact of w and
locus represented by (4981).
Hence the eight points
4982
with their
common
tangents
lie
of contact of the
lo
with the
two conies
on the curve F.
The reciprocal theorem from equation (4973) is
The eight tangents at the intersections of the conies envelope
the conic V.
4983
F=
4984
to the
the locus of a point from which the tangents
u, u' form a harmonic pencil.
is
two given conies
— Putting
=
we
get a quadratic of the form
in which the line y is
cut by tangents from o', /3', y'. Let the similar quadratic for the second conic
be a'cr + 2h'al3 + h'(30.
Then, by (1064), ab' + a'b
2hh' is the condition
that the four points may be in harmonic relation.
This equation will be
found to pi'oduce F
0.
Proof.
ad'
+ 2]iu^+bl3^' =
0,
7
in
(4081),
which determines the two points
=
=
=
4985
on
The
a', )3',
actual values of a, h,
h,
suppressing the accents
y\ are
CP'-\-Bf-2Ffiy,
G^y-^Fya-Ca^-Hy\
Af-\-Cd'-2Gya',
and similarly for a,
4986
h', h',
with A' w^ritten for A, &c.
If the anJictrmonic ratio of the pencil of four tangents
Jcuu'.
If the
be given, the locus of the vertex will be F"
given ratio be infinity or zero, the locus becomes the four
common tangents in (4981).
=
ANALYTICAL CONICS.
V ==
4987
is tlic envelope of a conic evciy tangent of
cut harmonically by the two conies u, u' ; i.e., the
equation is the condition that Xa+^/S vy should be cut harmonically by the two conies.
which
is
+
—
Proof. Eliminate y between the line (\, /u, v), and the conies n and m'
separately, and let AiC
stand
and A'a' 2ircily + ]l'ft2Hal^j + Bjy'
21111' produces
for the resulting equations.
Then, by (10G4), AB' -\-A'1j
the equation
0, which, by (4GGG), is the envelope of a conic.
=
+
+
=
=
V=
The actual values
4988
and similarly for A\ H',
4989
F'^
= 4AA'?n/
is
B',
B
H,
of A,
with
are respectively
for a, &c.
a'
a covariant (1629) of the conies u,
u'.
For the four common tangents are independent of the axes of reference.
C7=0
4990
V=
and
are both contravariants
(4973)
(1814) of u and n\
—
For ?7= is the condition that Xo + /j/3 + ry = shall touch the
and V =
is the condition that the same lino shall be cut harmonically by u and u'; and if all the equations be transformed by a reciprocal substitution (1813, '14), the right line and the conditions I'emain
Proof.
conic u
;
unaltered.
Any conic covariant with u and ii' can be expressed in
terms of u, u, and F and the tangential equation can be
expressed in terms of U, U' and V.
4991
;
4992
Ex.
(1).
— The polar reciprocal
=
Qn
is
Proof.
— Referring
n
F=
The
polar of
this
may
thing,
4993
^,
r],
u, n' to their
=
ax-
common
+ bij- + cz\
a(h + c)
x'
+h
^ with respect to «'
(c
self -conjugate triangle,
=
+ y- + z-,
+ c (» + h) z-.
^x + riy +
and the
It'
+ a)
is
of u with respect to u'
F.
.c'^
if
i^z,
condition that
is the same
+ caTi^ + abi,'^ = (4664), or, which
(bc + ca + ab)(x^ + y^+z^) = P or Gw' = P (4945).
touch u
Ex.
be written
is
(2).
hc£i'^
—The
enveloping conic
0//'
+ 0'// =
F.
V in
(4987)
may
also
INVARIANTS AND COVAlilANTS.
Proof.
(b
— With
+ c) X'+(c + a)
the same assumptions as in Ex. (1),
+ h)
fM-+ (a
r-
=
The
0.
V
689
in (-i'.To)
triliuear equation
is,
becomes
therefore,
by (4667),
(c
+ a) (a + b) X- + {a+b){b + c) if + (i + c) (c + a) £' = U,
+ ca + ab) {x- + if + z') {a b + c) (ax' + bif + cz") = P.
or
(be
4994
Ex.
hnes
-\-
— The condition that F may become two right
(3).
A A' (00' -A A') =
is
A'= B'= C'=
1
to Ex. (1),
be,
therefore, in (4981), a"
;
the discriminant
abc
(a
+ & + c)(&c +
To reduce
4995
By
|
0.
B = ca, C = ab, F= G = H=0,
= B + G = a(b+c), &c. Heuce
A of F = ahc (b c)(c + a)(a-\-h),
A=
Proof.— Referring
or
-{-
(4945),
ca
-{-
+ a?;) — ate =
|
the above, by (4945).
the two conies u, u to the forms
y will be the roots of the cubic
o, /3,
AF-eF-l-e7.--A'
=
(1),
be found in terms of x, v' and F, by solving
//,
the three equations
+ //- + ::'^= n, a,r-f-/3y^ + yr == u and (by
and
?:^
cif,
will
,1'-
4994),
a(i3
4996
to
Ex. (1)
+ y).r+/3(7 + a)/ + y(«+|3)^:-^=:F
:
Given
;t;-
+r+
-2(/
+ 2i^ + 3 =
To compute
be reduced as above.
0;
a;-
(2).
+ 2^H4;/ + 2a; + 6 =
;
the invariants, wef take
a
=
=
and
1
1
therefore
and
=
=
The roots of equa6
6, 6'= 11, A'= 6.
T.
Therefore (2) becomes 5X- + 8r^ + 9ZComputing P also by (4981) with the above values of A, B, &c., we get the
symmetry,
of
three equations as under, introducing z for the sake
Therefore (4938,
tion (1)
are
'9)
now
1,
A
2,
1,
=
3.
X-+ Y-+ Z-= a;-+ 7/+ 32-+ 2//z+ 22a;,
X- + 2r- + 3;;-= xr + ^f+ 6.-'-+ 4(/2+ 2zx,
5X- + 8YH9i^- = 5a!2 + 82/- + 22r + 16!/2 + 10za;,
The solution gives X=x + \, Y = -\-l, Z = I, and the equations in the
+ l)'' + 2 (^ + l)- + 3 = 0.
forms required are (x + l)- + (y + iy + l = 0,
\j
(a;
—To
4997
Ex. (2).
in a conic
u so that two
find the envelope of the base of a triangle inscribed
of its sides touch
4 T
a.
ANALYTICAL CONICS.
690
Let
« = X- + If -\- z' — 2yz — 2zx — 2xy — 2hkxy,
«' =
and
2fy3 + 2gzx + 2hxy,
X and y being the sides touched by u.
Then u + hi' will be a conic touched
by the third side z. By finding the invariants, it appears that 9^^ — 4Ae'
= 4AA'A;, whence k is determined, and the envelope becomes
Compare (4970).
The tangential equation
4998
of the
two
circular points at
infinity (4717) is
Proof.
— This
the condition that Xx+fuy + v should pass through either
x:^iy
c is the general form of such a line.
is
=
of those points, since
U=
4999
being the tangential equation of a conic, the
discriminant of h U-{-
TJ'
is
—
Proof. The discriminant of hJI-^- TJ' is identical in form with (4937),
but the capitals and small letters must be interchanged. Let then the discriminant be AA;HeA;H©'A; + A'
0.
We have
A
= A^
(4G70),
Similarly 6'
5000
= A'G,
=
= (BG-F') A' + &c. = A'ai\ + &c.
A' = A'\
e
If ©, 0' be the invariants of
of circular points X^
a parabola, and 9'
+
=
^t'^
(4998)
makes
;
it
any conic
C &c. =
G=G =
=
=
is
k'^A'
(a + b)
5001
The
U and
the pair
+ k (a + h) A + ab-h\
A
e'
{A'B'-m)
ab-Jv'; and A'
lows from the conditions (4471) and (4474).
;
= A0'.
then G
makes the conic
an equilateral hyperbola.
Proof.— The discriminant of kU+X^ + in^
For, as above, A = A^
Q = A'aA + B'bA =
;
(4668)
since
= 0.
A'
= B' =
The
tangential equation of the circular points
I,
rest fol-
is,
in
trilinear notation (see the note at 5030),
X-+/u,-4-»'"— 2/w,v cos
Proof
+ yir
= 0,
A—2v\ cosB—2\fjL
cos C.
shows that the perpendicular lot fall
from any point whatever upon any line passing through one of the points is
infinite.
Therefore, by (4624).
:
A.^'
in Cartesians,
5002 The conditions in (4689) and (4090), which make the
gcmoi-al conic a parabola or equilateral hyperbola, may be
obtained by forming 9 and 9' for the conic and equation
(5001) and applying (5000).
INVARIANTS AND GOVABIANTS.
5003
If
0'"'
= 49,
691
the conic passes througli one
the
of
circular points.
5004 When It in (4984) reduces to \^-\-ix\ that is, to the
becomes the locus of interseccircular points at infinity,
tion of tangents to n at right angles, and produces the equa-
F
tions of the director-circle (4693)
5005
The tangential equation
5006
And
two
factors,
if
it
—_Since
and (4694).
of a conic confocal with TJ is
the left side, by varying k, be resolved into
becomes the equation of the foci of the system.
+
/x^ represents the two circular points at infinity (4998),
(4914), is the tangential equation of a conic touched by
But these tangents
the four imaginary tangents of Z7 from those points.
and, for the same reason, in
intersect in two pairs in the foci of U (4720)
the foci of ]cU+y-+iu-, which must therefore have the same foci.
If
+ X^+fi^ consists of two factors, it represents two points which, by
(4913), are the intersections of the pairs of tangents just named, and are
Proof.
;^f/'_^\2-(.^2
\^
Q^ jjy
;
W
therefore the foci.
5007
with u
The general Cartesian equation
=
(4656)
of a conic confocal
is
k'Au+k {C{a^'-^y')-2G.v-2Ft/-{-A+B}+l
= 0.
Proof.— (5005) must be transformed. Written in full, by (4664), it
becomes (kA + 1) X^ + (kB + l) fir + kCi''. Hence, by (4667), the trilinear
equation will be
{(l-B + l)
W-l-F'}
a-
+ &c. =
F-
(BG-F')
a-
+ kCcr + &c.
= k-a^cr + ]cCa' + &c.,
and
so on, finally writing x, y, 1 for
ct, /3,
(4668)
y.
TO FIND THE FOCI OF THE GENERAL CONIC
(4656).
(First Method.)
Substitute in kU + XHA^'^ eithrr root of its discriminant
becomes re+ k(a + b) A + ab— h^ = (5000), and
5008
k-A'
it
The
factors (XxiH-/iyi-fv)(Xx.2+/iy2H-v).
foci are Xiyi and x.^ya, real for one value of k and imaginary
solvable
into
tioo
_
for the other.
Proof.— By (5006) the two factors represent the two foci, consequently
the coordinates of the foci are the coefficients of X, /x, y in those factors.
.
ANALYTIGAL CONTGS.
692
(^Second MetJiod.')
afocuii; then, by (4720), the equation of an
x)+i (») y)
(^
0.
Therefore substitute, in the tangential
or ^-hirj
(x-f-iy)
iy),
i, v
(x
equation (4665), the coefficients A
1,
and equate real and imaginarij parts to zero. The resultirig
equations for finding x and y are, with the notation of (4665),
5009
xy
^^cf'
1)6
—
=
=— +
=
—
=
/ti
=
—
iinaginari) tangent through that point is
= A[..-6+v/{4/r+(«-6)'^}].
2{Cn-Ff = ^[h~aJrV \^^^'^{^i-WW
= 0, and the coordinates
If the conic
a parabola,
2(0.r-G)^
5010
5011
5012
is
by
of the focus are given
(F2_,_ Q')
cv
(F^+G^) y
5013
Ex.
— To
= FH-\-l (A-B) G,
= GH-i (A-B) F.
+ 2xy + 2>/ + 2x =
find the foci of 2x'
0.
By
the
first
method, we have
a,
h,
c,
f,
g,
=
2,
2,
0,
0,
1,
=
0,
and
from which
'\
11
Hi
I'atic
for k
A
= — 2.
The quad-
is
= (2^— 3)(2A— 1) = 0,
= f or |.
hU+X' + = (-/ +
+ 2At>'-4A) +X' + m' = 0,
2X'-12v\-^i- + 9y' + 6fiy = 0.
A,
Taking
h
B,
-1,
(\
F,
3,
1,
/Li'
|,
or
Solving for
{
G,
—2,
^-
3.'-
:]
thrown into the factors
X, this is
2\ + /xy2-3 (2+ v/2)
V
}
Therefore the coordinates of the
2- 72
8
/rA'H4^-A + 3
therefore
J
v/2-1
5014
2A-/iv/2-3 (2-
2+72
,
8-
'
{
foci, after I'ationalizing
3-'
'^"'^
x/2) y
}
the fractions, are
72 + 1
-T-
Otlierwise, by the second method, equations (5010, '1) become, in
1)2)^
this instance, (3;i;
±2, the sohUion of which pro2, (3/y
duces the same values of a; and y.
+
=±
—
=
5015 When the axes are oblique, the coordinates x, y of a
focus are found from the equations
{C (.v+f/ c,oH(o)-F cosco-G]'
[Cy-Ff Hiii^o)
where I
ami J are
the invariants
= iA(\/i^-4./+2«-/)
= \A (y/2_4./^2«+/),
(4955) and (4954) respec-
INVARIANTS AND GOVAEIANTS.
The equations may he solved for
tively.
x'
y = J sin w,
693
= x -\-y cos
a>
and
mhiclh are the rectangular coordinates of the focus
with the same origin and x axis.
—
Proof.
Following' the method of (5009), the imaginary tangent thi'ough
the focus is, by (4721), £— .r-f- (»? ;(/)(cos w + i sin w).
The two equations
obtained from the tangential equation are, writing Aa for BG — F", &c.
(4668),
—
—A (a + h — 2h cos w — 2a sin^ w), XY = A (h sinw — a sin w cos w)
X = G {x + y cosc») — F cosu — G and Y = (Gy — F) sin
X^—Y'-=
where
;
u).
5016
If the equation of the conic
a.v^ -f 2hoci/
+
to
oblique axes be
hif -}- c
=
,
the equations for determining the foci reduce to
y
{cV-\-y
cos
a>)
_
cv(i/-\-cr
acos(o—h
cosw)
c
~
bcosQ)—h
The condition that the line Xa?+^?/
conic u-\-(X'x-{-fiy-\-vzY is
5017
U-\-(l)
5018
{fxv—ii'v, v\'
— v\,
Xji'—X'/j)
+ v^ may touch
= 0.
or
{A+u')U=n:\
20 = \' l\-\- [J^^v U,.
—
U
where
ab — h^
(4656, 4936, 74)
(4938)
(4674)
fj,'
Proof. Put a + \'^ for a, &c. in
from the first through the identity
A^ (^/ — /uV,
The second form follows
of (4664).
&c.)
=
the
UU' — lP.
F
+
= u,.,x-{-Uy,y u^,z, the polar of
Otherwise, let
(4659), then the condition that P' may touch u-\-V"'^
becomes, in terms of the coordinates of the poles,
5019
X, y,
z
5020
(1
+
"'') "'
=
(See 4657).
^^,x"^^,.y"^i>,,Z'
—
Pkoof. If we put ;«^.,, »,,., u,,, from (4659), for \, ^, v in ?7 to obtain the
and similar substitutions made in
condition of touching, the result is A?^'
{(\>^.x" -\-kc).
give A (0^,»" + &c.), therefore (50l8) becomes (l4-^'")«'
;
=
n
5021
The condition that the conies
u + {}!x + /t/ + v',<;)^
may touch
each other
u + {}!'x + /V + v'zf
is
(A+ r)(A+ V")
=
(A
± n)^
(49B8-74)
ANALYTICAL CONIC 8.
694
Proof.
— Make one of the common cbords
(X'x
+ ix'y + v'z) ± (\"x + V + v"z)
/u'
touch either conic by substituting
is
(A+
5022
of the
t7')(Z7'± 2Ii-\-TJ")
=
X'
(U'±
±
X" for X, &c. in (5018). The result
U)-, which reduces to the form above.
The condition, in terms of the coordinates of the poles
two lines, is found from the last, as in (5019), and is
(l+«')(l
5023
+
<'")
=
ll±{<l>...v"+<t>,!/"+<l>..z")V-
The Jacobian, J, of three conies it, v, iv, is the locus
whose polars with respect to the conies all meet in
of a point
Its equation is
a point.
(ha.^-\-h,i/-\-g\z,
a2^+hi/+g2^,
fh^+fhy-^g^z
Jh^v+b,ij-\-f\ z,
Jh.v-\-b,i/+f, z,
fh.v+b,2/-\-f, z
gi^'^-^fi!/+fi -»
g2^v+fz!/+c.,z,
g;^'-{-fij/+c, z
—
Proof. The equation is the eliminant of the equations of the three
passing thi'ough a point ^r)i, viz., ?t j.^ + ?t^7j + ««4
0, ViX + v^ri -{-i\i^ :=0,
0.
See (4657) and (1600).
wJ+w^f]-\-^tKi;
=
polai's
=
5024 The equation of a
conic passing through five
points ai/3iyi, 00/3.270, &c. is
the determinant equation
annexed ; and the equation
of a conic touching five
right lines X,/tiVi, X^^iaVo, &c.
is the same in form, X, ^, v
taking the place of a, /3, -y.
a^ B^ y^ fi
y y a a
/3
«! ^1 yl
Ayi
«2 ^2 yi
^lyz yi^i ^i^i
ctiA
ricii
y-iO^i
o.^^^
a! ^4 y\ ^174 74 a^
^4^
A75 7n%
ttoA
as 0:^ r' Ayij
a' ^5 75
=
0.
—
Proof. The determinant is the eliminant of six equations of the type
(4656) in the one case and (4665) in the otlier. By (583).
5025 If three conies have a common self -con jugate triangle,
their Jacobian is three right lines.
—
is,
Proof. The Jacobian oi
by (5023), x,jz = 0.
aiX^-\-'b^if-'rc^z-, a^x'
+ h^^^f + Cf-,
a.^x'-^-h.^^i'
+ c^z^
For the condition tliat three conies may have a common point,
Salmons Conic<, 6th edit., Art. 389a, and rroc. Land. ISlalJi. Sec., Vol.
p. 404, /. /. Walker, M.A.
see
iv.,
mVAUIANTS AND COVARIANTS.
A
5026
i(,
n, F,
Proof.
695
system of two conies has four covariant forms
conneeted by the equation
•/,
-{-Fun {&&'-3AA')-AA''u'-A'Ahi''
-^A'uhi {2AQ'-&')-\-Au''h (2A'0-0'2).
Form the Jacobian of u, u\ aud P. This will be the equation of
Compare the
the common self-conjugate triangle (4992, 5025).
—
the sides of
result with that obtained
by the method of (4995).
By
parity of reasoning, there are four contravariant
is the tangential equivalent of */,
where
forms t/, U' V,
and represents the vertices of the self-conjugate triangle. Its
and the invariants
square is expressed in terms of U, f/',
5027
F
r
V
precisely as J^ is expressed in (5026).
The locus of the centre of a conic
touches four given lines is a right hue.
which always
5028
— Let
= 0,
=
be the tangential equations of two fixed
is another
then, by (4914), TJ+kTJ'
The coordinates of its centre will be
conic also touching the four lines.
Proof.
IT
77'
conies, each touching the four lines
G±kG;_ ^^^ ^1±M^,
by (4402).
=
;
The point
thus seen, by (4032), to
is
lie
G-\-kC
G-{-kG
on the
line joining the centres of the
in the ratio hC'
5029
To
given
lines.
'.
two
fixed conies
and
to divide that line
G.
find the locus of the focus of a conic touching four
In the equations (5010, '1) for determining the coordinates of the focus,
The i-esult in general is a cubic
write A-\-kA' for A, &c., and eliminate k.
If 2, 2' be parabolas, S + /v2' is a parabola having three tangents in
curve.
0'= the locus becomes a circle. If the
common with 2 and 2'. If (7
conies be concentric, they touch four sides of a parallelogram, and the locus
is a rectangular hyperbola.
=
Note on Tangential Coordinates.
5030
in mind that a tangential equation in trilinear
the variables ai-e the coefficients of o, /3, y in the
tangent line la + mlS + nY) will not agree with the equation of the same locus
Thus, to convert
expressed in the tangential coordinates \ fj., v of (4019).
equation (5001), which, for distinctness, will now be written
It
must be borne
notation (that
is,
when
l^-\-m^-Yn'
— 'imn cos
into tangential coordinates,
A—'2nl cos
we must
B — 2lm
substitute,
cos
C*
=
by (4023),
a\,
&/u,
c»'
for
The equation then becomes
a-\'"+tV + cV-26ccos^/xv-2cacos5^X-2aOcos(7A^ = 0.
Put 2tc COS J. = b^ + c^ — a^, &c., and the result is the equation as presented
I,
m,
n.
in (4905).
Corrigenda.— In (4678) and (4692) erase the coefficient 2 and in (4680) and (4903) supply
the factor 4 on the left of the equation.
;
THEORY OF PLANE CURVES.
TANGENT AND NORMAL.
5100
Let
P
on
(Fig. 90) be a point
tlie
AP
curve
;
FT, PN,
ordinate, and normal intercepted by the <k
See definitions in (1160). Let /.PTX
axis of coordinates.
PG, the tangent,
= g,,
5101
tant/;
5104
Suh-taiigent
by(1403);
NT =
siuV'
= J;
Sub-normal
i/d\j,
cosr^
NG =
5106
PT=i/V'OT^),
pr=ciV(l +?/;).
5108
PG = i/x/(i^),
PG'
Let
Arc
OP =
AP — s.
5110
5113
5114
5115
sin
(l)
=
{d.vy-\-{duy
.
cos
=
<^
=
dr
j:*
(ds)\
tanV>^
^'-""^
lidercei)t6 of Normal
s.,
i/y^.
= uVii-\-4)-
r-\
r (Fig. 91), n
Then, by infinitesimals,
lie
= r—,
= g.
AOP =
,
J
_
^^^^
<^
OPT =
e,
=
^
;
(je
>'
dr
= x/(n-//a.
+ ^'""^
(1768)
.
OG = r~, OG =
Vj-.
(Fig.«JO)
69'
TANGENT AND NORMAL.
^^'''''
^^=^i^^PGN=^h:NPf-'l^-'^-
5116
se
Proof.— By
= ^/{r'+rl),
rOg
=
s„
sin^ and tan^
=
= y/{l+r%y
r8,.
(5110).
EQUATIONS OF THE TANGENT AND NORMAL.
of the curve beiug y =f{a') or u
equation of the tangent at oij is
The equation
=
0, the
^-^ = ^(^-.0,
5118
5119
or
ft/.,-^
5120
or
^w,+>;«,
= ^m^-i/^
= .r?/,+//«,.
=
=
(}>
{.r,
y)
(4120)
(1708)
liomogciicous function
i'» + f'«-i+ ••• +^o> where v„ is a
If (p {a;, y)
member of
of .« and y of bhe n^^ degree, the constant part forming the right
equation (5120) takes the value
5121
— v„_i — 2y„_2—
By
Euler's theorem (1621) and
The equation
5122
of the
(p
...
— (« — 1)^1 — ^%
= 0.
(x, y)
normal
at
xij is
THEORY OF PLANE CURVES.
698
OY ~p
Let
be
tlie
perpendicular from the pole upon the
tangent, then
5129
;>
5131
= r siu(^ =
/n'7^,
{nr-\-u'^-\
- ^!^^W^
-
(5112)
(40G4&5119,'20)
OS, drawn at right angles to r to meet the tangent,
is
called the 2^olar sub-tangent.
5133
Polar sub-tangent
= r-6,.
(5112)
RADIUS OF CURYATURE AND E VOLUTE.
Let ^,
be the centre of curvature for a point xy on the
curve, and p the radius of curvature then
J/
;
= p^
{.v-$)-^(^-r})ii, =
5134
i.v-iy^{^j-yjy
5135
(1).
(2),
l+2/l+(//->?)z/2.=
Proof.
sidering
— (2) and (3) are obtained
^,
T)
x,
con-
constants.
The following
5137
(3).
from (1) by differentiating for
^
are different values of p
= (1+^^
:
^Ft^^:
^
^.
5139
5141
=
5144
^
5146
PuooFS.
J-
(>''+/1)^
^
:!±
_
= = y>+/),^ =
— For (5137), eliminate x —
.sv
^_i^
(^+t<D'^
>•>',,.
l and y~n between equations
d),
1
W'
and (3).
(5138) is obtained from the i)rcccding value by substituting for y^. and
(2),
EABIUS OF GUEVATUEE AND EVOLUTE.
699
The equation of the curve is here supposed to be
the values (1708, '9).
0.
of the form
(x, y)
s.
For (5139) change the variable to t. For (5140) make t
ds (Fig. 92) be equal consecutive elements
For (5141-3) let
y.,^
=
=
;
;
PQ=QE =
;
of the curve. Draw the normals at P, Q, E, and the tangents at P and Q to
be drawn parallel
Then, if
in T and S.
meet the normals at Q and
Now
will ultimately fall on the normal QO.
and equal to QS, the point
is equal to the proupon
and
the difference of the projections of
x.,,ds)
(x,
ds
or QS
+
dsx, that of
Projection of
jection of TN.
ds
But
cos a.
dsx.,,ds
(1500); therefore the difference
a.
sin
Similarly pi/2s
cos a.
ds : p, therefore px.,g
FN
E
N
PN
PT
PT =
=
=
For (5144)
(5145)
from r
to
is
u by
(5146.)
;
change (5137)
obtained from p
r
to r
and
OX
=
PN
= TN
=
;
by (1768,
6,
= rr„ = -
''-^
TN
=
'9).
and (5129)
;
or change (5144)
= n'^.
In Fig. (93),
PQ = p,
PP'
= ds,
and PQP'
= dx^.
consecutive normals PT, P'T'
perpendiculars from the origin
PN, q for
putting p for OT
In Fig. (93), let PQ, P'Q be
tangents; OT, OT', ON, ON'
(5147.)
consecutive
upon the tangents and normals.
PT=
:
ON, and d^
Z
for
TPT'
Then,
= PQP',
;
=
we have
&c.,
= ^, QN=% and p = PQ = p + QN = p+p,.,.
dip
Eliminate cos
dp = r cos d\p and cos = r,
(5148.)
q
dtf/
(p
(p
5149
Def.— The
rvolnte of a curve
is
the locus of
its
centre
Eegarding the evolute as the principal curve,
of curvature.
the original curve
5150
(p.
.
is called its involute.
The normal
of
any curve
a tangent to
is
its
evolute.
Proof.— By differentiating equation (5135) on the hypothesis that ^ and
combining the result with (3), we
V are variables dependent upon x, and
1.
obtain i/^j/j
l r>
In (Fig. 94), the normal at P of the curve AP touches the evolute at Q.
curve.
given
the
of
normals
the
of
envelope
Otherwise the evolute is the
=—
If
5151
xij
and
^
^n are the points
P, Q,
i=f.f+!^.,= .p,
jS, =
the relations
^
and ns = dr,, then Qs = dp.
(5151) and proportion gives (5152).
Proof.— Take Qn
Qn, ns gives dp in
we have
,
= dk
=
f-
The
projection of
The evolute and involute are connected by the formulae below, in which r\ /, s in the evolute correspond to
r, 2^i s in the involute.
5153
5154
/)±*'
= constant;
p''
= r'-p';
r'
= r'+p'-2pp.
THEORY OF PLANE CURVES.
700
—
Proof. From Fig. (94), Jp = ±(?s', &c., s being tlie arc RQ measured
from a fixed point R. Hence, if a string is wrapped upon a given curve, the
(3155, 'G) from Fig. (93).
free end describes an involute of the curve.
To obtain
5157
tlie
equation of the evolute; eliminate x and
'6) and the equation of tlie curve.
from equations (5135,
ij
5158 To obtain the polar equation of the evolute eliminate
andp from (5156) and (5157) and the given equation of the
;
r
curve r
= ^(lO-
5159
Ex.— To
find the evolute of the catenary y
=
2/^=1 (e^-e''^)
^^y'-"'')
;
y.,^
=1
=
+ e"^) =
{e^-
2c
c
-^ (e'
^
;
+
Here
c' ').
so that equations
c
(5135, '0) become
(a;-0
From
these
+ (y-v)
we
find
equation of the curve,
^^-^'~''^
y=^,
we
=0
x
and
= ^— j-
1
+
t^ +
•/{yf—4c-).
(2/-v)
J
=
0.
Substituting in the
obtain the required equation in
i,
and
rj.
INVEESE PROBLEM AND INTRINSIC EQUATION.
An
inverse question occurs
tion of the abscissa, say
s
=
(p
when
(x)
;
the arc is a given functhe equation of the curve
in rectangular coordinates will then
5160
«/
The
be
= J v/(4- 1) d.V.
[From (5113).
an equation indebe the ordinary
equation, taking for origin a point
on the curve (Fig. 95),
and the tangent at
arc OP, and xp the
for x axis.
Let s
inclination of the tangent at P
then the intrinsic equation of
5161
intrinsic equation of a curve is
pendent of coordinate axes.
Let y
=
<l>
{x)
=
;
the curve
is
5162
where
= J sec
from tan =
s
x^,
is
found
t/».r^ (Ixfj
x^
<l>'{x).
;
—
ASYMPTOTES.
To
tion
701
obtain the Cartesian equation from the intrinsic equa-
:
Let
5163
between
s
= F(-^)
be the intrinsic equation.
Eliminate
t//
and the equations
this
,v
=
f
cos
xff
(Is,
y
= J sin
xft
ds.
5165 The intrinsic equation of the evolute obtained from
F{\p), is
the intrinsic equation of the curve, s
=
4^+*'=/,
(5154)
intrinsic equation of the involute obtained from
the equation of the curve, is
The
5166
s'
a constant.
= F(\p),
is the same for both curves (Fig. 94), ^ only differing
For
by ^TT, and s = ^ pd^.
cl\l,
ASYMPTOTES.
—
5167 Def. An asymptote of a curve is a straight line or
curve which the former continually approaches but never
{Vide 1185).
reaches.
GENERAL RULES FOR RECTILINEAR ASYMPTOTES.
Ascertain if y^ has a limiting value when
I.
If it has, find the intercept on the x or y axis, that
is, X— yxy or
xy^ (5104).
There toill be a.n asijmptote parallel to the y axis ivhen y^ is
infinite, and the x intercept finite, or one parallel to the x axis
Rule
5168
X
= 00
.
y—
ivhen y^ is zero
5169
Rule
and
II.
the
y intercept finite.
— When
the equation of the curve consists of
n^^, Si'c. degrees,
homogeneous functions of x and y, of the m*^,
so that it
may
be written
Hf Wi)+*'=- = »
)
(^)
—
THEORY OF PLANE CURVES.
702
put
y and expand <pUi+ -
^ix 4-/3/0?'
and make x
(1) hy X™,
&c.,
),
infinite; then
=
(^)
({>
Divide
(1500).
hij
determines
fi.
Next, put this value of /n in (1), divide hy x™~\ and malce x
=:
Should the
determines /3.
infinite; thus |3f/)' (/i)
-j^ (^)
last equation he indeterminate, then
+
gives two values for
When n
Rule
5170
and
/3,
<in — 1,
is
/3
so on.
= 0,
and luhen n
=
is
>m — 1, = oo
/3
.
he a rational integral equa-
III.
If (x, y)
discover asymptotes parallel to the axes, equate to zero
the coefficients of the highest powers of x and j, if those coefficients contain y or x respectively.
To find other asymptotes Suhstitute |itx-|-|3 for y in the
tion,
</>
to
—
To
original equation, and arrange according to powers of :k..
find fx, equate to zero the coefficient of the highest power ofx.
To find (3, equate to zero the coefficient of the next power of x,
or, if that equation be indeterminate, take the next coefficient in
order,
and
if
r^ CO
there is
tity,
=
Rule IV.
5171
and
so on.
where
i{0)
If the polar equation of the curve he v
makes the polar suhtangent v^Q^^q, a finite quanc
an asymptote whose equation is r cos {0 — a)
=
+ ^tt = f~^ (00 =
a
)
the value of B of the curve
when
;
r
is
infinite.
—
5172 Asymptotic curves. In these tlie difference of corresponding ordinates continually diminishes as x increases.
As an example,
5173
Ex.
1.
the curves y
— To
coefficient of
Putting y
7/^,
= fix +
a
(1)
coefficient of
value of
/J
(1
0,
ft
=^
= ± (3.c-2a).
(^')
H
—
are asymptotic.
=
(1).
4.c'^
1/
+ i.'x- + 2fiilt + rr)-4x' =
+a^-)-4
.
<}>
gives an asymptote parallel to the
=
in (2), the coefficient of »'
to zero, gives
are a^z/S
x\ 3
=
and y
axis.
becomes
(a + 3x)(x-
The
{'•i')
+ S.c)(x^ + f)
+ Sx =
ft,
(^
asymptotes of the curve
find the
(a
The
=
gives
V
becomes
(2).
= ± -^.
/i
-
-
Hence the equations
Substituting this
"J
± —~
;
of two
and
this,
equated
more asymptotes
SINGVLAUITIES OF CURVES.
Ex.
2.
— To find an asymptote of the curve
^.2
de
_
?•
tote
r cos
is
CO
d
,
=
and
lir,
rfl,.
= —a.
r cos 6 =: a cos 2d.
= —a.
Here
2d
— 2a sin 20
a cos 20 sin
dr
=
When
a^ cos
703
cos
Hence the equation
of the
asymp-
SINGULARITIES OF CURVES.
—A
5174
Concavity and Convexity.
curve is reckoned convex
or concave towards the axis of x according as yv/gx is positive
or negative.
POINTS OF INFLEXION.
5175
^4,
2)oint of Injiexion (Fig. 96) exists
where the tangent
has a Umiting position, and therefore where
mum or minimum vahie.
5176
Hence
7/2,,
must vanish and change
v/.^.
takes a maxi-
sign, as in (1832).
5177 Or, more generally, an even number of consecutive
derivatives of 7/
^ (,t) must vanish, and the curve will pass
from positive to negative, or from negative to positive, with
respect to the axis of re, according as the next derivative is
negative or positive.
[See (1833).
=
MULTIPLE POINTS.
5178
^
exists
when
multiple jpoint, known also as a node or crunode,
If
y^. has more than one value, as at B (Fig. 98),
{x, y)
be the curve, f and (j>y must both vanish, by
Then, by (1704), two values of v/,. determining a
(1713).
double point will be given by the quadratic
(}>
=
,.
J
+
<^2.^!.
5179
If
values of
the cubic
i>ij;,
y,.,
^2</3
'Pxy
Generally,
+
<^2,.
also vanish;
determining a
<l>Syyl
5180
2(^..^/..
triple
=
(1).
then, by
jw I nt,
(1705),
will be obtained
+ ^2y.ryl-^H,^.I/.r-^<l>S. =
when
all
the derivatives of
three
from
(2).
(p
of
an order
;
;;
TBIIORY OF PLANE CURVES.
704'
than n vanish, the equation for determining
be written
less
may
(a +
<p
+
And
-^
/^,
h
+ h) =
-, Qid^ + hdyYcp
h
®-^
=
(p„
-^ in the
dx
ai'e
small.
z)
{x,
y)
\
terms of higher order which vanish when h and h
=—
=
Then, by (1512),
point.
n
(put
= 0.
(=;rf,+rf.,)«.^G,.,y)
— Let ah be the multiple
Proof,
/^.
limit,
CUSPS.
5181
When two
5182
In the
branches of a curve have a common tangent at a point, but do not pass through the point, they form
a cusj)i termed also a spinode or stationary jJoint.
two values
of
first sjiecies,
^j.^x
or ceratoid cusp (Fig. 100), the
have opposite signs.
5183 In the second species, or raniphoid cusp (Fig. 101),
they have the same sign.
CONJUGATE POINTS.
A conjugate point, or acnode, is an isolated point whose
coordinates satisfy the equation of the curve. A necessary
5184
condition for the existence of a conjugate point
(j)y must both vanish.
Pkoof.
—
fore
is
— For
the tangent at such a point
There are four species
formed by the union of
(i.)
(ii.)
that
^^.
and
direction, there-
indeterminate (1713).
5185
it is
may have any
is
of the
trij^le
point according as
three crunodes, as in (Fig. 102)
two crunodes and a cusp, as
and two cusps, as
(iii.)
a crunode
(iv.)
when only one
in (Fig. 103)
in (Fig. 104)
real tangent exists at the point.
—
=
Ex.
The equation if
{x-a){,v-h){x-c)^ when
a -Ch <ic represents a curve, such as that drawn in (Fig. 07).
5186
*
Salmon's JUigher
I'lauc Curves, Arts. 39, 40.
UN:
(..
SINGULARITIES OF GURV^S.
= c the curve takes the form in (Fig. 98). But
= a, the oval shrinks into a point A (Fig. 99).
li a = b = c the point A becomes a cusp, as in (Fig. 100).
When
if,
705
b
instead, h
A
geometrical method of investigating; singular points.
5187 Describe an elementary circle of radius r round the
point X, y on the curve (p {x, y)
0, intersecting the curve in
Expand
r sin 0.
r cos B, k
Let h
the point x-\-h, y-\-h.
ii siny, ^y
by (1512), and put
^ {x-\-h, y-\-h)
thus obtain
cos y.
=
=
=
=
We
K
K sin (y + 0)-\-^
Bj
«/>,,
(<^2,,
cos^ d
+
2./,.,^
cos
sin
+
=
=
(p,^
sin-
=0
e)-\-—
(.')'
being put for the rest of the expansion
According as the quadratic in tan 0,
(^,.r
+
2(/),^^
tan
+
(^2i/
tan-
= 0,
equal, or imaginary roots ; i.e., according as
crunode,
i>ly
hxi^iy is positive, zero, or negative, xy will be a
a cusp, or an acnode. By examining the sign of B, the
species of cusp and character of the curvature may be deter-
has
real,
—
mined.
Figures (105) and (106), according as B and ^2^ liave
opposite or like signs, show the nature of a crunode and
;
figures (107)
—
and (108) show a cusp.
Proof. At an ordinary point the circle cuts the curve at the two points
7r — y.
But, if f^^ and ^^ both vanish, there is a
given by 6
—y, 9
Writing A, B, G for <po^, f-,^, (poy, equation (1) now becomes
singular point.
=
=
C'cos^^[tan-^0+^tane+||+^ =
(i.)
If
B^>AGy,
this
G cos^ d
(tan d
- tan
a) (tan 6
- tan /3) +
^=
of intersection with the circle are given
(Figs. 105 and 106.)
and the points
and7r + /3.
(ii.)
When
B^
O
(2).
may be put in the form
= AG,
we may
0,
by 6
=
a,
ft,
tt
+ o,
write equation (1)
27?
(7cos'0(tan0-tana)-+^
= 0.
If B and G have opposite signs, there is a cusp with a for the inclination of
the tangent (Fig. 107). So also, if B and G have the .same sign, the inclinaThe cusps exist in this ca.se
tion and direction being v + a (Fig. 108).
because B changes its sign when tt is added to 6, B being a homogeneous
function of the third degree in sin 6 and cos 0.
(iii.) If B^< AG, there are no real points of intersection, and therefore xij
is an acnode.
4 X
THEORY OF PLANE CURVES.
706
CONTACT OF CURVES.
A
contact of the n^^^ order exists between two curves
successive derivatives, y^., ... y^^ or Tg, ... r^g, correspond. The curves cross at the point if n be even. No curve
can pass between them which has a contact of a lower order
with either.
5188
when n
Ex.
— The
where x
= a,
=
curve y
<p
has a contact of the
(«)
with the curve y
=
(a)
(f>
at the point
n^^ order,
+ (x — a) ^'(a) -f
+
...
^—~-
ch"
(a).
—
5189 Cor. If the curve y =f(x) has n parameters, they
may be determined so that the curve shall have a contact of
=
—
iy^' order with y
(x).
contact of the first order between two curves implies a
common tangent, and a contact of the second order a common
radius of curvature.
the {n
(j>
A
Conic of closest contact with a given curve.
Lemma.
5190
—In a central conic
tan CPG = 4 -^•
S ds
(Fig. of 1195),
PCT=d, CrT=(p, GP = r, CD =
+ R:' = a- + b\ .•.rr, = -RR,
Proof.— Putting
(1211),
Also
Now
ii-;-
tan
sin
CPG
</)
=
a&,
by (1194),
= -cot^) = - ^
.-.
i2r0,
(5112)
=
r
But
wc
R,
have, by
(i.).
r''
p
=
^
ah
(4538),
= a&,
by (-5110)
— = ^,
rdg
ab
*^
by
(i.)
(ii.).
and
(ii.).
.
.-.
-V
S
^=
ds
^
=
tan
CPG.
ab
5191 To find the conic having a contact of the fourth order
with a given curve at a given point P.
If
be the conic' s centre, the radius r = OP, and the
angle v between r and the normal arc found from the equations
,
tail
v=
1
(/p
o -f,
as
and these determine the
conic.
vosp
r
=
1
p
(h
as
ENVELOPES.
707
—
P
the point of
Proof. In Fig. 93, let
be the centre of the conic and
contact.
The five disposable constants of the general equation of a conic
will be determined by the following five data two coordinates of 0, a common point P, a common tangent at P, and the same radius of curvature PQ.
PP', we have, in
Since V
POT, cW
POP', dx^
TOT', and ds
passing from
to P', di'
ds cos v,
d^P-dO. Now rdd
:
=
=
P
=
therefore
'
lemma.
-~-
=
= P'OT—POT =
— and tan v
= —
;
^'^
^°'
P
^
=
=
has been found in the
'^'
The squares of the semi-axes of the
roots of the equation
same conic are the
{Q+b'Sacy.v'-Oa' {lS+2b'-3ac){9-\-b'-3ac)
.v+7'29a'
= 0,
a, h, c
being written for
p, pg, p^g.
9(e^-2y
1-e'
from
_
~
The
eccentricity
is
found
(18+26^-3ae)^
9-\-b'-3ac
'
Also the equation of the conic referred to the tangent and
normal at the point is
=
Aa'+2Bay-{-Cif
where
A
= ^, B = -^,
op
c
= ^ + ^-Pf.
p
p
Ed. Times, Math. Reprint, Vol.
Wolstenholme will be found.
2i/,
vp
xxr., p. 87,
6
where the demonstrations by
Prof.
ENVELOPES.
5192 An envelope of a curve is the locus of the ultimate
intersections of the different curves of the same species, got
by varying continuously a parameter of the curve ; and the
envelope touches
5193
meter
Rule.
all
—If
the intersecting curves so obtained.
F (x,
envelope
betiveen the equations
a,
the
F
(a?, ?/,
a)
y, a)
is the
=
=
be a curve having the paracurve obtained by eliminating a
and
d^F (.i',
y, a)
= 0.
THEORY OF PLANE CURVES.
708
Proof.
tion of
— Let a change to a +
F (x,
a)
y,
=
and
F
F(x,y, a + h)-F(x,
The coordinates
h.
(:c,
a + h)
y,
_
y, a)
h
=0
_ ^^
(IF (x, y, a)
^^^^'
^^'
of the point of intersec-
satisfy the equation
(1404)
da
=
be the equation of a curve
connected by n — 1 equations,
then, by varying the parameters, a series of intersecting
curves may be obtained. The envelope of these curves will
be found by differentiating all the equations with respect to
5194
F{Xi
If
y,
a,h,c,
liavhig n parameters a,
a, h, c, &c.,
...)
b, c, ...
and eliminating da,
dh,
and a^b,
...
...
—
Ol95 Ex. In (2) of (5135), we have the equation of the normal of a
curve at a given point xy ^, >/ being the variable coordinates, and x, y the
By differ(x, y)
0.
parameters connected by the equation of the curve
entiating for X and y, (5136) is found, and the elimination as directed in
(5157) produces the equation of the evolute which, by (5194), is the envelope
;
=
F
of the curve.
INTEGRALS OF OUHVES AND AREAS.
FORMULA FOR THE LENGTH OF AN ARC
5196
s=^ds=
[v/(l+Z/i-)
d^^'
S.
= J \/l+iI%
(5113)
5200
= j v/K+2/?) dt = Jy(r+r^) cW
(5116)
5201
=
(5111)
fv/(>'^^;+l) dr
=
5203
Legendre's formula, 5
5204
The whole contour
r J'!''
= j>^H-
\
.,.
.
pd"*^.
of a closed curve
/»2Tr
=
\
pdxf/.
Jo
—
Pkoof. In figure (93), let P, P' be an element Js of the curve PT^ P'T'
tangents, and 07', 07'' the perpendiculars upon them from the origin
Then ch + P'T'-PT
TL, i.e., ds + dg
pd^ therefore s +
q.
;
=
PT =
= lpd\p.
all
But
qdij/
round the curve,
tion, or
dq
= 0.
= —dp;
=
;
;
OT=p
q
s
p^ + jjfdi}/. Also, in integrating
taken for every point vanishes in the summa-
therefore
P'T'—PT
Therefore
=
I
(?*•
=
I
pd\p.
.
INVEBSE CURVES.
709
FORMULA FOR PLANE AREAS.
=
t)e tlie equation of a curve, the area
1^ (,^)
y
bounded by the curve, two ordinates (x a, x h), and the x
5205
If
=
=
axis, is, as in (1902).
A
= rV
{^^)
dx-
With polar coordinates the area included between two
5206
radii (0
= a, B = ^)
Proof.
and the curve
is
— From figure (91) and the elemental area OPP'.
5209 The area bounded by two circles of
two curves = <^ (r), B = ^ (r) (Fig. 109).
=
rdrdd
4>
Here
r{Tp{T)
r
\
{^j^
(r)
-
radii «, h,
<!>
and the
(r)} dr.
Ja
(r)
— ^{r)}dr
is
the elemental area between the
dotted circumferences.
5211
and
The area bounded by two
its e volute
^
Proof.
radii of curvature, the curve,
(Fig. 110).
= \\p^dy\i = l\pds.
—From figure (93) and the elemental area QPP'
INVERSE CURVES.
The following
results
may be added
to those given in Arts. (1000-15).
be corresponding radii of a curve and its
F s, s' corresponding arcs, and ^, (p'
the angles between the radius and tangents, then
5212
Let
r,
r
inverse, so that rr'
=
= —r
-—7
ds
Proof.
— Let PQ
;
r
and
be the element of
<f)
ai-c ds,
=
<f>'.
P'Q' the element
els',
and
the origin.
Then OP. OP'
therefore
PQ
:
=
P'Q'
OQ.OQ', therefore OPQ, OQ'P' are similar
OP OQ' = r r' also z OPQ = OQ'P'.
::
:
:
;
triangles;
THEORY OF PLANE CURVES.
710
5214
K p,
p be the radii of curvature,
PEDAL CURVES.
711
PEDAL CURVES.
The locus of tlie foot of the perpendicular from the
upon the tangent is called ^ jpeclal curve. The pedal of
Rethe pedal curve is called the second pedal, and so on.
versing the order, the envelope of the right lines drawn from
5220
origin
each point of a curve at right angles to the radius vector
called t\ie first negative jpedal,
The pedal and the
5221
is
and so on.
reciprocal polar are inverse curves
(1000, 4844.)
AREA OF A PEDAL CURVE.
5222 Let C, P, Q be the respective areas of a closed curve,
the pedal of the curve, and the pedal of the evolute ; then
P-Q=C, P+Q = i jVv/t/f, 2P = C+iJ
Proof. — With
(93) and the notation of (5204), we have, by (5206),
=
therefore P + Q = U
+
P=i
Q = il
r'dxff.
figure
f
q'dxP
p-dij;,
Ci''
;
5')
^'Z'
5
I
'•'#•
=
A, we get
Also, taking two consecutive positions of the triangle OPT
SG + SQ-^P. Therefore, integrating all round,
OP'T'
=hA =
OPT-
[^clA
= = G+Q-P.
—
5225 Steiner^s Theorem. If P be the area of the pedal of a
closed curve when the pole is the origin, and P' the area of
the pedal when the pole is the point xy,
P'-P =
a=\
where
]3
^{j,^J^^f)^ax-hy,
cos Odd
h=\
and
9
being the inclination of
Proof.
— (Fig. 111.)
j9.
LM be a tangent, the point xy, perpendiculars
Draw SN perpendicular to OM, and let ON = p^
Let
»S'
OM = p
and
SB = 2/.
then p'
=
[/-# = ^{(p-piy-
i
= P+ ^
And
g
Pld^
2^&m9d6;
Jo
Jo
OS^- \p
-^
(x cos
#=i
+
7/
= twice the area of the
J
/f?^+
f
J
sin 0)
circle
p;^^/'-
clB,
m^^^
J
by (4094), and dd
whose diameter
is
OS.
= #.
.
THEORY OF PLANE CURVES.
712
5226
Cor.
1.
—
If P' be given, the locus of
.ri/
is
a circle
whose equation is (5225), and the centre of this circle is the
same for all values of P', the coordinates of the centre being
—a andJ —b
TT
TT
5227
let
Q8
—
Let Q be the fixed centre