A TREATISE ON THE INTEGRAL CALCULUS
VOLUME
I.
MACMILLAN AND
LONDON
CO., LIMITED
'MADRAS
BOMBAY CALCUTTA
MELBOURNE
THE MACMILLAN COMPANY
NEW YORK
BOSTON
CHICAGO
SAN FRANCISCO
DALLAS
THE MACMILLAN
CO.
OF CANADA,
TORONTO
LTD.
A TREATISE
ON THE
INTEGRAL CALCULUS
WITH APPLICATIONS, EXAMPLES
AND PROBLEMS
BY
JOSEPH EDWARDS, M.A.
FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE
PRINCIPAL OF QUEEN'S COLLEGE, LONDON
VOLUME
MACMILLAN AND
ST.
I
CO.,
LIMITED
MARTIN'S STREET, LONDON
1921
COPYRIGHT
GLASGOW PRINTED AT THE UNIVERSITY PRESS
BY ROBKRT MACI.EHOSE AND CO. LTD.
I
PREFACE.
AN
apology
is
due to readers
Calculus for tho
many
of
my
Treatise on the Differential
years of delay
between
its
publication
and that of the present companion volumes. This delay has
been due to several causes. In the first place it was due to the
very severe pressure of other duties.
several chapters of
what now
In the second place, when
constitutes the
first
volume had
been written, changes occurred in the regulations for the Mathematical Tripos and in the requirements of many of the class of
students I have come into contact with, and I was not sure
that such requirements were not already amply provided for by
I have been urged, however, by
other existent text-books.
time to time to continue the work I had begun years
and
to
put upon record the experience I had gained in the
ago,
teaching of the large number of advanced students it has been
my lot to meet. And I must also confess that in acceding to
this expressed desire, I have turned to this work with a sense of
pleasure and of relief from the distracting circumstances of the
great war through which we have recently passed.
In the preparation of the book for the press, I have endeavoured
many from
to collect together for the use of the reader all information
necessary to give him a good working knowledge of the subject,
both practically and theoretically, and to place before him this
information as clearly as possible, with abundance of illustrative
examples and instances of the application of the principles
explained.
To do
this as fully as I desired, it has unfortunately
been found necessary to enlarge the book beyond the ordinary
bounds of a text-book, and to divide it into two volumes.
Several of the matters treated of in the Second
subject of exhaustive treatises expressly
Volume
are the
devoted to the discussion
PREFACE.
vi
of those particular branches.
So that such chapters as are there
be found treating of Conformal Representation, Contour
Mean Values and Chances,
Integration, Elliptic Integrals,
Harmonic Analysis, etc., can only be regarded as an attempt
to
to put together in a convenient form for the reader the
most
important theorems and processes used in dealing with the
and merely as introductory and
no way exhaustive. The mode and sequence of treatment is
the same as that I have adopted in my advanced classes of students
earlier parts of these subjects,
in
during the last five-and-thirty years.
Such a book is necessarily to a considerable extent a compilation, and though some of the results and proofs are, so far as
I know, new, by far the greater part are to be found elsewhere.
have endeavoured to assign to their proper authorship as many
the results as possible, but
A
cases with certainty.
it is
very
difficult to
do
this in
I
of
many
teacher learns from his pupils and
from those he examines as well as from reading and research,
and one meets in this way with many proofs of the same theorem
;
it
be, in
may
some
they are due to the ingenuity of the
will be that such proofs, if not to be
cases, that
student, but in general
it
found in existing text-books, are due to one or other of the
distinguished body of teachers engaged at the Universities of
In such cases it is often
the Kingdom in teaching the subject.
desire
to assign the
however
much
one
it,
impossible,
may
authorship correctly.
A
large
number
of
works has been consulted, and I must
many authors. In parti-
acknowledge a great indebtedness to
cular, I
am
indebted for
much
information to the admirable and
exhaustive works of Legendre, Laplace, Lacroix, Jacobi, Serret,
Bertrand, Todhunter, Williamson, Boole, Cayley, Hobson, Forsyth,
Greenhill, Airy,
Chauvenet and
Glaisher, Culverwell and
journals.
I
am
others, as well as to articles
many more
also indebted to the
Educational Times for permission to
in various
by
mathematical
mathematical editor of the
make
use of some of the
very excellent examples on Chances and Mean Values,
to be found in that collection.
many
etc.,
The
early articles of
Volume
I.
have been so written that a
student already equipped with a knowledge of Graphical work
and Elementary Applications of the Summation- definition of
PREFACE
Integration
may
definition used
tion,
A
if
is
vii
begin at the second chapter at once, wKere the
that of the inverse of the operation of differentia-
he prefers to do
so.
number
of the examples are extracted from
Examination
and
College
Papers, and the source
University
of such examples is indicated when known.
Many others are
than anything
better
examination
define
These
new.
papers
considerable
the scope and extent of the knowledge expected of students
by the distinguished mathematicians engaged from time to time
in framing the regulations for such examinations and in conelse
ducting them.
My very grateful thanks are also due to the publishers, Messrs.
Macmillan & Co., to whose encouragement the appearance of
the book is in no small measure due.
They are also due to the
printers,
Messrs. Robert
MacLehose
at the
&
Co.,
and to
who have with
their Staff
constant courtesy
Glasgow University Press,
and unfailing care and patience carried through their part
piece of work which must at times have been far from easy.
of a
JOSEPH EDWARDS.
QUEEN'S COLLEGE, LONDON,
March, 1921.
PREFACE.
vi
of those particular branches.
So that such chapters as are there
be found treating of Conformal Representation, Contour
Mean Values and Chances,
Integration, Elliptic Integrals,
Harmonic Analysis, etc., can only be regarded as an attempt
to
to put together in a convenient form for the reader the
most
important theorems and processes used in dealing with the
and merely as introductory and
no way exhaustive. The mode and sequence of treatment is
the same as that I have adopted in my advanced classes of students
earlier parts of these subjects,
in
during the last five-and-thirty years.
Such a book is necessarily to a considerable extent a compilation, and though some of the results and proofs are, so far as
I know, new, by far the greater part are to be found elsewhere.
have endeavoured to assign to their proper authorship as many
I
of
it is very difficult to do this in many
with certainty. A teacher learns from his pupils and
from those he examines as well as from reading and research,
the results as possible, but
cases
and one meets in this way with many proofs of the same theorem
it may be, in some cases, that they are due to the ingenuity of the
;
student, but in general it will be that such proofs, if not to be
found in existing text-books, are due to one or other of the
distinguished body of teachers engaged at the Universities of
the Kingdom in teaching the subject. In such cases it is often
impossible, however much one may desire it, to assign the
authorship correctly.
A
large
number
of
works has been consulted, and I must
many authors. In parti-
acknowledge a great indebtedness to
cular, I
am
indebted for
much
information to the admirable and
exhaustive works of Legendre, Laplace, Lacroix, Jacobi, Serret,
Bertrand, Todhunter, Williamson, Boole, Cayley, Hobson, Forsyth,
Greenhill, Airy,
Chauvenet and
Glaisher, Culverwell and
journals.
I
am
others, as well as to articles
many more
also indebted to the
Educational Times for permission to
in various
by
mathematical
mathematical editor of the
make
use of some of the
very excellent examples on Chances and Mean Values,
to be found in that collection.
many
etc.,
The
early articles of
Volume
I.
have been so written that a
student already equipped with a knowledge of Graphical work
and Elementary Applications of the Summation- definition of
PREFACE
Integration
may
definition used
tion,
A
if
is
vii
begin at the second chapter at once, wKere the
that of the inverse of the operation of differentia-
he prefers to do
so.
number
of the examples are extracted from
Examination
and
College
Papers, and the source
University
of such examples is indicated when known.
Many others are
new.
considerable
These examination papers define better than anything
the scope and extent of the knowledge expected of students
by the distinguished mathematicians engaged from time to time
in framing the regulations for such examinations and in conelse
ducting them.
My very grateful thanks are also due to the publishers, Messrs.
Macmillan & Co., to whose encouragement the appearance of
the book is in no small measure due.
They are also due to the
& Co., and to their Staff
have
with constant courtesy
at the Glasgow University Press, who
and unfailing care and patience carried through their part of a
piece of work which must at times have been far from easy.
printers,
Messrs. Kobert
MacLehose
JOSEPH EDWARDS.
QUEEN'S COLLEGE, LONDON,
March, 1921.
CONTENTS.
CHAPTER
I.
NATURE OF THE PROBLEM. PRELIMINARY CONSIDERATIONS.
PAGES
ARTS.
1-8.
Fundamental Notions. Fluents and Fluxions.
blem to be attacked
9-15.
Newton's
Third
Lemmas.
1-3
Analytical
Notation
Expression.
16.
and
Second
Pro-
4-7
Illustrative
8-12
17-19.
Examples
The Fundamental Proposition
20.
Unknown Curve through
21.
Simpson's Rule
22-23.
Trapezoidal Rule, Weddle's Rule,
13-17
Specified Points
17-19
-
-
24-25.
Volumes
26.
Mechanical Integration.
etc.
-
19-20
-
21-22
of Revolution
22-25
General Review
26-28
-
PROBLEMS
28-39
CHAPTER
II.
STANDARD FORMS.
27-28.
Reversal of Differentiation
29-32.
Nomenclature.
40
-
Constant
of
Integration.
Inverse
41-42
Notation
by D~
33-35.
Laws
36-38.
n
1
Integration frfV, ar (ax+b)
39-42.
Forms
43-45.
TABLE OF RESULTS
-
46.
GENERAL REMARKS
-
satisfied
l
.
Integration cf Series.
Geo43-47
metrical Illustrations
,
<p
,
(a.r
(*)/(*+?>), <p'(x)fo(x), (yx)
+ 6)-
47-48
1
n
<p'(x),
F'(yx)y'(x)
49-50
52-53
54-56
56-66
PROBLEMS
ix
CONTENTS.
CHAPTER
III.
CHANGE OF THE INDEPENDENT VARIABLE.
PAGES
ARTS.
47-51.
Mode
52-54.
Case of a Multiple- Valued Function
55-58.
Purpose and Choice of a Substitution
59-68.
The Hyperbolic Functions, Direct and
of Effecting a
of the Limits
of Variable.
Change
Alteration
67-69
-
69-71
71-74
-
Pro-
Inverse.
76-84
perties
69.
The Gudermannian and
70-73.
As to Tables
77-79.
80-84.
Integration of cosec x,
1
(a cos x+ b sin a;)-
Integration of (a
cosec 3 *
2
a;,
cosecha:,
sechz,
T
2
-
-z T *,
(z
)
= ax 2 +2bx+c)
f^L,
*R (R
85-87
-
sec
2
;
84-85
Gudermannian, the Hyper-
bolic Functions, etc.
74-76.
Inverse
its
of the Inverse
-
-
88-89
-
2
+a
2
)
*,
(x
2
-a
various forms;
Fi
)
3
sec *,
,
U~Rdx
-
89-91
-
91-94
-
95
J
86-87.
jx(a-x)
=== = 2 cosh-
/
J
1
~ and other forms
-
Jx(x-a)
between the Integrand and the Integral
88.
Visible Relation
89.
ADDITIONAL LIST OF STANDARD RESULTS
-
95-96
96-97
-
PROBLEMS
99-104
CHAPTER
INTEGRATION BY PARTS.
90-93.
Integration
94-96.
Rule
97.
Forms
for
by
Parts.
IV.
POWERS OF SINES AND
Repeated Operation of Integration by Parts
e ax sin bx sin ex sin dx,
e ax sin p x cos* x,
e ax g j np
COSINES.
The Method and Rule
x cog nx e ^ c
.
105-107
-
108-109
-
1
-
1 1 1
^
99.
Integration of an Inverse Function
100.
Geometrical Consideration of Integration by Parts
101.
General Idea of a Reduction Formula
-
102.
m
m
Integration of x sin nx, x cos
-
nx
-
10
111-112
113
-
-
113-114
CONTENTS.
XI
PAGES
ARTS.
103.
n ax sin
n ax cos bx
bx, x e
Integration of x e
104-105.
Integration of
e ax cos
Integration of
xm (log x) n
106-111.
n
ax
bx, e
sm n
-
-
bx-
115-117
117-119
112-113.
A
114-126.
Powers and Products of Powers of Sines and Cosines,
with or without an Exponential Factor
Trigonometrical Process.
115
119-121
Multiple Angles
-
PROBLEMS
121-131
131-137
CHAPTER
V.
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
127-129.
Forms
-=-
~,
a z -x 2
.,,
x 2 -a 2
a 2 +x 2
138-139
130- 1 35.
136-138.
139-141.
142-143.
144-146.
147.
148-149.
150.
151.
152-154.
155-156.
157-159.
160-165.
166-167.
168-169.
Integration f
~
,
(
R=
various cases and
CONTENTS.
xii
CHAPTER
INTEGRALS OF FORM
VI.
r
f
J (a-\-b cos
^-
x+ c sin x) n-,
etc.
PAGES
ARTS.
170-179.
180-181.
Forms
6cosz
Ja +-ff
Forms
182-184.
185-187.
J!*.
Ja+osina;
f
da
--
.-,
f
f
,
J
a + ^_%ocosz+csmz
170-176
cfo
/'
/Ja+osmha;
da;
a+6
+ 6 cosh
J
,
i
a: a;
-f
176-178
x
c sinh
Forms expressed
Integration of the above
the Integrand
in
Terms
of
Reduction Formulae for
-
I
J(a+bcosx)
nr,
IT-
-r-
J(a+bsmx) n
fc
-
188-189.
Corresponding Reduction
Functions -
Formulae
190-193.
Integration of Fractions of
Forms
for
179-181
,
-
182-185
Hyperbolic
-
185-186
a+b cos 0+csin 8
a+6cos #+csin#
'
i
[ttj
__
+ 6j cos + c
x
sin ^) n
'
or the corresponding Hyper-
9(0080, sinfl)
bolic Functional
II(a r +6 r cos (9+c r sin 0)
Forms
187-188
-
i
194-195.
A
196-198.
IMPORTANT PARTICULAR CASES
199.
Cases required in Planetary Motion
200.
Illustrations
201-202.
The Forms f--^?-^- dx,ja+ocosa;
Method of Reduction avoiding the ordinary
189-191
Reduction Formula
different
203-204.
The
n
b
+ bcosx)
Forms
-
The Form
dx
206-207.
193
*?*** ..dx,
f,
2
J(a+ocosa:)
.......
*
leading
W*
J
191-192
-
194
to
I
mP
?
C
and
-~dx,
n
9
$n M
(a+bcos8)
HERMITE'S Integration
195-196
the
J (a-f6cosa:)
Reduction Formula for such Integrals
205.
-
-
si
.
[.
J (a
-
of
-
.....
[f^J^^dd, etc.
-
196-198
198
198-199
J Usin(d-o. n )
PROBLEMS
........
200-207
CONTENTS.
CHAPTER
VII.
FURTHER REDUCTION FORMULAE.
PAGES
ARTS.
208-210.
Summary
Reduction Formulae already found, and
of
General Remarks
211-216.
Integration of
/
xm
~1
-
Xp dx, where X = a + bxn
Avoid-
.
ance of a Reduction Formula in Three Cases
217-219.
THE Six CONNECTIONS
POSSIBLE.
The Rule
224-228. Reduction Formulae for
ft
/
232-237.
sm n Vd6
sin''
/
ft
and
8 cos?
/
sin"
coa n
/
The
same Form
-
215-222
222-225
dS
6 cos?
6dO
-
Introduction of the
d&.
of
213-215
220-223. Special Cases and Cases Reducible to the
229-231.
209-213
"
"
SmaUer Index + 1
208-209
Gamma Function
225-^29
229-233
-o
238-239.
Cases of
fx
m Xdx,
240-248. Reduction of /
249-257.
I
258-260.
X = a + bx+cx
2
233-235
-
= fx n (a + bx* + cx*)~ i dx
235-242
cos px cos n qx dx, etc.
p
242-250
-
-
250-253
xdx,
etc.
261-262.
fcospx/coBqxdx,
etc.
253-257
263.
n
etc.
jcos px/cosxdx,
257-258
lcosnxlcoa
n
264.
jcos
265-268.
[sin"
269-270.
/
J
271-274.
.
x/tfdx, etc.,
^
COS" X
f.'
pxlcosqxdx,
_j
_
fcc,
g2
"
etc.
l^>~dx,
-
<fa?
.
2
s/(l -a: )(l
PROBLEMS
258-259
etc.
f
-*V)
_
etc.
260-261
-
.....
261-263
___
263-266
2
2
7
7(l+aa:
)V(l -* )(1
.....
.
-*V)
-
266-274
CONTENTS.
xiv
CHAPTER
FORM
JF(x,
jR)dx,
VIII.
WHERE R
IS
QUADRATIC.
PAGES
ARTS.
275.
The Types
dx
j=
7
I
;
General Remarks
-
-
275
-
275-280
-^v
291-317.
X Linear, Y Linear
X Quadratic, 7 Linear
Case II.
X Linear, .7 Quadratic
Case III.
X Quadratic, 7 Quadratic
Case IV.
292-295.
Case IV.
296-300.
Reduction to Canonical Form
301-302.
Graphs
of the Transformation
303.
The Integration after Reduction to Canonical Form
The Integration without a Preliminary Reduction
276-282.
283-286.
287-290.
304.
305.
Case
I.
:
280-283
:
283-286
:
287-313
:
:
Preliminary Remarks
Comparison
rable
287-288
288-291
Formula
of the Processes.
291-294
294-295
-
295-297
Construction of Integ-
Forms
298
306-307. Various Forms of the Coefficients
299
308.
Connection between the Quadratics involved
309-311.
The Case
312.
313.
Illustrative Examples
Forms Reducible to Case IV.
314-316.
7
317.
Generalisation
318.
General Resume of the Position
aj/da -fcj/62
-
-
a Reciprocal Quartic,
299
300-304
-
304-308
308-309
etc.
-
-
309-312
-
312-313
-
-
PROBLEMS
313-314
314-323
CHAPTER
IX.
GENERAL THEOREMS.
319-320. Various Limiting
Forms expressed
as Definite Integrals
321-336. General Propositions and Geometrical Illustrations
337-339. ABEL'S
Theorem
340-342. BONNET'S
in Inequalities
-
-
Theorem
343-352.
General and Principal Values.
353.
Successive Integration
PROBLEMS
324-326
326-336
336-338
339-340
CAUCHY
....
341-351
-
351-353
.
353-360
CONTENTS
CHAPTER
XV
X.
DIFFERENTIATION OF A DEFINITE INTEGRAL
WITH REGARD TO A PARAMETER.
PAGES
ARTS.
354-355.
Differentiation with regard to a Parameter
356.
Geometrical Meaning of the Process
357.
The Case
-
an Indefinite Integral
of
-
361-362
-
362-364
-
364-365
358-359. Integration with regard to a Parameter
365-366
360.
Notation for a Double Integration
-
366
361.
Geometrical Interpretation
-
367-368
-
362-363. Successive Differentiation with regard to a Parameter
364.
Differentiation of a Multiple Integral
365.
Remainder
366.
Remainder after n + l terms
LAGRANGE'S Theorems -
n terms
after
PROBLEMS
-
-
371-372
TAYLOR'S Theorem
in
in
368-371
-
-
-
...
372-373
and
LAPLACE'S
373-375
375-382
-
CHAPTER XL
PRELIMINARY TO INTEGRATION OF M/NVQ, WHERE Q
QUARTIC. DEFINITIONS OF ELLIPTIC FUNCTIONS.
367-370.
Preliminary Considerations
-
371-374. LEGENDRE'S Three Standard
375-376.
Complete Values.
377-379.
Notation.
Forms
Integration
-
380-381. Elementary Transformations
382-383.
General Remarks
384.
The Complementary Modulus
385.
Transformations
386.
Inverse Notation
387-388. Illustrative
Examples
389-390.
The Pendulum
391.
LEGENDRE'S Formulae
PROBLEMS
A
383-385
386-387
Real Periodicity
Differentiation.
IS
-
387-388
-
389-390
-
390
-
390-391
-
391
-
392
391
-
-
392-395
-
395-398
.
-
399
400-402
CONTENTS.
xvi
CHAPTER
XII.
QUADRATURE
(I).
PLANE SURFACES. CARTESIANS AND POLARS.
PAGES
ARTS.
393-395.
Formula
Quadrature for Cartesians
for
396-397. Coordinates expressed in
Terms
-
Parameter
of a
404-405
-
398-401. Line Integral round a Contour
402.
A
403.
Illustrative
404-406.
On
405-408
409
Precaution
Examples
409-412
a certain Type of Problem
412-418
418-420
407-408. Polar Coordinates
420-423
409-411. Line Integrals
412-413.
Formula with x and tan 8
414.
A
different
for Coordinates
Interpretation
the
of
423-424
-
Area
Cartesian
424
Formula
415.
424-429
Illustrations
PROBLEMS
-
-
CHAPTER
s)
and
(p,
^) Formulae
437.
438-441
-
E volute
-
-
-
441-447
-
447-449
-
449-452
Polar Subtangent
453-454
438-442. Intrinsic Equations
...
443.
Inverse Curves
444-450.
Origins of Pedals of a given Area
PROBLEMS
429-437
(II).
-
422-430. Pedal Curves and Pedal Equations
431-432. Area between Curve and
"
"
433-436. Area swept by a
Tail
-
XIII.
QUADRATURE
416-421. (p,
403-404
QUADRATURE
...
457-465
466-472
(III).
Surface Integrals (Cartesians)
454.
Centroid of a Plane Area (Cartesians)
455.
Moment
-
457
XIV.
451-453.
of Inertia
454-457
.
..'...
-
CHAPTER
-
-
-
473-476
476-477
477-479
CONTENTS.
xvn
PAGES
ARTS.
456-457. Surface Integrals (Polars)
479-481
-
481-485
458-459. Centroids (Polars)
460-462. Trilinears and Areals
485-489
-
463-465. Corresponding Points and Areas
PROBLEMS
490-492
-
492-495
-
CHAPTER XV.
QUADRATURE
(IV).
MISCELLANEOUS THEOREMS.
466-472. STOKES'S
Theorem
-
Motion of a Rod in a Plane
480.
LEUDESDORF'S Theorem
A
-
496-501
501-503
504-505
Lamina
481-491. Motion of a Plane
492-504.
...
t
473-479.
in a
Plane
-
-
.....
505-510
General Theorem on the Motion of the Centroid of a
System
of
Moving Points
511-515
505 509. Planimeters
515-521
PROBLEMS
521-523
CHAPTER XVI
RECTIFICATION
(I).
510-515.
The Working Formulae
524-527
516.
NEIL'S Problem
527
-
517.
The Parabola
518.
WREN'S Problem
519.
Centroid of an Arc (Cartesians)
520-523.
Polar Formulae for Rectification
534-536
524-525.
Centroid of Arc.
537
526.
Moments and Products
527-529.
The Converse Problem
527-530
530-532
532-534
-
Polars
of Inertia
537-539
539-541
530-533.
LEGENDRE'S FORMULAE
542-545
534-535.
Arc
545-547
536-559.
Intrinsic
560.
CORNU'S Spiral
561-565.
Arcs of Pedals
of
an Evolute
547-566
Equations
PROBLEMS
566-567
68-570
-
-
-
-
570-576
CONTENTS.
XV111
CHAPTER
XVII.
RECTIFICATION
(II).
APPLICATION OF ELLIPTIC FUNCTIONS.
PAGES
ARTS.
566.
Scope of the Chapter
-
-
577
-
-
581-586
567-574.
The
575-581.
FAGNANO'S THEOREM
582.
Locus of Pointer which pulls tight an Inextensible
String which passes round an Oval
577-581
Ellipse
583-587. Theorems of
GRAVES and MACCULLAGH
587-588
588-592
592-593.
The Hyperbola
The Lemniscate
594-595.
TheLimagon
596.
Trochoidal Curves
-
597-602.
The Cassinian Ovals
-
603-608.
The Elastica
609.
Cotes's Spirals
610-613.
Bi-Polar Curves.
614.
Bi-Angular Coordinates
-
626-628
615-616.
GENNOCHI'S Theorem,
etc.
-
629-630
617-620.
A
-
588-591.
592-597
....
597-600
-
-
601-602
602-604
604-614
-
614-621
or Lintearia
622-623
-
623-626
Elliptic Coordinates
General Theorem
-
PROBLEMS
630-636
636-642
CHAPTER
XVIII.
RECTIFICATION
(III).
MISCELLANEOUS THEOREMS.
621-623. Arc of an Inverse Curve
624-632.
JOHN BERNOULLI'S
643-647
-
THEOREM.
An
Extension
633.
Areals and Trilinears
and
647-657
Application
-
634-635. Unicursal Curves
657-658
658-660
636-637.
Connexion between Quadrature and Rectification
661-662
638-641.
A
662-664
642-645.
Mr. R. A. ROBERTS'S
Class of Rectifiable Curves
646-648. SERRET'S
Mode
PROBLEMS
THEOREM
of Derivation of Rectifiable Curves
665-667
667-669
669-674
CONTENTS.
XIX
CHAPTER XIX.
MOVING CURVES.
PAGES
ARTS.
649-654.
The Instantaneous Centre and
655.
General Motion of a Lamina reduced to a Case of
656-659.
The Two Loci
660.
Difference of the Curvatures.
its
Loci
-
-
678-679
Rolling
of /
679-685
661-662. Difference of the Curvatures.
Analytical
BESANT'S Equations for the Fixed /-Locus
664-665.
Roulettes and Glisettes
672-673. STEINER'S and BESANT'S
-
Glisettes
688-690. y,
i
685-687
687
687-688
-
688-690
690-695
r
THEOREMS
by Normal
678-685. General Theorems
674-677. Area swept out
-
-
Area swept out by
666-671. Arc of a Roulette.
-
-
Geometrical
663.
686-687.
675-678
695-697
to a Roulette
697-699
699-703
703-704
-
705-709
Relations
691-700.
Curves on a Lamina touching Fixed Lines
701-705.
Isoperimetric Companionship of Curves
709-717
-
PROBLEMS
-
-
717-723
-
723-731
CHAPTER XX.
RECTIFICATION OF TWISTED CURVES.
706-711. General Formula.
712-713.
The Helix-
714-715.
A
732-737
Cartesians
737-738
-
738-739
Property of Geodesies
716-719. Cylindrical Coordinates.
Curves on a Cylinder
720.
General Polar Formulae
721.
Modifications for Sphere, Cylinder
722.
Rhumb
Lines
-
739-741
-
-
and Cone
743-744
-
....
723-724. p, r Formulae
-
Inversion
727.
Stenographic Projection
728-730.
Curves on Spherical Surfaces,
731.
To
732.
The Polar Curve
733-735.
Theorem
find sin
-
p
of
SCHULZ
745-747
-
725-726.
741-742
742
747-749
749-751
p, r
and p, ^ Formulae
751-754
754-755
-
755-756
756-763
CONTENTS.
XX
PAGES
ARTS.
737.
The Sphero-Conic. Quadrature and
BURSTALL'S Theorem -
738.
Artifices for the Construction of Rectifiable
V36.
763-765
Rectification
-
765-767
Twisted
Curves
-
739-744. Generalised Formulae
767-769
770-772
PROBLEMS
772-774
CHAPTER XXI.
VOLUMES OF REVOLUTION,
745-747.
Volumes
of Revolution
ETC.
-
-
748-749. Surfaces of Revolution
-
775-777
-
777-779
779-780
750.
Centroids
751.
Illustrative
752-759.
GULDIN'S Theorems
-
780-782
Examples
783-790
-
PROBLEMS
-
-
790-794
-
795-801
CHAPTER XXII.
SURFACES AND VOLUMES IN GENERAL.
760-764. Volumes.
765-771. Mass,
Moment, Centroid,
772-773. Surface.
774.
Cartesians-
Cartesians
805-808
-
808-809
Cylindrical Coordinates
775-776. Spherical-Polar Element of
777-779. Elements. of Surface.
780-781.
801-805
etc.
Areas on a Sphere
;
Volume
809-810
-
Cylindricals
Spherical Triangle
....
782-788. Solid of Revolution
789-791. Orthogonal Coordinates
792-793.
Plane Area.
-
Change
Volume Elements. Changs
800-803. Connection between 8V and
804.
Tetrahedral Coordinates
of the Variable
88, etc.
-
805-808. Revolution of a Twisted Curve
807-809.
Annular Element
810-811.
Generalised Coordinates
812-820.
Elliptic Coordinates
821-824.
Solid Angles.
825.
Illustrative
PROBLEMS
ANSWERS
of Surface.
-
The
-
827-832
832-835
-
836-839
835-836
Ellipsoid
-
839-843
843-845
845-850
-
GAUSS'S Theorems
Examples
818-825
-
-
-
815-818
825-827
of the Variable
794-799.
810-814
814-815
-
-
-
850-854
854-862
862-871
872-907
ABBREVIATIONS USED IN THE REFERENCES.
Ox.
Math. Trip.
Ox.
= First
or Second Public Examination,
Oxford University.
P. or Ox. II. P., etc.
I.
M.
J.
or Math. Trip. II. -Mathematical
I.
Tripos
Examination,
Cambridge University, Parts I. or II.
= Oxford, Junior Mathematical Scholar-
S.
ship.
Colleges a, etc.
To
indicate the sources from which
in cases
in
many
of the
Examples are derived
where a group of Cambridge Colleges have held an examination
common, the
references are abbreviated as follows
:
(a) =St. Peter's, Pembroke, Corpus Christi, Queen's and St. Catharine's.
(/5)
= Clare,
= Jesus,
Caius, Trinity Hall
and King's.
Emmanuel and Sidney Sussex.
= Jesus, Christ's, Emmanuel and Sidney Sussex.
() = Clare, Caius and King's.
I. C. S. = Examination for the Indian Civil Service and *Home Office
(y)
Christ's,
Magdalen,
(8)
ships,
(R. P.)
Grade
=Set
in problem paper to his classes by the late Dr. Routh, possibly
taken from examination papers or possibly original. Source
unknown
L.
E. F.
Clerk-
I.
-London
= Elliptic
C.I.=Calcul
References to
to the present author.
University Examinations.
Functions.
Integral.
Diff.
Cole,
the
are to
Differential Calculus.
3
author's larger
Treatise
on
the
CHAPTER
I.
NATURE OF THE PROBLEM. PRELIMINARY
CONSIDERATIONS.
INTEGRATION
is a reversal of the
operation of Differenof
the
a
function
of
x
when the differential
tiation,
finding
Thus the differential coefficient of x2 ex
coefficient is known.
1.
,
2
We
require a method of retracing our
T
x
steps, and having given the expression (2x + a?)e w e aim at
the formulation of a method of arriving at the original function
(2x+x
say, is
)e*.
,
x 2ex
The
result of integrating a function of x
of
the function.
integral
.
2.
is
called the
In the language of the early writers on the subject, a
was called a "fluxion." The original
differential coefficient
expression regarded as derived from the differential coefficient
was called the " fluent."
Thus, in Kinetics, if 8 be the space described by a particle
moving with a uniform acceleration / in time t, and with initial
= ut + ^ft 2 and the velocity at any time is given
=
by v u+ft. We obtain, by differentiating these expressions,
velocity u, s
,
dv
So
/ is
,
ds
the differential coefficient (or " fluxion ") of v with
regard to
u+ft
is
t,
the differential coefficient (or "fluxion") of s
with regard to
/.
Regarding u+ft and/ as the original quantities, their integrals
with regard to t (i.e. their "fluents") are respectively ut + %ft 2
i.e. s, and
ft -f- u, i.e. v.
,
B.I.C.
A
CHAPTER
2
3.
It will be noted that,
I.
as a constant quantity has
no
"rate of variation," all unattached constants, i.e. constants
which do not multiply variables, as for instance u in the
formula
v
= u+ft,
disappear on differentiation.
therefore expect constants to reappear
Thus
upon
We may
integration.
appears that the differential coefficient with regard
to time (or "fluxion") of a length, or distance, is a velocity or
rate of change of the length. The integral witli regard to time
it
"
In other words, the
fluent ") of a velocity is a length.
problem of the Differential 'Jalculus is, given any quantity
which is changing its value continuously, to find the rate of
that change whilst the problem to be attacked in the Integral
Calculus is the converse, viz., given the rate of change, to find
what the nature of the varying quantity must be.
(or
;
4. The general
Newton remarked
take notice,
of
character
of
integration
is
tentative.
necessarily
Method of Fluxions, " It may not be amiss to
that in the Science of Computation all the Operations are
two kinds,
in his
either Compositive or Resolutative.
The Compositive or
Synthetic Operations proceed necessarily and directly, in computing
Such
their several quaesita, and not tentatively or by way of tryal.
are Addition, Multiplication, Raising of Powers, and taking of Fluxions.
But the Resolutative or Analytical Operations, as Subtraction,
Extraction of Roots, and finding of Fluents, are forced
to
Division,
proceed
and tentatively, by long deduction, to arrive at their several
and suppose or require the contrary Synthetic Operations, to
quaesita
prove and compare every step of the process. The Compositive Operawhen
tions, always when' the data are finite and terminated, and often
indirectly
;
they are interminate or infinite, will produce finite conclusions whereas,
very often in the Resolutative Operations, tho' the data are in finite
Terms, yet the quaesita cannot be obtain'd without an infinite Series
;
of Terms."
5. We have illustrated the object of integration from the
fundamental equations of motion of a particle moving with
a constant acceleration and with a given initial velocity.
This
is
sufficient for the present.
But
it
will be seen later
that the reversal of the operation of differentiation will also
enable us to calculate with precision the areas bounded by
lines, the lengths of such curved lines, the volumes
contained by curved surfaces, the areas of such surfaces and
many other quantities which it is necessary to find in both
curved
Pure and Applied Mathematics.
NATURE OF THE PROBLEM.
6.
Before embarking upon
3
the general problem
of
the
reversal of a differential operation, it will be instructive to
the student to consider how such a reversal could be used
in such a problem as the discovery of the area of a space
bounded by curved lines.
The plan adopted
for this purpose is to imagine the area
divided into a very large number of very small elements
have then
according to some fixed principle of division.
We
some method of obtaining the limit of the sum of
to devise
elements
these
all
small,
when each
is
ultimately infinitesimally
and at the same time their number
And when
increased.
is
indefinitely
once such a method of summation
is
found to be applicable also to many
other problems, such as those already mentioned of finding
the lengths of specified portions of curves, volumes bounded
discovered
will be
it
specific surfaces, the positions of centroids, etc.
by
In some elementary cases it will be found that the
requisite summation can be performed by ordinary algebraical
or trigonometrical means.
But such processes will be
7.
generally tedious and almost always inadequate to the treatment of any but the simplest examples.
A
ing
fundamental theorem
how
this
will,
however, be established show-
summation depends upon the reversal of a
We
shall therefore, after a few illustrations,
differentiation.
confine our attention for several chapters mainly to the purely
analytical problem of reversing the fundamental operation
of the Differential Calculus, with the end explained in view.
And when
weapon we
the student
which the process
8.
is
well equipped with this powerful
more fully the uses to
shall proceed to discuss
may
To avoid constant
out the book
be applied:
repetition,
we may
state that through-
coordinate axes will be supposed rectangular,
all angles will be
supposed measured in circular measure, all
will
be
logarithms
supposed Napierian except where otherwise
all
expressly stated, and for the present all variables will be
supposed real and all functions will be considered continuous
functions of a real variable.
CHAPTER
4
9.
I.
NEWTON'S SECOND LEMMA.
In the First Section of the Principia
(Lemma
II.),
Newton
Theorem *
enunciates and proves the following
:
If in any figure Aab...kL boimded by the straight lines
Aa, AL and the curve abc...kL any number of parallelograms Ab, Be, Cd, etc., be inscribed upon equal bases AB, BG,
CD, etc., and having sides Bb, Gc, Dd, etc., parallel to the side
Aa
of the figure,
and the parallelograms aPbp,
bQcq, cRdr,
etc.,
completed then, if the breadth of these parallelograms be
diminished and the number increased indefinitely, the
be
;
ultimate ratios ichich the inscribed figure APbQcRdS ... kK,
the circumscribed figure Aapbqcrd
ykzL and the curvilinear
.
.
.
figure Aabcde ...hL have to one another are ratios of equality.
ABCDEFGHIJKLA
Fig. 1.
To prove
this statement it
may
he observed that the differ-
ence of the sums of the inscribed and circumscribed rectilineal
figures is the sum of the parallelograms Pp, Qq, Rr, ...,
as the bases Pb, Qc, ...
of these parallelograms are
,
KL
and
Kz and
;
all
equal
of their individual
their aggregate altitude is the sum
altitudes, the sum of these parallelograms
is equal to the
parallelogram Ap. And in the limit, when the bases
BC, ... are diminished indefinitely, the area of this parallelo-
AB
,
gram which has a finite
and indefinitely small breadth
anything conceivable, however small.
Hence the inscribed and circumscribed figures, and therefore
also the curvilinear figure whose area is intermediate between
the areas of these figures, in the limit become ultimately equal
becomes
less
altitude
than
*See Frost's Neivtoris Principia, pages
17,
18.
NATURE OF THE PROBLEM.
10. Newton devotes the next Lemma (III.) to proving that
"the same ultimate ratios are also ratios of equality when
the breadths of the parallelograms, AB, BC,
and are all diminished indefinitely."
This
proved in like manner, and
is
CD,
may
. . .
are unequal,
be established by
the student.
It follows that the limit of the
sum
of either the inscribed
parallelograms or of the parallelograms which make up the
circumscribed figure ultimately coincides in area with that
of the curvilinear figure itself.
11.
Analytical expression of the above result.
We
shall
now obtain an analytical expression for the sum of
such a system of inscribed parallelograms.
Suppose it be required to find the area of the portion of
space bounded by a given curve AB, whose Cartesian Equay = <f)(x), the ordinates
AL
and BM, and the axis of
the axes being rectangular, and all ordinates from A to E
being finite, and for the purposes of this article, increasing or
decreasing from A to B.
Following the method of Newton's Second Lemma, let LAI
tion
is
./',
n
be divided into
of length hj
ic.
OL = a,
and
OM=
L
/each
equal small parts LQ lt $%, Q2 ^3
let a and b be the abscissae of A and B,
Then
b.
Q,
Q. a
- a = nh.
b
Q 3 Q4
Fig. 2.
The ordinates LA,
points L,
QI} Q 2
,
...
,
Q^,
Qn _ M
Complete the rectangles
1}
AQ
L
Q2P 2)
etc.,
Qn _^
are respectively
,
2\Q 2
,
P Q^
2
....
at the
CHAPTER
6
Now
sum of
sought by the sum
the
these
of the
I.
n rectangles falls short of the area
n small figures AR^P^ P J2 2 P2 etc.
,
1
Let each of these be supposed to slide parallel to the #-axis
into a corresponding position upon the longest strip, say
i.e.
strip,
sum
Their
Pn-iQn-iMB'
is
less than the area of this
an infinitesimal of the first
is h and is ultimately an inand the length MB is supposed
then
in the limit less than
order, for the breadth
M
Qn _
finitesimal of the first order,
finite.
Hence the area required
therefore
n
the limit
is
sum
infinite) of the
of the
when h
n
is
zero (and
infinitesimal terms
of the first order,
sum may
This
be denoted by
+rh = b-h
g
b-h
^
or
4>{a+rh)h
a
a-}-rh=a,
where
$ or 2
denotes the
"
sum
"
between the limits indicated.
Regarding a-\-rh as a variable x, the infinitesimal increment
h may be written as Sx or dx. It is customary also upon
taking the limit to replace the symbol S by the more convenient sign
which is, as a matter of fact, merely only
j,
another way of writing the same letter, and the limit of the
above summation when h
written
is
diminished indefinitely
is
then
rb
</>(x)dx,
J a
and read as "the integral of <f>(x) with respect to x [or of
= a and x b"; or more shortly
<j)(x)dx] between the limits x
"the integral of <J)(x) from a to 6."
6 is called the
a
is
upper or
"
"
called the lower or
The sum
12.
"
"
of
(?i
+ l)
"
"
superior
"
inferior
terms of the same
"
limit,
limit.
series, viz.,
h<f>(a)
differs
term
from the above
h<j>(a
+ nli),
i.e.
merely in the addition of the
which being an infinitesimal of
series
h<j>(b),
NATURE OF THE PROBLEM.
the
first
order vanishes
limit of this series
may
when
the limit
is
taken.
7
Hence the
also be written
"b
In the same way, if in fig. 2, Art. 11, LQ lt Q^^ QzQs*
Qn^M are not necessarily equal, but are respectively h v h 2
//
h n the ordinates at the several points L, Q lt Q 2 ... Q n _i
...
3
13.
,
,
,
,
,
,
}
are respectively,
and the sum
of the inscribed rectangles
is
and the sum of the residuary areas AR^P^ P^R 2 P2 P2 R 3 P3
etc., is less than the area of a rectangle whose breadth is the
gn-atesfc of the quantities h lt h 2 A 3 ... h n and whose height is
the greatest ordinate of the given curve and as in the last
article, this sum therefore vanishes in the limit when h lt h 2
A 3 ... h n are each made infinitesimally small, provided that
the curve has no infinite ordinate either at A, B or between A
and B.
Hence the limit of
,
,
,
,
;
,
,
is
also the area of the portion
LABM described in Art.
11.
//
p h 2 k s ...h n may clearly be either
or
independent,
equal, or connected by an}^ arbitrary law,
that
provided only
they each and all become infinitesimally
14.
Tlie quantities
small in the limit
when
,
their
,
number
is
increased indefinitely.
These arbitrary infinitesimals will be chosen equal to each
other in general, and the series to be summed will therefore
be that of Art. 11.
15.
We
how
postpone
till
later in the chapter the explanation
is connected with the reversal of a
summation
and illustrate what has been stated as to the
finding of areas by a iVw elementary cases in which the limit
of the summation may be found by elementary processes without undue difficulty.
of
this
differentiation,
CHAPTER
8
16.
ILLUSTRATIVE EXAMPLES.
Ex.
1.
To
calculate
/
ce
mx
dx, that
II.
is
to find the area of the space
bounded by the .r-axis, the logarithmic curve y = ee mx and two ordinates
x=a and x b.
Here we have to evaluate
Lth=0 ck ma + ew <+*> + e rn(a+sh) -f
+ (+ J^i)
. , .
[e
where
b
]
= a-\-nh.
n
nmh
_
This expression
= Lt h ^che ma e-
c
.'.
of the
the area sought
1
() - 1],
.
by Dif.
Cat. (Art. 21),
equal to the rectangle contained by
line) and the difference of the initial and
dimension of a
E.g.
is
.
=1
now
if
inch and a = 0, b =1,
c
- (which
is
final ordinates.
= 2,
the area in question = 2(e - 1) = 2 x 1*71828... square inches
= 3'43656...
i.e.
Ex.
Shew
2.
a
little less
square inches,
than 3^ square inches.
that in the last result,
i.e.
y=ce
mx
,
if
A lt A Zj A 3
..
,
areas between
#=0
then ^j,
Ex.
3.
.4 2
,
and x = \,
^4 3
,
...
.r
form a
1
G.P.
and x
Calculate the area bounded
the #-axis and two ordinates
O A
2,
#=2
whose common
x=a
is e
by the curve of
and x=-b
N M
3.
etc.,
m
.
sines
(0<a<6<
V
m)
).
B
Fig.
and # = 3,
ratio
y=
be the
NATURE OF THE PROBLEM.
-6
Here we are
to evaluate
csin
/
mxdx,
Ja
that
Lt^ oG'A[sin
is
where
ma + sin m(ct-\- A) + sin wi( + 2A)
n/t = b
a.
f
sin
This
ovpr-oc.
^
ma + (n - 1 ) mA)c
,
,
Tt^ ^1,
.
.
sin
*)
^|
a ;r>
n
...
to
n terms]
mh
-^
T
sin
mA
cos
ma
c
Thus,
if
cos
mb
m
the limits are such as to take in one half wave length, i.e. the
x=0 to mx = rr, and if c=l inch, the area
portion above the .^-axis from
SOUght
is
COSO-COS7T
=
m
or
if,
say,
Ex.
4.
m=^
the area
Find the value
is
20 square inches.
C
of
b
3?
/
Ja
2
C
*dx\
3
cubical parabola c
the o;-axis
,
Here we have to evaluate
y=^
2
m*
that
is
the area bounded by the
and two ordinates x = a and x=b.
where nh = b-a.
NOW
and when n becomes
infinite this
Ex.
5.
We
have to evaluate
Fiod
becomes
/*
i <&;.
'
(a + 2A)*
+
'
'
'
+ 6J
'
10
CHAPTER
.
I.
and
_
a+h
a
and when h diminishes without
-=-
Thus the value
,
expressions,
Ex.
6.
/
entrapped between two ultimately equal
is
11
,
^dx
=-
,
b
have, to consider
-- =A and
In the
1
becomes
m
Integration of X from the definition, between limits a and
Here we
where
and
/*
limit, each of these expressions
?i
is
indefinitely large,
??i
+l
not being zero.
Differential Calculus for Beginners (Art. 13) it is
proved without
Theorem [which was purposely avoided, as it was
n
apply Taylor's Theorem to the expansion of (x+h)
the aid of the Binomial
then proposed later to
~\
tnat
zm+i
k=i
g
we have
^A=O
-
_1
1
=
= l+-i
z
Writing
_
'
.
= m + 1,
In this result put?/ successively a, a + A, a + 2A,
Lt h =
. . .
,
a + (w -
1)A,
and we get
NATURE OF THE PROBLEM.
new
or, adding numerators for a
denominator,
nuriierator
11
and denominators
new
for a
Lt
i.e.
i.e.
Lt h = Q h[a m + (a + /i) m + (a + 2h) m +...+(a+'^'lh)
m
]
=
in accordance with the notation of Art. 11,
rb
/
xm dx=
m+l_m+l
;
The
letters
a and
b
may
m
provided x does not become
When
represent any finite quantities whatever.
between x = a and x = b.
oo
taken exceedingly small and ultimately zero it is necessary
in the proof to suppose h an infinitesimal of higher order, for it has been
a
is
assumed that
in the limit - is zero for all the values given to y.
fs
When 6 = 1 and
ct=0, the theorem ultimately becomes
xm dx= -- ,
or
This result
may be
(w + 1) be
if
m+l
= oo
if
positive.
(ni+l) be negative.
written also
,
m+l
according as
is
The Limit
or,
which
tn
,
,
Lt n
^ n\-\nj
^fiV + f-Y'V
...+{-Y*l;
\n/ J
\nj
The
~
oo
case
,
according as
nm+l
-
from the former by
or
.
the same thing,
is
4- 1
oo
positive or negative.
Ltn=x
differs
or
,
m+l
m + l=0
when
i.e.
is
in the limit,
by
and
is
therefore also
positive or negative.
needs special consideration.
It
is
at once
derivable from the result
as a limiting form.
jm+i_ a
,-*
Ltm+l=Q I
->''"'
dx= Lt, ll+ i_o _
,
,
t
= log b - log a
'm +
l
(Diff. Gal.
Art. 21)
CHAPTER
12
I.
EXAMPLES.
1.
Find the values of
/
xdx and
^a
11
/
^a
x*dx, and interpret the results
geometrically.
2.
Find the area of the portion of the parabola x z = <iay cut
off
by the
latus rectum.
3.
Prove by summation that
rb
sinh#cfo;=cosh6-cosh a;
/
(a)
-'
1
=
/"*
/
(sin
m&
sin ma).
Ja
4.
axis
In a right circular cone of height h and semivertical angle a, the
divided into a large number, n, of equal portions, and planes are
is
drawn through the points
of division perpendicular to the axis, the
cone being thus divided into a large number of circular laminae. If x be
the distance from the vertex of any of these laminae, show that to the
first order of small quantities its volume may be written
7r#2 tan 2 a o>,
&x being the thickness of the lamina.
Find, by taking the limit of the summation of such quantities, the
volume
of the cone.
Show
also that the
where A and
5.
A
B are
quantity
y
volume
of a
frustum of thickness
the areas of the
is
two
T is
ends.
an unknown function of another quantity
When
x.
x has the values
5
is
y
8
10
12
14
16
3-2
3-8
5'0
6'5
found by observation to be
2-0
26
respectively, and the errors of observation
cent. ; draw the simplest continuous curve
its slope when .#=15.
also the value of x for
cannot be more than 5 per
which can represent y, and
estimate
Find
V
-
x
.
which the slope
Estimate the value of the definite integral
i
of the curve
/
Jii
17.
THE FUNDAMENTAL PKOPOSITION.
Let
<j>(x)
is
equal to
15
y dx.
be any function of a real variable
x, finite,
con-
tinuous and single valued, for all values of x from x = a to
x = b inclusive. Let a be less than b, each being finite, and
iVATURE OF THE PROBLEM.
a to be divided into
suppose the difference b
equal to h, so that b a
of tlie sum of the series
when
h
without
= yh.
It is
13
n
portions each
required to find the limit
diminished indefinitely, and therefore
a.
limit, keeping the product nk = b
is
n
increased
That this limit is finite may at once be made clear.
For if h(f>(a-\-rli), say, be the greatest term, the sum
which is finite, since by hypothesis
of x intermediate between b and a.
Let
for all values
be another function of x such that
\js(x)
differential coefficient,
We
<f>(x) is finite
i.e.
is
</>(x)
is its
such that
shall then prove that
7v^
/i[0(a)4-0
By
definition,
ami therefore
where a t
is
indefinitely
a quantity whose limit
thus
is
zero
when h diminishes
;
Similarly,
etc.,
(??
I)//}
=i/r(a + 7?./0
\/r{a
+ (7i
l)/
where the quantities a 2 a s ..., a n are all, like a 1? quantities
whose limits are zero when k diminishes indefinitely.
,
By
addition,
,
CHAPTER
14
I.
Let a be the greatest of the quantities a 1? a 2
that
+ an
A[a 1 + a 2 +...
Then
]
...
,
an
.
is
and therefore vanishes in the
Thus
The term
we
,
is
desire,
result,
it
-is
h</>(b)
limit.
itself also in
the limit zero;
be added to the left-hand
may
without affecting
it
fb
;
and
it
may
d>(x)dx = \js(b)
i.e.
\
Ja
hence,
member
if
of this
then be stated that
\/s(a),
where
The
result \fs(b)\Js(a)
is
frequently denoted by
MlFrom
function
appears that when the form of the
of which <j>(x) is the differential coefficient, is
this result
^fs(x),
it
summation
obtained, the process of algebraic or trigonometric
{6a
18.
The
(f>(x)dx
letters b
may
be avoided.
and a are supposed
We
denote finite quantities.
so as to let
I
shall
in the
<p(x)dx express the limit
Jn
infinitely large of \fs(b)
the notation
when
b
becomes
\ls(a), i.e.
</>(x)dx
{b
above work to
now extend
<f>(x)dx
= Lt b= A
we
<t>(x)dx.
shall be understood to
at
or
mean
NATURE OF THE PROBLEM
15
ILLUSTRATIVE EXAMPLES.
Taking the same examples as have been already considered otherwise
in Art. 16,
1.
ce
mx
is
the differential coefficient of
*
f cemx dx =
Therefore
the result obtained in Ex.
2.
csinmx
is
/
c sin
.'
the result of Ex.
'z
is
mx
.
- e ma
mb
^ (e
\
1, p. 8.
r
mx dx =\
L
b
c
-- cosmsc.
~\
m
=
~
c
m cosw.rJ a m (cosma
cos?6),
3, p. 9.
3?
3.
e
the differential coefficient of
rb
Therefore
m
X*
the differential coefficient of
5
4c*
c-
[bo?
'
Therefore
/
t
C2
Ja
the result of Ex. 4 of p.
-
b*-a*
dx
--
x
or
rb
Therefore
r
,
'
i
.
i~\b
- dx=\
i
i
=--r,
o'
L -~\
a,
arJa
9
Ja x-
the result of Ex. 5 of
.
9.
the differential coefficient of
4. -17 is
T
4c 2
p. 10.
Comparing these solutions with those
of the
same problems
of Art. 16,
the student will at once see the advantage derived from a use of the
fundamental proposition of Art. 17.
5.
-
is
CO
the differential coefficient of log x.
cb \
Therefore
- dx
/
''
Je
6.
+e~
x
is
= r log x ~i = log b
b
-e~
the differential coefficient of
Therefore
/
e
-x
^0
b
=
log a log
Q>
-la
L.
x
.
&
dx=Ltb ^ f- e-*~| = ( - er
a =0
L
-
00
)
-la
-e)= 1.
(
EXAMPLES.
1.
Write down the values
of
l
(1)
( xdx.
\afldx,
JQ
Jo
W
lafidx.
Jo
'
r-
(2)
I
.'0
/'-
sin.ro^r,
/
*
^
IT
4
/~
cxy&xdx,
.'0
I
sec 2 .rc?^,
Jo
^1^1^"' .Gib*' /.'i^^
and interpret each result geometrically
r4
/
sec
x tan x dx
.'0
'*
as the evaluation of an area.
:
CHAPTER
16
19.
I.
Geometrical Illustration of Proof.
The proof of the above theorem
may be interpreted
of Art. 17
geometrically thus
Let AB be a portion of a curve, of which the ordinate is finite
and continuous at all points between A and B, as also the
tangent of the angle which the tangent to the curve makes
with the x-axis.
:
Let the abscissae of
the ordinates
A
and
B be a and
b respectively.
Draw
NM
AN, BM.
Let the portion
be divided into
n equal parts, each of length h. Erect ordinates at each of these
points of division, cutting the curve in P, Q, R,
the successive tangents
lt PQ lf QR l} etc.,
. . .
AP
AP PQ QR
Z,
2
,
2 , etc.,
of the curve be
y
and
^(x), where
Now,
V/(a),
^'(a + h),
i//(a
tan^P^, imR QR
etc.,
+ 2fc),
z
lt
R RV
etc.
are respectively
PP
2
15
Q2 Q lt
clear that the algebraical
2
2
i.e.
Hence
= </>(x).
tanP2 4P15
P P, Q Q,
MB-NA,
is
\/s'(x)
<{>(a+2h),
the lengths
it is
and the
0(a-f ft),
h<f>(a), h<j>(a + h\
etc.
Draw
lines
parallel to the a-axis, and let the equation
Then
0(a),
are respectively
i.e.
,
2
sum
of
etc.,
etc.,
etc.,
NATURE OF THE PROBLEM.
17
Now, the portion between square brackets may be shown to
diminish indefinitely with IL For if R^R, for instance, be the
greatest of the several quantities P-f, Q^Q, etc., the sum
But
if
...]
is
the abscissa of
Q
<nR
l
R
t
be called x, then
LR =
2
^=
k\f,'(x),
7
2
= \[s(x) + h\/,'(x) + r& \//\x + Oh)
and
l_
(Di/. Cat. Art. 130),
so that
and
which
(b
(
an infinitesimal in general of the
is
first order.
Ltj^P^P, + Q 2 d 4- R.R, + ..'.) = ^ (b)
Thus
- \/,(a),
^^
Also, since Lt h =Jt,(/>(b)
20. Case of
= 0, we
have,
by
addition,
an Unknown Curve passing through a given system
of Points.
In a certain graph, such, for instance, as the graph on a temperature
chart, the temperature being noted at stated inter\ als, the following
table gives the corresponding abscissae and ordinates of eleven points on
7
the curve
.'
:
CHAPTER
18
The sum
I.
of the inscribed figures is
lx
[-879
+ '856+. + '639 + -600] = 7515.
..
clearly too large by the sum of the ten small triangular
shaped elements outside the area to be found.
The
first is
85
SO
75
70
60
9
op
FIG.
The second
is
too small
5.
by the sum
of
the ten
triangular-shaped
elements which are omitted.
The mean
of these
results,
viz.
7815 + -7515
=^
-7665,
will
be a much
be a little too small, because it omits the
very small areas which lie between the chords which join successive
points on the graph and the corresponding arcs.
closer approximation, but will
Hence, as a closer approximation, we
may
take
ra
/
Ji
udx=-1QQb square
units.
[From a finite number of ordinates it is impossible to assign the
equation to the curve, but it is customary to take the simplest algebraic
curve which satisfies the prescribed conditions. In the present case the
simplest curve to
fit
the data will be found to be
y= I
-
:
NATURE OF THE PROBLEM.
An
19
other curve of the form
^'
Io
- 2)
^
where <(.r) is any integral algebraic expression, would go through the
same points, but is much more complicated.
/'-'
on the supposition of the curve being
Tire true area
bv the
result of Art. 16, Ex.
shows errors as follows
In the
first
L
-^T,
5U_1}
= 1 - ---
~.
i.e.
3U
,
will be
or 7666...
,
found
which
:
estimate,
second
to be f.r
6,
?/
-
.,
mean
-
'0148 in excess,
'0152 in defect,
'0002 in defect,
i.e.
a 1*9
i.e.
a 2'0
i.e.
a 0'03
%
%
%
error in excess,
error in defect,
error in defect.]
SIMPSON'S RULE.
21.
If a curve be partially defined as passing through an odd
number of points whose abscissae are in arithmetical progression,
e.g.
(a,
and
if
the points
(a
yj,
+ h,
y 2 ),
(a
+ 2h,
the same assumptions be
yB )
...
made
(a
+ n=lh,
y
lt
),
as in the last article
as to continuity, etc., it is possible to find a very close approximation to the area of the curve, which is useful in many
practical cases, as follows
Consider first the case of the parabola
:
and
let a,
6,
c
be chosen so as to
O
make
<h,o)
Fig. 6.
Then
So that a = y t
,
b
=
whose equation
this curve
\
is
go through
CHAPTER
20
Now
the area bounded by the
ordinates y l and y 2
is,
by
I.
the parabola and the
ic-axis,
Art. 16, Ex.
6,
Q
**)
I
2c
l
3
we apply
If
arcs through
we have
etc.,
this rule to the case in question,
passing parabolic
the (1 st 2 nd 3 rd points), (3 rd 4 th 5 th ), (5 th 6 th 7 th ),
,
,
,
,
,
the following approximative rule,
+ 2/3
+ +4 +
+
2/3
2/4
,
viz.
'</5
2/5
+ 42/6 + 2/7
+ ...+2/- + 42/, _i
2
l
J
i.e.
o
(sum of first and
last -f- twice
+ four
This
it
sum of all other odd ordinates
times the
sum
of the
even ordinates).
known
as Simpson's Rule.
It will be noticed that
consists in the division of the area by an odd number of equiis
distant ordinates, and the substitution of parabolic arcs for
the actual but unknown arcs passing through consecutive
groups of 3 points.
Other approximations can be found. Thus we may take a
curve y = a-\-bx-{-cx2 -}-dxs to pass through 4 consecutive points,
or y
= a-{-bx-\-cx 2 -\-dxs +ex
points,
and so
on,
to pass through 5 consecutive
and thus build up similar rules. Simpson's
most cases gives a sufficiently close approxi4:
Rule, however, in
mation. for ordinary purposes.
(See
Examples
27, 28,
page
33.)
22. THE TRAPEZOIDAL RULE AND WEDDLE'S RULE.
The approximation previously adopted in Art. 20 of the
mean of the inscribed and circumscribed rectangles may be
expressed in similar manner, as
NATURE OF THE PROBLEM.
/'2/i+.?/2
;
/H
J
= jj-(sum
1
1
of first
all
which
2/3+2/4
2/2+2/3
^2~""f ~^~ + ~2~"f
the
and
,
'"
21
+ &"
,
last ordinates -f- twice the
sum^of
rest),
a convenient form, but not usually so accurate as
Simpson's Rule.
is
already explained, of substituting chords
consecutive
joining
points for their arcs, and as we are summing
a series of Trapezoids this is known as the Trapezoidal Rule.
It consists, as
Other Approximative Rules.
23.
Other rules will be found in Examples 27, 28 at the end of
this chapter, and in Examples 24. 25, 26, page 61.
A very convenient rule was given by Weddle, Math. Journal,
vol. ix., for the case where there are seven equidistant ordinates,
a ^ mutual distances h, viz.
2/i> 2/2' 2/3' -"'2/7
T TF%1 + 2/3 + 2/5 + 2/7 + 5 (2/2 + 2/4 + 2A>) + 2/J>
-f^x mutual distance [2 odds + 5 2 evens + middle].
3
i.e.
(Weddle'sRule.)
We
transcribe this for convenience, but the proof is one
It
most conveniently treated by finite difference methods.
found in Boole's Finite Differences, pages 47-48.
all applications of such approximate
"
formulae it is desirable to avoid extreme differences among
will be
Boole remarks that in
the ordinates."
Ex. Apply the Trapezoidal Rule, Simpson's Rule and Weddle's Rule to
bounded by the #-axis, the extreme ordinates and the arc of
find the area
a circle through the seven points
-
-1
-f
:
CHAPTER
22
For Trapezoidal Rule, Area =
I.
'86602 + 2'88562 + 1 '97228)
= ^(5-72592) = '95398.
Area = I18 (1 '73204 + 3'94456 + 11'54248)
= 1^(17-21908)
= 95661.
=
Area 2*0 (370432 + 14-428 10 + 1-00000)
= :95662
For Simpson's Rule,
For Weddle's Rule,
J-(
This area, being the area of that part of a semicircle whose centre is at
the origin and radius unity bounded by two ordinates ^r='5, #='5,
may be
seen to have
fore Simpson's
Rule
its
=
area correctly
/Q
+
= '956611...
,
and
there-
gives a result accurate to the last figure.
[See BOOLE, Finite Differences, p. 49.]
The approximation by Weddle's Rule does not appreciably
d.iffer
from
that by Simpson's Rule.
The Trapezoidal Rule
errs in defect
by
'00263,
i.e.
by about
'3
%
of the
whole.
DETERMINATION OF A VOLUME or REVOLUTION.
it be required to find the volume formed by the revoluof a given curve AB about an axis in its own plane which
24.
Let
tion
it
does not cut.
Fig. 7.
Taking
the. axis of revolution as the #-axis, the figure
be described exactly as in Art.
11.
may
The elementary rectangles
NATURE OF THE PROBLEM.
23
PiQz* P*Qv e tc., trace in their revolution circular discs of
equal thickness and of volumes TrAL^.LQ^ irP^Q^. Q^, etc.
The
formed by the revolution of the
several annular portions
AR^P^ P^^P^ P R P
portions
2
3
3,
etc.,
may
be considered to
a corresponding position upon
formed by the revolution
-that
the disc of greatest radius, say
Their
is less than this disc,
sum
of the figure Pn-^Qn-iNB.
slide parallel to the ic-axis into
i.e.
in the limit less than
an infinitesimal of the
first order,
for
according to the notation of Art. 11,
and is ultimately an infinitesimal of the first order, and the
radius
is, as in that article, supposed finite, as also all
the breadth
Qn -iN
is h,
NB
other ordinates of the curve from
Hence the volume required
therefore
n = co
)
of the
sum
is
as
it
may
to B.
the limit
when
7^
=
(and
of the series
+ 7r[0(a-h
or,
A
2
lh)] h,
be written,
Cb
2
fba
[(f>(x)]
dx
or
TT!
2
y dx,
Jo,
the equation of the curve being y = (f>(x) and the extreme
ordinates *x = a and x = 6, as in the article cited.
25.
ILLUSTRATIVE EXAMPLES.
The portion of the parabola y* = 4ax bounded by the line
Find the volume generated.
revolves about the axis.
Ex.
1.
x=c
Let the portion required be that formed by the revolution of the area
the axis, being bounded by the curve, the axis and an
A I'M about
ordinate MP.
(See Fig. 8.)
Dividing as in Art. % 24 into elementary circular laminae, we have
Vol.
=
re
r<-.
2
I
Jo
7ry
" o?.v=4a7r/
Jo
=%
=
1}
[Or,
if
.cr=4a7r
expressed as a
= 27rae- 2 (Art.
2>
cylinder of radius
vol. of
C2
PM and height
circumscribing cylinder.
series,
4?ra
= 27rac
>2
-5-
as before.]
A
If
16,
Ex. 6)
CHAPTER
2.
Find the area of the portion
the
ordinate through P.
double
being
Ex.
Area
I.
PAP'
of the
PAM=
same parabola,
'.."
>/4c = |.4 J/
=$
Area
:.
PP
of the circumscribing rectangle
.
MP
RPP'R'.
Fig. 8.
[Or we
may
ordinates.
Then
proceed thus
Let
Area
ON be
x=h z
3.
Divide
PA M= Lt
The portion
.
.
jj
n equal
portions,
and erect
x=- c.
A,
n
where /*=-,
=c'\/4arc, as before.]
+ y2 =
between ordinates x = h^
Find the volume of the frustum of the
of a circle a>2
rotates about the .r-axis.
into
c
the ordinate at
= 2^6'^
Ex.
:
2
sphere generated.
Let the portion required be that formed by the portion J\\P1 P2J.Y.2 of
the circle revolving about -V^V2 (Fig. 9).'
Here we are
to evaluate
(Art. 16, Ex. 6)
It
is
convenient for mensuration purposes to express this in terras of
the radii of the ends of the frustum and
its
thickness.
NATURE OF THE PROBLEM.
T be
Let
the thickness
/> 2
and
7^
Fig.
Then
Vol.
=
25
^-^i
y^
9.
- 27^ 2 - 2A 2 2 - 27j,A 2 )
1
+ 3^' +7*)
=
thickness x [3
of circular faces + circle
sum
on thickness as radius].
For the whole sphere
Cor.
EXAMPLES.
1.
Find the volume
of the
1.
a
formed by the revolution
of the prolate spheroid
ellipse ^72 + ^ = 1
about the
.r-axis.
o*
Find the mass of a rod whose density varies as the
the distance from one end.
I
p = J)'~ say, where
,
D and
c
m ih power
of
are constants.
J
Let a be ihe length of the rod,
GJ
the sectional area.
Divide as before into n equal elementary portions.
The volume of the (r+l) th element from the end of zero density
(o
a
n
.
Its density
J varies
from
D
-
c
m
/raY"
,
to
\
n J
D r(r+i)a~] m
-
c
m
n
I
therefore intermediate between
tn-L 1
c
m
m
"
1
(r-f
1)'
J
.
Its
mass
is
is
CHAPTER
26
and the mass of the rod lies between
m
m
m+1 m
D,,(oa l + 2 +L3 +...+(w-l)'"
--
and
when n
in the limit,
is
(ua TO+1
/)
assuming Art.
16,
Mass=
Ex.
f
a
/
Jo
3.
Find the position
[For the centroid
4.
;t?
and
increased indefinitely, becomes
c
[Or,
I.
m
'
m+
'
1
6,
n>v
m
D
(o
m+1
= m
D~ti>dx
m
c
m + =l
c
7
at once.
of the centroid of the rod in Question
= ^f-l, when m
is
2.
the mass of an element.]
Find the moment of inertia of the same rod about the lighter end.
[Moment
of Inertia
= ^wu?2.]
2
5. Find the area bounded
by the parabola 4y=.v the ordinates
and x = 4 and the .r-axis,
(1) by means of inscribed rectangles,
,
=2
circumscribed rectangles,
(2)
taking ordinates at distances
*1,
and compare the
by integration.
The sum of the inscribed rectangles
The sum
.t'
results
with that obtained
is
of the circumscribed rectangles is
The values
of these expressions are respectively (taking the squares
from Bottom ley's tables or summing otherwise),
4 '5 175, which is a little too small,
and
4'8175, which is a little too large.
Their
mean
is
4*6675.
The true value
6.
is
Plot the graph of
-
y=z
^
and mark on your figure the area repre-
ri
sented by the definite integral
dx
/
^.
Evaluate this integral by mensuration, and hence obtain an approxi-
mation for
26.
TT.
Note that
dx
1+^a2 ==-?-
tan" 1
.?:.
Mechanical Integration.
In a sense, any mechanical contrivance which performs
additions and registers the results is an Integrating machine
for
the particular class of function
to
which
it
may
be
NATURE OF THE PROBLEM.
27
Cash registers which record the day's takings, gas
water
meters, electric-light meters, all record the
meters,
amount passing into them. A slide rule adds up logarithms,
and thereby performs multiplications. Various forms of planiadapted.
meters add up the elements of area within a closed curve when
a pointer is made to trace the perimeter. The indicator of
a steam engine draws a work diagram and adds up work
elements, representing them by elements of area, from which
the Horse-Power of the engine may be deduced.
Such apparatus, however, though giving numerical
results
satisfactory for practical purposes, but subject to various errors
both instrumental and observational, fails to produce an exact
algebraical result,
and therefore
fails to satisfy
the mathe-
matician, however useful to the practical engineer.
shall have occasion later to return to the theory of
some apparatus of this kind. For the present it is sufficient
We
to
mention
its existence.
To sum up then
Integration,
I.
By
i.e.
;
we have
discussed
Four Methods
of
of finding
obtaining
By finding
which we obtain
II.
a function
*//(.<)
such that
^-
= 0(x),
from
III. By drawing the graph of y=(j>(x) and by some means
or other obtaining its area, by the Trapezoidal or Simpson's
or some other approximative rule, as, for instance, by drawing
on squared paper and counting all the squares within the
"
contour with a " give and take rule round the perimeter.
IV. By approximating to the
mechanical means.
It is obvious that III.
results,
the
contour by
and IV. can only give approximate
may approach a very high degree
though such results
of accuracy.
area of
CHAPTER
28
For exact
we have
results
Method
As has been
I.
to
Method
apply
I.
or
II.
leads to very difficult algebraic
seen,
or trigonometric summation, except in the very simplest cases.
Hence we are forced upon Method II. for exact general
This method
work.
and begin
to
we
I.
therefore shall in future rely upon
it in the next
develop the explanation of
chapter.
EXAMPLES.
If
1.
and
<'(0)
point be
a moving
the acceleration of
velocity be u
= <(0) = 0, show
/
that,
tne initial
</>"(0>
being the time from
a given epoch,
where
v
and
are respectively the velocity at time
s
and the space
t
described.
lOcosW and
the acceleration be
If
show that
=
10
the initial velocity be zero,
sn
0)
5
=6
--10
-jcos wr
or
where C
Show
a constant.
is
To what kind
that the "periodic time"
of
motion does
this refer
1
is
0>
If
2.
A
be the area bounded by a curve, the coordinate axes and
the ordinate at a given abscissa
A=
f*
y
\
x,
show that y =
What difference would it make
dx.
'
-',
,
and hence that
the measurement of
if
A
Jo
commences from a standard ordinate y whose
If
V
abscissa
be the volume of water in a pond, and
sectional area at a height x above the
-
bottom
A
is
XQ
t
the horizontal
of the pond,
show that
[A dx,
Jo
where h
3.
A
is
the depth of the pond.
large
number
of circular discs of the
,..
successive radii
2a
a
,
n
n
oa
,
n
same thickness - and
n
4
,
n
,
...
,
ci,
are threaded through their centres upon a straight wire
and
lie
with
NATURE OF THE PROBLEM.
29
Show that their total volume differs
their plane faces in contact.
of a cone of height h and with a for the radius of its base
from that
by the ultimately vanishing quantity
3n +
If
n= 1000,
show that the error
1
in taking this
of the cone is '1505 per cent, of the true
sum
as the
volume
volume.
4. Consider a sphere of diameter 2a to be divided into 2
thin
laminae of equal thickness by a series of parallel planes show that
;
the volume of the sphere
is
2Lt.
j>
rO
and that
this limit is
7ra 3
.
Obtain by a similar method the volume of the spheroid formed by
X2
the revolution of the ellipse
5.
Show by
the
2
+~=
'I/
1
round the axis
of length 2&.
method of summation that the volume of a parabounded by a plane at right angles to the axis
boloid of revolution
is
one half of the circumscribing cylinder.
Verify by consideration of the integral
4a
I
x dx.
i"
Jo
Draw on squared paper
(one inch squares divided into tenths
Divide one
convenient) a quadrant of a circle of radius 5 inches.
of the bounding radii into 10 half-inch divisions, and erect ordinates
6.
h
at each point.
Show
that the
Complete the inscribed and escribed rectangles.
sum
of the inscribed rectangles is 18'15 square inches
Also show that the mean of the inscribed and escribed
very nearly.
rectangles falls short of the true area of the quadrant
of a square inch.
5
by about
./'
7.
Rectangles of the same breadth and of areas
V^V
l/2cV
are set
up
Shew
V^Y
1
by side on bases in a straight line.
when n is very great, the sum of their
t
side
that
from that enclosed by y = x
Assume t to be positive.
Evaluate
t
~1
i
,
y = Q,
x=
areas differs
little
c.
r\n/
[I.
C. S.
KXAM.
1902.]
CHAPTER
30
I.
8. In the curve in which the abscissa varies as the
logarithm of
the ordinate, prove that the area bounded by the curve, the #-axis
and any two ordinates varies as the difference of the ordinates.
9.
to
Approximate
summation
f
the
3
10
I
integral
J2 X
dx,
regarding
it
as
a
(1) of inscribed parallelograms as in Art. 9,
(2) of
circumscribed parallelograms,
and compare with the result of integration.
[The results are 3*9724 and 4-1391, the reciprocals being taken
from Bottomley's tables. Their mean is 4*0557. The result to
three places of decimals as computed from 101oge f is 4 '055.]
10.
Draw
a sketch showing the curvilinear area which
is
repre-
sented by the definite integral
and evaluate the area approximately from the figure.
Without plotting, indicate roughly by dotted lines on your sketch
the relative positions of the curvilinear areas represented by the
definite integrals
no
fio
and
I0x-**dx
and
Itor^dx,
calculate the values of these integrals.
Calculate also
p.C.8.,1908.]
PN
th
the w power of
In any curve in which the ordinate
be
the abscissa, show that if any two ordinates
taken, viz. P l h\
and P 2 2 and two others, P 3^V3 and P4 ^V4 which are twice as far
11.
N
from the
,
,
P N and P^N respectively,
Area P3 P4 N4 N Area P^N^
i/-axis
as
1
2
1
Z
12.
Prove that the area
:
:
of the
y=
9^ square
13.
In
then
:
2 n+l
:
1.
diagram formed by
=
from
=
4
from
y
x2 - 10a + 25 = 4y from
x
is
v.
from
(0,
0) to (0, 4),
to (1, 4),
(1, 4) to (5, 0),
to (0, 0),
(5, 0)
(0, 4)
units.
the
construction
of
reservoir
walls
of
great height,
Kankine adopted the following plan
Taking a vertical ic-axis on which depths and ordinates are measured
:
NATURE OF THE PROBLEM.
in feet, the ordinates to the outer
following scheme
Depth
in feet.
:
and inner
faces are
31
shown
in the
CHAPTER
32
1
7.
A
of sines, y
wave on the sea
single
= a sin --
Show
.
is
I.
in the
form defined by the curve
that the quantity of water raised above
mean sea level contained in a length c pf the wave measured
on the surface at right angles to the direction of progression, is
#=
to x = b.
2abc/7r, the raised portion extending from
If c be 100 yards, b = 20 feet, a = 2 feet, and a cubic foot of water
the
portion of the
18.
weight, find the number of tons weight
wave higher than the mean sea level.
62|- Ibs.
weighs
Show
that
when n becomes
-
1.2 + 2.
Lit
3
3. 4
,.
is
the same as
the
infinitely large,
- -+
+
in
.
..+n(n+l)
.
Lt
n3
Illustrate geometrically.
19.
all
Show
that the limit
when
?i
= oo
of the ratio of the
sum
of
two and two together, of the first n natural
and that the limit of the ratio of the sum of
6
products, three and three together, to w is^ TV-
possible products,
4
numbers, to w is -J;
,
all
,
there be gas of volume v and pressure p below a piston in
20.
and occupying a length #. of the
a cylinder of sectional area
show
that
in
its
expansion, so as to occupy a length v + dx
cylinder,
If
A
of the cylinder, the
work done by the gas upon the
pA dx
or
piston
is
p dr,
the expansion continues so that the piston moves
=
a
finite
distance
through
say from x^x 1 to x y.21 the work done
and that
if
on expansion
is
x
y- log
Remembering that
find the value of this integral in the
=
,
two
cases
(2)
Find
Adiabatic expansion,
in foot-lbs. the
work done
of gas, initially at a pressure of
feet
pry
(2)
=
c'.
in the expansion of 10 cubic feet
1000
Ibs.
per square foot, to 40 cubic
;
( 1 )
:
pv = C'}
(1) Isothermal expansion,
pv =
According to the law,
According to the law,
l
pv
'
n
=
c
c'.
NATURE OF THE PROBLEM.
33
X
the graph of eft be drawn, prove that the areas bounded
the -axis and a set of equidistant ordinates are in
the
curve,
by
whose common ratio is the same as the
progression,
geometrical
21.
If
common
ratio of the tangents of the angles which the
tangents at
the ends of the successive ordinates make with the -axis.
Show
22.
ordinate
that the area bounded by a parabola, the axis and an
two-thirds of the circumscribing rectangle.
is
The
23.
about their
x2
circle
common
formed has for
its
+ y~ = 5& 2 and the parabola
Show that the smaller
axis.
y
1
=
4ax revolve
lens-shaped solid
volume
1^(5^/5 - 4).
&_! be a series of
x v x 2 XB ...
quantities taken between
=
x
that
n
and
when
is
made infinite, and the
n ), prove
b(
)
difference between any two consecutive terms of the series becomes
24.
If
,
,
,
=
a( z
indefinitely small, the limit of
is
0,
<f>(b)
Verify
x lt x.2t
this
...
,
-
where
4>(a),
is
the
in
case
and
the
series
[OXFORD, 2nd Public Examination, 1900.]
geometrical.
Plot the value of cos 2
25.
where f(x) = \ogx,
from
for 10 intervals
to
90, and
thus find as close an approximation as you can to
cos 2 a; dx
i
The true value
without integration.
is -.
26. If a cylindrical hole be drilled through a solid sphere, the
axis of the cylinder passing through the centre of the sphere, show
that the volume of the portion of the sphere left is equal to the
volume
27.
of a sphere
If
whose diameter
the curve y
is
= a + bx + ex + dx
2
the length of the hole.
3
pass through the extremities
y^ the distance apart being
the extreme ordinates, the curve
of four equidistant ordinates y 1 , y 2 , yz ,
show that the area bounded by
and the z-axis is
h,
[SIMPSON'S "Three-eighths' Rule."]
the curve y
= a + bx + ex 2 + dx 9 + ex4
pass through the extremities of 5 equidistant ordinates y lt y 2 , y 3 y 4 y5 at mutual
28.
If
,
,
,
CHAPTER
34
distances
h,
I,
show that the area bounded by the extreme
the curve and the
ordinates,
-axis is
45
[BOOLE, Finite Differences.]
29.
a parabola whose
If
the points
y^,
(a,
y 2 ),
(b,
(c,
/3 ),
find the area
show that
its
<(;) where a
why
these
32.
=
.
9
Jarfy
<
<
x
f(x)dx>
b,
I
and
if
both functions are
limits,
prove that
<j>(x)dx.
must
conditions
functions also be continuous
to 6
0.
range of the variable, including both
I
Explain
^ from 6 =
(2)
Jy&,
corresponding to the above limits for
>
and the extreme
x= 10(0+ sin
y = 10(1- cos 6)
(1)
If /(a;)
-axis
.
In the cycloid
finite for this
is
(x-a)(x-b)
tabulate the values of x and y for intervals of
Hence obtain approximate results for
31.
equation
bounded by the curve, the
ordinates y l and y3
30.
to the y-axis pass through
(x-c)(x-a)
(x-b)(x-c)
and
is parallel
axis
be postulated.
?
[I.
Must
the
C. S., 1905.]
Prove that the integral
dx
r N,
is
for all values of
n greater than
2,
nearly equal to O5.
[I.
33.
A
claret glass is 6 cm.
deep and
Its vertical section is
nearly parabolic.
its
rim
is
Calculate
5 cm. in diameter.
its
capacity in
to the nearest integer.
34.
[I.
Trace the curve y = xm ( 1 -
w = 0'5 and
2m
x)
from x =
C. S., 1905.]
to x
=
1
c.c.
C. S., 1905.]
for the values
Show that
the two curves on one diagram.
the area 'enclosed by the curve and the a;-axis diminishes as m
2.
Show
increases.
j-
35.
A
b inches,
L c s
^
1902
-j
cask has a head diameter of a inches, a
bung diameter of
and length
c.
Find an expression
for its volume,
supposing
NATURE OF THE PROBLEM.
35
that a section along a stave is an arc of a curve of sines, the curvature vanishing at the ends of the stave.
Evaluate the result when
36.
Find the value
of
a= 13, 6 = 17, c- 18.
4
f' z' (l
[I.
c. 8., 1902.]
[I.
c. S., 1903.]
-xfdx
Jo
two
to
significant figures,
(1) graphically,
(2)
37.
calculation.
by
Show, without integration, that
Q4.de
(5 + 3 cos
I
lies
between *644 and
2
(9)
'753.
[PETERHOUSE AND SIDNEY SUSSEX SCHOLARSHIP EXAM., 1917.]
,
o tan'i
Differentiate
/I
,
^ tan
B\
-
-
6 sin 6
J
and hence prove that the true value of / is about
1
1
(Take tan- J = -322 and tan~ J=-165.)
'68.
In a diagram of the work done by the expansion of steam in
Watt in 1782, there are 20 ordinates at equal
The respective lengths of the ordinates, of which
(unit) distances.
the first is one unit distance from the beginning of the diagram,
38.
a cylinder, given by
are
1,
1,
1,
1,
1,
-830, -711, -625, -555, -500, -454, -417, -385, -357,
333, -312, -294, -277, '262, -250, representing the steam pressure in
pounds weight per square inch as the piston arrives at a position
The
corresponding to the several ordinates.
of unit length.
(14
Ibs.
The steam pressure
is
initial
ordinate
is
also
supposed to be constant
weight per square inch), whilst the piston travels over the
and then the steam being cut off suddenly, the
first five divisions,
=
pressure is assumed to fall according to Boyle's Law (pv constant).
Show that the area of this diagram is very little more than 11 '562
square units, and that the mean pressure is '578 Ib. weight per
square inch.
"
Justify Watt's statement
whereby
it appears that only J of the
steam necessary to fill the whole cylinder is employed, and that the
effect is more than half of the effect which would have been pro-
duced by one whole cylinder full of steam, if it had been allowed
to enter freely above the piston during the whole length of its
descent."
[GOODBYE, On the Steam Engine.]
CHAPTER
36
I.
39. If steam at pressure p Ibs. weight per square inch be admitted into a cylinder of length a feet, and be cut off when the
- of
piston has completed
its
n
and the steam pressure then
stroke,
according to Boyle's Law for the rest of the stroke, prove by
the Integral Calculus that if the piston area be
square inches,
and there be no back pressure, the work done in one stroke is
fall
A
-
n
Show
also that the
loge en foot-pounds.
approximate result found by the method of
dividing the Indicator diagram as in the preceding question, and
assuming the cut-off to be at half -stroke, differs from the true result
by about
1 -5
per cent, of the estimated work.
[Assume
P-^log^,
log. 2
=
-69314718.]'
40. Steam is admitted into a cylinder at double the atmospheric
= 15 Ibs. wt. per sq. inch), and on the
pressure (atmosph. pres.
opposite side of the piston the pressure is atmospheric continually.
The steam
Divide the stroke into 20 equal
the
at
the beginning of each of these
parts.
Suppose
pressure
remain
uniform
until
to
the
portions
piston reaches the next in order,
and assume the fall of pressure after cut-off to be that of Boyle's
is
Law.
Show
stroke
is
cut off at half stroke.
that with these assumptions the work done in one
nearly 8466 foot-lbs. ; the area of the piston being 200
square inches and the length of the stroke 40 inches.
work diagram
41.
An
as accurately as possible
ellipse,
has a perimeter 20
whose major axis
ft
I
Vl -
is
6'16 sin 2 <d<
Jo
2
large scale the graph of \/l -0'16sin
following values, the angle
o-o
*
||
<
<
[Draw the
on squared paper.]
10 cm. and eccentricity
cm. in length.
as a function of
being in radians
:
0-4,
Draw on
</>
a
from the
NATURE OF THE PROBLEM.
'
Draw
42.
37
in
one figure the graphs of
,
,
,
C
MS
how they
From
are related
showing
the angle x being taken in radians.
;
^
the graph of
deduce the general shape of the graph
tC
Of
I
Jo
__L^
3
mate value
finding
its
of the integral
when
x
representation and draw the graph,
sufficient to describe the
43.
in
is
is
amperes
is
the approxi-
?
[Use a large scale of
from x = to x = 15. It is
large
say,
shape when x
is
[1 .0. S., 1907.]
negative.]
In an electric circuit of resistance
inductance.
What
proportions roughly.
R
L
ohms,
and
is
A current is
t is
measured
Also the voltage
and
The
J
=
in seconds
is
and Q vanishes with
given by
L = self-inductance.
45.
-"'
changing according to the law
-
where
self-
l-
Taking Q to vanish initially, prove that 6=ptf- -p^fl
and illustrate the growth of / graphically.
44.
the
The current
The voltage suddenly rises to a value V.
/.
The law of growth of the current is
V=RI+L-r
Express Q and
V in
t.
where
terms of
R = resistance
t.
figure shows the indicator-diagram of a gas engine
Scale,
80 Ibs per square inch per inch
Fig. 10.
CHAPTER
38
I.
which works on the Otto cycle. Estimate the horse-power
engine from the diagram and from the following data
Diameter of cylinder 9J",
of the
:
Length
of stroke 16",
Eevolutions per minute 180.
46.
Apply Weddle's Rule
[MECH.
So. TRIP.]
approximate evaluation
for the
of a
definite integral, viz.
or
r u dx=
x
ft
log sin
Odd
+ W4 + ^6* + 5 (% + MS*)
ft
to four places of decimals,
'
your result with the known value -log
47.
be
Prove from
finite real quantities
limit of
is zero,
(x
that
first principles
x
J.
and compare.
[BOOLE Fin Diff
if
such that, as n tends to infinity and x n to
Y + (x
x
2
}
-}-
49
+ (x
x
x,
the
2
)
then the limit of the sum
IS
[OxF. FIRST?., 1913.]
The
velocity of a train which starts from rest is given by the
following table, the time being reckoned in minutes from the start
and the speed in miles per hour
48.
:
2 min.
NATURE OF THE PROBLEM.
50.
Q
is
The
specific
39
heat of a substance at temperature
the quantity of heat required to raise
1
gram
t
is
of the substance
from some fixed temperature to t.
The specific heat of water (s, in joules) at a temperature
given by
f-
the following table
:
where
-^,
Ctv
of
t
being
CHAPTEE
II.
STANDARD FORMS.
27.
Reversal of Differentiation.
We now proceed
to consider Integration as the purely ana-
lytical problem of reversal of the operation of Differentiation.
In the Differential Calculus the student has learnt how
to differentiate a function of
character with
any assigned
o
/
regard to the independent variable contained.
y = \fs(x),
In other words,
methods have been there explained of
having given
obtaining the form of the function \}/(x) in the equation
-^
= ^'(x) = </>(x),
say.
we can reverse this operation and obtain the value
^(x) when \fs'(x) is the given function of x, we shall be able
If
of
to
perform the operation which has been indicated by the symbol
Cb
i.e.
</>(x)dx,
fba
I
\}/(x)dx,
Ja
(1) taking the function \ls(x), (2) substituting b and
a alternately for x in this function, and (3) subtracting the
latter result from the former thus obtaining
by merely
;
28.
We
shall therefore confine our attention for the next
few chapters to the problem of
this reversal of the operation
of the Differential Calculus.
The quantity
has been assumed to have any real value
whatever, provided it be finite we may therefore replace it
by x and write the result as
6
;
=
}<j)(x)dx
a
40
STANDARD FORMS.
When
is not specified and we are merely
of the function \^(x\ at present unknown,
the lower limit
enquiring the
whose
41
form
differential coefficient is the
notation
is
known
function
(f>(x),
the
f
I
</>(x)dx=\js(x),
the limits being omitted.
29.
Nomenclature.
The nomenclature of these expressions is as follows
The function <j>(x) whose integral is sought is termed the
"integrand," and the result \js(x) is termed the ''integral."
:
*
or
<j>(x)dx
V^( a )
ijs(b)
i
is
called the "definite" integral of
Hmits a and
<f>(x)
between the assigned
b.
f*
<p
(x)
dx
or
\{s
Ja
where the lower limit
undetermined,
is
is
called a
I
(x)
\fs
(a),
assigned and the upper limit
"
corrected
is left
"
integral.
or
(/>(x)dx
any specified limits and regarded merely as the
reversal of an operation of the differential calculus, is called
without
an
"
indefinite
It is
"
or
"
uncorrected
"
integral.
customary to read the expression
integral of
(f>(x)
<p(x)dx"
And
I
</>(x)dx as
"the
with respect to x" or as "the integral of
the process of obtaining \fs(x) is called In-
tegration.
30.
Addition of a Constant.
It will be observed that if
of
\fs(x), it is
<f>(x)
be the differential coefficient
also the differential coefficient of \ls(x)-\-G,
where
C is any constant whatever, that is to say, a quantity which does
not depend upon the variable x for the differential coefficient
of such a quantity with regard to x is zero.
(See Art. 3.)
;
Accordingly,
we might
write
<j>(x)dx
CHAPTER
42
II.
This arbitrary constant is, however, not usually expressly
written down, but will be understood to be existent in all
cases where the lower limit of the integral is not expressed.
Different processes of indefinite integration will frequently give results of different form for instance,
31.
;
=
j=
dx
sin" 1 ^
is
or
J x/1
for the
expression
is
..
J13F
We
cannot infer that sirr^a?
expressions.
What is really true is that sin" 1 ^
cos" 1 ^ are equal.
cos- 1 ^ differ by a constant, for
either of these
and
and
the differential coefficient of
"1
i
1
So that
i
J
,
v
_L
x
^=
i
cos" 1 x.
dx = sin-^+O,
or
l
the arbitrary constants
32.
x"
C and
C' being necessarily different.
Inverse Differential Notation.
In agreement with the accepted notation for the Inverse
Trigonometrical and Inverse Hyperbolic functions, we might
express the equation
i,
as
or
or
and it is not infrequently useful to employ this notation,
which very well expresses the interrogative character of the
operation
we
are conducting.
STANDARD FORMS.
GENERAL LAWS SATISFIED BY THE INTEGRATING SYMBOL
33.
OR
\dx
I.
43
j?.
from the meaning of the symbols that
It is plain
vdx
is
or
a;
But
C being any
arbitrary constant.
II. The operation
number of terms.
For
if
u lt u 2 UB
,
I
\
-j-
of integration is distributive for a finite
be any functions of
u^ dx-{- \
x,
u 2 dx-\- us dx \
\
and
therefore, omitting additive constants, i.e. supposing the
lower limit to have been assigned and to be the same in each
case,
I
u^ dx
+
\
u 2 dx+\ us dx = (tti+^+Ua) dx.
+
1
u z dx
I
Similarly,
u^ dx
I
I
u s dx =
I
(?e x
-\-u z
us ) dx.
If the lower limits in these several integrations are not
the same, the left-hand member of the equation may differ
from the right-hand by a constant. It is in this sense that
the equality sign
III.
is
used.
The operation
of
integration
is
regard to constants.
For
if
.
t
= v, and
a be any constant,
d
.
du =
A=a^
av.
commutative with
CHAPTER
44
So
II.
that, omitting additive constants of integration,
au = av dx;
I
a
or
v dx
I
\
av dx,
which establishes the theorem.
Case of an Infinite Series.
34.
In the case of an infinite series of real quantities,
U=u +u +u
2
l
of
s
+...+u n +...
which the terms are connected by a
have
to GO,
definite law,
we
shall
still
rx.2
Cxn
Udx=\
u s dx+...
u^dx-\-\
to
oo
=
F, say,
JXi
JXi
provided the series
U itself,
and the
series
V
formed by the
integrations of the separate terms, are both absolutely con=
vergent within a range of values of x, viz. x = b to x a,
say,
where a
>
6,
between which quantities both limits of
integration xl and x2
lie,
a
For
series
R
let
U and
that
is
>x >x >
2
l
b.
and 8 be the remainders
V,
after
n
terms of the
i.e.
CX2
.+
\
un dx+S.
J Xi
Then, by supposition, both JH and S vanish
indefinitely increased for all values of x between
J*
x
R dx,
for
it lies
when n
a and
b,
between R'(xz
is
and
x^)
i
and R"(x2
x 1 ), where
R
R
f
and R" are the greatest and least
from a to 6, and which
as x changes continuously
are quantities vanishing in the limit.
values of
Hence,
F-S= \U-R)dx= \**Udx- ^Rdx
J
JX
J
l
and when n
is
Xi
indefinitely increased,
Xi
(Art. 33, II),
STANDARD FORMS.
If
45
then a function $(x) can be expanded in a power-series
r = =o
as
(/>(x)
= VJ-X',
}-
from x = b
the series
x = a, we can write
to
Jr
= co
f
0(z)t:f;=yM r
r=
*i
if
_
.'.-.
r=
F|
a
2A rx
r
[Art. 16, Ex. 6],
r
where
for
absolutely convergent
being
=
>x >x >b
z
l
;
be absolutely convergent, so also will be
r+1
_ ^ r+1
and
Under such circumstances,
therefore,
we may expand
before
integrating.
35.
Geometrical Illustrations.
We may
illustrate these facts geometrically.
Fig. 11.
Let the graph of y = (j>(x) be represented by the curve
Let the coordinates of a fixed point P on the
P.
CP
curve be x
y Q let x, y be the coordinates of a current
on
the
curve, and let A be the area of the figure
point
P X NP. Let x increase to x+8x, and in consequence let y
become y+8y and A become A-{-SA. Then 8 A is the area
of the strip PNN'P* between two contiguous ordinates NP
and N'P', and lies in magnitude between y Sx and
,
,
P
?
and therefore
j
lies
between y and
CHAPTER
Hence, in the
when
limit,
II.
Sx
is
made
indefinitely small
we have
dA
dx~ V>
A=
Hence
So long
as the lower limit
of the area
ordinate
may
NP
and the case
,
unassigned the reckoning
is
from any arbitrary position of the
start
is
that
of
the
"indefinite"
integral.
When
the lower limit
x=ON
is
reckoned from the ordinate
Jx
(f(x) dx,
and
0) the area is
assigned, say
to any arbitrary ordinate
NP
is
then " corrected."
0A*
When
both limits
ON
d>(x)dx
{ON
ON
and
ON
are numerically assigned
is "definite."
n
N
x
Fig. 12.
If
all
there be several curves (a finite
number
of them,
and
continuous, and none of the ordinates infinite within
the limits of integration),
y = Fl (x), y = Ft(x), y
= U1}
= u2
,
= F3 (x)
=U
B,
viz.
the curves
P
P,
Q
Q,
R
R,
and a curve be derived from them by the algebraic addition
of ordinates so that
),
viz.
the curve
S
S,
STANDARD FORMS.
then the distributive property
II.
47
of the integration symbol
asserts that
Area
7W\TP+area Q # #Q+area
Again,
if
a curve be given by the equation
y
= F(x),
i.e.
PQ P,
curve
and a new one be derived by increasing
the ratio a 1 so as to have an equation
all
the ordinates in
:
y = aF(x),
i.e.
curve
$
$, say,
the commutative rule III. asserts that
Area SQ
If
N NS = a x area P N NP.
the lower limit be
assumed
not
the
same
each case, as
in
the figure, the stated results would, instead
of being equal, differ by constants which depend upon the
positions of the initial ordinates in the several cases.
36.
By
in
Integration of
xn
Differentation of
.
>ytt+l
--
^n+l
--= xn
/-7
we
obtain ^
(as has already
been seen, Art.
x n dx =
%
;
j
+an
-
dxn+\
n-j-1
16,
Ex.
.
Hence
6).
arbitrary constant.
Thus the
x
may
rule for the integration of any constant power of
be stated in words;
Increase the index by unity,
(dx,
and divide by
ldjc or
i.e.
J
the
new
index.
CHAPTER
48
The case of
37.
II.
1
a,
.
remembered that x~ l or
It will be
coefficient of log e x.
1
:
is
:
the differential
Thus,
x' 1 dx
or
I
- dx =
log e x.
'
This therefore forms an apparent exception to the general
rule,
J
x n+l
x n dx
n 4-1
however, only apparent. For we may deduce the
logarithmic form as a limiting case. Supplying the arbitrary
constant (7, we have
It
is,
z+i7&4-1
where
A = G-\
not contain
H+ly
and
limit
when n+l=Q,
takes the form logc x (Diff. GaL, Art.
an
arbitrary
constant,
^=-i
-
_pr,
does
And
that
j^
-
as
G
is
contains
it
together with another
% n dx = log x+A.
This has also been seen in Art. 16, Ex.
38.
21).
we may suppose
a negatively infinite portion
arbitrary portion A.
Then
i.e.
/-y.n+1
Taking the
x.
an arbitrary constant,
still is
6.
n
l
In the same way as in the integrations of x and x~ we have
- (n + l)a(ax
aild
+ n dx =
J (ax b)
and therefore
^ + \a
and
[Although
to x,
[-T--
J\/^2
we
I
dx
is
really one
shall often find
---cfa;
2
printed as
symbol indicating integration with regard
/^-^^ printed for convenience
/-= =-,
v^ + a
J
2
etc.]
as
STANDARD FORMS.
39.
We
now
are
49
in a position to integrate
any expression
of
the form
ax-\-b'
where
of
indicates
<f>(x)
any
rational integral algebraic function
x.
This can be done in two ways
(1)
By
ordinary division of
<f>(x)'
rv
:
by
</>(x)
ax-\-b
form n
0-f
,,
.
in the
ax+b
we can
express
W
R
ax+b'
where Q
consists of a series of descending
is independent of x.
powers of x and
R
Every term is then integrable by the foregoing rules, and
the result will be partly algebraic and partly logarithmic,
D
the last term being
log (ax +b).
QJ
should be entirely algebraic
vanish,
i.e.
(f>
(
-- j^ft
is
or that
The
condition
obviously
(j>(x)
that
R
that
it
should
should contain ax-\-b as
a factor.
(2)
A second process would be to put ax -\-b = ay,
i.e.
x=y
and then
ax+b
Then expand
tf>(y
--
)
ay
in descending
powers of
pressing the fraction ultimately in the form
is
a series of powers of y and
Thus
T
,
is
is
+
thus ex-
T/,
R'
,
where
again integrable.
R
Q'
u
is
independent of
y.
expressed in a series of powers of
together with a term
each term
R
Q
/
(
x + -)
being independent of x and
CHAPTER
50
II.
fx^+x3
Thus, in the foregoing case, J
dx, putting
.r
#+2
Hence
The
results are of different form, but of course equivalent, except that
they differ by a constant.
also to be observed that since the differential
coefficients of [<j>(x)] n+l and log </>(x) are respectively
40.
It is
(+l)|>(z)]V(*)
and
^
<P\
Xl
we have
The second
may
of these results especially
be put into words thus
is
of great use.
It
:
The integral of any fraction of which the numerator is the
is log (denominator).
differential coefficient of the denominator
41.
For example
/cot
/
tan
:
dx = log sin x.
xdx=\j -^
sm^
x dx = -
42.
More
~ S1U '*
I
J
J
cos
x
of
generally, since the differential coefficient
s
we
clearly
dx = - log cos x = log sec x.
have
Thus, for example,
'
/ ^
J
1
-.
=
+ sin 2 x
51
jXAMPLES.
Write down the
1
i.
x~.10
#
,
2
(tf^
indefinite integrals of
_5
T
7
7
- 10
n
I
,
i,
u,
^r o
3/~^5
.
,
:
I
r=,
^/jr--,
,
2
+ bx + c)(a.r+ bx-i + c)
"
1
'
1
rt-^-'
a?
1
(a-x^ (a-x
:<
5.
Calculate
6.
Calculate
/
(a +
2
2
6^) 36^
^
for the values
a = 2,
6
= 5.
,
8.
9.
What
10.
be 2 foot-seconds per second, and
will it come to a stop ?
If the retardation of a particle
its initial
when and where
velocity lOf.s.,
Given pv = constant, and that p = 40 when v=10, calculate
does this integration
mean
Calculate
between limits
(a)
and
(rf)
3 and
1,
4,
(6)
1
and
2,
(e)
4 and
5.
Explain the signs which occur in the
11.
Write down the
_i_
^6
6z
Integrate
Uj
dx
/"
J
(
1
+^7uiF^'
2
and
Illustrate
3,
by a graph.
:
+ 6e*).
/
^.gto ^i
/^g"*
/.v
(c)
results.
indefinite integrals of
x M
1
(ax* + 6e ) (pa^'-
(i v)
13.
pdv.
/
?
C
.
f
l)
(")
/
cofc
2 ^'
/
__
J (i
dx^
+^)(tan-^)'
^-'i
(
.....
i;
iij )
/
1
/'(sin- ^)'
J
i-a?
1
,
'
CHAPTER
52
14.
w Jf-^
x log x
Integrate
(i)
fx
^
(ii)
'
'
II.
)
log
# log # log log a? (log
__
l
r
x represent
log log log
a?
log log
dx,
n
log log #)
<fc?
f
J xl(x)V(x)
where
A-
log #, the log being repeated r times.
. . .
now be perceived that, the operations of the
Calculus
Integral
being of a tentative nature, success in
will
Integration
depend in the first place on a knowledge
It will
43.
of the results of differentiating the ordinary simple functions
which occur in Algebra and Trigonometry. It is therefore
necessary to learn the table of Standard Forms which is now
appended. It is practically the same list as that already
learnt for Differentiation, and the proofs of the facts stated
members of the several
was printed on page 46 of the Author's
There are a few additions, as we are
Differential Calculus.
now specifically considering Integration. The list will be
gradually extended, and a supplementary list will be given
in differentiating the right-hand
lie
The
results.
when
list
the results have been established.
44.
PRELIMINARY TABLE OF RESULTS TO BE COMMITTED
TO MEMORY.
J xndx
(3)
(5)
I
=
e*dx = e*.
(4)
cosxdx = siux.
(6)
(7)
m\
(y )
I
f
I
sec 2 x dx
= tan x.
sec x tan
JB otic
7
/i
A\
f
=
(8)
fsina;
5
I
J cos-
J
(10)'
cosec x cot
dx =
?
a;
J
f
I
cot x
dx = log
e
sin
7
x
cos
.
9
2
J sin
\
\
(12)
lsino;cZa;=
cosic.
J
J
x.
aa;
a;
x
cosec 2 x <r =
= sec
7
=
dx
(Art. 41.)
a;.
cosec
x.
cot
a.%
STANDARD FORMS.
dx
= sin" 1 x
,x
,
.
53
cos" 1 -.
or
a
dx
2
/T
It
CbX
rt\
I
It
is
x
,
1
1
or
a
1
=
=.
a help to the
i
memory
1
a
/dx
~2
must be
be tan" -, which
1
difficulty in
it
,
is
5
a
1.
of zero dimensions.
in
.
to observe the dimensions of each side.
of dimensions
remembering
X
1
f
linear,
/
dx
~T 2
2
of zero
is
prefixed to the integral.
of dimensions
is
#
cosec" 1 -.
There could, therefore, be no
dimensions.
.
covers" 1
or
For instance, x and a being supposed
integration
,
,
X
vers" 1 -
x
a
cot" 1 -
or
a
a
2
a
^
,
=-sec~ 1
._
Jxjx*a
(16)
45.
,
a
dx
f
tr\
(15)
= 1 tan"
-for
1.
Thus the
Hence the
result
of.
integral could not
There should, therefore, be no
which cases the factor - appears, and when
does not.
Also, so long as we are dealing with the trigonometrical functions,
whenever the result begins with the letters " co," it must be with a
The reason is obvious ; the cosine, cosecant, coversine
their inverses are all decreasing functions as x increases through the
first quadrant, and their differential coefficients are negative.
The rule of the " co " does not apply to the hyperbolic functions.
negative sign.
and
EXAMPLES.
Write down the
x
1
,
a
indefinite integrals of the following functions
x
x+a
Bin', cot*+tan*,
=P xjM^v" Jn
J
x2
xn ~ l
*+
^ ^+4'
.^-4
:
CHAPTER
54
7.
Write down the
indefinite integrals of
3
Write down the
I
(ii)
(v)
J(ax*
/
(e
+ a)n e* dx,
:
dx
f
....
(iii)
+ bx + c)"(3c
indefinite integrals of
,
x
cotxsec xdx,
f
l+x*
:
2
[(xt + a^^dx,
(iv)
8.
sm~ xcosxdx,
/
(i)
II.
dx
FlSffr'
9.
Evaluate
(ii)
(i)
J*
the graph of 64 (#-2) (#-3) (2# -5), and show that the
area between .r = 2 and # = 2'5, bounded by the curve and the #-axis, is
10.
Draw
32
r
~i2.5
2
(#-2) (#-3)
2
,
i.e.
=2
square units.
Verify by multiplying out and integrating each term.
11.
Write down the values of:
0)'
r6
Tl e2a;^_ e4a;
(iii)
I
e
JQ
12.
0^
(iv)
c?^,
cosh (log x) dx.
/
Ja
Evaluate
r?
t-s
(i)
cos^o?.^,
/
/
(ii)
.'o
/"4
13.
/"as
cos 2.r
/
(iii)
J
/
(i v)
c?.r,
(cosh
x + cos ^7) dx.
sec
tan x dx,
Jo
Evaluate
/IS
(i)
14.
cos 2 ^o?.r.
^'o
r^
sec 2 ?i.rc?^,
/
J
I
(ii)
a?
*'
Evaluate
,. N
(i)
7
c
\
(iv)
n
n
Cx
- -a-
,
I
J^-a dar,
f & j
J
dx.
Cx5 - 2x+l
-
n
,
(n)'
^
i\
(v)
f
/
)
x
-dx.
x-a
.
(in)'
fx*-foc* + Ilx-6
^2 _ 3, + 2
I
x-l
/
J
,
efo,
^
,
The processes of. Integration being necessarily of a
tentative nature and founded upon a knowledge of the
forms obtained by differentiating the known functions46.
algebraic, logarithmic, exponential, trigonometric or hyperbolic,
or the inverse forms, it will be realized that many expressions
may be
little
written
down which
are not the differential coefficients
functions or of any combination of them. A
consideration will show that this is necessarily the case.
of such
known
STANDARD FORMS
If the inverse sine
had never received the consideration of
mathematicians, the expression -j=
=
of
coefficient
same way,
would have been the
x2
vl
differential
55
In
something so far uninvented.
the invention of a logarithm had not
1
preceded the necessity for the integration of or the integral
the
if
,
- would have been
lacking and have presented
jO
of
Hence
will
it
be seen that
it
is
difficulty.
only certain classes of
algebraic, trigonometrical, exponential, logarithmic, or hyperbolic functions, or the corresponding inverse functions, that
admit of integration in finite terms. Some functions there
are which admit of integration in terms of an infinite series
though such series may not be otherwise expressible as the
expansion of any known function. For example,
j
x lw
vl
is
not the differential coefficient of any
known
function.
But
supposng
oo
1
+ Li3
.
3
^
\
dx
.<*'*
an
infinite series, not capable of summation, but nevertheless useful for
approximative purposes, supposing x to be a positive proper fraction, if
such arithmetical approximation be required.
And
to
go back to the case of
by
the failure to integrate
f
dx=
\
xn
-y,
integrals of
it,
a?' 1 , it is
also clear that as
by considering
it
a case of
there would have been a gap in the
powers of
list
of
x, viz.,
the properties of a function which had x' 1 for its differential
coefficient could not long have remained undiscovered.
For
if
F(x) stand for
I
j
.
- dx,
x
we must have
= xdy + ydx
~
f(t+tH
J
=l(w),
XIJ
XIJ
CHAPTER
56
II.
which constitutes the fundamental theorem of logarithms
and indicates how an addition may be used to perform a
multiplication when tables of F(x) have been constructed.
dO
In a similar way, the expression f*
where k
2
2
AT sin
Jo vl
-.-
is
a constant
< 1,
presents itself in the consideration of
Now
problems geometric and kinetic.
.
Vl
2
/c'
=
many
is
not
sin 2 #
the differential coefficient of any combination of algebraic,
exponential or circular functions. Hence, this is a case in
This is an integral where necessity for discussion has
point.
arisen prior to a knowledge of the expression of which it is
the differential coefficient.
Calling
V
it
u,
dO
/l-A?sin 2 0'
We
the upper limit
the amplitude of u, and write
l
=
and
u
&Ta.~
Thus u receives a name.
16,
inversely
is a function whose
leading properties we propose to
call
<
= am
It
<f>.
discuss later.
EXAMPLES.
1
.
Write down the
indefinite integrals of
,
9
v
a cos x -
p-9i q+ r 9-ri r+p r-pi p + q
(3)
YlTf
V^
9
VaTf
:
(4)
b sin
X
tan" 1 i
(5)
e^* + e^*,
(6)
cosa;(l
(11)
/..v
,,
(13)
tanz(l+secz),
(15) (cos x
(16)
-
sin x)('2
x
atmx + bcoBX + c'
(H)
+ sin 2x) sec 2z cosec 2
o;,
+sinx)
STANDARD FORMS.
2.
!/(*) = 5-^, prove
(a)
that
57
<fe
=
y
I6
]///(*)
r
f (x)dx = a -7o
r
f(x) means fff
.
.
l+
I-
fi
J
+ br
1
z2
,
where
2i
that
,
dx
log,
B
(b);
v
1
- a
f(x), the functionality symbol / occurring r times.
Show by expansion
3.
,
.
-
where
4.
Prove by Differentiation or Integration from the Binomial
n
of (1 +x)
where n is a positive integer,
Expansion
(a)
lCl
(b)
l.2
(c)
(d)
(/)
,
!
l
2
^+2
3 2 <73
+
.
.
.
+ n*Cn = n(n +
1) 2"-
2
,
.
4.
1.2
A,
,3
2.3
3.
__2.3.4^3.4.5
__ _
,
4.5
n
2
1.2.3
,.3.4
0)
C2 +
1^ + 2^ + 3(72+.. + (TI+ 1)^=2"-^ + 2),
-^
(i];
2
"^
(n+
l)(
+ 2)(n + 3)
5.6
r^^-'JT^^ + s:^^""- 4
"
=2
!
3"
+
2)
2(n +
3)'
CHAPTER
58
Prove from the expansions
5.
that
sin x
I
dx =
1
of sinx
- cos x and that
cos
I
and cosz
in
powers of x
xdx = sin x.
Jo
Jo
6.
II.
Prove from the expansion
of
exp x that
exp x dx = exp x
(x
[exp x
= e x]
.
-00
7.
Prove that
fn\a*-l
.
fn\x*-l
-=
+
i
8.
Show
that
9.
Show
that
10. If
11.
I
\
r
)
V"
QW-2
^
J_
S
T-
considering
of the series (p
I
(x
-
and q being
p
a} (x
-
q
b) dx,
1.2
independent of
12.
x.
Verify by differentiation that
(1)'
/0\
I
o
1
)
0"-3V
^
O
Jl+*4
f
._
1
4v/2
,Jl-X
3
'
JVi^?[i +
show that the
positive integers)
1.2
is
/
)
L
Tf
"
be a rational integral algebraic function of x, show that
^>(a;)
By
/
r
os
g
l-W2
difference
STANDARD FORMS
13.
where
14.
<f>(x)
B = hJB
r
Show
may
<}>(x)
= A Q x n + A^xn ~ + A 2 x n ~ 2 +
l
If
+ Ar
r_l
that
also
1
.
...
59
+ AM prove
Write down the values of
.8
Q
that
B
,
1
,
Bz
.
21-4 das for rational integral algebraic forms of
fi
Jx
be expressed as
Prove that
-
15.
Prove that
16.
If
2
-*
f*
*2
(t.e.
Bessel's function), prove that
(1)
J
\*
[Jn ^(x)-Jn+l
(2)
Jo
17.
Prove that
^-i(l
o
when
1
^(a,
/3,
y,
a:)
-^"^-^(-v,
\
-, /^
+ 1,
A1
denotes the hypergeometric series
|
l.y
1.2
yy+1
1.2.3
77+17 + 2
[I.
18.
Assuming that the speed of the current
bank follows the law
C. S., 1898.]
a river at a
in
distance x from the
the breadth of the river and V Q and k are constants, find
by integration how far down stream a man will be carried who rows
4 miles an hour, pointing the boat's head always straight at the
where a
is
the width of
opposite bank, so as- to cross in the least time possible
the river being half a mile, the banks being straight and parallel,
and the speed of the current being 2 miles an hour near the banks,
:
and
3 miles
an hour in mid stream.
[I,
C. S., 1905.]
CHAPTER
60
19.
Find the moment
II.
of inertia of a rectangle of sides 2a,
2ft
about
a line joining the mid-points of the opposite sides of length 2a.
The section of a ship at the water line is 120 feet long. If the
middle line be divided into six equal portions, the ordinates of the
boundary of the area at the middle points of the segments are given
by the following
table
:
Distances from end
10
STANDARD FORMS.
22.
OK is
the diameter bounding a semicircle of radius r, P any
PQ an ordinate to the diameter OK. If x denote
point on OK,
arid
the length OP and z the area which
dz
, d?z
interpret -=- and -=-*.
2
.
,
61
PQ
cuts off the semicircle,
,
dx
dx
Find a curve for which the area bounded by the curve, the axes
x and y and the ordinate at a distance x from the axis of ?/,
a2ta
V
23.
[I.
From
of
is
C. S., 1902.]
the equation
*>
!(<* +]%<**)-
where a and A are constants, find y in terms of #.
The value of a being 2 feet, and of A 10 feet, evaluate y when
z is 30 feet.
[I. C. S., 1910.]
24.
axis of
is
Denoting by A the area between the curve y=f(x) and the
x, from the value zero to the value a of a, show that, when f(x)
a rational integral algebraic function of the third degree,
where
yQ
Compare the
result given
by
this rule
with the true value, taken
to three places of decimals, for the curve y
and 0-5 of x reckoned in radians.
values
25.
[I.
the
C. S., 1912.]
Verify that the area of the curve
between the limits x = h and
the
= sinx, between
sum
x= - h
is
equal to the product of h and
of the ordinates at
and
x= -
In the case of the curve
y=
verify in like
equal to
A + Bx + Cx + Dx* + Ex* + Fx*> =/(),
z
manner that the area between x = h and x= -h
{5/(W3/5) + 8/(0) +
5/(
-
Av/3/5)} A/9,
[I.
26.
Find the
differential coefficient of
is
C. S., 1913.]
CHAPTER
62
and deduce that the sum
series is less
than
What would
to
e?
II.
of the first five terms of the exponential
by the quantity
be the corresponding result
of to five terms?
the series were taken
if
n terms instead
[I.
c. S., 1913.]
27.
Weddle's Rule for finding, approximately, the area bounded
by a
curve, two ordinates, and a base forming part of the axis
" Divide the base into six
is
equal parts and draw the
of
x,
:
of division, making, with the extreme
Of these ordinates add the first, third,
and seventh, and five times the second, fourth and
ordinates at the
points
ordinates, seven in
fourth, fifth
all.
sum by
Multiply the
sixth.
one-twentieth of the base."
Prove that the rule gives the true result when the
and 1 and the curve has any of the forms
y
where a
is
= a,
y
ax 2
y
,
a constant and n
is
= ax*,
y = a(x- 1/2 )
limits are
n
,
an odd positive integer.
j^
Find, by the rule, the value of 6 x
I
Jo
Check the
decimals.
28.
from
Show
that the
result
by
l+"
integration.
[I.
work done by a gas
v l to v 2 according to the Adiabatic
to seven places of
c. S., 1911.]
volume
in altering its
Law
is
7-
= 1), show
that this becomes
If
the law be Isothermal (7
If
a gas expands isothermally from state
v 1 to state p.2) V2
p lt
(Operation
then expands adiabatically from state p>2
v.
2
,
to state
p3
,
I.),
v3
(Operation
then contracts isothermally from state p 2 v z to state _p 4 ,#4
II.),
,
(Operation III.),
then contracts adiabatically from state p, v to state p lt v 1
(Operation IV.),
each
(1) find the amounts of work done by or upon the gas during
of these four operations, drawing a graph of the whole cycle,
pf
changes
;
STANDARD FORMS.
(2)
show that the work done
measured by
(Pi v i
(3)
~Pz vz) lg
that
(4) that,
whole cycle of operations
is
(th e adiabatic portions cancelling)
= v2Vi
;
;
writing
the above expression for the
[This cycle of operations
heat engine.]
dQ be
If
~
fljflg
= Pi
29.
in the
63
>
Pf? =Pi v y =
i
work may be written
is
known
as a Carnot's cycle for a perfect
the whole heat absorbed by a body of uniform
temperature whilst its temperature changes continuously from 6
to
+ (10, and if be a function of the independent variables which
<f>
define the state of the
body and such that
called the
Entropy of the body (Clausius).
a graph be drawn to represent 6 as a function of <,
the area between the graph, the <-axis and the ordinates corresponding to the initial and final states represents on some scale the
<
is
Show
that
if
heat absorbed.
In the case of a perfect gas satisfying the law
the Thermodynamic Equation
where Cv
is
^ = const. = R, assume
the specific heat at constant volume, and show that in
changing from state 6 lt
v,,
</> x
to state
2
,
v2
,
</> 2
,
Taking the temperature as a function of the entropy and simultaneous values of
and 6 as given in the following table
<
:
<
CHAPTER
64
[There
is
II.
a brief sketch of the fundamental formulae of Thermo-
dynamics on pages 56 and 57 of Solutions of Senate House Problems for
1878 which may be found useful. Students may also read Tait's
Thermodynamics or Parker's Thermodynamics for detailed accounts of
the theory ; other useful books are Zeuner, Theorie Mtcanique de la
Chaleur ; Briot, Thdorie Mfaanique de la ChaleurJ]
30.
In the case of a saturated vapour, if C" be the specific heat of
i.e. the heat
imparted to one gramme of the saturated
the vapour,
it constantly in the saturated state when slowly
the
compressed
temperature rises one degree Fahrenheit ; C that
of the liquid from which it is derived at the same pressure and
temperature, L the latent heat, then it can be shown that
vapour to keep
till
dL L
C==C+ d0-Q'
where 6
the absolute temperature.
specific heat of the liquid at constant pressure,
which, as liquids are practically incompressible, is so nearly equal
to C that no appreciable error results from regarding them as
is
Let Cp be the
identical.
Then Regnault has shown experimentally
and latent
heat, viz.
L+\J273 Cp d0,
is
that the
sum
of the free
not a constant as had been
supposed by Watt in his earlier experiments, but is a function of
the temperature 0, viz. putting = 273 + & and / being the number
of ergs in one calorie (41,539,739-8 ergs or about 3 foot-lbs.), he
obtained the equations
(1)
Cp d6 = J(a. + pe' + 7 0">\
L+
J273
(2)
cp =j(a'+p0' + ye'i)j
experimentally, determining the constants
vapours.
Using these
(1)
data, prove that
^+C
p
[
a, j3,
y
;
a', /?',
y for several
STANDARD FORMS.
Show
that the integral equivalent of the equation
dx
f*
is
+x + x 2
i
Jo
I
65
Cv
+
Jo
z
dii
.
\
i
..
dz
[
..+!
.
*
i
=
form
of the
xyz+afaz+i
where
S^.
a, b, c
If
are certain constants.
the variables
y,
a;,
[Oxr.
I.
P., 1913.]
z be so related that
xy = F(z\
show that
J x i F({c)diB+.|
J Vi
For example,
and
x
if
F(y)dy+\ F(z)dz = xyz-x l yl zr
J *i
+ y + z = 0,
:
show that
/FT^
v/T
fy
7
[BERTRAND,
33.
If
y=
I
eP
iax
expand y in powers
dx,
(7a^c. Int., p.
of x as far as x 5
34.
Prove that
g.
Integrate
where
36.
TI
^
f
- 2n ~ 2
a^
(
I
-
^)"( 1
-
Prove that the
<j>(x)dx
-
L p>
^
n
I.
[Oxr.
P., 1917.]
fifth differential coefficient of
x*(x<}>(x}dx
is
+ 6x 2 (x^(x)dx - 4xf x s <j>(x)dx +
[Oxr.
24</>(a:).
37.
P., 1911.]
cx) dT,
a positive integer.
is
.
I.
[OxF.
383.]
{x*<t>(x)dx
I.
P., 1917.]
fcos8fl-cos7fl
1 + 2 cos 50
Integrate
J
38.
If
between
f(x) and
and
x,
F(x) be two functions continuous and
such that
^1-
finite
F(t)dt,
Jo
obtain their expansions in ascending powers of
E.I.C.
E
x.
[Oxr.
I.
P., 1915.]
CHAPTER
66
f
39.
1
IT.
dx
= 0'288
-^
(L +X)(Z +X)
Prove that
,
r
J
nearly.
[MATH. TRIP.
40.
If
y
1
= a'2x + c,
2
19
express in terms of y the differential coefficier
of the functions
with regard to
I.,
\og(ax +
xy
y),
x.
Hence evaluate
I'
-
and
\ydx,
and prove that
t
[MATH. TRIP.
41.
I., 19.
Prove that
{\og(a+6h)-loga}
can be expressed in the form
-
6 {log (a
+ h)- log a}
xdx
-
*)
f
Jo
Deduce that in calculating a logarithm to base 10 by the method
of proportional parts from tables which give the logarithms of all
5
4
the error is one of defect and cannot
integers from 10 to 10
,
amount
to
JIO'
8
/*,
where
/z
42. Integrate
W
l-f
I
sin 6 *
\
cos(4?i+l)T
l+cos0
n being a positive integer.
= log10 0=
-
seven-figure tables'?
~
1
'43429.
7?d0.
+ cos 6
~
2 /sin
Is this negligible in
[MATH
6
TRIP. Pt.
and show that
~"26 ~~
sin
sin 3(9
II., 1919.]
CHAPTER
III.
CHANGE OF THE INDEPENDENT VARIABLE.
47. It will frequently facilitate integration if we change
the independent variable a? to a new variable z by a suitable
choice of relation connecting the two.
Let x = F(z) be the relation chosen, and
\Vdx
or
let
\f(x)dx
be the integral to be transformed.
u
Let
Then
**=
ax
1
Vdx.
V.
or
1
."gtan
48.
Thus, to integrate
/
=
Instead of
tan~
x
.r = z, and
then
-
A'
-
writing
x
2 rf^,
tan- 1 ^ = 2 or
f
= tan^,
%
let
it
would be a
~~J^~^-
the integral = [ez dz, at once
= tf = gtan
67
1
a^
little
shorter to take
CHAPTER
68
49.
III.
In the practical use of the formula
after having made choice of the transformation x = F(z),
usual to make use of differentials, and instead of writing
it is
T^>
we
shall write the
and the formula
same equation as
dx = F'(z)dz,
will thus be reproduced
by replacing dx
the integral \f(x) dx by F'(z) dz, and the x by F(z).
Gal, Art. 156.)
1
(See Diff.
x
-1
%dx, after putting tan ^
1 -\- x
/gtan
may wnue
dx
and the integral becomes
/
e
2
in
= 2, we
,
^=
When
the integration is a definite one between specified
the
limits
for z will not in general be the same as
limits,
those for x. But supposing a and b to be the inferior and
50.
superior limits for
x,
those for z must be such that whilst x
ranges once over its values from a to b, z passes once and
once only through the corresponding range of values for z,
viz.
from
F~ l (a)
to
^~ 1 (6), where x = F(z)
is
the connecting
formula.
51.
The transformation
/(*) = *'(*)
Let
Then,
of the indefinite integral
if
the limits for x be a and
b
f /(*)
Now, when x = a,
and, when x = b,
Ja
=
f
Ja
^'(*) dx
2
= .F-^a)
z
= F~ 1 (b).
6,
is
CHANGE OF THE INDEPENDENT VARIABLE.
69
Also
and
/*-!(&)
CF-*(b)
whence
J
J F~^(a)f{F(z)}F(z)dz=\
ft
%
F-^(a)d
So that the result of integrating f[F(z)]F'(z) with regard to
between limits F~l (a) and F~l (b) is identical with that of
integrating f(x) with regard to x between the limits a and b.
z
52. Case of a Multiple- Valued Function.
must be noted that F~l (a) and F~l (b) may be multiplevalued functions of. a and 6. Thus, for instance,
It
sin-1 ^ being the
same thing
as
mr +
n 7T
l)
(
-~>
where n is any integer whatever, is a multiple-valued function.
The question will thus frequently arise as to which of a
l
l
variety of values of F~ (a) and F~ (b) it is proper to take as
the limits in the transformed integral.
If, however, we remember the connecting formula x
F(z)
and imagine x continuing its march in a continuous manner,
always increasing from the value of a to the value of 6, then,
1
l
starting with any of the values of F~ (a), say a, F~ (x) is to
on
in
manner
to
occasion
a
from
the
continuous
a
first
change
1
which it takes up the value F- ^), or /3 say, increasing along the
whole march from a to /3, if x and z increase together, i.e. if
= a to # = 6, or decreasing along the
F'(z) be positive from x
whole march from a to /3 if x and z are such that z decreases
as x increases, i.e. if F'(z) be negative from x = a to x = b.
Then a and /3 are the limits for z which correspond to a and b
respectively for
53.
we
x.
For instance,
let it
be required to find the value of
assign the positive sign to the radicaf^l
# = sin#, we have -jj.cosO.
or
$,
according as
+*Jl
x*-=
And
-X L
.
By
cos B.
/
the transformation
the indefinite integral
+cos Q or
=" where
I
is
/
(l)dO
CHAPTER
70
When x= 0, = siu-
= wr.
1
In the march of # from
positive.
1
-
6 = s\n- l l=>-
When #=1,
III.
to
1,
.
W
increases from
and
to
6
9 decreases from
lie
,
2i
Generally,
if
If
sin
6
is
start at (2ra + I)TT, 6
it is
is
negative.
to
must decrease, as x
and therefore must
increases,
I)TT, where sin 6 is zero,
proper to take our limits, either
or
TT
or
2?r to
or
3;r to
^,
to -,
,
,
sin
increasing, 6 increasing
6 increasing,
sin
increasing,
increasing
sin 6 is
1.
;
sin 6 increasing, # diminishing
sin
is 1 is
1.
to(2m+ I)TT - -, where
to
always
is positive.
and -^
Oi\J
increasing from
pass from (2m +
Therefore
-^
6 starts from 2w?r, the next occasion on which sin
= 2wi7r + - and
at
is
in the second quadrant,
to
TT
integers.
lie
g-,
If the terminating radius of
m being any
n and
to 1 and
passes from
in the first quadrant,
sin
terminating 6
If the radius
\
TT
;
;
diminishing
;
etc.
But we have noted that +\/l
if
cos
be positive, the
sign
if
-#2 =
cos 0, the
cos 6 be negative.
+
sign to be taken
Accordingly,
etc.
54. It will perhaps make the matter clearer
formation formula be drawn in such cases.
In the present
case,
,y
= sin
referred to
$,
if
a graph of the trans-
x axes
is
a curve of sines
CHANGE OF THE INDEPENDENT VARIABLE.
whose axis
the 0-axis cutting
of the arcs, viz.,
is
to 1 along
it
at 0, Z,
M,
N...
;
x
71
increases from
any
L to A,
to A,
MtoB, NtoB,
etc.,
e
Fig. 13.
and the limits are as stated
:
if
OA
along
\_&~]~
Q
,
along
LA
[
- 0T,
Purpose of a Substitution.
of a substitution
55.
The purpose
(1)
0J^
,
etc.
dx
dd
dx
dO
dx
dV
MB
along
two-fold.
is
Given an elementary known
integral
to construct a
more complex one, and thus extend one's knowledge
of integrable forms.
(2)
Given an integral which does not fall under the list
of fundamental forms, to reduce it to such form
if
And
it
possible.
must be noted that
often happens that though
it
may reduce to a simpler form, that a further
or
further
substitutions, may be necessary before
substitution,
the integration can be effected.
one substitution
56.
As an
illustration of the first.
r
Beginning with the known result
/
tf x
--
-j
2
= tan~
1
^, let
us put
72
Then
whence
or
As an
illustration of the second, let us try to get
back from
/=
of
The presence
y are powers
of
of
y*dy combined with the fact that
4
y suggests that we should put
y*=z
and
t
all
remaining powers.
/
'
Then
The fact that the denominator is a reciprocal function
2
equidistant from the ends equal) suggests a division by z
(i.e.
coefficients
.
2
is
which
then written as
is
seen to be
,
.
1+
The form
K)
of this suggests further that
we should now put
,+i=,
Z
for then
/ now becomes
(1
1
f
^
\
dz = du.
du
4J
57.
Choice of Substitution.
proper choice of substitution can
No general rules can be
be
the
result
of
only
experience.
but
learn
the
student
something as to the proper
given,
may
course to be taken from observation of the worked-out cases
It will be obvious that a
which follow and from the accompanying remarks.
CHANGE OF THE INDEPENDENT VARIABLE.
Ex.1. Evaluate
Let.2;=22
/
Here
Ex.
2.
cos >Jx d.v.
dx=2zdz.
then
;
^CQS'Jxdx=
i
-cos z. 2 zdz = %
<
I
l
J z
*fx
it
7=
I
was desirable
/
have
x=l we
have
>dx.
-7
Vl+tf2
dx = sec 2
then
When x=0 we
dd.
= 0,
^=
sec
-^|
se c^
=
L
sec
(9
\
It
is
dz=2 sin z = 2 sin\/^]
to get rid of the irrational form of the angle.
Jo
;
cos z
I
J
Evaluate
Put x = tan 9
73
|
Jo
4
2
'
dO =
sec
6>
tan
rf0
=sec--secO = \/2-l.
4
noted that when \/ +^ occurs in the integrand,
or x = acotd or ^r=asinh2 are likely substitutions, for they
2
to be
#=atan#
2
rationalize the radical.
When
2
N/.
-a2
occurs,
#= a sec 0, x=acosecO
or
#=acothz,
are good
substitutions.
Ex.
3.
Evaluate
Let xn =anz~ l
;
then
_1
and
C
J
=
__ j_n
Note that xn = a n z~
l
is
dz
r
na J
=
.Id
,
x/1
-sm~
na
1
n
1
1
2
na
-
cases
generally a proper substitution in
occurs.
Also,
or
#n = an
or
a n cot0 or a n sinli2 for
= an
or
a n cos0 or
.r
n
a^sech^ for
might be used.
When
would be
/v/.
useful.
a n sec6*
or
a"cosec#, or
when
74
'
Ex.
CHAPTER
When \l%ax
4.
x2
occurs,
#=a(l
a useful
trial is
#=
i.e.
cos$),
x
/=
Thus, to evaluate
III.
dr
/
~
\/2o#-.r2 = asin
dx=asm0dd,
;
/ = [a(l-co0)aBi
.
a
J
sin
= a f(l-cosO)d0
=a(<9~sin(9)
X
=
Ex.
When \fa-x
5.
xa cos 6
avers
or
l
a
\l
V a+x
occurs in the inteqrand the substitution
will often be found useful, or perhaps better,
/= x
Evaluate
j
Jj~
x
a cos, 20.
dx.
Let jc=acos20; then
= - 2a 2 /"(cos 2(9 tan
2 siu
cos 0)
dO
= - 2a 2 f cos 2(9 (1 - cos 20) dO
2a 2o/Y cos
/
a
(
2
--
-
)
J sin 2(9
6>
2
2
\
= a_
1+COS4^\
- dd
W -sin4^\
8 /
_ 4 s i n 20 + sin 40
(4(9
= ~ [2 cos"
1
4 L
- - - N/^TP +
a
a
?|
a* V^T^"j
J
-- -
/-=2
= a cos _, x-+ v(^P 2a)^Va
-^2^
2
.
1
ii
58.
When an
/V
as sin"1 -, cos -1
a
CC
.
2
inverse function occurs in the integrand such
/V
a
/v
,
tan -1
a
/y>
,
vers" 1 -,
a
it is
a sin 0, a cos 0, a tan 0, or a vers 6,
work with the direct functions.
x
usually helpful to put
as the case
may
be,
and
CHANGE OF THE INDEPENDENT VARIABLE.
75
other forms of substitution will occur in due course,
Many
but what has been said will
suffice for
present purposes.
EXAMPLES.
1.
Put x*=z.
Evaluate
dx
Put #-1=2;
-
x
dx
2
J 2e
*
Put
+ 2<r +l
t
e*
,
etc.
z
=
l-z
dx
@j
2.
dx
-= dz
x-l
Put e*=z.
L
@
/.
>/ cosh
2
?;
Evaluate
Jo
Draw graphs
3)
to illustrate these
Find the values
two
integrations.
of
(ii)
Interpret the meaning of these integrations
-&=$*.
^.Integrate
Put
Xft Integrate
Put tan
Integrate
^,.
sec^osec^^.
J\cos^-dx.
#
8ll
Integrate
Jf
6
[ST J OHNX 1883 .]
.
Put tan *=.
[TRINITY, 1883.]
Integrate
9.
Integrate
(
r e atan
(v /)
-
^
1+^2
.,
J
r
</./.
(vi)'
. in
-i
o=
/
J
,-
f/./-.
iJiap
(Vll)
/
J
s n
-i
-j-
.7:
cos
;r
-cos~.r
^ c?^
CHAPTER
76
10.
Integrate
11.
Show
that
*
where
59.
III.
THE HYPERBOLIC FUNCTIONS.
To avoid complexity of form in many integrations and to
secure symmetry in the results of integrations of expressions
of similar algebraic form, it is customary to make full use of the
hyperbolic functions and their inverses. (Biff- Calc., Art. 23.)
By analogy with the exponential values of the sine, cosine,
tangent,
the exponential functions
etc.,
~x
PTP
'
2
2
are respectively written
sinh x
cosh
}
tanh
x,
etc.,
x,
more shortly as shx, chx, thcc, etc.
further analogy with the inverse circular functions,
or sometimes
By
if
we
it
= sinh #
cosh#
or
or
tanhx,
etc.,
write the inverse hyperbolic functions
x = sinh-1 u
cosh-%
or
or sometimes as
This notation
Professor
Calculus,
sbr
is
1
^,
or
ch-
tanh-%,
1
^,
th-
1
etc.,
respectively,
^.
now commonly adopted by modern
writers.
George Greenhill (Chapter on the Integral
1888) indicates it as being common amongst
Sir
American writers, and as being frequently employed by
writers on Applied Mathematics. The earlier notation used by
Bertrand,
viz.
sect sin
is
far too
hyp
x,
cumbrous
sect cos
hyp x,
for free use.
sect tan
hyp x,
etc.,
CHANGE OF THE INDEPENDENT VARIABLE.
at
77
The properties of these functions are now usually discussed
some length in books on Trigonometry [see Dr. Hobson's
Trigonometry, pages 303-316]. It is therefore unnecessary to
repeat them here fully. But for the convenience of students
who have not already sufficient familiarity with their use, we
we
reproduce those of the elementary properties which
require for the immediate purpose in hand.
60.
Elementary Properties.
We
clearly
shall
have
cosh 2 z
e-*\ 2
smh 2 #=((& +
^
/e x -e~x \ 2
=1,
J
(
J
=1
;
= cos 20
;
= 1 + cos 20
;
2
2
analogous to cos 6 + sin
x+ sinh 2 x = (\
cosh 2
~
L
=
J)
-
+\
(
Z
)
/
= cosh 2x,
2i
2
analogous to cos
2 cosh 2 x = 1
whence
sin 2
+ cosh 2x,
2
analogous to 2 cos
2
analogous to 2 sin
0=1
cos 20
= 1,
analoous
tanhx =
coth x =
e
x_ e -x
e x -\-e~x
e x -f
e
x
e~x
e~x
=
r
-,
coshx
= cosh x
.
,
sinh x
,
tan 2
to sec 2
analogous to tan
analogous to cot
= --^;
cosO
=
cos
-=,
sin
tanh x
etc.
It is
unnecessary to point out methods of proof or analogies
and the following results may be proved by the
further,
;
CHAPTER
78
III.
student as exercises, and will form a convenient
for
list
reference
sinh (x
+ y) = sinh x cosh y + cosh x sinh y,
= sinh x cosh y cosh x sinh y,
cosh (# -f y) = cosh x cosh y + sinh # sinh y,
cosh (z y) = cosh # cosh y sinh # sinh y,
sinh 2# = 2 sinh cosh x,
cosh 2x = cosh 2 x + sinh 2 x.
= 2cosh 2 z-l
= 1 + 2 sinh 2 x,
sinh (x
y)
a;
tanh
2
1
sinh
a?
a;
+ tanh
2,
,
x
^ cosh
+ sinh y = 2 sinh
^
9
,
|
etc.
Xs
s
a;
5
X
X
= a;+-n+ ^- +
3
5
should be remarked that such expressions as sin0,
is complex, i.e. of the form u + iv, do not
cos^, etc., where
come under the heading of the sines and cosines defined
61.
It
geometrically in the early parts of trigonometry.
are re-defined now by the exponential values
alO
sin
etc.,
6,
for
standing for
any value
-
_ a^
10
sin i x =
I
cos
0,
f
standing for
i
= /!,
sinh
x,
= cosh x,
tan x = tanh x
coth x,
cot ix =
cos i x
i
--
^tfl
;
of 6 real or complex.
Then writing 0=ix, where
They
i
}
i
^
-
16
:
CHANGE OF THE INDEPENDENT VARIABLE.
79
Also the ordinary formulae of trigonometry can be proved
from
we have
these definitions, viz.,
cos
2
+ sin
= 1,
2
sin (0 -f 0) = sin
cos
sin
</>-}-cos
<f>,
etc.,
and the
and
restriction of the reality of
is
removed.
Then, having proved the addition formulae for the sines
and cosines from these definitions, we have
sin (u
+ iv)
sin
u cos iv+ cos u sin tv
= sin u cosh v+
cos
1
u sinh v,
etc.
62.
Inverse Hyperbolic Functions.
We are, in the Integral Calculus, more particularly interested
in the inverse forms.
.
Let us search for the meaning of the inverse function
.x
1
,
sinh-
Put
x
= sinh
.
Then
a
y=
e~y
cy
~
S**t*
;
?y-2- e y-l=0
a
)
u_
a
and remembering that
e
t2X7r
,
= cos 2A-7T
we may,
where X
f
is
sin 2\7r
an integer,
=1
to retain generality, write this as
'-2iAi
a
or
x\/a? + xz
,
CHAPTER
80
III.
Now
log
*
a
\
= 2(A7r-log(-l)-log
= -.'2X'-l)r-log
for
n and
\'
being integers.
y = fjuw + (
Thus,
where
/UL
is
an
- 1)M log X+ ^W
a
integer.
The "principal value"
of y is then log
-ix
-
^ 4- #
2
,
and
it
K
is
usual to take this as synonymous with sinh" 1 -, omitting
the generality obtained by the addition of unreal constants.
63.
Similarly putting
cosh" 1 - = y,
x
-
and
and omitting as before the generality derived from the unreal
constants,
we
shall take the solution
y = log
viz.
the
as cosh"
"principal
1
value"
x+JUF-a*
of
y
with
the
positive
sign
T
-,
a
and therefore cosh" 1
i
a
synonymous with
,
log
is
to
be understood
as
CHANGE OF THE INDEPENDENT VARIABLE.
64.
tanh- 1 - = y
Again, putting
y
x
-
= tanh y =
a
ezv
and therefore
and omitting
81
=
a
x
,
generalities as before,
a+x
,x 1
--i-^log
a 2 ^ ax
,
65.
??
coth- 1 - = \ log
a 2*
x ~~ a
Similarly,
,x ,
,
sech- 1 - = log
.05
x
cosech- 1 - = log
a
,
66.
,
x
We shall therefore consider
sinh-1 -
as
meaning
cosh-1 -
as
meaning
,x
tanh -1
a
as
meaning
as
meaning
&
2
meaning
log
1,0;
,
-,
coth"1
,
,
.
x
v
i
a
.
as
ct
1.
los
x+a
---
*x
a
city
a?
>
meaning
T
log
x
&-f \/a 2 -h
x
2
Periodicity of the Hyperbolic Functions.
These hyperbolic functions are periodic.
is
2
.
as
cosech-1
x+Jx
log
.1,- log^
x
a
sech- 1t
log
,
,
67.
.
But the
periodi-
= an
integer),
imaginary.
For, since
we have
e
lXn
E.I.C.
i
sin XTT
=
(
1)
x
,
(X
cosh(a;-hX/7r)=
Similarly, sinh
whence
= cos XTT
=
1 )* sinh x,
= tanh #.
(a?-f X<TT)
(
tanh (x + X* TT)
F
CHAPTER
82
III.
Thus, the periodicity of sinh x and cosh x is 2?,
that of tanh x and coth x
is TTI.
Also
sinh
=
/>0
_g
= 0,
^
-~
= gO
x,
[
cosh
sinh
*TT
=
COSh
f7r
= COS7T=
= 1,
tanh
= 0,
etc.
1,
etc.
cosh- 1 (
Again,
= log (
z)
+ \2
z
2
1
= -log (Z+JZ 2 = COS
- z) = log ( -
=
l,
I/
tanh-!(
sinh" 1 z,
- z) = Iflog 12 = - 1,log
v
etc.,
analogous to the properties of the circular functions,
cos" 1 (
z)
=
cos- 1 z
+
TT,
sin~ 1 (
z)
=
sin" 1 2?,
etc.
68.
Geometrical Interpretation.
Let a rectangular hyperbola x2 y z = a 2 and its auxiliary
circle be drawn
then any point on the hyperbola may be
;
represented by either of the parameters
$,|
x
or
or u
a cosh
cosh u,
|
y*asrahtM
by putting
CHANGE OF THE INDEPENDENT VARIABLE.
and u are connected by the equations
Hence
sec 9
tan
or
= cosh u
= sinh u.
P
be the point
(or u) on the hyperbolic arc AP;
to the auxiliary circle.
the
the ordinate,
tangent from
Let
PN
83
N
NT
Fig. 14.
Then obviously the
Hence, the angle
Also,
since
obvious, since
The area
abscissa
NOT is the parameter
0.
ON*-NT* = a?, it follows that NT=y,
y = a tan 9, as also NT= a tan 9.
of the portion
NA P of the hyperbola
[u
a sinh u a sinh u
Jxn ydx=\Jo
.
sinh z udu
{u,o
a 2 /sinh ~~
2u
Also, area of triangle
a 2u
_ a 2 sinh 2u
\
"
~*
ONP = J
2
sinh
2?^
--= 1= a cosh t6 a sinh u = a---j
.
,
.
Hence the area
= o?u
-q%
of the hyperbolic sector
a2 9
,
analogous to
-^r
'-
e
.,
.
OAP
T
tor the circular sector
as
is
CHAPTER
84
III.
This indicates the meaning of u,
u
viz.
CA P
2 area of hyperbolic sector
-
.
a2
with the hyperbola from which these
termed hyperbolic functions.
For other properties in connection with this figure, see
Greenhill's Chapter on the Integral Calculus, p. 27, or
Hobson's Trigonometry, p. 309, and an " Essai sur les Fonct.
It is this connection
transcendental functions are
Hyperboliques,"
1875, cited
by
Mem.
d.
I.
Soc. des Sciences Phys.,
Bordeaux,
Greenhill.
cosh u = sec
Since
and
.*.
we have
= tan 0,
sinh
i
tanh
u=
7
-
= sin 0-
sec 6
coth
u = cosec 0,
etc.,
which express functions of u
in terms of u,
we
terms of
have
obviously
sin = tanh u,
cos
sech u,
tan
= sinh u,
= cosech u,
cot
69.
in
0.
Again, expressing
etc.
The Gudermannian.
The angle
which may therefore be variously expressed as
0,
sin" 1 (tanh u), cos" 1 (sech u), tan" 1 (sinh u},
sec" 1 (cosh u), or cosec" 1 (coth u), is called
cot" 1 (cosech u),
by Cay ley the
"
Gudermannian
of u* (Elliptic Functions, p. 56), and
denoted by him by the convenient notation
"
u = gd" 1 0.
or inversely
Then
sin 0, cos 0, tan
Again,
log tan
fir
(
\-
he denotes by sg u, eg u, tg u.
0\
)
LJ
hs
= log (sec + tan 0)
= log (cosh u sinh u) = log eu
-f-
*So named from Gudermann, who
(Cayley, p. 44).
specially
discussed
this
u.
function
CHANGE OF THE INDEPENDENT VARIABLE.
Hence, gd u
is
such that
/
log tan (\4
j
or
Differentiating
f)\
+ - = gd"
which
6,
logtanf^ + ^J
dO = log tan
sec
J
and
1
)
2t/
(sec
$d$
is
a degenerate
the same thing.
+ tan#), we
get
Hence,
(|
form
is
or log (sec
as the differential coefficient.
sec
85
+ 1) = gd-
of the
1
0,
more general integral
c
I
.
I s/1
-^ sin
Tables of the
70.
inverse
.
,
2
values
Gudermannian
Fonctions Elliptiques,
of
0,
vol.
of
u = log tan T + SJ)* ^ e
f
-
^ ne
are given by Legendre, Theorie des
to 12 places of decimals for angles
ii.,
in the first quadrant.
They will be found to seven places at
in
intervals
Hobson's
Trigonometry, p. 316, and to five
degree
places at degree intervals in Greenhill's Elliptic Functions,
p. 16, whence it is easy to extract the values of u correspond-
ing to any angle 0, or the value of 9 corresponding to any
given value of u, and hence from the relations cosh u = sec 0,
sinh u tan 0, etc., we can find the values of the hyperbolic functions coshu, sinh
it,
etc.,
the intermediary angle
secants, tangents, etc.
any values of u by the use
means
of the ordinary tables
by
p. 15,
71.
to
of
In the absence of direct tables of the
hyperbolic functions this will be the proper
putation to follow in numerical calculations.
Report
of
for
mode
of
com-
See Lodge's
Brit. Assoc. 1888, and remarks by Greenhill on
Elliptic
Functions*
Unless extremely close approximations are required
will be sufficient to take the values of log tan
f
it
T + Q) from
"The Smithsonian Institute of the City of Washington publishes a set of
Mathematical Tables of the Hyperbolic Functions, by G. F. Becker and C. E.
van Orstand.
The Harvard University Press publishes Tables of Complex Hyperbolic
Circular Functions, by A. E. Keiinelly.
and
CHAPTER
III.
the following graph, which indicates the march of the func= to = 90. There is not much deviation
tion from
from a straight
line
from
=
to
= 45,
but beyond that the
function begins to increase more rapidly, passing from 4'7413
at 89 to oo at 90.
For the first part of the graph, obviously
the ordinary rule of proportional parts will give a fair approxi-
y
3*4
CHANGE OF THE INDEPENDENT VARIABLE.
Let us illustrate the use of the graph
72.
Fig. 15.
If, for instance,
we
should of course be
e
_ e -i
-
,
u
lies
1
/.
1= tan 49
To check
35' =1*1744,
By
proportional parts,
0-45 ~
_ -1186
1293
5
from
35'
+ 45
'
-49
35'.
the tables of natural tangents.
this,
e-e~
which shows an error
73.
it).
between '8814 and 1'0107,
=4
sinh
from the tables (which
and therefore we can check
tan" 1 sinh '8814 = 45,
tan" 1 sinh 1 "0107 = 50,
tan" 1 sinh 1 '0000 = 6.
i.e.
or the values tabulated in
require the value of sinh
i
87
There
of
l
2-7183 -'3679
about
2'3504
'0008.
also a useful table giving the values of various
is
ei
eJ, e*, ^...^10; e ^ e l, ci, el
e* ei, ei, ei, ei, e*, el, et, eA,
Bottomley's tables, p. 56, which will be convenient in some
viz.
powers of
e,
;
}
in
E.g. (extracting the values
cases.
from these tables)
If great accuracy be required it will be necessary to use the 7, or
perhaps, in cases, the 12-figure tables, but such extreme accuracy would
but seldom be required in practice.
EXAMPLES.
Establish the following results
1.
/
3.
/
5.
/
6.
I
7.
:
xdx
cosh #cfo?= sinh #.
2.
/
sinh
sech 2 ;r dx = tanh x.
4.
/
cosech 2 .^ dx
p
.
,
2
cosh x.
- coth x.
dx=\ sech x tanh tf o?^ = - sech x.
O?^P
=
*dE*d
/
cosech x coth
8.
xdx=
cosech x.
(zgxdx=*gx.
9.
j
-tgA
CHAPTER
88
Integrals of cosecx and sec
74.
x*
Let tan
=
;
a?.
then, taking the logarithmic differential,
1
2 tan
III.
x
sec 2
x
~a# =
7
2
dz
-dx - = dz
i.e.
,
z
.
z
since
^
Thus
I
cosec x
dx =
\
J
J
In this result put x
And
sec
That
I
is,
sec #
we have
75.
z
log tan
~.
dy.
y dy = log tan Qj + | )
dx or
I
-
Jcosic
J
as
J
then dx
-^-{-y\
&
J
= log z
=
sin x
= log
tan (-r+ ~
\4
)
2/
= gd'
1
seen before.
From
we may
this result
infer the integral of
dx
f
acosc
J
For
where
^ = \/a
2
-f 6
2
and tan a = T
o
dx
f
}
;
1 f
a cos x + b sin
dx
x~ Rj sin (z -fa)
4f
Rj
x-\-a
76.
The
f
J
integrals of cosecha?
j =
cosech # aa;
f
dx
J
smh x
.
,
and sechx give no trouble.
=2o(
x
j e
d xe~ x
= 21f -^e^
2a;
J e
,
- dx
1
CHANGE OF THE INDEPENDENT VARIABLE.
f
I
sech x dx =
f
\
J
dx
,
cosh x
=2
f
1
J 1
ex
,
-
+e
52x
dx = 2
(
89
de x
\
J
= cos" 1 le^
2x
= 2 tan" 1 ex
or
= cos" 1 ( tanh x)
= cos" 1 (tanh a;) + const.
= sin" 1 (sech x) + const.
77.
Integrals of
The
differential
Thus
d
(
CHAPTER
90
To
I.
Jd
N/o _ x
1
Integrals of
78.
find
2
I
III.
x\ -Jtf+x2
^ pu
2
\/#
,
2
^ _ a sjn
a2
.
.
dx = a cos
= 771 a sin
.
g?Va**
or
a2
.acos$ + -^ 6
or
a
.
2
To
II.
find
2
2
\
\/a -\-x dx,
put # = a sinh 2; then
dx = a cosh 2
Then, since
1
-f-
cfa
f
i.e.
dz.
= cosh 2 0, we
sinh 2
,#
a
.
sin" 1 -.
2
have
= -1
.
,
a sinh
.
a cosh
a2
+ -^-0,
a
/Ja
J
or
and
in the latter form,
if
the integral be indefinite,
we may
drop out the a in the denominator of the logarithm, as this will
only add a constant to the whole.
III.
To
find
I
v^e
2
a?dx, put x = a cosh z\ then
dx = a sinh z dz.
1= sinh 2 z,
2
Then, since cosh z
I
-Jx*
a 2 dx = a?
\
sinh 2 z dz =
I
-^-
=
4
sinh 20
(cosh 2z
--^- = 5 rtsi
2
-Z
I)dz
_)
,
CHANGE OF THE INDEPENDENT VARIABLE.
__
-a a
xjx ---
.
2
2
Jx 2 - a*dx =
2
%
Y COS
z
z
91
'a;
a
"
a
and the a in the denominator may be omitted, as
before, if the
integral be indefinite.
[This last integral has already appeared in Art. 68 in
finding the area of a portion of space bounded by a rectangular hyperbola, an ordinate and the a>axis].
79.
From
we may deduce the integration
tan x = t, sec 2 xdx = dt, we have
of sec 3
Art. 78
For, putting
3
Jsec
xdx=\J \/l -M 2 dt =
1
h H sinh"
s
2
a;.
1
2
or
i.e.
I
sec3
xdx = -^ tan x set x + lg ( sec & + tan a;)
1
or
1
,
1
,
,
=-tanxseca;+^
sm x
1
,
log
,
z
+ sin x
/TT
sin
.
x
x\
--f- logtani T + s ).
2cos 2 z 2
\4 2/
or
,
=TT
Just in the same way, putting cotic = c, cosec 2 #cfo=
we have
I
cosec** dx =
=
2
-jVi+C
cVl+c 2
^ cot x cosec #
1
COS X
1,
4
1 cos
1
We may now
ic
,
1
+ COS X
1
cos x
,
slgt
&
2
7^ is
a;)
x
an o'
deduce from Art. 77 the integration of
[dv
where
1
x-\- cot
^ log (cosec
lop*
2 sin 2 a?
-
80.
(?C=-
Jyy
a quadratic function of
x, viz.
dc,
CHAPTER
92
CASE
a
I.
When a
is
III.
Positive.
positive
we may
write this integral as
If
dx
__ __
a
a
which we may arrange as
dx
^ C
a
\
J
2
2
/7~, 6\ - 6 -ac
+
V(* a)
1
C
Va
I
-ST-
dx_
/7~
1
,
ax
,
ac-/
V(* + J + ~^-
J
2
according as 6 is greater or less than ac,
of the integral is therefore (Art. 77)
+
-= cosh~ 1-7,=^
2
ac
Va
Jb
6\ 2
,
and the
real
form
oa; +6
-= 8HjOi~*-2:====,
62
v'oc
v-
b
I
or
.
,
,
<
2
oc.
according as 6 is >> or
In either case the integral may be written in the logarithmic
form
_
-j= log (ax + 6 +
JaJax 2 +2bx + c),
vo
1
the constant
-r^log^-oc
log (ax + b
-j=
<*
v
+ N/oS),
being omitted.
cosh" 1 z = sinh-
Also, since
i.e.
V
2
,
,
1
and
1
,
.
1
j= cosh-
Va
+b =
-ax
====
V6
-r= sinh" 1
and
7
2
ac
1
=
-^
.
-7= smhVc6
V6 -ac
1
r= cosh- 1
^^
2
.^
>
81.
or
<
CASE
in*
> ac)
2
(6
is
<oc),
positive and
ac respectively.
II.
a* Negative.
0*2;
f
If
o^
2
06
which forms may therefore be taken when a
b2
/7
(D
the
Then our
integral
integral
,
,
2
may
_ f
A\
o be negative, write
be written
r
^
T^/rTT
JV-^ +z^ + z
2
CHANGE OF THE INDEPENDENT VARIABLE.
-A
or
.
I
\
-_+
lAc
93
dX
l
J \~~Z2~
^/y_A
sin" 1
or
or,
v^
omitting a constant,
1
j==a cos" -^2
ac L
1
1
Also, since cos" z = sin- ^/l
s6 2
To sum up then;
we have
~%
2
,
we have
" 1.
cos
82.
sin- 1^ = cc
for
v&
>J
the results
vo 2 ac
appears that when
ac
it
R = ax +2bx+c
2
:
,
\/- a
a negative,
dx
ac,
a
positive,
or
< ac,
Jac-b*
and the
real
Ex.
83.
We may
or
it
form
to be chosen in each case.
is
dx
1.
Integrate
dx
= 1;=
write this
V2J
may be
Integrate
f
j=-.
-
Jv/4
This
may
1
p=
.
.
be written
and therefore
is
j=
^sin\/2
1
^^,
V41
which may be written as
1
,
2\/2
,
sinn"" 1
written -7= cosh"1 -j-= >/2.r2 + 3^ + 4
rejecting the constant-y-z log
2.
=
v(.r
or
Ex.
f
T=-I
\/23
CHAPTER
94
2
In exactly the same way,
84.
positive or
of Art. 78.
when a
is
III.
I
\/ ax -}-2bx -\-cdx,
when a
is
negative, can be deduced from the results
It appears then that the general rule in all cases of
dx
I171
or
where R is quadratic, will be, " Divide out the coefficient of x2
and then complete the square, and then make use of the suitable
standard form."
form
85. Functions of the
be integrated
may
.
first putting Ax-\-B into the form A(a# +&) + /*, which may
be done either by inspection or by equating the coefficients,
and we obtain
by
Ax+B _A
ax+b aB Ab
""
a
a
R
1
~
rp,
_
A
The
integral of the first fraction
is
d
*JR,
and that of the
second has been discussed in Arts. 80, 81.
More general forms, such as
or
where / and
cj>
are rational integral algebraic polynomials in x,
are to be discussed later.
,
the student
\/R
fclx
may
be
preferable to that given,
e.g.
should observe other forms into which the results
thrown.
For some purposes a 'double angle result
3
I
J
ax
r
*Jx(a^x)
But we may throw
usng
is
ax
laT~fa
this into the
form 2 sin
l
z
by making
z
2
'-
and
CHANGE OF THE INDEPENDENT VARIABLE.
95
Then
(2)
f
J
-x
(3)
(a
the ordinary form
;
but writing this =2sinh~ 1 2,
i.e.
>x>
b).
cosh -1 (22;2 +l),
*
if
or
J
and so for other
87.
if
a
>
6,
< 6,
cases.
Of such the following forms are
particularly useful
:
and the others can be derived from
these forms as shown above.
a
a
88.
It will be noticed also in
the integral of Art. 81,
many
cases, as, for instance, in
viz.
that the Jft of the integrand reappears in the integral.
did not do so when the result was arrived at as
1
/"OQ~~ *,
\s\J&
/
a
ax+b
i
J
It
CHAPTER
96
III.
but was made to do so by the subsequent transformation
cos" 1 z
32
sin" 1*/!
Examine the
.
dx
f
t
d*
f
J
,
2
va
2
x
x
i
A
given as sin
I
This could be written
earlier integral (Art. 44)
.
.
=cos -i^
a2
-x
\
a
.e.
a
So
dx
also
f
J
a
_dx_ = cosh -iN/a' +a
2
could be written as
f
J va?2
f
i
-i
Similarly
+a
2
c^a;
2
-a 2
J
vR
Jx/a;
^_
a
1- =
1
= cosh" ,&
smh"iJx
,
.
a
= = sinh~
could be written as
,
a
,
.
2
7
1
x2
(R
&
z
a?
TL
a2 ).
And though these forms are obviously not the simplest forms
of the various integrals, it is frequently desirable to adopt
them, as they exhibit a visible relation between the integrand
result of integration.
The simplest forms are those
tabulated to be remembered in the two lists of standard forms,
and the
Arts 44 and
We
89.
89.
are
now
in a position to
make our
list
of
ADDITIONAL STANDARD FORMS.
.
1
cosh xdx = sinh x
and
2.
I
sech 2 x
and
1
3.
r^
J
cosh 2 a?
J
smh 2 #
4.
dx
tanh x
dx = cosh x.
I
sinh x
I
cosech 2 x dx =
dx = sech x tanh x dx =
I
sech
x.
J
dx
\
cosech x coth x dx =
cosech x,
coth x.
t
CHANGE OF THE INDEPENDENT VARIABLE.
5
dx
f
I
yj&+a*
dx
f
6.
= log x +
\--r-
+ a* = smh _
tf
,
.
-
.
,
T
l
= log + N/a:
se
,
t*
2
-
x
-.
a
'
2
97
= cosh- ,x-.
.
1
7.
8.
f
9.
-
2
a
a---
x-x^
-
a 2, dx =-
,-^j
2
N/a;
7
<L
10.
dx
-7=
(
cosh
-
,
x
.
U,
t
x
-=
ljx(a-x)
12.
13.
^
Vet
JV^-a) =200811-^.
f
.
cosec
I
a? (/a:
=
I
J
J
-
seccr^
14.
sm x
= lo^ tan -.
dx
N**
X =logtan (j
f
J COS
_~
a
2
n
1C.
17.
I
dx
dx
.,
Ja-
2
,,
^
,
\
_
sin
1
,
a;)
_j
CT
sc
x
'
I
1
-=
-
LI
a
'la
-W
1
,(a;<a) = ^
a-
= log (sec + tan
<*>
(x >> a) =
/
-
"
cosech'
-
.,
,
se
=
.
f
f
1
=
xja + x 2
xja'
Jx--a
18.
"9
~~7~^
J
-1*7
dx
f
1 o.
+^)
^/
z
1
x
1
= -tanh
a
x
i
coth" 1
1
#
a
-.
a
.
customary to obtain 17 and 18 when wanted, rather
than to commit them to memory. They will be discussed
It is
later (Art. 127).
CHAPTER
98
III.
EXAMPLES.
Write down the integrals
4.
^
5.
rc
of
,
(ax
-
- 2bx + c
b) (ax*
.
a
"
'
7.
cosec nx,
9.
-|
,
(cot x
sin x
Deduce
1 2.
Deduce
Show
(1J.
f
I
I
-
5
a sin x +
1
+ # cos a;'
.
-- -
4 cos 3 a:- 3 cos a:
-j
sin 2a;
+ cos 2a;
-a
---~,
l-tan 2 a;
,
b cos
a;
+ d cos
a*
'
'
c sin
a;
^
cosec x dx = log tan -
sec x dx
by expressing
= log tan 7 +
9 ) by
(
(i)
putting
sina;
=
(ii)
putting
sec
+ tan x = z.
that
Integrate
a;
+ tan x).
1
10.
11
- 4x 2 + 4.T + 5.
,
I
f
\
a:
,
sec x dx = cosh" 1 (sec x).
costidO
sin 0x/l
J si
-
sin 271
cosec x as
CHANGE OF THE INDEPENDENT VARIABLE.
99
GENEEAL EXAMPLES.
If
1.
APE
diameter
Lt
PN an
be a semicircle, centre 0, and
ordinate to the
AB, and P'N' a contiguous ordinate, show that
AW = circular measure of
^t^ -
the angle
OPN
t
N
the summation being from the centre to any point
and B, and NN' being indefinitely diminished.
Find
2.
the-
area in the
first
between
quadrant bounded by the axes of
ordinates, the ordinate x
= x and
the range x =
= a on
the curve x
l
2
=a
2
co-
6*
g.
the #-axis be divided into n equal
of
h
and
portions
length
rectangles be inscribed in the Newtonian
limit
the
of the area of the last of these rectangles
examine
manner,
If
when h
to x
is
to x
Find the whole area from x
indefinitely diminished.
= a.
3.
Find the value
4.
Evaluate
of
_____
x
2x
+ ae dx
>Je
I
[R. P.]
^
1884>]
[OXFORD SECOND PUBLIC Ex.,
1880.]
[L
*
a
+x)dx
(1
+ 3x +
J.
[COLLEGES
4)
j8,
1891.]
x-l
;iv)
(v)
dx.
j (x
2
+ 2x-
fJ
idx.
2
(.T
[TRINITY, 1892.]
1)'
+
2x
+ 5) T
[MATH. TRIP.
===
f/7/v.
v tt x
may
(i)
or as
2 cos->
(ii)
2 sin'i
2v/a
or
as
(iii)
2 tan" 1 A/
=5,
where
,
1887.]
be ex-
CHAPTER
100
6.
If
R = ax
+
2br.
+ c and
III.
-ac
K-
-r-,
show that
a
1
tanh1
or
/\/
according as a
7.
Evaluate
is
-a
positive or negative.
(i)
|.-^=
6
-
(Put a
6?#
:( (*41)V**-1
(ii)
9.
Integrate
.
= tanft)
(Puta? =
.
[0xF L> 1888
.
CHANGE OF THE INDEPENDENT VARIABLE.
can always be rationalized (provided
tion) by one of the substitutions
u
=
CUj.
a rational algebraic func-
is
_ 1 + if
=
2
u
'l>i
YT^
7-^37
/
s
Find the relation connecting x and
'
T^v-
vr^a
[CoLL>
f*-
Show
J
15.
^'
z*(^-$fdz =.-.
Jj
v
-
f
sin 6
LUx.
JU,
I.b8.j
dO
rj
-'
(
,
n
,
r
...
cos 6>Ja eos 2
J c
-f
26 sin 6 cos d
+
26 sin
1
2
J sin ^v/rt cos 6>
(v)
+ c sin 2 ^'
dO
f
cos
+ c sin 2 0'
^
f
J sin
0JauQ&0 + b snT2 ^ + c
[TRIN., 1888.]
Integrate
r
^_
I,,-'
17.
C. S., 1889.]
Integrate
f
16.
]
^ O ^ Q.
1
*
that
a> 189(X
being given
y,
[I.
14.
101
(a)
;
^
,;>
'
(x
pj
2
;ii)
''
}(a + ba?)J&=&'
- &x+
13)
c/r,
first
|
putting
z2
^
/
)
2
Evaluate
/
- 6x + 13 - y.
[TRIN., 1888.]
directly,
second
(Draw a graph and explain
by
fully.)
K
(b)
2
Evaluate
(ax
-
'2bx
+ c)
r/,r,
J
and explain l>y a graph the result when 26 2
Obtain the same result by substituting
"2l.f-
+c=
//,
taking
= 3ac.
2
b'
< ac.
:','>
Also obtain
(us-
I
- 2bx +
c)
dx
l>y
this substitution,
Jo
your limits
IS.
for y
by means
of the graph.
Point out the fallacy in the following argument:
J'
explaining
CHAPTER
102
But putting x =
1
When x= ,
17
=
When
dy
dx=
-,
|.
2
_ f
dy
for,
as the result
is
1,
,
y
y
III.
1
1,
y= y=
1.)
U
(fa
numerical, the letter used in integration cannot
affect the result.
19.
Point out the fallacy in the following reasoning:
We
have,
if
we put
a
= -
x=
e*,
z
dx
But when
1
we have
,
d
and these two
20.
results
do not agree.
[R.P.
Prove that
,gd(Jgd.)-is
and show that
[CAYLEY
3
gd u = a t u + asu + a5 u& +
if
1
gd" u
then will
= a^i - a^v? + abu5 -
. . .
,
,
....
X
21.
If
l
gd~ x = x + S3
show that
+ ^5
S_S
Sp+l = S +
that
l
and
^,,
- /TlX^
+
\gj8n-*
and that
^3 = 2,
S5 =
/^\o,
+
7i7T
-
+
cos
(^4)^-4
16,
5^
= 272,
^-
= sm HIT
.
^9 = 7936,
etc.
Calc., Art. 573, etc.]
i/.
22.
,
-^-
Integrate
c
by putting
+ = (a - c)0'2
or
(c
- a)z 2
,
a > or < c.
according as
Taking the case ft>c, consider the same integral with a + da replacing a, subtract the original integral, divide by da, and take the
limit
when da
is
indefinitely diminished.
CHANGE OF THE INDEPENDENT VARIABLE.
103
fix
dx
r
Hence obtain
J
(a
+ x)*(G + ap
Jdx
..
(a+x)(t+xy
2
2
2
- a2
)(z -c )*
;
(a?
(i)
if
rt>c,
if
a<c.
'
(ii)
c
24.
Show
(x-p)*
~^T
^
2
</
2
and
25.
yvoa;
-f
2 to
^'
[MATH. TRIP.,
n +*
that
where
G
c?a;
=
r
1
2
7
2
_
1878.]'
TTJTI
1
O/
~?
i
)'
^)
= ftp 2 + 26p + c
-f c
= (ap + b)x + bp + c.
[COLLEGES, 1901. ]
F(x)
If
prove that
[Ox. J.
26.
Integrate
M. Sen.,
1904.]
-
(i)
J 1
-x 2
fl
-X 2
\/l
+x*
'
27.
Show
that
if ^(:/-, //)
be a rational function of x and
y,
can be thrown into rational form by the substitution
"
X+
yx
4=
2"
+S
Hence show that
Jf
28.
-
(l^
VI +2
Show that
2
.
if
/*'(./, //)
l>o
2
any rational integral function of
F (x, Jax' + 2bx + c) <lx
2
and
y,
CHAPTER
104
III.
can be thrown into rational form by any of the substitutions
(1)
-ax 1 + 2bx
(2)
Jo**
+ 21
=z2 (x
+ '2bx + r = 0.
x
(3)
where x l1
x.
2
are the roots of ax
2
x
x
)
[BEKTRAND,
C.I., p. 39.]
Apply each of these methods to the integration of
xdx
f
showing that the result
in each case reduces to
2
v/^ -6.T
as derived
9Q
Ziu.
by the method
+ 8 + 3 cosh-
(x
-
3),
of Art. 85.
~3
Tf
II
^n^T
n^ii
y.
Oil* JJ
J->
dx
show that
1
it"
= 1-
-
dy
-- .
f
and hence obtain Cardan's formula
^
for the solution of a cubic.
[J.
30.
cosxd.r
f-
Evaluate
J
o
M. SCH. Ox.]
-
1
^-,
Sin-a COSnZ
and deduce the expansion
2a
sin 2a
2
= 1 + -2 sin 2 a + ^
4
3.5
sin 4 a
+
.
.
.
,
TT
where - > a > 0.
I.
[OxF.
31.
Show
P., 1915. J
that
dx
_
_
,
.-
^^ Ltill
_l
x
~~
"
r~
i
[EULEK, C.I.,
Integrate
32.
Show
f
iv.]
t
JHthat the integrals
dx
Jl
-{B
w
m2a;
r
I'-'"
1
and
x
f
m ~ dx
l
Jl
are reduced to the integration of rational fractions
m
2"
substitutions 2x - I = u^x
and 2z m - 1 = ?r' n
by the respective
1
.
[LEXELL, ^c^es
r/e
Petersbourg, 1781,
ii.
;
LACROIX, C.D.,
ii.^p. 65.]
CHAPTER
IV
INTEGRATION BY PARTS.
AND
POWERS OF SINES
COSINES.
INTEGRATION BY PARTS.
Let u and
90.
differentiations,
iv
and
dru
Thus u" stands
.
,
,
on with
u
,
,
w
s
,
be functions of
and
x,
let accents denote
with respect to x.
suffixes integrations,
for
and
:
T
dx"
f
w.,
for
f f
I
L
J
~|
\ivdx \dx, and so
J
etc.
d
dw
du
= u -=+ w-^-j- (uw)
dx
dx
dx
mi
,
Then
which we may write as
(uw)' = ^lw
uw=
It follows that
or
I
This
Let
may
uiv dx -f-
wu' dx
1
wn
dx.
be put into another form.
u = <j)(x) and
Then the above
i.e.
wu'.
f
\
mv' dx = uw
w'
i.e.
I,
w=
f
J
-f-
rule
~j-}
I
may
= \js(x) = v,
\js(x)
dxi\.
be written
i
/
JJ^'U
)c
'
say
f
U
QiQj //'/* 77^
'?/'?)
105
I
'>y
f
'
oj
fl'T
f
;
so that
CHAPTER
106
or the
I
and
two functions
</)(x)\ls(x)dx
= \fs(x)
i.e.
I
i
be interchanged, and then
may
\fs
I
IV.
uvdx = vul
\ls'(x) \
I
dxf
<j>(x)
I
0(#) dx
dx
;
j-
v'u^dx.
I
Thus, in integrating the product of two functions, if the
integral be not at once obtainable, it is possible to connect
the integral
<j>(x)\/s(x)dx
with either of two new integrals,
r
>
(x
viz.
those of
J{^ (x)dx \ dx
J
I
J
and supposing that the integral of one of the two factors
0(x), \fs(x) is known, one of these new integrals may be more
easily obtainable than that of the original product.
The
91.
rule
be put into words thus
may
Int. of Prod.
:
= 1 function x Integral of 2 nd
\fs
- Integral of [ITifF. Co. of 1 x Int.
st
.
st
Ex.
92.
Here
/
it is
x sin nx dx.
important to connect
in which the factor
x has
possible /.#sin
if
There
been removed.
u=x
and
v = sin
or
u = sin nx
and
v=x
it
we
nx
;
will be observed that in the connected integral
take the
first alternative.
,
u
J-iien
X)
u '_i1
cos
,
v
v\
sin7W7,
Thus, by the rule,
[
JXB
- f
c s
=
l\r
n#~]
~n~]~J
--x]^
lu'v^dx,
Hence the removal of x will be
been differentiated, v integrated.
if
nx dx with another
a choice as to whether
is
we put
but
of 2 nd ],
xcosnx
+
,
n
x cos
~ nx
1 f
- cos
/
nJ
sin
nx
t
I
L~
~o
cos
n
nx dx
nx
u has
effected
INTEGRATION BY PARTS.
93. It is to be noted that unity
the factors to aid an integration.
Thus
or as
/
it
log
may be
.'.
dx =
1
1
107
be regarded as one of
may
x dx
log
=xlog.v-
I
x - dx
=x\ogx-
\
I
written = x log e ( J
dx
.
Repetition of the Operation.
94.
The operation of integration by parts may be repeated as
often as may be considered necessary for the evaluation of the
original integral.
f
rni
Thus
/
J
i
/ A
.\ /
xA4 sin nx dx=(,i
) (
-
\
COS ?W? \
-
n
-
)
/
/
J
/
/
n\
t
/
3
(4 r )(
\
t
COS
7
- dx,
- ?k\
11
I
.
//(4
snj;\
8
2 .,)
J(4.3.
(
+
J
.
dx
^)*=(4.S..*)('-S~)- /(4.3. 2.
Then adding and subtracting
alternately,
cos
?wA
sn nx
,.
^
/
The student
until the
will note that
whole operation
no arithmetical simplification
complete.
The
if
I
.r
4
sin
nx dx = P cos tix + Q sin
p. .^4.
.
where
,
^3
^
' = 4^-4.
2
is
total operation is
simplification be postponed to the end.
obviously have
liable to error
We now
is
7.
3.
nx,
4.3.2.1
.'-'
2-*4
.
attempted
much
less
108
CHAPTER
IV.
The General Rule.
95.
It is
obviously possible to formulate a general rule for the
And such a method is most serviceable in
repeated operation.
practice.
The rule
I
is
uvdx = uv - u'r z + U"VB - u'"v
+ ...+(_ 1)- %<-i>t>n
l
+ (-!)
where u
(n
~ l)
u with n-l
written for
is
tu (n) v n dx
t
accents,
i.e.
the
<>-l)
th
differential coefficient of u.
For
\
J
J
uv dx
= uv
u'v l dx
= u'v 2
u"v2 dx,
u"v 2 dx
= u"vp
u'"v 3 dx,
u'"v s
dx
u''i\
l
= iif'Vi etc. = etc.,
dx,
u""v dx,
,
t>-%- dx= u^v^ - [ ?>-%_, dx,
2
^-%.., dx = u^-^v n -
(
u^v n dx.
Hence, adding and subtracting alternately,
uv dx = uv l
Ex.
each
1.
uv 2 + u"v 3
Thus applying
term
u'"v^
this to the last
+
...
'-
4-
(
example (Art.
1 )"
l
u
(n
~ 1}
vn
94),
being derived from tlie preceding by the simple rule of
and integ. 2 nd " and connecting by alternate signs. When
"diff. 1 st factor
one of the factors
a rational integral algebraic polynomial,
the
successive differentiations.
mately destroyed by
is
it is ulti-
INTEGRATION BY PARTS.
E x.
2.
/
- mxm ~ 1 ~, + m(m -
x m e ax dx = xm&
a
J
^
- m(m - l)(m 96.
If
I
a-
109
}x
m~3
m^ e
as
+ ...+(- l) m
~,
f^
!
one of the subsidiary integrals returns to the original
may be utilized to infer the result of the
form, this fact
integration.
Ex.
and
i
e
I
ax
cos bx
dx
Tence,
P=
if
I
eax sin
bx dx
.
,
sin
,.
+
Q=
whence
F = enx__ a sin bx
^os bx
br e ax
aLa
^sin6.r +
-'
.
.
.(ii)
cos bx dx,
b
n~]
+ -P
a A
~\
U
a^J
,
,
cos bx
+^
Or we might have written equations
= e" x sin bx,
= e ax cos 6.r, J
ax
.
b cos
-
rtr
e
I
i)
sinkr ---
bx
---
..................... (i)
sin bx dx.
cos bx
-
a* + 6 2
n
V-
cosbxdx
a\_a
and
a
ax
e
Q=
breax
bx-
/
aj
and
x
P= e"a
+-
cos bx
a
J
le ax
sinbx-
e^sinbxdr^
/
(i)
and
(ii)
as
~\
}
r
\\
P and Q as
e niay write
and then solve
follows
P and
for
0.
:
1
l
Jj
forms which are fre<juently useful and which are derivable at once from
the formula for the n th differential coefficient, viz.
'
d'
1
e
1
t/./"
ly putting
And
e
"X
n=
cos (bx + c)
the angle
tion,
by
sin,
cos
70 ,"
bx = (a 2 + b-)* e a *
.,
r
sin/
b\
bx + n tan- a)
cos\
7
is
\ve
shouhl be led to expect.
the same as to multiply
it
to divide out
1
-.
!<\>r
\ f <t-
by
tan"" -, the effect of integration,
angle by tan"
1
[X>(/f.
1
must be
,
-1.
what
this is
ax
by the factor Va'
+ b~
Co/c., Art. 93.]
to differentiate
+ lr and
which
2
if
,
is
to increase
the inverse opera-
and
to diminish the
CHAPTER
110
And
in this form, viz.
it is
e
fl
sin//,
bx + c - tan- 1,\
(
)
\
*/
,
+ b 2 cos
/ e'
*S
n
+
COS (bx c]
'
dx
most
is
easily
remembered.
In cases of the form
97.
e?
ax
2
l
that the integration of
x
IV.
e ax sin1
sin bx sin ex sin dx,
p and # being
'
e ax
x cos? x,
smp x cos nx, etc.
positive integers, the trigonometrical factor
,
must
be expressed as the sum of a series of sines or cosines of
multiples of x by trigonometrical means, and then each term
first
being of form
Ex.
98.
mx
e?*
cos
1=
1.
Now
sin 2.r cos
I
x=
/= \\e
/.
Ex.
can be integrated.
x
e
x
(sin
Now
sin 2
^?
x dx.
3# + sin x)
(sin 3.r
/*=* sin*
2.
sin 2.r cos
ff
;
+ sin x) dx
cos* a? <fo?.
= ^ 1 cos 4:) cos x
= ^e (2 cos x - cos 3.r - cos 5.r)
cos 3 ^ = J sin 2 2.r cos ^
(
;
.
.
/
e
3*
sin 2
^ cos 3 x dx = ^6
I
x - cos 3.r - cos 5.r) o?.r
e?*(2 cos
r4-(^^4)^'^^
16 Lv/10
Ex.
3.
37
V
Integrate
/
J
v/o 2
^
2
Va 2 -.r2
<fcc
=
.^
cfcr?
\/a 2
4/
\
3 N/2
by
" Parts."
-^2 - J .r
^ <Jar-x* dx
2
/-a
J
[Note this
step.
^34
-(a
^
2
A
\'a*-
Some such rearrangement
is
_
frequently necessary.]
,,,
a 2 - ^2 + a 2 sin- 1
|
v a 2 - ^2 dr,
INTEGRATION BY PARTS.
whence, transposing and dividing by
f
-
2,
Wa o^
v a* - x* dx = ----n
/
2
7
,
a2
f-
/
A
J
Ill
,#
sin- 1 .
A
,
OL
which agrees with the result of Art. 78 obtained by the method of substitution of a sin 6 for
./.
The method of Integration by
that
whenever a direct function
ately
99.
"
Parts
<j>(x)
"
so also can the corresponding inverse function
4>(x)dx can be found, so also can
For, putting
1
</r (x)
(f>~
which establishes the
l
I
(x)
(j>~
(x), i.e. if
dx be found.
dx = <>'zdz.
and
l
I
l
<f>~
= z,
x = <>z
Hence
shows immedi-
can be integrated,
(x)
dx=
I
z^'(z) dz
rule.
100.
Geometrical Consideration.
This
is
no more than might have been anticipated from
geometrical considerations.
Let PQ be any arc of a curve referred to rectangular axes
be (x y ) and of Q (xlf y^).
Ox, Oy, and let the coordinates of
P
,
Let the equation of the curve be y = <f>(x)', or if x, y be expressed in terms of a single variable t, let the equations of the
curve be
and
let tQ and
^ be the values of t corresponding to the values
XQ y and x lf y lf of x and y respectively.
Let PN,
be the ordi nates and PJV,
the abscissae
l
,
QM
,
of the points P, Q.
area
But
Then
PNMQ = rect.
area
OQ-rect.
PNMQ =
Jxo
area
Also
rect.
QM
plainly
OP -area PQM Nr
y dx = Jx
l
</>(x) r/jr,
PQM.N^ J^xdy= ^[%-%) dy.
OQ = x y
l
1
and
rect,
OP = x y
Q
CHAPTER
112
= <X2/i-^o2/o)-
Thus
J
f*i
x
TV.
<j>(x)dx=(x l y l -x y
f?/i
J
%dy,
(1)
UK
p/i
)-\J
l
<j>~ (
3/0
Fig. 16.
Hence the dependence of the one integral upon the other is
obvious, and to establish the possibility of calculating the area
PNMQ is to establish incidentally
the possibility of obtaining
the area of
l
Further,
J XQ
and
I
ydx =
^
v du=\
xay =
u (ir =\f' u
and
X-1
So that the equation
i
Jt
1
'Efii/n
(1)
may
v -7,- dt = uv
di
L
==
i^v
<*V
14.
dt
I
\
be written
I
Jt
dt
7
(It
7
2/1
7,
v
Jt
J(
u -JT dt,
at
and thus the general rule t>f integration by parts
is
established
geometrically.
The meaning
of the process
is
therefore this
:
In cases where
a difficulty in finding the area PNMQ, we may find
and deduce the former result from
instead the area
1
1
there
is
PQM N
the latter.
INTEGRATION BY PARTS.
113
EXAMPLES.
integrate by parts
x2 e ax
1.
xe**,
2.
.rcos.r,
jp&inhx.
x^er*, .rcosh#,
,
>5
.?
cos2 r, # 2 cos 2 .r,
.v
t
3. e x sin 2.r,
ex
^log.f, .r'Mog.r, ^"(log.f)
5.
e"* sin
6.
e'
7.
Evaluate
A-,
%x sin
.r,
x sin # sin
e"5* cos x sin 2 # cos
2.rsin3,r.
3.r.
u
2
F
r
I
1
j.
r
x d.r,
sin
P
5
y- cos
I
Jo
8.
cos
.r
,
(log j-f'.
ax
?U
e
sin
sin
sin
,
p.v sin 5-^' cos ru.
px
gvr
x
sin pa- sin ^.r cos'2 r.r, (? cos p.r cos ^.v cos- (p +q)x.
4.
Ij:
2
*
sin 2 .r, e 3 sin 3 x cos
x d.r,
I
.r
2
cos 2# o?x
-'
''
Integrate
\woT afdx
l
lsin~ l .vd,r,
t
\
x* $\n~ l x dx,
lxteaTl xdx.
Reduction Formulae.
101.
not infrequently occurs that a function which it is
desired to integrate is not immediately integrable or reducible
It
by substitution
whose integrals
it
one or other of the standard forms
have been committed to memory. But
to
in such
may happen
a case that the integral
may
be
connected in a linear manner with the integral of another
function, or with the integrals of other functions, which are
simpler or easier to integrate than the original function.
Such a connecting formula is called a Reduction Formula.
Thus an integration by parts makes one integral depend upon
a second integral, and is a Reduction Formula.
Many Formulae of this type will be found and used in
subsequent chapters.
We
102.
integration
have seen how a repetition of the process of
by parts
will enable us to calculate the integrals
Sm = xm sin nx dx,
Cm = xm cos nx dx.
I
We
I
"
"
Reduction Formulae for these
in
of $,_,, C w _ 2 respectively.
terms
S
C
m
integrals, giving m
have
at
we
once
Integrating by parts,
propose to construct
,
and
Ow =
sn nx
m
---
xm -
71
VI
E.I.C.
H
CHAPTER
114
IV.
Thus,
m
nx
cos
n
n
,
^
m~
Thus,
nx
-
when
~~ ml^
~~
~~~~"
sin
Cw =
and
w
m _ l cosnx
"1 " ,1
t
---m (m
cos no;
1)'
the four integrals for the cases
w=
and
m=l
are found, viz.
of-
o =
sin
I
cosnx
j
nxdx
~
o,
f
=
C
,
x sin nx dx =
I
x
=
x cos
I
l
I
cos
j =
nx dx
cos nx
nxdx=x
nx
n2
-\
>
71
J
siunx
sin
H
n
J
C =
(
,
,
/C*
others can be deduced by successive applications of the
above formulae.
This illustrates the use of a reduction formula. But for
m
m
expressions like x smnx, x cosnx it is ordinarily better in
practice to apply the method of Art. 95 at once and avoid the
all
successive substitutions.
EXAMPLES.
Write down the integrals of
1
.
/
x^e* dx,
I
a? sinh
rf
I
f*
3.
I
Jo
dx,
I
x b cosh 2 x dx.
rf
rf
x3 sin x dx,
2.
.r
x^s'mxdx,
x 3 sin 2 x dx,
j
x^co^xdx.
|
cosh.vdx.
n
rs
r
I
Jo
?>
Jo
^(a 2 cos 2 x + b 2 sin 2 cc) dx,
rf
5.
I
Jo
/
4.
Px
/"*
I
x 4 sin x cos x dx.
I
e sin
a;
cos 2
x dx,
p
I
Jo
.r
sin
/
a;
3
x sin
log
cc
dx,
2.r sin 3.r
x tan
j
dx.
*
x dx.
INTEGRATION BY PARTS.
115
The Determination of the Integrals
103.
xn c ax gj n x (lx
fr
xn
^
J
may
be at once effected.
For remembering:
,
e ax
cos
fsin
where
xn
r
= Ja
1
gax s j n
-\-
bxdx =
7
^^_
___
= - we
,
^x s n
^
j
/T
(bx
v
cos
r
and tan
b~
e ar sin
0),
have
0)
.,-
e * sin
6 arr sin (6x
^-
= ear (P sin bx Q cos 6ic),
or
where
xn
xn
n
Q = ~ sin0
/
w
~l
~
(^1)^- COS 30
Xn ~^
iC
2~sin20 + w(n
1)
...
,
n ~^
3- sin
30
....
Similarly,
a:K
.'.-"
104.
f?
cos
60? cZa;
= eax P cos bx + Q sin 60?
{
}
.
Integration of
<
(
H
=
jgW
We may now
CQS n l)X
fa
f
^=
f
n
express cos
Lax sin n
^
(J
,
f
and sin n foe in a series of
bx and then integrate each
foe
cosines or sines of multiples of
ti'rm by Art. 96; or -we may obtain formulae connecting
witli 6 n _ 2 and Sn with Sn _ 2 thus
r
:
,
tgax
^
e ax CQS n l)X
= &X cos w ox
a
=
x
(l
_'
r
.
(
()S
n
r ear
-\
a La
-
\
ci >s"
e ax
fa _|_
"l
w ^ COS"" 1 6x
sill
ox sin ox
~e ax {cos n bx
(n
l)cos
n~2
bx
Rm 2
Cn
CHAPTER
116
e 035
a
IV.
ax
cos n 6#
+ >ih[e
a L a
sin
6ic
-e aa; {wcos n 6#
1
=(l+'Jl^O
a2 / n
a
\
cos
n
L)cos ~*bx}d
(n
g
e a* cos*1
"1
6# sin
Hence
-
'
Similarly
*-,
And
eaa;
as
a sin bxnb cos bx
,
e aa;
c?ic,
--n (n
sin
?>ic
I
^ic,
8 X and C x ) can be written down
eax cos
n
where n
105.
bx dx and
is
e ax sin
I
n tic
a positive integer,
Ex. Integrate
/
e
x
sin 5
# cfo?
Let cos.%-f isin#=y; then
/.
/
5
e* sin
i
:=_
sin 5
r
g* (sin 5.^
(ii)
5.^
- 5 sin 3.^+10 sin
"(5.r-tan-
1
5)
hy "reduction."
x=y
(see Art. 132).
10t sin 3.^ + 20i sin
5 sin 3# +10 sin
.-)
x
;
a?).
dx
j- sin(3.r- tdii
vlO
=
Proceeding with the reduction formula, a l.
.
.
sin x - 5 cos x
-
,
CQ,
"
by the multiple angle" method,
(i)
x dx
/
S
successive reduction.
by
2t sin
x = -j (sin
is,
dx can be completed, in any case
= 2i sin 5.r
.'.
e ax cos tie ^a? (that
(Art. 96), the integration of
(ii)
(i)
,
5.4
\
b
= l,
n=6
t
INTEGRATION BY PARTS.
S3 = <?, sin 2 x
.
Similarly
.,
x-3
sin
cos
.*;
3.2
+
,
^
ry
117
,
and
sin 4
x (sin x
5 cos x)
5.4(
106. Integrals of form
integer and
7n =
m not equal to
Integrating by parts,
^
for log
and proceeding
which is
I
1
a;" (log A*)
I
H
x
dx,
3.2
n
.
being a positive
1.
we have
"
Writing
sin A-- 3 cos
.
-
~
w
f
m + i } *" (log X)
"" 1
<fa '
35,
in this way,
xm log x dx,
we
ultimately get
x m+l
.
i.e.
7
7 I
m+1
down
to I lt
xm + l
T
Hence
r
~m+lL
__n
m+r
+ n(n-l)
n(n-l)(ti-2)
(m
t
the definite integral
(m>-l),note that
107.
If
xm+l (\ogx) r = Q
and that
I
a;
w
n
(loga;)
^
be required
when x=l and r>0,
Lt x=vv m+1 (\ogx)r = Q.
[Diff. Calc., Art. 474,
Ex.
3.]
CHAPTER
118
IV.
Hence
and
finally,
Hence
.
...(3)
+1.
which
is
When
108.
also directly obvious
m=
1
from
result
(2).
,
The reduction formula
established
by integration by parts
We
"
may point out that this could be obtained by the rule of the
n
m+l
smaller index -fl" of Art. 217 by putting
differenand
(\ogx)
tiating, but in this case there is no advantage in using this method, as
P=x
the same formula
109.
We may
is
"
immediately written down by parts
add, in passing, that
/
~
terms except when
in
In that case, we have
1.
/
1
J .'log.i
dx = log (log #).
In other cases put x = &.
->.i
~ - dx -
/'
Then
J \og.c
/V"*
J/
/\i('t+i).//
e*dy
>/
-
=
/
J
n
- dy,
and expanding the exponential, we have
and the integration
is
as above.
dx cannot be integrated
o
finite
"
expressed as an infinite
series.
INTEGRATION BY PARTS.
110.
H
1
the form
of
Integrals
/
a;"
(log x) dx,
119
where n
is
a negative
integer, may be reduced to the above form by vising the reduction formula
in the reversed form, and writing n for n
1,
/
Tli us
'
2
(log-*)
But as these expansions are not
little
finite in expression,
they are of but
practical importance.
however, where in is negative and n is
can
be
positive,
expressed in finite terms by the reduction
111.
Integrals,
formulae, and present no difficulty.
3
(log
-*)
1 (log ^)
-
*T" (log -r) 2 dx
-
3
3
L 2~ log^g
~
3.2.1
"
NOTE ON A TRIGONOMETRICAL PROCESS.
We
112.
return to the Method of Multiple Angles already
introduced in Arts. 97, 105.
The process
of expressing sinp xcos ? aj in multiple angles is
a matter of Trigonometry. But for the convenience of the
student
it is
required in
briefly indicated here, as it will be extensively
follows.
what
Remembering that
(cos x
+
1
sin x) n
= cos
r
nx + t sin nx (Demoivre),
CHAPTER
120
cos x 4- 1 sin x = y
let
IV,
then
;
cos x
fsin# =
,
t/
cos
nx +
1
nx = y n
sin
if we
require,
we proceed thus
8
say, sin
Thus,
sin
8
^=^r (cos
sin
nx =
~,
2ismx=*y --
n
,
,
16
cos
fix
+ 56 cos 4x - 1 1 2 cos 2.r + 70
8.^-8 cos 6^ + 28 cos
4^-56 cos 2#+35).
then ready either for finding the n th differential
or for integration, or for expansion in powers of x, as may be
x thus expressed
coefficient,
t
in a series of sines or cosines of multiples
.?
'
sin 8
nx
:
= 2 cos Sx and
cos
U
= y+
Thus
of x,
arid
is
required.
If
we required
sin 6
x cos 2 x,
say, in a series
of
sines
or cosines
of
multiples of x, then
2 6 i 6 sin 6 #. 2 2 cos 2
.r
= U- ~ -Y
= 2 cos
sin 6
and
x cos 2 x =
.
a'
and
is
f
- 8 cos 6.r + 8 cos 4x + 8 cos 2.r - 10,
- cos 8# + 4 cos 6# - 4 cos 4x - 4 cos 2# + 5
1
1
1
2
,
etc.
such examples to remember that
coefficients may be quickly
binomial
reproduced in the following scheme
1
j-
J
It is convenient for
the several sets of
(See the next article.)
v
ready for integration,
113.
8.'?
(#+*),
:
POWERS OF SINES AND COSINES
121
each number being formed at once as the sum of the one immediately
it and the preceding one.
Thus, in forming the seventh row,
above
1+5 = 6,
+ 1 = 1,
5
10+10 = 20,
+ 10=15,
etc.,
multiplying out such a product as the one in Art. 112, we
6
2
and all the work appearing
t) (I+t)
only need the coefficients of (1
will be
and
in
,
Coefficients of
(1-0
Coefficients of (I-*)
Coefficients of
(
1
- t)
6
(1+0
H- 1)*
ft
(
are
1-6+15-20+15-6 + 1,
are
1-5+ 9- 5- 5 + 9-- 5 + 1,
1-4+ 4+ 4-10 + 4 + 4-4+1,
are
each row of figures being formed according to the same law as before.
The student will discover the reason of this by performing the actual
multiplication of
b
...
hi
which the several
+ a, a + b,
Similarly,
if
1
+
t,
coefficients in the result are
the coefficients in
b
+ c,
(1+)
c
+ d,
4
...
2
(1
t)
.
were required, the work
appearing would be
1+4+6+4+1
1+3+2-2-3-1
1 + 2-1-4-1 + 2 + 1,
and the
The
last
row gives the
coefficients required.
coefficients here are
1-0 = 1,
formed thus
:
6-4 = 2,
4-1=3,
4-6=-2,
etc.
POWERS AND PRODUCTS OF SINES AND COSINES.
Any odd
a
positive poiver of
integrated immediately thus
To
Odd Integral Index.
Sine or Cosine with Positive
114.
sine
or
cosine
can
:
integrate
I
sin 2n+1
x dx,
let cos
x
c
;
Hence
sin^+ia;
I
dx= -
f (1
n
c-)
dc
.
.
sin x
dx =
dc.
be
CHAPTER
122
"
nc3
5
1) c
n(n
"
IV.
,
"T7?~lH
3
siii'ic
Similarly, putting
}
2
therefore cosxdx =
= 5, and
ds,
we
have
=
f
r
f
(1
J
<4in3
olll
x-y
1
wsinS
/r
JU
'r
Ju
*
^
,.
*j1n2+l/v
olil
ut/
5
Products of form
115.
rsLLl.
j-y
3
1
sin'
M
.
cos 7
p
,/-,
or
an odd
being
(f
positive integer.
In the same
way as before, any product of the form
cos^x admits of immediate integration by the same
method whenever either p or q is a positive odd integer,
sin^'ic
whatever the other
may
Thus, to integrate
cos
xdx=ds
Isin^'cc co* 2 l+l
'
and
and expanding as
be.
I
sinj) x cos 2n+1
""
'~
2
p+l
When
cos*
a;
x dx =
1
s' ( I
I
s2)
n
"~"
i g a negative even
integer, the expression
2^ + l
admits of immediate integration in terms of
r
tan x or cot x.
For, put tan x =
t,
n being positive and
Thus
x cos 2
=
(^ 4[
ds,
before,
_
116.
Let sinx = s: then
xdx.
and therefore
2
4-
xdx = dt, and
integral.
xdx=( tan'' x cos^+ 2
"-^ ^+'
sec 2
n - 1 (7
2
^+ 4 4-
a;
dt
- . .
4-
=
\
n~l
t
p
(1 4-
Cn -i
tan*+ 5 z
t
z
)
n ~ l dt
let
POWERS OF SINES AND COSINES
we put
if
Similarly,
123
cot x =
cosec 2 x ix
then
c,
dc,
and
sin"
-
x cos'' x dx =
\
cot? x
sin''+''+
2
-
x dc =
I
{
1
c' (
+ c~) n -
1
dc
J
This result
is
the same as the former, arranged in the
opposite order.
117.
Use of Multiple Angles.
sin"r,
cos'1 x,
sin''
x
.
cos'1
x,
where p and q are 'positive integers, either odd or even.
To sum up then, when, in suVx, p is odd, or in cos'J x, q is
odd, or in sin #cos''x one of the two p, q is odd, the best
method of procedure is that of Arts. 114, 115.
But when both p and q are positive even indices, this
method cannot be adopted, for the series used are not
?>
terminating series.
We then express the function to be integrated as the
sum of a series of sines or cosines of multiples of x, whicli
can be done in all cases by the method of Art. 112, or in
simple cases without having recourse to that method.
then have
sin'' x,
cos'' x
or
sin'' x cos'1 x
expressed in the form
2Asinnx
and each term
or
S-
be integrated at once, giving
may
-
cos nx
v
2*An
.
n
sin nx
v
^j-a^n
.
Ol'
n
as the integral.
110
n,
118. Ex.
/
V
1.
A small
mall even \
index.
Ex.
o
/
2
J
/"
2.
index.
x
/"l+cos2^
-dx =
.
7
cos-.fflU'=
)
c(
/AsmallodclX
\
/"
/
J
J
s3
--\
-
sin2.>;
--
C 3 cos .v + coa 3x
3
x dx=\ dx = 7 sm x +
J
3
=
/".,
/
(1
1
.
,
/
or otherwise
.
sin
- s 2 ) as =sm .v --
o;
-
_ sin
L*
3.<;
We
CHAPTER
124
Ex.
/A
\
( eos4 #e*;
3.
/
small even \
iiidex.
/
J
2
Ex.
4.
^
/
(I + i cos 2^'
.= %x + j sin
2^-
+
1-
4
cos 4^)
+ gL sin 4#.
fw&xdx.
Let cos x + 1
sin
x=y,
etc.
= 2 cos 8x - 16 cos 6^?+ 56 cos 4x - 112 cos 2.e + 70
in
Ex.
/A
112.
J
large even\
7
(A. index.
\
,
dx
dx
powers we adopt the method of Art.
for higher
.
("'
I
/
J \
/
But
+ *"**+
- \*dx=
= //l+co.arV,
=
119.
IV.
8x
fsin|*#P=-
5.
f (l-c
28 sin 4x
6^;
2 4
)
c/f=-
J
J
large odd \
index.
)
8 sin
=
sin 8
Find
/
Ex. 6.
(Both indices even.) J
.t'
/
J
;
-
2.i-
+
3o,J
(l-
6cos 5
-COSA' +
56 sin
4cos 7
.v
.r
-
g
cos 2 .t* dx.
Then, as in Art. 112,
/
2 8 t 8 sin 8
x
2 2 cos 2
.
[and the working of the multiplication
Coefficients in
= 2 cos lOx /
sin 8
8 /
1\
\y +
-1\
2
J
is
1-8 + 28-56 + 70-56 + 28- 8 + 1
1-7 + 20-28 + 14 + 14-28 + 20-7 +
s
(l-t) (l+tf 1-6+13- 8-14 + 28-14- 8+13-6 + 1]
(l-O
8
s
Coefficients in (l-*) (l
Coefficients in
1\
x = f y - ~\
1
+
1
2 cos 8^ + 26 cos
6.v
- 16 cos 4x - 28 cos 2# + 28
;
x cos 2 x dx
=
1
rsin
1
rsinlO.v
10^;
6 sin
13 sin
80;
3 sin 8^
+
,
gal
1Q
4
-
6^'
13 sin 6^
8 sin 4^
14 sin ^x
POWERS OF SINES AND COSINES
Find
Ex.
7.
(One index odd.)
I
sin 8
x cos 3
.
*V.r.
J
I
# cos 3 x dx =
sin 8
/
sin 8 x
(
sin 2 x)
I
d (sin #).
em
sin 11
x
in 9
sin
/>
9
Ex. 8.
(An exponential
125
11
2*
/Vsi
e sin 6 x cos 2 a? cfc
/
factor.)
-^
2x
e'
I
[cos 8.r
- 4 cos 6.r + 4 cos 4.v + 4 cos 2.r -
5]
dx
(Art. 112)
2*
_
rcos(8^-tan"
1
1
os(6.v- tan' 3)
4)
2'L
Ex.
9.
Consider
/=
exponential factor and \
trigonometrical factor \
/An
/
sin tix cos 3
e'
-,
~
a
+2
^ sin 2 x dx.
.'
iruc, in which M is not I
/
necessarily integral.
As
2 3 cos 3
before,
x 2 2 t 2 sin 2 .^ =
Coefficients of
(l+O 3
Coefficients of
3
.'.
sin
cos 3 x sin 2 x
nx cos 8 x sin 2 ^- =
=
1 \3
+ ~J
+ O 3 (1 -
^ [2 sin nx cos
(7i
- -\
1
J
2
.
1+2 + 0-2-1,
s
1
^ (cos 5# + cos 3.r
^5 [sin
/
\v
14-3 + 3 + 1,
(1+*) (1-*)
Coefficients of (1
/.
/
(y
5.*r
+ 5).r + sin (?i
+1 -2-2+1+1
2 cos
;
1
.i
)
;
+ 2 sin n.v cos 3.r
5)# + sin
(?i
4 sin ??# cos ^]
+ 3).r + sin (n
- 2 sin (71 + \)x - 2 sin (?i wlience
_
/
e* sin
1)^]
nx cos 3 A* sin 2 x dx
sin
e
{ (71
- 5).r - tan"
(7i- 5)
sin
3).^
2
(?t
- 5)}
+l
sin {(n - 3).y - tan" (n - 3) }
+ 3).t? - tan" (?i + 3)}
2
2
l
V(7l-3) +l
V(7i + 3) +
~
tan"
r
tan+ ] )'
^^ + )> _ 2 ^J^ 7 ]^
+
1
+
+
x^w^
J(n
1
1
{(TI
1
1
1
1
1
}
1
)-'
)-
1
;
CHAPTER
126
120.
IV.
Integral Powers of a Secant or Cosecant.
Even
powers of a secant or cosecant are even
and come under the
negative powers
head discussed in Art. 116.
positive
of a cosine or a sine,
Thus,
/
= tan
sec 2 #&*;
f(l
.
+ tan 2 #) d ta
= tan x +
I
see 6#
dx =
I
(1
+ 2 tan 2 .r + tan 4 .r) d ta
2 tan 3 y
and generally
sec 2n + 2 # tto= f(l
+ t 2 )n dt, where t = tan x,
~ tan 3 A-
.
tan 5 .r
,
Similarly,
/
dx
cosec2 #
cot x,
<=-/<'
-cot^--
cot 3 ^
3
etc.,
and generally
121. Exactly in the
same way
I
sec^sc cosec*
ic
dx
can be integrated when_p + ^ is a positive even integer, either
in terms of tan x or of cot x.
This has been done already in Art. 116, for it may be
written
I
where
p
q
is
in-%
dx,
a negative even integer.
POWERS OF SECANTS AND COSECANTS
127
Odd Powers.
odd positive powers
122.
of a secant or a cosecant, we
have to adopt another method, because the Binomial Series used
would be non-terminating.
But
for
We now
By
proceed as follows
:
differentiation,
and
(n +
1 ) sec n+2
(n -f-
1)
=
nsec n x
n
-r-(tanixsec x)
n cosec n # =
cosec n+2 ic
j- (cot
cix
x cosec n x)
;
whence
nx
+ n\$ec n xdx
(A)
and
(n+ 1)
f
cosec n+2
n
Hence, changing
sec n x
dx=
to
n
,
n
J
(
l
cot x cosec"
cosec n x dx = ---
I
sec x
= log
dx
~2
x
l
?i
J
Now
+ n r cosec nxdx.
2,
tana?sec"~ 2 ic
=
7
dx
cot x cosec n j?
~
tau(
cosec x dx = log tan
^
n
2f
n
Ij
n
n
2(
\
,1
eec n
~t
xdx
i
cosec n
~2
x dx.
1J
+ - J = gd-
1
^,
.
Hence
j^toJS***^
/"
f
S6C ^'
dx =
7
tan
./
x sec ~^
3
-IT
tan
(see
(-+)
Art 79)
3 tan x sec x
taU
+ 4 ~ "T~* ~ + 31,
4 2 log
.
.
/TT
(4
+
,
.r\
i>
etc.,
and generally
tan
a;
w
sec n
1
~2
x w - 2 tan x sec n - 4 x
--f
.
nl
(>?
2)(/i
n
3
4)
tan x sec"
''.ii
~"
(n odd).
CHAPTER
128
IV.
The same formula would equally apply if n be even, except
that it would terminate differently, viz. the last term would be
,
tan *
(n-l)(n-3)...o.3
In the same
way
I, ^ dx =
cosec
3
7
_
cot x cosec
x
1
x
,
+ - log tan -
cot # cosec Vr
,
3 cot .? cosec x
:
3
1
and generally,
cot x cosec n
~2
nl
_(w
2) (w
z __n
2 cot x cosec n
n3
nl
4) cot
~4
x
x cosec n -6 # _
n-o
~~(n-l)(n-^)
- (n-2)(n-4).v;4.2
cota;
(Meven )
-
'
as explained above, if n be even we should not in
general employ this method, but that of Art. 120.
But
Since positive or negative powers of secants and cosecants are negative or positive powers respectively of cosines
123.
and
sines, it will
appear that so long as p is an integer, whether
positive or negative,
P
;>
Jsin iC?x,
can be integrated.
|CO9
2(,
Also
it
Isec^rfa
1
,
100860*2; <2z
appears that
I
smpx cos qx dx
can
always be integrated directly if p and q are positive integers
even if one of the two p or q be negative or
fractional, the integration can still be directly effected if the
other be a positive odd integer. And further, this integration
can be directly effected if p + q be a negative even integer,
even though both p and q may be fractional.
;
also that,
For other cases of
integers, a
Art. 228).
I
sin^ic
cos^ dx, where
reduction formula
is
in
p, q are negative
general
required (see
POWERS OF TANGENTS AND COTANGENTS.
129
124. If the student has any difficulty in reproducing the
formulae of connection marked (A), they may be obtained at
once by integration by parts thus
:
f
I
j
sec n+2 xdx=
4-9
f
I
sec n x
d tan
a;
j
dx
-j
= sec n xtaiiic
\nsec n xta,u 2 xdx
= secn x tan x
n
I
f
And
.
(sec
n+ 2 x
sec n x) dx
f
n+2
similarly for
(n+l)\cosec
|cosec
n+2 xdx
=
ic^x,
cosec n xcotx+n\cosec n xdx.
125.
Integral Powers of tangents or cotangents.
Any
integral
powers of tangents or cotangents
readily integrated.
For
I
ta,n n
xdx=
Itan n
"2
2
a;(sec
x
l)dx
r
f
=
I
n-l
And
since
an
f
I
we may
tan n
~2
ic
dx.
J
tan x dx = log sec x
iax\ z
xdx
2
a;
1)
J(sec
dx = ta,nx
x,
3
4
5
integrate successively tan ce, tan x, tan ic, etc.
Thus we have
I
tan 3 .r
dx =
tan 4 ^flte=
J
f
3
^-
g
tan 6 .? o?^r =
tan # +#,
42
tan 4 .v
f
lta.n*xdx=
j
- log sec #,
~
/tan
tan
5
a;
tan 2 .#
tan
=
hloersec^,
3
^
h
O
5
etc.,
E.I.C.
}
I
tan
x - x,
may
be
CHAPTER
130
IV.
and generally
?i-3,7i-5 -.;.+(_,*!*
n-l
n-l
2
+(-l)
log sec
x
(n odd)
(n even).
126.
Similarly for cotangents,
coin xdx=
I
~2
cot n
1
2
oj(cosec
I
n
whilst
1
I
Thus we have
1
cotn
-2
#cfcc,
1
cot xdx = log sin
cot 2 ajcfo =
l)dx
ic
x,
(cosec
2
l)dx=
o;
cotcc
successively
/cot
cot3 ^7 a^7
C
=
2
#
,
log sin x,
3
cot x cot #
^^= -T+
+ *'
cot
dx = ---2
cot o;a^7=
^ ^S sm x
o
cot
J
/cot
6
5
/
4
2
^7
a;
^;
'
t
and generally
/
71-
71-
71-
Hence any odd or even
positive or negative
(;t
even).
power
of a
tangent or cotangent can be integrated readily.
EXAMPLES.
an
2n+1
sin
sin 4 cc, sin 5 #, sin 8 ^, sin 9 z, sin a;, sin
a:,
Integrate sin
doing those with odd indices in two ways.
2
1.
2.
sin
6
o;
3
,
Integrate
,
sin 2z cos 2z,
sin 3 z cos 3^,
sin 4 x cos 4 z,
sin 3 x
INTEGRATION BY PARTS.
3.
Integrate
cos
f
4.
Evaluate
cos 2 #cosec 4
~,
V
f
2
xdx,
5.
2
Integrate sin az cos &c,
6.
Show
7.
-v
r
sin 4 z cos 4z
cos 6xdx.
Jo
smSxcos 3 ^,
sin
nx cos 2x.
that
sin x sin 2# sin
Show
3xdx= -
-
cos 2x
^
^ cos 4# + ^ cos
6z.
that
,..
f
mx cos wa; c?a: =
,
.
si
sin
I
(i)
^ '
cos(m + w)^
-r^
cos(m-n).c
- ~.
2(m 71)
J
f
,
I
(11)
.
sin
m
sin
sin (m + w)a;
sin (m - ri)x
nx ax = ~-^7v ----^--^-.
^--
'
2(m-n)
f
x
I
(111)
,
cos
Deduce from
mx cos nx ax =
(ii)
I
and verify the
8.
z,
i
I
Jo
Jo
I
f
cosPxdx,
I
Gosec 2
sec 2
fl
sii\
I
.
131
and
and
m
(m + n)x
^
^-.
integration.
mx sin nx dx and
mx cos nx dx
cos
I
are both
Jo
Jo
zero so long as
sin
Icos 2 mc?ic,
by independent
sin
I
(m - n)x
^-H
^
the values of
sin 2mxdx
results
Prove that
(iii)
sin
and n are integral and unequal.
are equal integers their values are each equal to
But
if
m and
n
-=.
GENERAL EXAMPLES.
i
1.
-n
,
i
.
Prove that
f
I
J
2.
d*v
,
u -^dx
dx 2
2
dv
du
f d u
u^--v-j- + \v -^-^d
dx
dx J dx 2
7
-
Perform the following integrations
(i)
I
cos- 1 ^ dx.
f
(111)
(v)
I
I
a:
3
(ii)
a;
Icos- 1 -^.
f
tan" 1 .^ dx.
x sec x tan
:
(iv)
dx.
(vi)
J
\xsecxdx.
I
(ra
+ b) log (ca; + ^?) dx.
J
xoP(viii)j^
v
[ST. JOHN'S,
1886.
/
I
^
<
1
[Ox.
II. P.,
1889.]
CHAPTER
132
/
ism" 1 *// -f
\
(ix)'
x
i
Va + x
J
cos- l
(xi)
j
3.
x log x dx.
(xii)
J
v
(n)
1
ficsin" ^ 7
I
-rdx.
J(l-z
.....
fa3 sin" 1 a
J(l-a
)*
I
&
-TT^
1
f p m ts^n" 1 *
<">
-
(i+*
T^t
/
(v)
r
&
2
P^mtan"
-
(iii)
l(
*
-dx.
{^mtan-2
)*
1
1
*
JO^-
^
gnJtaii-
^
\
^x
+
(TI
=
a positive integer).
J(i+^f
Integrate
(i)
(ii)
(iii)
\xe
f
2
bx
e
cos ax dx.
rta
sin^^.
[a 1888.]
l^sin 2 ^^.
Integrate
(i)
ie
e
ax
ax
(ii)
\
(iii)
I
e"*
I
<s
a:r
(iv)
(sin 5x
+ cos bx) dx.
(sinh 6
+ cosh bx) dx.
sinh bx cosh ax dz.
cosh ax sin
foe rfic.
2
f
(v)
(vi)
I
(vii)
I
cos (b log -
cosh
(
dx.
J
Hog - J dx.
6 sin ^ cosh (cos 6) dO.
I
(viii)
3* sin
J
[
!?
7.
7
dx.
(m)
2
Integrate
l
p mia.n~ x
6.
i
n
dx.
x
f aaia _ilx
dx.
\e
J
(i)
5.
2a ~ x j
ax.
\ 4a
/
i.Tsin" 1 /!/
J
~
^l a
f
\
/
(x)'
.
Integrate
v
4.
IV.
1891.]
Integrate
M
x
f
I
-
ice*
,..s
Trzdx.
(11)
;
'}(x'+iy
fcosh a
+ sinh a sin a
_
'
1+cosa
J
s/TT^<fc.
I,
- sin x ,
.....
f , 1
(m)7 10*=1-cosa da.
+ sin x
-da.
1+cosx
f xr 1
le ^
J
,
7
,
,
J
da
f
'
J
1
f
J
'
[MECH.
Sc. TRIP.]
1
e
(vii)
+e x
-^
[Ox.
I.
P., 1890.]
INTEGRATION BY PARTS.
8.
133
Integrate
(i)
P., 1888.]
[Ox.
I.
P., 1889.]
II. P.,
1887.]
[MATH. TRIP.,
1882.]
[Ox.
(iv)
|
(v)
[ST.
JOHN'S, 1884.]
[ST.
JOHN'S, 1888.]
J
(vi)
j
(vii)
I
(viii)
1
2
2
Ja + x dx.
[ST. JOHN'S, 1888.]
e^z 2 sin (to + c) dx.
[COLL., 1892.]
(a
f a*
(ix)
9.
I.
or 2 tan" 1 x dx.
I
(iii)
[Ox.
+
)
(1- **)*<&.
I.
[Ox.
P., 1890.]
Integrate
(i)
(ii)
10.
I
x e"* sin to sin ex dx.
I
xe
ax
sin to sin 2 c# dx.
{(
Show
that
if
u be a rational integral function
a
<?'u dx = a^'
I
u-a
f7
+ a2
-j-
6?a;
[
where the
//-i/
?/
-
7
f/.r
2
of x,
d^iJ
- a 3 -,- , +
3
c?^
I
h,
J
series within the brackets is necessarily finite.
[TRIN. COLL., 1881.]
11.
If
u=
I
e
ax
cos to
t/a;,
v
=
tan -1 -
u
and that
12.
2
(a
+ b~)
Evaluate
I
\
e
ax
sin to
das,
prove that
+ tan" 1 - = to,
a
2
(u
x 2 log (1 - x 2 )dt,
111
and deduce that
82,
[a, 1889.]-
CHAPTER
134
13.
Integrate
14.
Find the value
IV.
f sec*
fsec^0cosec*0d<9,
sin
dd.
of
J
U n
\ dx
X
~
'
^/^'
[7, 1890.]
Evaluate
15.
Crd 3 u/dv
3
]\_dx \dx
dw\
d z vfdw
du\
d 3 w/du
dx)
dx\dx
dx)
dx*\dx
dv\~\j
dx) J
[7, 1890.]
Establish the following formulae for integration by parts,
x, and accents denoting differentiations
16.
u and
v being functions of
and suffixes integrations with respect to
n accents
x,
and u (n) denoting u with
:
(i)
I
uv dx = uv l -
i
(ii)
j
2
(uv) (dx)
u'v^
+ u"v3 - u'"v +...+(-
= uv2 -
2u'v 3
1)
n~l
u (n
~ l)
+ 3u'\ - 4u'"v5 +... + (-
+(-
1
n
)
n
[
u^vn+l dx + ( -
I)
vn
1
n-l
)
n
dx
[
nu (n
i
~l
\
+l
u^vn dx.
[a, 1888.]
17.
If
u be a function of
are respectively denoted
n accents, show that
x,
by
and
and integrations
and (n) means
differentiations
accents and
suffixes,
[ST..
18.
If u,
v,
JOHN'S, 1889.]
w
differentiations
be functions of x, and accents and suffixes denote
and integrations respectively, show that
(vuj)\
(wu)\
(uvyw-i
-
-
(vw)"u2
+ (vw)'"u 3 -... + (-
1
(wu)"v2
+ (wu)"'v3 -...+(-
1
(uv)"w 2
+ 0>)"X
-+(-
w -1
)
f
1
)"~ f
1
I)"'
1
[ST. JOHN'S,
l
INTEGRATION BY PARTS.
135
Prove that
19.
f1
X2
X
~d = l-
l
-
gi+Tji
o
X*
44
& -"
55
+
etc.
[MATH. TRIP., 1878.]
20.
Find the value of
21.
Prove that
f
i
x*
I
dx correct to
decimal places.
five
[J.
M. SCH. Ox.,
1904.]
2^i*
Ix
3. 5
o
+
(2a
M
2
)
2 2
3.5. ...(2?i-l)*
f*
Jo
22.
Find the sum
x
of the series,
5
y and
23. If
following
f
(i)
supposed convergent,
x9
x7
1.3.5
+ =3.5.7^5.7.9K
be functions of
z
x,
etc. to QO
[COLL., 1881.]
and u = yz' -
zy',
prove the
ar* (y'z" - z'y") dx = - y- 1 (1 + y'zur 1 ),
zy'*)
can be reduced to that
[ST. JOHN'S,
.
Show how
24.
.
:
the integration of zy~ l u~ 2 (yz"
of y~'2
(ii)
PUB., 1899.]
I.
[Ox.
the method of integration by parts
may
1886.]
be applied
to find
where /(a;)
a rational algebraical expression of the
is
th
?i
degree.
Prove that
1876 ,
Prove that
25.
I
(cos x)
3
N N
I}
2,
N%,
...
n having any
26.
H
dx
may
be expressed by the
series,
5
being the coefficients of the expansion
real value positive or negative.
(1
+
a)~*~,
and
[SMITH'S PRIZE, 1876.]
Prove that
sn
-
CHAPTEPv
136
Express the
27.
infinite series
1
as a definite integral,
Show
28.
2
w
(*
IV.
and
1.3.5
1
1LJ5
1
find its value.
II.
[Ox.
P., 1902.]
that
sin
cos wzc.
2x
-(m
in
j
1 )
sin 4#
r~
IT 2
2
f
-
sin 2mx.
,
_1
where
m is an
integer and
A
is
independent of
x.
[COLL.
a,
1885.]
value depends on that of A.
[MECH. Sc. TRIP.,
1899.]
Evaluate the integral
29.
Ia
o
x
sin
^"r
and draw curves showing how
Prove that
30.
if
ft
its
sin
~V(^
+ A) ^
*
=
y=f(x) and x <j>(y)
between any corresponding
then,
2
4
are equivalent relations,
limits,
Hence, or otherwise, prove that if tan
dx
_
/5
= \/l - c tan a,
-
[Ox.
31.
may
Prove that the remainder
R
II.
P.,
1886.]
in the series
be written as a definite integral
[COLL., 1881.]
32.
Show
that the integrals
f /(*) &*,
z
n
f(n \z) dz
are connected
thus:
Jo
arid that if
one can be integrated the other can also be integrated.
[BERNOULLI.]
INTEGRATION BY PARTS.
33.
137
Integrate
f { (2?i
+
1) cos
(2n +
and prove that when n
)0
is
+ (2n -
2) cos (2w,
-
f) 6} (cos ty*d0,
a positive integer,
*
cos
(2-M-
+
<9
i)
o
!*
(cos
OY d6 = 0.
II.
[OXFORD
34.
Find the sum
PUB., 1913.]
between the axis of x and
of the areas included
the arc of the curve y = x sin (x/a) from the ordinate x =
ordinate x = mra, n being any positive integer, odd or even.
[OxF.
,jM
when n
<lv.
- x2
o N/2a
{2a
is
any positive
Showthatf
jclog(l
+x)dx = % (I - 2logf),
I.
P., 1911.]
integer,
[OxF.
36.
to the
I.
P., 1916.]
and prove that
Jo
this is less
37. If
than
Tn =
\
^x 2 dx.
\
ta,n*xdx,
[MATH TMp
?
^
PART
show that (n-l)(Tn + Tn . 2 ) = tan"-
1
1913>]
^.
Jo
Given that
f
TT
tan 5
= 3-141592...,
Ioge 2
= 0-693147..., show
n
^ = 0-09657...,
that
= 0-1 1873...
1
[MATH. TRIP.
K
r JL
38.
Prove that
Jo
sin" 1 ^
,
dx
-|
2
(1-Z
=
T
TT
~
4
1
3
Find the area
A
I.,
lo
o S- 2 '
)-
TRIP. L, 1917.]
between the curve
=
y a (sin x +
J-
sin 3x
and the axis of x between the limits
+
-J-
sin
and
5)
TT
;
and the volume
obtained by rotating this area about the axis of x.
Prove that 4 V=ir-aA.
[MATH. TRIP. L,
40.
Show
1915.]
.
[MATH
39.
.
Jo
Jo
V
1913.]
that
[MATH. TRIP., PT. L, 1916.]
CHAPTER
V.
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
127.
Integration of
-i-
*
and
Either of these forms
-i
be
should
Fractions, which can be done by
dx
1 f/ 1
1
\ ,
dx
thrown
into
inspection.
a
= -1 tanh- 1 -X
or
f
dx
jx2
1 f /
a
<
x
a).
1
1
2aJ Vica
a?
=
or
(x
v
a
a
Partial
ce+a
ce
coth- 1 -
or
a
a
tanh- 1 -
(x
ic
>
a).
The
Partial Fractions are so simple that the results are not
usually committed to memory.
128.
These inverse hyperbolic .forms should be compared
with
dx
= -1 tan,
a
,
1
x
a
= -1 cosa
,
1
a
-.
*J
138
== -1 sec-
,
1
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
The three
results are
:
{dx+
-=
Jdx
-52
a
a-52
5=
2
1
,z
-tan- 1 - or
a
a
3
-tanh- 1
^
a
1
j.
1
35
2
--l,
cot-
,
a?
^
J^
139
i
i
#
-
x
=--coth-'1,
i
a
.
/
1
a?
-.
a
\
(&<o),
i
^
\
(*>a),
ax
--1 tanh"
,
or
^
129.
Extension of above
In the same way, a and
,
,
1
a
-.
rule.
ft
being
real,
dx
130.
Integration of
Since ax2 + bx+c can always be written as
or as
\(
_|_
taking the
AY_k
first
2
~~
1'
I
ff
[(
or the second according as 6 2
the rules of the former article apply.
<
-L
4>ac
Y2_
or
CHAPTER
140
V.
Thus
131.
CASE
I.
dx
/(
dx
b \2
4,ac
4ac-6 2
2
or
,
2ax+b
-
cot" 1
.
V4ac-6 2
,
o
\/4ac
62
\4>ac
132.
CASE
II.
dx
ax'2 -\-bx+c
a
b2
4>ac
4a 2
2a/
- 4ac
coth"
or
.
v 62
,
which
T
cosech- 1
or
is
or
a real form
i
if
da?
=
a
6 2 -4<xc
/
.
b \2
^4ac - ( 2aa? + 6)
= etc.,
.
tanh" 1
which
is
a real form
if
2ax + b
< v6-
'
4ac.
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
Of these several forms the
133.
real one
is
141
to be chosen in
The general forms are equivalent, except
a
constant
which may be unreal.
by
each numerical case.
that they differ
Another Method.
134.
As the
factors in the second case are real, say
x l )(x
a(x
x2 ),
the usual proceeding is to write the work as follows without
the formal completing of the square in the denominator
:
f
}
-
dx
=1f
~
ax 2 + bx + c a J (x
dx
1
T;
x2 )
x^) (x
f
I
dx-
-- _
1
-4
x 2 )jxx l
a(x 1
==
dx -_
f
I
xj J x
a(x2
x2
fl+
log
.
Cv (
JU-t
~~"~
U/o
OUn
Qv
)
Other forms of the above results.
135.
Other forms of these results may be exhibited. For instance.
R = ax2 + bx + c, and 4ac 2 = 4aV= 4aV 2
taking
then
.
2 tan- 1
?;
%ax + b-= sin- ./ K
-=
.
1
^/4oc^S
(
\
;
2ax 2
+ b-\
s-.-r
ax*+bx+c/
= sin-
-
,
1
/
(
\
/c
d~
7
dx
W
.
6
and
whence
^f^
or
10
the real form to be chosen.
136.
Integrals of expressions of the form
'
'
can be obtained at once by
throwing
where
A, /z
are constants to be found
R
px+q into
;
the form
CHAPTER
14:2
V.
for then
r
*
=
f
px + q
7
= fX-R'o+ M
dx
?
R
J
Jax* + bx + c
1
7
i
\
I
7
dx=\
dx
and the second member of the right
This transformation
137.
It
is
side has been discussed.
one very frequently required.
be performed either by inspection, or by comparing
may
coefficients.
(i)
By
inspection,
^ +g3
(ii)
By comparing
(2
coefficients,
Thus
j
Saz-ffr
_P^f
7
.
/
c?x
pfr\ f
<iic
ax" + bx+c'
It is essential that the numerator of the first partial fraction
shall be the differential coefficient of the
denominator^ and
that the x's of the numerator of the given fraction are thereby
exhausted.
-i(**Htt
a
+2
-
2.r+12
7 r
V2
/"
^A~I^
J 35 +
2^- ^ 2
//
=
\/2
9
~^
/"
J (7-o?) (5 +
10
1
11
"
1
j
dx
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
143
to be noted in such examples as the two preceding
form of the result is real for all real values of x
5 and +7.
in the second the form given is only real if x lies between
For values of x > 7 we should write it
This difference
is
;
in the first the
and
;
x
for values of
< - 5,
These three forms
by unreal
differ
constants.
EXAMPLES.
xdx
i
7'
'
xdx
dx
'
,
x+1
[
j
(x +
xdx
q
Jz z + 4x
xdx
l)dx
'
NOTE ON PARTIAL FRACTIONS.
139. In the author's Differential Calculus (p. 72) a Note
to be pursued in the case of
was inserted on the methods
finding the
when
it
?i
th
an algebraical fraction
to resolve the fraction into its simple
Differential Coefficient of
was necessary
or partial fractions.
with some additions
now
necessary to repeat this Note,
and alterations, as success in the integraIt is
tion of complicated rational algebraic fractions will depend
upon the ability of the student to obtain the equivalent partial
fractions with facility.
Moreover, many subsequent articles
will
depend upon the general theory.
140.
f(x)-
Let^~
be the fraction in
its
lowest terms which
is
<p(x)
to be resolved into its
simple
f(x) and
<f>(x)
being
component or
supposed
rational
partial fractions,
integral
algebraic
CHAPTER
144
functions of
x,
V.
the coefficients being real and,
unless the
contrary be stated, rational.
Then if the degree of f(x) be not already less than the degree
of
(f>(x),
we
can,
by ordinary
in the
division, express
where a x n -\-a l x n - l + ...+a n is the quotient, and
remainder, of lower degree than <p(x).
Hence the integration of
f(x)
^
*-4:dx
<>x
.
is
a n cc n+1
n+1
form
-%(x) is
the
n
-+% xn
l
and we only have to attend to
I
^7-^ dx.
Hence we may confine our attention to the case when f(x)
of lower degree than
is
Also
efficient of
141.
(/>(x).
we may, without
loss of generality, consider the co-
the highest power of x in <p(x) to be unity.
proved in Theory of Equations that
It is
if
0(ce)
=
be a rational algebraical equation of degree n,
n roots, real or imaginary,
that
(2)
imaginary roots occur in pairs,
(1) there are
a*/3,
etc.
be repeated.
Then the general form of is of the nature
Any
of these roots
may
where we have taken the case of
(1) a real linear factor
occurring once only ;
a
real
linear
factor
(2)
occurring p times ;
a
unreal
(3)
pair of
factors, each occurring once
;
pair of unreal factors, each occurring q times.
Any other factors which there may be in must be of one
(4) a
or other of these categories.
We consider these four cases separately.
And
as
we
are going to suppose that
fix)
Q4
is
a fraction in
<p(x)
its
lowest terms, none of the factors described above will be
factors of f(x) also,
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
142.
I.
x
a
= (x
<f>(x)
factor
Let
7
fraction
partial
corresponding to the
occurring once only.
a as a
tain x
Let
To obtain the
145
^
f
.
Then
a)\lr(x) for short.
factor,
,
=
x
and
\Is(a)
-an
x-a^^(x)'
[-.,
\^(x) does not con-
does not vanish.
[,
assumption
iustifiable if
(x-a)\Is(x)
we succeed in finding A, supposed independent of x.
f(x)
\(x]
Then 4^~4= A + ir^( x a ) ^ s an identity and true for
~
values of
all
x.
Hence putting x = a,
,\ \
= A.
Therefore
(xa)\}s(a)
(xa)\/s(x)
Hence our rule
<l
to find
A
\[s(x)
is,
Write a for x in every portion of the fraction
except in ike factor
And this process
(xa)
may
-
v '
Jf(x}
)Y\
)
itself."
be applied to every partial fractiofi
which only occurs once.
<j>(x),
corresponding to a factor of
Moreover, since
)
and
= (xa)\f,(x),
\//(a) is finite,
Hence we may
<t>'(x)
0' (a)
.'.
also write
= (x
=
A
(a).
*//
in the
form
,,
,
2
1
^-2 2(^-3)
'
2(^-1)
Thus, here, three partial fractions must occur.
For
if
there were a fourth fraction
,
say, the
No
denominator of their
sum must be (.r - 1) (x - 2) (x - 3) (x - 8), which is not
Hence we have obtained the whole expression,
p.i.c.
1$
others can occur.
so,
CHAPTER
146
x5
Ex.
2.
Here the numerator not being
,-..
7
V.
(x-a)(x-b)
of lower degree
than the denominator, we must divide by the denominator.
will then be expressible as
-
3?
(x-a)(x
where
A and B
Since
A
=-
t
-&'
,
A -B
= x + (a + b}-\--x-a x~b=,
1
a)[x-\-a + b] + A-\
j-
and similarly
B = b-a
,
---1
(^~~
O
OC
x=a we
a \ putting
get
.
We may here stop to remark that A
rule
result
are to be found.
-- = (x
3G
b)
The
"Put x = a everywhere except
and
B
can be written
--
in
x
a
down by
the
itself" just as well in the
yA
yA
rx as in
r - (x'+a + b).
(x-a)(x-b)
(x-a)(x-b)
This remark is general, and will usually save much trouble.
original expression
r-;
-^
Thus
.
---~
-JL
=x+(a + b)+-^-,
b-a x-b
a-b x-a +~
nz
-.
(x-a)(x-b]
i
7,3
i
.
Let the roots of xn =\ be a, ft, y, ... and F(x) a rational integral
algebraic expression of degree lower than n then, by the second rule
Ex.
3.
;
of Art. 142,
xn -
where the summation
This
may
for all the roots.
is
be also further expressed as
m~ l + ... +
m
K(m < n), then, since the
F(x) be written as Ax + Bx
th
th
sum of the r powers of the ?i roots of unity is zero when 0<r<n,
we have
If
By
=
taking F(x} = x and putting x e, deduce that
--
sin
(w-2)^
= --1
\
n
nx
'
sin
-=(n-i)
-
2
2 sm n
/
?-7T
r =i
cot
\x
\
nr\
--
n
).
J
[MATH. TRIP., PART
144. II.
Next suppose
the factor (x
II.,
and no more, so that we may write
= (xa)r \js(x) where \js(a) does not vanish.
<p(x)
be repeated r times
Put
xa =
y.
1919.]
a) in the denominator to
RATIONAL ALGEBRAIC FRACTIONAL FORMS
Then J-^\ =
<(z)
any means
-\
-.
or expanding each function
\
^( a +y)
y
in ascending
147
powers of
by
y,
1
y
Divide out thus
r
:
etc.,
and let the division be continued until y r
remainder.
Let the remainder be y r x(y)Hence
,
-1
A+
+fr-i
r- 2
is
a factor of
the
x(y)
i
-i
,
Hence the
partial fractions corresponding to
"
"
determined by a long division sum.
145. Ex.
(i).
Take
Then
_
.
the fraction =
r
(x
a)
are
Put *-l =y.
i
3/3
Therefore the fraction
^++^
3
1
146.
Remarks.
In practice it is desirable to perform the division by the "detached
"
coefficients method, and the above work appears as
(1)
iJ
-I
CHAPTER
148
V.
(2) In cases where there is but one other linear or quadratic factor in the
denominator </> (x) and that not a repeated one, this process vr\\\. finish the
whole operation.
~,2
i
g~.
The fractional
3 + 3 + 3-
1+f
+1+
1
3-
~*-|
-i-t-4
o
the fraction =
Hence
3
and
is
Ex.
5
2(^-l) ~4(^-l) 3
4
3
+
-I
-I
4
3
1
~2(^-l)
o
-I
+
+
2
^1
1
1
2
2(^-]) ~8(^-l)
x-3
8 1+tf 2
'
then ready for integration.
(iii).
(X
In such a case we find the three partial
.
j-
L)
(X
2)
fractions corresponding to 37
or
1, and then, either from the remainder
2
beginning over again, the two corresponding to (x 2)
.
expanding out f(a + y) and \]s(a + y)
separately, as shown above (which is however usually best in
Instead
147.
practical cases),
of
we may expand 77~ZT
\
as th u g n
it
were
Taylor's theorem, or otherwise, which shows a
Cv
theoretical form for the several coefficients, (7
F(a+y) by
compact
C2
,
...
,
,
of Art. 144.
Thus
,
So that
y
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
148.
Nothing has been assumed
149
so far as to the reality of
the several roots, a, 6, etc., of <f>(x) = 0. Hence the rules
obtained equally apply for unreal or for real roots.
If
whether
a, b, c
p + q + r+
be real or unreal, so that
the degree of </>(#),
a result of form
we
obtain,
...=n,
by methods explained above,
and imagining these fractions to be reduced to a common
\
f( x
denominator and added up to get back to the form Q-\, the
coefficient of
xn
~l
is
obviously
The integral will be
A p _ + Bq _ + Cr _
l
l
x
...
1 -\-
.
a
1
-'O
etc.,
ie. in general
149.
partly algebraic and partly logarithmic.
The conditions necessary that the
integral should be
purely algebraic are clearly
A p _ = Bq _ = C
l
r.l
l
= ...=0
>
number the same as the number of different roots of
~
=
0.
But the coefficient of x n in/(x)/0(x) has been seen
0(x)
and
in
l
to be
A p _, + Bq . + C
l
and
this
r_
l
+...,
must vanish when the above conditions are
satisfied.
CHAPTER
150
V.
Hence the index of the highest power of x in the numerator
must be at least 2 less than that of the highest power of x
in the denominator.
If
a, 6,
then the number of different roots of
be
and
say
than the degree of <p(x),
c,
.
. .
,
k,
;
if
<j>(x)
= Q,
the degree of f(x) be lower
we must
viz.
by
2
necessarily have
= ... =
and one of the k conditions, A p _^
cluded in the others, and there are then only k
B^
=
'^4
}f(x}
(p(X)
0,
1
must be
in-
independent
to be entirely algebraic.
150. III. Consider next the case of an irreducible quadratic
**
(*-)* + /32
,
not repeated, occurring in the denominator, 0(x),and let
Then the
partial fractions of
fM',
.
i.e.
of
corresponding to these unreal factors, are
f(a-t)
or,
as
separating out the real and unreal parts of
P + iQ,
a
.
^
o\
these partial fractions are
,
....
which
whe
nd
and
is
,.
form
of
P_
4
i
4
Z = 2P,
Lx +
M
which are both
real,
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
151. IV. Case of the factor
a)
+ /3
2
repeated r times.
= [(x-a
<t>(x)
Let
Then
will be possible to write
it
For this
equivalent to determining
is
contains x
effected
Pr
and Q r so that
,
f(x)-(Pr x + Q r
so that
-i.e.
2
(,/;
151
a
and x
1/3
a
-\-ifi
as factors,
and
this will be
P r and Qr such that
by taking
,
and
if
when
4v^
o\
VK+p)
becomes
+
J.
.B,
then
5
Pr a + Q = ^l
r
and
r==
i.e.
separated into real and unreal parts,
/3
Thus
Pr Q
,
This being
r,
^ r=
and
P
r /3
= J9,
5a = Afi-Ba
~^~
^8
and therefore ^r are determinate.
so, it is
obvious that
Xrfr)
[(x-a)
can
itself
be expressed as
and by continued repetition of the argument we get finally that
f(x)
_
P
P^X+Q^
P X+Qr
r
and the values of the r pairs of quantities,
Pr and Q r Pr _, and Qr _ l9 ... P t and Qv
,
,
are successively obtainable as described.
The general forni of the result is thus established.
mode
of finding the numerical value of the P's
laborious, except when r is small.
But this
and Q's is
CHAPTER
152
152.
^rr4
now
It
V.
appears that the general result of putting
into partial fractions,
where
<
(x) is, say,
2
2
(x-a)(x-b^(x +px+q)(x +rx+sf,
the last two factors being irreducible to real linear factors,
and f(x) is any rational integral function of x of any degree,
will be of the form
Jfix)'
^~
= an
integral algebraic quotient
xa
Px+Q
R x+S
l
This
is
l
the general typical form of the result. If other
(x), other partial fractions will occur in the
factors occur in
result.
153.
For
But
all
<
others will be of the types exhibited.
The integration can therefore be effected.
The integrals of the algebraic terms are
(1)
of type
[#dxAt s+l'
JA
(3)
The
dx
integral of
is
is
cZ,
WThe integration of
I
I
-,
ff&
I
/>V
^log(x
-7
I
Q
a).
dx has been
effected
dx can be
effected
in Art. 136.
(5)
The integration
by means
of
of
I
-^
a reduction
^
formula, as
will
be
explained in a subsequent article.
Hence we may then regard the integration I'TT-^^ a s
complete whenever
nrW
is
a rational algebraic function of
x.
In practice, when irresoluble quadratic factors are
present in the denominator we may first of all determine the
154.
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
153
partial fractions corresponding to the real linear factors, single
and repeated.
Then,
that not repeated,
if
it
there be only one quadratic factor, and
will appear without further trouble in
f(x)
But
the remainder of 3~4.
there be several such factors
if
*(a)
or a repeated factor, we may subtract the simple partial
fractions when obtained and then after simplification discuss
the remainder.
Use of "Undetermined or Indeterminate Coefficients."
often with advantage apply the method of " indeter-
155.
We may
minate
coefficients."
When
the fraction has been reduced
numerator
degree n
by
division
till
of lower degree than the denominator,
at most, and we get, as in I.,
is
1
A
fix) _
^{(x
a
x
of
Rkx+Sk
Px+Q
B.
the
i.e.
s
b)
we have, upon multiplying up by <t>(x) an identity in which
the right-hand side is of degree n
I and consists of n terms
when arranged in powers of x, and the left side is of degree
n 1 at most, viz./(x).
Now
the
<f>(x)
is
number of
of degree 1-fA-f 2-h2yu,
quantities
A, (B lf
is
1
which must =n, and
-f
B
z ,...),
A
+
(P,G,),
(R lt
2
4-
8lt RZt S2>
...)
i.e.=n.
2^,
Hence, upon equating coefficients of the n terms on the
right-hand, side to the corresponding coefficients in f(x), we
have just enough equations to obtain the n quantities, pro-
vided that these equations are
all
independent.
But as we
have established otherwise a means of finding these quantities
we may infer the independence of the equations obtained by
equating coefficients.
156.
Many
of the coefficients, or
all,
may
be found by the
substitution in the identity of numerical values for x.
Obviously
any number of equations of this kind could be obtained, but
only n would be independent. The most suitable values to take
for this purpose will be such as will make one of the factors
z
x a,
x* px
rx s vanish, for such values
q or x
xb,
+
+
+
+
would cause many of the terms of the identity to disappear.
CHAPTER
154
V.
In substituting roots of x z + px + q, viz. a*/3 say, only one
Then the real and unreal parts on
root need be substituted.
each side of the identity may be equated.
All the J5's and A, i.e. X + l of the quantities, can be found
by the easy rules given above (Arts. 140 to 147). Hence X + l
of the equations obtained
independent of the others
by equating coefficients
when the values of A, B
will not be
l
,
B
z
,
...
#A
,
which have been found, are substituted. But there will still
remain 2 + 2/x independent relations from the equating of
The substitution of a root of xz +px+q and of a
coefficients.
2
root of aj + riC+5 = with the equating of real and unreal parts
will furnish four other relations and reduce the number of
independent "equated coefficient equations" to 2/x 2, which
The
are linear and to be solved in the easiest way available.
student will perceive that in practice it will be best to combine
several methods to determine the coefficients and to use
redundant equations to check numerical results.
157. If none but even powers of x occur in both numerator
and denominator, we may put x 2 = y, and thereby reduce the
In such fractions, the quadratic factors
becoming linear by this substitution, their occurrence may be
labour considerably.
termed pseudo-quadratic or
* +1
quasi-linear.
Bx.l.
This
is
of
form
Putting, then, or (or y)
=z - 9,
-+ -+(!+*
"5
3s 2
3z
25
5
2
_?i
25
-5 +z
25
3
1
3
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
155
1
The
partial fractions are of
A
x-l
Multiplying up
form
Dx+E Fx+G
Bx + C
'"
we have
the identity
Putting #=1,
l=50A.
Putting x=
l
Putting
i,
^=2
t
;
= (2^ + (?)(2 -l)(-3);
4*'+ =
F- G =
l
,
...
Equating
= (Bt + C)(t-l)9
coefficients of
#6
t
t
,
Equating absolute terms,
-4^-^ = 1,
111
1
158.
Case
denominator
when the numerator
is
E=^
x+l
"T"
/r*
;
8
^+1
re'
__9!
l
1
15
an odd function of
(.
x
and the
even.
<fc,
and putting
is
whence
z2 = y
t
takes the form
I=
%
and the factors in the denominator which were quadratic
factors in x are linear in y.
1
2
1
log
.
1
+
^+T 2^Ti
x2 - 1
1
1
CHAPTER
156
V.
159. Case when the denominator is odd and the numerator even.
The same process may be adopted.
lhus
/O0
(3
The
2
.
partial fractions are of the
and the integral
The value
(
ai
2
-
2
)
(a 2
The denominator
(i - a r ) (a 2 -
- ar 2
)
...
r ) ...
(ar_i
Taking the case when 1?
6, this denominator
2
- ^ r2
may be
- ar
2
3
,
Z),
)
(ar+1
- a r2
- a r ) (a
r+2
D = (-l) r- b -*(r-l)lbn-
and
= a 1} we
in the
r
r
(n-r)\
ff
l]
have
I
giving for this case the partial fractions
a1
2r=n
V
-
2n -2
2
f
r
1
and the integral
O
r=ii
/
V
_
r2l+2
'
1
2
- ar2
)
- ar
)
...
A.P.,
(an
-
with
common
6.26. 36...(w-r)6x
;
Ar = (
... (a,n
say, is
where in forming the product of the factors
term (a r + a r ) has been supplied
l
)
form an
...
,
2
written as
(a r+1
)
-2)6...26.6,
If 6
< n.
1
-tan
(a r_,
factorized
difference
and
5
is
2
r
.
form
2
is
Ar
of
- -
j
where a
9
9w 29
^ 7-*-.
9T
2
2
2
2
2
+ a 1 ow
)(2 + a2 )(z + a3 )...(2 +aw )'
r^~.
2
\q-r-\
I
(n + r)l(n-r)\
lower line the missing
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
161.
Obviously we should also have in the same case
+l
z-
=
__
9
-
a
2n-2q
fr=n
/
V
I
(
Taking the case
t-i
i
2 2n
!
-1
r
L
dz
\0-r
\
-2j
(n + r)\(n-r)lz
t
.
162.
157
0^
= 2,
6
= 2,
and therefore a r = 2r,
(2n)g^ _ 2
2
+2 2
7i
+ arz
-,.
)t
,.
z
2
(2^
1
- 2)^+ 2
22 +(2?i-2) 2
"
"
t
(
V
+ ...+(-!)--C^,
and
its
integral
-l'**-
7
1
1
(2^)
2
+1
tan- 1 ^-
- 2(7
1
(2?i
+ ... + (163.
And
similarly,
instead of 2^, the
and
its
2n
[
if
-
+1
2
2)
1
I)'
-
1
tan" 1
^C^^+Han-
1
......
(A)
the index of z in the numerator had been
same work shows
integral
(7
(2w)
2
f2
log (z* + 2W)
-2
'l
(71 (2n
- 2) 2
+s
log {z
2
+ (2n - 2) 2}
.
...(B)
CHAPTER
158
Taking the case
164.
1
!
= !,
/O r
1
= 2,
6
and therefore a r = 2r
_ I \2<vH
_il__ 2n-l/7n 2"- 2
-
(2*
*..,
l)
its
2g+1
-
2
and
V,
1.
1
X
_ 2n _!
12
2
integral
2n ~ 2
-t)l 2
n- 1
2
-
^-^
2M
- 1 (7
1
(2w-3) 2} tan-
1
^-1
.
And
165.
for
2
I2)t>>
the integral will be
_
^2 + (2yt
(C)
1)2 ]
^-
2
I)
}
.
2n ~ 1 C'
-
-
1 (2?i
3)
29+1
+ (2w - 3) 2 }
2
log{
-
+ ... + (- ir-^^C^l^Uog^ + l 2 }].
***
Consider the integral
166.
Here
f(x)=x
m
X
-
f #2n
9n - . w n
2a ^p cos na
+ a' n
n
2n
^(^)=^ -2a ^ coswa + a
J
(m < 2n)
.
2 'l
,
=
r=r-lp
n
(Art. 142)
-
/
9r-7r\
Lr -2aa;cos(a +
L
n
\
~
n
n- n
=
2rca;
a cos na).
<^)'(^)
2
r=o
1
(.t;
aH
Let
-- = A
2r?r
,
Y.
71
The
factor
and gives
x z - %ax cos x +
= (^? - ae' x ) (a; - ae
~ lx
),
rise to the partial fractions
Iff
<p
NOW
2
!
<p(cie*)
=
\ae
C.V\
)
x
IV
ae A
I
//
<p\a,e
Qn-l L^n-lW/
tnv
^
e v
Sna"
'*(*
- IV\
*)x
ae
^
coswa)
-
1}
/
"1
+a2J
-
,
(D)
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
Hence the two
partial fractions
*(--i)x
- |I| -
1
e i(>-m-l)x-i
x- ae*
na
sin
2wa af|
x-ae~^ J
sma
r2tsin(?i-m)x - 2#tsm(?i
waa 2 "-" 1 L
# 2 - 2atf cos + 2
1
2i?i sin
:
159
-m-
1
)
x~]
J
'
r2asinxcos(w-m-l)x-2(ic-acosx)sin(w-m-l)x
~~ "l
- a cos 2 + 2 sin 2
J
(ic
x)
x
2w sin waa 2 """*" 1
x rn dx
*-
.
5in
/
I
V
aH
2rrr\
n
J
2rrr^
Iii
the same
n
p~l
way x /(x
results are given in Exs.
167.
Ex. Calculate
Here
(Art. 166)
The
an)
may
be integrated.
The
39 and 40, pages 166 and 167.
dx
f
P=-->
m==0
>
7l
indefinite integral is
^-acos
37T
3
+cos
_
sin 2J8
-
J cos
L
acos/3
2
ft
log (a;
- 2a
sin ^8
+ a 2 + J cos /^ log (x 2 + 2rt# sin /3 + a 2 )],
)
CHAPTER
160
and taken between
The
indefinite integral
1
2a? sin
An
168.
and
limits
26 L
may
Sm ^
V,
oo
also be written as
,
B
Zax cos
2
-
COS
2
0^11 2# sin B~\
^ tann
2
2
J
integral of the form
P
b&
-,
can always be integrated as follows
Let
I
J& U
4"
be the
I
w
*C
s,
and
let
- = T and
q
V
/^
q and
L.C.M. of
:
/r <v
5
tt't*/
Then
1 o^"~
k/V
^
I
s
=^
L
rl
ty
tl/xy.
+^
s
and the expression to be integrated is now rational, and when
expressed in partial fractions each term can be integrated.
flf!
Ex.
J
%
dx
(Let
=
#=26 .)
Q^ dz
I
= 4^ + 2^-3^-4 log (1 +57') - log(l -^+^)
+ 2V3tan~
In exactly the same
169.
way
can be effected by putting a
q and
s,
2^-1
1
^
the integration of
+ fix = z when
and more generally that of
l
is
the L.C.M. of
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
where
for,
f(t)
and
putting a
$({) are
+ /3x
any
161
rational algebraic functions of
t\
becomes
z\ as before, the integral
and the integrand being now rational and algebraic, we can in
any such case proceed to put it into partial fractions and then
integrate.
EXAMPLES.
Integrate with regard to x the expressions in the following seven
groups
Linear unrepeated factors
1.
W
-,,*
/ill)>
(V
(
<V-l)(*3 -4>
(*-l)fo\
2
Ma; -9) (a; 2 -16)'
/
^
^j^lg;^^
iC
/~
MV^
(V
2)
N
n)
(-l)(*-S)(*-5)
,
/i,.\
2.
i\-
3*-*?
( vii,
.
:
n'
(V iii)
T1
(j6-o)(s-i)(a!-e)
(z-a^a;- b l )(x --;,)'
^^i
/>
Linear repeated factors
_..
t
iC-|-l
:
^
(a;-l)(+l)'
/...,
(lll)
(v)
x+l
^fa-1)*'
2 (z
a:
3
(vii)
[I.
3.
Quasi-linear occurrence of factors.
Powers
C. S., 1900.]
of x all even
:
t
x^dx
)
a./-
2
+&
.
dx.
\
In the last
two
c,
c^,
e,
/, g,
h
x* (caj
may be
+ rf) (ra2 +/) (^2 + h)
ppqsidered positive.
CHAPTER
162
V.
Numerator an odd
Numerator even, Denominator odd
Quasi-linear factors.
4.
even, or
dx
.
+
x1
f
[
dx
f
'
f
\
5
Jz'-6a: +llz3-6x'
f
7
^^
f
J (ax*
Quadratic factors not repeated
5.
Denominator
^
2
<">
.....
function,
:
+ bx + cf + (x 2 -bx + cf
:
dx
f a; 2
1
I
rOJC.
+
JX +
x
(ill)'
4
,
.
--^x2
f
x
+
1
1
r
1-34
J
(iv)7
>
r dx.
+l
(v)
Linear factors repeated.
6.
Quadratic factors not repeated.
dx
W
dx
;..v
9
2T~2
i
T'
(
u)
wT~
71
dx
2
)
;
+ 4)'
2
(a;
dx
dx
/
*T7
(
*
\
+
J)
dx
...x
9\
9
2
(
da;
.v
*
'
\
2
(a;+ 1)
/
*3^
^
^\/7i
i
(
_jfe_
'
2
(2x-3) (4x
)'
7.
Eepeated quadratic factors
dx
X7!??*Va:
:
+ l)dx
7
(V^V'
f
8.
Evaluate
Evaluate
x
(x
rl
Vtan^d^
and
>
|
Jo
Jo
9.
.
(1V)
f
I
_
(i)
f^
,.
^
4
Jo cos x
i
+ cotfx sin 2 x + sin 4
+ rt)
(.
+ &)
(**+<??
7
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
163
cos x dx
10.
Evaluate
i
:
J
r
,
J
x*dx
2
2
2
(& + **)(& + b ) (x + c )
o
+
12.
Show
r-
(l+smz)(2 + smz)
that
f
-
TT_
2 (a
-f 6) (6
+ c) (c + a)
*
-y^
2+
.
^r
[7,
13.
Show
1891. J
that the sura of the infinite series
can be expressed as a definite integral, viz.
And
hence prove that
A
log,2).
[OXFORD, 1887.]
25 dx
...
14.
Integrate:
X
( )
[COLLEGES, 1882.]
[ST. JOHN'S, 1881.]
+a?)dx
[COLLEGES, 1882.]
[COLLEGES
15.
a,
1891.]
Prove that
[ST. JOHN'S, 1881.]
16.
Prove that
p+r+
(x-a)-
Or
-6)"
+r
-p + r+l
Pr
p+
(b-a)
-q + r+l
1
where
Pr
and ^r are the
respectively.
coefficients of z
r
in (1
+z)~
p
and
(1 4-2)"'
CHAPTER
164
V.
dx
17.
Integrate
I
(i)
(DX
ox/\x
[MATH. TRIP.,
1878.]
[OXFORD L,
1888.]
dx
(ii)
Jxdx
(iii)
e
18.
Prove
(i.)
[COLLEGES /3, 1891.]
[TRINITY, 1882.]
[TRINITY, 1895.]
xttx
19.
[
Integrate
Prove that
^11jpn
+
3
J.c
+r
i
+
+
U~T7
'''
to GC
= -i r
T -
^
,1
log 2
J.
[COLLEGES, 1896.]
20.
f (v/cot
Integrate
=
1
x - N/tan x) dx
^
+ 3^ sin 2x
-.
r
^
Ie0
a
[COLLEGES
/3,
m
1890.]
T
tan" 1 A/
{/~2
(ii)
22.
23.
I
xAi
2
25.
2
^x.
_
+ \/6 2 +
c/a;
<fo.
[MATH. TRIP.
Integrate
Integrate
[J.
Evaluate
Integrate
,
1898.]
[COLLEGES, 1896.]
If COS
I
f
)x{x-a)
,
M. SCH., Ox.,
1904.]
4/r
i
24.
,
t-dx.
5
[ST. JOHN'S, 1892.]
iC
n being a positive integer
[ST. JOHN'S, 1882.]
n
C/x-b\*
l(
dx.
j\x-a/
7
26.
27.
Integrate
Integrate
)
[COLLEGES
a, 1885.]
[MATH. TRIP.,
1895.]
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
28.
Sum
assuming
165
the series
to be convergent.
it
Deduce that
1
'
F73
j_
23
1
+
'
375
J_
+
2^
1
'
5T7
m^~4~T6
F + '" 7-/-_I_Al
J_
[I.
29.
g
C. S., 1899.]
Prove that
1111
7T
[COLLEGES
30.
x-2 log
(1
log(l
Evaluate
l.u'
I
-Xx 2
8
2
9
3
"
[COLLEGES
no* + a*)"
Integrate
1888.]
dx,
)dx,
1.5'2.7'3.9'
1
,
)
111
and deduce that
31.
'
a, 1889.]
efcc.
Prove that
[MATH. TRIP.,
32.
Show
fir
33.
Show
+
Show
1
ic
r=rn
1
that
_3"2^(1
34.
w
W
a rational integral algebraic expression of a finite
number of terms.
that
2
2
31
n(-/i-l)32
2"+
-23)*" 2^r^2Tr"
1
1
dx
L
c< 1,
if
a;
.
3
,
_
2
c)(l -c' a:)
...
to co
z4
1
h
f
w
7
l
is
the
sum
of the
to co
Tn + 2 ^.
/7-r
w
(x-a )(x-a )(x-a
Ig.n+2
35.
where llr
1896.]
that
2
\
s
7
)...(x-an
'>
}
homogeneous products
r at a time of
CHAPTER
166
36.
Show
V.
that the part of the indefinite integral
fix}
1
which becomes
infinite
when
,
= 0,f and $ being
when x = 0, is
x
rational integral
functions of x which do not vanish
1
/(0)_1.
X
[Ox.
I.
P., 1901.]
Show
that when a rational fraction is decomposed into.
"
"
or
fractions, the decomposition is unique.
partial
simple
37.
its
th
F(x) be a function of the (n-l) degree which assumes
=
the values u lt u 2 w 3 ... un when x x x lt x 2 ... xn respectively,
38.
If
,
,
,
,
,
,
show that
(x
I
2
(x-xj(x-x3 )...(x-x n )
-x1 )(x2 -x3 ...(x2 -xn
(x2
n
(Xn
39.
Prove that
if
)
-
p<n +
Xj (Xn -X 2 )...
r=
= lOf \X
Xn-l)
n-l
/
_._r
CL)
-
l,
1
sjy>
(Xn
-f-
/
2?*/?7T
COS
,
l^fif
J/Y.P
27-7T
x
r =-o-
-2
xr^
sm
>,
frf
"nwr,
tan
n
acos
l
asin
if
and
= log (x - a) + ( -
1
p
)
log
r=
^ =1
a)
-
cc-acos
ft
S
-r
-200; cos
cos
-2
(a;
sin 52^ tail-'
w
n
a^
a sin
n
n be odd,
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
Prove that
40.
p<n+l,
if
x-acos (2r- 1)-
V
+2
167
sin(2r-l)^tan-i
/9
1
i
'
11
if
= -
and
2
-
cos
(2?*
r=i
1
z 2 - 2ax cos
w log |(
)
3-7acos(2r
r=f
2
+2
sin (2r
-
1)^
r=1
4
1
.
(2r
-
1)
- + a2 1
n
-2
(2?-
}
1.)-
tan- 1
a sin
n be odd,
-
1)'
if
be even
-
n
Prove that
fj r
1
A
=
r=w-l
v^v
2w 2./
{Xi"i"^
^ COS
/
?'7r
n
.
.
tanh- 1
cos
\
t
+^
+ sin
rir
tan"
[MATH. TRIP.,
42.
Show
43.
(i)
1884.]
ATT
^
that
Show
=
dt
that the remainder
integral function /(a:)
by
(x
-
2
c)
+ b2
V
left after
dividing the rational
is
r
r +l)//'\
4.
"J'
where /
(ii)
(s)
(c)
If /(#)
denotes
and
does not contain
determine
finite
that
is
divisible,
^^.
and <j>(x)
show that it is possible to
the constants P and Q in such a manner,
</>() are rational integral functions of x,
(a;
2
c)
+ lz
values for
as a factor,
f(x)-[P(*-c)
without remainder, by (z-c) 2
CHAPTER
168
(iii)
Apply the
last result to
show
V.
prove
(or
in
any manner) that
f(x)
[(X-C^ + WY<1>(X}
can be expressed in the form
Pn (x-c) + Q n
X (x)
r being a positive integer, \ (x) a rational integral function of
Pn
and Q n constants.
and
[I.C.S., 1892.]
TC
It
A A
44:.
x,
\*^V
/
=J
(/)
/
(
Tv^v
+ "^
#\
i
)
1 1
^ are rational polynomials of degrees m + n, n m,
= 0,
respectively, show that if Jt a 2 a z ... a be the roots of(x)
where
n-l
T7
',
/,
</>,
t
a.
considered
,
,
;t
,
be determinable from
all different, \j/(x) will
x,
av
='0.
o2
Also determine ^(x) when /(a;)
A*
45.
46.
T
=
,
has equal roots.
[OXFORD
I.
P., 1913.]
[OXFORD
I.
P., 1917.]
4.
Integrate
Prove that
v
(i)
fcos
n be a positive
(n - 2p) 6
I
J
if
.
Sin?ifl
1
=^
71
4^
integer,
2prir
dv = - > cos-^
T/1
,
/
?'TT
\
71
log ami P
71
log cosec
47. Integrate
and prove that
(i)
JV^,
(ii)
sm
x dx
(a;
-i-
(
6
-
a)
7T
(iii
4'
(iv
[MATH. TRIP.
.J
Obtain the rational part of
I
I.,
1917.]
.
^
[MATH. TRIP.
II., 1915.]
RATIONAL ALGEBRAIC FRACTIONAL FORMS.
49.
169
Prove that
2n
2n
(1 -f x)'
+
__
-
(1
2n
x)'
where a r = (2r+ l)-/4?i.
Write down the values
~
a r cos 2u 2 a r
'^ sin
^
sin (2/i
-
1
x2
1) a r
+ tan-a r
[Oxr.
'
II. P., 1899.]
of the integrals
xdx
Show
50.
that
X dx
_
o(<t + a;r
(
,(
_, ) H
=
according as
n
is
[Cf.
'
//
that
JL
f
?t
1 [
n(n+l)
1.2
52.
Show
ji ll
J_1 4
1
= ao
'
!!
a:"- 1 J
1
1
1
[
r =l
if
'
or to A
t
1912.]
(a
-
n
x)
2
2 Ly*"
1
-n(n+l)...(2-2)
.^- 3 J
1.2...(w-l)
[MURPHY, Cawift.
z
1
g
y*
T'r., vi.]
p<q,
\
r
'/-/*/-
rx-\
(
i{(i_
= ^^
^
-/ M
&n
COS
?'7T
[TODHUNTEE,
Deduce that ifp<q,
J Sings
=
JL fj_2 J_
W~
w1
8
'-3[y*-
that
r i
-^
n be even and x + y=\,
if
l
y"-
=
WOLSTENHOLME'S Problems, No.
+ x) n -
I
Show
A
to
1
of the integral
1,(a
51.
ATT
odd or even.
Write down the value
(lf
ATT
.
.
summation extending from A =
the
f
J /"
^ (
IT
2^^^
fef J
COS?-TT
?-TT
I.C., p. 38.]
CHAPTER
VI.
T-TJT
x+
+
Ifl'T
b cos
(a
170.
f
etc
-
Integration of forms
dx
dx
f
+
(dx
,
COS
t>
((-
or
we may
sin
^+
b) cos
(a
P
^j
CASE
I.
a2
>b
etc.
write a-\-b cos x as
2
,
/v
^v
cos 2 ~
sin 2 -
^ + (a
2
^
j
2
6) sin -,
-f
tan 2
L
^
-
r_2^'
-+
^6|a
2
da;
If
f
2
-P
6cosz
Ja +
Thus
+6
2
+ 6) cos
(a
-..
/
/v\
i.e.
-
ja
iC
/
a f cos 2
ffa?
f
'
171.
n
c sin-w>
x)
-
a
................ (1)
i
this
becomes
tan
x
a -b
2
_2
in a
a sin
- r tan -- tan- ^M-^r
1
tan- 1 ( tan
\
-.
/
tan H
"1
170
),
2/
,
where
b
= a cos a.
VARIOUS STANDARD METHODS.
Tins
may
be written in other forms
:
2 tan' 1 z
since
we may
cos" 1
-.
"^
.
>
write the result as
-
1
i
a
--
"
or
^2
^
*1
e.g.
cl
b,
+
-
2
,
1
1
If
,
.
.
2
<6
2
a
b
,
(1),
X
2
6'
x/aHh^cos
,
etc.
writing the integral in the form
\b-\-
X
a
x
tan 92
we have by
-
Art. 127,
+a
=
Vbb^a +
an
x
2
log-
M
-i
= =L=iog
7
/,yo""
fl
x/^>
+ +vi
,
y
x/6
+a
x/6
a tan ^
_^x
a tan ^
_-
x
a
a;
COS
1
2
.
a tan
a
]ot
*
a;
cos
4<j
= aseca.
a?
sin #
j
2
wliere 6
a;
V. + y6cos^
2
^
\
,
form
+ cos x
-{- cos a cos
I
a7 tan
in place of the
1
2
sin- 1
or
II.
7
X
.
bsm^
x/a
CASE
1
:
r
i
Ja
2
cos a
.
:
2
Further forms are
z
-rtan 2 fr
2
-== cos- 6 +7-5a cos x = 1 cos"
a + ocosa; asma
6
^/a
1
-7= 2
172.
171
a
CHAPTER
172
By
2
Art. 64, this
may
also be written as
Ib
x
a,
== tanh" 1 \ Y - tan -2
Vfe-fa
2
or
2_ a 2
we may
.
./
a,
tanh- 1 tan s tan
\
1
a tan a
= cosh"
2 tanh- 1
or, since
VI.
1 -
2
x
TT
2
^,
further exhibit the result as
still
a
b
.
"
,
or
1
7=
2
^/fe
and
.
,
.
= cosh- 1
a
2
b 4- a cos x
a + 6 cos x
j
,
^.e.
-
1
- cosh" 1 -1
,
in other but equivalent forms as in Case
173.
We
dx
therefore have
,
a tan a
4-
cos
cos #
cos a 4- cos
I.
,
a;
VARIOUS STANDARD METHODS.
1
The integration
75.
of
X
-
1
-
.
.
173
reduced to the fore-
is
=
going forms by the substitution x ^ + y, when
f
]
dx
a + b sin x
f
dy
}
a -f b cos y
2
=
la
tan- 1
-
b,/x
TT\
-
r tan (^
y
we have
J
1
= COS"
1
snce
a cos a
where
b
+
1
sin a sin
2
-.
tanh" 1 \ /6-a,tan
\0 a
-.
a2
Jlr
-
\//>
1
or
a2
!og
1
6 4- a
V/rr^
,
a cot a
17G.
= a cosec a,
We
might
COSll" 1
with
+
~
x
tr~~
a tan ( 5
V6
J
f
^\
I
)
snce
:
+
x
;
,
sin a
other forms.
many
I
)
~
-+- -
J
TT\
T
4/
\2
a tan
sin a sin
1
sin a
also treat
/#
(-^
^
+ a 4- V&
cosh- 1 -
or
6
ce'
= a sin a,
or
where
+ since
sin a
.
POS
.
-
-
a + 6 since
independently.
-
}dx
,
we
j
a + b sin x = a (cos 2 T + sin 2
^j
Thus,
d
dx
t
\
ii
+a
-f
26 sin - cos
^
write
CH AFTER
174
and two cases
as
arise
VI.
a>
before, viz.
Art. 127;
when we apply
6,
x
dx
f
J
a 4- b
2
^tan^ + 6
.
sin x
or
showing the
result
in
different
forms from those already
given, but of course differing from them only by quantities
independent of x. The student should consider this state-
ment and
little
ifc
is
Extension.
Rcos(x
deduce
y),
may be written
and tan~ 1 y = r, we may
Again, since b cos x -f c sin x
where
= \/b 2 -\-c*
21
1
.
csin x
J
from
---^
I
course,
f
-
ja + bcosx
proceed independently, at our pleasure.
J
a matter of some
ingenuity.
177.
as
reconcile the results, as
dx
a + 6 cos x -\- c sin x
=
y)
2
a
1
R,
xy
tan-^
~^-==,tan-
.if
1
or
or
_t
.S
.
-*
x -y
JR + a + jRa tan
^.e.
or
cosh" 1
.RH-acosx
s~~
a4-/ccosic
with other forms.
a2
+ acosiC y
V/*
B
s+,
y
=
y
'
or
we may
Adopting the former
we have
}a + R cos (#
,
/y
,
^~
VARIOUS STANDAUD METHODS.
And
175
these of course include the forms of Arts. 171 to 176
when
as particular cases, viz.
The reduction
178.
f
has the advantage of
=
or b
= 0.
form
to the
J
c
dx
a + b cos x
making the
integral depend
upon the
integration of
treatment throws the integration
the independent
whilst
upon the form
f
J
dx
ax 2 -f 26x + c
'
and involves the completion of the square
Illustrative Examples.
179.
Ex.
/
in the denominator.
1.
dx
cfx
dx
cfx
r
_
f
J 3 + 5co3#~
}
J
_/
3
\
<,x
cos- --f
2
I
2/
/
+ 51
./
cos*
V
N
2
a;\
-sin 2 2/
9
)
dx
/
8cos 2
J
.,x\
sin 2 -
-2 sin 2
|-
-
4TTT^'
(iC
tan 2 -
"
2
l^
4
+ tail f
8 " tMI
.
Ex.
l^^/l
2
I
2.
dx
f
C
dy
r///
j3 + 5cosv
= i cosh-*
.
where x =
,
.
.
,
5
- 3 cos x
r
6
Ex.
3.
dx
f
SIMILAR RESULTS FOR HYPERBOLIC FUNCTIONS
177
Again,
{dx
dx
C
_
a-\-bsmhx~
I
a cosh
]
<,x
2 -
.
smh
(
\
,
9
2
x\
^
A)
)
<
x
,x
+ 20 sinh ^A cosh -L
,
.
cUanh?
a
\a?+W
(
J~~tf~~r
mh
b\
x
2~a
>
/
JL^tanh- 1
and other forms
will be exhibited later.
Similarly, in the general case,
f
dx
J
a + b cosh x+ c sinh x
a (cosh 2 1- sinh 2 ?} + b (cosh 2 ~ + sinh 2 ?
^
\
L
\
LJ
sech 2 - dx
i a H- 6 + 2c tanh ^
(a
cZ
ot
p
a
6
I
fa + &
c2
6)
tanh 2
tanh
tan
!_/ tanh x
,
2
gr
6
a
I
^
cZ
c2
(a-\-b
J \6
a
tanli
(b
2
a)'
a -6) tanh- -c
or
v//y2_ a2_ c
C.E.I.
2
)
L)
c
ab
cosh?
+ 2c sinh -'LA
t
CHAPTER
178
But we notice
also that just as
written
R cos a
b
and tan a = T, we
r>
and
6
by putting
c
2
a
= R sin a, where R = \/b' + c
2
c
z
write
may
a -f b cosh x + c sinh x
62
a-^bcosO-^osmO may be
a+R cos
.
by putting
=N
VI.
a + R cosh x + y
as
= 12 cosh y and
c
=R
sinh
62
if
y
>c
2
,
where
and tanh^,-, or as
sinh # + y
by putting
&
= .R sinh y,
c
= R cosh
y,
R = Jtf^
where
tanh y = - when 6 2 <c 2 and therefore the case
c
,
may
and
b*
be regarded
as one of the previous ones or vice versa.
181.
A
Another Method.
further method of treatment will
we remember
that these hyperbolic functions
are merely functions of a real exponential.
Taking the general integral in this way, we have
be obvious
if
dx
J.- -|-fr
cosh
(
+ c sinh #
J
x
2a + b(e x -}-e
2e x
+ c(e x
)
e x)
dx
de*
or
(6
+ c)
2
giving the forms
2
or
--
.
tan" 1
-
V
.
.
=
Comparing with the
coth
;
=
2
x/6'2_ c 2_ a
1
*
+
c)e*
(6
._- -i
v
+
.,.
if
tt.
results of Art. 180,
that the integrals of such expressions differ
.
if
it
>a
70 ^
62
2
+c
2
79
62
will be
much
remarked
in appearance
INTEGRATION IN TERMS OF THE INTEGRAND.
179
Integrals
according to the method adopted in integration.
of the same expression, however, can only differ by a quantity
or unreal) which does not
(real
contain
x,
anc^
will be
it
a useful exercise to deduce one form from another; and,
as has been said previously, this will sometimes require some
ingenuity.
182. The Integration expressed in terms of the Integrand.
Far more symmetry, however, will be obtained in the results
if we attempt to express the integration in terms of the
integrand, as
we now
These integrals
proceed to show.
be deduced from the form
may
dx
which
is
A > 0, B
.
z
> A C (Arts. 80 and 81
),
01
the case
A < 0, B2 <^AG
being omitted because the radical
in the integrand becomes unreal in that case.
The rule is to substitute y for ike integrand
and
integrate in terms of y.
in
all cases
This method leads to remark-
and expresses the result in terms of the
and yields new forms for the integration.
Thus, considering the general case, and writing
able
symmetry
integrand
of form,
itself,
dO
y
where
we have
'
a+bcosO + csmO
6cos
+ csin = -
a;
J
and therefore
b sin
0c cos
=
.
CHAPTER
180
VI.
Squaring and adding,
->
7 2
6
+c
,
2ft
l
?=>
2
[-
2
y
2/
Hence
=
=
r
where
1
f/"
0.
increase together, -
and
it
be a case where 6
6
as
quadrant
*
,,
,
increases, that
I
J
-+
^
:
ve
is -f
we
increases
is -f
and
use a
cos
e.</.
;
in
+,
in
is,
TV.
which
in
for
/.
diminishes;
y increases.
supposing 9 to
lie
-
-,
-^,
in the
first
throughout the integration, we should use the
183.
I
.
bsuiO
a
if
by examining whether y
to be determined
is
throughout the integration,
In
4^
COS
increases or decreases with
provided
^
= a + 6 cos + c sin ^.
The sign
If y
^
v
cosh" 1
-
< <^
the
first
-
5
a + bcos6
quadrant
sign.
In the same way, to integrate
(
1
J
where
-
dx ~
r^6 cosh
\T"
^-j
x
a+
x + c sinh
-
f
or
I
,
.
,
6 cosh
say,
1
ra + b cosh x + c smh x
we have
,
y d%,
,
x + c sinh x = --- a,
y
b sinh x
+ c cosh x
-,
2
2/
-J^*
c^x
Squaring and subtracting,
62
and taking the case
increases
;
c2
6
and
c
/^?y\
---11-7-
1
2a
2/
y
a 2 = -:,-
2
>
y \dx/
both positive,
^/
decreases as x
INTEGRATION IN TERMS OF THE INTEGRAND
dy
f
y ax=
{?
181
I
1
^^ cosh" -
-=
cosh'
V +c -6p.
2
2
,
-
1
^==2 2 -^-+ const.
x/& -c
,
2
if 6
2
c2
and
'a?
_cos" 1
-
v
...
v/6
^- + const.
__
2
,
-c 2
if
=
or
wlicre
184.
1
7/"
.
.
smh
l
= a + 6 cosh x -\- c sinh ic.
Hence we get the following
putting b or
c =
-,=
)a + bcos6
1
dO
>Jb*a'
a^b
results
by
or
Jip
+ acosO
72z
(b
a+bcosv
fr-fasinfl
.
70
sin- 1
dx
(o-<a
.
a + bsm9
JaPb"
_1
cos" 1
r
a + o cosh x
;
2
s //
a2
/722
a + b cosh
=
(
b
>a
2
)
2
),
9X
2
)
ic
-= =cosh.^o + acoshx
1
,
!
1
a+ 6 sinh a;
ff?a?
-
7
-I
or
2
>a
a+6sin$
a^
=
>a
a+bcos6
2
1
,_
Ja + 6sin0
.b
.
cosh" 1
.
=
or
r
particular
in the general results of Arts. 182, 183,
(10
r
a2 +
2
Va + 6-
.
,
,
sinh" 1
b
-
--
+ asinhx
o+osinha
The symmetrical form of the several results was given
(without proof) by Greenhill in his Chapter on the Integral
,
p. 34.
CHAPTER
182
When a =
we
VI.
arrive at results obtained earlier in other
forms, viz.
= cosh"
7i
cos#
=
^
1
(compare Art. 74)
(sec 0)
coslr^cosec
6),
J sin 6
dx f
= cos " 1 (sech x),x
J cosh x
,
-77
smhz
{dx
= sinh" (cosech x)
and from the general
f/
-
-j
f
-r
W
=
~fi
:
;
6cosha? + csmhaj
sinh" 1 (cosech x)
1
;
results
"r^
=^
=
secn
Hftf*""
*Jb
2
"
1
,cosxc sinho;1
c*
c
\/b*
1
= --T------cosech"
Vc 2 -6 2
. ?>
1
cosh
a;
79
.
62
if
>C 2
2
+ c sinh
a:
J&-W
if
or again,
fn.n-i.^
=
- -=r^ coth-
or
1
^-.fea
A
W 6._6
e*
if
if
2
?>
<c
2
,
forms which the student should compare with those previously
obtained.
185.
Reduction formulae for integrals of form
A^ = a + 6r
/w =
(dx
^-,
where
,
Bin
Let us consider the case
We
shall connect the integral
^
Put
P= a
with another,
viz.
a + b cos x
b cosic
+T~
That is, to form P, since is introduced into the
[Note.
numerator of the integrand of / and the index of the
denominator is lowered by unity.]
,
VARIOUS REDUCTION FORMULAE.
dP ~
_ cos x(a + b cos x) + b
Thus
dx
(a
+
b cos
(
cos #)
I
xf
^
7
6
__
tt
a+^
r + r(
b^
,
/
7
,
cossc)'
b
(a
+ b cos x)
2
183
2
+ 6 cos
(a
2
a?)
'
a
I
a2 -b*
1
b
a + bcosx
b
(a+bcosx)
2
'
Therefore integrating,
&
sinz
Hence
72
a + bcosx
b
b
sin
=
and 7 X has been given
r ~
\s\JtJ
2
according as a
is
a + 6 cos x
sin
2
/>'
,
cosh" 1
a
2
_
cr
a
/^2
Again, in the general case,
.b
A
+ acosx,
a + 6 cos
.
r>
if
Then
dP _ cos a;(a + b cos x) + (n>
(a
+
,
i
.
/y2
,,
^^ /j2\
cc
6 -f a cos x
sm
P=
x
, 2
,
<^ 2
oa^^rT-rr
a + 6 cos -(
x
J-n
put
.
-^^J
dx
f
F
,
5 CO s
.
.
2\f
e.g.
+ a cos x
_j_
2\$
a
a;
a + 6 cos
b
2
f^va
n
/
.
_ \^\_/kJJ.X
.
a
sinx
a
fe
6
c/-
,
x/^2_ a
I
a2
180.
or
V
1
2
greater or less than b
b
T
/
1
a + b cos x
b'
forms in Art. 173,
in various
,b
+ a cos x
l
Jd'
x
1) 6(1
cos 2 SB)
b cos x) n
A+B(a + b cos x) + C(a + b cos x)
z
)-
CHAPTER
184
VI.
0=-^=^, 5 = ? + 2^(n
giving
Hence, substituting these values and integrating,
a2
since
2
ft
a
-
,
The reduction formula
is
then
b
,
2n
since
2
2
~(n-l)(a -b )
(a
a
3
+ bcosx) n ~ l+ n-l
a 2 -//2
71-2
Thus, as
1
and 79 have already been found
7j
2
?i
-i-
J
we can
in finite terms,
successively deduce the values of 7 3 7 4 etc.
It will be noted that In is in this case shown to be dependent upon tivo integrals of lower order, viz. 7 n _ x and 7 n _ 2
,
,
,
when n = 2.
except
Also, the result of Art. 185 could
n=2
putting
Generalization of above method.
187.
AS
1
J
I+y
have been obtained by
in the present result.
-
-
rn reduces
:
1
7
to
(a-f6smcc)
X and
f or
I
J
,
-
~r~^
r on
n
(a -f 6 cos y)
substituting
dx
J (a -\-b cos ic+c since)"
may
be written as
y^tan"
1
^,
it
is
I
^~
^-
/
'
.-=
.-=-,
n
J[a + 72cos(cc
where
R = \/6
2
2
-f c
and
y)]
usual to refer these integrals to the case
considered in Art. 186.
We
may, however, establish a reduction formula independently for each case.
Taking
Let
PS
dx
!={
cos x
- 6sin
(a -f 6 cos
+ c sin x) n
'
x -f c sin x) n ~ l
i.e.
if
7)^a+6coscc-f c since,
P j^i
VARIOUS REDUCTION FORMULAE
__
Then
bcosx csmx
_
_________
dx
dP __
__
a (b cos x + c sin x)
_
_
//IT
n -1
cos x
(b
.
1
b sin
(v
\
+ c sin #)
2
l)[6 4-c
(n
# 4- c cos a-) __2
;
v
185
'_
2
2
n
(a 4- 6 cos x 4- c sin #)
+ 7?(a 4- 6 cos x + c sin a;) + (7(a + 6 cos x-{-c sin #)
J.
2
^
rt
(a 4- b cos ic 4- c sin x)
G
where A, B,
are constants to be determined so that
B + 2aC=-a,
(7=71-2,
whence
B=
z
A=(n-I)(a*-b -c
2
),
(2n 3)a, (7=ii
Therefore the proper reduction formula for 7n is
2.
6 sin ic 4- c cos ic
n
(a -f 6 cos x 4- c sin x)
~l
= (n-l)(a*-b*-c*)ln -(2n-3)al n_ + (n-2)l n
We note that when n = 2, the last term disappears, and
1
792
b
2
,
(a
c
<n
2
)
LT = aLT +
_. 2
.
-
bsinx + ccosx
.
-.
-.
,-y
r,
+ b cos x + c sin x)
tfo
6 since + c cos x
79
vf
T
-=
--W
c )!.
h^/,,
6
c
sin
cos
a
x+
4- 6 cos x + c sm x
cc)J (a +
(a
-
9
2
2
,
'i.e.
(a
^
7
.
1
the real form of /j being selected from the various forms in
Art. 177.
now having been
Also 7 X and 7 2
deduce 7 3 74
,
,
etc.,
successively
by
found, we can proceed to
aid of the reduction formula
established.
188.
Corresponding
formulae
for
the
case
of
Hyperbolic
Functions.
In like manner reduction formulae for
dx
f
J
(a 4- 6
may
cosh x) n
C
dx
J
n
(a + 6 sinh x)
'
be constructed.
f_
'
dx
] (a 4- b cosh x-\-c siiiha;)
CHAPTER
186
VI.
As the last includes the iirst two as particular cases,
consider that one in particular, and proceed as before.
PEE
Put
-+ - - + x
b sinh
~.
j
;
x 4- c sin
o cosh
(a
cosh x
c
~.
j
we
Then
(b
cosh x + c sinh x) (a + b cosh
dx
a (b cosh
#+ c sinh x) + (6 cosh
1
(TI
)
[(6
(a -\-b cosh
__A + B(a + b cosh x + c sinh
(a
+
aj
a;
+ c cosh #
+ c sinh ic)""
As before, the
Hence
last
(b"
2
c
)]
2
1
a:)
b cosh
a;
)
say,
(
n -l)(b z -c?),
a,
= (2w
the proper reduction formula
a?
-f c
+ csinh x)
+ C (a + 6 cosh x + c sinh a
+ c sinh x) n
A = (n-l)(-a 2 + b z -c*
cosh
-fc cosh
n
2
k sinh
(a -f 6
sinh #)
sinh z) 2
cosh # 4- c sinh #) 2
A + Ba+Ca =
=
where
whence
-f c
(n
l)(b sinh
n
b
cosh
a; + c sinh
+
z)
(a
</*_
~
And
aj
3)a,
C=
(n
2).
is
1
term disappears
6 sinh
9.
in the case
n = 2.
x + c cosh x
a + b cosh
a;
+ c sinh x
the real form of I I being selected from the various forms
shown
in Art. 180.
now known, we
I l and 7 2 being
deduce successively 7 3 74
,
189.
,
etc.,
by
can proceed as before to
aid of the reduction formula.
Special Cases.
or c = 0, or two
notice also that, putting a = 0, b =
of them, in these reduction formulae, we have a mode of
We
reduction for such expressions as
n
(sech
r
x dx,
I
'
cosech n ;r dx,
r-r-
J (6
J
dx
n
(6 cos x + c sin x)
'
r-,
cosh x + c sinh x) n
etc.
INTEGRATION OF VARIOUS FRACTIONAL FORMS.
Fractions of form
190.
a+b
cos
x + c sin x
x
The numerator
A(a
1
4(denr.) + J3(diff.
&i sm x + c
b1
sinx + c cosx)
l
+C
of denr.)+G',
co.
1
Bb l = c,
Acl
l
i
form
cos x
i
ai + &i cos x + c i
and the integral
thrown into the form
C.
fraction then takes the
D
a?
Ab + Bc = b,
Aa 1 + C=a,
by taking
which determine A, B and
~~
sin
of this fraction can be
COSX + G! smx) + B(
-\-b l
i.e.
The
187
si n
a i + &i cos # + c i
^
is
dx
I n
al + 61
and the
has been evaluated.
last integral
Extension of above Method.
191.
In the same
a+
--
way
J
-.
b
+b
cos x
r-
Z>
cos #
+
"1
6 X sin cc""
be arranged as
+ c^ sm x) n may
:
l
-
cosx + cs'mx
-,
(Oj
t
sin x -f Cj cos x
(r->i cos x -f Cj sin x)
-
-
j
cos
sc
+c
x
n
sin x) n
The integrals of the first and last fractions may be deduced
by the reduction formula of Art. 187, and that of the second
fraction is
D
n
1
(a x
+
&]_
cos x + c a sin a?)"- 1
192. Case of Hyperbolic Functions.
Exactly in the same way fractions of the forms
a + frcoshz + csinha;
a+b coshz + e sinh x
!
may
t
cosh
cc
+ Cj sinh
n
(^ + 6 X cosh a? + Cj sinh x)
cc'
be integrated.
193.
If
+b
Further Generalization.
II(ar +6 r cos
+ cr sin 0)
r=l
stands for the product of
n
factors,
some
of
which may be
CHAPTER VI
188
repeated, and of which the one exhibited is a type, and
(j>(x, y) be any rational integral algebraic function of x and
if
y,
the integral of
(ft(eosfl, sinfl)
II (a r 4- b r cos
4- c r sin 6)
r=\
can
now
be found.
and
For expressing cos
in terms of
sin
/}
= tan-,
the tangent of the half angle, and writing
where p is the degree
homogeneous, and ^(0
of <j>(x,y) in x
is a rational
function of
2p at most.
Also
t
of degree
ar 4- b r cos 6 4- c r sin $ = a r 4-
fr r
and y> not necessarily
and integral algebraic
- 4- c r
-
2
^
^,
whence
II
K+
b r 4- 2e r * 4- (Or- b r ) V]
r
U(ar + b r cos0 + cr sm6) = -^-
r=l
-.
V
1
"I
l
)
d9 = T^r-72
also
Hence
0(cos
^, si
II (ar 4- &r cos
and supposing a r =f=b r
of x(/) in
<,
t.g.
any of the values of r, the degree
lower than that of the denominator,
for
2p, is
is 2(^4-171)4-271, i.e. 2p + 2.
This expression may then be put into partial fractions, some
A + Bt ,.
C+Dt
,,
which
The proper reduction formulae for such cases will be found
The integration can now be effected.
The reader may consider for himself the effect of a r = b r for
in the next chapter.
any value
or values of
r.
A REDUCTION OF DIFFERENT NATURE.
194,
A
Method.
different
To obtain
189
integrals of form
do
dx
r
and their particular cases, we may avoid the reduction formulae
referred to, and proceed as follows, using a reduction of
different nature.
Consider the
first of these.
Case
f
_
Taking
dO
a + b cos 6
+ c sin
_
_
(
ydO=
c
/0
fe
/
V o 2 -f-c2
=_
(^2
J^
and
u
a_
-jv + ?=tf'
2
62
1
whore
Ln
c-ay
_x
cc
752+7,
+ c -2_ a2
y- = a + bcosO + csin6 and
x/ 6
+ c >a
2
2
(Art. 182),
(
i
a?
-__ (^T^
f
2
cosh
+c2_ a2 ) -2-J
u- a)
'
71
1
dw,
d6
(a
+
b cos
n
-f c sin 0)
1
Wo may
then expand
2
(\/6
+c
2
~
cosh
?
each term, finally substituting back for w
!
a)*
its
and integrate
value
oshr
1
i.e.
cosh- 1 -7==
/6
2
^_|_
;
c
2
2_ a2
-i
-.
r <*li
a 4-
La+6cosO+csin^
the proper sign having been selected as indicated in Art. 182.
CHAPTER
190
Case b z + c 2
<a
VI.
2
.
dO
f
a+b
cos
6+c siuO
i=.
x
-
-
===
COS" 1
du
and
(a-
-b~
&)y = a
9
9N
9
/, j
vo
+c2 cosw;
cosw
do
r
l 6
'
'
J
(d+b cos 0+ c sin
-
1
^
222'J
-
f(
2
195.
a
-J
cos
w
_
In exactly the same way, from the three forms
1
(where y* = a -\- b cosh x -\-csmhx)
dx
a -f b cosh x-{- c sinh x
u
where
1
~
g-(a +c -6
2
2
CC
1
_1
62
>a + c
>3
2
;
2
)?/_
"
or
where
we
obtain respectively,
Case 6 2
r
>a
^x
J (a
^^
~* (\/6
2
I
where a
=
a -j- b cosh x
2
c cos
n~l
u
a)
.
+ c sinh
a;
du,
62
<c
1
A REDUCTION OF DIFFERENT NATURE.
1
Case a
>b > c
2
191
2
,
dx
J
(a+ b cosh z-f- c sinh
x)
n
1
a-fr-c
-
a 2 4- <?
where a
=-
2
cosh u) n
a+b cosh x
-f-
c
du,
-
b2
J
~l
i^.
2
= V6
v-,
smh x
-
2
c cosli u.
Case
dx
f
}(a -{-bcoshx-{-c
si
2^1
2
+c
2
2
I
a
(
~ Vc
2
6 2 sinh w)*
1
"1
J
*
fr
2
|
where a ---
2
1,2
6 2 sinliM.
,
a+ 6 cosh x+ c smh
I
196.
IMPORTANT PARTICULAR CASES.
The particular cases (according as 6 or c =
wmulae, and which should be worked ab
in the general
initio
iudent) are
d
l
U
2n-1
1
^r^
2
(6
-a
n
I
(6
H
6 cos w) n-1
(a
62
^<>\
2
O}
<a
du,
c^w,
2
cosh u
J
,
n~l
a)
du,
)
,
^
\a-\-bsmO
2n=!
I
---J
(
a
~ b sm
s
by the
CHAPTER
192
r-
jT
n
f (a+bcoshx)
=
r-
,
I
o^-T(6
u
j
VI.
n~l
u
cos
-
/6+acoBhi
r =oos
Va+ocoshx
,
2n-i
2
fc
r-=
J-.(o+6smhx)"
.
.
,
,
(fl
I
fr
cosh
71
-^y
5-:
I
--i
(a
b sinh u)
H~l
i).
J
du,
(<f+J
b-\-as'mhx =
,
We
\
r
\a-}-ocoshx
:
197.
K
/
du,
si)
/6+acosha;
-,
I
=1
"1
\
ppJ
have the further
results,
.
.
a+osmhx
\
b
du,
a)
from putting
\
,
.
sinh u
/
a=0
and
in the above, viz.
I
see"
dO
cosec n 9
where 6 = sec" cosh u
=1 cosh"" u du,
1
1
d6 = cosh"" u du,
= cosec"
where
1
I
1
;
cosh u.
j
Hence
1
either integral
may
~ sinh(n -3)t6
rsinh(^-l)^
~2^L
be expressed in the form
^T~
Ol
+
.
.
.
+i
^^3~
n -l
C1L-iU
_ ir sinh(?t
,
91- 5
or
-f
N- 1
^ sinh
a
a"
(Compare the forms
198. Further, if in the results of Art. 196
in Art. 122.)
we write n
1
=
in,
we have
(6
cosh u
- a) m
L^ - \( a +b cos
v
a
J
)
J(aetc.
Several of these results are given in Greenhill's Chapter on
The geometrical significance of some
the Integral Calculus.
of these transformations will appear later.
A REDUCTION OF DIFFERENT NATURE.
193
Cases required for the time in an Elliptic Orbit.
199.
The
cases of
\.
i
J(a
r-
nt
where a=l,
6
= 6, n = 2,
are
required in the theory of Planetary Motion in finding the
time in an assigned portion of an elliptic (or hyperbolic) orbit.
We may
either quote the results
independently as follows.
If
<l,by
Art. 171,
dO
and
from Art. 185, or proceed
~
I
_" x g
+ cosfl
e>l,
if
Taking e<
dO
1,
du
1
1
and
j
ic
(orby Art
196)>
T
The time
is
e cos
1-e
for a planet measured from passing Perihelion
expressed by this integral as
d
where
n
is
a certain constant (see E.
Dynamics of a
'
If
Particle).
J.
Routh, or Tait
It follows that
nT= u
&
e>l,
dv
B.I.C
I
ecoshi>
Steele,
e sin u.
1
CHAPTER
194
VI.
In practice, each example should be worked ab initio.
For example, suppose we require
200.
_
r
A
Putting 5 + 3 cos #=-,
3 sin
rfj
L_
5
+ 3cos^
.a;
= -5
We
5
take the + sign, because, as x
+ 3 cos x decreases and y increases.
dx
[
Thus,
dx
Then
increases
f
in
the
dy
J
1
=
1
,3 + 5008.?;
--h const.
sin" 1
4
5 + 3 cos x
=
1
3 + 5 cos x
cos" 1 h const.
7
4
5 + 3 cos x
=
- n + const.
.
---
.
3
T
+ 5 cos A' =
cosw,
+ 3cos.r
where-=-du,
4
5
~
"
16
_ (5
~
- 3 cos u) 3
16 3
(
dx
_
- 3 cos u)3 du
(5
+ 3 cos #) ~2 14 J
when # = 0, COSM=!,
*=1, and when
4
[for
'
J
call this
o
4
"
(5
X
x-
TT,
cosw=
1],
jr
=
^
.
2
385r
3
T(5 + 3
.
5
.
3 2 cos 2 u)
du
first
quadrant,
VARIOUS REDUCTION FORMULAE.
The integrals
201.
Lr =
sin m
--*
f
I
)
#?
,
a + bcosx
dx
1T9 =
and
sin m
f
I
J
integer.
Consider the
first, viz.
I
J
The case
m be
If
771
=1
say,
,
^4 ax
(a H- b cos
a?)
m
is
when
a positive
dx.
7
a + o cos x
-
obviously gives
=2k + l,
odd,
dx =
- --
x
,
-,
can both be integrated in finite terms
6 sin x
195
j-
log (a +6 cos
x).
put a + bcosx = z, and therefore
cfo.
/r n " /z-a\ 2 T 7
<**
1
r
sin-*'*
J
a + 6j cos a;
Thus,
,
-dx=
every term of which
of
is
-r) J
I
i
&*/
integrable wlien expanded in powers
0.
If
77i
be even,
= 2k,
say,
2*
xa + o cos x
Jsin
and
dx=
7
f
(
J
k
1
a+
cos 2 x) j
-dx,
o cos 05
the numerator be
expanded in descending powers
and then divided by b cos x -f a, we arrive at an
expression of form
if
of cos
a?,
where the
X's are
Hence, in
numerical
all cases,
I
J
terms.
coefficients.
-.
a + b cos x
v
he same argument applies
to
m,
1
dx can be integrated
I
7
I
let
\
%/
202.
If
In =
I
j--
7
^ a#.
-ho cos x)/
'^- dx, there is a reduction formula con-
necting In with /_! and I n _.2
can be effected in finite terms.
To obtain
sm m #
f
in finite
.
Hence
this reduction formula,
all
such integrations
put
sin m+1 #
~~
(a
[i.e.
increase
the
decrease that of the
+ 6 cos x) n ~
l
index of the numerator by unity
denominate by unity].
and
CHAPTER
196
VI.
Tnen
dP _ (in 4-
1
)
sin m x cos
x (a +6 cos
dx
(a
.
where
,4
x) -f
(nl) b sin m x (1
6 cos x)
-f-
cos2 x)
n
+ Ba + Oa = (TI - 1) 6,
2
(n
1)6,
giving
B=(2_ TO _ 3 )|, 0=*
A = -(n-lfi=-,
j-
Hence
sm m+
x
(a
x
7
+ 6 cos ic)"-
.
.
1
\
.
IV
A
6
q-
/
,.
-*
7
>l
,/9
i*tV
II V
|^ \
6
a
_o\
Oy
/
r
JL *,
f
6
H
+ -^-7
n_
6
and the reduction formula required
.
2n
in
?n
of
2,
is
?i,
3
+2
a
1
which the formula of Art. 186 is a particular case.
since 7,L and 7 9 have been shown integrable
O
And
,.
in
finil
m
is given, we can use the reduction formu]
terms when
just established to find successively 7 3 7 4 etc., in terms
,
1^
,
and 7 2 and thus integrate them.
,
Again, Integrals of form
203.
sin''
cos q
d9
terms,
p
c sin?
are always integrable in finite
cos 9
and q being positr
integers.
For
(1) if
p
be odd,
=2& + l,
i
i
,
where
dO
c
= cos 9,
VARIOUS REDUCTION FORMULAE.
197
and after expansion of the numerator in descending powers
and division by bc-\-a, we get a series of powers of c
of c
and a remainder
respect to
(2) If
p
A
=, and each term
integrable with
is
a+bc
c.
be even, =2k,
cos" 6
sin"
= (1 - cos 2 0) k cos* 0,
which, when expanded in descending powers of cos 6 and
with
divided by 6 cos 6 + a, gives a series of powers of cos
^
-
a remainder of form
and each term
^,
is
integrable
with respect to
by Arts. 117, 173.
And the same argument holds good for /', 73 except that
the remainders to be integrated involve such terms as
',
*
d cos
d cos
(
^
cos
~
or
__
~*
g
dO
.
f
dO
"
'
according as jp is odd or even, and such integrations have been
already considered.
204.
We may
then obtain a reduction formula for
r _
Let
f sin"
cos
Jfi
sin^fl cos^fl
(a+bcosO)
Then
^
n -1
'
CHAPTER
198
VI.
where
A = (n
whence
1) a
B = (n q2) (3np q5)^
and the reduction formula
is
^y^Al'n+BI'^+CI'^+DI'^,
from which
can be expressed in terms of three integrals
I'n
and ultimately made to depend upon
whose integration has been discussed.
of the next lower orders
I\, 7' 2 7' 3
,
205.
,
General Conclusion.
From what has been
said in Art. 204,
will
it
now appear
that any integral of form
f/ (sin
ft
cosO)de
can be integrated when n is a positive (or negative) integer,
and f(x, y) is a rational integral algebraic function of sin 0,
cosO; for /(sin 0, cos0) is then the sum of a number of terms
f
A.smp Ocosq O.
tyP e
206.
HERMITE
to integrate
(Proc.
Lond. Math.
of form
Soc. 1872) has
/(sin
a^sin^
sin(0
where f(x, y)
dimensions.
Q,
cos 0)
a 2 )sin(0
a3)
f(t, 1)
(the
a2 )
factor
.
=
.
.
a,.
...
an )
(
ar
the above coefficient).
V-
a n )'
sin(0
any homogeneous function of
is
For by the ordinary rules of partial
(*-
shown how
any expression
x,
y of (n
1)
fractions,
_/(OT,
1
1)
{ (a r -a 1 )(a r -a 2 )
being omitted in the
...
(a r
a n ) t-a r
denominator
of
HERMITE'S FRACTIONAL FORM.
a l = tan a x
= tan 0,
t
Writing
CL2
,
\>
an )
a r ) being omitted in the denominator.
the factor sin (a r
r/(sin
cos 0)
-
0,
V-
Thus,
^
/(sin or, cos a r )
\
-~7
?
a 1 )sin(a r --a 2 ) ... sm(a r
>
.
sm(a r
^
= 2jAA r log tan 0-ar
a
dO
1
.
nsin(0-a r )
I
i
207.
(i)
we have
Thus, for example,
sin 2
a) sin
sin (x
x
6) sin
(.r
(x
c)
sin 2
_ _
sin 2
C
1
(.i
-
(/
)
sin
(.r
x
- i) sin
c) sin
a)
^
(^
- c)
in(fl-6)sm
Similarly,
T
,
J sin (x
- a) sin (x - b) sin (.r - c)
.v-a
-2
cos a
,_
~
= 2<
/
\ log
/
,
'
sm(a-6)sm(a-f)
(i'i)
'
(x
.
sin 2 a
.
1s
=
(ii)
a
b) sin (a
sin (a
J sin
tan
.
Hence adding,
dx
i'
sn
-
.r
#
sn
sn .-
.#-
^
=2
-7
-
sin (a
1
.r
EX-^
- 6)
sin
/
(a
-~~\ log tan
'
-a
2
c)
,
'
(iv) or subtracting,
cos 2.1
2.v dx
tic
I
1
J sin
.t
a ) sin(.rsin#- 66) sin (x
./;- c
cos 2a
^2j
(v) Jt is
,.
7
log tan
- b)
sin (a - c)
x
v
.
7
sin (a
a
.
TT
2
'
easy to show that
sin (x
sin x
- a) sin (x - 6) sin
=V
JL ~^
(x
T
sin (a
C
- c)
sin
s
J sin
a
i.
)
x->
.
7
sin (a
a
:
\
)
,
sin (^ - A) sin
=X
/
1\
T
- c)\ cot (x
- 6)
sm (a
sin./(./
becomes
a r)
II sin(0
Ar =
this
etc..
7 ,
^
/(sin0, cos0)_
where
= tan a.
Ar
199
-
s '"
r:
c)
(./
"
;
/
i
:
b) sin (a
- c) log sin (x
x
'
a
}
CHAPTER
200
Vf.
EXAMPLES.
1.
Integrate
,.v
(i)
L
+
frtsintf
ftcostf
dO
~d&a
\
,
Jcsm# + ecos0
fa + ffsinfl
[a, 1883.]
Wi
v
cos0-sm0'
[I.C.S., 1880.]
'
'
j
a
+
dx
f
'
J
AND MAGD.]
[TRIN. H.
cos
b
cosa + cosz'
[I.
a cos x
C.S., 1889.]
+ b sin a*
[COLL. 1876.]
,
I.
[Ox.
.
....
{
-
'
J3(l
dx]
P., 1889.]
dx
f
- cosx
sin x)
[a, 1881.]
____
\/2rfa;
f
j2v/2
+ cos4-sina;'
[Ox.
+ btanx
P., 1888.]
a
= tan ^x to
the integrals
I.
[Sx. JOHN'S, 1888.]
(xi) Apply the transformation
t
idx
3
Hence or otherwise,
+ 5 cos x
these
evaluate
integrals
to
the
nearest
and JTT. Prove in any way
hundredth, when the limits are x =
that the second is the greater of the two integrals, when taken
and
between
(xii)
-
dx
f
)o
[MATH. TRIP.
JTT.
I.,
1913.]
Prove that
x+ at-x
n
1** lri
adx
(
1
Jta+a*-*
s=
~r"
~7
* 8
^
2
the positive sign being taken for the radical in each of the subjects
of integration.
[MATH. TRIP. II., 1913.]
2.
Evaluate
[I.
C. S., 1589.]
[I.
dx
dx
l11
'
J
2
+ cos x
C. S., 1879.]
[ST. JOHN'S, 1882.]
(iT)
Cr^
'2a
cos x
+ a?
[I.C. S., 1888.]
3.
Show
dx
that
Jo
and integrate
1
- cos a cos x
fcos
= ~ cosec a,
a cos x
z
+
1
J cos a + cos a
(fo.
[Ox. II. P. 1889
TRIN., 1887.]
,
;
INTEGRALS OF DIFFERENT FORMS.
4.
Evaluate
5.
Evaluate
Disintegrate
202
10.
and
Evaluate
INTEGRALS OF VARIOUS FORMS.
17.
203
Integrate
d$
f
,..
(1)
..
r
f-
(U)
Jl5sin 0-16cos0'
rcot0-3cot30
(111)
- tan
J 3 tan 30
2
x
l
J
,
+ sina;
...
,.
P., 1888.]
'
(IV '
+ bcosx) 2
} (a
18.
I.
sin 2x dx
f
,
*
[0 x.
[a,
1889.]
Integrate
,.
f
(i)'
I
0/31
cos 20 log
&
J
+ sin 0,.
cos
cos
s
-
=
sin
- cos 6
fsin
I
.
^dO.
(n);
,
,
dO.
^sin 20
J
i
(iii)
fjl
Vcos0l + c
J
19.
+ sinx ^jj-si
/1
\ 1-sinz' 2-si
f
Integrate
J
20.
Integrate
Jsin
- cos
sin
+ cos
(sin
21.
Integrate
/
(\)
+ cos 2 0)Jl + cos 2
R
7
U/l+smzcte.
T
,
26.
f
Integrate
^
x 2 rfx
-.
x + cos xy
f
sec.rcosecx
J
log tan x
Integrate
Integrate
27. Integrate
f
J
tjl+x*
J
1-r2
I
J
x sin x - cosh x ,
dx.
1 - cos x
J (x sin
iii)
(iii)
Jvl + sina
f sinh
tan x
f
dx.
I
(ii)
J
25.
du.
+ cos 4
sin x
f
:
23. Integrate
24.
!
cos 2
Integrate
f
J
09
+ sin 2
cosT0
.
J (1
22.
0) Vsin
,
dx.
.
r
v& +
==<&.
tan^ a;
CHAPTER
204
T
28.
f sin
,
x
f sin
j
r-dx,
Integrate
x
f sin
,
,j-dx,
3z
\
J sin '2x
VI.
J sin
x
,
--dx.
sm 4z
J
and prove that
f
an d
f-jn
-^-TTZ
2
sin B
Show
that
-
=-7
sin 2 a
-
.,
2
J sin
;
s
sm 2 a
rf#.
can be expressed in partial fractions of tvDe
J*
sm nOH
COS0
- sin 2 a
1_
-
sin 2
according as n
r>n<s ft
.
I
sin 2 a
sin 2 6
'
an odd or an even integer and can thereby be
is
integrated.
30. Integrate
5
.....
J
Show how
31.
sin 3
f
x^
,.
'
(m)
.
+ ftco.)
(a
f^
J
.
to effect the integration of
,
-ttTsm 2nx dx,
fcos^z
p and n being
.
(IV)
v
f cos
J cos
x
nx
integers.
32. Integrate
I
,
dx,
[
cot (x - a) cot (x
-
e,
1883,
AND COLL.,
1879.]
[y,
1891.]
3) dx,
J
and show that
I
-
cot (x
a) cot
(x-b) cot (a; c)dx
2 cot (a -
i)cot(a
-
c)logsin(
-
a).
[TRINITY, 1891.]
33.
I
Show
sin x sec
that
(a;
a) sec (x
TT -
sin (p
r
-
6) dx
[cos
L
a cosh" 1 sec (#- a) - cos
3 cosh" 1 sec (x - B}\
a)
[TRINITY, 1889.]
34.
Prove that
x sec x sec (P -
x)
dx = fi cosec
/3
log sec
j3.
f/3
[OxF.
II. P., 1901.]
INTEGRALS OF VARIOUS FORMS
that, if a
Prove
35.
J
Q
~
*V +
^WO 2 x
(acos
I I*.
f3
f*J
and
j3
2
0111
sin
x)"
ft^/
205
be positive quantities,
+l
~
2
I
nl
\da
d/3.
[a,
Prove
36.
that,
P=
and
where a 1?
where
...O B
a.,,
^ =a
II (1
- 2a r
37. If c
be
r
than a sin
6,
independent of
6.
less
cos
(9
+ a r 2 ),
denote real quantities, then
/(l/ a r)/'( ar) anc^
numerically greater or less than
r
1884.]
if
==
~^> or +^> according as a r
1.
is
[ST. JOHN'S, 1886.]
show that the
coefficient of c
m
in the
expansion of
where
38.
^w
is
Show
[COLL., 1892.]
n be a positive integer,
r=
. sin 7ta
2
n6 cos na .
^T
cW = v
h-r
a
a
sin
cos v
sin a
cos
r=1
fcos
that
if
1
sin
-
?
^
.
sin (?i
-n a.
r
[HEEMITE.]
39.
Prove that
40.
Show
that
II cot(^
fn
- ar
)dO=0 cos
i
r
the factor cot(a r
41.
(1)
n
2^
+
Show
-a r )
r
r
log sin ((9
-a
1 )cot(a r
a.2 )...COt(a r
being omitted.
o r ),
- aw
),
[HERMITE.]
that
a tan - -
dx
(2)
-
i
.
^ = COt(a
where
w_
Differentiate with regard to
a,
+
11- sin x
-
1
1
+ sin$"
1
CHAPTER
206
Deduce from
and
(1)
la
,
tan" 1 A /
(2) that
+
\ a -
is
independent of
43.
1,
&
>
ab
1,
Integrate
e
(i)
Integrate
-
1
1
1
sin x
:
+ sin
tan" 1
\-
x
/-^
Jo? -
+ ao1 - a cos x - b sec x
+
T
I
1
,
[OXF.
(xcosx + sinx).
+
[C. S., 1898.]
I.
2.T
3
+ 2x+
sin (x
2
-
p)
+
(z
(ii)
+x+
I)
sm (x -
7)
l)/(x-
I.
P., 1917.]
I.
P., 1918.]
l)~*.
3
[OXF.
.
-.
Deduce from the identity
the expression for sina; as an
46.
x
a tan - -
-
sm (x - a)
J
45.
x
(x*
(iii)
44.
1
fdy
1
<
1
and verify your conclusion.
x,
i
where a
VI.
infinite product.
[Oxr.
II. P.. 1887.]
Evaluate the integrals
2
;
1
iii-
(in)
~-w-
-f
ax log x
[MATH. TRIPOS,
47.
Show
1885.]
that
x2
f*o
+ a(a-l)
a: sin
(arsi
a sin x - x cos x
-
x + a cos a?)
2
dx
a;
sin
a?
+ a- cos 2
[TRIN. COLL., 1891.]
48. Evaluate the indefinite integrals
...
f
(sin
{(x(ii)
*/
J\rr^
l\
t
49.
Integrate
(i)
(ii)
x + cos
2
x)
1) cos
#-(: +
1
a?
)cos
\(x
-( +
!) sin
-
+ Jl+x
=
\t
J (x sin
2 r
)
2
'
a?}
[COLLEGES, 1886.
;
'dx.
x + cos x)^2
[ST. JOHN'S, 1882.]
INTEGRALS OF VARIOUS FORMS.
(1+z
J
207
2 2
[Ox. II. P., 1899.]
)
fi-
(")
51.
Prove
that,
if
1-COS 2
n be an
[COLL.
a, 1891.]
integer,
+ COS a CO3 X
Jcosnx
dx
1
and deduce the value
6>
= TT cosec a (tan a - sec a) n
,
of
cos
rj#
,
2
[COLLEGES 7, 1891.]
J*o (1 +cosacosa:)
52.
From
considering the integral
r
Jo a
* ne
- cos de
0'
show that
Cos4(x
~
24
1.2
= 2" (sec
53.
Prove
that,
<
if
/
r sin
\
/
Ti^tan-
tan
1
Jo
54.
Show
<-
- tan
and n be a
n
a)
sec" a cosec a.
positive integer,
<A
9\
a
\d<{>
=
TT
[(sec a
- tan
a)"
-
(
1
)"].
^
that
T^
vuV^ne-cosn^n
f sin
J
7T
a
a
nO
dO
where
^8 r
1
C S6C ft log
sin J (ft
4-
SiTjl
= a + ?!^.
55. Discuss the
integration of
(a)'
where p and
56.
With
J sin
,
'
q&
sin qQ
J
,
q are positive integers.
the help of the substitution x~ l
= V/ a -
1
,
or otherwise
prove that
_.
12
_,
:'.'
[MATH. TRIP., FT.
II., 1920.]
CHAPTER
VII.
FURTHER REDUCTION FORMULAE.
"
"
have already been
have gathered some information as to their nature, mode of construction and use.
The nature of these formulae is that a connection, in gener*
linear, is found between two or more integrals, so that whei
all but one have been found, the remaining one can be inferrec
208. Several
established,
Formulae of Reduction
and the student
209. It will be useful to
occurred.
They
will
summarize those which have
are as follows
alread:
:
1. The rule for integration by parts, Art. 90, and for continued integration by parts, Art. 95.
2.
Reduction formulae for
I
xm (
"
'sin\
J
}
nx dx,
Art. 1
w
e ax
}
Vcos/
f/sin\
4.
Reduction formulae for
I
5.
Reduction formulae for
I
\
bx dx,
Art.
xm (log x) n dx,
sec n xdx.
Art. 10(
w
Jcosec
a?t/aj,
Art. 120
etc.
t&u n xdx,
Art. 125,
|
6.
Reduction formulae for
U
185 to 199.
7.
Reduction formulae for
IT
to 203.
208
dx
n>
-f 6 cos x-\- c sin x)
;,
etc.,
Arts. 20:
FURTHER REDUCTION FORMULAE.
209
210. General Remarks.
The subject of the present chapter
will be the construction
of such further reduction formulae as may be necessary for
present or future uses in the book, and a general indication to
the student of the mode of procedure to facilitate their speedy
It will be noted also that two distinct modes of
have
been exhibited
procedure
production.
:
(i)
That
same
of integration
by
parts, or,
what comes
to the
thing, a proper choice of "P," with a differentiation and subsequent arrangement of the result as
a linear function of the expressions whose integrals
are to be connected, as exemplified in Arts. 185
to 188.
(ii)
A
change of the variable, taking the integrand
itself,
or some function of
new
part of it, as a
Arts. 194 to 198
We
shall also
as arc
(xm
211. Integration of
-l
reduction formulae
is
,
m*When
vl
.
n
p
a
be
in a finite scries
.
integration,
and no
:
-\-p is
an integer
(i)
:
...;
Positive.
XT
,.
(n) Negative.
pox'tti
r<>
formula
integer
is necessary.
we can expand
n
(a-\-bx
Y
by the binomial theorem and integrate each
term.
Thus
/v.JH-fn
m
K. I.e.
n
Positive.
in
n
direct
X = a+bx
(i)
an integer: ,.'
AT
(11) Negative.
is
In other cases a reduction
If
when
and
a positive integer.
is
m
-
of
required
When p
n\\j\.
When
I.
the general cases
for
X*dx, where
In three cases this admits
212.
exemplified in
for the particular cases
convenient to avoid their use.
it is
I.
as
complete the discussion of such integrations
to be considered, both
which
or of some essential
variable,
reduction formulae are required
in
it,
m-\-n
CHAPTER
210
VII.
If p be fractional or negative, the binomial expansion is
non-terminating, and therefore the integration after expansion would not express the result in finite terms.
Expansion
therefore in such cases should not be resorted to if avoidable.
M
213. II. Let p =- where r and s are integers, and s, at
s
commensurable fractional
positive (which covers all
least,
or negative values of p).
=a
Put
X
f
f,
yT- l XPdx =
\
bn
bnj
Hence when
(i)
is
n
a positive integer^, a
finite expressioi
be found for the integral by expanding this binomit
integrating each term, and finally substituting back for z il
may
n s
value, viz. (a-\-bx )
(ii)
And when
.
is
a negative integer or
zero,
be put into partial fractions by the rules explained ii
Chapter V., and the integration can then be effected in finil
may
terms.
214. III. Again,
f
\x m
-l
we may
(a+bx
n
and therefore by case
.
write the integral
-
}*dx
r
as
4-
\x
II. this is
an
integer, positive or negative,
integer negative or positive,
n
)*dx,
...
m-\
be
(b+ax-
integrable in finite terms
rn
s
-
-i
^~
and
the
i.e.
if
m
ii
r
-
f-
be a\
proper substitution u
FURTHER REDUCTION FORMULAE.
b+ax-n = z
s
a finite expansion
to
leading
,
211
if
|-S
71*
111
negative integer, or to partial fractions
if
be
a
/Vl
+-
be a positive
integer or zero.
215.
To sum up
Case
a positive integer
p
I.
:
Expand.
:
Substitute a-\-bx n =
Case
--an
II.
integer
:f',
may
Substitute aa?- n
an integer:
Case
III.
210.
Illustrative Examples.
1.
p a positive
Consider
+& = s*;
7= />(l+.r') 3 d.*-=
/"
4.
I
'
20
13
6
27
a positive integer.
Here -*=
Let
'1
<
l+.r'^r
5
/.
;
a negative integer.
'cmsider
1=
f
.r~ 8
(
1
+.v')
Following the rules of Arts.
l_l_
9
1
-!"*" 9
Here
dr.
1
lf)f)-15G,
ir
ii
- 1.
-"'
we may express
^ + 1+51
1
r
-.
then
expand, or partial fractions,
as the case may require.
integer.
__
2.
the case
require.
1YL
\-p
then expand,
or partial fractions, as
:
*++
22
+ e'^
+ 1+1
*
-
3
as
CHAPTER
212
VII
whence
In the last term, put
z
+-=
tan
-- sec 2 dO
.'.
;
4
.2s+l
22+1
= ^-^
tal1
AT + QlT
1
.r~ 8
Hence
J
where z= Vl +.r7
+
4.
7i
.
a positive integer.
S
Consider
/=
f #*
'
(
1
+ .r 8 )*
c?.r.
Here
-+ 71
S
o
"j-
+-=1
.
-i
Then
Let
which can be put into partial
fractions.
can be avoided by the substitution
2
f sec
flfe=
sec
2
= sec
tan
-ton-*-0-
f
In this case, however, the lahom
0, and then
cosec 33/3,7/3
dv
cot
/.
,
where cos
.
= -1 =
/= - fcot
.r-
cosec
+ log (cot + cosec
0)J
,
FURTHER REDUCTION FORMULAE
a negative integer.
h
5.
213
Consider
1=
I
^
(
1
+
5
.<'
)~
^ du.
Here
?i
/
+ -o = rO
p
O
= - 2.
1
7- /V(l +.t- )"^' ^.
6
5
dx=1 + .v- = s
6
Then
Put
;
~~3
8
8*
217.
1
3
THE Six CONNECTIONS POSSIBLE.
When
X = a.+bx
integrable
by one
that,
+
n
U"
arid
1
"
1
^^
is
not
of the foregoing rules, it
4
by integration by
immediately
may
parts, it can be connected
be shown
with any
of six other integrals.
o
Thus, for instance,
m
and by different modes of treatment
six integrals, with any one of which
can
l)c
that the
linearly connected, are
f
y.m-ijp-1
L'n~n-l
f x 'n~n-l
that
we may show
is,
the index
ci
[ xm
x>
Xp ^
-l
X +1 dx,
{
X P+^ dx,
of X can
be decreased or increased by 1,
leaving the index of x unaltered
the index of x can be decreased or increased by n,
unaltered
leaving the index of
;
X
the index of x can be decreased
increased
by
1
;
;
by
n,
and that
of
X
CHAPTER
2 14
VII
the index of x can be increased by
decreased by 1.
or,
That
is,
n and
y
X
that of
either index can be increased or decreased, leaving the
other unaltered, that of x by n, that of
by 1
X
;
the one increased and the other decreased in that
or,
way
(but not both increased or both decreased at the
same operation).
The
connection may be put into the
form
following handy
Let P = sc x+1 -<T'* +1 where X and
are the smaller indices of
x and X respectively, in the two expressions whose integrals
rule for effecting this
:
,
//.
dP
Find -=.
are to be connected.
Rearrange this
if
necessary
as a linear function of the expressions whose integrals are to be
connected.
Integrate, and the connection is complete.
In the rearrangement
a-\-bx
n for
X, or
.-
it
may
for x n
be necessary to substitute
as
,
may
be required for the
particular case in hand.
The rearrangement can always be performed. It will be
unnecessary to integrate by parts. The advantage derivable
"
will
from the use of the rule of " The Smaller Index + 1
it will enable us to connect at once with the particular
one of the six possible integrals which may appear desirable.
be that
218. Proof of the Rule of
For proof it is
Thus to connect
(x m
put
P = xm Xp
"
The Smaller +1."
sufficient to verify the rule in
~l
X
p
dx
with
{x m
-l
Xp -
each
case.
l
dx,
.
_,dX
dx
dx
,
"
(note the rearrangement as a linear
function,
Xp
etc."),
apnx
m~
l
X
p-1
.
FURTHER REDUCTION FORMULAE.
Hence,
P= (m+pn)
*-' J? dx =
or,
m+pn
td*~
+
l
X
p
dx-apn
fa;"
-- a--^-'
215
1
-
1
A'"- 1
dx
;
dx.
The advantage in this reduction lies in the fact that the
v
index of the often troublesome factor
may be lowered if p be
positive, or raised if p be negative, and by successive applica-
X
tions of the
same formula,
if
necessary,
we may
ultimately
reduce the integral to one which has been previously obtained,
or which can be managed with greater ease.
219. List of the Six Connections.
The student should verify all six connections by the above
rule, and also by integration by parts.
They are
as follow
(i)
:
m+pn m+pnj
'
(2)
(3)
*
(4)
am
am
m-n
(5);
(G)
(
771
J
771
m
We
have written
1 as the index of x in the
primary
This
is
integral.
merely for the convenience of making the
several coefficients on the right-hand side smaller and more
compact than they would be with an index m.
220. Special Cases.
The
i
.
case
where
m + pn = Q
comes under the
heading
,
f>
= integer,
already discussed (Art. 211), and needs no
reduction formula.
The case p = integrates at once as also the case n = 0.
The case ^ + 1. = integrates by partial fractions.
;
CHAPTER
216
The
VII.
m=
needs no reduction formula, coming under
the heading of Case II. Art. 213, (ii).
When the student is convinced of the truth of the rule in
all cases, the six possibilities of connection and the method
case
of connection are all that need be remembered.
That the increase or decrease
"n
X
at a time," whilst that of
expected, since
An
221.
X = a-\-bx n
integral of
only "1 at a time,"
is
to be
.
form
{xn+pr
can be written as
in the index of x should be
is
-l
(a+bx
q
-J
r
')
dx,
or as
and therefore
is
reduced at once to the form considered.
222. Integrals of form
dx
are obviously included in the same rules, as there has been
110 limitation as to the signs of the indices in the formulae
discussed.
Illustrative Examples.
223.
Ex.
1.
Find the value
We may
and
connect with
this last is a
of
/
/=
(x
2
/
+ a?fdx, and
this again with
j
(.r-
standard form.
is to be used more than once, we will connect
A. the reduction
with
arf-
l
dx.
Let P=,
Then
dP
+ n (x* + a 2 - a 2 ) (
(note
this
preparatory
performed mentally)
step,
which might
be
FURTHER REDUCTION FORMULAE.
is
(which
now arranged
217
as a linear function of the tivo expressions whose
integrals were to be connected).
P=(n + l)l
Integrating,
Putting n
+ arf dx - no*
1
(.c
(
and then ?i=3,
5
te=rf +
* + **
!
d*
and
f
and
I
Thus
.
2
(.i-
f (rf
,..i
J "
)
+
dx =
,
xLv2 + a 2 )*
^
a2
+
.#
smh" .
,
,
1
+ t<?f dx=^.x (.f2 + 2 f + -^- d\v
o
5
.
a
/
<>
,
.
2
(.t-
'?\
7r
+ a?f
5.3
This result might have been obtained more quickly by substituting
6 and using the reduction formula
tf
whence
ive
+
get
^.f = a6
f
which gives the same result as before.
F,.\.
-1.
Find the value of
First connect
Put
P=xl
/
(x
z
/=
+ a 2 )~^dx
/
-
dx
with
/
-
sec"0rf0 (Art. 122),
CHAPTER
218
VII.
Putting n = 5 and tben?i = 3,
and
This again would have been shortened by the substitution
2
is specially suited for functions involving *JjcP + a
which
dx
f
j
TllUS
(^
2
+
-r=-7 fsec
1
)t
=
is
0d0 =
2
-g-flsec
#
*J
2
=
which
.r
= atan 0,
.
I
V
COS 3 #
c0
i((l-sin 2 <9)c*sin0
*
4
(sin
V
0--Y
3
where
sin
/'
0=
,
V
the same as the previous result, though in a different form.
/n
(x
integer.
C
1
-i
Let ^
2
+a
2
)^dx, n being a positive odd
!.-.+. j^
Since
n-
n-
1
etc.,
r
(Ex
.
1})
1
~~"
a2
.
,x
we have
P
OJ
rtl
/4>
_ O\
ca 4 P,,_.+
+
Ex.
4.
7
-^- ct
^~r/
Find the value
of
/=
n
~
|-
l
PI
-,
2
?z
being a positive integer
-
Let
aSince
we have
i7(n-^T
- 2)
a
^
2
P _.
n
I*-***]
a2
n-2-.^
n-5 I
etc.
...
a
n -^
FURTHER REDUCTION FORMULAE.
When n
is
an odd positive integer, we ultimately arrive at /3 and
,
a*
n-2
+
a-
-
-.
a2
(w-2)(w-4) a 4
(rc-3)(n-5)...2
P3
where
Ot-S)(n-4)...8.1^i'
In the case when n
_
2m ~
^
f
1
an even integer,
is
Pm
m~
2
A
+
2m-3
'
/>.
S-
"
a;
2
+
^
2
2
)
say,
*"
JL^
a4
-2) (2m -4)
(27?i
= 2m
Pym_i)
2m -3
2wi-2* a 2
2w
and
QO
/^.g
#
1
(2m-2)(2w-4)(2m-6)...2a
In integration between limits
~
(2m -3) (2m -5)
(2m-2)(2m -4)(2?-6)
(2TO-3)(2m-6)...l
r
a6
(ra-2)(?i-4)(?&-6)
(*
7
219
w-i ta
'
,
g>
,
n = (2w-3)(2w-5)...l
2m
Jo
(2m-2)(2m-4)...2
'
_1^
" a2
1
TT
'
1
2'
M. Bertrand* shows a very ingenious deduction from
putting a = l and
this result, viz.
#=-=,
V7)l
_L f"
Vm
w
^
Jo
Take the case when
^
(2m-3)(2w-5)...
1 TT
A lr~(2-2)(2*-4)~2
2'
^
?- is indefinitely increased
then
;
Hence
and by Wallis's Theorem (Hobson, Trigonometry,
~
2. 4. 6. ..(2m
-2)
i.8.6...(8m-8)
become
p. 331),
and
infinite in a ratio of equality.
^_TT
"
Consider also 7m =
f
*Jm
r
x m e~
BERTRAND,
dx,
Calc. Dif. p. 130
\/7T
m being a
x<i
Jo
*
"~~~
:
positive integer.
see also Hall, D.
and
I.
O.,
p
330.
CHAPTER
220
VII.
Integrating by parts,
/*=-5
52
r& ^(-Zxe-
Jo
and
o4-i
ar n+1 e-* cfo=
Note
but
also that
,2
if
7
2)...4.2 = ?i!
-^,
2vrF^~
2w.(2tt
^ being a
the integration extends from
oo
to
positive integer,
+00,
x>"-*
/
J-eo
any positive element of the integrand in the third integral there
always an equal negative element.
for to
is
,
Ex.
5.
Calculate the value of
/
_____
x m *JZax
x*dx,
m
being a positive
integer.
We
proceed to connect
m
\^
I
i.e.
Let
x
^ax-x
m+
i
dx with { x m -\/2ax - x z dx,
n
*(2a-x}*dx with f x ^(2a-x)^ d.v.
P=xm+ *(2a-xf-,
according to the rule; then
= (2w+l)flKBm
*
(2a -#)*-
Hence
f>
(w+2)
/'a;
iaf
B
JO
(w + 2) z
m+
*(2a -.r)*.
FURTHER REDUCTION FORMULAE.
.-.
Im =
if
221
xm J :2ax-.r-dx,
I
Jo
2
2?>i+l
m + 1 2m -1
2m + l
_
,.20
7
to find
Now,
or
2m - 1
m+
,
A-
2m - 3
'
3
=
3
5
.
'
m
}
'
'"4
= a(l
\'2ax-x*dx, put
/
3
J
3*
-coa
mj
'
0).
= asin0^
2
*/2a: - x = a sin 0.
Then
c?a:
and
Also,
2i-3
~~
2m -1
when x =
=
we have
when x = 2a we have O = TT.
;
Hence
/ =
_
~
EXAMPLES.
Prove that
m+p
(p-m + l)
b
m
_
___
+
''
\,.'*(a
4
.
X
'
'
a
b
1
xm ~
l
xm
m-2
_
'
'
a2
_ml~
.r
2
m-3
a3
/(g
x
J
r>
1
m-1
bx)
[(a
aJ
/
J
p
x
J
[HKRTRAND.]
f//?
^
2p-l /"__<^_
_
+ ft.r)"+ ~2a^(a + fe.r*) p+ 2ap J (a + fer 2 )*
1
(<
/'
.r" ( /.r
;'
J (a
H- 6.1-")
"+
~
n+1
^c
?j-3jt?
3
3ap (a + ft.r
+l
''
3r/p
)
n
r
J
'
1
jv
(
a
__
+ /u-3 )*'
,
'
[BERTRAND.]
and evaluate
./"
,
f
.r
3
(^T6^p
dx
r
r7-'''
I
.,(,
CHAPTER
222
^n
/
3
a
.r""
J a + Zw*
~(w-3)6
a?"" 4
f
b) a + bx*
^
cfa
VII.
4jp
jp
-
'
1
dx
/
jf
} (a + bxty^~ ap(a + bx*y
lap J
+&
(
and equate
224. Reduction formulae for
I
smp x cos q x dx.
s
Integrals of this form also conform to the rule of "the
smaller index +1," explained in Art. 217.
Connection can be effected with any of the following six
integrals
:
I
sin^'"
I
sin
I
sinp
2
cosmic dx,
# cos q
?)
~2
a;
by the following
~2
x dx,
cos Q+2 x dx,
rule
1
sin ?)+2 ic cos 9 a? dx,
I
sin1
\
sin ;i+2 a; cos^" ^ dx,
'
a;
cos q+2 xdx,
2
:
P = sin A+1 ajcosM+1
Put
indices of since
whose
^
where X and /m are the smaller
JC,
and cos a? respectively in the two expressions
integrals are to be connected.
Find
-y-,
and rearrange as a
linear function of the expressions
whose integrals are to be connected.
always be performed.
This rearrangement can
Integrate, and the connection is effected.
of these connections might be effected
Each
by integration
by parts, but the advantage to be gained by the present rule
is the same as has been explained in Art. 217.
For example,
let
sin7 x cos 9 x dx
'
1
us connect the integrals
and
1
sin*
7
" 2
x cos'7 x dx,
1
FURTHER REDUCTION FORMULAE.
Let
P = sin
p-1
a;cos +1 a;.
-j-=(pI)siu
dx
= (p
sm
l)
p - 2 xcos q+2 x
p~2
p
1) sin
(p
223
(q+
sin 2 x)
xcos q x(l
~2
cosmic
0?+?)
siup x cos q x.
last two lines of rearrangement as a linear
p
q
p ~ 2 xcos q
function of sin x cos x and sin
x.]
[Note the
Hence
P=
smp ~ x cosmic dx
2
(p
1)
I
(p-{- q)
I
sinp a; cos^x dx
and
fsin
siup x cos q x
-
dx
p " 1 a;cos 9+1 a;
plC.sinp ~-x cos x dx.
q
p+q
\
P+qj
225. List of the Six Connections.
The student should note carefully the
possibilities of
con-
nection for IsinPzcos^da;.
The
indices of
either sinsc or cos
a;
may
be increased or
diminished by 2, the other index being unaltered
or, the
index of the one lowered by 2 and the other increased by 2.
;
Writin
s for sin
x and
c for cos x,
the six connections are
:
JoP
sc"dx=-
(6)
Each of these should be
verified
P=
by the student by means
where X, /m
of the rule
sin A+1 a;cos' 4+1 ic
given, viz. "Put
are, etc. ... ," and also by integration by parts.
>
*
224
CHAPTER
VII.
226. Special Cases.
When
p-{- r/
= 0,
the integral
ltan p .T^ie, and
is
is
integrated
the reduction formulae of Art. 125.
by
When p+l=0,
q
9
,,
j = (cos Xj = - f cos a;
sm p xcos q xdx
dx
^-d (coax ),
\~.
cos z
f.
J smz
J 1
and then we write cos x = z, and use the method of partial
,
2
fractions, or proceed as in Art. 228.
W hen
T
-
smp xcos q xdx =
-,
J.
fsm p z
I
p
sin z
,.
U--^-^dtsmx),
sm
f
,
dx=
.
2
Jl
Jcosa
and then we again use partial
x
z
fractions, or proceed as
in
Art. 228.
227.
is
odd,
an
The student
or
easier
is
again reminded that when either p or
negative even integer, there
when p + q is a
mode of procedure
we have
Also that in an\
(Art. 114).
method of multiple angles when the indices
are positive and integral
and in general this will be a rnoi
of
method
speedy
obtaining the indefinite integral than the
of
a
reduction
formula. The results, however,
employment
will be necessarily produced in a different form by sue!
case
the
;
processes.
228.
now
We
must
also notice that, in the formulae of Art. 225,
Hence w<
or
p
q, or both of* them, may be negative.
have reduction formulae for integrals such as
either
sin^a;,
f cosmic
dx
f
,
"
'
'
and to these the "multiple-angle method" of Art. 112 woul<
not apply, by reason of the non -termination of the binomii
expansion used for the purpose of conversion.
Thus, putting
q for q in formula (5) of Art. 225,
J~
cos q x
Putting
p
Ccos q x
dx = -r
(q
for
p
2l)cos
~l
dx.
q1
x
x-
J
Jsin''a;
l
in formula (6),
cos q
-,
q
(p^-l)sm
p~
q
1 fcos 5
I
l
x
p
--.--
"2
^
.,-
-x
IJsm'
Jsin''~-#
1
,
dx
FURTHER REDUCTION FORMULAE.
dx
q for q in (2) and
p and
for
p
Putting
~
^^cco^x~
('
l)sm
(p
/
^
1
xcosq
-l
1_ ^~~rt ~- p xcos q
sm
\
1
l
l)
(q
p-1
1
l
(4),
p+q
1
x
2 C
1
p
Hi"
x
225
J
sin p
p+<?- 2.f
ql
1
I
dx
z cos%
-2
<fa
oi
sin
etc.
If,
of
p=l
however,
form
dx or
I
J
cos x
f sin
fsin
Then
p
a;
cos x
J
for integrals
these reductions obviously
x
we may put q=
dx,
J C
rp,
i.e,
j =
dx=
--$in
$i
p~l
p
x
p
be an even integer, or at
respectively log tan
Similarly, for
f
s
+T
I
~" s i n
,
cos x
J
OOS^
'T*
-:
smx
ff*r)Q^
finally arriving at
fcos
2
ic
j
J
dx
x or
lg sec # n ^ ne ^ wo cases.
=
sm x
in formula (3);
1
~"
I
OOS^~"T*
J
sin x
/
dx,
:
1
7
fcosa;
,
dx or at
I
J
be odd, giving
p
if
I
i
C?X=
q
-r-
,
dx,
sin x
/y
i.e.
229.
log tan
The
--f-
cases
cos x or log sin x as the case
fr
when p
or q vanishes,
sm n x dx
and
I
i.e.
may
be.
the integrals
co
J
are of primary
importance.
sin n
Connect
Let
P = sin'
~1
l
~T
f
\sm
J
n
with
I
sin
iccos x, according to rule; then
dP =
.',
<fce
(n
1) sin
(n
1) sin
xdx=- sin*"
1
a?
n
dx
-
)
-dx, put p
I
(1),
- dx,
fsi
I
in formula
1
and repeating the operation, we presently arrive at
if
fail.
p
I
Art. 225.
in these results,
-- dx,
sm
\
J
In the case
=l
or q
n~2
n~2
cos
sin n a;
iccos 2 x
ic
nsiii n x',
n
1
n
f
.
\sm
J
n
-*xdx.
CHAPTER
226
VII
Similarly,
I,
cos n z dx =
siniecos'
1 "" 1
n
^
1 f
n
n
cos n
~2
x
dx.
J
230. To calculate
w
Cn =\
and
Jo
Since sin n
than
2,
~1
when n
ajcosa; vanishes
at both limits, x =
is
an integer, not
less
and x = ^> we have
713
5
71
ct
etc
-
If
7i
be
et'eTi
,
>n
If
be odd
71
comes to
this ultimately
we
_7i-l 7i-3
~~^r~' 7i 2
3
5
1
TT
"'6' 4' 2* 2
similarly get
4 2
7i-l 7i-3
S n =- ^c'y
2
71
3
5
71
r
K
and
sin x dx
since
=
cos x\
Jo
\
Jo
5" =
we have
1
Ti
'
-JT
= 1,
Ti3 '" 4
^2
2
'
'
3
5
T
H
cisely the
n even.
We
same
This
COS CC dx has >r-eJ^o
value as the above integral in each case, n odd,
may
be shown,
from other considerations.
thus have
1
or
too,
n
3
n5
= n-1 w-3o "n-5
~i
3
1
^ 2
-
q
1
>
^
n odd.
FURTHER REDUCTION FORMULAE.
231.
written
227
The student should notice that these formulae are
down most easily by beginning with the denomi-
nator.
We
(n under
n
then have the ordinary sequence of the natural
numbers written backwards,
l)x(w
2
under n
4 under n
5)... etc.,
if
n
be even, and writing a factor
under 2)
if
n
stopping at (2 under 1)
or stopping at (3
3)x(n
be odd, with no extra factor.
(3)
where
= 1T53
~4L6 4
1 TT
2
6421
27 5 3_T
(4)
2*
sin^cps
s 11
2^
etc.
EXAMPLES.
1.
tin
1
Prove that
indices
, Prove
fteiw^r
th
6o^A diminished.
^^,
- 2 cos 13
2 4 6 8 10 /
33579
=
<j>
11
llA
c^>
12
13
where
+ cos 15
12 14 \
+ 13
157
;
CHAPTER
228
4.
VII
Prove that
1
d
9Jsln*0cos*0~(?-l)sm^- 0cos
[
1
~
1
p + q-2
q-l
l
l
(p-l)BU*?- Ocoa*- 6
-^
/sin^0
cos
du =
sin^~ ^
;
\
H
J
_dO_
sin*0cos-
p-l
J
sin"- 2
2
~0
cos 9 0'
dO
sin^*""^
2
cos u
/
J
I
f)
f __
t
0"
7.
...
+
+ '" +
where
and
c
s
and
stand respectively for cos
sin
2.4...2n
1.3...(2n~
[BERTRAND.]
0.
8.
+
c
being respectively cos
and
and
1.3. ..(2/i-l)
C+ 1.3. ..(2^-1)
-i
sin 0.
[BERTRAND.]
Prove
9.
(a)
where
10.
c
/cosec>^=
= cos ^, 5 = sin
-^-^1
^^
^.
Prove that
(a)
(
J
1
sin
27i(9
-2
2n-2
sin (2n
1.2
2w - 4
FURTHER REDUCTION FORMULAE.
2w-3
1.2
Bin
229
(2n- 2)0
-1)
sin (2?i
-4)0
2^-4
1.2
2n(2n-l)...(n + 2) sin_20
2
1.2. ..(Ti-1)
7*-l)... (TI+
1)01
1.2. ..
[BEBTBAND.]
INTRODUCTION OF THE GAMMA FUNCTION.
For what follows we shall require a new function r(^-J-l),
which will be defined sufficiently for present purposes by the
232.
equations
This will be enough to find
positive integer, or of the
its
value
whenever n
2&+1 where
form &
,
k
is
is
a
a positive
integer.
For instance
= 5.4.3.2r(2)=5.4.3.2.ir(l) = 5!,
This function
it
is
more generally
called a
later
Gamma
function.
and investigate
its
We
shall define
properties.
For
in
the present, it is
temporarily introduced to secure facility
the rapid evaluation of a class of integrals to be discussed.
CHAPTER
230
VII.
be noted that the products of the first n odd numbers
- 1 ) and of the first n even numbers 2 4 6 ... 2?& can be
of this function, for
in
terms
expressed
233. It will
1.3.5.7...
(2?i
.
2M-l
27i
-3
2ft
2
2
-5
1
.
_
"
2
>
,
/~
2
2
2
4-9
9
and
284. To investigate a formula for
sin^O cos q OdO,
p
and #
being positive integers.
Let this integral be denoted by f(p, q)
f rin'0
J
we
have,
CASE
Then
I.
;
then since
** eft a _n^co^^ g-1 f ^.^
p+g
p+gj
if
p and
g be positive integers and
p
not less than
Let p be even, =2m, and ^ be also even, =2n.
/(2m, 2n) =
2 ra )
J^^/(2m -2,
(2m-l)(2m-3)
,._
v
and
P cos- BdO=
(0,2n)=/o
2?i
-1
-^
^ -^ ^
2?i
-3
1
TT
o r x_[1.3.6...(2m-l)][1.3.5...(2-l)]
~2.4.6...(2w+2n)
2
/2m+l\
V
2
2"
2
TT
2
2,
FURTHER REDUCTION FORMULAE;
CASE
Then
2m, and q be odd,
Let p be even,
II.
= 2?i
231
1.
-2, 2w-l) = etc.
___ M
-3)...l
-
i-3)...(2?i +
_
l)'
fc--'
.....-.....
a*
~1.3.5...(2
y
+ 2-l
/2m + l\
2
CASE
Let
III.
In this case
we
2w) _[2^_._6^(2m-2l
/(2m- 1
But
this
- 1, and q be
odd, =2m
obtain similarly
p be
for then
[~sm
.3.5 ...2
deduced at once from Case
also be
may
even, =2w.
p Qcos'1
OdO=
7
[ cos^^sin'
J*
Jo
II.
^ (
by putting
l)d<j>
TT
=
I
sin q <f> cos
Jo
so that
f(p,q)=f(q,p)-
Hence the
result
CASE IV.
Let p be odd, =2??i-l, and q be odd,
is
again
4
-(ft+8-ir(g^
=2w-l.
8
^"
(8.+8)
TT
TT
and
f(
1
2>j
1
)
=
/
.'o
/a
.sin
cos 2rj1 @ dO
QH
|
Jr\
=
2?i
'
*n
~
i)
CHAPTER
232
VII
[2.4.6...
(2m-2)][2. 4. 6. ..(2^-2)3
~
235. Hence, in every case
and
it
will be noticed
we have
the same result,
that the
1"
z
denominator
the
is
sum
of the
viz.
-fl occurring in the
^
and
the ^"1"
in the
numerator.
236.
As
it
has been assumed that
particular cases
p = 1, p =
p
is
not
<2
we must
separately.
if
Hence, this case conforms to the general
i
rule.
TT
(n-l)(n-3) ...
n(n-2)...8
1
TT
2
(n odd).
In the case n even, the above result
may
be written
consider the
FURTHER REDUCTION FORMULAE.
and
in the case
n odd, the
result
is
x
Hence these
cases also
233
21
/
conform to the general rule
sin*0cos ?
6W =
o
may
therefore be
assumed in
all cases
p and
where
q are positive
integers.
very convenient formula for evaluating
237. This, then, is a
quickly integrals of the above form.
Thus,
2.7.6.5.4.3.2.1
57T
// however,
multiple of ^,
2t
we
and an
than
the limits be other
mitstf ^TicZ ^Ae indefinite
a reduction formula or by
integral
integral either by
method of Arts. 114-117
the
before inserting the limits.
238. Integrals of form
Im. P
= (x m
X
p
X = a + bx +
dx, where
[This form obviously includes
all
caj
2
.
such cases as
x
etcl
I.
Consider the case
Put P=(6+2cB)-l'
1
'.
when
m
0,
i& J
0j
p
==
I
Arp
CHAPTER
234
r/P
Then
VII,
- 2cX* +p(b+2cx?X*-
1
This reduction /a^s w&en 2p-fl=0, but in that case the
fdx
Ja+bx+cx
t
The formula
2
,
and has been considered
(A) will finally reduce the integration of
to that of something of the form
and
If s
1.
Hence, in
k
is
all
=
I
X
s
dx,
where
cases
where p
is
integral or of form
Xp dx
between
,
where
Zi
X dx can be effected.
p
I
J
2&+1 we
9
be a negative integer or of form
p
s lies
I
or J, the integration can be written down.
a positive integer, the integration of
If
in Art. 80.
can
apply the same formula to lower the index in the denominator, viz.
f
I
CtX
-^L
-
or writing
p
p
--
(b+tox)X>
-
J
-j-^
2
(6
-
4ac)
Z(Zp + l)cf
-
-TJ-^
2
p (b
I
^\_
U/J,,
4ac) J
for p,
dx
II.
Put
Next, consider the case when
m=l,
P=
and the
last integral
This reduction
has been considered.
fails
when p =
l.
i.e.
/1|P =
FURTHER REDUCTION FORMULAE.
But
this case
I
is
I
_l
=
\
-^,
J a-\-bx-\-cx'
235
and no reduction
is
required.
fjrx
Then
p
~=p(b+2cx)X
u>X
-
dx,
put
P = Xp
.
1
= P[bx+2(X-a-bx)]Xp -
1
x
x p Ub
2p^2
-dx = ^-+ -
x
JYP
chat
is
I-i.p
= ^+zI.p-i+aI-i.p-i
240. In the case
/n =
f
)Ja+bx+cx2
dx, put
P = x- ja+bx+cx
l
2(m
2
.
TO _ 2>
I)(a
_
1N
PD = m _l)
a/
.
.-.
(C)
(2m_
_ 2 _p
,
7n
(
1),
L^Jrm
which connects Im with Im _^ and /m _2 (unless
XT
r
Now
7j
=
xdx
f
I
1
.
f//
/
m=
0).
=
A
7>_L9/~r
\
dla;
J
1
.
__
c
and 7
is
discussed in Arts. 80, 81.
5
'
'2c
CHAPTER
236
VII.
241. III. In the general case /,,,,=
xm
we have
~2
\x m X p dx, since
a+bx+cx 2 = X,
Xp+l = xm ~ (a+bx+cxz
2
)
Xp
,
and therefore
I m -2, p+i
let
Again,
P=x
m~l
= aIm _ 2tp + bIM
X
p+ l
.
_ l<p
xm
fx
(E),
XP+ i_ (m _ 1)Im _ 2(p+i = (p+1)6 7 m _ iip + 2(p + i) c / mp
We
m ~2
(D)
Then we have
Eliminating I m - 2>P between (D) and
-i
+cImtP .............
.
(F)
thus have, collecting the results,
X
p+l
dx
(D)
(m + 2p + 1) c
= xm f xm
X
p
l
(xm
p
dx
Xp+l -(m-l)a txm - X dx-(m+p)b {xm~ 3
2
p
l
p
dx,
(E)
dx
2(p+I)c
or,
X
writing
2c,
2(2>+l)cJ
p
for
which the index of
p
to
adapt them
to the use of cases in
X is negative,
TWl-2
f
***
'~1dx,.
...(D')
FURTHER REDUCTION FORMULAE.
(m
2p
237
+ l)c \^p dx
~dx,
irn
JH-1
f~m-2
1\
(m
//yy,.
/
(E')
\
'
242. Remarks.
The case of
p=
1,
in
which formula (F)
a+bx-\-cx
But
in this case
reduction
is
we
fails, is
dx.
2
proceed to partial fractions, and no
required.
Equation (D) (p positive) expresses one integral in terms
at the
of three others, with a lower power of
of
x
and
of
higher
powers
introducing
expense
X
;
Equation
(D') raises the
power
of
X in the denominators.
reduce to integrations with the same
of
but lower powers of x.
powers
two integrals, in both of which
with
connects
Equation (F)
is
the index of x is lowered, whilst that of
raised in one integral and remains the same
Equations (E) and
(E')
X
X
in the other.
Equation
(F')
plays a similar part for the negative index of X.
243. Integrals of form
%x,
or
obviously come under the heading discussed, after trans2
formation, by making px -f- q = y, which transforms a-\-bx+ ex
to the form A -\-By-\-Cy 2 ,- where
2
Ap = ap 2
bpq -f eg
and
\(px+q)
becomes
-
m
m
2
,
(a-\-bx
+ cx
{y (A
(A+By + Cy
and similarly in other
cases.
2
)
n
2
)
dy,
CHAPTER
238
The
where
particular cases
6
VII.
=
m~
heading of those discussed as \x
244. Integrals of form
1
or c
(a->r bx
=
r
)
come under the
p dx in
Art. 217.
dx
In =
-1,(q -\-px) n ^Ja -\-bx-\- ex*
be regarded as coming under the head of those discussed
in "Art. 241, for the substitution q+px = y immediately reduces
may
them to that form. But as this form occurs very frequently
and is of considerable importance, it is desirable to consider it
independently.
j
p ^ ija -\-~bx + cx*
.
1
Then
dP
b+2cx
(q+px)
n
2 (q -\-px) n
q+px)*
2
= qb 2(n
= Zqc +pb
jup + 2vpq
where
\-\-ij.q-\-vq*
l)pa,
2 (n
1 )pb,
from which we obtain
\=
2(n-I)(ap
And 2P = \In -}-/uiIn _
That
-{-vIn_ 2 is
bpq+cq*)/p
>
the formula sought,
is
-
The
1
2
1
p-cq T
~J~
where n =
c
}
n~ 2
p
l is given in Art. 287, whence / 2 can be
found from the present formula, in which the coefficient of
7 n _ 2 vanishes when n = Z. Then / 3 7 4 ... can be successively
case
,
derived.
,
FURTHER REDUCTION FORMULAE.
The
245.
239
integral
dx
=H (px+q)
be written as
may
.
where /
is
the integral discussed in Art. 244.
This therefore constitutes a reduction formula for
But both
this integral
,
Mx + N
C
=
Jn
.
and the more general integral
dx
~J(
more conveniently evaluated by
are
of the constants involved, q in the
differentiation with regard to one
case, G in the other, as explained
one
subsequently (see Art. 364).
246. The integrable cases.
Denote I m>l >= \x m X p dx for shortness by (m,p).
The
special cases
(0,
-1),
(0,
-*),
(0,i),
(0,1)
simple elementary integrals whose values have been discussed.
Formula (A), which connects (0, >) and (0, p 1), will therefore continue
the series both ways and yield
are
all
(0,
2),
(0,
f),
(0,
(0,
|),
3),
(0,
),
etc.,
<2
i.e.
where k is any integer.
Formula (B) connects
*)
(0,
or
(o,
(1,
p) with
(i,
**>.
(0, p),
and therefore contributes the
integrals
wl it-re k
is
any
Formula
(
1,
and
as also
(C) connects
1) are
?*J),
(-1,
2),
+),
(-1,
(-1,
~
),
(-1,
p-l); and (-1,
(-1, p) with (-1,
simple cases already discussed
(-1, +i),
-1,
(i,
integer.
(-1,
3),
L
(-1,
+),
J,
and
etc.,
4), etc.
are contributed where k
;
-)
;
is
any
integer.
CHAPTER
240
Formula (D) connects
(in
2,
p + 1),
(wi
/O
(X,
Z\
*)>
(3,
fc),
i.
1,
jt?),
(,
(0,jt?),
(I,j0),
(2, j?)
(!,/>),
(2,/>),
(3, _p)
are connected,
O
^
I
^
;
i
>
2
fa
are contributed,
>
(4,
2
etc.
Formula (E) connects
therefore also
(w
2, p),
are connected,
etc.
.
VII.
(wi-2,/>), (wi
- 2,
(
;
),
(
- 2,
I,/?),
(m,/?)
;
),
are contributed,
etc.
Hence
all integrals
{x m
can
X
be integrated
2 k -4-
1
-*
,
p
of form
X=a
where
dx,
in finite terms when
and m, k
p
is
of .form
k or
are integers positive or negative.
EXAMPLES.
247.
1.
Taking
f
j
dx
_ b + Zcx
I^~
dx
p rove
H
2(2y>-l)c f dx
pkX*
b + 2cx
2c C
pk
j T*>
where
X=a + bx + ex* and k = 4ac - 6
d>OG
O "4~
C3C i
,
dx
3c \
L
/
2
OC
J.
6c 2 r dx
Iv/C
\
4j\jC
i
(JiOC
[BERTRAND.]
2.
Show
that
if
/=
Cxdx_
I
-^o?^,
1
,
then c/M + 6/n _j + a/w _ 2 =
b
f
dx
,
and prove
FURTHER REDUCTION FORMULAE.
fx ~dx = x
z
Deduce
b
/
^
log^H
,
b
2
'2ac C
I
241
dx
~y>
[BERTRAND.]
3,
Prove
dx
I
x
,
z
b
[dx
and deduce
2ax z+
a 2x
2a 3
log
'dx
X+
2a3
J
X'
[BERTRAND.]
(The value of
J
4.
If
in each of these results is given in Art. 80.)
occurring
-y
-A
I
X=a + bx+cx
2
prove that
,
n-l
5.
Prove that
X = x + ax + a
z
if
dx
b
xm
C
[BERTRAND.]
2
,
f
J
6.
[ST. JOHN'S, 1889.]
Prove that ifX = x* + x+l,
dx
C
.
(a)
= 1,lo x
J
2
1
1
1
z-3^ + 2^-vi
2
.
tan
dx _1 +2* 2(2y- l)f<to
ay J x*'
-.api*^
r
"Jz+
7.
Show
(
l)
that
f
2
J (x
if p be a
dx
+x+l
positive integer
and
X=x
v+l
(2p-p/2\
1
(2p
-1
)
(2p
- 3) /2\
(2y-p(2p-3)...3.1 /2\^
/>(/>-!) ...2.1
-3...3.1
\3/
Z
2
'
I
(b)
J
74,
z
gj-j-j,
-
being a positive iinteger.
/2\
(n-l)
(2n-l)(2n-3) \3/
(n-l)(n-2)
_J?LZiL
!
23
-^f
/2\ 2 2 s
/2V-
(2n-l)(2-3)...l \3/
E.I.C.
1
3
p(p-l)(p-2) \3/ Z^-
1
p(p-l) \3/ X*-
1
2 2 "-^n
^
J
CHAPTER
242
/=
248. Reduction of
X=a+bx + cx*
2
Let
dP =
mi
Then
.
(
dx
n
3)x
-
n
P=x
-*r^
i
l)cx
(n
= dx.
and put
}
VII.
n
,
Jbx-\-2cX
S
s
)
n
+(n
2)6a;
249. Integrations of
i
(
)
cos
I
px cos n qx dx,
sinp^cos
(iii)
including
n
g^^,
-^dx
I
cos n
t
qx
There are two
J
(
ii
)
(iv)
I
cos pa? sin n qx dx,
\
etc.
classes of reduction
formulae for such
in-
tegrals.
We may
I
or
connect
cos
px
cos n qx
dx with
I
cos px cos n
~z
qx dx,
we may connect
I
and the
cos
n
px cos qx dx with
71
q ) x cos
"1
qx dx,
like with the other three cases.
250. First,
(i)
cos (p
I
we
Let I n =
I
consider the former class of reduction.
cos px cos n qx dx.
Then
7
/B
-
-^
= sin ^px cos" qx + nq
.
= sin px cos n oaH no
.
p
p
-fl
{
(n
f
I
.
.
sin pa; cos"" 1 qx sin go;
\~
L
cos
px cos n ~
'
p
l )qcos n
~2
qx(l
l
,
aa;
qx mn qx
cos 2 g^+^cos"^}^
FURTHER REDUCTION FORMULAE.
= sin px cos" qx
nq cos
*-
no 2
^-
-\
P
1
\
px cos"- qx sin qx
(n
1)
C
1
cos
px{
243
cos"" 2 qx
+ n cos" qx} dx
;
J
2^y2
\
-
n(n-l)q
*.
e?
Now
= COS W _
!
cos n <705
=;cos n
[
dx cos px
x
-
-
sn p# cos (/#
QX p
-l
-
sin IMS cos qx
7)
qx"
Tig cos^?x
o
2
a
^
^nq cospx
2
sn qx
sin
oa;
~-
cos 2 px
Hence the reduction formula may be written more t compactly as
,
d cos n qx
cos 2 joic
n(nl)q2
By successive reduction, the power-factor cos"^ may be
reduced either to cos^cc or to unity, when n is a positive
integer, and the integration can then be completed.
If n be negative (=
m), we can, by solving for / n_ 2
>
express the same formula as
,
--~
cos 2
d SQC m x
x
z
-m22
,
k
and therefore a reduction formula for
.
ramished.
Similar
(ii),
(iii)
work and remarks apply
and
but
(iv),
it
consider
(ii)
Let I n =
\
cos
px
n
sii\
qx dx.
Then
/T
px ain n
= sin ~
qx.
nq
f
\
.
sin
also
them
detail.
251.
is
to the other three cases,
desirable to
is
^ dx
]cos m qx
,
n~ l
qx cosqxda
px sm
.
1
,
in
CHAPTER
244
-p
sin f)x
-
.+
nq f
sm n qx -p L
.
-- vx
cos
VII.
.
siu n
~l
p
qxcosqx
1
-
= sinm sin n yaH
nor
.
--T
cos
I
n zz
n
sm"- 1
| cos px
^{(w
=
1
sin"- 2
)
^
.
(^ sm ^x
,
2
^ cos gx
n sin n <?#} $r;
sm ?x + w ? cos
^ cos qx)
-n(n-l)^/
.
r _-
'
n=
d
sin
joa;
sin
ga;
4- ^g cos px cos
ffff)
sm n qx ~~n(n
2
Let I n =
(iii)
s (j?
cos z px
,
252.
'j
n_2
jsin^aj
cos n qxdx.
Then
7n =
=
^- cos n qx ---
'
--px
cos
p
P
J
/.;
(1
2
x
cos n qx
-{
I
cos
n ~l
qx sin qx dx
--nq fsin p# cos n
-
p
p
(nl)q cos n
-
2
px cos
~2
qx ( I
cos 2 <p;) 4- q cos n qx} dx
-
1
w g \
cos"- ^.
-I =1- J n
^- (p cos px cos + ^g sm ^?x sm
7
--
n
~
n^.-1/rr
<p?
^ COS ^ COS gX + ^ Sm ^ sin
-
f - ri*f dx
~~
s
ga?)
;
FURTHER REDUCTION FORMULAE
/= Isinpx sin n
253. (iv) Let
?,45
dx.
</x
Then
=
/
^-~sin w ax-|--
I
1
cospx sin"" ox cos </x dx
- sm n qx-\ nq-\Tsin ^vx sin"- 1 ax cos qx
cos vx
fsin
.
/
DX
,
((nI\asm n - 2 ax(lsm 2 qx)qsm n qx}dx
~]
;
J.
n 2 q''\,
/n
.*.
.
.
(1
\
}I n
sin**- 1
^.
2~(P cospx sm qx
=
nq sin px cos qx)
2
w(n-l]
/
t.e.
The four
254.
d siu n
p n q dx sin px
=
2
2
results are therefore
fcos
Jcos
cos
n
px cos ax dx =
p
px d cos n qx
^
^-5
f
n 2 q 2 dx
cospx
2
^
2
n(n-I)q
\
pi
sm n gx dx
?
2
_p
f
I
.
sin
,
n
px cos qx dx =
J
px 8in n
flfa;
aa;
=
px d
~-z
2 2
a dx
n^q^
p
sin n ax
cos
c?
cos n ox
sin pa;
2
^ x cZ
-^
n 92 q*9 dx
=
7l(?i
1)0
-Vz~2
3L
sin" qx
:
sin
"
2
px
i
dx
x
*
4
2
J
?i
~
Isin
sin
(
q
sin 2
p
2
n /f- |cospxcos
2
cos px
2 2
2
p -n q
2
2
f
js
.
px
w -2
,
gxax.
2
/
CHAPTER
246
That
is,
A
if
stands for the
In =
i.e.
power-factor,
VII
factor
first
\APdx, we
and
P the
second, or
have, in all cases,
or
-m for
Writing
where n
n, for the cases
is
negative,
we
write this as
may
m(m + 1 J9 /_m_ = A~-P--(p
2
2
2
Such formulae are more particularly useful for negative indices
For if the integral sought be, say,
factor.
255.
of the
-
power
cos
/
the "multiple angle" process for
reduction.
sin 6 3#
Thus,
:.
cos
=
4x sin 6 3^ =
^
(sin 15.r
6
[(sin
4r sin 6 Zxdx,
sm 5 3#
will
be more convenient than a
- 5 sin 9# + 10 sin 3#)
19^+sin llx) -
;
5 (sin 13# -f sin 5^)
+ 10(sin7#-sintf)],
and the integral
is
Il~cosl9#
~2"5 L 19
cosll^
5cosl3a?
5 cos 5^
10cos7.r
~^LT~
13
~5~
~~T~
4^2?
".
/COSs
ange n to
n
t
dx
this process
t
in the second of the
10 cos x~\
~l
Therefore
useless.
is
J'
formulae of Art. 254.
Then
1-
/cospx^
sin n+2
dx=
d secpx
-7w
n(n+l)q* dx sin qx
p 2 -'n z q [ cospx T
dx
*-;
n
n(n+\}q* j sin
2
cos 2 px
'
*-
^
,
whence
4r
,
sin 6 3^
/cos
rcos4^
_
~
cos 2 4^
J sin^3^
13. 5
cos 2 4# d sec4x
~
v 1.2.3 2 dx S3^
C cos 4^7
3 4 3 2 J sin 3 3#
.
.
,
sec4#
d^
3 4 3 2 rf^ sin 3 &K
.
.
7.1
_
1
.
2
.
r
cos 4a
3 2 J sin"3^
whilst
Tcos4a:
J sin 3a:
1 C (
dx = ,
(
I
-
-.
2 J Vsin
a;
\
1
sm
:
3a:
,
4 Bin x] ax
/
x I
1
'3x
= -log
tan- - ~ log tan
+ 2 cos a;;
.
.
,
:
we
FURTHER REDUCTION FORMULAE,
247
hence
cos 4x
cos 2 4#
,
d
13.5
4#
sec
d sec4#
cos 2 4.#
(
/cS
= etc.
256.
For the second mode
Art. 249,
we may connect
of reduction,
Ipin that
,
mentioned above in
n
\cospxcos qxdx or one
is
with
of the other cases
Or With
Ip-q,n-l
cp for
Ip_ Z q n-2,
t
To shorten the expressions we
shall use the notation
sp for
smpx, etc.
The mode of procedure is the same in all cases, viz.
Put P = the power factor x the complementary function of
cospx,
:
the other factor.
Differentiate
and
rearrange.
(i)
P = sp c
Lett
l
Then
-j^=pcp c q
n
.
q
n
nqsp c q -^sq
n
= c q n - l [(p + 114) cp c q - nqcp_ q
.*.
P = (p-\-nq]\cp c q n dx
(ii)
p,n
T Of
jLjet
p2
Then
nq
l
= \Cp8q
^'
x
]
;
\cp _ q
c
-
Q Q n
"?>"</
^
= sq n ~
l
[j(p
-\-
nq) cp s q + n*l sp-q\
r
>
r
-flSfl""
(iii)
P = cp c
Let
3
3
Then
-y-
=
psp c q
-
.*-
^3= -
n
n
q
.
nqcp c q
n~ l s
q
1
dx.
248
FURTHER REDUCTION FORMULAE.
249
257. Avoidance of a Reduction Formula.
For integrals of the
I
it is
under discussion,
classes
n
cos
px cos qx dx,
viz.
etc.,
often convenient to avoid a reduction formula altogether
n is a positive integer, when we shall require to
so long as
n
put the power-factor (cos qx in this case) into cosines or
of
as
seen
in the example in Art. 255.
sines of multiples
qx,
in
Art. 112, the formulae required are:
Proceeding as
n
(l\
y-\
=etc.
)
= 2[cosn0+C
1
cos(n- 2)0 +
^= o
where
=
or
-5 if
-
t
2
2"-
-
1
1)""'
(
sm0 = cos nO -
C2 cos(n- 4)0+.
..
be even,
ft
n-l n+l
n
cosfl if
(A)
n be odd,
.........
(B)
t
2
n
C^ cos(n
- 2) 6
n
M
\
'
ifnbeeven;
(C)
n-l
2
)
sin"
= sin nO - nC sin(n-2)6
l
sin0
2
2
if
= qx,
2 n cos n qx cos px = a
n be odd. (D)
Then taking
r
series of
form 2S/< r cos r x cos px, say,
i
and
taking due account of the final terms.
Similarly we may proceed in the other cases.
The formulae
(A), (B), (C), (D) can be readily reproduced as
explained previously in Art. 112 for any particular value of n
for which they may be required.
CHAPTER
250
Ex.
/
2# sin 8 5# dx.
sin
=
.'.
VII
2 cos 6<9- 12 cos 40 + 30 cos 20
-20;
taking #=5#,
sin 6 bx sin
2# =
~g [2 sin
2# cos 30# - 1 2 sin 2# cos 20#
+ 30 sin 2# cos 1 0#
= -5
.
_
.
t sin
~ sin 28 r 6 sin 22<r ~ sil1 18<r
+ 1 5 (sin 1 2# - sin 8#) - 20 sin 2^],
32 ^
-*'
cos32.r
,
-,
20 sin 2#]
(
)
/cos 22^
cos28o;
cos
258. The Integrals
f cos
/1X
(1)
.
In case
I
13
J sin
^
I
(4)
J
,
-dx.
smp a?
f2cosa;cos(n
l)x
f2cosajsin(n
"
l)x
- cos(n
-*
2)x
7
(2),
(3)
and
(3),
nx
I
;
\
sm x
sin(n
- 2)z
,
~~
*n_2,2?
sin
,
-
_fcosnx,
dx = fp=
p
}
-
(4), let
cos
j,
fsin^a?
/yix
(1),
For cases
T
in
f
J
,
c?o?.
~ "ln\, p-\
In case
/nv
(2)'
-t
-d!.
Ccosnx
/0
(3)'
In case
nx
J cos^a?
2 sin x sin (n-
,
dx
-
;
s
J
-,
In case
jn
'
p
(4),
_rsmwx, _C2siuxcos(n
I)x-{-sm(n2)x
'
J sin^a;
~J
sin^cc
_.
FURTHER REDUCTION FORMULAE.
The
cases
and
(1)
reduce to lower order
therefore,
(2),
251
integrals of the same form.
The cases (3) and (4) reduce to lower order integrals, but in
each case the forms are partly interchanged.
may
It
be worth noting that in the form
J
cos p x
dx we
might as an alternative method express cos nx as a series of
powers of cos x and integrate each term by methods already
discussed.
n be odd
If
.
dx
v
be
may
treated
by
similarly
expressing sin nx as a series of powers of sin x and integrating
each term.
n
be even sin nx contains a factor cos x and the integral
immediately obtainable e.g.
If
is
;
4>x
-r- 5
-,
sm x
Jsin
7
=
dx
f
\
J
-4-
sin x(l
t
- 2 sin 2z)
TT
sm x
cos x
'-
5
dx
3sm 3
ce
since
Similar remarks apply in the other cases.
259.
Ex.
1.
-
-
fcos3.r
fcosbx dx=2_ fcos4.r
7
dx-
I
J
sCQS>X
7
I
J
I
COS 2 X
J
COS 3 X
- dx= 2o F 2 f -cos 3.#
7
L
=4
J
/
J
cosx
2
(4 cos *
,
dx
C cos
-
I
2x
- 3) dx - 2
f
~|
,
C
cos
3
(2
f (4
- 3 sec 2 #) dx
-*v +2 tan a; - 4.
-
;
or otherwise, and more readily, without a reduction,
/*
cos
5.2?
~
6
J cos
x
dx=
7
ri6cos6.r-20cos 3^ + 5cos^
I
J
cos 3#
=M8(1+ cos 2z) -20 + 5 sec
= 4 sin 2x - 1 2x + 5 tan
or,
7
2
1
- sec z #) dx
-
= 4 sin 2.r - 12#+ 5 tan x
3^
=-dx
dx\-\J J cos ^
cos 2 .*-
J
7
dx
x}
dx
as before.
CHAPTER
252
-2 I -^<fo+
?d*=
sm ^
VII
S
Ex.
2.
f
J
(
J
2
] sin .*?
3
X
z*(**d
sinx
2#
sin
f
dx
+ fL - 2 -;
J sm*#
=
4
I
(cot #
- 4 sin x cos #)
C cos
8
c&e
I
cosx
C
7
-f-
/
J
.
sm 3, #
cot x dx
x
j
+ sm*x
, -dx
J
I
.
2
=
, rt
16
8 sin 2 ^ -
=
or otherwise,
2
and more
-1
2 log sin
x
2
\ cosec #
;
readily, without a reduction,
rcosSz, = ri-12sin 2 o;+16sin 4 ^ ,
a sm x
dx
r-^3
3
sm
J
I
x
J
sin
= - ^ cosec 2 x
260.
Integrals
12
Csin n px
^
ITn =\
cospx
n
_
~
~sinpx
,
I
2
x,
as before.
px dx.
-JTn = Ccos
\-j
n
cospx
j
=
og sin x + 8 sin
n
-j
rsin
=
1
dx,
)cospx
n
.
*
cospx sin
n -2
^a5
dx + 7 n _2
;
-f J n _2
j
_~
siupx
=
I
sin pa? cos n
-2
^>aj
c?ic
Also since
/!
=
I
tan pxdx=- log sec
7 2 = f (sec px
J1 =
J2 =
J
- cos px) dx = -
cot pxdx = - log sin ^x,
f.
I
>#,
(cosec px
.
,
smpa?)(te
^^ +^
log tan
(^ +
,
ax~]
J
FURTHER REDUCTION FORMULAE.
253
we have
/y^
vjJ
siu 2n
Csm 2n px /YO*
i
COSVX
cospx
1
I
-*-
'
r
Cv**/
~l
px^
-*
.
zn
\
sin 2n
;
2n
L
smB px
px
>i
~3
_
..
-j
sinpx
_
,
2n
coapx
2
2?i
sin
4
m--sm ^
px
+
z
-
,
,
log sec px,
co
d
COS 2n+l p aj
d/X
Integration of
261.
x ,
fcos ^
px
cos ga?
(i)
+ -S2n2^- + ."
2n
f sinpx
sinp
We may
fcos
dx,
x
px
r~A
(/^
J sm
regard
p
}
f sin
7
dx,
J cos
if
p, q be fractional,
= - and
-1
-
be reduced to the forms
where 8
is
the L.C.M. of
Let x = Sy.
I
sin
I
cos
J
sin
other.
respectively let
,
-2
-
1
and
qx
s2
and
Rv R
z
are integers.
Then
/,cosAK,\
^cos. x
(P
-^ ;)
I
s1
sin
*v
s2
and
J
and prime to each
1
s1
sn
f si
(/a?
q as integral
/v>
For
x ,
px
^-dx,
^cos
,
(*)
Hence we only need to consider the case where p and q are
integers.
The signs
of p
and
q are also immaterial to the discussion.
CHAPTER
254
if
Again,
the
G.C.M.,
p and
and
VII.
q were not prime to each other,
p = Gp', q = Gq, and
let
x=
let
C
let
G
be
Then
8
f sL ^'*)
\
COS ,~
,
i
-,
^ sin
si n (Gqx)
d>'>
I
COS
,
-j
,
(qy)
^/ oiti
sin **'"
.
.
-1
where p' and
f
~~0r
x
are prime to each other.
shall need only to consider the case where p, q
are positive integers, prime to each other
Therefore
(ii)
</'
we
Supposing p>q.
Since
cos px -}- cos (p
sin
px -f sin (p
2g)#=
2cos(p
q)xcosqx,
x=
2 sin (p
q)x cos qx,
2q)
we have
pq
(
j
cosqx
rac
cos ^
.
fsm
dx=
7
--
(7)cc
g
>
'
dx=
'
g
jp
,
sm
s
J
(P-ti x
cos
S[npx dx=
qx
[
j cos
1A-+ fcos(
Hf -2q)x
O cos(i9
^2
go?
cosqx
J
s{u
(
J
(P~ 2
sin
qx
(r2)? + fsin(y-2g)g ^
2
p
Jsingx
g
J
sin qx
Hence, by a sufficient number of reductions of this kind, we
can reduce the integration of
cos,
x
;
^
sm v(px)
.
cos
i sm
to that of another integral of the
COS
Px
sm
cos
j sm qx
where
P lies
between q and
q.
same form,
say,
FURTHER REDUCTION FORMULAE.
255
no limitation upon our method in
the discussion of such integrals in assuming p
q.
to each
and
We
then,
take,
integral,
prime
q
p
positive,
(iii)
need
not be
both
and
The
case
even,
other, and p
q,
p
q.
considered, being a reducible case as shown.
Hence we
shall introduce
<
<
Now
if
n
CHAPTER
256
VII
Finally,
(iv)
a cos a dx
,
sm 2 z
/tan x\
-M - - )
Vtan a/
.
;sm
sm 2 a
sin a cos x dx
-
x cos a
sin
5
r*r\d&nr
t
i
sn x
.
1
=tanh- 1
.
sm 2a #
sin 2 a
dsc
,
-.
\sm a/
--
/cos
r-= coth-M\r*r\Q #\),
_ cosa
Vcos a/
,
,
,
:
f*r\c*&*-*
.-*
/
and
since
f cos
2
,
- dx
x-l
Hence
x
=
log tan ^
cosx
cos
f
I
,
2'
--
TT
-.
Jl-si
in all such cases the integration can be performed.
numerator
It is not essential that the
.
sin
(px) should
vz
be
might be expanded in powers of cos x or sin x,
may be. But the factorization is convenient, presents no difficulty, and as a rule is simpler in application, as
it indicates in factorized form the values of the constants
f actorized.
It
as the case
occurring in the partial fractions.
262. Ex.
Find the integral
Let# = 6y.
1=1
^^dx.
Then
dy =
^~
and
/
cos3y
by the
first
reduction formula, (Art. 261,
sin 2
ii).
7T
sir
"
9 71
sin 2
Also
dy
sin 2 y
sin 2
cosy
"
9 71
sm 2 ^
i/
2^ C
in1
2
o
I
7T
I
^4jI
cosydy
(
sin 2
2
2
2
^-sin yj(sin ^-sin yj
sin 2
sin 2
ft
sin
2
- -sin 2 ii
o
cos?/
sin 2 ^-sin 2 y
o
dy
FURTHER REDUCTION FORMULAE.
7T
.
sin 2 .
sin 2 -.
sin 2
257
-sm ,7T
.
2
-- sin -7;
sin 2 ?
-sin 2 ?
-^^^^
sin
2
cosec
TT ^
,
sin
.
-tanh-i
7;
?/
I
-
? -sin 2 ?
Sm
6J
obvious arithmetical simplification is postponed, so that the
general process may be exhibited and made clear.
Simplifying the arithmetic, we shall finally get
So
far,
%x
^^
,
=
_,
.
x
12smg-
/cos
263. "integrals of form
n m?
7
TJ=- dx,
cos
fcos
w
a? ,
fcos
r-^
ax,
sin^'
J
a?
fsin
n
a?
J cos
Csin n^
px
,
dx,
.
a?
sin a?
J
where p and n are integers, n being positive.
These are generally integrated as follows
,
ax,
:
First put the power factor in the numerator into the
of a series of cosines or sines of multiples of px, say
form
? (rpx).
24 r csin
*
s
v
We
are then to integrate each term, viz. expressions of type
COS
.
sin
.
x
(rpcc)
v
*
i
.
sm
,
dx
cos
.
(ce)
'
v
by a reduction formula, a case of Art. 261
k
J cosce
dx =
7
O cos(/c
2-
k
J since
kx
cosx
fsin
A'cc
,
dx =
2
,
_.
2
since
fsin
l)x
sin
-^
J
:
2)a;
since
7
dx,
2)ce ,
fsin (k
dx,
cosce
J
1 )ce
fsin
l)cc
1
--
fcos(/u
-HI-
1
(Ic
k
viz.
cos a;
cos
J
1
cos (k
ic
dx=
1
(ii),
hiJ
\/v
^
2i}X
-,
dx,
since
which obviously follow from the trigonometrical formulae
cos
Jcx
+ cos (k
2)ce
= 2 cos
etc.
E.I.C.
ce
cos (k
l)ce,
CHAPTER
258
VII.
5
,
-J
fcos 3^ dx.
Ex. n
Consider
i?
I
cos x
J
We
y = e 3ix
have, taking
5
2 cos
5
COS
,
= etc. = 2 cos 15# + 10 cos
3j; = y + (
CtOG
1
_
/
COS
I
._,__.
24 \
COS.!'
..
O COS
lOi2?
r
.
lx)
\&OC
_i
+ 20 cos 3#
9.*-
}
COS &3C\1
|
COS X
COS X
COS#
)'
But
2
14^
-<***-
cos 15#
2 sin 12.r
sin
,
~~ "~
~~ ~"
2 sin
cos 9.r
,
cos x
cos 5 3*'
8#
2sin2.r
,
"~
2^
~"
2sin4jp
2sin6.r
2 sin
2 sin 2.r
~
~~ 30
2
,
Ct$ ===
'
T2 sin
1
,
2sinl2.r
14^7
^^aMJ-iT"
~T2
8 sin
2 sin I0.c
+ "To
6.r
8 sin 4^
~6~
264.
~
2sin4r
6^
2 sin
2sin8A'
_
'
cos 3^;
2 sin lO.r
+ ^10"
-i2
+
1
T"
8 sin
8.^7
~8"
2 sin 1x
~2~
-6#].
Integrals of form
Ccos n px
n
px j
-^=
rfa?,
cos </#
fcos
J
n w^
fsm ^7
dx.
7
dx,
cos
j
J sin g,x
qx
a similar manner to those
qx
These are dealt with in
previous
Csm n ^px
7
r^dx,
sin
J
of the
article.
First expressing the
we reduce
power factor as
the integration in each case to that of a series of
terms of type
cos
,
x
sin (PX)
,
cos
.
sm
and proceed as explained
5
Ex. Integrate
r
/=
/~COS
-
5.2?
/
4#
J cos
-dx,
.
x
(qx)
in Art. 261.
,
dx.
We have, taking y = e6^,
2 5 cos 5 5x
/.
=y + -
= etc. = 2 cos 25# + 1
_
,
irfcos25.^,
dx + 5
/= -r
I
2 4 LJ
cos
4.v
/"cos 15^-
I
J
cos
4.1'
cos
,
1 ;xr
dx + 1
+ 20 cos 5.?
rcoso.r,
-
/
j cos 4.r
rf.r
;
"1
J
.
FURTHER REDUCTION FORMULAE.
259
The reduction formula
Ccospx, _2 sin (p
p q
cos (p
cos
f
q}.v
J cos qx
J
%q)w
<l
,
x
gives
25.r
7
=
dx
cos 4.27
/cos
2 sin 21 A-
cos
f
21
2 sin 2 U-
2 sin
2 sin
,
cos 5^7
dx=
7
r
I
J cos 4^;
1
cos 9#
2 sin bx?
3.#
,
-cos
I
J cos
.
;:
fo
f cos ( ~
~~3~ + J
2sin.r - /"cos(
-
7
/"
1
r
-
~~5
2 sin 3^
2sin3A-
1 Ix
_22 sin 11.27
~TT~
-
*
/*
and
f
13
21
cos 15^
1 5x
1 SA-
J cos4A^+/:
2 sin 2lA'
cos4A/cos
7
ax
13
21
'
7o;
1
eos4r
J
cos 4^
J
'-dx.
Hence
=
j.
j
i
4
2"
5iii
^
^ji^
iu sin
OIAI i_5.7?
2T~
L
13
1
i^
1*'?/*
sin
iu sin
0*2^
o^f
^u sin
3
5
11
i
1
where
H
1
cos
A-
-5 cos x +10 cos 3x
cos
,
f
4,2?
40 cos 3 x - 34 cos x
3-20sin 2 .r
.
7T
sin 2
o
.
sm
2
W
.r
)
/
I
.
sin
2
7r
.
.Sir
.
,
sin 2 -- sin 2 -^-dx
37T
.
sm.r
-^
o
\
,
cos 4.r
j
\
I
/
3-20sin a 5
C0 s.r
sin 2
1
=-
-^-sin
sin 2
2
\
-^"
\
_ F/ 3-20 sin 2 7T
2v2L-\
sin 2 r
3
o
sin 2
.
^-sin
o
2
.^
N
cosec
'-
O
8^
tanh"
1
.
/
r,
2
7T
3?r\
sin.r ~\
3?r
-^-SOsin'-g-Jcowc-g-bmB-'
J cos
sln
j-
Tj
4#
jrro LiJ
1
sin
21.^7
-
sin 13.r
1 3
+1
1
sin HA1
f
/o
-win 3
+ o- sin 5.r - o5 sin 3.r + 10 sin
8)
sn.r
1
'
7r \
on
- 2081
"1
.
.r
eo
"8 tanh " sinTJ" ^
i
,
1
CHAPTER
260
VII.
265. Integrals of form
Csmp x
)
x
q
Ccos p x
-.
dx,
j
xi
l
J
~
,
f
dx,
sux
7
- dx,
afl
yfl
f
p
)sm x
)cos
+ p
p
- dx
x
-
,
(p
19
Therefore, provided q=f=l or
2,
(A)
This formula will be found useful in evaluating certain
%
Jsin^x
p^q
where
and where p and q are either both odd or both even
integers
both
dx, in the case
>2
where p
for in this case the right-hand side vanishes at
;
limits.
We thus
< q>2
266. In the
have
(see Chap. XXVI.)
same way, in the second
,
~~
cos^x
p
case,
supposing q =f=
fcos^- 1 ^ sin x
-
I
or
2,
,
^ sn ic
1
fcos^a;
(p
l)cos^-
2
ic(l
cos 2 #)
,
-Ig,
ij
[(<?
2) cosic
pxsinx]. (B)
FURTHER REDUCTION FORMULAE.
,
267. Again, in the third case,
= Ir#? COSQC P X dx #2+1 cosec^x
q
=
c
17
+ TT
X<1+1
I
cosec 1
*
a?
cot a?
c?a;
-
r #<z+2
nj
cosec J) aH
261
-flL
s cosec^a? cot x
dx
P
-(q
fX
+ l)(q + 2))
= aj5 +1 cosec p+1 [(g-f 2) sin x + px cos x].
I
(C)
And
268.
=
finally, in the fourth case,
r
T)
#2+1
xisecp x dx
sec'' a;
\
=
q+l
i
-^
secp x
q+l
c
I
xv +l SQCP X tan x
dx
q+l]
^
sec^a?
tan x
q+liq+2
1
f
7
2
2
1
q+%]
(D)
be seen that formulae (C) and (D) could have been
derived from (A) and (B) by changing the signs of p and q.
.
It will
269. Integrals of form /_
as the case q
^n =
I
J
r~
C08M JC
=1
=
I
n
J cos a?
in Art. 265,
dx = cos x
I
J
x
=x
sin a;
cos n+1 x
x sec n x dx, included
.
coscc
-\smX-
:
+
cos n+2 #
J
x
-=
cosn+1 a?
Jsin
1
I
J
dx
f
cos n+1 a;
or
be treated thus
may
COS n+1 X
dx
1
n cos n x
j
dx
.
(n
+ l) f\x
'}
cos 2 a; ,
=
dx
cos n+2 x
1
CHAPTER
'262
VII.
Therefore
= nx&mxcosx
n+1
1X T
(w-f l)/n+2
ln + 2
n cos
= nxsinx
changing n to n
or,
n
cosx
n1
-
T
>
n
'
2,
n2
cosx
2)xsinx
_ (n
j
\-n! n
x
j
Now,
I 2 = x sec 2 x
I
dec
= x tan
a;
-flog cos #
and
T
/!
=
j =
x log tan
x sec x dx
f
I
Thus, 7 4 Z6
,
But Z3 Z 5 Z7
,
which
is
270.
...
,
,
(
-j
+ ^\
-
i
fi
I
J
j
log tan
(IT
(
x
+ 9 \ dx.
'
7
,
)
can be readily written down.
ultimately connect with
...
,
/""
not expressible in
Similarly, if
finite
j
log tanf^-f- -J dx,
terms as an indefinite integral.
7B =f -^- dx
J sin x
or
f^
t
we have
_
= r si
sn x
=
.
-.
x
,
f
-:
cos a?) - n+1
hlcosa;
'sm cc
.
(
dx
J
=
.
cosx)/
(
x
f
cos
a?
eosx
-r
1
,
or,
+ 1)/+9=-
changing w to n
7
'
'
wx cos a; + sin x
2,
2
,
Noting that
72 =
and
7j
=
I
r
I
cc
cosec 2 x
c?x
=
x cot x 4- log sin a?
x
x cosec x dx = x log tan ^
sin 2 ic
sm n+2 x
ic
1
(H
(w
1
7dx
sm n +1 x +1Jsm n+1
r
ic
log tan dx,
7
dx
FURTHER REDUCTION FORMULAE
can be successively written down, but
x
f
which connect with log tan ^ dx, cannot be
clear that 7 4 7 6
it is
,
that 7 3
75
,
,
...
263
,
...
I
,
expressed in finite terms as an indefinite integral.
It is also obvious that these formulae (1) and (2) could be
reproduced by taking
p = (n
J.
cosx
2)x since
- ;
cos"- 1 ^
,
and.
+ siux
_(n2)xcosx
sm n - l x
;
~
and rearranging the terms.
respectively, differentiating,
271
J:>
Jr
/
Reduction formula for
.
fx
2n
x'l-^TCT^*''
n being
Let
integral.
R = (l-x' )(I-k x
Then
2
dP =
2
2
),
and put
2^-
2n-4
Hence
P=
T 2n-2
P = (2i
'
^-
71-1
*=/;..
:;,;+2
[Serret, p. 44, Torn,
Cole. Diff. et Integral.]
ii.,
By successive reduction I n may be made to depend upon 7
and I I by putting in succession ?i = 2, 3, 4,...; and 7 7 1
,
,
which are respectively
^2
dx
f
f
x/r^ x^^
2
are
the integrals
and discussed
When n = 1
known
Jv
as
Legendrian Elliptic Integrals,
later.
,
/
2
II
^
/r^2 vr
= x- JR + /_,
1
,
CHAPTER
261
When n = 0,
Jc
2
VII.
I = --arVJK + 2(1+ /j 2 )/^ - 3/_ 2
and putting successively n= 1,
/_2 /__ 3 etc., in terms of I and J1
,
,
272. Obviously, if
2,
etc.,
we
,
can calculate
.
we put x = sin6,
sin 2 "
f
~l-fc
=^
and the same reduction formula
Thus
7n
J
VI -k* sin
applies.
/ n=
and
2
can both be connected linearly with
f
-. ------
dO
f
and
and the
si
JVl-
Jx/l-& 2 sin 2
latter being
we have connected
each of In and 7_ n with
x/l-^sin 2 6> ^6-
and
P=x2n ~ s \/R, we
might have
required by means of
which presents no difficulty.
273. Instead of starting with
proceeded to form the connection
integration
by
parts,
#=
Thus
l
j= and integrate
Multiply by
2i
J,v.2
But the
left side
=xZn ~ 5 jR-(2n-3) (x^-\
FURTHER REDUCTION FORMULAE.
.e.
x->
= n-
;
n
-w
the result already obtained.
274. Reduction formula for
dx
dO
(
J
where a? = sin ^.
Let ^-(l-a; 2 )(l-^ 2 a; 2 ), as before; then
Pllf
Put
Then
a;
V/^
,
)
^
[1-2(1 + A:
2
wlicre
whence we obtain
a*A=
(2n
2)(a+l)(a + A;
2
),
265
CHAPTER
266
P = AI n +BI n _ + CI n
Then
and I n
VII.
l
connected with three integrals of the same form, but
Also, the formula is true whether n is positive
is
lower order.
or negative.
N W
and
is
'
first elliptic
Legendre's
XL and XXXI.),
integral (Chaps.
dx
and
Legendre's third elliptic integral
is
Cv \
*
and these
I
) \
nv
,\
/
and
w
tvJL'
/*
~
1^9
:
I
-
n
T
i
^v
iv
-1-
7^9! I
tfu
"
/-.
-I
&'/)
/J v(l
are, respectively,
Legendre's
first
and second
elliptic
integrals.
These integrals 7 7 15 /_! are therefore all known. Their
We thus
properties will be discussed in the proper place.
have a means of connecting I n with them for any integral
,
value of
n, positive or negative.
The same reduction formula obviously must hold
for
dO
asin
which
is
w
2
6>)
x/l
-& 2 sin?#'
only another form of the same integral.
EXAMPLES.
If
1.
uptq =
A'^a^ +
I
-y-q
where A,
of
.
ft,
obtain
reduction
formulae
for
the
integral
dx of the forms,
,
A', B' are constants
and R, R' are algebraic functions
[MATH. TRIP.,
1896]
FURTHER REDUCTION FORMULAE.
2.
267
Prove that
cos
(a)
=
2n
<j>d<f>
[
~ tan
cos
<
2
-
+
"<
(l.
i)
cos
I
2n ~ 2
<
</<,
[TRINITY, 1891.]
2n+1
f sec
(i)
3.
^ - ^ tan
<
sec
<j>
271
f
2
1
[I.
C. S., 1886.]
[I.
C. S., 1886.]
(a^
Investigate a formula of reduction applicable to
where
and n are positive integers, and complete the integration
?w
m=5,
5.
A r ~)| s^ "" * dh
2n-fl
2
r
,
2
if
^H-
Prove that
2n+l
4.
'1
?i=7.
JOHN'S COLL., 1881.]
[ST.
If <j>(n)= a
3
1
P rove that *
r^^yi
(a
-t- u:
j
W = I^n ^ ^(
~
-i
^
!)
[B. P.]
6.
Investigate formulae of reduction for
(a)
*
dx.
(,;)
j
J(a
W
f
2
+x2 )t
77^77 <k-
and obtain the value of
7.
8
l
3
-
1)"
(ic
[COLLEGES, CAMB.]
Investigate a formula of reduction for
2n+l
f
x'
j
J-J^
I
and by means of
1
1
this integral
1.3
1
show that
1.3.5
..
2TT76
1
3. 5. 7. ..(271
+
+ 1)'
also the series
1
1
'2n
2^ + 8
2. 4. 6. ..27i
~
Sum
1
<
+
1
'
1
2
2n + 3
T
1.3
1
'
2
.
4
2n + 5
T
1.3.5
1
'
2
.
4 6
.
2n
+7
ad
in
[MATH. TRIP.,
'
1897.]
CHAPTER
268
Find a reduction formula
8.
VII,
for
n
x dx
Show
that
2. 4. 6. ..2?i
where a lt a 2
,
...
Prove that
9.
X+
X + "' + 1.8...(2n-l)
l;.lj^;
2
2.
2. 4. ..27i
are the binomial coefficients.
if
un =
I
sii\
2n
[ST.
"I
JOHN'S, 1886.]
xdx,
Jo
un
then
and deduce
f
f
J
271-1
1
/I
sm 2n a:^- -^ W+1M-+-T
2
ITI
.
n+
(2-l)(2-3)
-^r+---
TT7
/
7i
Ti(w-J)
(TI-
l)(n-
2)
(2n-l)(2ro-3)...3
[MATH. TRIP.,
10.
1
f
X4m+i
}
1 1
.
l^-^fi _l-3.5...(2m-l)
Vr+2
2. 4. 6.
Find a reduction formula
6
fax
where n
is
TT
'
..2m
4
~
2. 4. 6.. .2m
3. 5. 7...
2m +1
1
'
2'
for
COS X CLX)
a positive integer, and evaluate
fax
12.
1878.]
Prove that
[OXFORD, 1889.]
Find formulae of reduction
for
n
(1)
(2)
Deduce from the
latter a
\x sinxdx,
I
e
n*
n
sin #
6&c.
formula of reduction for
n
I
cos ax sin x dx.
[COLLEGES
7, 1890.]
FURTHER REDUCTION FORMULAE.
Show
13.
(in
269
that
+ n) (m + n -
2) f sin
m9
dd
cos"
= (m-l)smm+1 0cosn - 0-(?i-l)sin m 1
1
0cos w+1 <9
J[TRIN. COLL. CAMB., 1889.]
Show
14.
2
m
I
cos
that
mx cos m x dx
m(m -
sin 2x
where
15.
m is
sin
1)
1.2
2
4$
sin
a positive integer.
Show
2mx
2m
4
[COLLEGES
a, 1885.]
that
-4). ..4. 2
(4m-l)(4m-3)...
being a positive integer.
16.
[OXFORD, 1889.]
r
Evaluate the integral
e~
nx
m
cos xdx,
being a positive integer.
17.
Prove that
[COLL., 1886.]
if
Im n=
I
,
cos"* a; sin
(m + n) Imi n =
cos
nx dx,
m o? cos nx
+ mlm _
li
_!
and
[BERTRAND.]
==
um >Mn =
18. If
4
I
\
cosmic sin
TO
Jc
um
prove that
Hence
find the value,
=
when
h
m is
a positive integer, of
r cos m # sin 2m#
I
Jo
19.
If
prove that
Imtn =\cos m xcosnxdx,
/,,=
_^J^_
rfjr.
[7, 1887.]
CHAPTER
270
VII.
and show that
(-
1)1
J7T
C08 XCOSrKcdx =
n
-^^
t
[BERTRAND.]
being a positive integer.
21
.
If
m and n be positive integers, and if m + n be
even, prove that
2
[COLLEGES, 1882.]
jr
22. If
Icosm zcos7i;ecfo be denoted by /(m,
?i),
show that
o
[OXFORD, 1890.
23.
Prove that
if
?fc
be a positive integer, greater than unity,
1
p
24. If u m n
,
= U^cosec^a:^, prove
?
that
{m sin x + (n-
2) a; cos
a;}
[MATH. TRIP.,
prove that
</>(7i
+ 2) -
+
(27i
l)<fr(n
Show
then
1)
+ w 2 <#(w) = 0.
[R. P., ST. JOHN'S COLL., 1881.]
that
Prove that
C/ n
if
2 Z7n+1
27.
+
1S9C.]
x n eFdx
-/f ===>
-a;
1
26.
1889i
if
+
=f
I7n (2ro
Jovl
-
1)
- 2w
3
<#(m)= f^(a: +
/"_!
3aaj
= 0.
[COLLEGES
/3,
1887.]
+ c)-*dx,
+
(2m-l)^>(m) + 3a(2m-3)(/>(7/i-2) (2m-4)cc/)(m-3)
= 2a:
l
-2
3
(x
+ 3a + c)*.
[TRINITY, 1886.]
28. If
prove that
2
2
=
(w + w )tt Ml
m(m -
l)wm _2
+ w. [OXFORD I. P., 1900.]
FURTHER REDUCTION FORMULAE.
.
,
29.
Prove that
f
IT m =
if
271
sin m xdx
p
,
2
Jo
(l-Fsin ;^
2
2
I
m
(m-l)k
-(m-2)(l+k )Im _2 + (m - 3)/w_ = 0.
then
4
[TRINITY, 1890.]
30.
Obtain a reduction formula for the integral
In =
2
(a cos 6
+ 2h sin
+ b sin 2 6)~ n dO
6 cos
I
in the
form
n+ 2-
(2n+
+ 6)/w+ + 2w/n
l)(a
i
<*/
1
'
d0 2
2?i
[MATH. TRIP.,
2
31.
e
Show
that
f.W
i
S-8e +3e
,xo
.
4
1898.]
TT
s
T^J
being less than unity.
[ST.
JOHN'S COLL., 1885.]
InS rEs?&.
sma;
30. i f
J
-
prove that
(w-
1
- /H _2
) (/
)
= 2 sin (w -
1
) x,
and hence that
fJ
3
'
:
.
^
si
=
sma:
A'
If
if
2
be odd,
= a + bx n + cx z>l
and
/.,
,
m
p = \x Xr Jx,
prove the existence of reduction formulae of the nature of
and
find the values of the fifteen constants.
34.
Show
that
(a
can be reduced to the integration of
f...
Ja + ftB
a
+ ca5*
and
f
Ja +
and integrate these expressions
;
ff+ o ,
6
2
p being
4
(6>0, 8t>4ae).
integral,
[BKRTRAND.]
CHAPTER
272
35.
Show
that, if
VII.
dX
?lEE
'
'
x)
(10
>
xm+l
1
i,
n-i
(logz)"[OXFORD,
V/
36.
Find reduction formulae
I
(a)
I.
P., 1889.]
for
tanh w
sm n x
(ft)'
c
w/
J (a
37. If
+ be* + ee-*"'
r, where X=ax* + 2
Jm =
amlm + (2m -
1)
blm_ l
+ (m -
show that
1 ) clm _ 2
= xm ~ l*JX.
[p,
1891. ]
38. Establish a reduction formula for
where
Z=
4
aa?
+ te 2 + c,
in the form
showing that
X=
-3)
39.
Show
m
n
that, if
=
\
s\n m 6cosnO dO,
Jm n =
sin m 6 sin
si
nO dO,
Jo
Jo
,
M
then
77 f
where
m is
a positive integer
be used to find the values of
;
and point out how these results can
and Jm>n
[C. S., 1896.]
/,. n
.
FURTHER REDUCTION FORMULAE.
40. If
T be
273
a function of x such that
'
= A + 3BT+3CT 2 +
prove that
d
I
dT\
3
(n-l)A
(n-2)C
T n _z
(2n-5)D
2Tn_ s
,
Apply the
result to investigate a reduction formula for
Cdx_
]
By
Tn
a consideration of the case where (7=0,
D=Q
(or in
any
other way), obtain a reduction formula for
f_
}(a
41.
dx
+ 2bx + cx*) n
'
[I.
where n
is
a positive integer.
[COLLEGES
T-ttf/T)
/rV'r
rtH~*"I/7*
"[^t/
JjJ\Jj
'^/J
.//
fba
show that
2n,,,
= (2n -
l)(a
+ b) un
__ }
wise),
By applying the substitution
prove that the definite integral
a, 1890.]
It'*',
-2 (n
1
)
abu n _ z
[OXFORD
43.
C. S., 1897.]
Prove that
.
I.
x = acos> 2 @ + bsin z 6
PUB., 1912.]
(or other-
x n dx
b
( a *J(x -a)(b- x)
a rational integral function of a
evaluate it when n = 3.
is
T~
I
b
when n
cos 2nx
*
where n
is
^
a positive integer and
Consider the case
when
s
an integer
and
P., 1913.]
I.
P., 1915.]
7
and
,
'
W'2/|
-^
the lower limit
is
negative.
[OxF.
K.I.C.
;
I.
efo,
v-g
obtain a formula of reduction connecting u n
Hence, or otherwise, evaluate
w
*
is
[Oxr.
fcos2rw;
un =
44. If
and
s
CHAPTER
274
45.
multiplying the inequality
By
and by
VII.
sin
2
1^2 sin x - sin 2 x by
'% and integrating between
and
JTT,
sin 2w
~1
aj
show that
2. 4. ..2?i.
+ 3)(2?i+l) ark*
>
4?i + 4
2J
1.3...(2w-l)
r(4yfc
'
\
[MATH. TRIP. L,
The expression
-
46.
where
l>a>0,
efficient
of
an is
-
.-(1 -
expanded in ascending powers of
Prove that
denoted by un
is
and the
a,
co-
.
Und
-
[MATH. TRIP.
J
--^-.--l)x dx,
= fsin(2?i,
.,_
TP
47. If
1915.]
sn
I
vn
sm
J
I.,
1916.]
its
value
I.,
1914.]
~sm*nx
f si
=
-:
J s
prove the reduction formulae
n (sn+l - sn )
= sin 2nx,
- vn = sn+l
v,
il+l
and show that if vn be taken between the
is fynr, where n is an integer.
48. If
A 2 = cos 2 ^/a 2 + sin 2
^
2
P
and that
f
^=
2
1
3/x
lated
If
UH =
\
by means
AU
n
sin
iraft{3(a*
m
2a 2 6 2 }/16,
e^
(1-eV)'
x(a + bcosx)~
n
dx,
3(1
-e 2 )^
prove that
[6,1883.]
Un
can be calcu-
of a reduction formula of the nature
+ B Un _^ +
CUn _s = sin m+ X
and determine the constants A, B,
oO.
+^ )+
dp
1
JTT,
[MATH. TRIP.
JO
4
2
Jo
49.
-
;
and
find
,
Tt_
and prove that
limits
Prove that
r
dx
=
2
J
c
(a
where A denotes 4 (a 2 -
-
c2
)
2cx
+ b cos x)~n+
^
;
C.
2n\
=
^ ^
+
?i!?i!
+ or)
and
,
is
supposed positive.
[TRIN., 1887.]
CHAPTER
FORM
275.
(F(x,jR)dx,
The integration
VIII.
WHERE R
QUADRATIC.
IS
of expressions of the type
dx
can be effected in
X and
which
all cases for
F
are rational
integral algebraic expressions of degree not greater than the
second.
There are four Cases
:
III.
X and F, both linear.^
X quadratic, F linear./
X linear, F quadratic.
IV.
X quadratic,
I.
II.
F
The general substitution
tion in all cases.
__
= ^'
PutA" = -.
y
Put -^ = y or y 2
X
quadratic.
Y=
-
yory
2
.
will effect the integra-
But the simpler substitutions noted,
in Cases I. and II.
viz.
=
X
and
Case
IV., in
- in Case III., are better.
y
which we employ the substitution
is
or f.
much more troublesome, but includes
the
\Vo shall, in all cases, assume the radical
276.
CASE
Let/=f
I.
X linear, Y linear.
*.
275
previous ones.
F to be real.
V
:
CHAPTER
276
VIII.
V Y= Jpx + q
Putting
dx
ax + b = -(y*-q) + b.
and
Thus I becomes
2
-
~
1
Jay
2
which being of the standard form
2f
-
dy
.
1-9-^9
2
\2
a)y
is
viz.
immediately integrable,
X
a\
according as
-^
is
Cv
,.
where X 2
>
:
-
= bp~aq-,
a
--,
or
a\
X
positive or negative,
"
Ja(aq
Jaq -bp
bp)
2
*Ja(bp
=
or
aq)
2
.
'
r
P
with other forms, the real one to be chosen in each
case.
Another Method.
277.
form shows how the factors of the integrand are
involved in the result of integration, and indicates that the
The
last
/V)O*
substitution
stitution
If
we
|
-
sy
-4
= y2
mentioned above as the general sub-
would have led directly
elect to proceed in this
to this result.
way,
viz.
putting
2=
*
we have
\px+q
"
dx
ax + b/
2
y
dy7
y
.
(ax-\-b)(px + q)
bpaq
2
QJ.
aqbp
dy
g.
y'
y
2
,
THE FORM
by
vr
Now #
When
bp
-
2
p
aq
q
%2
-
is
2
and px-\-q = (bp-aq)y
^~' or
9
ay*p
p-ay*
ay
positive
dx
r
}
277
2
_
f
>/6p aq
(ax
J
dy
\/p
-a
dy
sin~
or other forms.
When
bp
-
aq
is
negative,
=
___!_ =
f
-bp
J
sjaq
dy
2
*Jay
-p
dy
-
a
"z/
9
or
a
sinh-i
/_
V
or other forms, the real
form to be chosen in each
278.
Illustrative Examples.
Ex.
Integrate
1.
/=
(
J
Also
Again,
if
we put
and
\/2
cte^
case.
CHAPTER
278
which
is
VIII.
the same as before, but exhibits the result as a function of both
the factors of the integrand.
Ex.
2.
I
Integrate
Let
~{
N/2~-^=?y
Tl'^^"
(
= - 2ofc/
.*.
;
= log + !- = log <N2-#+l
.-==
,,
?/
An
-
or other forms.
-
279.
;
,
Extension.
The same
viz.
substitution,
integration of
jY = y,
will
suffice
for
the
f^dx
jvr
}
where
JT,
Fare both
algebraic function of
For
and
if
if
linear
Y=px+q = y*,
^rfi-42^
and
then
is
(j>(x)
x.
z=
-
any
rational integral
and
pdx = 2ydy;
be expanded in descending powers of
2
^/
and
then divided by ay 2 + (frp
of y, we
of form
n
ag) till the remainder is independent
have to integrate with regard to y an expression
being the degree of
integrable, after
</>(#)
and each term
in x'
}
which operation y
is
at once
is
to be written
back as
280. Ex. Integrate
Writing <x/+2=y, we have
^|=^ and x =f~^
5+
by
so that
division.
THE FORM X-
l
279
Y~bdx.
Thus
if
V+^
-
=/ (V
- 5+
--L^
the logarithmic form be preferred.
Forms reducible
281.
to Case
I.
The student should note the variety
the case considered, viz.
dO
sin
.
J
put cos
\/cosec6d&
---
/
C
,
"r
M
a
/
"^=>
cos t}
+q
+ o)\/pcos
/"
,.
(2)
by a proper sub-
For example,
stitution.
J (a
of forms reducible to
X linear, Y linear,
_
0*coir**.
i.e.
^
.
putcot^=.r,
^N/pcos^ + gsin^
cos
6>=.i;,
ff^ootrT1
i.e.
OdO
/n
+ qsm z &'
9
L cos + J/sin 6 dO
2
^put cot ^
^
.r, i.c,
separate into two
cotr'W.i
1
r
I
J
c^i?
-
integrals,
put log * =
dx
J
/.
A-
i.e.
= ey
.
.'log(^)x/lo g (c.^)
17)1
1
put # = v-.
-
6/^
aa,-
etc.
282.
Ex. Integrate
s
1+2 cos 4
I
(1
I
J (sec
-+
sec
2
4
6>
sin
+ 12 cos 2 6)\/l +3cos 2
2 tan ^isec 2
/.
=
.
2
put cot ^ = o; in one and
tan 2 #=,y in the other.
acos 2
1>ut
(6)
^
2
0+12\/sec
6'
cos*
OdB = d>j
;
1
tan
2
+3
B8ec-0d& = -
-
C
2j
+ l) + 2
ay.
"
+ i3 x/?/ + 4
(y
(
2
'
.
CHAPTER
280
Now
\y + = z;
put
/.
4;
VIII.
y = z*
dy =
4,
146
z*
--15s +
146
_
_
o
CASE
283.
Let
J
, + 4)
.
(fatn
146
^
tf
JT auadratic,
.
(ax
2
tan _
+
5
II.
7=1
u
+ ox + c) vpx +g
F linear.
dx, M and
JV
being constants.
The terms Mx-+-N now existent in the numerator do not
introduce any difficulty and make the result more general.
The same substitution being made, viz. /F=y, we put
ax 2 -\-bx-\-c reduces to the form Ay* -\- By 2 + C
Mx +
reduces to the form M'y^+N'
N
and
Thus / takes the form -
yTo ^ dy.
p)Ay*+By* + C
Now
of the
.
/n
Ay* + By
2
284.
can be thrown into partial fractions
-{-C
is
integrable
by
it is
,.
.
the integration
(f>(x)dx
f
I
,
J (ax + bx
2
x=
i.e.
-|-
c) *Jpx
and
P
o,,ox
+
155
evident that the same substitution will effect
+q
rational integral algebraic function of
ax*
earlier rules (Arts.
Extension.
Further,
.
.
form
and each portion
and 156).
,,
.
+c
reduces to the form
,
x.
where
,.
.
<f>(x)
.
is
any
For when px -+- q = y*
t
THE FORM X~
l
281
Y-bdx.
where n is the degree of (x) in x and therefore, by division
and our rules for partial fractions the integrand may be
;
2+
expressed as
,2n-4
and each term
is
x,
4.
_M^_.
integrable.
/T
Again,
where
P
lr-* 4.
,
and ^ are any rational integral algebraic functions of
be seen to be integrable by the same substitution,
becomes
<
may now
for it
and the new integrand can be expressed by partial fraction
methods (Art. 152, etc.) in the form
R
and integrals of the expressions of the first four terms can be
obtained by the rules given before, and the integral of the
last by aid of the reduction formula established in Art. 238.
285
.
Ex.
1.
Integrate
/=
(.
} (^
Putting *Jx + 1 =y, we have
-d
f+
_
f
+ ar + 3)^+1
.
x
=2
/
dx.
^/j
^/P^^-J^+TV^+T)*
.
u ,->2Z+l + -i tan-'
\/3
Put
.
8
\/3
J.v^T=y,
N/3
.:
-j=
...
^= -4_tan-W3^H
2
\/3
\/3
*=*2dy
and
^
CHAPTER
282
7_ 9
+.V
a 2
VIII.
-5(l+7/2)-37.
)
""+^-Td+yj-ao
4
r
?/
-3?/
2
y
-
-41
J
286.
Forms reducible
to Case
II.
The student should again note the variety of forms which
may be brought under the foregoing rule by suitable substitutions and integrated.
Thus
/ 1 \
(1)
[
I
'
pv
-
*=r.
+ b sin
cos 6 + c sin 2 0) Vp cos
a cos 2 6/
+ g sin
___
+ B\/cos&
+ b sin # cos + c sin 2
vWsin
/"
J
vain
2
(a cos
Q^#
6^
6>
V
(9
_
+ g sin "0
cos
sejDarate
'
into
integrals.
^^cot- 1 ^
two
Use
in the
one and
in the other,
similarly.
/
J
/
5)
.iVn^ +
A + /A
cos
2^;
.
+
i/
sin 2.r
vjt?
287.
CASE
The proper
Let
A"
III.
cos
x + q sin .r
dx
lAjwnx + B*Jua&x _
J a + 6co
from
linear,
substitution
is
Y quadratic.
now
dx
r
T
X =-
dj'
Putting ax-\-b = -,
we
have,
by logarithmic
u
dx
ax-{-b~
1 dy
a y
differentiation,
,
THE FORM X~
Y-ldx.
l
283
and
offer
r
Hence the
integral has been reduced to the
known form
dy
+ 2'By+C'
-if
which has been discussed in Art.
80.
Ex. Integrate
Let A
1
1
=-. and therefore
^-
Hence
288.
Forms reducible
to Case III.
Again we note the varieties of integrals which may be
reduced to the present form by a suitable transformation,
for instance:
(1)
/( a cos
T/3
(2) (
J (a cos
'
8 + b sin 0) \lp cos 8 + q sin
2
sin
/'
cos
OdO
cos
+ r sin 2
put cot-$=#.
(*>/from
/i
etc.
(2)
CHAPTER
284
VIII,
289. Remarks.
now appear
It will
that any integration of the form
(x)
(ax + b)Jpx
can be
effected,
function of
dx
2
-f qx
+r
being any rational integral algebraic
</>(#)
For by
in the
J division we can express
ax + b
x.
',
,
.
form
Ajxr-i+A^+A^-jt
...
+^ n_ z+^ n +1
M-v,
u>.>i/-|-0
where w
is
is
the degree of 0(x) in x,
M
the remainder independent of x.
the quotient, and
We have thus reduced the process to the integration of a
number
of terms of the class
Ex m dx
and one
of the class
r
The
latter has
Mdx
+
(ax
b)Jpx* + qx + r
been discussed in Art. 287, and integrals
may be obtained by the reduction formula
of the former class
of Art. 240, viz.
l) rlm _ 2
,
ql^ + mplm
is,
2m
mp
r
1m
Ex. Integrate
division
C
//
~J
(
2
=tf-xr+x--\--
^
x+l
XA
p
I
J
.
By
J
m~
x m dx
,.
I m standing for
that
+
-
x+l
x
W^+i~^
;
2
,
,
THE FORM X-iY-kdx.
285
Then
-
--
1
-
/2 =-
-sinh- 1 ^,
by the reduction formula (w = 2),
.r
T
/a-
/r4 =
and
--
.
3 r
--- 3
= a?*+l
-
-^
1
31.
,--^:
^.-W
-4
4/2
for the last part of the integral, viz. 2
~~
/
J
dx _
dy
;
/
(.^+l)\/^
.,
2
+l
>
put
#+ 1 = -
;
y
f
dx
f
'
J
-- + 1
(
1
'
Thus
T
*
*
-~
'
dx
- - sinh- 1 .? - V2 sinh" 1
290.
-
Extension.
Further, we are
of the form
now
in a position to effect
any integration
dx
where </>(x), x( x ) are rational integral algebraic functions of
and all the factors of -%(x) are real and linear.
For putting 140 to 146,
^
x,
into partial fractions, as described in Arts.
CHAPTER
286
VIII.
Hence the integration can be performed when we can
integrate
x m dx
dx
f
'
dx
f
j% b*Jx +x+r'
z
The
first species of integral
the formula of Art. 240.
J
(xc)
n
2
s/px'
+qx+r
reduced as already explained
is
by
The second species was discussed in Art. 287.
The third species can be reduced as explained
in Art. 244,
or obtained from
dx
by n
1 differentiations
with regard to
c,
as will be explained
later.
EXAMPLES.
Integrate the following expressions
4.
:
Prove that
f_^_ = 8itan_ j
1
/<* + c)Va?
according as
5.
\
c is
positive
Integrate
cos
^ c
Vc
+
(cos
(a cos
2
v -c
v.i-
- \/ - c
\ -c
+
or negative.
sin 0)
/*in~0
2
>/r
or
+ 3 sin
+ Jco* 6
- 6 2 sin 2 0)
-===.
V2 cos
[C. S., 1904.]
(a
^/cos (
> > 0).
ft
^
tan 20
7.
Integrate
9.
Integrate
8.
Integrate
1
[ST. JOHN'S, 1890.]
THE FORM X-
l
Y-$dx.
1
civ
= cosh-
1
(2x
- 3) +
cosh[ST. JOHN'S, 1883.]
12.
Integrate
13.
Integrate
14.
Integrate
[a,
,
1887.]
1890.]
[COLL., 1892.]
15.
Integrate
dx
x*dx
dx
[MATH. TRIP., PT.
291.
The
CASE IV.
integral
is
A" quadratic,
now
of the
I.,
1920.]
Y quadratic.
form
dx,
where a linear factor has been inserted
in Case II., for the same reason.
Before beginning the integration
preliminary remarks
292. (1).
cases, a
We
293.
(2).
For
(a) if
the following
of the subject of integration
is
for
shall consider later, as in previous
numerator which
function of
we make
:
The numerator
the present linear.
in the numerator, as
is
any
rational
integral algebraic
x, viz. <j>(x).
The
cases
612
^o
1cl
and
6 22
= a 2c 2
rational us regards x,
considered.
are excluded.
+ 26^ + c2
becomes
and such integrations have been already
& 22==a 2 c 2> t ne expression *Jafl 2
CHAPTER
288
would be resolvable into
VIII.
partial fractions either of the
form
and the forms
of integral resulting have already been considered in Articles 287 to 290.
294.
be regarded as positive without loss of
any case in which this is not so, we may
the signs of the factor a 1x 2 + 26 l x+c 1 and finally
(3).
may
!
generality, for in
change all
change back the sign of the result when the integration has
been effected.
,
Hence we assume:
positive.
(4).
>a2c2
may
b^
We shall assume the subject of integration real. If
the expression a 2x 2 +26 2 x+c 2 has real factors, and
be written
295.
622
a^
(1) a^ positive, (2)
,
= a z (x
\)(x
where X 1 <X 2
X 2 ), say,
In order that the radical should be
real,
confine both the limits of integration to
between
either
or between
If 6 2
2
<a c
2 2
,
and Xi
and -f oo
30
or between X 2
X x and X 2
1
,
this case
may
is
negative.
+ 2& +c
2
the condition a 2 positive is all that
radical may be real for all values of
gration in
a^ is positive,
when a2
2
therefore
...
.
,
,
the factors of a 2
we must
lie
when
v
)
.
is
x.
are unreal, and
2
necessary that the
The
limits of inte-
therefore be any real quantities
whatever.
296.
(5).
REDUCTION TO A CANONICAL FORM.
LEMMA. Any three expressions of the forms
Mx+N,
a 1x 2 +26 1x + c 1
,
a& 2 + 2b&+c 2
can be in general simultaneously thrown into the forms
where'
,
respectively.
2
are
linear expressions
of
forms x
xv x
REDUCTION TO A CANONICAL FORM.
289
In order to do this it is necessary to determine the eight
quantities (x lt x 2 ), (P, Q\ (p lf q^, (p 2 g 2 ); and we have eight
linear equations to find them, viz.
,
+?i
x
i
Pi
+Vix2
Pi
= a i>
= -A,
=
+ ?2
l\
P +Q =
a z>
^>
It follows that
1,
1,
<*!
r
*l,j
r
*l>
h
u
\
X*
X* 2 ,
C,
1,
=
a.
1,
= 0.
and
X*
X*,
X
which
or Y are
assume x l not equal
Also, as the consideration of the cases in
perfect squares
to
xz
is
to be rejected,
we may
.
The determinants give at once on
division
by x z
x lt
QA
= J
) +c
.(i)
i.e.
whence x 1 and x 2 are determined, being the roots of
where A, B,
C
are the co-factors of
a, b, c,
in
= aA+bB+cC.
av
That
is,
p
is
given by
b l}
ct
-P> P
!>
2
= 0.
a2
The remaining
b2
,
,
six quantities are
c2
found at once by solving
the equations
-bv
or
or
-N,
which give
(Pi* 9i),
B.I.C,
(P*ft)
T
(P,
Q) respectively.
CHAPTER
290
VIII.
be remarked that the equations (1)
reproduced immediately from the functions
o 1 a; a +26 1 a;+c 1
a^+2b^+c2
the
rule
by
simple
It
297.
may
may
be
,
:
"For x 2 write
for 2# write (a^H-a?,)
x^;
and leave the
coefficients unaltered."
-~
= j~
^2
^2
In the case when
Now
equation (2) has one root
therefore the general theorem of our Lemma
must receive separate consideration.
infinite.
fails,
and
the case
298.
(6).
In this case,
viz.
&
= 7^,
0-
the three expressions are:
Mx+ N,
and putting
x-\
ai
=g
they are
and therefore are simultaneously reducible
i.e.
the same as
299.
(7).
if
When
we put
-1
de>
2
=1
= ^ = -^,
OQ
Ca
to the forms
in the former transformation.
the two quadratic functions arc
the same function, and the integral takes the form
I=
f
Mx+N
\
Ia
--
and a reduction formula may be used
f
M'x+N'
to connect with
dx;
which has been considered before (Art. 85). Or the integral
/ may be deduced from the latter integral by differentiation
with regard to c.
300.
Ex.
1.
ILLUSTRATIVE EXAMPLES.
In the case
_
- 46.r+ 103) Vl Lr2 - 70^:+ 155
23(^ +^2 + 103 = 0,
35 (xl 4- x 2 ) + 1 55 =
)
;
and therefore
#j
= l, #8 = 5
(the order
is
\
whence
J
immaterial).
'
x
l
REDUCTION TO A CANONICAL FORM.
p + = 7,\
=
Also
q--
= 11,1
= 35,
or
p,= 5,1
or
or
23,}
Pi=
giving
3
,\
<? 2
And
Ex.
the transformed result
2.
=3,)
=11, J
f
or
is
=
6,
291
P=
=
j
1,1
2.
J
therefore
In the case
3^2 - 3(Xr + 7
we have
= 5(#-5) 2 + 6.
Putting .r-5 =
301.
present
,
the transformed result
Taking the general case then, we suppose
a^a.
d = xx
where
~ ^i = ^i
^2
so that
Also,
is
we
V
y = -g,
lt
xz
i=xx
an(i
^^i
2
=
are to use the transformation
.
i-e.
=
and
dy_
(
(
Now
T 2~ X
r l)\
X
[
felC2
y
and
-':;)
for the
CHAPTER
292
a lt a 2
bi,
when
VIII.
sO
b2
expressed in terms of the original coefficients.
The points on the graph
of
where the ordinate has a maximum or a minimum
i.e.
the " turning-points," are given
|=
0,
66 = 0,
by
i.e.
value,
by
and are therefore at ^=0 and 2 = 0, *- e a^ x = x l and # = # 2
and the values of the corresponding ordinates, viz. y 1 and y 2
-
are plainly
yi
= 2*
and
V2
We
shall suppose
2
.
the graph such that x = x 1 gives the
maximum, and that 052 >a?1
ordinate and x = x 2 the
Again, clearly y = a
the y-axis where y
is
an asymptote, and the curve cuts
= -.
It cuts the aj-axis
if 6 2 2
Q
if 6 2 2
in unreal points
It cuts the
.
i
in real points P,
i.e.
where
>a c
<ac
2
2,
2 2
.
asymptote where
a 1a; 2 -|-26 1ic-|-c 1 ""a^
*.e.
i.e.
a 1b 2
where
a?
=
,
Pi
fc
minimum
=^
;
lajCo
- -J
--a ^~ = ol^B
TT
2a l^
b2
2 Ci
a 2b l
20
at a point J? at a finite distance from the y-axis, unless
a 2 b L = 0, a case for the present excluded.
There are three cases with which we are concerned, i.e. in
which some portion of the graph lies on the upper side of the
aJ-axis, otherwise ^/Y would be unreal.
REDUCTION TO A CANONICAL FORM.
y
(1)
293
M
Fig. 17.
(2)
(3)
+ 00
Fig. 19.
302. These are typical cases.
It will be seen that
we have
2 >ic 1 and the turning-points both on the right-hand
side of the 2/-axis, i.e. x l and x 2 both .positive.
The student
taken
will
cc
have no
difficulty in
making the necessary
modifications
any particular case in which the numerical values of the
several constants are given.
It is to be observed that p t and
for
q l are necessarily
both positive, for a x has been taken positive,
and the roots of
a 1 o; 2 +26 1 x-fc 1 =0,
i.e.
CHAPTER
294
are imaginary
also that
;
p 2 and
VIII.
q 2 cannot both be negative, for
to be real.
Moreover, p 2 js the positive one,
as
the maximum ordinate.
being regarded
unreal values of X*jY (i.e. X^-Jy) are to be excluded,
is to be regarded as positive, it will be clear that we
for Pz/pi
As
and
is
X
shall only be concerned
which y
is
with those portions of these graphs in
and the
positive,
limits of integration of
'Mx+N
XjY
must be such as
as
make
this true.
In Fig. 17, y
limits
In
to lie within the boundaries of such regions
Fig.
is
may
y
18,
oo to o?= -f-oo and the
positive from x=
therefore be any real quantities whatever.
,
is
OQ = X 2 );
x
(
x=OP
negative between
therefore the limits
(
=X
and
1)
may be anything
or between X 2 and -fco, both
between -co and X^
limits to lie in the same region.
In Fig. 19, y is only positive between x =
and x = \ 2 and
must
both
lie
between these values of x.
the limits
\
303.
We
,
The Integration after Preparation.
now in a position to proceed with the integration of
are
j m tM*+%
wKich we shall, to begin with, suppose
formed as explained to the form
have been trans-
to
/==
Y
Putting -y
also,
y,WQ have dx =
X
~~
1
2
^y
^j^
',
&!__*
Pi
y-2-y=f-
.,
'
A
*K
2
.
9-2
JX_
1
/
K
the
\
~~
~^~
ITT A/
^y
I^T^
'
17
I
signs
of
the
ambiguities being
governed by the signs
^ ^2 anc^
(?i
INTEGRATION AFTER THE PREPARATION.
+
both
i.e.
ifcc 1 <ic 2 <x,
vc
first
,
second
typical case
i
x^<x <x 2
if
a;
<x
1 <ic 9
,
.
we take x t <x z <x and both
x: =
and note that x z
ve
-f-
ve
both
As the
ve
295
~
^
if
signs positive,
expressed in terms of the
original coefficients.
Substituting in the integral
2AT
dy
COS -,
if
y x be
/F_ Ti
ve
-f
;
or,
1
^
V^^-x^L v^
>
,
if
I"--
yx be negative.
And the suitable
modification
results as to signs
of radicals
numerical case which
304.
may
is
to be
and
present
made
in these general
form in each
reality of
itself.
THE INTEGRATION WITHOUT A PRELIMINARY TRANS-
FORMATION.
be preferred to pass directly to the integration without the preliminary transformation, we
proceed as follows
If it
:
/3
r
J
Let
Then
=
--=
CHAPTER Vllt
296
where J
is
the Jacobian of the
two quadratic expressions
a 1 x z -{-2b 1 x-{-c l)
J=
=
2a 1 o?+26 1
,
\
-- =
Hence
^2
'
Let x lt x 2 be the roots of the equation J = 0, and y lt y^ the
corresponding values of y. Then the points (x lt yj, (x 2 y.2 )
,
"
are the " turning-points
of y, i.e. the points of maximum or
minimum ordinates of the graph. Let yl be the minimum
maximum.
The equation giving x lt x z
ordinate, y2 the
where A, B,
C
i.e.
J = 0,
is
obviously
Bx+A = 0,
Cx 2
i.e.
,
are the cofactors of
a,
6,
standard
c in the
determinant
2>
2>
and we may write J= +4(7(^
=
Again, any straight line y p
in
x^
2
when
or (x
/uL
= yi
or
'
x 2 ).
x^)(x
will cut the
two points which are coincident
Hence,
(x
C2
in the
graph
two
01
cases
jm
=
y1
y2 the numerator must contain
z 2 ) 2 as a factor, and the equation
must have
in these cases equal roots.
Hence, the necessary values of //, viz. y l
roots of the quadratic
and
/2
,
are the
INTEGRATION WITHOUT THE PREPARATION.
297
and
[a
tf
supposed positive, y 1
<,
2/2
>a
y intermediate between
,
'
i
yl and y2 Figs. 17 and 18].
,
Thus
VS/
2/i>
the signs of the right-hand sides
being both positive,
the first positive, the second negative,
both negative,
xl < #2 < x
2
l
we have
Mx+N
2X* ,
r^r~T- dy
= +^ cosh-
'
1
A/ -2.
V
2/1
=
+ J sinh- y _- y
1
-#
+6? cos- 1 A/
^
,
if
yi
be positive,
2/2
^ sin-
1
^,
if
^
;
< x <x
if x <x <x
taking x < x < x
x1
2
l
Substituting in the original integral, and
as the standard case,
if
if
be negative,
2
;
.
CHAPTER
298
where
F
and G are constants,
VTIL
viz.
has been seen above that C(x 2 x^ = *fB 2
The suitable modification is to be made in these general
results as to sign of radicals and reality of form in each
numerical case which may present itself.
for
it
305. Comparison of the Processes. Construction of Examples.
Considerable arithmetical simplification accrues from the
treatment shown in Art. 303, but of course at the cost of the
reduction to the canonical form.
initial
The method there shown
method of construction
indicates a
of such examples, for the values of p lt q l} p 2 q 2 P, Q, xv x 2
are there all at choice, due care being taken that p lt ql are
,
,
both taken positive, and that p^, 72 are not both negative, as
explained in Art. 302.
[See a paper by Eussell, cited by Greenhill, Chapter on
the
Integral Calculus.]
306.
Various Forms of the Coefficients.
The two
coefficients
for since
jui
= yv
or with (x
by comparison
into various forms
(x
2
xj as a
when
/x
=
?/ 2
of coefficients
and
so
Also
aJ> 1
when
we have
factor in the numerator
x 2 ) 2 in the numerator
=
and
:
^-^
a fraction with
is
may be thrown
= K=C
(Art. 301).
,
VARIOUS FORMS OF THE COEFFICIENTS.
whence we have the following modifications of the
in Art. 303, viz.
P
P
P
Mx +N
2
,
I~~C~~
v=S)
similarly for the coefficient involving Q.
307.
It
_
coefficients
:
P
And
299
Convenient General Form of the Result.
appears
then that
minimum and maximum
y l and y% be
if
respectively the
ordinates of
y== ax'
corresponding abscissae, and if Mx-\-N be
written in the form P(x x l )+Q(x x 2 ), then the integral
and x v
x.2
the
= CMx+N
)~xJT
can be written, amongst
many
other ways, in the convenient
form
2/2
according as yl
p
Where
provided
308.
is -|-
ve
or
A^Z^?
ve
,
A-
and
a^a^
Remark.
It is further to
be noticed that the two quadratics involved
in this discussion, viz.
2
((^ib,2
a.jbjx
c 2 a l )x-\-(b ] c.2
(c^
are transformable the one into the other
transformation
by the homographic
a ^ _|_ ^
JL
The one gives the ordinates
of the turning
points.
6^) = 0,
'
J.
y 2 ), the other the abscissae (x v x^
[See Salmon, Higher Algebra, p. 173.]
(y v
CHAPTER
300
309.
A
VIII.
Special Case.
It remains to discuss the case
viz.
when
we have
_!==
so far excluded,
-l.
In this case the asymptote of the graph of
= =),
= -*,
viz.
does not meet the graph at a finite distance from the ?/-axis,
and one of the two turning points has disappeared. It has
been seen that the expression can, however, be written
where
and
Also
-p
^=0
obviously
is
= + ve by
Art. 294,
= + ve by
Art. 294,
=
=
gives the turning point, viz. x x l} y
=-
.
The only forms
of the
y^\
and y l
graph with which
we
are concerned are the four following. Cases, in which
the graph lies entirely below the #-axis, give rise to entirely
unreal values of \/TT Note the symmetry in all cases of the
graph about an ordinate through the turning
Fig. 20.
point.
A SPECIAL
2
b
CASE.
301
positive
>c
Fig. 21.
+ 00
a
positive
Fig. 22.
with corresponding forms
if
a 2 be negative,
viz.
+ 00
GO negative
Fig. 23.
In the case a 2 negative, 6 2 2 <a 2c 2
below the x-axis.
,
the graph
is
entirely
310. When the graph cuts the z-axis, as in Figs. 21 and 23,
at points P, Q, *JY is unreal for the value of x intermediate
between
and Q, i.e. intermediate between the roots of
P
CHAPTER
302
VIII.
a 2 be positive, but real for such values of x, if a.2 be negative.
Hence, in the first and third cases (Figs. 20 and 22) the limits
if
of integration may be any whatever in the second case (Fig. 21 )
both limits must be in the region from
oo to the smaller root
;
+
of the quadratic, or in the region from the larger root to
<x>
or in the fourth case (Fig. 23), 2 being negative, <JY is unreal
for all values of x which are not intermediate between these
Thus
roots.
sidered
And
in the fourth case the integration is only to be conlimits lie intermediate between the* roots of
when both
the whole range
have
x=
oo
The
,
&22
x= + oc
<a2c
2
,
\/Y
is
unreal for
.
also
splits
first falls
under the
=P
2
_P
P
or
2
to
/=
or
ve
in the fifth case, viz.
We
;
up
into
two
integrals, viz.
class discussed in Art. 277,
and
_2_
2
2 v
the real form to be chosen.
For the second
integral,
according as
or <y v
y
>
or
A SPECIAL
303
CASE.
1
= Q
the real form
to be chosen.
or
=
or
Q
Hence we have
(1)
(P&-Pl)
i
+"'
P2
2/1
r
and
(3)
'
+
forms to those obtained when
again the coefficients
311.
We
substitution
obtained
its
would be
by the
,
and
be expressed in various forms.
may
two integrals,
to
Art. 277, for
integration
note that the
been referred for
,
sn -
-
results of similar
ve
first
2
;p 2
of the
+2 2 :=
2
f
2/
'
which has
which the
might equally well be
substitution
the same as used in the second integral. Upon this substitution being made in the integral 7, we get a result of form
i.e.
CHAPTER
304
In the
first
of these
In the second
we
substitute for *JX
we form
r~Y
and substitute for
VIII.
I
.
5- its
-
2Ml/ iL
y= -^xJy&l
yl
value, viz.
value in terms
its
^2^
1
A/-^
.
>
j
1
>
/
Vft-y
?i
as shown.
312.
Ex.
1.
Illustrative Examples.
Consider the integral
-
.
X
(a)
without reduction to the canonical form,
(b)
'
first
reducing
it
as in
Art. 295.
^=i^Ti
<> Puttin *
1
2^-1
dv
_x(x
\)
_x(x
1)
~TF" ~x^~'
The turning points are given by # =
-t
and #=1.
If# = 0,y = l; if#=l,
2
_ ~ 2.r
2
2~ 2X
1
assuming
If /I
2
WJrfy
^
N/2
-cos-"V
o
n
.
,
+V2COSU-
1
T
Ar ._ Ztr+l
ILLUSTRATIVE EXAMPLES.
The graph
305
of the transformation formula in this case
is
shown
in Fig. 24.
Fig. 24.
And
the signs selected refer to values of
x> 1, and
we have
= - cos- 1 "Jy + <\/2 cosh" >/%
1
/!
if
A n A 2 be the limits and both > 1.
x lie between and 1, we have
If
,
and
and we shall have
72 =
if
- cos- 1 \f - -s/2 cosh- 1
A 2 both lie between and
x lie between - oo and 0,
1.
Aj,
If
*JX
1
and
VTand we
have
shall
r-
/3 =
if
A 1} A 2 be both
L
_
+ cos- 1 *Jy - \/2 cosh- 1 >/2?/
iA 2
'
J^!
negative.
on cue side of a turning point, say #=1, and
and the other, A 2 on the opposite side, i.e. A 2 > 1, the
integration should be conducted from the lower limit to the turning
point with the corresponding result, say /2 and from the turning point
to the upper limit with the result I
v
one
If
limit, AI, falls
0<Aj<],
,
,
(b)
Next
let
us transform to the canonical form
.dx
before integration.
<!<>, by the rule of Art. 297,
1
1
:5.
.r
/
-j
2
- (xi + X2 ) + I = 0,
- (x + .r2 ) + 1 =
l
E.
I.
C.
8
;
i
= 0,
= 0,
CHAPTER
306
and
Pi
+
31
VIII.
= 3,
.
\
dx.
<%
:.
gives
y=
gives
y=
I= n r*> \**^
vy
-2
,
dy
-
iwr
1
i.
=
cos-1 \y + N
f
cosh" 1 \2?/
,
as before.
Ex.
Let
2.
As
a case where
1
=^
,
consider the integration of
ILLUSTRATIVE EXAMPLES,
Now
and
(1
X
307
10
Therefore
Also, at the
maximum
=0
ordinate,
And
and y=9.
9 -.y= 9
N/X
.
jr
f
= VlO
/
N/9
ve
taking ^7>3, ^e. ^ as +
Therefore, in the second integral,
.
^-6^+105
2
The graph
3
of the substitution formula,
Qx-x*
y ~^ 2
is
shown
in Fig. 25,
O
Fig. 25.
attaining its maximum value 9 when
values of x
except such as lie between
.y
.r
= 3, and
and
6,
being negative for
all
and as we confine the
CHAPTER
308
integration to real values of
that both
x
as
lie
\/Y the
limits of integration are to be such
within the region from
increases through the
when we take
=H
*
to
value
-7=
VIII.
3.
Also the sign of
6.
changes
Hence the signs adopted above
apply to values of x between 3 and 6
and 3 we must use
For values between
,-
j=-
E-=
and make
N/9-p
the corresponding change in the sign of the part of the result dependent
thereon.
Forms reducible
313.
As
to Case IV.
in previous cases, Arts. 281, 286, 288, attention is callec
form of integrals deducible from the case
to the varieties of
X quadratic, F quadratic.
just considered, viz.
w
Thus
sm
/^i
26 sin 6 + c
dO reduces
Lx + M
dx
sin
if
,
,_.
0=x.
,
reduces to
d>6
(2)'
v
to
L +N
Lx
Mdx
dx
ax* +
L sin + Mcos + A
7
in
dO,
(3)
v
;
similarly.
J
L sin + Mcos 6
(4 >
by .putting tan 0=.r.
v '
(5)
similarly to
(1).
similarly to
(2).
tLaxk
asmh 2
;
J
7 '}
I
J
(8)
-
Lsinhu + Mcoshu
C
(
d
,
,
2
a smh
s
/=
=
If in
we put x
1=
t
.
~,
x+
f
J
N
(a^ + 26
dx=(
1
--^Jdz,
c
cosh 2 ?*
by tanh u = x.
,
-,
,
+ 26 sinh u cosh u +
du,
dx
,
SPECIAL QUARTIC FORMS OF
where d
d2
l ^
/=
f
F
_
/F
.
.
.
2
(z
{_ __
J (a^ + 2b lZ? +
respectively
;
so that
I
)
dz
(Jfe
2
by the substitution
Y be
any
0,
and
JC,
t
by the substitution
+
z
+ - =x.
^-
d^-^z + aj)
F=a
314. The Case of
Let
2
1
-
Similarly
integrates
+2
be a "reciprocal" quadratic function of
we can integrate
reciprocal quartic expressions in s
(9)
c2
(
if
Hence,
+ 2a l5
are written for c l
309
7.
=x.
z
Reciprocal Quartic.
reciprocal binary quartic expression
= ax*
Then 7 =
I
reduces at once to the form
'-
-
x
J
JY
Jdz
\/Quadratic'
by the substitution x + a; =
-
z,
whence (1
V
as
;|
2
dx = c
/
Y = x4a(x*+^
For
2
2
]
J
-, where #=2(26 -3ac+a
2
.
/=
i
dz
r
N/al
//
I+
JV(
26\
i
X
2
or
r
V-al /x
if
or
-cosh-
--,
1
-T-cosV a
1
xjK
,
if
^be - ve
if
a be
,
-.
);
&
'
/
2
,26;
jVaiv)
T)
which, by Arts. 80, 81,
or
2
Y
CHAPTER
310
Note that
K be
if
VIIL
positive, the factors of
terms of z are real
if
and that
315.
aY+Kx*
A
expressed in
;
K be negative, unreal
is
F as
;
a perfect square.
Similar Case.
In the same way,
if
3
2
Fj = ax* -f- 46x -f- Qcx
4>bx -f- a,
the integration of
II
=
I
For
-
== can be
-
F=
1
effected
-
by the substitution x
= z.
;
az 2 + 46^
r/
s
L\
+ 6c + 2a]
26\ 2
+
46 2
2-l
/
^
|,
where
^ = 2(26 -3ac-a
2
2
)
;
1
or
or
V
if
F--,
a
a be
expressed in terms of z has real or unreal factors, as
ve
ve
an d aY -\-K x 2 is a perfect square.
K! is + or
In the integrations of these two articles, since the final form
also, F!
l
>
exhibited
is
arrived at
function of F, or of
/
[e.g.
it is
.
l
by the conversion
of a function of z into a
Y v in which process a square root is extractec
2
.<M-2o = cos- 1 L1 - -(az + 2b)' =
sin- 1
,
y
,
desirable to check
by
^
,
\
,
etc.j
direct differentiation the sign of al
numerical results obtained.
SPECIAL QUARTIC FORMS OF
311
Y.
Other Forms.
316.
The substitutions
,11
= -,
x-\
X
11
=-
x
X
Z
Z
respectively reduce
x2
2
dx
1
[x
+l
_dx
'
and Yl denoting the same quartic functions as before.
[See Green hill's Chapter on the Integral Calculus, p. 41.]
Y
For taking
x-\
00
= - we have, differentiating logarithmically,
%
1-1
x2
X-\
x2
7
dz
1*-
-7-
l-e
1
xdz
,
dx=
-^+I
^'
X
and
a;
2
!
eto
_
r
dz
f
~
J
whose integral can be written down by Art.
And
similarly, if x
x
80.
= -,
z
xdz
X2
1
dx
,
and
whose integral can be written down as before.
The
integrals
2
(x
dx
-!)
r
a;
2
x
dx
+l
z
}a l x +b lx-a l
a;
2
1
x
dx
Cx
2
-\-l
x
JY
dx
x*+I a 1z 2 +& 1z+a i jy>
fa;
are reduced to forms
already considered
and are therefore integrable.
stitutions,
by the same sub-
312
Similarly,
CHAPTER
VIII.
can
effected
Y^
if
Y
=a
l
the integrations of
2
dx
Cpx
q
J
>
Cpx
px
'
V
z
q
dx
^
'
7F
]px*+q
be
by the
respective
substitutions
dx
Ex. Consider the integral
7_
Here
and putting
Put
^+-x = 2,
^-1=^,
.:
/=
zdz
/
J (z 2
zdz=wdiv;
1=
I
dw
p<s/6
J
tan" 1 -=
V5
i
I
^
v5
1
\lw^
/
=~j=sm~ A/
i
1
317.
Summing
It will
can be
now
For
if
~,
.
up.
where
and
\js
are rational integral algebraic
x.
\
~
T^ -4- T^ -I- 1
be clear that any integration of the form
dx
C(f>(x) _
effected,
functions of
\/5
be put into the form
GENERAL CONSIDERATION OF THE
POSITION.
313
on Partial Fractions, then of the
as explained in the chapter
resulting integrals
x n dx
r
is
2
]jax +bx-\-c
reducible to a lower order
240,
and
by
Art.
integrable.
dx
has been considered in
dx
reduces
1
Art. 287.
/
by the method
'
r
J (x
-/3)
x/^+^Tc
of Art. 290.
has been considered in
(\'x-\-/uf)dx
'
(Ax*+Bx+C)J'ax*+bx+c
Art. 291.
r
'
}
is
best got -by differentiation with regard to C of the result
where s = l, as will be explained later. This
for the case
method may
318.
also be adopted in
(3).
GENERAL CONSIDERATION OF THE POSITION.
We
have therefore now completed the integration of the
most general function of x of form
A + BjR
C + DjR'
where A, B,C, D are rational integral algebraic functions of
x of any degree, and J? is a rational integral algebraic function
of
x of degree 1 or 2.
For rationalizing the denominator,
C*-D*R
C + DJR
AC-BDJt (BC-AD)R
H
P
where P, Q, M,
Jp
and
if
jj-
</.';
M
N are rational
is
integrable
I
integral algebraic functions of x.
by the methods of partial
be put into partial fractions,
.=
I
^r,
dx
fractions
;
can, as has
CHAPTER
314
VIII.
been explained, be expressed as the sum of a finite number of
such terms as have been discussed in the present chapter, and
each term may then be integrated.
Hence the theory
is
now
of the integration of
complete, where
R
is
linear or quadratic.
And
it
will
be noted that the integration has been in all cases effected in
terms of the known algebraic, logarithmic, inverse circular or
inverse hyperbolic functions.
is of higher degree than the second, it has been
When
seen that in some special cases the integration can still be
R
terms of the elementary functions, but for the
is cubic or quartic,
of the cases where
discussion
general
we shall require the elliptic functions, and in general for forms
of higher degree than the fourth, we should require the
of
effected in
R
R
functions
known
as hyperelliptic.
GENERAL EXAMPLES.
1.
Obtain the following integrals
(i)
(iii)
t(l+x)-*x-*dx.
{^(l-Zx + x^dx.
:
(ii)
i(l
(iv)
f(l
*)~*dx.
x/1
1
2
.
Integrate
+x
dx.
*Jx
+ x 2 + x*
(i)
[BARNES SCHOL.,
3.
Show
1887.]
that
-ac
where p
lies
between the roots of a +
,
'2bx
+ ex = 0,
2
supposed
J
real.
[TRINITY, 1886 and 1891.1
GENERAL PROBLEMS.
4.
Show
315
that
dx
f
and =
5.
Prove that
~
=CQ gh-l
I
3X2
V2
l
2
~ 2X +
1
-2g+l
.
+ 2cos
Ti
6.
Integrate
l
where a < b <^
7.
Integrate
(i)
J
(fl2
f
m)
8.
^^
^^ _
[ST. JOHN'S, 1888.]
(G"
[ST. JOHN'S, 1889.]
__
dO
si
J sin
6>
2
N/cos ^
+ 6sin 2 ^ + c*
[TRINITY, 1888.]
Find the values of
sin x
I
(cos x
(ii)
J;cos (x
dx
+ cos a) x/( cos x + cos/?) (cos x + cosy)
[?
1890.]
+ a) v/cos (x + P) cos (x + 7)
fr
189
r
9.
_
+^_
(x + b)dx
-
]
99
-dx,
Integrate
J
2
(a
-ax +
2
4
)
(a
+ a% 2 + x*)*
transforming by the substitution
x2 + ax + a 2 = y2 (x2
-ax + a 2 ).
[a, 1884.]
-
10. Integrate
(i);
^
f
J
'(x
-l)(x- 2)J(x -
3)(x
-
4)'
f
7
2
J(;c
y
.
x
(iv)
-
-2^ + 2
f
I
)(x- b)(x-c)(x-d)Jx-e
-
[COLL., 1892.]
CHAPTER
316
x4
f
11.
Integrate
I
(i)'
Vllt.
1
dx
.
JX JX4+X 2 +1
2
[COLL., 1901.]
l
v
(x~ (x*-l)
4cfa.
Jv/^ + z 2 *!
,
.
(n)'
Show
12.
W
sin
that
(a;
sin (x
,
-a) ax
1
= cos a cos",/cosaN
+ a)(
.
,
~/&inx\
sin a cosh" 1 (
)
-I.
\sm a/
\cos a/
[COLL., 1901.]
13. Integrate
- 2x*
(i)
J
...
J
z
TT-^
/7-/f
+ ^4 ).s/(v
./v 1 + z 4 (
X
1
f
-
tan^) (6
2
cosh -i
Show how
+ a)
where n
is
any
[
P., 1900.]
MATH TRIP "
-
Show
^
1886
'
J
(
sin
v/siaa
a;
that
(n
-l_ T)'?*^
fJT
r
I
J (a
+
2bx
+ CX 2 }*
[a, 1890.]
T
=
-.
(n. 4- 9.hr.
-1-
^2^^
Prove by
J the substitution
y
A
2
and AC -
= (ax2 + 2bx + c)/(Ax2 + 2Bx + C),
B
2
are positive, that the integral
(Mx + N)dx
2Bx
+ C]
)(Ax +
(
2
becomes of the form
Aj
,
and A 2 are the roots of the quadratic
P.,
[COLL., 1892.]
+ g) (ax 2 + bx + c)% dx,
positive or negative integer.
C
and P.
I.
to integrate
(fx
where
[OxF.
- tan^)*'
dx
f
where
- ax a?)Jaa+&
Prove that
J si
'sinsin^(2
18.
vol. iv.]
)
dx
J (a*
i/
.-;,.,
,v
2
Evaluate the integral
14.
17.
a;
__
2
16.
[R R]
rfjB
(ft
15.
/1+ce
f
ii)
2
^-^ ^j^fA
f1
,
are definite constants.
rm
[TRINITY, 1889.
.
GENERAL PROBLEMS.
_
2
V + *V(* +
2
317
Integrate completely the function
+3
x
2
pa
19.
Prove
-
+ 49
26.r
[MATH. TRIP.,
- tanh 2 a dx = -
(1
- sech
,
+
^n
T* dx.
J(l+cos^) x/l+cos^ + co^'
and evaluate
-
J-l(a
22.
is
Show
that
{
2
1891.]
a).
_- _
2
20. Integrate
T
Wx + 17) V4z
2
>/tanh a
I
Jo
1
-
[ST.
LP>
18W.J
p
im
JOHN'S, 1882.]
....
+ C2Z 2 )v/l-Z 2
dx
.
Jx-ajAx* + 2
transcendental unless
Aa 1 + 2Ba + (7 = 0.
M. SCH. Ox.,
[J.
1904.]
Establish the results
f
.
and
(11)
'
I
-
J (x
dx
1
x-l
dx
-
.
.
- 2)V(3 + 2x - 2x 2 )
= sin" 1
(x
-
2)
[CoLL.
23.
Show
that
__Jl~^
dx
_,
("1
- 2ax +
- 2bx + V]
a*)(l
24. Describe the steps
of a single variable,
Prove that
if
in cyclical order,
is
.r,
whereby the
=
*2-ab
1 -^F
'2
a, 1890.]
V, 1884.
integral of a rational function
can be obtained.
the sign of summation refer to the suffixes
the integral
1, 2,
3
a certain constant multiple of
.1
:;
[MATH. TRIP.,
25.
Determine the degenerate form of the
1896.]
elliptic integral
ds
ll>tt>h
,
when
.s\,
is
made
to coincide with
^
or with ss
I
.
[!NT. ARTS,
LONDON.]
CHAPTER
318
26. Prove,
VIII.
by
* means of the substitution
;
x- 8 =
X-a
2
?/
J
that
,
'
=
or
[!NT.
27.
a
x
y
ARTS, LONDON.]
Prove that
1
dx
[MATH. TRIP. L,
28.
Show
1912.]
that the integral
dx
f_
JW3
is
rationalized
otherwise, find
Prove that
2
by the assumption z=(l
its
if
+^/ )/(3
2
),
and hence, or
value.
m
be a positive proper fraction, the value of the
above integral when taken between limits
as
-^
when taken between
limits
+m
2
=
and
3
-:
2
+m
is
the same
and
m(2
[MATH. TRIP.
29.
I.,
1910.]
[MATH. TRIP.,
1878.]
Numbers 1900-1903
for a
Prove by means of the substitution
a-x a-d c-y
x-b
that, if
m be any
cy
b
positive quantity,
d*
> b > c> d,
and a
r
f(a-
Jh \
a-d
b-c
r
~~
e
^T?^
(
(a
*j
^^^^^^^^.
I
(a-x)(x-d)
Jd
,
a-d
I
tAi'-.
(b-x)(c-xy
b-c
[See Wolstenholme's Mathematical Problems,
)
group of similar examples.]
30.
By
the transformation
f px^
px + - = >j2pq/z,
-q
J px + q
integrate
dx
'
1
JpZx* -f if
[Cf
.
E DLER, C. I.
, i v. ,
p. 22. J
GENERAL PROBLEMS.
31.
Apply the transformation
+
x-
2
= 2/z2
,'-
....
J
to integrate
x z dx
f
(11)
319
a
[EULER,
32.
Show
that
_
2 d6
33.
\vill
Show
__
cosO
_ tanh
l-
C.I., iv.j
costf
tan~
that the transformation
reduce the integration
~
x m l dx
1
-
to the form
f
um~ 1 du
J1
[EULER,
34.
(i)
Show
C.I., iv., 53
and 56
;
PEACOCK,
p. 305.]
[PEACOCK,
p. 309.]
that
2
f
J (1
and
e*(3-* )
f
(11)
integrate
e
x
l+noJ-i-a*
V
1
-
*'
1
7
...-^dx.
J
35.
fa =
2
aj)>/l
Integrate
x
^ 1}
f2-3x
J 2T3^
IT+x
....
j
Vrt*
1
(H)
5
5
fsin (9+2cos
cos sin 40
J
6>
d6
'
[ST. JOHN'S, 1881.]
36.
Show
that
[HALL, I.C.,
37. If F(x, y)
be a rational algebraic function of x and
that
F(x, \/l
+ a;-) (a; + v/1 +x-) v
^
J
may be
integrated by the transformation
2 = sinh(log#).
p. 325.]
y,
show
CHAPTER
320
38.
Show
V11J.
that
f(cos20f cos 0^0
(i)
i
2
\/sin
39.
Show
a-
cos 20)s/cos 20
sin 2
+ r-^snr 1
+ \/sin 2 a- sin 2
Vcos
^
sin 0).
,)
that
af^^q:^
(i)
40. If
3
.
= ~sin 0(3 + 2
<#>
(cc)
=
1
13
-T
L
^
+ - a 2 x2 +
-
.
4
x4 +
.
.
.
,
show that
4:
.
(i)
20
4
*~
11\2
1
1/1. 3\ 2
1/1. 3. 5\ 3
=
l
[ANCLIN.]
41. Integrate
.,
sin20d0
f
J v/sin
4
+ 4 sin 2
cos 2
+ 2 cos 4
:,
J,
42. If
/
be the Jacobian of two quadratic functions of a?,
2(b 2 x
2 ),
show that
if
1^
= 0,
it
2
=
have no positive
-
rJ
MjMj
^i
a
+ c2 )
roots,
c')
dx = 2 log
& a-i-2.
7
^
2Cl
then
viz.
GENERAL PROBLEMS.
43.
By means
(a
I
of the identity
P
r
321
+ sin 2 z) n cos xdx\ (l+a
sin 2 ic) n sin x dx,
Jo
Jo
prove that
nn
1
2
sin
= (1 + a)" - 2
//
3
~
o
(1
+ a)"- 1 + 2"
o
.
(1
o
+ a)"-
[WoLSTENHOLME, Problems, No. 1929; WIGGINS, #. ^imes, No.
44.
Show
(i)
13323.]
that
a n + n C,*
-^ an~i + n
1.3.5
)n
x
-
(1
l
P+5
1.3.5
"
45.
(i)
2*3-1
f
(iii)
a6 + 2a^ -
a;
2
+1
+ 6)-3J3 +<< _
&
'
[Ox.
I.
P., 1903.]
Integrate
Prove that
ff
r
*
E.I.C.
frt
^)
Integrate
J
(ii)
(P+^)(P + *)(P +
(1
+
sin
X
cos
3
6>)
AQ
CHAPTER
322
.
46.
(i)
(ii)
Show
VIII.
that
Evaluate
["^(a + h-x)"-
dn
1
'
(.!/!
J:
How
a*
could your result be applied to the summation of series
[a,
1
1886.]
47. Discuss the integration of
where / denotes a rational integral
48. If F(x)
[LACBOIX, C.I.,
V-/_
_ 7-^ where
is
ii.,
p. 35.]
x,
show
1
k
is
Tl
1
U
2
the coefficient of - in the product
^
f+i
-iv/l-z2
"'
the
of
be a rational integral algebraic function of
jp/aA
or where k
function
algebraic
quantities indicated.
'
1 + 1^3
a3
2
.
4
'
1+
ci
"I
'
5
'
'
'
J
the constant term in the expansion of
[ST. JOHN'S, 1891.]
-
-
,.
(*-!)*
[COLL., 1892.]
49.
If/() be an
arbitrary algebraic polynomial of degree
?i
1,
and
where
A
is
a constant, then
[LOND
]>>*.<*><fa-o.
50.
Prove that
a
f
i
-arfa;
7
Jo cos
51.
Show
that
if
cos (a
r
=
-a)
a
.
.
UN ; V ,
.
10
sin a
SCO a.
[COLL., 1896.]
a be less than unity,
r
J
1
xsinxdx
~
+ a 2 COS 2
tan" 1 a
a
[a, 1891.]
GENERAL PROBLEMS.
323
-pJb*x
(ii '
)
53.
From
~
[5,
f
.
COS<
JCO
[ST. JOHN'S, 1885. J
the definition of a Bessel's function,
_J^J fl
2*P(+ 1)L
1881.]
x
-
*
2(2/1
+ 2)
+
2
.
viz.
__
4(2^ + 2)(2 +
4)
1
'"_}
derive the results
sinx
- cos x
1
X
[COLL., 1896.]
J
Xs
54.
Integrate
-
(i)
'
1
(ii)
-s/1
+ 3 sin
a;
cos
a;
sin 2
+2
a;
cos 2 x
1
(iii)
55.
Show
+ sin x) (2 + sin x)
that
f
\
.
7"
si
sin
x
sm nx dx = sm 2 nx
J
where the form of the function
.,,_
~
w2 - m 2
where
is
2
(n
-
-
sin
defined
m2 ){n 2 - (m - 2)
7i;c
by the
h
,
J
relation
m_2
}
2
2
}
{n
-
(m
-
2
4)
}
a positive integer and n not being of the form
r is a positive
integer
not greater than
-~
(m -
Draw graphs
(
of the transformation
2
x2
2?-),
.
[MATH. TRIP.,
56.
1897. ]
Z
2
m ){?i2 - (m - 2)
2
-
,
f<f>(sinx)}
7
-j- \
\
ax
w(m-l)
~
+ 7-0
m being
(f>
d
[MATH. TRIP.
formula
+ 2b 2 x + c 2 )?/ 2 = a^2 +
corresponding to those of Arts. 301 and 309 for
1897.]
CHAPTER
IX.
GENERAL THEOREMS.
Various Limiting Forms expressed as Definite Integrals.
319.
The
definition of
an
integral, viz.
where b = a-\-nh may be expressed as
ti
f*
=
a,/
b
a
n
and can be used for the evaluation of a certain
limiting forms.
Ex. Find the value of
Lt
This
may be
F
2
I
32
22
n2
.
written as
r_l
n n
and taking
& n as x and n
as'
=Jlog.2.
In the same
320.
where h
-
n
may
way
be evaluated.
324
~]
class of
GENERAL THEOREMS.
325
i
Let
M=(<f>(a)<Ai
then logw = -{log0(a)+log0(a+A)-f
and therefore
if
we
write
and
is
f
J
Hence
where A
[see
Di/.
6
^
a
dx.
Lt^{<f>(a)<f>(a+h)<f>(a+2k)...
6
a
=
Cede., p. 6,
Ex. Find the limit
Ex.
when
Calling this expression u,
and
a)-=dx,
(b
the limit of log u
...
Ltlogu = J
71
3].
= 00
of
-flog <j>(a+rh)
CHAPTER
326
IX.
EXAMPLES.
Determine by integration the limiting values
following series when n is infinitely great
1.
of the
sums
of the
:
n 7-7
+ 1
+ n T+^2 + n + 3
n
n+
n'
[a,
1884.]
n
-*-
32
\/2-l 2
V4tt
-
[OXFORD, 1888.]
- 32
J^
+ ""
J
[CLARE, ETC., 1882.]
Ic
being a positive integer.
2.
Show
[ST. JOHN'S, 1886.]
when n
that the limit
is
increased indefinitely of
(n-mfi
-
+
...
+
v
5-^-
Oil:
Zl'K,
'/(,
Z
11'
[COLLEGES, 1892.]
3.
Find the limit when n
is
indefinitely great of the series
fn-l
4.
Evaluate
".
5.
r
x
L^a^-l
l
,
J4a*n-l
l
x
i
,
i
*j6a 2n-l
f
.
^2a% 2 -l
Evaluate
8
+ ...+
[C. S., 1901.]
GENERAL THEOREMS ON INTEGRATION.
321.
Various Propositions.
There are certain general propositions on integration, many
of which are almost self-evident from the definition of integration or from geometrical considerations, the truth of some
of which the student will have noticed for himself, but which
require to be definitely stated. It will be assumed that all
functions occurring in the following theorems are finite and
continuous between the limits ascribed, unless the contrary
be specified
:
GENERAL THEOREMS.
rb
322.
for
I.
rb
(j>(x)dx=\
if \Is(x]
327
(j>(z)dz,
be such that
d
and therefore such that
d
each integral is equal to ifs(b) \/s(a).
In other words, the result being necessarily eventually independent of x or z, it is plainly immaterial whether the letter x
or the letter z
used in the process of obtaining the indefinite
is
integral previous to the substitution of the limits.
323. II.
For
if
(
\l/(x)
<j>(x)dx= V <t>(x)dx+( </>(x)dx.
be the indefinite integral of
the left side
and the right side
which
is
<f>(x),
is
is
the same thing.
Further,
it is
0(z)cfo=
equally clear that
0(z)cfo+
0(z)dz
<j>(x)dx+
...
+
d, e,f, ... k are any real quantities which lie in the
from
a to b for which 0(x) has been assumed to be
region
finite and continuous.
where
c,
Let us illustrate the fact geometrically.
CHAPTER
328
IX.
= cj)(x),
Let the curve drawn be the graph of
and
let
the
equations of the ordinates
x = a, x = c, x = d,
be
x = k, x = b
...
respectively.
Then the above theorem
in
expresses
integration
the
obvious fact that
Area
N.N^P, =
Area
Nflff^ + Area N N3P3P2 +
+ Area N N P P
2
'
5
324. III.
b
a
J a
<j>(x)dx=-{ b
[
6
6
5
...
.
<f>(x)dx.
J
For, with the same notation as before,
the left side
and
An
is
the right side
tyfi)
is
{\fs(a)
ty(a)
^(b)}.
interchange of the limits, therefore, changes the sign
of the integral.
a
325. IV.
<t>(x)
dx=
Jo
For
if
we put x
f
Jo
a
(j>(ax) dx.
X, we have
if
x = a,
if
x = 0,
dx=dX',
X = 0;
X = a.
and
Q
X O
Fig. 27.
Hence
dx=-(\t>(a-X)dX
=
I"'t(a-X)dX, (by III.),
Jo
(j>(ax) dx,
(by
I.).
GENERAL THEOREMS.
329
Geometrically this expresses the obvious fact that, in
esti-
mating the area 00' QP (Fig. 27) between the y and cc-axes,
an ordinate O'Q, and the curve PQ, which is the graph of
y = (j)(x), we may if we like take our origin at 0', O'Q as our
Y-axis and O'X as our X-axis, as it cannot affect the result,
whether the elements of area are added up from left to right,
or from right to left.
a
326. V.
Jo
For,
by
<j>(2a-x)dx.
<j>(x)dx
II.,
a
l
Jo
and
and
if
<j>(x)dx=\
\
J
Ja
term we put x= 2a X, we have dx =
when x = a, X = a\
when x = 2a, X = 0.
Y
in the second
dX,
Q
y
N
O'
Fig. 28.
Thus the second
integral on the right side, viz.
=- [<j>(2a-X) dX
J a
<j>(2a-X)dX
<j>(2a-x)dx
f
J
(by III/
(by I);
<j>(2a-x)dx.
o
o
The geometrical interpretation is, that if we are estimating
the area 00 'QP (Fig. 28) between the y and x axes, an ordinate
O'Q, viz.
x= 2a, and the graph of y = (j>(x), viz. the
curve QP,
we
CHAPTER
330
IX.
may if we like take Ox and Oy for our
ONRP, NR being the mid-ordinate, and
axes for the portion
O'X, O'Y for axes in
the second portion, thus finding each part separately, and
then adding together, a fact obviously true.
327. VI. Plainly,
if
<j>(x)
be such that
<f>(2a-x)
this proposition takes the
= <j>(x),
form
2a
f
J
and
if
(f>(x)
be such that
N
O'
Fig. 29.
In the
NR
first
(Fig. 29),
double that of
case there
is
symmetry about the mid-ordinate
and the whole area OO'QRP
in such a case is
ONRP.
Y
X O
Fig. 30.
In the second case
cuts the aj-axis at
N
<f>(a)=
<(a),
(Fig. 30), viz.
i.e.
<^(a
where
0,
and the curve
= a, and
though
GENERAL THEOREMS.
ONP, O'NQ
the regions
331
are equal in absolute area, the second
Ja (p(2ax)dx,
which
is
referred to
o
area O'NQ), for all the
(7 Fas axes, represents
a
ordinates are affected by
negative sign.
Hence, the algebraic sum of the two is zero, the one
O'X and
(the
cancelling the other.
There
is
now symmetry about
the point N.
328. This principle is very useful in the integrals of the
trigonometric or of any periodic functions.
n
Thus, since sin .^ = sin' (7r-,r),
l
r,
sin".rcfo?=2
/
And
since cos 2n+1
so also since cos 2w .r
#=
fl
/
.
sin*
Jo
./o
-cos 2 " +1 (7T-.r)',
= cos 2n (TT-X],
rfk
cos 2n xdx =2
/
cos 2n .t'cr.
Jo
We may
express these propositions in words, thus
:
To add up all terms of the form sin n xdx at equal into TT is to add up all such
definitely small intervals from
terms
from
to
^ and
double the result.
For the second
quadrant sines are merely repetitions of the
si lies
first
quadrant
in the reverse order.
Or geometrically, the curve y = siu n x being symmetrical
about the ordinate x = -~, the whole area between the ordinates
and
TT is
double that between
and
-=
Similarly, the second quadrant cosines are repetitions of
the first quadrant cosines with opposite signs, and therefore
a term of form cos 2n+1 xdx in the
first
quadrant
is
cancelled
by the corresponding term in the second quadrant, but a term
co&n xdx, the index being now even, is duplicated by the
corresponding term in the second quadrant.
Similar
remarks and geometrical illustrations apply to
other cases and for \\idcr limits of integration.
CHAPTER IX
332
Mao 2n+1 x dx = 0,
Thus
Jo
for the third
and fourth quadrant elements cancel those from the
first
and second.
sin 2n .rfl?.r
=4
|
si
I
cos 2n fl .ro?.r=0,
'
/
r
n .rcfo
and
=4
so on.
329.
VII.
A
Periodic Function.
(>x
If
na
fa
f
(j>(x)dx
Jo
= n\
Jo
For, drawing the graph of y =
of
/
an
(p(x)dx.
(j)(x),
clear that
it is
it
infinite series of repetitions of the part lying
the ordinates
OP
,
(a
= 0), and
and therefore writing
N^,
(x
= a),
consists
between
(Fig. 31), for
x-\-a for x,
= etc.
Also the areas bounded by the successive portions of the curve,
the corresponding ordinates and the ic-axis are all equal.
T3a
C2a
fa
(p(x)dx=\ a <p(x}dx=\ (j>(x)dx =
2a
J
fa
and
JO
fa
r2a
Jo
O
J'na
tb(x)dx+
<p(x)dx=\ d>(x)dx+\
= n\f
Qtc.
J
J
...
+ J(n-l)a(}>(x)dx
<p(x)dx.
N,
Fig. 31.
2
2n
Thus, for instance, since sin .r = sin "(7r +
sin
JO
2M
.r^=4 Jo
x},
_
r-ir
C-l-r
si
w
1
O
_ Q
1
-,-
-5- a2 'I
2w-2
2
GENERAL THEOREMS.
VIII.
330.
Arbitrary Change of the Limits.
P
In estimating
d>(x)dx, the limits
Ja
I
to p,
q,
333
be altered arbitrarily
may
provided x be transformed linearly in a suitable
manner.
Take
x= A+Bg.
a = A+Bp,)
Let
A
and
,
,/'[ whence
b=A+Bq,j
t.C.
B
be chosen so that
^
AA = aqbp
1
qp
,
B=
b
a
qp
,
CHAPTER
334
of the origin a distance
the x-axis
direction
-
-
IX.
in the
positive
q-p
direction of
this quantity be positive, or in the opposite
This alteration in the graph leaves
negative.
if
if
number of units of area in the portion of the graph
considered unaltered, the effect being merely that of drawing
the graph on a different scale, the ordinates being altered
the
in the ratio -
q-p
,
whilst the breadths of the elementary strips
are altered in the inverse ratio, leaving the areas unchanged.
IX.
331.
If
(f>(x),
finite functions of x,
between a and
b,
be single-valued continuous and
of which the latter retains the same sign
\fs(x)
then
rb
Jba
(x)
where
Now,
=$ () ja,
\js
(x) dx,
Then
(j)(x)\[s(x)dx,
by the
definition of
an integral
(Art. 11),
of all the expressions
be the greatest and <(
$()
and
(x) dx
a<g<b.
Jba
let
\js
2)
the
<j>(a)\fs(a)+<j>(a+h)\Is(a+h)+
least.
...
+<j>(b-h)\[s(b-h)
>4
Hence
<p(x)\lr(x)dx<(/>(^1 )\ \J/-(x)dx
n
and
>)
Ja
^'(x)dx,
rb
and therefore must
where
<p(g) is
=^(^)
intermediate between
\]s(x)dx,
Ja
</>())
a value of x somewhere between a and
and
b.
has been assumed that ^(x) is positive for the range
If ^(x) be negative throughout, the order of the
fee:*! a to b.
inequalities is reversed, but the final result remains the same.
It
GENERAL THEOREMS
332.
and
Cor.
I.
335
case of this theorem write
As a
(j>'(x)
for
</>(#),
1 for \js(x).
Then
a<fo
= 0' ()
dx = (b-a)<j>'(),
1
j*
-0 () = (&-)*'();
or putting
b=a+h and g=a+9h,
where #
is
a positive proper
fraction,
subject to the condition that </>(x) and </>'(x) are finite and
continuous functions of x for the whole range of values of x
from a to a+h.
[See Diff. Cede., Art. 139.]
333. Cor. II. If
a < a?
< 6, it
(j>(x)
has a
finite
f
7=1
follows that
J a
finite, for if
0()
value for
values of
all
cc,
&
a and
(j>(x)dx is finite if
be the greatest and
<()
6 are
the least of the
values of <(#), / lies between <j>(i)(ba) and 0(
is therefore finite.
2)
(&#), and
u lt u2 us ... be all single-valued functions
and continuous for all values of x between a and b,
and if the series u -\-u2 -\-u3 -\-u4 -}- ... to an infinite number of
terms be convergent for all values of x between these limits,
and f(x) the limit towards which it converges, then the series
334. Cor. III. If
,
,
of x, finite
l
pX
*X
u
J a
is
l
f
dx+\J a u dx+\J
2
also convergent for values of
x between a and
b,
and con-
-X
verges to the limit
f(x) dx.
Jo
[This theorem has already been
proved in Art. 34 from a slightly different point of view.]
Let R n be the remainder after n terms of the given series,
Then
rx
Cx
u
J a
l
Now, by
greatest
a to
6.
fx
rx
fx
dx+\J a u dx+\J a u3 dx+...+ J a R n dx=\J af(x)dx.
2
supposition,
and
\
Rn
least values of
Let R'n and R^ be the
x changes continuously f: ^m
is finite.
Rn
as
CHAPTER
336
IX.
/.x
R n dx lies between R'n (xa) and R'n (x~a).
Moreover, R n vanishes by hypothesis when n is indefinitely
Then
J a
increased,
Ja
whence R'n and R"n
R n dx
also vanish in the limit
;
vanishes in the limit.
fX
Hence
/X
X
Ja
J a
u 3 dx-\-
t*j(ic-H
t^efaj-f-l
...
J a
fX
converges to the limit
f(x) dx.
Ja
[SERRET, Calcul
Integ., p. 108.]
335. Cor. IV. If a continuous function f(x) can be expanded
powers of x convergent for values of x between
in a series of
and
a,
2
4) + ^ z + ^ 2 x +...,
1
say,
A x+
then
is
also a continuous
A^ + A^ +
and convergent
- -
series
tending to the limit
X
[Cf. Art.
f(x)dx.
{
34]
o
336. Cor. V.
dx
\*f(x)dx= J
Jo
convergent between the same limits for which Maclaurin's
series, which has been used, is convergent.
This gives a means of expressing an integration by means
of a series.
337.
LEMMA.
A THEOREM
of the first r terms,
and
S{.
DUE TO ABEL. If -Sr be the sum
sum of the last r terms of the
the
series
each term being real and
the same sign, and
2 and
and
2'
and
finite,
but not necessarily
if
or
be the greatest and least values of
Sr
</
be the greatest and least values of
/SJ.,
,
all
of
GENERAL THEOREMS.
337
if a lt O2 O3 ... a n be n positive finite quantities arranged
in descending order of magnitude, and if
and
,
then
we
and
if
,
have
shall
a lt O2 o3
,
a^E
an be arranged
...
,
> S> a^j\
in ascending order of
magnitude, then
For
S = a^ + a u + a u3 +
2
+ a n un
. . .
3
2
= a, (S,) + a,(S, -S,) + a,(S -S,)
+ +a n _ (Sn _ -S n .
= S (a a + S (a ~a )-}-S (a a
3
...
2)
1
l
2
3
2
3
1
l
3
i)
+ .-.+S n,
and
aj
and
2
><r[(a
l
3
,
...
aw _i
n
>
a
-a + (a - a + (a
2
2)
$<!
i.e.
if
a2
,
and
3)
S>a
3
(a n _ 1
l
2
-
are a ll positive quantities;
-a,) +
i.e.
l a- )
. . .
a I
l
l
+ (an _! -o n + a n
)
>S>a
]
,
l o-.
In the same way, writing the series from the other end, and
a w a n _!, an _ 2 ...
be in descending order of magnitude,
i
,
,
This theorem in inequalities is due to ABEL.
We note also that if a 1; a2 a3 ... an were all negative, the
same theorems would still hold, except that the inequalities
,
,
would have been reversed,
al
338. X.
viz.
< S < an
a n 1!
and
<r'.
Applying Abel's inequality theorem to the case of
the integral
p&
<f>(x)\f,(x)dxt
J a
where
(f>(x)
and
for all values of
\/s(x) are finite and continuous functions of x
x between the limits a and 6, and (j>(x) positive
and continually decreasing throughout that range, and writing
0(a+/0, 0(0+2/0,
a2>
a3
0(o),
respectively for
a lt
,
-
0(6-/0
an
...
and
wx
for
B.I.C.
,
w2
w3
,
Y
,
...
un
,
,
CHAPTER
338
and taking the limit when h
is
IX.
indefinitely small,
we have
Jba cj>(x)\fs(x) dx,
f^a \fs(x)dx,
where
,
2
are the limits corresponding to the greatest and
least values of
and
I
for different values of
\js(x)dx
between a
Jfl
6;
f
a
Ji
6
r
\f,(x)dx>\ <j>(x)\/,(x)dx><l>(a)\
a
\/s(x)dx,
J a
J
ft
Jba <p(x)\ls(x)dx
for
some value
Similarly,
= (f>(a)\
\fs(x)dx
ja
intermediate between a and
of
if </>(x)
fb
Jb
\[s(x)
b.
be a continually increasing function,
dx
fi'
>\J
fb
(f>(x) \[r(x)
dx
>
</>(&)
J
'
\f,(x) dx,
2
fb
where ^,
'
2
are the values
greatest or least,
where
is
which make
of
I
\/s(x)
dx
and therefore
intermediate between a and
6.
339. From the last remark of Art. 337 it appears that the
same theorem will be true when (f>(x) is negative throughThat is, that provided <p(x) be continually positive or
out.
=
=
continually negative from x a to x b, and <j>'(x) retains the
same sign throughout this range,
C
b
<p(x)\/s(x)dx
a
= <p(a)\ft \fs(x)dx
Ja
fb
= ^(b)\
J
is some
according as </>'(x) is negative or positive, where
9 is
and
i.e.
a
where
value of x between
6,
g=a-\-9(ba),
some positive proper
fraction,
GENERAL THEOREMS.
340.
If
A
<j>'(x)
Theorem due
339
to Ossian Bonnet.
be negative,
sign in the interval
i.e.
0(z) decreasing, but 0(#) changing
from x = a to x = 6, and therefore cf)(b)
negative and 0(a) positive, write
then
x'( x ) is
negative and x( x )
= 0(6)
is
positive
b.
^(x
a) {
f
J a
from a to
J a
f >M
[0(6)
)
l
f
Ja
f
>(*
Ja
j
^
r&
^
Jfa
(a?)
da +
(6)1
^
(a;)
J f
341. Finally, if (/>'(x) be positive, i.e. tf>(x) increasing, but
changing sign in the interval between a and 6, and therefore
(f>(a)
negative and 0(6) positive, write
<j>(x)-<j>(a)= x (x)',
then x( x )
ig
positive
and x( x )
is
positive
from a to
6.
W [J V
ffr
J
J g
Hence, in all cases where the differential coefficient of (f>(x) is
a continuous function, retaining one sign between the limits,
though
0(s&) itself
may change
sign,
CHAPTER
340
IX.
some value of
intermediate between a and
and
continuous throughout.
being finite
This theorem is due to OSSIAN BONNET.
for
342.
XI.
(o 1
(i)
b,
<j>
and
Since
+o,+V.
<t(a 1 6
we have upon
1
b1
putting
= 0(a),
=^(a)
and taking the
(ii)
)
limit
If
and
when
&
is
indefinitely small,
a lf
a,,
a3
&!,
62
63
,
,
,
...
an
...
6W
,
,
be two sets of positive quantities, both in descending or both
in ascending order of magnitude,
2ar 2aJ b r
a s )(br
[for 2aras (ar
And
it
(i)
that
and
(j>'(x)
follows as in
tinuous, and
positive,
both negative from
x=a
to
^
If 0' and
is reversed.
7
\/r
2a;l 2a r b r <t
b s ) is positive].
0(cc) and ^(x) be finite, conand ^(x) be both positive or
if
x=b, then
Ja
Ja
are of opposite signs the order of the inequality
GENERAL AND PRINCIPAL VALUES OF AN INTEGRAL. CAUCHY.
Modifications.
343. XII. The Definition of Integration.
In our summation definition of integration, as
which has been denoted by
f
we have assumed
and continuous and single- valued for the
whole range from x = a t^> x=b.
a and b to be both finite quantities,
(1) $(x] finite
(2)
GENERAL THEOREMS.
This definition will
and
fail
when
341
these conditions are not satisfied,
will require modification.
have also (Art. 18) extended our notation so as to let
We
f
J~
stand for the limit
<{>(x)dx
when
b is indefinitely increased
<f>(x),
with a similar extension
J
of \Is(b)\js(a)
when
where
'
j
The subject of
the lower limit becomes infinitely large.
has been so
however, in all
integration
(j>(x),
to
be
understood
cases,
finite, single-valued, and continuous
for the whole range of integration from a to b, whether that
itself, viz.
range be
finite or infinite.
GENERAL AND PRINCIPAL
Infinities of the Integrand.
344.
CAUCHY.
VALUES.
When
far,
becomes infinite between the limits of integration,
say
point ic=c, where a<c<6, and nowhere else
between a and 6, our definition holds
(x)
<j>
at the
from x = a
from xc-\-tj
and
where
e
and
as small as
are
>/
we
two
to x
e
6,
positive quantities
which
may
be taken
please.
Jba (j>(x)dx
^e=oM
rj=OLj a
This limit
x=c
to
may
be
is
now
to be understood as
<J>(x)dx+{
J C+T,
finite,
infinite,
meaning
<f>(x)dx\.
-J
or
of
undetermined
value.
GENERAL VALUE of the Integral.
CAUCHY has named the limiting form
PRINCIPAL VALUE of the Integral, viz.
It is called the
When
the
which
A
rj
=
may
e,
be
similar
finite or infinite.
modification
obviously be necessary
<l>(x),
derived,
of
the
original
definition
when
will
the subject of integration, viz.
attains an infinite value more than once between the
extreme limits of the integration,
viz.
betweert a and
6.
CHAPTER
342
If the infinity of
stood to
IX.
one of the
occurs at
<p(x]
mean
_
Lt e= o
Jb
6_ e
\
<f>(
(j)
(x)
dx
is
limits,
say
to be under-
x ] dx.
J a
Again when the upper limit
(j>(x)dxto
I
is infinite
we
shall understand
mean
Ja
1
e
f>(x}dx
and when the lower limit
Jb
</>
(x)
dx
to
infinite
is
we
shall
understand
mean
ac
Jbi
<f>
(x) dx.
e
When
the integration
the integration
I
(f>(x)
J-
is
from
to
+00 we
%
c/>(x)dx,
shall refer to as its General value
;
i.e.
d\}s
where -r
e
and
other
rj
;
shall consider
dx to mean
*
Lt e =o
which we
oo
=
being small positive quantities independent of each
and when
^
= e we
shall refer to
i
Ltt e=0
=0
as
its Principal
value
;
dx
<>x
<(>(x)
i.e.
345. Geometrical Illustrations.
and let OA = a, OC = c,
<j>(w),
=
Then at C (x c) there is an asymptote parallel to
The graph may be such as to approach the
^/-axis.
Let a graph be drawn of y =
OB = b.
the
asymptote from opposite sides at the same extremity (Fig. 33),
In
or from opposite sides at opposite extremities (Fig. 34).
the first case there is no change of sign of (j>(x) as x passes
GENERAL THEOREMS.
343
through the value c. In the second, <j>(x) does change sign.
Let the inscribed rectangles be drawn as in Art. 11.
Let
P r Nr and PgN 8 be the ordinates at distances pe and qe on
opposite sides of the asymptote; then it is clear that
f
Cauchy's "General Value" of
&
<[>(x)dx is
the limit of
J
area
where
e
is
indefinitely
definite,
^^.P^iarea N BP n P
s
s
,
where N r C, CN S are
indefinitely decreased,
decreased in such a manner as to retain a
i.e.
but arbitrary ratio to each other,
viz.
p
:
q,
whilst
,
\
/
$\
R/4
O
\
\
A
N r C N,
x
B
Fig. 33.
"
the " Principal Value is what this becomes
ultimately vanish in a ratio of equality.
when
N C, CN
r
S
This treatment in either case excludes the area bounded
NN
Pr r sPs ao Pr in Fig. 33, where <j>(x) retains the same sign
or the difference of the areas
r C<x>Pr
s C(-<x> )P 8
S
r
by
where
N
(j>(x)
when both
N N
,
N
,
changes sign as x passes through C, as in Fig. 34,
ordinates
NP
r
r
and
NP
S
S
are
made
to approach
indefinitely closely to the asymptote.
There is no advantage in prescribing beforehand the relative
N
N
S P 8 are made to approach
speeds at which the ordinates r P r
the asymptote, viz. by making the approach in the ratio of
some definite but arbitrarily chosen quantities p, q.
We
,
CHAPTER
344
IX.
leave the choice of these relative speeds
and thereby retain command of the
ordinates are
made
till
after integration,
in which the
mode
to close up.
A
O
Fig. 34.
In understanding
f
6
I
(j>(x)dx to
mean
J a
fb
-j
aC-e
where
-=P
we
"
e,
rj
are
two
(f>(x)dx-\-
positive quantities,
in our investigations of
V
take
p = q,
1
that
is
e
=
(j)(x)dx
we can
\,
ultimately
make
the "General Value," and
>/,
we
shall
have
if
Cauchy's
Principal Value."
346.
drawn
When
the inscribed and circumscribed rectangles are
in the Newtonian manner (Art. 11), the pairs in
immediate contiguity with the asymptote are in area
e(p(ce), e(j>(c) and
[Fig. 35]
GENERAL THEOREMS.
345
The circumscribed rectangles are numerically greater than
the inscribed ones. They are of infinite length (j>(c), and of
infinitesimal breadths e and
respectively (Fig. 35).
tj
"
"
quantities until we
the nature of <(c). If the orders of the infinitesimals
be higher than the order of the infinity <(c) their limits
These areas then are
undetermined
know
e,
Y\
If of lower order their limits are infinite.
But, in
the latter case, if $(x) change sign as x passes through the
value c, we may be only concerned with the difference of
these infinities, which may be finite.
are zero.
347. If
way
in
(j>(x)
which
becomes
it
does so
a point x = c, the general
the
by
vanishing of a factor in its
infinite at
is
denominator.
Let
where F(x) contains no factor xc, and
(x-c)'
therefore retains the same sign as x increases through the
value c, and n is positive.
We
are
only
this function in
concerned
to
discuss
the immediate vicinity
the
of
behaviour
of
the asymptote.
CHAPTER
34G
we may take our
Therefore
IX.
x=c
limits a, b so near to
same sign throughout, and
F(x) retains the
if
A
and
that
B
are
the greatest and least values of F(x) in this interval,
/&
Cb
(h(x)dx
is
intermediate between
A
^,-p
rn
~.
\
J
Hence we may confine our discussion
will be convenient to
to
I
and
B
And
r-.
-.
Ja(z-C)
w
it
push forward our origin to the point (c, 0),
with the asymptote, and we then
so that the y-axis coincides
have to discuss the limit of
~
e
dx
,^dx
where a =c
a,
'
This expression has the value
When n
(a)
and
is
is
<
1,
< n < 1, the
i.e.
limit is finite, viz.
is
then
Principal Value."
the summations,
viz.
independent of the limiting value of
both the
The
"
General Value
first
and
and
7--^,
"
"
and the
elements
last
being respectively
This
-.
in
e
l
~n
and
l
rj
~n
(n
< 1)
vanish
independently of each other.
(b)
which
If
is
n> 1,
the limit to be discussed
infinite in general,
dently and
inequality.
ultimately
Hence the
But when n
is
when
e
vanish in
"
is
and
ij
any
diminish indepenarbitrary
of
is infinite.
^ A 4- 1
.
^
ratio
"
General Value
odd or of the form
that of
,
(X and
JUL
being
GENERAL THEOREMS.
integers
e,
i]
>
and X
347
when
the infinities will cancel each other
/m),
ultimately vanish in a ratio of equality.
is therefore finite, and
The Principal Value
=_J^r_J
n-lL3"j8
when n is odd
2X-4-
or of the form -
When
,
>
(X
/x),
2X
..
(X
,
>
and
infinite if
n
is
yu).
w = l we have to discuss the limit of
~
or putting x
1
-^
an even integer or of the form
(c)
L]!
a"- 1
1
=
dx
in the first integral,
Q
- + Lt
log
i.e.
log a
rj
This limit depends entirely upon the mode of approach of
the ordinates
S P S (Fig. 34) to the asymptote, and is
rP r
N
,
till
that
is settled.
When -=-, where
p, q are
undetermined
N
P
chosen, the limit
8
is
'P
+log
log
-,
finite
any
and
upon the choice of p and q.
When p and q have been chosen
is
quantities
be
to
arbitrary, depending
equal, that is
when
e,
rj
vanish in a ratio of equality, the limit becomes log
Hence the General Value
is
Q
Principal Value
If
is
log
n be of the form
an arbitrary quantity
the
-
^-
,
becomes unreal when x
negative and the
first integral is unreal, from
Excluding this we are then only concerned with
*dx
i
r l
.
sn
~
;
,
^.e.
*
r
a to
is
e.
CHAPTER
348
which
is
and =
real
^ Qn-i
and may be referred
IX.
^
n<~L, and
to as the Principal
infinite if
Value of the real
> 1,
part.
We
next consider the case when the infinite value of
occurs at one of the limits, say b.
348.
<p(x)
fb-e
fb
I
n
<p(x)dx
then to be interpreted as Lt e =o
is
J a
which
Let
is
called the
(h(x)
=
"
fix]
/,
.
(x
xb,
I
<f>(x)dx,
J a
n,
Principal Value."
where
fix) does not contain the factor
b)
and therefore does not vanish when x = b\ and
let
n
be
Then,
positive.
if n be
(a)
<
1 and if we can find some quantity y between
a and b such that throughout the range of values of x from
y to b the numerical value of f(x) does not exceed some finite
quantity A, the Principal Value will be finite.
/6
For
(p(x)dx=\
J a
The
first
/&-e
/y
e
of these
J
two
<j)(x)dx+\
a
Jy
integrals
the numerical value of the second
<p(x)dx.
and in the limit
not greater than
is finite,,
is
moreover
the limit of which,
when
e
= 0,
is
=
(y
l
b)
~n
and there-
fore finite.
however, n > 1, and if we can find some quantity y
between a and 6, such that throughout the range of values of
x from y to b the numerical value of f(x) is greater than
(b)
If,
some finite quantity B throughout this range of values of x,
and if f(x) preserves the same sign throughout that range,
the Principal Value of the integral will be infinite.
GENERAL THEOREMS.
349
For, as before,
<j>(x)dx,
the
the two integrals being
first of
finite.
Cb-e
But the numerical value
of
Z
e
=
(#)
dx
is
greater than
the numerical value of
dx
fJ y
which becomes
(c)
infinite
w = l, and
if
Lastly,
when
vanishes.
e
if,
as in the last case
(6),
such a
quantity y can be found as there described, the numerical
rb-e
value of Zf e=0
<f>(x)
dx is greater than the numerical value of
Jy
e
rb-e
dx
and
x-b'
f^~
,
j
the numerical value of which
}ba (j)(x)
349. To
sum up
dx
is
is
y
^.
i=-&
=1
infinite,
ssiv
and therefore the
in this case, also, infinite.
these Statements.*
y between a and
If it be possible to find a quantity
b
such that
the numerical value of (j>(x)(xb) n that is/(x), does not exceed
some finite quantity
throughout the range from y to 6, and if
,
A
Cb
n< 1, then
the Principal Value of
I
Ja
(p(x)dx
is finite.
If it be
possible to find a quantity y between a and b such that the
numerical value of <j>(x)(xb) n does exceed some finite quantity
B
n
throughout that range, and if c/)(x)(xb) does not change
sign throughout that range, then if n < 1 the Principal Value
fba (j)(x)dx
will be infinite.
)bviously a similar rule holds for the lower limit by reversing the order of integration, i.e. interchanging the limits.
(
*
Sorret, Calcul Integral, p, 100,
CHAPTER
350
A
350. (a) Consider
fa
j=..
I
Here the subject
IX.
of integration, viz.
fle
We have to consider
Lt e =Q\
Q
Let<ft(.r)
=
^_
infinite at the
upper
2
fa
-===.
Then <t>(x)*JT^x =
.
is
,
vl-^^
limit.
,
/
which
is
<
for the
1
whole range < x^. 1 or for any part of it, and the index of the factor
Ia? is ^, which is < 1. Hence by Art. 348 (a) the Principal Value is
finite.
*
It
is
j&
i.e.
(ft)
Z^e =o sin- 1
of course obviously equal to
1
e =o{siir~
(l
- e) - sin"
T
Consider
1
1
0}
= sin-
_^l_
,
- sin~ 1 = -
!
.
.
-
cf integration, viz. -
Here the subject
1
.*;
,
is infinite
at the upper
(l-a*>*
limit.
Let
<
(x}
=
h
Then
.
</>
(,v)
(!-#)* =
,
which
is
< -^
and does not change sign for all values of x from x = to a=l or for
any part of that range. Also the index of the factor 1 x is f, i.e. > 1.
Hence, by Art. 348 (6), the Principal Value of this integral is GO
.
351.
-dx,
Consider
where
,
Ltx =Q
"When x
viz.
p
is
0<n<l.
to approach zero indefinitely closely, the integrand,
increases numerically without limit. Take a quantity
zero and l-n, so that p is positive and
1.
Then
,
<
at x=e~p, vanishes at
whilst numerically decreasing to zero as
less
always numerically
than
p. 103.)
made
n
$(x) = \ogxlx
lying between
less
/.,
logx
^ = cc.
X
xp+n <f>(x)=xp logx has a turning point
is
(Serret,
than
.
x diminishes from
Moreover p
+n
^'
= 0, and
e~ P to zero
a positive index
\s
1.
Hence, by Art. 349, the Principal Value of this integral
is finite.
>
352.
Suppose that
\vhen/(.r) becomes
| ft
oo
Lt * =
f(x}dx has a value which
at
x=c.
(a
<c<
**/(*)<**
+
b.)
is finite
Then
and determinate,
this value
/(*)<***
must be
.................. (A)
GENERAL THEOREMS.
whatever
of p
:
the ratio of p q, and if this limit were not independent
General Value would not be determinate.
may be
this
q,
:
The Principal Value
The
351
the case
is
when p = q = l,
difference of these expressions
A and B
is
.'c+g
and
must therefore vanish whatever the
this limit
ratio
p
:
q
may
be
if
b
f(x)
dx
to
is
Cauchy*
have a
calls
finite
and determinate
value.
such integrals "Singular Definite" integrals [Integrates
which the subject of integration becomes
same time that the limits differ by an infinitesimal.
In order that p and q shall disappear, the first integral must be independent of JD, the second of q, when e is indefinitely diminished.
For example, in the case
definies singulieres], viz. those in
infinitely great at the
'
,
where a < c < b
;
here
and the
limit
when e =
Similarly for
is
/
*+
(
integral
See
/ is
zero
fa
fc+qe
and independent
- the
limit
is
of p.
independent of
g,
and the
x -c]\
determinate.
Moigno, Calc. Intig.
pages 128-135
91-107
<?./.,
Bertrand,
pages
p. 117, for further
information as to General and Principal Values.
Serret,
Williamson, Int.
Calc.
Calc.,
Int.,
;
;
;
353. Successive Integrations.
Successive integrations of a function
terms of single integrals.
Let u be any function of
Then
may
be expressed in
x.
will
where
*
Serret, Calcul Integral, p. 107.
D = dx
-j-
CHAPTER
352
FOF
IX.
V u== JLJ u
\\\
yj5
= x \u dx
l
^
and the theorem
is
I
xudx
v _l
therefore true
when n = I.
Also, integrating each term of the stated result, assumed
for the moment true,
,
r^.n+1
Ju
1
J.
H
rx n
1
J.
JL
JU
n+l
x n+l
i
--^xu ^
D n uJ
*\_n D
C1
,,
1
l
i
\
r n-l
1
1
,v.7i+l
^l^-D
[
and
n
n
"
_/'
1
rn
-i\n+i
/
-i\n+ii
v
_ nn
*'
L(
Hence, the right-hand members of the several brackets add
to
i\
/
iH-l
Therefore, multiplying
D
by
--...-i.e.
if
the theorem be true for the operator
integrations,
tions;
it is
true for
n+2
,
which establishes the inductive
i.e.
for
proof,
^v
i.e.
for
n + 2 integrafor
we have
GENERAL THEOREMS.
353
shown that it is true if n = I, whence it is true for n = 2,
and generally.
The theorem 'shows that a repeated integral such as
etc.,
udxdxdxdx
I
can be expressed in terms of single integrations of
I
u dx,
This theorem
is
I
xu dx,
I
x 2 u dx,
I
x 3 u dx.
given by Todhunter, Integral Calculus,
p. 72, q.v.
MISCELLANEOUS EXAMPLES.
1.
Integrate
(i)
L
..,
f
r^
:
logxdx
-
[)
J a?(l- log
Prove that
2.
-
*
[I,]
z)'
-
^
5
= 4c
-=TI4
,
(o>c).
-=- can be
fdx
ax
Find a reduction formula for
w = 4.
-3
case
se
4
I
cos
made
mx ain n x dx, and
rT
to
apply
-,
depend
it
to the
^
[L.]
dx
Evaluate
'
Prove that
5.
|
jo
can be
made
to
u-rdx
ax
depend upon
^ uw
&
f
v
I
Jo
Hence show that
/(.T)
-
,
aa;.
be an arbitrary polynomial of degree
r/WPnW^ =
then
where
if
^
-j
a, ft
E. I.C.
0,
are the roots, considered real, of the quadratic
CHAPTER
354
6.
Prove that the
function acos(nt
and
effect of the
+ c)
operation p-rj
+ q on
to multiply the amplitude' a
is
to increase the angle nt
Write down the
IX.
effect of
+
by tan"
e.
a periodic
by
1
.
the operation
and generally, of the operation
d2
d
on the same periodic function.
When f = ax2 + 2bx + c,
7.
1
yJa
- = -p.
ch" 1 /
,
s/ac-J 2
>/5
LONDON.]
prove that
yJa
/2
1
,
y
Idx
[!NT. ARTS,
1
-[= sh-
Va
v/6
1
-c
or
-
,
l i/*J
4
-i===.war
/-a
a
,
*Jb*-ac
the real form to be chosen, and deduce the value of the integral
in the degenerate case when a = 0.
[INT. ARTS, LONDON.]
i
8.
Find the limiting value
of
(!)*/, when n
Find the limiting value when n
the sum of the n quantities
is infinite
9.
71+1
."IT"'
71 + 2
~n~'
71
is infinite.
+3
^T'"
of the
n ih part of
71 + 71
~^T'
bears to the limiting value of the ?i tb root of the
product of the same quantities the ratio 3e 8, where e is the base
of the Napierian logarithms.
[OXFORD 1886, and I. P., 1911.]
and show that
it
:
10. If na is always equal to unity, and n
show that the limiting value of the product
(l+a
4
){l
+
4
(2a)
}^{l+(3)
i
4
}
{l
11.
Show
that the limit of the
3
7i
when n
12.
7i
Find U.
71
-
C
;,/-
271
-
[OXFORD, 1888.]
of
n terms of the
series
nz
3
._
is infinite, is
sum
indefinitely great,
+ (4) 4 } i ...{l+(7ia) 4 }
**.
is
is
^
GENERAL THEOREMS.
13.
Find the limiting value, when n
355
of
is infinite,
i
tan
{TTIn
.
tan
2rr
Sir
.
tan ----- tan
.
2n
'In
[OXFORD
14.
Show
when n
15.
is
increased indefinitely,
Find the
when n
limit,
is
is tffw.
is
a:
<
(n-l)x-\
secI,
+
...
n
case
when
is
j
x>.
Find the limiting value of
2 log n - log [(1 + ?i 2 )(2 2 + 7i 2 )"
when n
17.
[COLLEGES, 1896.]
.
Examine the
16.
2
.
.
.
(2?i
indefinitely increased.
Show from elementary
indefinitely,
1
)],
[OXFORD
considerations that
11 + "' +
+ +
3
approaches a
P., 1903.]
indefinitely increased, of
If sec -x + sec 2x +
n
n \I
n
where
I.
that the limit of the product
3
when n
P., 1900.]
increases
i
n~ logW
intermediate between | and
finite limit
I.
1.
[ST. JOHN'S, 1884.]
18.
If f(x)
=f(a + x), show that
Cna
and
1
fa
f(x)dx = (n-l)\ f(x)dx,
Jo
J
illustrate geometrically.
9.
Prove that
|
[OXFORD
%() dx = Jo
[
(a
</>
Jo
i.\
and show that
xsin n x
p"
-
f
TT
,
x)
=
dx
I.
P., 1888.]
%
sin M ic
7
n
^-dx,
2
a
l+cos 2^-dx=-2j l+cos x
= 1 and when n = 3.
integral when n
(1)'
J
aiid evaluate this
ft,
(2)
I
Jo
20. If
(l+sin
loeV
& l
2
2
*)'
.
+ ism
,
2
,
dx
=
TT,
log
2.
<(z)= -<j>(2a-x), show that
[COLLKOES.1886.]
CHAPTER
356
21.
provided
22.
~
<j>(x)
remains
Prove that
from x =-- a to x =
if
^z=
I
when x
finite
~
C c (j>(c-x)
C c <l>(x-b}
Prove that
IX.
<(z), ^(x),
j\dx,
vanishes.
be continuous and
$'(%)
</>'(-)>
[ST. JOHN'S, 1883.]
finite
b,
rb
dx =
<fi(x)\l/'(x)
</>
Ja
where
is
+
{a
0(b
^)}[i/'(6)
^(fl)ji
a positive proper fraction.
/T
23.
Prove that
24.
Show
a
/JT
#/(sin x}dx
I
=~
-a
/(sin x) dx.
I
[ST. JOHN'S, 1883.]
,
that
f /*$*(' -*)fa- [*/(*)*n (
~ x dx
)
c
a
where
25.
f
n
(x)
Show
th
means the
that, if
differential coefficient of f(x).
?i
^ (x)
=1
(a)
<#>'
(2a
-
x)
[7, 1893.]
<fo,
Jo
then
^(2a)
26. If
/(ic,
2^ (a) =
-
2
[^(a)]
symmetrical in
y) is
a;
-
and
<(0)<(2a).
y,
[TRINITY, 1895.]
prove that
b
f
l-x)dx =
xf(x,
1
f(x,
[
a:)
dx.
"Ji-b
Jl-b
[COLLEGES
27.
Examine under what
a, 1889.]
limitations the formula
a
r<j>(x)dx=
j
<^(x)dx+{
<f>(x)dx
holds good.
Show
that
f
J_
to
4>(x)dx.
(x + ~}<i>(x--}-=2r
X
_
\
X/
X/
\
J
oo
[MATH. TRIPOS, 1884.]
28. If
A^l + l + l +
1
Sm=
show that when n and
to a limit F,
m
An
2
1
+ +
4
...
+
^,
1
- + S'
are both infinite
_
and the
Rm = log 2 + log
^.
ratio
n
:
[COLLEGES
m
tends
a, 1888.]
GENERAL THEOREMS.
Show
29.
that
sin f,
1
j
A
lt
357
-Sin.,,
e ax
n
cos
+ a)x =
(bx
v
cos
/,^
e ax -
(bx +
-,
(bx
and explain how the
)
ft /
being arbitrary constants, and also that
etc.,
b\
a-n tan" 1 -
\
it
be written
may
+ a} +
latter operation is to
be conducted.
TT
30.
Il =[\og(l+a sm*0)d8,
If
l
Jo
II = -r log ( 1 + 04) + 7J/2
show that
,
jr
where
f log
f-'-.
(1
+ a. sin 2 B) d6
Jo
and
4(1
Hence show that
2
4(1+ a r+1 ) (1 + a r ) = (2 + a,-)
where
31.
Show
that
if
?i>l,
i
i
tanh
How
is
dx<-(\+
log n).
n
v
nx
Ji
32.
-
r
~
[OXFORD
I.
P., 1911.]
the equation
n
|
Ja
to be interpreted
Illustrate
f(x)dx=f(b)-f(a)
when f(.r)
is
not a single-valued function
?
your answer by evaluating
c
a2
J
where a and
b
are real and n
is
a positive integer.
[OXFORD
33.
Remembering
that
means the
I
I.
limit tended to b
by
first
of the
two
positive quantities
second to infinity, prove that
I
I
as
Je
Jo
the
P., 1912.]
if
e,
?/
tends to zero, and the
ft>l, the value of
n
ax
(a cr
-e-*)x
n ~ l <!.>'
Jo
is
/ero
if
n>0, but not
if
n
= 0.
[OXFORD
I.
P., 1917.]
CHAPTER
358
34.
If f(x)
IX.
be any function of x which can be put into partial
fractions of the
A
form
-,-
a-
then will
5,
-x-
V-
1
prove that
1
that
2
f% (a
36.
Show
[R.
R]
= (ab)^,
b
0<b<a, a
b<b l <a l <a, a -b <^(a-b).
= a - b tan 2 0,
(a + b) tan cot
35. If
Show
= ^(a + b),
if
l
l
<
+ i 2 sin 2 <9)
cos 2
fi
L-"
2
d# =
I
i
(a^cos^ + ^sin
*
2
^)
d<j>.
[MATH. TRIP., PART
II., 1915.]
that
w
f
tan- 1 (sin
sin
0}
dd = ~
(x/2
-
1).
[MATH. TRIP.,
37.
Show how
where E(x,
to evaluate \JR(x t y)dx,
rational algebraic function of the coordinates
.r,
conic.
38.
that
if
J
.T </.r
a 2 - cos 2 z
o
~
7T
.
'
2aja -
HIT! O* /7l'
xsinxdx
r*
f*
Prove that
<TT
2
[
1
Integrate
I
,
3,
2
TT
TT
-.--
XF
J
-
p
-
1890 -]
JOHN'S COLL., 1882.]
when
c lies
n
[Oxp.
I.
P., 1889.]
between a and k
[R. P.]
Prove that
Jx
n
(2
-
w
a?)
dx = 2 2
;l
(1
-
x)
n dx.
Jo
o
[Oxr.
43.
-
.
/--
tan~ V2.
~^- + cos2x
v/2
dx
-^
ri>
42.
JOHN'S, 1891.]
Prove that
J
41.
y of a point on a
2
2
r.
40.
any
a be greater than unity,
f'
39.
y) denotes
[ST.
Show
1882.]
II. P., 1886.]
Prove that
tr
TT
2
{?-
2 cos x $(sin 2x)
p-r
dx=\
<f>
(sin x) dx.
J2a
[ST. JOHN'S.]
GENERAL THEOREMS
44.
Show
that
=
6
f
a2 -
'
the equation of a
45.
c2
I
+C
J a -c V2(tt
is
359
+ r2
-
2
2
)7'
j
dr
---
-
2
(a
C
- r*
2 2
)
circle.
[MATH. TRIP., 1882.]
Find the integrals
sin's tan"
46.
[ST. JOHN'S, 1887.]
Prove that
r
log (1
+
tan a tan x)dx
= a log sec
"
[COLLS., 1896.]
Jo
47.
Evaluate
[HALL,
[THIN., 1891.]
.....
f
Ul)
/
1-S
6*
JnWl Vl^2^
2
,
f
.x
/
'
J
p- 1?
J
e2
sec
(V1)
(T+StS^
J
a;
11-i
show that
provided n
a
>
Show also
where
/(ft)
49.
Show
a;
,
~
x
[TRIN., 1884.]
(I-^'
In = 2n(2n -
-
l)/,,^
n(n
-
l)/,,_ 2
,
1.
that
In = -^i
{/(a) sin a
+ g (a) cos a }
and g(a) are algebraic functions
with integral
[HALL, LC.
cosec
log tan
.
[TRIN., 1891.]
2
7. (7.]
.T^^
f
-)
dx
(
\
(V)
^
,
,
of a, of degrees }> n,
coefficients.
[TRIN., 1892.]
that
2
f^
+
l
^
J ^Ti TH^T^
=
cos _iWa~2r
!
v/a^2
[HALL,
^rr
fl
(
7.6'., p. 3l>iJ
^
and
p. 346.]
CHAPTER IX
360
50.
Show
that
1
f
Jo
51.
Prove that
1
52. if
prove that
/1\ 2
11
CHAPTER
DIFFERENTIATION,
X.
UNDER AN INTEGRATION
ETC.,
SIGN.
354. Differentiation of a
Definite
Integral with regard to a
Parameter.
A
definite
value
integral
is
nature independent of the
in terms of which the
its
by
the
of
variable
particular
integration is effected, and its value depends upon any
other quantities which may occur in the integrand or in the
limits.
First, let us consider the differentiation
with regard to
c of
Cb
the integral
u=
I
Ja
<f>(x,c)dx,
and independent of
a
to
b.
change
with regard
finite
shall suppose also that <f>(x,c) is
as also its differential
for the range of values of x from
c+Sc, suppose that the consequent
to c
When c changes to
of u is to u+Su.
u + Su =
Then
each
b are
and continuous,
single-valued, finite
coefficient
We
c.
where a and
rb
</>
\
(x, c
-f Sc)
dx
Ja
rb
and
u =\ [0 (x,
c
+ &)
<j>
(x, c)] dx.
Ja
Now
0(
where the accent represents differentiation of
regard to c, and 6 is a positive proper fraction,
written for c after the differentiation
3u
^=
T
Su
Mjc=<>-,
_
=^r=o f
is
b
,.
.
performed,
.
(x,c+9Sc)dx=
361
(J>(x, c)
6
f
1
with
c+9 Sc being
i.e.
30(a?,c),
'.:.
-dx.
CHAPTER X
362
a and b be
355. Next, let
also functions of
c.
+6&
(j>(x,
c+Sc)dx
+&a
cb+sb
Su =
and
\
Ja+Sa
rb
<f>(x,c -\-Sc)
dx
\
Ja
$ (x,
c)
dx
ra
Jb+Sb
b
<j>(x,
c+Sc)dx+\J
(j>(x,c-}-Sc)dx
a+Sa.
Cb
+ J a [<p(x,c+3c)-<{>(x,c)]dx.
Now
b
Jb+Sb
and
<j>(x,
c+ Sc) dx = <j>(b+0 1 Sb, c+Sc) Sb
fa
Ja+Sa
(j>(a+9 2 Sa, c
(j>(x,c+Sc)dx=
where O l and
2
+ Sc)Sa,
are positive proper fractions.
^
Also
(by Art. 332)
oe
has been discussed in the last
article.
Hence, dividing the expression for &u by Sc and taking the
limit,
when
Sc is indefinitely diminished,
and the conditions under which
above, viz.
(}>(x,
^
and
c)
'
this is true
are single-valued, finite and
range
x=a
a case of the theorem on partial differentiation,
Diff.
continuous functions of x throughout
to
xb,
This
have been stated
the finite
inclusive.
is
Cole., Art. 1GO, viz.
du_^u
'du
da .du db
dc
'da
dc
'dc
ob
dc
356. Geometrical Meaning of the Process.
We
next examine the geometrical meaning of this differen-
tiation.
Let
a/3, a/3'
be the respective graphs of
DIFFERENTIATION,
UNDER INTEGRATION SIGN
ETC.,
363
Let the ordinates of both curves be drawn at the points
viz.
Aay,
respectively.
Let
B(3S'
be any other ordinate, and draw
NQP
Then
aS, /3R parallel to the z-axis.
We
by the area ABpa.
&SP,
A'ya',
;
f
6
Ja
</>(x,
c)dx
is
represented
have to differentiate this area with
ft'
regard to
c.
When
dependent upon
^
c,
c is
increased to c+Sc, a and b being both
area AB/3a
Ajto
area AB/3a
=^
T4
6c=0
is
changed to
area
- A'B'fi'a
A'B'/3'a,
area
and
AB8u
-s
fiSya 4- BB'/3'S'~ AA'a'y
-
Now
.
-i:
Also
OC
CHAPTER
364
X.
-
T AA'a'y
T .AASa~\-aSay'
*--^LtLt
and
Sc
Sc
'
c
-
Sc
Now,
~~
area
T
/BRfl'S'
area aSa'y
the terminal ordinates A'a,
if
Aa and
B' ft'
Bfi are
,
finite, as supposed, the portions /3R/3'S' and aSa'y' are both
of the second order of infinitesimals, for their breadths and
greatest lengths are both first order infinitesimals and therefore, when divided by Sc, they still remain of the first order
;
of infinitesimals
and disappear when the
limit
is
taken.
.db
.
^
dc
The student
will see that the truth of this
not be asserted without further examination
ordinates
became
of the figure
infinite, or
if
da
dc
theorem could
if
any of the
either of the
graphs were discontinuous, or if either graph were cut by an
ordinate in more places than one for any position between
the extreme ordinates of the portion considered.
When
true,
one of the limits
the theorem may
needed in each case.
is infinite
but special consideration
is
still
be
357. If the integral to be differentiated with respect to c
i.e. the limits not stated,
say
be "indefinite,"
u=
where
A
is
A
</>(#,
c)dx-\-A,
an arbitrary constant, then
du_
and
I
fd0(x,
c)
,
^
being an arbitrary constant as regards
x,
?
- is
also
DIFFERENTIATION,
UNDER INTEGRATION
ETC.,
an arbitrary constant as regards x
result as
where A'
SIGN.
365
and we may write the
;
,
du
is
an arbitrary constant.
358. Integration of a Definite Integral with regard to a Parameter.
Take the
u=
integral
\
J a
$(x
where a and
b are not functions of
Then, by the previous articles,
^
I
I
<p(x, c)
dc \dx
dx,
c)
t
c.
=\
<-/>(, c)
C
b
<l>(
\
dc dx
x c)dx = u;
>
Ja
.'.
r
i.e.
I
(j>(x,
c)
I
u dc =
I
<j>(x, c)
dx \dc =
fI
^ (x,
c)
dc dx,
dc
dx.
Supposing that instead of an indefinite integration of u
require a definite integration between c and c, say, regarded
as independent of a and b, then we shall have in general
359.
we
f
that
is
</>
(x, c)
dx] dc =
the order of integration
For putting
then
Jc
$(x,
0(a;, c)
and
sEllj^*
c)
c)
(j>
is
immaterial.
dc=f(x,
dc\ dx
=
dc dx
=
=
r
ic
c
r Cb
LJa
dx,
/ (x
>
c)
dx
6
4>fac)fa
t
Ja
o
c)
b
\
also
say,
c),
f(x,
^c\
]
(x, c)
Cb
<f>(xf
c)dx\dc=\Ja <}>(x,c)dx.
dc dx
CHAPTER
366
Hence both
X.
<j)(x,c)dx \dc
and
(j>(x, c)
dc \dx
|
have the same
c=c
may be
This theorem
fc
c
J
Hence they are
.
c,
and
equal.
written
1*6
(/>
J
with regard to
differential coefficient
both vanish when
(x, c)
dc dx
fb Cc
=
I
(j)(x, c)
I
J a J
it
dxdc,
c
and expresses that the order of the integrations may be
changed. The theorem presupposes that the limits of integration c and c are independent of the limits a and b, and
also that </>(#, c) remains single-valued, finite and continuous
for all values of the quantities x and c between or at their
limits.
360. Notation.
The notation
of this "double integration" calls for expla-
It will be noticed that
nation.
J
c
LJ a
(/>(x, c)
dx \dc
we have
as
written
(j>(x,
J
c)dcdx,
J a
c
inverting the order of the dx and dc. The order of writing
these symbols does not appear to be universally agreed upon,
some authors adopting the opposite
order.
For the sake
of
we may state that throughout this book the righthand element and the right-hand integration sign refer to
the first operation, the left-hand element and the left-hand
clearness
integration sign refer to the second.
f
1
CVi
Thus
<p(x,
y)dxdy
will
mean
that
JzoJj/o
to be integrated with regard to
constant, between limits y=y^ y=y\>
(1)
<j>(x,y) is
(2)
That the
A
is then to be integrated with
x between limits x = x and x=x
l
notation whiqh carries
there
is
keeping x
result obtained
regard to
when
y,
any
its
own
fear of confusion,
f*i
J Xn
explanation, and used
is
CVi
dx\J
dy<j>(x,y).
|/n
.
DIFFERENTIATION,
ETC.,
UNDER INTEGRATION
SIGN.
367
361. Geometrical Interpretation.
Writing y where we had
to establish the theorem
c in
c/>(ic,
c)
and dy for
dX
f []a^ ^ } dlJ= la LL ^' ^
X)
dc,
we have
^^
Imagine the rectangular space bounded by
x=a,
x=b,
y=c
,
y= c
up into infinitesimal rectangles by two families
of straight lines, the first set being equidistant from each
other and parallel to the x-axis, and the second set being equidistant from each other and parallel to the i/-axis, the distance
to be divided
between consecutive
lines of each family
R'S'
being infinitesimal.
CHAPTER
368
Then,
X.
we sum the strips from y=c
when &y is indefinitely small,
if
the limit,
to y
=
c,
we have
in
we first sum the elements <f>(x, y)SxSy along the strip
we
have in the limit, when Sy is indefinitely small,
RSS'R',
But
And
if
if
the limit,
And
we sum these strips from x = a
when Sx is indefinitely small,
to x
=b
we have
in
as the order of addition of these elements
is obviously
perceive that these two results must be equal.
Hence the truth of the theorem, provided <j>(x, y) be finite for
all points of the rectangle.
immaterial
we
362. Successive Differentiation.
Having
established the equation
.
c
we can differentiate again and again and successively obtain
the second, third, etc., differential coefficients with regard to c.
The successive results however, in general form, rapidly get
complicated. Thus, for instance, we have
which reduces
an expression with seven terms.
Similarly, the third and other differential coefficients
be found when necessary.
to
In particular cases there
may
may
be considerable simplification.
DIFFERENTIATION,
UNDER INTEGRATION
ETC.,
SIGN.
369
important results can be derived from these
and new forms deduced, by differentiation or
integration with regard to letters which have been regarded
363.
Many
theorems,
as constants in a previous integration.
Ex.
1.
For example, taking the case
(a
^
we
> b)= -*:
[~
-s/aT^L.
tan-i
tan fT( Art.
A/^|
2_Jo
ya + b
v
= - T^=
Va 2 -6 2>
have, upon differentiation with regard to a,
\
3 /
TT
=
3a \Ja* &)
dx
f
Jo (a + bcosx)
2
'
'-f> - ;-
.
Differentiating again with regard to
T
Tra
?
+ b cos a;) 3
with regard to
r
5
a,
3
I
afo?
(a
J
or,
171)
6,
cos x
dx
Jo
j" a'
Hence,
+ b'cosx
/
Jo
Generally,
dx
f.
f:(a + b cos #)"
Ex.
2.
Clearly
je
(w
dx =
-
1
)
!
e
-^>
Also
a
[Int. Calc.for Beginners,
Show
Ex.
that these results are identical.
3.
Starting with
e~
/
ax
dx = -
Jo
we have
/
a e -**dx
Jo
hy n
differentiations with respect to a.
E.I.C.
2A
'
Art. 213.]
2.
CHAPTER
370
Ex.
From
4.
X.
such integrals as
dx
dx
C
I
I
J
we can deduce
.
dx
C
.
,
dx
C
.
*
J
<**
or
Vo.r2 + 26^- + c2
(3)/
<&
or
by
1 times with regard to
respectively differentiating the first ?i
or once with regard to c,
n l times with regard to
once with regard to c 2
or the second
or
integral has been found (Chap. VIII.), and this
be more convenient than the employment of a reduction forDifferentiation with regard to other letters, p, a, 6, a lt b^ a% or
will often
mula.
6 2 , will give other integrals.
For example, by Art. 276 (supposing bp>aq, a and p
2
.;-.
aq)
\'a(bp
la
Vp
positive),
J> px + g
ax+ b
Therefore
and
.
etc.
5.
If
$ = \/(
2
+ A)(6 + A)(e + A),
2
2
ry_j
Jo
._
.
We have
\
2
L_
&2
+A.
2o?0
~^=
+A
prove that
MV
'i
c-
+A
A/
c^A
^~
/.
c 1}
,
when once the primary
Ex.
q,
the integral in question
is
r/2rf_l.\
W<*A Aj
Jo
Similarly
n
2
Jo
abc
DIFFERENTIATION,
UNDER INTEGRATION
ETC.,
SIGN.
371
and the above equation may be written
37
37
97
_
W+ WT^5f""iK:
1
For several useful illustrations of such
on the attraction of ellipsoidal
Routh, vol. II., pp. 100-101.
integrals,
shells, see
which occur
DIFFERENTIATION OF A MULTIPLE
REGARD TO AN INVOLVED CONSTANT.
364.
It will
in
problems
Analytical Statics, by E. J.
INTEGRAL WITH
be sufficient to take the case of a multiple integral
of the second order.
1=
Consider
I
dx\
Jxo
where
xv y y l are
c, a?
not involving x or y.
,
,
Let
(x,
J
where x
is
all
<f>(x,
y, c),
functions of some quantity
y, c) dy
= F(x,
t
but
y, c),
regarded as a constant in this integration, so that
dF(x, y,
l
then
dy
Jyo
</>
(x, y, c)
c)
_
dy = F(x, y v c)- F(x, y
,
c)
= v,
say.
J Vo
I=Tvdx.
Then
Differentiating
by the
dl
rule of Art. 355,
f^cto
,
.
where v and v l are the values of v
x and x l respectively.
Thus, substituting for
~, we
dxn
dx,
when x
have
receives the values
CHAPTER
372
This
may
X.
be written in the more compact form
"'I
A
similar
process
may
be applied in cases of Multiple
It is to be understood that all
Integrals of a higher order.
limitations with regard to the nature of (j>, and the range of
integration, which correspond to those described in Art. 355
for the case of a single variable, are supposed to be assumed.
365. REMAINDER AFTER n TERMS OF TAYLOR'S SERIES
EXPRESSED AS A DEFINITE INTEGRAL.
Let f(x) be a function of x which is finite and continuous
j
throughout the range of values of x, from x=a to x=a r h,
as also all its differential coefficients as far
Let
x= a-\-h
z
as/^x).
be an intermediate value of
x, (
<
k).
fh
Considering the integral
or
I
f (a -\-hz)dz,
we may
r
h
Jo
f(a + hz)
~\
(1)
integrate directly as
(2)
apply the rule of continued integration by parts
(Art. 95), viz.
z3
^/
~"1
O
I
O:
./
=f(a-\-h)f(a),
DIFFERENTIATION,
Hence the remainder
R
By theorem
for
some value
mainder
after
UNDER INTEGRATION
n terms
SIGN.
373
is
=&=vif~lfM(a+h
- z) dz
-
IX., Art. 331, this is equal to
of
be written f=(l
hn
Hence
ETC.,
Rn
(
lying between f=0 and =h, which may
is a positive proper fraction.
0)h where
/(n) (a-|-$^), which
is
Lagrange's form of re-
(see Diff. Calc., Art. 130).
366. REMAINDERS AFTER (n + 1) TERMS IN LAGRANGE'S
THEOREM AND IN LAPLACE'S EXTENSION, EXPRESSED BY
MEANS OF A DEFINITE INTEGRAL.
It is easy to
find
an expression for the remainder after
in Laplace's extension of Lagrange's theorem
Art.
Calc.,
518).
(Diff.
Lagrange's theorem states that if z = y+x</>(z) and u be
any function of z, say /(z), then the expansion of u in powers
(n+1) terms
of x is
and Laplace's extension states that
if
z
=
and contains the former as a particular case.
Take then z Fty + xi^z)}, and consider the integral
<S
J
where
c
v ;
Jf'(v)
.
dv
CHAPTER
374
We
X.
shall write </>Ft for </>{F(t)}, etc., to avoid the multi
plicity of brackets.
Putting
7i
= 0, we
have
Again, differentiating I n with regard to y (Art. 355),
dl
-[x<j>Fy](f'Fy)(F'y)
Putting n
...
1, 2, 3,
etc.
successively in this result,
;
whence
= etc.,
and
f(z)=fFy+x(<t>Fy)
(fFy)
+
y
The remainder sought
is
therefore
DIFFERENTIATION,
UNDER INTEGRATION
ETC.,
SIGN.
375
This includes, as a particular case, the remainder after
terms in Lagrange's theorem, when z=y-\-x<f>(z), viz.
"
Rn
-
1
s
-
[y +x<t>(t)
-w
(t}
dt>
by Prof essor Williamson (Encyclopaedia Britannica, "Infinitesimal Calculus," 151) as due to M. Popoff (Comptes Rendus,
1861), the demonstration of which by M. Zolotareff, quoted in
cited
the Encyclopaedia Britannica,
is
similar to the above.
GENERAL EXAMPLES.
1.
Prove that
a^xn dx =- a- 1
\_\n
1
+
l
+pa*'
)
*-
-y + q
\n+l
arid verify
the result by performing the integration
A
be the area bounded by a parabola and
2.
(la),
If
prove
fa
(1)
(2)
by
differentiating the integral 4
by
first
1
Jo
first.
its
latus
rectum
_
>Jaxdx with regard to
a,
integrating and then differentiating with regard to at
dA^lQa
da"
3.
Apply the method of Art. 355
3
to prove that
cVf
d
and explain geometrically each step of the process.
Obtain the same result by first integrating and then differentiating
the result with regard to c ; and also geometrically.
w-8.-a.-i,
provided a be positive.
5.
6.
Show
If f(x
that
[TRINITY, 1888.]
~^_ f(x
+ c) =f(x)
+ c)dx = 2/<-(2c).
for all values of x,
^
1883>]
[a,
1887.]
show that
f(y + az)dy
1Co
is
independent of
z.
CHAPTER
376
_ _^
X.
Prove that
7.
f*
dx
2
J
8.
2
(a cos'
z
+ /^sin 2
Prove that
if
u=
~
1
TT
2 n+i
(-!)/! 3 *
[w
>Y
1
\a 3a
l
m_i
-"'
,
'
djg/
"
(a
n
+b
>l
F{(a
\
)
J
-
b)x} dx,
/3
where F denotes any function, /3 and a being independent of a and
and n being a positive integer, then
b,
[OXFORD, 1886.]
9.
If
o
where
c is
a function of u and
prove that
#,
{
dx~
10.
'dc
X
>
C
<
[5,1885.]
u
If
where a and
11.
~^
/3
are functions of x and u, prove that
Comment upon the application of the rule of Art. 355 to the case
d
f
*
$(x)dx
da] -a v/ft 2 - x'2
Prove that in
'
this case the true result is
x<j>'(x)dx
12. If
-
we have
Do you
-7a
'
* F/ ~ x
(
consider that this formula
^(a)
If so, to
^^ ^^' ^
= a,
/(a)
=
what extent and
and
in
<j>
~^
fails in
(6, a)
dF
"^
-
1
^
the case in which
=
what respect 1
-*- f
\cos
-
cos a
DIFFERENTIATION,
Prove that in
ETC.,
UNDER INTEGRATION
SIGN.
this case
du
Make any remarks
sin 2 S
sinaf^
dO
that occur to you as to the reasons for the
assumes in
peculiar form which the general formula
this case.
[e,
13.
Show
1884.]
that the equation
ceases to hold for x
14.
377
= Q.
[MATH. TRIPOS, 1897.]
Find a curve in which the abscissa of the centroid of the
area of that portion bounded by the curve, the coordinate axes and
an ordinate is proportional to the abscissa of the bounding ordinate.
[COLLEGES, 1878.]
15.
A
vessel in the
and a
axis
flat
form of a right circular cylinder with vertical
is filled to varying depths with liquid
horizontal base
If the depth of the centre of gravity of the
of varying density.
liquid be always
71
of the immersed portion of the axis,
show that
2-n
the density varies as (depth)"- 1
.
16. Find the general equation of all solids of revolution for which
the distance from the vertex of the centroid of a segment made by
a plane perpendicular to the axis, is proportional to the height of
the segment.
17.
Find the form
[TODHUNTEB, Integral
of the curve for
Calculus, p. 198.]
which the area bounded by
the curve, the coordinate axes and an ordinate is such that the
moments of inertia of this area about the coordinate axes are in a
Constant ratio.
1
S.
A
body moves from
rest at a distance a
towards a centre of
Show that the time of
^traction varying inversely as the distance.
n
will
a maximum when
the
a
be
between
and
describing
space
fia
fi
[/dx\
It
nuiy be assumed that
I
-j-. }
^
~\
(i
<x
\dt]
[TAIT
log
-
.
x J
AND STEELE, Dynamics of a
Particle.}
CHAPTER
378
X.
19. Find the density of a parabolic plate as a function of the
abscissa in order that the distance of the centroid from the vertex
as the square root of the length of the plate.
may vary
20.
Find the equation
by the ordinate
at
[a ,
1881.]
of the curve such that the area included
point, the axis of x
any
and the curve
in a
is
constant ratio to the area included by the ordinate, the axis of x
and the tangent.
[MATH. TRIPOS, 1882.]
21.
Prove that
Under what circumstances
will
,'- x be independent
I
J
[ToDHUNTER,
m
2'2.
tan
1
Int. Calc.]
-K =
sm u
s
i+^
2
j
""
Tr
It
of a
\/a
o
= x*
sin
-=
/I
/
i=iin
2
verify that
!
J
^
^
^_ ^Jo
f v/l-x8
f
i
2
O
\/i_tan^
s
inV
--''[*
J
,
Jl - tan^O sin^
\
[MATH. TRIPOS,
23.
1896.]
Prove that
cosbx
o
a
2
00
_
,
cosbx
f
7-^
2
1
,
+ x*
J
o
(a
,
,
2
+ a:2^dx-Sa
)
-
C
9
2
\
J
o
cosbx ^
o
o^^ =
2
2
(a + x )*
_
0.
1884.]
[e,
24. Verify that
o
satisfies
i
the differential equation of the hypergeometric
-(i-)g + {r-(^+iM|-^=o,
when
ft
>
25. If
and 7
u=
>
ft.
i*
{ e
cose
Jo
{A+E\og(xsin
2
verify that
d u du
x~r^ + -r
CiX
CLX
-
2
q xu==Q.
2
6)}de,
series,
f
*
DIFFERENTIATION,
Prove that
26.
satisfies
=-
cos <f>Jx 2
+
(x
\
^Jo
n
-I)
d<j>
that the differential equation
d2u
..
379
SIGN.
the equation
Show
27.
y
UNDER INTEGRATION
ETC.,
c
,
u=
,
is satisfied
by
a
asmz + bcosz
f
\
y=
28. If
I
_
dz.
X+3
Jo
Write down the complete
b
[Sx. JOHN'S, 1883.]
solution.
Jxe nxcose cos{v/7 log (Jxsin
(9)
+ a} dO,
Jo
prove that
29.
-
rfx
-
2'
-
Vy
-5
v
V
.
[Sx. JOHN'S, 1889.]
Prove that
f*
I
2
(cosh
a;
2 "- 1
2
- sinh x cos
a</>
</>)
^cf>
f'
=
I
J
Jo (cosh x
Prove also that
- sinh a; cos
~
2^+1*
2
<#>)~
if
P=
- sinh
(cosh x
I
a;
cos 4>)^ d<f>,
Jo
[a,
30. If
-=1 cos^^sin^cos"^)^,
Jo
X
~+m
z
d y
equation
2
n2 x 2n
~2
prove that y
satisfies
.d
1886.]
that
the limits are given
by
e
ux
VU
l
+
provided
the
= 0.
[a,
\'erify
1886.]
2
a^ 2 - a 2 ^ = 6 2
.
M
)
0, satisfies
the equation
+ ( a o + V) =
2/
>
[SWTZBR, OeWc,
vol. liv.]
CHAPTER
380
X.
be positive
32. Verify that if x
*
*(f>
and
if
a;
_
-l
g2)f
fa
+ C2
f
e*<
_
(J2
02)1-1
^
be negative
n
-i
r
*-i
f
?
2
2
J,
solves the differential equation
d2u
X -y-sz +
dx
33.
du
o
iCM
ft -j
dx
= U.
[PETZVAL.]
Prove that
A
I
/i
sm
J,_
cy
COS a
arc cos
7/1
7:^6
sin 6
/t
= 7T
-^ (I -cos a).
2
V
o
[TRINITY, 1886.]
34.
Prove that
f
1
,
J
35. Establish the
1
+ ax
1
- ax
xjl -x z
known
dx
~
.
_l
[OXFORD
:
1888.]
[MATH. TRIPOS,
1883.]
result
,&=^
ffi
and hence prove that when n
is
.
a positive integer
111
1
I
2
__
22
32
111
a function of
36. If the operator A, applied to
changing a to a
that
of
+
1,
and subtracting the
6
A
has the effect
original function,
show
6
f
I
a,
<(z, a)dx
Ja
where a and b are independent
Prove that
--
f
=
I
A<(z, a)dx,
Ja
of a.
f
-
1
V
1
[BERTRAND,
>
C./., p. 1S. >.]
DIFFERENTIATION,
u=
Given
37.
+
*
Jo
~
ETC.,
dx, differentiating twice
rx
d-u
da*
But
this
indeterminate
is
UNDER INTEGRATION
2
cos ax
SIGN.
381
we have
~ dx.
,
l+X
Q
when x
Discuss the validity
is infinite.
of the differentiation.
Gal. Int., p. 181.]
[BERTRAND,
38. Is it true that
If
why
riot,
not
1
Evaluate each side separately and compare the results.
[BERTRAND, CaL Int.,
39. If
P + iQ = <j>(x + iy),
Examine the
case
show that
in general
+ iy) = e~-^ x + ^, taking a = 0,
1
<
(x
p. 181.]
= and b = oo
a
.
[BERTRAND.]
40.
(l-ajsin
ff/(fc)-(2-aJ)*|
Jo
2
^^,
show that
f
1
ir
=
^|^ i(2-z)-H
Hence show that
"*
.
= to
oo
c
as x increases from
to
1
,
/(*) increases from
.
v/2
41.
[C. S., 1898.]
Prove that
"
"
du
f
Jo
[
du
Jo
p
Jo
du
.
.
.
["
Jo
duf(u)
=
f
-Lp,
*/ *Je
\*~
there being w integration signs in the left
(7
- )-
member
V
of the equality.
[R. P-]
42.
Show
that
^ {J!
* (* +y * c)
J!
rfaj
^}
" 9</>(3c) ~
8</>(2c)
+ ^ (c)
[OxF.
'
II. P., 1890.^
CHAPTER
382
43.
Show
X.
that the quartic function
can, in general, be expressed in three different
two squares
ways
as the sura of
P + R^, where
P = aT* [(ax + 5) + 3 (ac - V) - 2A]
2
2
5=
and
A having any one
fl~
f
A~*[2 (oa? + 5) A + a*d Sale +
of three determinate values
2ft
8
],
A T A 2 A3
,
,
.
.
-y
J^
A = (A 2 - A3 ) (A 3 - Ax
where
)
(Aj
in the
form
- A ).
2
[MATH. TRIP.,
44.
Show
1897.]
that
1
s,n + r-l
[i.
as.,
1892.;
CHAPTEK XL
PRELIMINARY TO INTEGRATION OF
>
WHERE
Q IS A RATIONAL QUARTIC. DEFINITIONS
OF ELLIPTIC FUNCTIONS. ELEMENTARY CONSIDERATIONS.
367. In
many problems
of both pure
and applied mathe-
matics, such as the investigation of the length of an arc of
an ellipse, or of a lemniscate, or the time of a finite oscilla-
an ordinary simple circular pendulum, integrals occur
which the integrand contains a square root of an algebraic
function of higher degree than the second.
tion of
in
Now
where
the integral
Q
is
["/TV
the general biquadratic function
a
cc
4
-f
4a x x3 -f 6a.2 cc 2 -f- 4a3 # -f
4
,
cannot in general be integrated by means of the circular, inverse
circular, or inverse hyperbolic functions, though it has been seen
that for particular values of the coefficients this may be possible
for no such function is known which will, on differentiation, give
;
-=
rise to the general expression
as its differential coefficient.
Hence, in discussing such an integral as this, we are in a
position similar to that which would have occurred if we had
====
1*
= before the inverse circular
2
ffl
Ja+bx+cx
had
^ = would
been discovered.
or inverse hyperbolic functions
integration even of the case
f
dx
I
J \J i
383
x
The
then have pre-
CHAPTER
384
XI.
sented a difficulty. And the necessity for the consideration
of such an integral would have formed a suitable startingpoint for the investigation of such functions as would have
-,
x2
vl
or,
mere generally,
for their differential
.
2
\la+bx-\-cx
coefficients.
And
built
the whole theory of such functions could have been
up from
368.
For
this starting-point.
instance, let F(x)
=
Then F(0)=0.
Let x and y be two variables connected by the equation
-_
_
jf=grji=^r
F'(x)dx+F'(y)dy =
The integral is F(x) +F(y) = constant = F(z), say, where z is
0.
i.e.
the value of y
when x
vanishes.
But multiplying by
and we can integrate
xjl-y 2 +{x
J
V
Vl
y
2
>/l
this
x 2 Vl
by
y
2
,
parts, viz.
dy+yjl-x 2 +(y
I
.
A
Vl
dx = constant
x2
= C,
and the part under the integration sign vanishes.
2
z
Hence, x\/l y -\-yJl x = z, say, where z is the value of
y
if
x vanishes.
Hence we have the addition equation
we then choose to write
symbol) for F, we should have
and
if
sn-
a;
1
or writing sin-
sin- 1 (a supposed
sn-^= sn
^^^ and sm- y =
1
<j>,
unknown
INTEGRALS.
ELLIPTIC
385
and we should thus have .arrived at one of the fundamental
propositions of trigonometry, and could have built up the
general theory.
Such
is
actually our position with regard to the integration
M
N
are
and
-^ -j=, where
J** vQ
rational integral algebraic functions of x, and Q is a rational
integral algebraic polynomial of degree higher than the
second, say the quartic
of [-7=, or,
more generally,
I
i\'Q
Q = a^ + 40^ + 6a 2 z2 + 4a3 z + a4
,
and the absence of knowledge of any function which, upon
differentiation, would give a general result of this kind long
barred the progress of geometers.
was natural that after having exhausted the discussion of integrations which could be expressed algebraically
or by means of logarithms, or by inverse circular functions, that
369. It
in terms of arcs of a circle, that investigators should turn
their attention to such expressions as could be integrated by
means of arcs of an ellipse or a hyperbola. Thus Colin
is
Maclaurin, in his Fluxions, vol.
/->
discusses "the fluent of
=," or as
-/-^
2jxx
written
Jx dx
;
^JVz 2 1
1 f
?
i.e.
it
1742,
would now be
1
x dx
If
,
Art. 799, of date
ii.,
j77*
<r|-7= 2
*J\/x(x
,
which he expresses as
1)
the arc of a rectangular hyperbola of semi-axis unity, viz.
drawing a tangent at the vertex A of the hyperbola, centre C,
and a
circle
with the same centre and radius x cutting the
A
tangent at the point
M, then
the hyperbola at E, arc
letting the bisector of
AE-~\j=
J *Jx\x
,
ACM cut
which we leave
1)
to the student to verify.
870. The real starting-point of the general theory of such
integrals, which have been termed Elliptic Integrals, from
their intimate connexion with that curve, may be taken to be
Fagnano's discovery* that upon every
it
is
possible to assign in
an
infinite
ellipse or
number
*Fagnano, Produzioni matematiche,
K.l.C.
1>| 5
torn.
i\.
of
hyperbola
ways two
CHAPTER
386
XI.
whose difference is equal to an algebraic expression, and
that the lemniscate "jouit de cette singuliere
propriete, que
arcs
peuvent etre multiplies ou divises algebriquement,
arcs de cercle, quoique chacun d'eux soit une
ses arcs
comme
les
transcendante d'un ordre superieur."*
371. Definitions. Various mathematicians,
Euler,f Lagrange, J
others, turned their attention to this matter, and
Landen and
much progress was made. But the chief advance was due to
the investigations of Legendre, first in his Memoires sur les
Transcendantes Elliptiques, 1793, and, after a long interval, in
his Exercices de Calcul Integral, 1811.
In this last work he
treated the general reduction of the integral
Pdx
JQ'
where
P
is
any
rational function whatever of
x,
and
Q
is
the
quartic function
showing that in all cases the integration may be made
depend upon that of three fundamental integrals, viz.
U(0,k,n) =
"*
where
_
A
\/l
to
k* sin 2 0,
which he calls the " Elliptic Integrals of the First, Second and
Third kind respectively," k being a real constant quantity less
than unity, called the modulus, and n any constant whatever.
372. Legendre in a footnote, (pages 18, 19) of the Exercices suggested
names for these functions, but it does not appear that the names were
generally adopted, except as to the initial letter E and II still used for.
the second and third. He remarks
"Ces fonctions r^unissent un si grand nombre de proprietes, que
:
*
Legendre, Exercices de Calcul Integral, 1811.
Euler, Novi. Com. Petrop., torn.
J Mem. de Turin, torn. iv.
f
vi. et vii.
Math. Memoirs, by John Landen, 1780.
LEGENDRE'S STANDARD FORMS.
seront plus generalement connues, on jugera sans doute
leur imposer un nom particulier, et de designer la
elles
quaud
necessaire
de
fonction de c et
est x,
ou
387
<
egale a
J-^> cornme on d6signe
1'arc
dont
le
sinus
nombre dont
le
le logarithme est y. 'II semble qu'on caracen lui donnant le nom de Nome,
bien la fonction
F
teriserait assez
parce que cette fonction a la propriete de regler tout ce qui concerne
la
comparaison des fonctions elliptiques. Peut-etre conviendrait-il en
meme temps
E
de donner
les
noms d'Epinome
et de
Paranome aux fonctions
que constituent les deux autres especes."
et II
373. Legendre established addition formulae for each of
these functions analogous to the trigonometrical formulae for
sin($0), cos(6(j>), whence their whole theory may be
deduced, as for the ordinary circular functions of trigonometry, and their numerical values calculated and tabulated
This having been done, they
for definite values of k and n.
are available for numerical use, as in the case of the circular
and inverse
circular functions.
<Q
374. All three of Legendre's standard forms are compre-
hended
in the
one formula
fQ ^A
TJ
rl
\
\
I
-J-
I
JJ oin2)
bill 17
~D
~.
[~~\9
The
/7/3
U/U
.
^^
cases are
A = 1,5-0,
n = 0,
A=l, 5=0,
375. The
"
H=F(9,k),
H=tt(9,k,n).
Complete Values."
The Eeal
Periodicity.
The function
obviously goes through
from
to 27T,
in the second
the
first,
values four times, as 9 increases
cycle.
quadrant are merely repetitions
passed through in the reverse order.
It is clear
and that
all its
and then repeats the same
then that
The values
of
those
in
CHAPTER XI
388
We may
HJ
call
the quarter period of the integral H.
L
In the case of the
r?
integral
=
JQ
I
.
real quarter period
first elliptic integral,
is
of F(9,
this
"
complete"
denoted by F, or K, and called the
k).
"
"
Similarly, l and IIj are written for the complete integral
of the second and third kinds respectively, i.e. when the limits
E
are
and
and
-^,
E(0, k) and II
E
1
,
IIj are
the respective quarter periods of
(0, k, n).
r
Wl-
do
do
1
& 2 sin 2
=K-F
1
analogous to cos"
^^^
sin" 1 ^
In this respect these integrals resemble the length of the
arc of an ellipse, or of any oval symmetrical about two perIn fact, as will be presently shown, one
pendicular axes.
them, E, represents the length of an arc of an
measured from the end of the minor axis. And
of
this particular fact that led
functions.
It
will be
noticed
that the
numerical until the values of
tions of
Jc
and
Legendre
Jc,
to style
them
ellipse
it
was
Elliptic
"complete" values are not
n are assigned, but are func-
n.
376. It is not the object of the present chapter to discuss
of
elliptic functions at length, nor to establish the mode
reduction of
I
=
to one of the above canonical forms.
These
matters, as well as the addition formulae, will be postponed
The present chapter must be regarded
for later treatment.
as an introductory description of such functions, so that the
student will gradually grow accustomed to their use in cases
that
may
appear in treating of the rectification of
and other curves.
ellipses
JACOBI'S ELLIPTIC FUNCTIONS.
389
377. The Jacobian Notation.
s\
7
In the integral u
x
is
usual to call the
and write
u,
it
as
with the usual notation for inverse functions
in accordance
Thus
If
it
.
the amplitude of
superior limit
and
=
am-J.J^J^.
= sin 0,
we have x = sin am u, which
x2
Similarly, \/l
VI The quantity
is
abbreviated into
= cos = cos am u,
abbreviated to cnw;
=tan# = tanam u,
abbreviated to tnw.
& 2 sin 2 #, which
\/l
be written A(6 ), (mod.
put 0, k in evidence
I
k),
or A(#,
/<;)
we have called A, may
when it is necessary to
;
which
is
dn u =
Thus
The names
as spelt,
dn u.
further abbreviated to
i.e.
Aam u = A0 = \/l
of these expressions, sn
each letter read
,
u en u, dn
)
u,
are spoken
off.
378. Differentiation.
From
the integral
Hence we can
Thus
itself
-dO
cos #
d
d
-j-cnu =-=-cos9=
du
du
d
/-dn u =-- N /l
,
It follows
may
= -7=
differentiate each of these functions.
_d sn w zs d gmfl
d
-7/5
,
^7 2
.dO
-=- =
du
sin
.
.,.
sm 2 6
=
>
=
A:
sn
2
sin
uduu,
9 cos 6> d0
=
=
A:
2
sn
wen
that any expression involving such functions
be differentiated by the ordinary rules of differentiation.
CHAPTER
390
XI.
379. Integration.
Conversely,
we can
integrate various forms involving such
functions.
Thus
I
I
en u dn u du
sn
sn u dn u du
en w,
snucnudu=
M,
dn
u.
f
380.
The elementary
are
transformations
merely those of
'ordinary trigonometry for single angles.
Thus
cn 2 w=cos 2
sn 2 u =sin 2
=l
#=1
=l
#=1
sin 2 6>
sn 2 ^,
cos 2
cn 2 w,
=1 ^sn
tn u
sn u
--
ctn u
,
cnu'
2
^,
en u
1
= cot arn u = --- =
snw tnw'
etc.
J
o
v
which exhibits the quartic nature of the
radical.
f}/
,
fXov/(l-z )(l-& 2 z 2
a;
and
u
may
2
= sn u, (mod. k)
sn^x, (mod.
or as sn(w, k)
;
k)
then be written as
)
;
or
sn~
;
1
(a;,
k).
382. The earlier authors treating of this subject, Legendre,
Euler and others, regarded the direct integral u as the function to be studied, and 9 as its inverse.
The course followed by
all
later writers,
Ferrers, Cayley, Greenhill and others,
direct function and u as its inverse.
is
Abel, Clifford,
to regard 9 as the
JACOBI'S ELLIPTIC FUNCTIONS.
383.
am- 1 ^,
fc
= 0,
The inverse nature
-x
Wl=
is
expressed in calling
it
dx
_i
JoVT^P~
A/p
a
whilst
u
conformity with the simple case where
arid this is in
viz.
of
391
J
(flT
vd-xw-^r
811
"1
'*' *>
384. Complementary Modulus.
It is desirable to introduce
k' is
called the
a
new quantity k such
that
complementary modulus.
385. Transformations.
Each
of the functions, sn u, en u,
dn u, tn u, can be expressed
in terms of the others.
If
sn.u=x,
cnu=>Jlx
tn u
If en
u = x,
=
2
= Jl
sn 2 u
sn u
sn u
r-
<lx
sn u
= \/l
tn u
=
x
2
cnw
2
cnu
-dr\ 2 u
If
If
tnu=x,
sn u
=
x
'l+x
tn
2
1
1
u
+ tnV
_J
'1
+ tnV
CHAPTER XL
392
386. Inverse Notation.
With the inverse notation the same formulae would be
written
sir-
1
x
x
= en-
1
vlz = dn2
^2 = dn
=
l-
1
X2
-i
Similarly
- .1 /
(
2 0- cos
- cos 2/A
2/3
V/cos1- cos ^73
V~
>
25
.
/I
'
V V
.
/sin 9
\
'
^, sin/5).
I
^l^sh" 1 !-
2
/
\siii
3
388. Illustrative Examples of Reduction to the Legendrian Form.
1.
/EE
Consider
Let .r=6sin
\*
^o
v(a
2
-^2 )(6 2 -a;2 )
(a?
< 6 < a).
^,
=
bcosOdO
/"
~Jo NA a8-62 8in a ^62 co
if
^
,
"Jo
=-1
a
2.
Consider the case
Put x = cos
0,
l-n'O
b\
\o' a/).
sn" 1.(x
T
(
,
/=
'
J* v/1-.r4
-'Wft*).
ILLUSTRATIVE TRANSFORMATIONS.
393
-sin 6 dO
f
mod.
;
x = cn
d
9
de
~T^J
v2
/v/2,
"&" \*'VT
3.
Consider
/=
,
where a <
b< c < x.
|
a = (c
Let x
~
2
a) cosec'
8.
r
2
cot & dO
a) 2 cosec
2
2
6 { (c a) cosec 6-(b- a)} {(c - a) cot
(c
Jo \/4 (c
- a) cosec
2
dd
dd
r<>
I
'
*Jc-a
A /
A/
>
6 = am
(*Jc
1
-a I)
-
a.
b
c
;
2
- a sm 6
mod. \ ^T^>
-'y
V^-^-^ = 810^ = 811
(\/c-aI)
1
:.
4.
snI=~r
vc-a
i
1
/A /cV \/
'.f-a
;
(
,
. lb-a\
\/~
>c-a/
Consider the case
Put
.r;
+ A = (1
+ A) bos?_.
Thus,
_
-
T
~J+
- A + (1 + A)
(1
+ A) 2 cos sin
+ A) - U
cos a <} {( 1
c/>
1
CHAPTER
394
If
XI.
#=cos2# and A = cos 2/3,
/ W2sn-i
(-,
cos/?)).
~v
Vcos/}'
(Art. 387.)
Similarly,
These integrals are useful
5.
in the rectification of a Cassinian oval.
Consider the integration
Putting #=csin
-
c
o
x<c<a,
t
0,
:->
6.
Consider the integration
/=
fa
lx^
" -5
A/
/
-
Jc
_ of
- c- dx,
1
where
-,
x>c> a,
Here we may put
-
c2
x=
Then
vc 2
rt
c2
and
=
/*
I
sec j a)
i
u
,
- a2
)
sin
<u
-a 2 sm 2 a>
Vc'
-a2
w
-a2 sin 2 a>
cr2
c
a)
a
Gr2 sin 2 w
c2
,
cf
..
2
jo
w
2
sin'
cos 2 w \c 2
<^w
.
a
cosw
2
i
cos 2 a>
,
Sec2w
=Jo
=J
sina>
sec 2 toVc2
_
= tantov c2 -
2
= tantox/c 2 -
2
/
,
^^2^
^^^^^
-
-a2 sin a)^,-J
2
sin2to+Jo
sin 2 to+
r (c
I
Jo
2
2
2
2 2 (c
~
-sin to)
/v2 ---j-.2
sin g2 w
vc
)
-
THE PENDULUM
REAL PERIODICITY.
-
395
2
)
c/
Jo
tan
to
v/c
2
a'1 sin 2
w
-I
oSin 2
A/1
*
c2
F \ w,
(
c
-)
c)
c^
(
\
o>,
-
).
cr
the integration needed in the rectification of a hyperbola.
7.
Reduce the integral
~
to
Legendrian form, taking a
Write b tan 6 = a cot x-
> b.
Then
Hence
e
V (a 2 + c 2 + (6 2 + c 2
)
)
tan 2
cosec Y
^
Jo
[a~
2
[a
- (a 2 -
-
(a
2
2
V) sin x ] V(a
2
+ c2
)
6 2 sin-
-
- 6 2 ) si
r__
JO
(
T-WJ
1
an integral of the Third Species.
This integral is needed in the rectification and quadrature of a spheroconic.
389. The Simple Pendulum.
Dynamical
illustration of the real
periodicity of F.
Consider
the
Let
finite
oscillation
of
a
simple
circular
pendulum.
angular displacement of the rod
from the vertical at time t, a the extreme value of 0, in the
be
tin-
CHAPTER XL
396
mass of the bob, a the length
zero-velocity
of
the rod.
The
this case cuts the circle described
in
bob at two points A, A' between which the bob
The energy equation is
2
\ ma?6 = mg (a cos 9
r>
U
giving
being measured from the instant at which the bob
through its lowest position.
t
Let
sin
^
=
sin^
sin
2 sin
.
.
cos
d<f>
dO =
.-.
.
,_
t -
.-.
A
la
/
^= A/-am~ J
v
^.e.
^
;
sin ^),
(mod.
v
^x
lg\
-
y
of
by
the
oscillates.
a cos a)
o Ct
line
REAL PERIODICITY.
When
viz.
and
T
t
is
9=
is
a,
and
*
.
.
<f>
=
THE PENDULUM.
= 0, and
,
397
the time to this point,
given by
the quarter period of the whole time of a complete
for
it
Writing
l
appears that the function"
K
oscillation.
F
is periodic and has a real
period 4<K.
the " quarter period of the integral F"
F
l
or
.a=
rr
K=
where k = sm
For an indefinitely small
r=?A/-,
2 v
Thus
K
is
called
viz.,
.
2
oscillation a is infinitesimal
and
the ordinary formula for a small oscillation.
g
390. Complete Revolutions.
Case of the pendulum making complete revolutions.
Line of zero velocity
In the case
(>
when
the line of zero velocity
is
at a height h
2a) above the lowest point and does not cut the
circle
CHAPTER XI
398
described by the bob of the pendulum, the velocity of the bob
is not exhausted when it arrives at the
highest point of its
path.
The rod then makes complete revolutions and does not
oscillate.
In this case the energy equation
\
ma2 $ = mg [h
2
al
is
cos 0]
;
a
2qhf 1
= -~a2
^"' v
(
\
2a
h
.
2
6>
2
-j- sin -.
d9
J
Let
=
The time
and
*frW
of a half revolution
is
given by
-T
2a
391.
LEGENDRE'S FORMULAE.
Legendre gives (Exercises,
199) a
p.
list
of results connect-
ing various integrals at once by elementary means with the
first two standard integrals of Art. 371, viz.
da
a
Jeo &
These we
= F(0,k),
ft*
I
bd9=E(9,k).
Jo
may usefully reproduce for reference, and they will
furnish a useful set of examples for the student to verify.
LEGENDRE'S FORMULAE.
399
EXAMPLES (LEGENDRE).
Prove the following twelve results
** sing cos
:
[Putting
a
little
P= --
-r
2
reduction, &
- and
differentiating, we obtain, after
dP
k' 2
-^=A
-r^,
then integrating
we
obtain
the result stated.]
sin 2 9
dO
'-).
-k' 2 F).
i_
F
2
l
= ^(Atangk
2
- 2Atan^+.F-
=
=
12
Atan0+jP
?,
A sin g cos g +
392. Further discussion of Elliptic integrals is reserved till
Chapter XXXI. Enough has been written to explain their
nature,
and the student
when wanted
will be able to
employ the notation
in the intervening chapters.
CHAPTER
400
XI.
EXAMPLES.
1.
By
J
putting
a?
J
u=\
=^
.
+ sm
1
r
.
,
,v
TJ,
shew that
dd
= _K\
/,
-r-
,
I
a
0-
/^
6
(
and that
2.
- sn
(wv/2)
/
l+sn(ws/2)
\
=1
l
nearly.
2 2 .4 2
2 2 .4 2 .6
Prove that
if
...]
n be
<
1.
Establish the truth of
1 \
--
(a)
2
cnw/
en M
sntiy
V
en u + sn
v
(c)'
.
,v
'
/
\
snwcn
sn u
-I
1
6.
1
= -,=
sn
V2
Prove that
+
5.
-M
F^O, TV)= 1'574745 very
g|
4.
i.e.
x/2/
Prove that
and that
3.
1 \
7^, (mod.-pr);
sn u
it
(
\sn u
en
en ?*\
\ /sn w
H
J \cn u sn w/
\ / 1
1
snujf"
--cnttlf
\cn u
J
1
-cnw_
"~
/
1
+ en w
\sn u
1
cnw\ 2
'
sn 5/
Prove that
(1)
1
v '
(2)
)
= 1.
>
EXAMPLES.
401
Prove that
7.
(
1
-T-
)
=2
sn 2 w
w,
udnudu =
s
(2)
sn u en u dn
1
u
cnw
Jo, =
au
1
fOS
cc
'
,
'-
7
fa en u +
t
!
N
=
I
Prove
8.
2snucnv = sin (am % + am v) + sin (am w - am
- am
3 en u en v = cos (am u + am v) + cos (am w
v).
?),
By
9.
putting x
= a cos
show that
0,
_j
-
^
f*
10.
Prove
._
Jav/^-a
11.
By
=
putting x a^l^
dx
12.
Prove that
13.
Prove that
14.
Prove that
(1)
-j-(snu
(2)
f (*2 sn
15.
+ en
n
w)
u + en
Draw graphs
tt
,=
4
-,
1
,
/a;
/a
pcn-M-,
V*
W2
1
-
-7
V
show that
1\
=
a2
1
_
pn""-*
~_
I
2
'
N/2/
= w(sn u + en u)*~ l (en. u-snu) dn w,
)"(#cn w sn w)dn w
of y
J_
= A0 and
y=
-\-h)
^=
showing that the former
an undulating curve lying entirely below the line y=\ and
the other of an undulating line lying entirely above the line y=l.
Take the cases k* = J and k z = J.
consists of
Show that the areas bounded by these curves, the z-axis,
the y-axis and any ordinate at a point whose abscissa is 6 represent
E(6)smdF(6) completely. Examine what happens in the limiting
cases & =
E.I.C.
and k=l.
2c
CHAPTER
402
16.
Show
XI.
that the complete elliptic integrals of the First and
Second Species may be expressed as
where f(a,
b, c,
x) is the
*
17.
Show by
hypergeometric series
a. b
x
and E(8,
E and F alternately,
1
- 3& 2 ^F
for the complete functions
n
v1
l
:
differentiating F(6, k)
Hence, eliminating
and
aa
~ + 1 bb +
F E
lt
Mx^i+Lli^^
H
k
dk
*>^F
with regard to k
show that
sin
l
k)
CHAPTER
XII.
QUADRATURE
(I).
PLANE SURFACES, CARTESIAN AND POLAR EQUATIONS.
393. The process of finding the area bounded by any
defined contour line is termed Quadrature, or, which amounts
to
the same thing, Quadrature
is
the investigation of the
of a square which shall have the same area as that of
the region under consideration.
The closed contour may consist of a single curve or of a
size
system of several arcs of different curves or straight lines.
As we shall, in most cases, have to form some rough idea of
the shape of the curves under discussion so as to be able
properly to assign the limits of integration, the student
should be familiar with the rules of procedure adopted in
the tracing of curves for the various systems of coordinates
by which they may be
defined, Cartesians, Polars,
etc.,
and
for such information
may be referred to the author's treatise
on the Differential Calculus, Chap. XII.
394. It has been already shown (Art. 11) that the area
bounded by a curve whose equation is y = <f)(x), any pair of
and the z-axis, may be considered
ordinates, x=a and
as the limit of the sum of an infinite number of inscribed
rectangles; and that the expression for the area is
xb
ydx,
or
I
(f)(x)dx:
was assumed that (f>(x) is a finite and continuous funcx, which does not change sign between these limits.
In the same way the area bounded by the curve, two given
and
it
tion of
abscissae,
y=c
and y=d, and the
403
?/-axis is
I
xdy.
CHAPTER.
404
XII.
If the angle between the coordinate axes were
90, we should have the expressions
sin
co
I
or
ydx,
Ja
sinw
to
instead of
xdy
Jc
for the area.
395. Again, if the area desired be bounded by two given
and y=\js(x), and two given ordinates x=a
curves y
(j>(x)
and x=b, it will be clear by similar reasoning that this area
=
Fig. 40.
may
be also considered as the limit of the
sum
of a series of
If PQ be the
rectangles constructed as indicated in the figure.
of
of
the
ordinates
between
the curves,
any
intercepted
portion
and &e the breadth
is
a
side,
of the elementary rectangle of which
the expression for the area will accordingly be
PQ
or
where the same assumption is made as before as to (j>(x) and
to x =b, and,
\js(x) being finite and continuous from
must
retain
the
same
moreover, (j>(x)\js(x)
sign throughout
the integration, i.e. the curves must not cross each other, and
has been assumed >\/s(x) throughout.
<j>(x)
xa
396. Case
when
the Coordinates are expressed in terms of a
Parameter.
We
have regarded x as the independent
variable.
If this is
not so the formula can be modified to suit the circumstances.
CARTESIAN EQUATIONS.
AREAS.
405
Suppose the curve defined by the equations
and that the values of
ordinates are t and 9
corresponding to the
t
initial
and
final
.
l
Then
ySx=\lf(t)^'(t)
St
to the first order,
and
in the limit
it being supposed that the integrand remains finite and
continuous throughout, and that as t changes continuously,
the point
2
increasing from the value ^ to the value
,
(x,
0)
to
(6,
once,
also travels continuously along the cc-axis from (a, 0)
0) without going over any part of its course more than
and always
in the
same direction
397. Case where the Arc
is
of increase of
x.
the Parameter.
curve be the independent variable, being
measured from some definite point on the curve, then at a
If the arc of the
point at which the gradient of the tangent
dx = cos \[s ds, and we
may
,
is
write the expression
\//-,
I
we have
y dx as
h
or
the limits of the integration with regard to s being the values
of s corresponding to the beginning and end of the arc, and
supposing that ycos\ls does not change sign.
In the same
way we may
I
1
x -j- ds,
ds
write \xdy as
or
\x sin \lr
ds.
)
398. Area expressed by a Line Integral round the Contour.
Let the formulae \ycos\]sds, \xsm\jsds be applied to the
evaluation of the area of a closed curve consisting of a single
oval.
Let us suppose
measured from any point on the curve in
such a direction that a person travelling along it in the direction
.<?
an increase of s has the area sought always to his left. Let \]s
be the angle the tangent makes with the positive direction
of the se-axis.
Let APBQ be the oval in question, and let
of
CHAPTER
406
AL,
BN
arc
APB
be the
in
tangents
the figure,
XII.
parallel
the
?/-axis.
changing from
is
\/y
to
-
-
2t
and
^
to
from
cosx//-
and cos
^,
A to
In the arc
is positive.
\//-
is
we
\fs
is
obtain the area
to
+2
,
changing from
Integrating then
negative.
B, through P,
BQA
In the
|
y cos \[r ds
ALMNBPA taken
from B to A, through Q, obtains
taken negatively. Hence, to obtain the
positively, whilst integration
the area
whole
BQALMNB
area, it is necessary to take our
formula as
-j
y cos \^
<fo
integration round the whole perimeter in the counterclockwise direction.
in
In the same
way and under
area will also be given by
the same circumstances the
x sin \]s
-f-
cfc.
j
This is the conventional mode of measuring s. If we
measured in a clockwise direction the signs would both be
reversed.
399. Precautions.
If the curve cuts itself once,
having a node, as in the case
from an inspection of the
in
accompanying figure, that,
travelling completely round
the whole curve, the directions in which the two loops are
travelled round in continuously progressing in the direction
of the increase of s, are one clockwise and the other counterclockwise, and therefore, in conducting the integration completely round we get the difference of the areas of the two
of a lemniscate, it will be clear,
LINE INTEGRAL ROUND THE CONTOUR.
407
loops with either formula, and in the case of equality of the
loops the total line-integral of xsm\]s, or of ycos\/r, round the
complete curve will be zero. If we require the absolute area
Fig. 42.
enclosed we must therefore treat each loop separately and
add the positive results.
If in travelling continuously round the perimeter of the
closed curve there be several nodes and several loops, we
shall see in the same way that the total line-integral of x sin \js
or of ycos\/r, will give the difference of the areas of the odd
and even loops.
400.
The student should examine the truth
of the result in
Fig. 43.
figures of other shapes
say a horseshoe-shaped closed curve,
such as shown in Fig. 43.
CHAPTER
408
Let
ABCDEF
XII.
be the points at which the tangents are
if
BN 2 etc., be the ordinates,
lf
parallel to the y-axis, then
AN
,
the integral
-
1
y cos \Js <fo yields
-area Atf^tf+area
-farea
ie.
DEN 5 N- area
the closed area
ABCPDEFQA.
401. If y be continuous, but
-/-
discontinuous at points on
the boundary of the figure, as at A BCD in Fig. 44, the integration must be conducted along each of the portions into
D,
44.
r.
which the perimeter is divided by the
same rule holds, as before, viz.
area
ABCD =
fB
I
JA
discontinuities, but the
PC
y 1 cos
x//- x
dsl
\
JB
y 2 cos \]s 2 ds 2
PA
JDc
JD
I
PC
JBA
JDc
x1 sin
\}^ l
ds -f
I
JB
x 2 sin \js 2 ds 2
PA
XzSm\ls z dSz+\ ^ 4
JD
is
conducted, and
s lt s 2
,
s 3 , etc.,
rf5 4
,
which the intealways being measured
suffixes denoting the several portions along
gration
smv^ 4
DISCONTINUITIES IN
"
409
jg-
"
Here the limits
along the perimeter.
the
are
denoted
of the integrals
by
points A, B, C ... of the
perimeter successively arrived at in a continuous progress
in the
round
same sense
it.
402. If
(/>(x)
x = c,
has an infinite ordinate between a and
b,
has been explained that the infinity can be
say at
excluded by taking
it
fb
I
(f>(x)dx
to
mean Lie=0
FfC-e
-1
f&
<j>(x)dx+\
I
(j)(x)dx
.
will, in general, change sign in passing
value and the graph reappear from infinity at the opposite end of the asymptote, it will be desirable
to consider the areas on opposite sides of the asymptote
As, however,
through an
</>(x)
infinite
separately, and, after evaluation, add the positive results toThis is of course the same precaution we have had
gether.
to take in Art. 395, in stipulating that
(f>(x)
does not change
sign between the limits, which would mean that part of the
curve was above the x-axis and part below, so that careless-
ness in this respect would lead to a result which would
represent the difference of the two portions of the area
required instead of their sum.
403.
Illustrative Examples.
Find the area bounded by the
#=e, x=d and the #-axis.
Here
1.
ellipse
# 2 /a 2 +3/ 2 /6 2 = l, the ordinates
in -i
*_
a
sin -i
c
\\
/J
a result obtainable without integration by reduction of the ordinates
of the auxiliary circle in the ratio 6 a.
:
For a quadrant of the
expression
whole
becomes
5~-
ellipse,
2
we put d = a and c=0, and
-K or ~
"
the above
gi y i n g "*ab for the area of the
ellipse.
Find the area which lies in the first quadrant and is bounded by the
#2 + ?/ 2 = 2rt.F and the parabola y z = ax.
The curves touch at the origin and cut again at (a, a).
The limits for x are therefore from # = to x = a.
2.
circle
CHAPTER
410
The area sought
is
XII.
therefore
Putting x = a(l -cosfl) in the
first
"
v/2^^7 dx = Pa
2
/
Jo
2
d6 = a 2
sin 2
Jo
*
ii
22
=
,
4'
^
Fig. 45.
as of course
of radius
a
;
might have been written down, being a quadrant
and
Thus the area required
of a circle
is
Find the area
3.
(1
of the loop of the curve
)
.rU'
(2) of
2
+y = a(.r -y
2
2
)
),
the portion bounded by the curve and
its
asymptote.
<*.#
Here
To
2
a+x
trace this curve,
(1)
It
(2)
No
is
we observe
symmetrical about the
real part exists for points at
(3)
It has an asymptote
(4)
It goes
(5)
It crosses the^-axis
(6)
The shape
which x > a or
<
- a.
x + a = Q.
through the origin, and the tangents there are y
when x = a, and
at this point
of the curve is therefore that
~
shown
.r.
is infinite.
in the figure
(Fig. 46).
and a, and then
Hence, for the loop the limits of integration are
double the result so as to include the portion below the .r-axis.
For the portion between the curve and the asymptote, the limits are
x= - a
to
tf
= and
double as before.
ILLUSTRATIVE EXAMPLES.
411
For the loop we therefore have,
Area
=2
/
Jo
x\> a + x dx.
For the portion between the curve and the asymptote we have,
Area
The meaning
=
._.
2
^7
.
.
/
/
a
J~
la
A/
*
x
of the negative sign is this
before the radical
in
y=js^-'-
,
we
,
dx.
a+x
:
In choosing the
4-
sign
are tracing the portion of the
curve below the x-axis on the left of the origin and above the x-axis on
Fig. 46.
Hence, y being negative between the limits
be expected that we should obtain a negative result
the right of the origin.
a and
if
we
0, it is to
evaluate the expression,
*=o
Lt^ydx.
x~a
Therefore
we prefix
the
-
before the radical before integration to ensure
a positive result.
To
integrate \x
t
"7^<te= -
Thu
And
^^r^dx
+*
Area
of loop
=
put x = acos
T
/
and
.'.
dx= -a sin
OdQ.
CHAPTER
412
<"
.
F
Att-^j
f
A
XII.
IjB^dx* -]
Again,
f
a* /'(cos
0- cos 2 0) rf0
and the area between the asymptote and the curve
With regard to the latter portion of this example, it is to be observed
that the greatest ordinate is an infinite one. In Arts. 11 and 394 it was
assumed that every ordinate was finite. Is then the result obtained for
the area bounded by the curve and the asymptote rigorously true ?
It will be noted that the factor
(a+x}% which occurs in the denominaand gives rise to the infinite value of y has an index < 1 and positive.
Hence (Art. 348) we infer that the principal value of the integral is finite.
Let us examine the case more closely, and integrate between - +
and 0, where is some small positive quantity, so as to exclude the infinite
ordinate at the point where x=
a.
tor
We have as before
where
-a + = acos(;r-8),
so that 8
cos- 1
(
is
1
a small positive angle,
-Ck
This integral
is
6
when
4.
8 is
made
sin20->-
9
r,.
to diminish
TT\
sin28~l
r
without limit to
J
Here, solving for
zero.
of the curve
a 2 (x2 +/ 2
is
/7T-8
s
L-^-vT-J*
indefinitely closely to the former result
Prove that the whole area
where y l
).
,
then
-2--4-i
and approaches
viz.
= a4
is Tra
2
.
y,
the ordinate of a parabola and
y 2 that
of a circle of radius a.
COMPOSITE CURVES.
The area
of a strip parallel to the y-axis
and
413
of breadth o>
is
f
and the
circle,
total area of the curve is 2
=7ra 2
J
y 2 dx,
the same as that of the
i.e.
.
Fig. 47.
404.
The
last
will suggest to the student that
example
2
y = (#)\/a
the curve
2
# be drawn,
<f>
constructed by means of
two
y l = (j)(x)
may
it
if
be regarded as
curves, viz.
and
2/ 2
= ^Ja
1
x* ,
the latter being a circle and the ordinates of the resultant
curve being the sum or difference of y l and y 2 viz.
,
and as in the parabola and circle of Ex. 4, the closed curve
formed will be divisible into strips of length (2/ 1 + 2/ 2 )~"(2/i~2/2)
and breadth Sx, and therefore of area 2y 2 Sx.
Hence the area in any such case
the same as that of the circle.
This curve,
<f>(x)
if
.
2
written in rational form,
being supposed rational.
are =7ra 2
is
1
2
y 2 dx = Tra and
,
~a
is
is
And the areas
of all such curves
CHAPTER
414
XII.
Similarly, for curves of form
which are clearly to be constructed as
and
consist of closed curves of area
-n-a
2
;
or
2
more generally
still,
y =f(x) be a closed curve whose area
another curve can be constructed from it of form
if
i.e.
y
2
- 2y0(z) + [0 (z)]
2
is
A, -then
-f(x) =
whose area is also A.
For the areas of corresponding elementary strips parallel to
the y-axis are for the original curve and the derived curve
respectively,
and
which are equal, and therefore their sums are equal
also.
Similarly for
405. In Art. 395 it is shown that the area between the
two curves y = <j>(x) and y = \fs(x) and a pair of ordinates
be that y = </>(x) and y = \js(x) are different branches of
the same curve. This is really what happens in the various cases
It
may
considered in the last article.
Ex. Consider the case of an ellipse
406.
ax2 + %hxy + by 2 = 1
If
y
.'.
l ,
1/2
,
h2
<ab.
are the ordinates for any abscissa x,
the length of the strip
is
EXAMPLES.
And
the area
415
is
?/ 2 )
d.v,
between ordinates
&\
and xz
,
y
Fig. 48.
or for the whole ellipse
v ab
h2
x area of
circle of radius
EXAMPLES.
1.
Obtain the area bounded by a parabola and its latus rectum. A
drawn between the vertex and the latus rectum,
series of ordinates are
parallel to the latter,
viz.
.?=(-)
a,
where
that they divide the aforementioned area into
2.
= l,
(a)
(b)
(c)
(d)
(e)
,
,
y
ellipse
= - <Ja*~- .r 2
2
The hyperbola
#?/ =
a and b both >0; first,
if
y=xe*,
(1)
(2)
In what ratio
is
parts.
.r-axis,
and the
specified
from x =
to
x = h.
from # =
to
x = li,
to
x = li
from
,
#=1
from x = \/o*^P
to
> 1)
(h
x = a.
the hyperbola be rectangular;
the angle between the asymptotes be w.
The curve
Obtain the area
Show
...n-l.
from x = a to x=b,
,
second,
(/)
'
The logarithmic curve y = ex
The logarithmic curve # = log.#,
The
3,
:
y = c cosh
The catenary
2,
n equal
Obtain the areas bounded by the curve, the
ordinates in the following cases
3.
r
if
from
#=0
to
x=h.
bounded by the parabolas y* = 4ax, #2 = 4//
bounded by the parabolas y 2 = 4a.r, 2 = 4/>//.
this area divided
;
.i'
by the common chord
in each case
?
CHAPTER
416
Find the areas of the portions into which the ellipse X2 la 2 +y 2 /b 2 =l
=c
(1) by the straight line y
the
two
lines
straight
(2) by
y = c, x=d, supposed to cut within
4.
is
XII.
divided
the
ellipse.
Trace the curve X2 i/ 2 = a 2 (y 2 x2 ) and find the whole area included
between the curve and its asymptotes.
5.
t
6.
Find the area between the curve y 2 (a -f x) = (a - x) 3 and
7.
Find the area
asymptote.
of the loop of the curve
y*x + (x + of (x + 2)
= 0.
curves in which y oc xm and two in which
show that its area is
quadrilateral
Two
8.
its
y<x.
xn form a
;
where
(#1,3/1), (#25^2)5
(%>
are the coordinates of the corners
#3)5 (^4> #4)
taken in order.
By means
9.
triangle formed
[TRINITY, 1891.]
of the integral
by the
\ydx taken round the contour
of the
intersecting lines,
show that they enclose the area
(&i
-6 3
2
)
-
2
!)
!
(
3
- a2
'
)
[SMITH'S PRIZE, 1876.]
10.
A four-sided figure is formed by the three
y
and the axis
enclosed
11.
and
by
of x.
2
a2 = 0,
ax -\-
Prove that
its
parabolas,
area
is
12a 2 and
the chords of the area.
,
equal to the area
[COLLEGES a, 1886.]
is
Find the curvilinear area enclosed between the parabola y i = kax
its evolute.
I.
[OxF.
P., 1889.]
12. Show that the area cut off from a semi-cubical parabola by a
tangent is divided by the tangent at the cusp in the ratio 64 17.
:
[OXFORD
13.
(i)
Find the area
af=x*(a-x}.
(ii)
II. P., 1889.]
of a loop of the curve
Find the whole area
[I.
C. S., 1882.]
[I.
C. S., 1881.]
of the curve
aY = a x
2
2
-x*.
EXAMPLES.
Trace the curve a 2#2 = ?/ 3 (2a
14.
that of the circle whose radius
y),
417
and prove that
is a.
[I.
the curve a*y 2 =x5 (2a x), and
15.
to that of the circle of radius a as 5 to 4.
Trace
its
area
is
C. S., 1887
prove
that
equal to
AND
1890.]
area
its
is
Find the area of the curve
16.
v (z? + 1) = x?
from x =
1
x = 1.
to
[Sr. JOHN'S, 1881.]
17.
(i)
(ii)
and
its
Find the area between y
Show
Show
is
irc(a
its
=
and
its
asymptote.
+ b).
that the area between the curve
asymptote
is
a 2x
3
that of a circle of radius
[ST. JOHN'S, 1889.]
a.
18. Find the area between the axis of x, the hyperbola
and the line y =x tan a, where
|>t>a
A
If
#2/a 2
,2S
x =a cosh
is
the sector ial area
,
11
*
.
i.
p.,
mi.]
any point on the hyperbola
= b sinh,2S
,
,
air
X2ja 2 -y 2/b 2 =l,
[0x
P
the centre, and
2
2
y /6 =l, prove that
be the vertex,
where
II. P., 1903.]
[Ox.
y^x = a
and
%3
that the whole area between
asymptote
(iii)
2
.
-,
,
ao'
,
A OP
[MATH. TRIPOS, 1885.]
Find by integration the area lying on the same side of the axis of
x as the positive part of the axis of y, and which is contained by the lines
19.
Express the area both when
y
is
.r
is
the independent variable and
the independent variable.
20.
when
[COLLEGES, 1882.]
Prove that the area of the loop of
a<fi-y)(x-ty)-y>
b
g.
[C OLL. ^ ( 1891.]
Find the areas of the two regions of space bounded by the straight
y = c, and the curves whose equations are
21.
line
=
c2 ,
4c 2
22.
and
.
[I.
C. S., 1891.]
Prove that the area contained between the curve
its
asymptote
E.I.C.
is
3-\/3.
[Oxr.
2
I.
P., 1901.]
CHAPTER
418
23.
XII.
Prove that the area of the curve
2
)=0
[MATH. TRIP.,
24.
Find the area of one loop
1893.]
of the curve
4
z
y - ?/ + x = 0.
[COLLEGES
1
a,
885. ]
Through the cusp of the evolute of a parabola, a line is drawn
Show that it divides the area between the
perpendicular to the axis.
25.
parabola and the evolute in the ratio 17:5.
26.
and
Show
its
that the ordinate
[C. S., 1896.]
x = a divides the area between y 2 (2 - .r) = .r 3
asymptote into two parts in the ratio
37T-8 :37r + 8.
407. Sectorial Areas.
When
[MATH. TRIP.
1912.]
I.,
Polar Coordinates.
the area to be found
and two
directions,
radii
is bounded
by a curve rf(9)
drawn from the origin in given
vectores
we may
divide the area into elementary sectors
SO, as shown in the figure. Let the
with the same small angle
Fig. 49.
PQ and the radii
OPV OP ,... OPM at
area to be found
be bounded by the arc
vectores OP, OQ.
Draw
radii vectores
2
equal angular intervals, so that
POP = P.OPg =
l
. . .
=
Pn., OQ = SO.
the successive circular arcs
Then by drawing with centre
PjJVj, P 2 # 2 etc ^ may ^ e afc once seen
of the sum of the circular sectors OPN, OP^, OP^N2
PN,
>
->
,
etc.
SECTORIAL AREAS.
P^N^Py /yV 2 P3
occupy new
etc.,
,
419
For the remaining elements
the area required.
is
POLARS.
may
be
made
rotate about
positions on the greatest
so as to
say OPn-iQ, as
plainly less than this
sector,
Their sum is
indicated in the figure
and in the limit when the angle of this sector is
sector
indefinitely diminished its area also diminishes without limit,
;
provided the radius vector OQ is finite.
Now the area of a circular sector is
2
x circular measure of angle of sector.
Thus the area required = iLtSr2 #, the summation being
J (radius)
A
conducted for such values of
as lie between 9 = xOP and
A
A
= xOPn_ i.e., xOQ in the limit, Ox being the initial line.
A
In the notation of the integral calculus, if xOP = a and
l
A
=/3, this will
be expressed as
dO
It is
9
(3
assumed that f(0)
or
is finite
and continuous from
=a
to
inclusive.
and the origin be
find
to
the
whole area are
the
limits
of
it,
integration
and STT, viz. the extent to which a radius vector must
408. If the curve consist of a closed oval
within
rotate about
to
sweep out the whole area (Fig.
Fig. 50.
50).
Fig. 51.
If the origin be on the
perimeter of the oval, and if it be
not a singular point, the limits will be from
a to +?r-a if
the tangent at the origin makes an angle
a with the tf-
as
shown
in Fio\ 51.
CHAPTER
420
In this case,
if
XII.
the initial line be an axis of symmetry,
sufficient to integrate
from
to
~ and double
it is
the result
(Fig. 52)
Fig.
52.
If there be a loop and the origin be a
singular point on
the curve at which the tangents make an angle 2a with each
Fig. 53.
other,
and
if
the initial line be an axis of symmetry, the
and a and double
limits for the area of the
loop will be
the result (Fig. 53).
409. Another Expression for an Area.
Let
(x,
a curve,
y)
be the Cartesian coordinates of any point
(x-\-Sx, y-\-Sy) those of
Fig
(r,
0),
Also,
P
to
(r
an adjacent point
Q.
P
on
Let
54.
+
Sr, 0-\-SO) be the corresponding polar coordinates.
shall suppose that, in travelling along the curve from
we
Q on an
infinitesimal arc
PQ, the direction of rotation of
OTHER FORMULAE FOR AREAS.
OP
the radius vector
is
421
counter-clockwise, and that the area
to be considered is on the
hand
left
to
a person travelling in
this
direction (Fig. 54).
Then, to the
first
order of infinitesimals,
= sectorial area OPQ
x,
1
y,
001
x + Sx,
y
+ Sy,
1
Hence, another expression for the area of a sectorial portion
bounded by a definite portion of an arc is
of a curve
z
or
_
the limits being the initial and final values of s, corresponding
to the portion of the sectorial area to be found.
Obviously we might take any other independent variable,
say t, and supposing the curve expressed as
and that the values of t, corresponding to the beginning and
end of the arc, are ^ and t 2 respectively,
sectorial area
If the
=4 fV(0^'(0-/'(0^(0}^.
curve be a closed curve and the origin
and
the limits for 9 are
2?r,
1 f
area = -
In the same case,
if
yx)
P
within
it,
2*
r 2 dO.
we take the formula
or
t must be such that the
point
and once only, completely round the curve.
the limits for
lies
and
(x,
y) travels once,
410. If the origin lies outside the curve, as the current point
travels round the curve, we obtain sectorial elements such as
OPiQi
(Fig. 54), including portions of space
such as
OP Q
2
2
,
CHAPTER
422
shown in the figure, which
are,
lie
XII.
outside the curve.
These portions
however, ultimately removed from the whole integral
1
f
-\(xdy-ydx),
when
P
the point
travels over the element
.
sectorial
OP 2 Q
element
creasing and SO
is
2
is
P Q
2
2
,
for the
Fig. 55.
reckoned negatively as 9
de
is
negative.
411. Precautions.
If the
curve cross
itself as in Fig. 56,
the expression
taken round the whole perimeter, no longer represents the sum
For draw two contiguous
of the areas of the several regions.
radii vectores OP l OQ^ cutting the curve again at Q 2 P 3 Q 4
and P 2 Q 3 P 4 respectively. Then, in travelling round the
,
,
,
,
,
curve continuously through the complete perimeter, we obtain
and negative
positive elements such as OP 1 Q 1 and OP 3 Q 3)
elements such as
Now, taking
OP 2 Q
all
2
OP 4 <? 4
and
.
these elements positively,
= quadrilateral P Q P
1
1
4
Q4
quadrilateral
P$
2
>
2
3$3>
in integrating for the whole curve we therefore obtain
the difference of the two regions instead of their sum.
and
Similarly,
integral
=
I
if
(xdy
the curve cuts
itself
more than
once, the
ydx) gives the difference of the
sum
of
OTHER FORMULAE FOR AREAS
423
the odd regions and the sum of the even regions. Thus, to
obtain the absolute area bounded by such a curve, we must
take our limits for each area separately and obtain the absolute
It is
area of each region, and then add together the results.
Fig. 56.
obvious that in curves consisting of several equal regions, or
the area of any one, and
loops, it will be sufficient to ascertain
then to multiply that area by the number of the loops.
412. Another Form.
we
If
write
-v, we
x
have
xdy
and accordingly we
may
If,,
This
one
is
ydx
xz dv,
transform the formula
1
into
*
equivalent to a choice of new coordinates, of which
the Cartesian abscissa and the other, viz. v, is the
is
tangent of the polar angle 9.
In using the formula, x is to be expressed in terms of v and
the limits of the integration so chosen that the current point
,
(x,
a,
for
y) travels
/3
v.
from the beginning to the end of the arc, i.e. if
9, tana and tan/3 will be the limits
be the limits for
CHAPTER
424
XII
In using this formula, however, care must be taken not to
It must be remembered
integrate through an infinite value of v.
that
v
= tan$
and becomes
when 0=~,
infinite
any odd
or
7T
multiple of
For example,
413.
2
2
.r /tt
.
-^
2
-fy /6
2
= l,
if
we apply this method
we have
an
to the area of
ellipse
putting y/.r=v,
1,
Area = i
and
2
between properly chosen
from
to
oo
limits.
Hence the area
.
of a
+ v2
the
in
Now,
first
quadrant =
quadrant
=
~z
ail(i
7~i
v varies
therefore
the area of the ellipse = 7ra&.
It will be noted that the formula
is
equivalent to half the
sum
of
ds and
\x-jj
-
-
j
y ^-ds, each
which has been shown to represent the area
integration follows the complete perimeter.
of
414. It
may
also be
worth the student's notice
and a pair of ordinates x=a, x = b,
viz.
to
=
C
A=
that the problem of finding the area bounded by y
cc-axis,
when
the
remark
(j>(x),
b
the
(f>(x)dx,
Ja
is
manifestly the same as that of finding the mass of a rod
of length ba, and of
<j>(x),
x
be
measured
the
rod.
For the mass of
any shape
along
a length Sx of the rod is (x) Sx, the limit of the sum of such
expressions being required, when Sx is indefinitely diminished,
of small section but of line density
if
<^>
Cb
between limits
x=a
and x =b, that
is
Ja
415.
1.
(x)dx.
Illustrative Examples.
Obtain the area of the semicircle bounded by r = acos# and the
initial line.
Here the radius vector sweeps over the angular
=
to
=
.
interval from
ILLUSTRATIVE EXAMPLES.
Hence the area
425
is
=
1*
y--
2
=^
/
*
~8~~'
T
V
2*- (radius)-.
2
2
2. Find the area of the lemniscate r =
cos2#.
Here the axis is a line of symmetry the tangents
;
at the origin are
Fig. 57.
The area
is
therefore
4x
3.
.
cos Stf rffl = 8o
Find the area
With
of the pedal of an
ellipse with regard to the centre
the usual axes and notation, the
equation of the pedal is
and
4.
Find the area of one loop of the curve
The curve
r
= asin 30.
consists of three equal loops, as indicated in the
figure
Fig. 58.
CHAPTER
426
XII.
The proper limits for the integration extending over the
are 0=0 and 0=~, for these are two successive values of 6
r vanishes
Area
of loop
=
tt
|J
=
2
loop
which
sin 2 30cZ0
sin 2
/
b
</>
(&f>,
where 30=^,
---
5.
for
:
.*,
The
first
total area of the three loops is therefore
Find the area
-7-
-
.
of the curve
x = a cos 3
,
Fig. 59.
Upon
elimination of
in the figure.
There
we have
^,
is
'
=1
+
symmetry about both
>
ancl the sha P e is
axes,
rs
l2abl sin 4 ^
Jo
_
2F(4)
or
we may
use the formula
..
2.3.2.
and the area
snown
ILLUSTRATIVE EXAMPLES.
427
which gives
1
'
2J
= 66
r?
/
.'o
T
= 66
/
sin 2 1 cos 2 *
eft
Jo
as before.
6.
Find the area of the loop of the curve
(1)
(-2)
(3)
There is symmetry about the line y=x.
There is an asymptote x+y = a.
By Newton's rule, the form at the origin
is
that of two
semicubical parabolas y 3 = 5au,' 2 , .t-3 = 5ay 2 .
The shape is then as shown in Fig. 60.
Fig. 60,
The polar equation
is
sin 2
..
sin 5
As
there
is
cos 2 9
+ cos5 0*
= ~, we may
symmetry about
take limits
double.
TT
Area
of loop
=2
.
A 25a 2
.
/
Jo
or,
putting tan
=
t,
,
(sin
5
0+ cos6 0y*
to
-.
and
CHAPTER
428
Otherwise
-
and integrate
(5a)
2
this curve is unicursal
;
J
\*-y-\dt
t
XII.
and we may write (putting
;
and
with limits
which gives
oo,
2
+ *5)3
(1
_
_
=
o
2
ytx)
Jo
(1
+
*
6 2
2~'
)
as before.
EXAMPLES.
Find the areas bounded by
2
= a 2 cos 2 $
6 2 sin 2 #, the central pedal of a hyperbola.
1.
r'
2.
One
=
loop of r asin 40.
Also state the total area.
3.
One
loop of r
= a sin 50.
Also state the total area.
4.
One
=
loop of r a sin nQ.
total area in the cases,
Give the
5.
The
portion of r
n even
(i)
bounded by the
a-e
= (3 + 7
0=&
= a?
(0 = a to # = /?).
Any
sector of r*0
7.
Any
sector of the reciprocal spiral rO = a
8.
The
r=a(l - cos
Prove that the area
2
(.r
12.
if
a> b
(ii)
;
a
if
Find the area included between the two loops
r
11.
(0 = a to
Find the area
<b
= ( a 2<p3 + J2 y 3)2
a
1
s
of the closed part of the
_
=
[Oxr.
quadrant
J
obtain the two
of the curve
= a(2cos#W3).
in the positive
+y2)5
= ft).
0).
The Limacon r = a + bcos 0, (i)
areas of outer and inner portions.
9.
10.
radii vectores
(y<27r).
6.
cardioide
n odd.
(ii)
;
(2 + 6
Show
[7, 1899.]
).
Folium
cos
sin
-
r\
>
a an ^
c
=
being positive.
[COLLEGES, 1881.]
2(2>i+l)c'
14.
C. S., 1884.]
that the area of a loop of the curve
1
is TTTTT
P., 1889.]
2
[I.
13.
I.
of the curve
Trace the curve whose equation
r4
= a 4 sec
is
tan
0,
J
and find the area between the curve and any
drawn from the pole.
pair
of radii
vectores
[TRINITY, 1882.]
PROBLEMS ON QUADRATURE.
15.
Trace the lemniscate r2 = a 2 cos2# and
show that the area
of a loop of the latter
is
429
its first positive pedal,
and
double the area of a loop of
the former.
Find the areas of each of the two small lozenge-shaped portions
to the two loops of the pedal.
common
16.
Show
and the
that the area contained between the curve
circle r
=a
is
three-fourths of the area of the circle.
[OxF.
17.
Find the area
between
curve r
the
a(sec
equal to
19.
0-2acsin 0cos
+ a2 sin 2 0)=a 2 e2
irac.
[I.
C. 8., 1879.]
Find the area of the curve
r
20.
its
Prove that the area of the curve
r 2 (2c 2 cos 2
is
P., 1888.]
[ST. JOHN'S, 1881.]
asymptote.
18.
I.
+ cos 0) and
=3
cos
+ a cos 30.
[MATH. TRIP.
,
1
882. ]
Find the area of the loop of the curve
r 2 = a -6 cos
= and 0=52
between
GENERAL PROBLEMS ON QUADRATURE.
(CARTESIANS AND POLARS.)
1.
Find the area bounded by
x~
+ 3/2 = 4a 2
+ y- = 2ay
x2
,
and
x
= a.
[H. C. S. ]
Also the area of the loop of the curve
2
Iy
(a
and
2.
b
= x 2 (a -x)
both positive).
[I.
Find the whole area
C. S., 1882.]
of the curve
..o-iX-^
2
2
a'
+x
[
'
3.
A
4.
Find the area included between one
LC
-
8.,
1885;
COLLEGES, 1892.]
2 =
ax cuts the hyperbola # 2 - y 2 = 2a 2 at the
and
the tangent at P to the hyperbola cuts the
points P, Q-,
at
Find the area of the curvilinear triangle PQE.
It.
parabola again
parabola y
curve a 2 ?/ 2
= a 2 (z2 +
?/'-')
Find the whole area
and
its
of the branches of the
asymptotes.
of the curve
x*
+
4
?/
= a 2 (:r 2 + y-).
[COLLEGES
a,
1
887. ]
CHAPTER
430
5.
2
Trace the curve
a'
)f
XII
= X s (2a - x), and
whose radius
equal to that of the circle
prove that
is a.
[j.
area
its
is
c. y., 1887.]
Prove that the whole area of
6.
2
(x
+ a 2 ) f + 3a*y + 2a 4 =
is
(3
Find the area
7.
-
y
-
2
2v/2)7ra
[COLLEGES
.
p, 1891.]
of the loops of the curve
a:
4
-
a?f +
bW =
when
62
> a2
.
[OXFORD
I.
P., 1902.]
Find the area bounded by the cycloid
8.
y
and the straight
Show
9.
= a (I - cos
line joining
+ y* = axy
P
that as
at its area
is
varies
t
^
on the Folium
of
can be expressed as
at
-
10.
cusps.
that the coordinates of a point
Descartes x s
Show
6),
two consecutive
at-
5
from
-3-
*
to GO
P
traces out a closed loop,
.
and
[COLLEGES, 1896.]
Prove that the area
of either loop of the curve
[7, 1893.
11.
that in that part of the curve (x + y- 3c)xy + c* = for
and the
positive, the area between the curve, the axis of x,
Show
which x
is
ordinate which touches the curve
12.
Trace the curve
and show that the area
is
2
Jc
[ST. JOHN'S, 1886.]
.
+ x*y = a x\
2
y*
of the
segment which
of y and the straight line whose equation
is
y
lies
=x
is
between the axis
^
a 2 log
2.
[COLLEGES
e,
1883.]
2
ordinates of the hyperbola xy = a are determined
the condition that the area included by any pair, the curve, and
13. Paiis of
by
the
re-axis is
constant
a constant ratio.
;
show that the lengths
of
any such pair are
[OXFORD
I.
in
P., 1888.]
PROBLEMS ON QUADRATURE.
Show
14.
431
that the area between the curve
2
x(x + f-a>) + ^a\/3 =
and
its
15.
asymptote
Show
and the
16.
2
is 7m'
.
[ST.
JOHN'S, 1892.]
that the area between the inner branch of the curve
positive parts of the
two axes
is 7ra 2
/3v/3.
[ST. JOHN'S, 1888.]
Prove that the whole area of the epicycloid generated by a
rolling on a fixed circle of radius a
point on a circle of radius
the area of the fixed circle in the ratio of 15 to
17.
2
(x
+
1
is
8.
Find the whole area of the curve whose equation is
2
= 0.
4- a) (x + y
a) + X f
[COLLEGES,
) (x + y
2
7/
8.
Find the area
1
886. ]
of a loop of the curve
x 4 + y 4 = 2a 2xy
19.
to
Find the area cut
off
from an
[OXFORD
.
ellipse
by a
I.
P.
,
1 888. ]
focal chord.
[COLLEGES
a, 1883.]
by the equiangular spiral r = ae
from the space bounded by any two fixed lines through the pole are
20.
Prove that the areas cut
ecoi<L
off
in geometrical progression.
21.
Find the area
given radii vectores
[OXFORD!.
of the curve r
= aBe
be
P., 1900.]
enclosed between
and two successive branches
two
of the curve.
[TRINITY, 1881.]
22.
fl
Find the area
= 0and
23.
of the loop of the
curve r = aOcosti between
=
|.
Find the area
[OXFORD
II. P., 1890.]
of the curve
(r
- a cos
6)2
= a 2 cos 20.
[COLLEGES
a, 1887.]
= 3axy is
24. Show that the area of the
loop of the folium x* + y*
divided by the parabola y 1 = ax in the ratio 5 4.
In what ratio does the line x + y = 2a cut the loop in the above
:
folium.
25.
[OXFORD
I.
P., 1889.]
Find the area included between the axis of y and the curve
2
- 2aj(f/ + 1 = a* - 3^ + 3,
y + 2y
)
the curve being supposed to stop at the node.
[ST. JOHN'S, 1884.]
CHAPTER
432
XII.
Determine by integration the area
26.
27.
(i)
Find the whole area enclosed by the hypocycloid
x*
Prove that the area
(ii)
of the ellipse
+ y* = a*.
[OXFORD
P.
I.
1888.]
,
of the locus of intersection of pairs of
tangents at right angles for this curve
is
ira~.
[MATH. TRIPOS, 1888.]
28. Prove that the locus of the points of bisection of the intercepts on the normals of a cycloid between the cycloid and its base
divides the area between the cycloid and its base into two parts in
the ratio 7
:
5.
[OXFORD
is
area of the loop
between the
II. P., 1886.]
+ if" = (2n + l)ax y when n is even,
odd, n being a positive integer ; and prove that the
Trace the curve x
29.
and when n
is
n+l
n
+l
n
,
(2n+l)~.
Prove that
this
is
also the
area
branches of the curve and the asymptote.
infinite
[ST. JOHN'S, 1882.]
Find the whole area contained between the curve
30.
2
X*(x
and
its
+ y 2 ) = a 2 (y 2 -x?)
asymptotes.
[OXFORD
Find the area bounded by the
31.
x = b cosh M,
circle
I.
tf=acos0,
P., 1887.]
?/
=
y = b sinh u
and the hyperbola
that area being taken
which lies within the circle and on the convex side of the hyperbola, and b being less than a.
[TRINITY, 1888.]
32.
(a)
A SJ A
Show
that in the Archimedean Spiral r = aO,
if
A
lt
A^
be the areas of the inner loop and the successive heart4
shaped figures formed by the convolutions of the curve
(b)
,
...
In the Reciprocal Spiral rO = a,
if
A A2 A
lt
,
s
...
be the areas
of the successive closed loops,
33.
Find the area
of the loop of the curve
(x
34.
At
all
points
cosh(?0cota)
of
+ y) (x 2 + f) = '2axy.
the
lines are
first
negative
[OXFORD
pedal
of
I.
P.
the
,
1890.]
curve
drawn making a constant angle a with
PROBLEMS ON QUADRATURE.
433
Show that the area bounded by any pair of such
the tangent.
the curve enveloped and the first negative pedal is
lines,
2
2
^{l + (m -l)cos' a},
A
is the area of the corresponding portion of the first
where
negative pedal bounded by radii vectores from the pole.
[COLLEGES a, 1891.]
Find the area
35.
of that portion of the loop of the curve
r2
which
is
=p cos + q sin
not enclosed by the curve
r 2 = b + a cos
0,
0.
a family of such curves be taken, (by varying p and q), such
that this area is constant, show that the envelope of the system is a
If
curve whose equation
is
r2
36.
curve
Show that the whole area
r^ = ttcosf# is f 2 \/3'
C
37. In a hyperbola,
axis
=c+a
and
P
any point
is
(x,
cos
38.
Show
[COLLEGES
p, 1889. ]
enclosed by the outer line of the
[COLLEGES, 1876.]
the centre,
A
the end of the transverse
on the same branch of the curve as
y)
prove that twice the area of the sector
and a
0.
CAP
A
;
is
that the area contained between a hyperbola, any tangent
asymptote which bisects the part of the
line parallel to the
tangent intercepted between the curve and the asymptote
and
is
39.
constant.
[TRINITY, 1886.]
Prove that the area of the curve
s
_l ap*(l-p^
ap
~
TTTO.
40.
[MATH. TRIPOS, 1882.]
Show
that the area cut off from the ellipse
ax2
by the
line lx
+ my=l
+ 2hxy + bf=l
is
a ft (9
where
a, /3
-sin0 costf),
are the semiaxes of the ellipse
and
~
cos=
[COLLEGES, 1892.]
E.I.C.
2E
CHAPTER
434
41.
XII.
Trace the curve whose equation
is
and prove that the area between the curve, the axis
tangent parallel to the axis of y
(2n
|4?i
v
42.
Show
of x
and a
is
-
1
-
log 2n).
[ST. JOHN'S, 1885.]
that in the curve
= sec 20 log (2 cos 2 6)
curve and the lines 0=
r2
the area between the
2
^TT is (|^r) .
[ST. JOHN'S, 1886.]
43. Find in integral form, and completely, the area enclosed
between two confocal conies and two given radii from the centre.
[TRINITY, 1881.]
44.
Prove that the area
2
2
pieces of the ellipse x /a
bola z 2/a 2 - if/p*
=1
(
<
the two equal and similar
are cut off by the hyper-
of each of
+ y z/b 2 =l which
a) is
[ST. JOHN'S, 1887.]
45.
Prove that the areas of the two loops of the curve
- Sar + 9a 2 =
r 2 - 2ar cos
are
(327r
+ 24v/3)a 2
and
(16n-
-
24v/3)a
2
.
[MATH. TRIPOS, 1875.]
46.
The area between two tangents
an equiangular
to the
is
same convolution
of
one another, and the curve,
spiral at right angles to
p/ + i (/_/) cot 2 7
,
where p, p are the perpendiculars from the pole on the tangents and
y is the angle of the spiral.
[COLLEGES, 1882.]
47.
xs
A
circle
with centre at the origin cuts the loop of the Folium
If the angle subtended at the origin by the
+ y$ - 3 axy = 0.
common chord
equals
-
2^1
2 tan- 1 -5
2*
+
,
l
prove that the area between the loop and the
circle is
[COLLEGES, 1885.]
PROBLEMS ON QUADRATURE.
The
48.
a
moves along a
centre of a circle of constant radius a
AB in its
drawn
AP
tangent
straight line
plane,
between the locus
of
P
and
its
fixed
A
and from
a fixed point in the line
Show that the area included
to the circle.
is
435
asymptotes
is Tra
2
.
[MATH. TRIPOS, 1882.]
Show
49.
that the curve
has three loops, whose areas are
a 2 (-|;r
respectively.
Show
50.
(|TT
_
a^Vr-fv^)
[COLLEGES, 1892.]
that the area of the Cassinian
7T
\/6
f7
_
but
-A ^3),
2
+ 2v/3),
2
is
r
4
- a 4 sin'J
</>
& 4 cos 2
I
Jo v/a
4
<^=r,
-6 4 sm'2
provided
d(f>,
b
> a,
when a>b.
<
[MATH. TRIPOS,
Prove that the area
51.
with respect to the focus
where a and
e
ellipse
and the eccentricity
of the
is
are the semi major axis
[COLLEGES, 1892.]
ellipse.
How
52.
1883.]
an
of the first negative pedal of
do you interpret
Find the area
this result if e
of the curve
<
\
1
whose Cartesian equation
is
[MATH. TRIPOS,
vx dx, vx
1896.]
being the real root of the cubic
Ji
[COLLEGES, 1872; R. P.]
~>1.
Fiiid the area in the first
quadrant bounded by the axes of
coordinates and the curve
,
, x
smrr" 1 a
.
taking
a, b, c all
positive.
.
,
. 11
+ sinn" 1 y = c,
o
[I.
C. S., 1897.]
CHAPTER
436
55. Trace the
whole curve
xY = c
< b < a,
where
XII.
and
2
(a-x)(x-b)
whole
find its
)
area.
c. S., 1898.]
[I.
56. It is given that the abscissa ON and ordinate NP of a point
on any arch of a cycloidal arc are a(6 - sin 6) and a(l - cos 0). NP
= 2a, and the rectangle ONKA is
so that
is produced to
NK
K
Prove that the area included by ON, NP and the
completed.
arc OP never differs from three-fourths of ONKA by more than
3rt
2
~g~>/3
and
;
find for
what positions
of
P
the difference vanishes.
C. S., 1912.1
[I.
57.
Trace on squared centimetre paper the curves
taking a
=
10 cm., and estimate the area of a loop of each curve.
Prove that
,,
I
.
,,, q
dt
=T
dt=-
=- r
.
of a loop of the second curve. Find also
Give each area to the
the area of a loop of the first curve.
nearest square centimetre when a is 10 centimetres.
and hence calculate the area
[C. S., 1913.]
between the two curves
58. Obtain the area contained
r2 cos20
= 4a'2 cos 4
r 2 cos20
and
= a2
.
[Oxr.
59.
Show
is
x2
I.
P., 1914.]
= axzy*
if
2
equal to a /! 4.
60.
P., 1912.]
that the area of the loop of the curve
x7 +
is
I.
[Oxr.
Prove by any method that the area
and find the area.
independent of
Prove also that the straight line
of the ellipse
,
+
3y
2
= $y
into
two areas which are
4vr
61.
- 3>/3
:
STT
y=x
divides
the
ellipse
in the ratio
+ 3v/3.
[Oxr.
I.
P., 1916.]
Trace the curve
r cos
and show that the area
= a sin
3#,
of a loop is
2
|a (9v/3
-
47r).
[MATH. TRIP. L,
1919.]
PROBLEMS ON QUADRATURE.
62.
Show
that the curve r = a(2 cos 0-fcos 30) has three loops,
^-
the area of the larger loop being
5*
two smaller loops being
63.
Show
a2
'"^
2
,
.
and the areas
of the
[MATH TRIP L>
1916
-,
that the coordinates of any point on the curve
2
y (a
may
437
+ x) = x'*(3a-x)
be taken as
= a sin
x
30/sin
y = a sin 30/cos
6,
0,
and prove that the area of the loop and the area between the curve
and its asymptote are both equal to 3\/3a 2
[MATH. TRIP. I., 1915.]
.
64.
Show
that the area of the loop of the curve
in the positive
65.
quadrant
is
ira2
[MATH. TRIP.
.
Having established Simpson's Rule, that
I.,
1920.]
still
using
if
y = y(x)=a Q
fV* =
then
Jo
prove that
if
y(x) also contains a term a 4 x* the error in
Simpson's Rule
is
\
f20
a 4-
[MATH. TRIP.
I.,
1920.]
CHAPTER
XIII.
QUADRATURE
(II).
TANGENTIAL POLARS, PEDAL EQUATIONS AND PEDAL
CURVES, INTRINSIC EQUATIONS, ETC.
416. Other Expressions for an Area.
other expressions may be deduced for the area of a
plane curve, or proved independently, specially adapted to
the cases when the curve is defined by systems of coordinates
Many
other than Cartesians or Polars, or for regions bounded in a
particular manner.
To avoid continual redefinition of the symbols used
state that in the subsequent work the letters
x,
y,
r,
0,
have the meanings
of Curvature in the author's
s,
assigned to
417. The (p,
s)
p,
\]s,
0>
we may
p
them throughout the treatment
Differential Calculus.
formula.
i
Fig. 61.
be an element Ss of a plane curve and OY the perpendicular from the pole upon the chord PQ. Then
Let
PQ
438
TANGENTIAL-POLAR CURVES.
and any
439
seetorial area
the summation being conducted along the whole bounding
arc.
In the notation of the Integral Calculus this
is
This might be deduced from the polar formula at once.
A=
For
where
<
is
the angle between the tangent and the radius
vector.
418. Tangential-Polar
Aain
we have
Form
(p,
ds
= -=>
since
Area =
If
~ \pds
4j
a form suitable for use
,
\//-).
dtp
If
= ^If/
= ^\ppd\j^
\plp-\*J
when
*j
\
the Tangential-Polar
(i.e.
p,
\js)
form of the equation to the curve is given.
This gives the seetorial area bounded by the curve and the
initial and final radii vectores.
419. Caution.
In using the formula
care should be taken not to integrate over a point, between
the proposed limits, at which the integrand changes sign.
If
such points exist the whole integration
is
to be conducted
in sections along each of which the sign of the integrand is
permanent. The results for the several sections are then to
be taken positively and added together.
inflexion is passed
and changes
When
P + J~YZ passes through an
a point of
infinite value
sign.
420. The Case of a Closed Curve.
When
the curve
simplification.
is
closed
the formula admits of some
CHAPTER
440
XIII.
For integrating by parts
Hence
Area
=
\p
-~
-f
In integrating round the whole perimeter the term between
square brackets,
viz.
^
p -~
disappears, for
it
resumes the
had when we return
starting-point after integrating round the contour
same value as
curve.
421.
OA
l
it
originally
to the
of the
Hence, for a closed curve,
Ex.
1.
the initial
Let A 1 CA 2 be one
line.
foil of
Then p vanishes
if
the epicycloid
-5^ = 0,
p = Asm
TT, 27r, ...
J2v//-
and
.
Fig. 62.
Therefore, for the area bounded by OA n OA 2 and a
the kite-shaped figure OA-^GA^O in Fig. 62,
viz.
foil of
the epicycloid,
PEDAL CURVES.
441
Thus, for the whole cardioide, which is a one-cusped epicycloid formed
as the path of a point attached to the circumference of a circle of radius
a rolling upon an equal circle whose centre is at the origin 0,
p = 3asin ^
And
Ex.
the area
2.
(See
.
Diff. Calc., p. 345.)
is
Otherwise, the cardioide p
3a sin
^o
is
a " closed
"
curve.
Let us apply the second formula
^
The whole area = -
I
( 9a 2 sin 2
i/r
Putting
^ = 3$,
this
^-a
=
2
cos 2
and
this case.
~)fltyr
taken between limits
^ = 371-.
becomes
'
422.
If
as before
'
Pedal Curves.
be the tangential-polar equation of a given curve,
the angle between the perpendiculars from the pole
p =f(\]s)
s is
upon two contiguous tangents, and the area of the pedal
curve
may
be expressed as
,
p,
x//-
being the polar coordinates of F.
.e.
CHAPTER
442
XIII.
%
423. Ex. Find the area of the pedal of a
on the circumference (i.e. the cardioide).
circle
with regard to a point
Fig. 64.
Here, if OY is the perpendicular on the tangent at P, and OA the
diameter =2c, it is geometrically obvious that OP bisects the angle AOY.
A
Hence
calling
AO I
7
,
^,
we have
for the tangential polar equation of the
circle
Hence Area=
and
TT,
and the
-
/
4c 2 cos 4
result
is
^- c?^,
to be
where the
limits are to be taken as
doubled so as to include the lower portion
of the pedal.
Then
424. Area bounded by a Curve,
its
Pedal and a Pair of Tangents.
be two contiguous points on a given curve Y, Y' the
corresponding points of the pedal for any origin
(Fig. 65).
Let P,
Q
Then since, with the
;
usual notation,
PY =
..-.
ttvf
triangle
,
the elementary
bounded by two contiguous tangents PY, QY', and
the chord of the pedal
quantities
YY',
is
to the first order of small
PEDAL OF THE EVOLUTE.
443
Hence the area of any portion bounded by the two curves
and a pair of tangents to the original curve may be expressed as
Fig. 65.
and
is
the same as the corresponding portion of the pedal of the
the perpendicular from
upon the normal
evolute, for
at
P
PY=
(Fig. 66).
Pedal of Evolute of a Closed Curve.
In the case of a closed curve, then, the equation
425.
admits of two interpretations.
^-\Z
Q
Fig. 66.
Let
be the pole,
of the evolute, P,
Q
AP
an arc of the
BQ an arc
on
the
curve
and the
corresponding points
closed oval,
CHAPTER
444
evolute,
XIII.
OY, OZ, perpendiculars from
on the tangent and
normal at P.
Then the
Y
locus
is
Z
the pedal to the curve, the
Hence the equation
locus
is
the pedal to the evolute.
cty
= area
of
expresses
(A)
That the area of the pedal of the
= area
of the oval
+
oval
(B) That the area of the pedal
= area
426.
of the oval
oval
the area of the region between the
+ the
and
its
pedal.
of the oval
area of the pedal of the
evolute.
Additional Results.
Further, since
area of pedal
and
= area
of oval
+^
area of pedal =
^
we have upon addition
2 x area of pedal = area
of oval
+^
= area
of oval
+-
= area
of oval
+^
or
(^4)
^
I
ds,
2jp
i.e.
the area of the pedal of a closed curve with regard to any
origin within
This result
1 (V
it
exceeds half the area of the curve by 7
may
the integral
ds.
be regarded as giving an interpretation for
/
jy>
2
Jr
<fy
an expression which figures
427.
2
I
or
j-ds,
in the discussion of roulettes.
Geometrical Proofs.
These facts
may be established by elementary geometry thus.
Let Pv Q v Y v Z be the contiguous positions to P, Q, Y, Z on
l
the respective
OY l at N.
loci,
and
let
YP,
YP
l
l
intersect at
T
and YP,
PEDAL OF THE EVOLUTE.
445
Then
A GYP- A 07^ = (A OYN+ A ONP)
- (A ONP +ANY T + quadrilateral OPTPJ
1
= sectorial
OY Y^ sectorial
-quadrilateral OPTPV
area
area
TYY
l
Fig. 67.
And summing
A OZZ^ = A
and
.'.
or
i.e.
for a closed oval,
TY Y to the first order
SOFY^SOZZ^areaof oval
= ST T Y' + area of oval
l
;
,
area of pedal of evolute, or area) =area of pedal of oval
between pedal and oval
area of oval.
J
428. Ex. 1. As an illustration, consider the central pedal of
evolute of an ellipse.
Area of pedal of evolute = area of pedal of ellipse -area of ellipse
= x- (a 2
-J-
62 )
irab
the
CHAPTER
446
XIII.
Ex. 2. The pedal of a circle of radius c and centre C with regard to a
on the circumference is r = c(l + cos#), a cardioide. The evolute
point
of the circle is a point, viz. the centre.
As the current point P travels
round the circumference of the
circle once, the
path of Z, the foot of
Fig. 68.
the perpendicular upon
for diameter)
twice.
PC
travels
The pedal
round
its
path
of the evolute
is
(viz.
a circle on
OC
therefore the twice
described circle of radius -.
2
And
area of cardioide = area of circle radius c + 2 x area of circle of radius a
429.
Pedal Equation
When
equation,
(p, r).
the relation between p, r
is
given,
i.e.
the pedal
we have
J_(
rp
r
,
dr.
J*s
This again gives the sectorial area between the curve and a
definite pair of radii vectores.
Again care is required in the use of the formula to avoid
through a value of r for which sec< changes
i.e. when
sign,
changes from acute to obtuse, as it will do at
r
where
has
a maximum or minimum value. If such
points
the
points occur,
integration must be conducted separately
for each of the portions into which these points divide the
perimeter and the results taken positively added together.
integration
PEDAL EQUATION.
Ex.
430.
1.
In the equiangular spiral
=H
Ex.
2.
Find the area
1
p = r sin a, and any
2 2
dr = -r (r
V 2
r, ) tan
,
/
rcosa
2>,
447
sectorial area
,
4
of the lemniscate
a.
'
P = ^-
A=^
Here
Taking limits from
r
=
to r
= a, we
get a result
.
This gives the area of half a loop.
The whole area is four times this result, viz. =ar.
Note, that if we integrated through the maximum without change of
to r =
sign of the radical from r =
again, we should obtain a zero
result
i.e.
the difference of the two halves of the loop instead of the
sum
as desired.
431. Area included between a Curve,
two Radii
of Curvature
and
the Evolute.
In this case
we take
as our element of area the elementary
Fig. 69.
triangle contained by two contiguous radii of curvature
the infinitesimal arc Ss of the curve.
To
first
order infinitesimals this
notation as before.
is J/o 2 S\Js,
and
using the same
CHAPTER
448
And
XIII.
area required
r
i.e.
=I
or
or other forms adapted to the particular species of coordinates
in use.
For instance, for Cartesians
_ dy
_ d*y
~dx' y *~'dx*''
or for Polars
1
?,
= ^,
tl/f
i
where
r1
= -^-2
\A/
r2
I
,
etc.
Fig. 70.
432.
Ex.
the circle
is
1.
The area between a
(Fig. 70)
circle,
an involute and a tangent to
AREA SWEPT BY A
Ex.
2.
"TAIL."
The area between the tractrix and
The tractrix is described in
similar manner.
its
449
asymptote
is
found
Calc., Art. 444.
Diff.
portion of the tangent between the point of contact and the .r-axis
constant length c.
T
in
The
is
of
r
Fig. 71.
Taking two adjacent tangents and the axis
of<d;as
forming an elementary
triangle (Fig. 71),
Area = 2 i
f
^"W
-2
7TC 2
433. Area swept by a "Tail."
In exactly the same way as in the last example we may find
the area swept out by a "tail" of length varying according
to
any specified
of contact.
law measured along a tangent from the point
Let the length of the
distances 0(s),
E.I.C.
<J)(s-\-Ss)
tail
be
(J>(s).
Let
P Q
lt
measured along the
2l
l
be at the
tangents
at
CHAPTER
450
XIII
contiguous points P and Q respectively from the points of
Then the area of the triangular element bounded
contact.
by the two contiguous
tails
and the arc
PQ
1
l
to the first
is
order
and the area swept out by the
If
(j)(s)
a constant
closed oval of
= c,
tail is
continuous curvature
a circle of radius
=-
Area swept
= ?rc2
,
c2 d\//%
I
viz.
and
for a
the area of
c.
If the tail be of length equal to the corresponding radius of
curvature, the area swept out
=
I
P
2
^V
o
I
P ^s
-
434. If lengths be taken along the normal drawn outwards,
and
specified in the
same way,
viz.
<(s), the area
the original curve and the locus traced
or
if
the distance
</>()
is
be on the inward drawn normal
()} ety.
between
AREA SWEPT BY A "TAIL."
451
435. Parallel Curves.
If,
be constant = c, a 'parallel'
traced, and the area between a curve
in this case (Art. 434),
to the original curve is
its parallel will be found
and
</>(s)
from
and for a closed oval of one convolution surrounding the pole
2
s being the perimeter of the oval, the
this becomes cs
ire
positive sign being taken for exterior parallels, the negative
If the normal makes n revolutions
sign for interior ones.
,
before returning to its original position, the area swept over
by PPl will be numerically
CS
2
H7TC
.
436. General Case.
More generally, let us construct a new curve from a given
one by measuring a distance a along the tangent from the
point of contact, in the direction of measurement of the arc,
and a distance /3 through the extremity of a, parallel to the
outward drawn normal at P, and let the point at which we
arrive be called
Q
;
a, /3
not necessarily being constants.
Fig. 74.
if x, y be the coordinates of P and
those of Q, and
tj
be the inclination of the tangent at P to the initial line,
Then
if
\\s
= x+a cos \!s + (3 sin
\//-,
rj
= y + a sin ^
Then dg = dx+(d a co9\!s+d/3 sin \ls)+(
a sin
ft
cos
^ + fi
\fs.
cos
>//
CHAPTER
452
XIII.
-f ft sin
\Js)
{dy -f (da sin
+ (a cos
(?/
+ a sin
/3
x//-
cos
\js)
\/r
+
(
\ls
xdyydx-\-{(a cos ^-\-/3 sin \//-)
(a sin
a sin \fs
d/3 cos
\/^ )
-f/3 sin
{dx -f (da cos
a sin
^+/^ g n Vr
(a cos
i
(a sin
)
(^ a s i n
^
(for
/8
\/^)
cos
+ ^/5 sin
\/y) d\//d\//-
^sdft cos
cos \/r)
x//-)
\/r
cos
cos\/r)
-f-
-fd/3 sin
-f /3 cos
y (da cos \/r
sin
x//>
sin
j8
X//-
d/3 cos
\]s
(^ft
\//-)
cos
dx =
ds+(/3d
x ( a sin \/^
a term
{
dx(a sin \^
\
i.e.
that
is
/3ds
/3
cos
x//-)
$ cos \/r )}
-\-dy(a
o
having
been added and subtracted in
the
arrangement.
^
be the corresponding sectorial areas
Hence, if A and
radii
out
the
vectores
OP and OQ,
by
swept
[ ] being between limits corresponding to the
and
ending of the arc traced by P.
beginning
If the curves be closed this term disappears, and
the portion
This formula of course includes the foregoing cases.
Thus, for parallels a = 0, /3 = c, and the oval being closed,
as before.
POLAR SUBTANGENT.
437.
453
Polar Subtangent.
The area bounded by any portion of a given curve, two
tangents, and the corresponding portion of the locus of the
extremity of the polar subtangent
where
For
OT
given by
T =r
T
if
is
be the polar subtangent corresponding to a point
Fig. 75.
P, the point of contact of the tangent,
notation
and
Area swept by
we have with
the usual
PT =
I
jj
ds
d9
CHAPTER
454
XIII.
the limits being the initial and final values of 9 for the
arc specified.
For a closed curve
this area therefore exceeds twice the
area of the original curve bv
438.
Intrinsic Equation.
When
the intrinsic equation
is
given, viz.
the area bounded
by the curve, an initial tangent, and an
ordinate from any point of the curve to the same, is given by
C$ Cx
=\
/'(x)/'
I
Jo Jo
M
cos
si
w
X sn
being assumed that the integrand is finite and continuous
and does not change sign within the limits of integration.
it
This
is
merely a transformation of
-
For
= sin\r
and
y=\JQ s
dx = cos \^ ds =f (\fs)
Also
Hence
A=
/'(^) cos
Jo
This
may
^
j
cos
f(u>) sin
Jo
clearly be written
A=
JO Jo
/' (x) cos
X/'M
i
w dwf d\]s.
455
INTRINSIC EQUATION.
439.
Ex. Taking as a test the case of the circle
= a2
I
= a*
I
9
=
cos
I
1*
2
/
Jo
r
cos
x
cos
x (-1
cos
x
sin
cos
w
T
- cos
x
O
T
N
x
Fig. 77.
which may readily be verified otherwise.
44-0.
Closed Oval.
a point on the circumbe a closed oval and
for
measurement of s,
the
the
ference, viz.
starting point
we may obtain the area of the whole curve by integrating
If the area
-
\ycos\Isds round the whole contour, and our formula
be written
^
4=1
may
ro
I
/' (x)/ (a)) cosx sin
wc?xc?ft>,
the integrand being supposed finite and continuous throughout, and the curve s=f(\}r) having no singularities.
Closed Oval.
Another Form.
Another form may be given for the area of a closed curve
whose intrinsic equation is s=f(\js).
441.
Fig. 78.
= 0, we have at any
Measuring s from the point at which
ri where the inclination of the tangent to the initial
tangent is ^, and the element of arc ds v
\//-
point
^7
=sm X
CHAPTER
456
f
x=\ cosx^-
.-.
Jo
=\
xdy
.*.
area of curve
=
,
rf*
V r J\
1
.
,
J VA./
I
\
AV
V
LJo
JQ
we may
write
1 f
A
r
'
it.
2'
^=9
1
f
/'
^Jo Joo
it
si
y dx) taken round the perimeter,
dy
^ \(%
i
as
co
Jo
ydx
p
2J
or,
f*
sinx^lJo
Jo
.'.
XIII.
/W/ (x) sin (^-x)^^X'
/
being understood that the
first
integration is with regard
to \Js, and then the result
to x. considering >//- a constant, from
from to 2-rr with regard to \js.
Also
Joo
may
/
be integrated by parts, and becomes
~
cos
for/(0)
Hence the
may
2
1 f ' f*
or
it
result
=9^Jo
Jo
be exhibited as
/
being understood as before that the
regard to
x from
to
= 0.
first
integration
442. If the curve be not closed, and the limits for
from \]s = a
to
is
with
\js.
^ = /3, we
find
by these formulae, a
\fr
are
sectorial
INVERSE CURVE.
457
area bounded by the arc and two specified radii vectores, viz.
= /3, and
from the origin to the points where \js = a and
^
=
Jf' J
*f
443. Inverse Curve.
If the points P, Q be contiguous points on a curve,
their respective inverses, k being the constant of
and
P', Q'
inversion
Fig. 79.
and
the
the pole,
curve,
we have
for
any
sectorial
element OP'Q' of
new
to the first order
= -s
-
-2
89 to the
and the area of any
v
bein
first order,
sectorial portion of the inverse is
the radius vector of the oriinal curve.
Ex. Thus the area of the inverse of Ax* + By* = a 2 (x2 +y 2 ) with regard
to the origin is
T
i?)r
be noted that this amounts to performing the inver-
It will
sion
first,
formula
44-4.
Let
<''
and then finding the area as
-1-^
-
\r'
2
d9, so that our
J
k* f 1
is
of but little additional convenience.
Locus of Origins of Pedals of given Area.
be a fixed point.
Let p, \]s be the polar coordinates
of the foot of the
perpendicular
OY
upon any tangent
to a
CHAPTER
458
P be any other fixed point, PY = p
P upon the same tangent. Then the
and P respectively as origins, are
Let
given curve.
XIII.
l
(
perpendicular from
of the pedals, with
21
fs
wv
p^pr cos (y\Js) = p
and j9 is a known function of
Hence
2A t =
=
tpi'd^raBi \(p
\yPd\ls
-f
Now
2
&
1
/j
xcos\]s
cos 2 \lsd\ls-}- 2xy
d\is,
x cos
y sin \/r,
>//
2
y sin
\fs)
d\js
2y \psm\fsd\Is
cos
>/r
I
2\psm\ls
sin \^
c?\/r
cos 2
d\fs,
si
+ y*\ sn
^^,
taken between such limits that the whole pedal
will be definite constants.
A and
P with
\f/.
2x \pcos\jsd\fs
cos \/r
the
areas
^
CvIlL
taken between the same definite limits. Call these
be the polar coordinates of
A! respectively. Let r,
regard to 0, and x, y their Cartesian equivalents.
Then
1 ),
Call
-2g, -2f,
and we thus obtain
2A l = 2A + 2gx+ 2
them
a,
2h,
respectively
b,
is
etc.,
described,
PEDALS OF GIVEN AREA.
P move
If then
in
must be a conic
locus
A
such a manner that
is
1
constant,
its
section.
Article 342,
By
rt
cos 2 \//- d\]s
fA
xl
(
>
sin 2 \fs d\ff
ab
i.e.
Hence
>
rt
cos
i
j
j
\/^
"p
sin
\//-
d^ t
,
2
//
.
this conic section is in general an ellipse.
Moreover,
its
459
centre being given
its
by
position is independent of the magnitude of A r
these several conic-loci
for different values of
l
A
Hence
will all be concentric.
We
shall call this centre Q.
445. Closed Oval.
Next suppose that the
and that the point P
and 2?r.
is
a closed oval curve,
Then the limits of inte-
original curve
within
it.
is
gration are
Thus
a
cos 2 x//- d\ls
I
= TT
and
h
p2T
=
J
cos \//- sin \Js
sin 2 \Js d\fs
I
Jo
d\]s
=b
o^
= 0.
Hence the conic becomes
that
is
a circle whose centre
is
'*
1 f
1 f
pcos\js
d\js,
T^Jo
Now,
and x
if x,
at the point
2'
^Jo
ps
y be the point of contact of the tangent,
~- sin \!s,
= p cos \ls
...
by projecting
cos +,
= p COS
I
\//-
dx/r
dp
,
d\ls
p, -*y
coordinate axes
I
[p sm \Js] -\-
\
;
viz. Q,
upon the
CHAPTER
460
XIII.
and
r
\y
=
d\js
f
f
= 21fpsii\\js d\]s,
\psm\lsd\]s-\-[pcos\Is]-}- \p sin\/r a\/r
J
J
J
J
for the portions in square brackets disappear in integrating
round the whole curve.
Hence the
coordinates of the centre of the circle
may
be
written
x
-ds,
1
\xd\jr
where-
\yd\js, or *%
^TTJ
\d\ls
,
is the curvature
P
at the element ds.
or ^
\-ds,
ATTjp
446. Another Determination of the Centre.
If the original curve be regarded as a material curve of uni-
form section
u>
at each point,
and with a density proportional to the curvature
= k-, say, the mass of each element Ss is k- o> <5s,
p
p
and the formulae
_=
of Statics
show that the centroid
of
any arc
of this curve is
given by
- ds
~
{k
-u>ds
Cl
-
Jp
JP
(k
-ooyas
ooyds
\x d\l/
p
'
or
J
'
,
f
I d\ls
ds
J
- as
\-ds
\yd\ls
~
'
ck
fi
JP
JP
-wds
- ds
r
\d\!s
J
Q, which is the centre of these loci, is
with the centroid of a material wire of fine uniform
bent into the form of the original curve, and having
Hence the point
identical
section,
a density proportional to the curvature at each point or,
which comes to the same thing, having uniform density and
cross-section infinitesimally small but proportional at each
;
point to the curvature.
PEDAL OF MINIMUM AREA.
44-7.
Connection of Areas.
The point Q
having been
origin from
to
Q.
thereby be removed.
\p cos
where
p
the area
found,
\$s
and
d\]s
T2,
the pedal whose pole
oval
2^4 1
is
our
transfer
\psm\Isd\Is,
now measured from
is
.of
us
let
The linear terms of the conic will
Thus Q is a point such that the integrals
is
2A 1 = 2H+ax2 -}-2hxy+by 2
and
461
= 2II-(-7r(x2 -f y 2 )
both vanish, and
we have
Q,
for
if
any
II be
other,
in the general case,
in the particular case,
when
the
closed.
,,
.
The area ofe the conic
rp,
.
is
~
27r(A l-
(Smith, Conic Sections,
II)
-.
Jab-h
-
2
Art. 17 J.)
Thus, in the general case,
Jabh
rr
-- x area
5
AA = II
-f
o
2
c
*
I
1
And
where
conic.
^7T
in the particular case of the closed oval,
r is
values of
the radius of the circle on which
A
I}
i.e.
P
the distance of
448. Position of the Point
Q
P lies
for constant
from Q.
for a Centric Closed Oval.
In any oval which has a centre the point Q is plainly at
that centre. For when the centre is taken as origin, the
integrals
p cos \js
d\Js
both vanish
and
when
I
p
sin \//-
d\Is,
the integration
i.e.
is
^
x d\js and ^ y d\js,
performed for the com-
or,
plete oval, opposite elements of the integration cancelling
which is the same thing, the centroid of a material centric
;
oval curve for a law of density, which varies as the curvature
at each point, is
obviously at the centre of the oval.
449. Origin for Pedal of
When Q
is
Minimum
taken as origin,
2^!
it
Area.
appears that
= 211 + f (x cos ^ + y sin V')2 d\]s.
Hence, as the term \(x cos\fs
positive, it is clear that
A
l
+ ysm\ls)
can never be
2
d\lr
less
is
than
necessarily
IT.
CHAPTER
462
Q
XIII.
therefore the origin for which the corresponding pedal
is
curve has a
A
450.
minimum
Statical
area.
View
of the Case.
be the origin, QRS the closed oval, OY the perpendicular from
upon a tangent to the curve. Let P be any
other point, and f2 the centre of gravity of the curve, QRS
Let
having a density at each point proportional to the curvature.
Fig. 81.
A
theorem by Lagrange (Routh,
states that if
heavy
and if
m m w
lf
particles at
P
2
,
3
,
Q I} Q 2 Q 3
,
small section
...
,
and
Q
their centre of gravity,
theorem to our curve of density
and total mass \koo, say,
this
u>,
P
it
,
Statics, vol. i. Art. 436)
be the masses of a system of
be any other point, then
Applying
Now
...
P
1
1
Cr 2
-the area of the oval by
4J p
pedal with regard to
-.
P = ~1 oval
1 f
-f-
*^
and
uniform
has been proved in Art. 426 that the area of the pedal
of a closed oval exceeds
.'.
k
-,
pedal with regard to
Q = ^ oval
-f
-
PQ
J
2
ds;
I
f)
~ ds;
ds.
A STATICAL VIEW OF
and
Xkco
= mass
of curve
Ck
-u>ds
=
I
ip
.'.
463
= ku>\f ddr =
<
J
X-27T.
P = pedal
pedal with regard to
.;.
CASE.
P = pedal wijh regard to &+-r
pedal with regard to
.*.
~HE
Hence we are led by
statical
with regard to
a
considerations to the same
result as already obtained, viz. that the loci of the origins P,
of which the pedal curves of a closed oval are of constant
area, are concentric circles, their centre being the origin of
the pedal of minimum area and the centroid of a fine wire
bent into the form of the original oval, and having uniform
and a density varying as the curvature.
cross-section
Illustrative Examples.
Ex.
1.
Find the area
of the pedal of a circle
within the circle at a distance
Here
c
n = 7ra
2
.
A = Tra 2 + Jra 2
Hence
2.
a Iiraa9on.
i.e.
Al
and
Ex.
with regard to any point
from the centre
.
l
Find the area of the pedal of an
from the centre.
ellipse
with regard to any point
at a distance c
In this case, II
is
the area of the pedal with regard to the centre
f
=2
Hence
^i
r
=
Ex.3. The area of the pedal of the cardioide r = a(l-cos#) taken
with respect to an internal point on the axis at a distance c from the
pole
is
3r_
T
Let
(/>
P
be the pole,
perpendiculars
0F
the angle Y.2 OP
2
and
and
(5
[MATH. TRIPOS,
the given internal point
/'F,
upon any tangent from
OT=c
= 2A-2c
then
;
(
p^p-ccos^,
;
1876.]
p and p the two
and P respectively
l
;
and
'CHAPTER
464:
Now,
in order that
between limits
</>
XIII.
p may sweep out the whole
= and
and double.
<f>=
p = OQ sin
Y.2
we must integrate
pedal,
Now in the cardioide (Fig. 82).
QO = OQ sin J xOQ.
Fig. 82.
For
Hence
O^T
or
ot/
-
So
Hence
\s
= TT
i
Q)
f*
i
and
-<Hy
-=
= 2sm-3^
i
/</>
2a cos 3 J cos
o
r^
= 12
Jo
-J.
r
2
L4cos
(^
dd>
= 4a x 3
r^
/
<
^ cos
v/o
-3eo s
~i
-,J*
3
5
'
4
Also
Finally,
r*~
26^
2
r
3
.'0
5
3
1
TT
P
JQ
ISrra 2
3
ILLUSTRATIVE EXAMPLES.
Thus,
Al
465
=
o
Ex.
Let A, B,
4.
C
be any three points and
P a fourth
ABC
point whose
is regarded as the
areal coordinates are x, y, z when the triangle
To find the relation of the areas of the pedals of
triangle of reference.
any closed curve with respect to A, B, C and P.
Let [A], [B], [C], [P] represent the areas of the pedals.
the areal coordinates of
12,
the centre for the pedal of
Let X, F,
minimum
Z be
area.
Then
B
R
C
Fig. 83.
Now
(Ferrers' Trilinears, p. 6) the distance
from
#, y, z to
X, Y,
Z
is
given by
Pft 2 =
-a?(y- Y)(z-Z)-W(z-Z}(x-X)-c*(x-X)(ij- Y)
= -a*(0- Y)(0-Z)-b 2 (0-Z)(l- X)-c 2 (l- X)(0- Y)
= - a 2 YZ - WZX - c*X Y+ b*Z+ c Y,
BW - b*ZX - c*X Y- a2 YZ+ c*X + a*Z,
2
2
2
(7ft = - c*X Y- a YZ- WZX + a F+ b*X
Pft 2 - xA ft 2 - yBW - zCW = - tfyz - tfzx - c*xy.
and
/Ii2
2
1
;
.-.
2
2
is the equation of the circumcircle,
S=a?i/z + b zx + c xy, S =
to
minus
the
equal
square of the tangent from the point (a*, y, z)
to the circle S=0 if the point lie without the circle, or to the rectangle of
Now,
and
if
*S is
the segments of any chord through
meaning for
E
T.C
3,
.r,
>
20
y, z if
within.
Therefore with this
CHAPTER
466
XIII.
PROBLEMS ON QUADRATURE.
1.
\Jr
1
Interpret geometrically
2
-p
2
dp
in the case of the
curve
J
r-f(f).
"_
Prove that the value of \*Jr 2
whose semiaxes are
2.
a, b,
-p 2 dp,
taken
and whose centre
Use the pedal equation
an
of
round an
all
ellipse
the pole, is TT (a [OXFORD I. P.,
is
ellipse, viz.
^-
2
.
b)
1903.]
= a 2 + b 2 -r2
^
to
of the portion of an ellipse included between
the curve, the semi-major axis and a central radius vector ?*, is
show that the area
ab
a, b
being the semiaxes of the
[COLLEGES, 1882.]
ellipse.
Find the area of the part of the ellipse p (2a - r) = b2r included
between two focal radii vectores drawn, one to an extremity of the
minor axis and the other to the nearer extremity of the major axis.
2
3.
[OXFORD
4.
Find the area included between an
bounding
radii of curvature, the
ellipse
and
its
I.
P., 1889.]
evolute and
one coinciding with the major axis
and the other inclined at an angle
of
-j
to
it.
[COLLEGES, 1884, AND ft 1888.]
Through every point of an
normal to the ellipse and equal
5.
point.
Show
ellipse
a line
that the area of the curve thus obtained
2ab
6.
Show
drawn outwards
is
to the radius of curvature at the
is
[COLLEGES
a,
1891.]
that the area of that part of the evolute of an ellipse
>
(eccentricity
-= which
2/J
lies
outside the ellipse
is
\/
2
dp
3/o)
[COLLEGES, 1882.]
7.
Find the area
of the pedal of the curve
the origin being taken at x
= \/a2 - b 2
,
y=
0.
[OXFORD
I.
P.
,
1888.]
PROBLEMS ON QUADRATURE.
8.
Show
467
of the space between the epicycloid
2
pedal curve taken from cusp to cusp is \irA B.
that the area
p = A smB\j; and
its
[COLLEGES, 1878.]
9.
Show
that the area between an epicycloid and the arc of the
between two consecutive cusps is
fixed circle included
b are the radii of the fixed
where a and
and
rolling circles respec-
[COLLEGES
tively.
Show
circle is that of
an
ellipse
with semiaxes the radii of the two
[OXFORD
10.
cusps
Show
is
a, 1884.]
also that the area of the corresponding sector of the fixed
that the p-^ equation to a cycloid
taken as origin
I.
circles.
P., 1913.]
when one
of the
is
p = 2a(sin $ - $ cos
\fs),
where a is the radius of the generating circle; and find the area
between the curve from cusp to cusp and the corresponding arc of
the pedal with regard to a cusp.
r?
[OXFORD
II. P., 1903.]
11. Show that the area bounded by that portion of the
= a? sin |0, which lies in the first quadrant, the terminal
and the corresponding portion
Show
tangents,
extremity of the
is
polar subtangent,
12.
of the locus of the
cardioide
3a2 (10 -
Sir)/ 16.
[MATH. TRIPOS,
that in the curve in which the area
1896.]
bounded by the
curve and the radii vectores from a certain fixed point varies as the
square of the length of the bounding arc, the radius of curvature
varies as the projection of the radius vector on the tangent.
[COLLEGES
13.
The pedal
a, 1891.]
of a cycloid
with regard to any point on its axis
A and cuts the tangent at the cusp
find the area between it and the chord AQ
and prove that
meets the cycloid at the vertex
in Q-,
;
this area is least
when
the origin
is
the middle point of the axis.
[ST. JOHN'S, 1883.]
14.
An
elliptic
straight tube
;
wire
is
pushed
prove that the area of each loop
semiaxes.
in
find the equation
one plane through a very short
to the locus of the centre,
is
*(a
-
2
b)
,
where a and
b
and
are the
[COLLEGES, 1886].
CHAPTER
468
A
taken on the normal drawn outward at a point
Prove that if PQ is
of a catenary, the parameter of which is c.
15.
P
XIII.
point
Q
is
equal to the length of the arc of the catenary measured from the
vertex to P, the area between the locus of Q and the catenary, and
bounded by the normal
at an angle
\l/
to this,
and by another normal inclined
at the vertex
is
2
c
2
(tan
\//
+ tan ^ -
tfr).
[COLLEGES
7, 1882.]
/j
Prove that the pedal
16.
of the cardioide r
= acos 2
to the cusp consists of two closed regions of areas
being external to
sisting of the inner loop and
B
^ with respect
A and B, A conA and bounded
1
by the outer
line of the curve
and such that 2 A
+B=
K
_
2
.
9
[COLLEGES
*
Prove that the area of the pedal
17.
of the curve
+y
7, 1899.]
o* with
respect to the point (a, 0) is five times as great as the area of its
[OXFORD II. P., 1899.]
pedal with respect to the origin.
The tangent
18.
at a point
P
of a lemniscate cuts the curve again
Prove that the middle point of QR is at the same distance
from the nodal point as P and that the equation to its locus is
= r* 8 + 4 4 x2 a
at Q, R.
io(
f)
{a
Show
(a
r2
where
that
it
r2
can be written
= a 2 cos
f 0.
Trace the curve completely, and prove that the portion corresponding to the upper half of one branch of the lemniscate divides
the other branch into two parts whose areas are in the ratio of
6 - 3\/3 3v/3 - 4.
[ST. JOHN'S, 1884.]
:
19.
Show
that the area of a loop of the curve
/7T
y/3+1
.
aV2 (^ ~ log
^j
e
[MATH TRlpos>
The tangent at every point P of a
produced to Q so that PQ is constant. Find
How is the
locus of Q and the original curve.
20.
(i)
if
the curvature of the
sometimes
first
curve
in the opposite direction
given number of times.
is
;
closed finite
1882i]
curve
is
the area between the
result to be explained,
sometimes in one direction,
(ii)
if
the curve cuts
[ST.
itself
a
JOHN'S COLL., 1881.]
PROBLEMS ON QUADRATURE.
469
A
straight line of constant length c is drawn from each point
of a closed oval curve making a given angle a with the normal at
21.
Prove that the area of the curve traced out
that point.
is
line
end of the
$ + ire2 k cos a,
where S
is
the area of the given oval curve and
is its
I
by the
length.
[COLL. 7, 1893.]
that the area of the polar reciprocal of a curve whose
given in rectangular coordinates is
Show
22.
is
equation
w
x,
V-
X
dx
y being the coordinates of a point on the original curve.
x2
v2
=1
Apply this to find the area of the ellipse '-^ +
p
[COLLEGES, 1886.]
The area of a given closed oval curve is A the bisectors of
internal and external angles between tangents to it which meet
23.
the
\
at a given constant angle
and A*
24.
of the
show that
;
A
A
l
COS
2a envelop curves whose areas are
9
2
>."'-
a+A
2
SU\ 2 a
= A.
A
l
A
[COLLEGES
7, 1888.]
Prove that for any closed curve which has a centre, the area
locus of intersection of tangents at right angles, and the
area of the locus of intersection of normals at right angles differ by
twice the area of the curve.
[MATH. TRIPOS, 1888.]
being a fixed point, OP a radius vector of any curve, OP
2
produced to Q so that OP PQ = a and A is the area between
the locus of Q and the given curve.
If A' be the area of the inverse
25.
is
.
of the curve
show that
If
,
with respect to 0, the constant of inversion being
is independent of the form of the curve.
A-A
a,
and
a point on its circumference,
bounded
the
locus of Q, the circle and
any part
by
vectores from 0.
[ST. JOHN'S, 1891.]
the given curve be a
circle,
find the area of
two
radii
A
26.
circle rolls on the outside of an oval curve, the pedals of
the curve, of the locus of the centre of the circle and of the
envelope of the circle are of areas
that
A 2 - 2A-L + A Q
Show
that
A A A
,
lt
respectively; prove
2,
depends only on the rolling
circle.
the area of the oval curve, of the locus of the centre
of the circle and of the
Sv $ respectively,
envelope of the circle be
if
,
S<,-2S1
+S
.
[TRINITY, 1878.]
CHAPTER
470
One
27.
of the curved given
XIII.
by the equation
d (d
1
cuts the axis of x twice at the angle
the curve and the axis is
fd*
Prove that the area between
a.
a2 {tan a sec a + log(sec a
A
curve concave to the axis of x
+ tan a)}.
I.
[OXF.
P., 1912.]
such that the product of
the ordinate and radius of curvature at any point is constant and
28.
equal to
c2
(The Elastica, or Bent Bow).
value of the ordinate
is
curve crosses the axis of
Show
is
string
2c sin
where a
-,
2c 2 sin
of the line x
is
the angle at which the
x.
[Ox.
The
I.
bow and
P., 1903.]
the bow-
a.
that the area of a closed curve, which is the envelope
is the value of the
t/> =p,
integral
cos^ + y sin
taken completely round the curve.
30.
maximum
Prove that the
that the area which lies between the
Show
29.
is
integral
-
1 f
~
fdv
/?
(
\
[MATH. TRIP.,
1898.]
2
+ nP d^
)
is
taken round a closed curve,
n being taken equal to tan ^ or to - cot \j/ according as the one or
Show that the value
the other is numerically less than unity.
t
from the area
of the integral differs
of the curve
by the sum
of
the squares of the perpendiculars from the origin upon the tangents
at the points where the integral changes form.
[MATH. TRIP., 1898.]
31. In the cycloid
prove that the conic locus of points with
which the area of the pedal is constant, is in general
and find the point lor which the area of the pedal is a
regard to
a
circle,
minimum.
[Ox.
32. In a catenary,
AO,
A
PN perpendiculars
is
the vertex,
upon the
P
33.
Show
may
that the area of the
first
be obtained by the formula
1
P., 1900.]
any point on the curve,
PY a tangent and
directrix,
Show that the area of the figure
perpendicular to it.
double that of the triangle YNP.
p =f(r)
I.
NY
ONPA
is
positive pedal of the curve
PROBLEMS ON QUADRATURE.
where the
letters
p and
471
r are the pedal coordinates of a point
on the
original curve.
Apply this method to find the area of the caro^ioide, which
first
2
positive pedal of the circle r
34.
is
the
= ap.
Employ the formula
1
f
pr
-\-jJ==dr
to find the area of
To what curve does
35.
1
this pedal equation
In the epicycloid
where a and
-
-
2
p
2
= a2
belong
r2
- a2
^
a~%
c
?
>
are the radii of the fixed and rolling circles
respectively, obtain a formula for the area of any sectorial portion
with centre of the sector at the origin.
Hence show that the
area between one
36.
the curve and the fixed circle
When a<b the conchoid of
2
2
2
3?f = (a + y) (b -y )
has a loop.
37.
foil of
Find
Nicomedes,
or
r
is
viz.
=
its area.
Let S be the focus of a parabola, SP V
The latus rectum
vectores of lengths r v r2
.
SP2 two
is
Prove Lambert's expression for the sectorial area
focal radii
4a and
SP P2
1
,
P P2 =
l
c.
viz.
T [-<->*].
where 2s = r^
Show
4-
r2
+ c.
that the segment cut
off'
3
38. In
the form
by a
focal chord of length c
is
A*
the case of the Cotes's spirals, whose equations are of
i
A
show that the area of the sectorial portion bounded by the curve
and the radii vectores r-^ and r2 is
CHAPTER
472
Examine
XIII.
in detail the particular cases of
the equiangular spiral ;
the reciprocal spiral ;
and (v) the cases which reduce to the polar forms,
(iii), (iv)
u = a cosh nd
u = a sin nO respectively.
u = a sinh nO
(i)
(ii)
,
,
is
,
The tractory
39. Riccati's Syntradory* is generated as follows.
an involute of a common catenary of parameter c, starting from
PT is a tangent at any point
the directrix of the catenary at T.
Q
that
The
locus
of
such
QT=c'.
produced
the vertex.
Show
of the tractory, cutting
a point
on
PT
or
PT
the syntractory.
that the areas between the two branches and the directrix
are
Q
is
TT
40.
(k
and
P
is
If
+
A
be the area of the
2
l){(z
V the
'
Helmet,'
+ ka*)y* - 2ay(a? - z2
volume formed by
its
)
}
+
2
(a
- a2 ) 2 =
,
(k
* -
1),
revolution about the y-axis, prove
that
A=
[For the
part of the example, and for several others of
first
similar character, see Wolstenholme's Problems, Nos. 1886 to 1870.]
*
Comment. Bononensia, Tom.
iii.,
1755.
CHAPTER
XIV.
ETC.
QUADRATURE,
(Ill)
SUKFACE INTEGRALS, AKEALS, COERESPONDING CURVES.
451.
Use of Second Order Infinitesimals as Elements of Area.
"Surface Integrals," Centroids,
For many purposes
to use for the
it is
etc.
found desirable, and often necessary,
element of area a second order infinitesimal.
we desire to find the mass of the area
bounded by a given curve, the #-axis and a pair of ordinates,
where there is a distribution of surface density over the area,
Suppose, for instance,
not uniform, but represented at any point by <r = <f>(x,
where x, y are the coordinates of the point in question.
Let Ox,
Oy be
whose equation
the coordinate axes,
is
y
_
//
AB
any arc
y), say,
of the curve
\
j\x),
{a, /(a)} and {b,f(b)} the coordinates of the points A, B upon
Let PN, QM be any
it, A J and BK the ordinates of A and B.
contiguous ordinates of the curve, and x,x + Sx the abscissae
of the points P, Q.
Let R, U be contiguous points on the
473
CHAPTER
474
XIV.
+
ordinate of P, their ordinates being y, y Sy,
suppose Sx, Sy to be small quantities of the
and we
first
shall
order of
smallness.
Draw RS, UT, PV
RSTU
parallel to the
cc-axis.'
Thus the area
of
Sx Sy, and its mass may be regarded as
to the second order of smallness.
(f>(x, y)SxSy
Then the mass of the strip
may be written
the rectangle
is
PNMV
and in conformity with the notation
may
of the Integral Calculus
be expressed as
between the limits y =
and y =/(#).
In performing this integration with regard to y, x is to be
regarded as constant, for we are finding the limit of the sum
of the masses of all elements in the elementary strip PM,
parallel to the ^/-axis, for which x retains the same value, i.e.
are finding the mass of the strip PM.
If then we search for the mass of the area
we
strips as the
the ordinates
AJKB,
all
such
above must now be summed which lie between
A J, BK, and the result may be written
which may be further written as
b
r (/(*)
'0
.11.
*
the limits of the integration with regard to x being from
x = a to x
= b.
Thus the mass
of the area
AJKB
for surface density
(j>(x,
y)
f/X*)
452.
Notation.
This will be written
Cb Cf(x)
Ja JO
<j>(x,y)dxdy,
the elements dx, dy being written in the reverse order to
that in which they occur in the previous expression, and it
SURFACE INTEGRALS.
475
remembered that the right-hand one refers
integration, and the left-hand one to the second.
be
will
first
already been stated (Art. 363) that
book
If
we
to
the
It
has
throughout the
shall
adopt this order.
we put
(T=(j>(x, y)
= 1,
the result of our integration will
be to find the area.
Area =
Thus,
ffr
f/(*)
dxdy
Ja Jo
=
\
Ja
or, in
ydx, as before
the case of the area being bounded
y=
I
by two
y = \fs(x), as
(x),
Area = P
;
(
f* *i
curves,
in Art. 395,
*
dxdy
JaJ)
Ex. If the surface density of a circular disc bounded by x2 +y 2 =a? be
given to vary as the square of the distance from the ?/-axis, find the mass
of the disc.
Here we have
2
[MX
for the density of the element Sx 8y,
and
its
mass
is
therefore
and the whole mass
will be
I
be y =
I
2
p,x
dx dy.
y = ^a^-x^ for the positive quadrant
result must then be multiplied by 4,
for the distribution being symmetrical in the four quadrants, the mass is
four times the mass of the first quadrant.
^ ~*
Mass = 4 ("
Thus,
p&dx dy
The
and
limits for
for
y
will
x from x = Q to x = a.
Putting
a:
= a sin 6
to
The
and dx = acos6d@, we have
Mass = 4/xa 4
1
prin
cos 2 6 dO
CHAPTER
4:76
XIV.
Other Uses of Double Integration.
453.
The same
process
may
be used for
other purposes,
many
which we give a few
illustrative examples, which will
serve to indicate to the student the field of investigation now
of
open to him.
Ex. Find the statical
moment
of a
quadrant of the
ellipse
about the y-axis, the surface density being supposed uniform.
Here each element of area .8x8y is to be multiplied by
surface
its
density cr (which is by hypothesis constant in the case supposed), and by
its distance from the y-axis
the sum of such elementary quantities is
then to be found over the whole quadrant. The limits of integration will
;
be from *
v=Q to yv = -*Ja 2 -x 2 for y
a
;
and from x =
to x = a for x.
Thus
we have
^*-
fa
Moment
\
x'
"
\
a
x dx Ay
Jo
<rba*
where
M
is
the mass of the quadrant,
i.e.
TTCtb
T"
454. Centroid of a Plane Area.
The formulae proved
in Analytical Statics for the coordinumber of masses 7%, 2
3 ... at
m m
nates of the centroid of a
points
(&!, 2/J, (z 2
,
y 2 ), (x 3
x=
,
,
*/ 3 ),
2^'
etc.,
,
are
*!
We may
apply these to find the coordinates of the centroid
of a given area on which. there is any proposed distribution
of surface density.
Let (7 be the surface density at a given point, which may
be either a constant, as for a uniform distribution, or a given
function of x and y. Then the mass of the element SxSy is
o-tefyand
\\trxdxdy
\\a-dxdy
CENTROIDS AND MOMENTS OF INERTIA.
477
Similarly,
-
_"
the limits in each case being determined so that the summation will be effected for the whole area in question.
Ex. Find the centroid of the
quadrant of the example in the
elliptic
last article.
It
was proved there that
o-ba?
,,4a
I!
and
/
I
a-dxdy- mass
.*.
x=
Also
ll(Tydxdy
.
Hence the coordinates
-
of
quadrant =
=u
[[|j'
Moment
dx
= 1&
of the centroid are
^
O7T
455.
M;
4a
,
=O7T
of Inertia.
When
every element of mass of a given body is multiplied
by the square of its distance from a given line, the limit of
the sum of such products is called the Moment of Inertia
with regard to the
Ex.
1.
line.
Find the moment
of inertia of the
quadrant of an
ellipse
about
the y-axis, again taking uniform surface density
Here we have to multiply each element of mass, viz. o-SxSy, by
and then integrate
Moment
x2
,
as before.
of Inertia =
I
I
<rx2
dx dy
**fiif**te
L Jo
b
<r
SjGXflr^b
a
aj
<T -
o
,
this integral
integr
example
having been worked out in the
of Art. 452,
CHAPTER
478
Ex.
Find the moment
2.
2
y = 4ax, bounded by the
XIV.
of inertia of
the portion of the parabola
and the latus rectum, about the #-axis,
at
each point to vary as the nth power of
surface
the
density
supposing
axis
the abscissa.
Here the mass-element
is px,
n
8x 6y,
/x
being a constant, and the
moment
of inertia is
where the
We
y are from y =
limits for
to 2\/o;r,
x from
and
for
^
8a?x n+ '*djc
to a.
thus get
Mom.
of In.
=
/
My 3
ax n
I
x dx =
[
Again, the Mass of this portion of the parabola
is
given by
=
jf jJ
Thus we have
Moment
of Inertia
about 0# = -
3 2n + 5
EXAMPLES.
1.
In the
first
quadrant
of the circle
x2 +y 2 = a? the
surface density
varies at each point as xy.
Find
(i)
(ii)
(iii)
2.
y
z
the mass of the quadrant
its centroid,
its
moment
of inertia about the #-axis.
Work
=ax
out the corresponding results for the portion of the parabola
bounded by the axis and the latus rectum, the surface density
9
varying as xPy
.
3. Find the centroid of a fine rod of uniform sectional area and of
which the line-density varies as the ?i th power of the distance from one end.
Also its moment of inertia about that end, about the other end, and about
the middle point.
4. Find the centroid of the
triangle bounded by the lines y=mx, x=a
and the #-axis when the surface density at each point varies as the square
of the distance from the origin.
Also find the moment of inertia about the
y-axis.
5.
Find the centroid
(i)
of
bounded by the
ax^
either of the areas
parabola y^
circle
2
(#-a) 4-y
2
= a2 and the
POLAR SURFACE ELEMENT.
(ii)
the centroid of the area bounded by the parabolas
y = 4o#, # =46y;
the centroid of the area bounded by
2
(iii)
479
#
2
2
= 4o#, y = 2#,
the surface density being uniform in each case.
6.
Find the moment
(i)
(ii)
of inertia of a triangle of
about one of its sides
about an axis perpendicular to
uniform surface density
;
its
plane through an angular
point.
456.
Polar Coordinates.
For polar curves
it is
Second Order Element.
desirable to use for our element of area
a second order infinitesimal of different form.
Let OP,
OQ
be two contiguous radii vectores of the curve
Let 0, 9 -{-SO be the vectorial
initial line.
r=/(0); Ox the
Draw two circular arcs
angles of the points P, Q on the curve.
R U, ST cutting the radii OP, OQ, with centre and radii r
r-\-Sr respectively,
and
let Sr,
SO be small quantities of the
first
order of smallness.
Then
area
RSTU=sectoT OST'-sector
=r SO Sr
And
to this order
ORU
to the second order.
RSTU may
therefore be considered a
rectangle of sides Sr (=RS) and r SO (=arc RU).
Thus, if the surface density at each point R(r, 0) be o-=</>(r, 9),
the mass of the element
is (to second order quantities)
RSTU
a-r
SO
Sr,
and the mass of the elementary sector
OPQ
is
CHAPTER
480
the summation being effected for
XIV.
elements from r=() r=f(0),
all
~]
err
aW)
dr
SO,
o
which integration
is to be
regarded as constant; and
limit
of
the
sum
of
the
the
taking
elementary sectors for
infinitesimal values of SO between any specified radii vectores
9 a and #=/3, we get the mass of the sectorial area OAB
in
7(0)
or,
as
we have agreed
Obviously when
457.
Ex.
1.
to write
it
(Art. 363),
crrdOdr.
JaJ
cr=l this formula gives the area of the sector.
Find the mass
of a circular
lamina of radius a in which the
surface density at each point varies as the n th
on the circumference.
point from a point
Taking
power
of the distance of that
as origin, and the diameter through
equation of the curve
as the initial line, the
is
Fig. 86.
R
Then we have
distant r
for the density at a point
a constant. The mass of the element
is
where
Hence the mass
/u,
lamina
is
_-5t.r(2oCo. *)<
n+2
Jo
rc-lw-3
n
n
is
odd or even.
2
n- 2 "'3
n-ln-3
or
n
n
0, <r=/ir
n
RSTU=^r
of the circular
rCOS0
according as
from
n^2'"
ITT
2 2'
(r868r).
CENTROIDS.
POLARS.
481
Ex. 2. If the moment of inertia were required about a perpendicular to
the plane of the lamina through 0, each elementary mass fj.r n (r 86 o>) is to
be multiplied by r2 before integration. The result merely changes n into
for the value found for the mass,
n + 2 in the former work, and writing
M
n
Moment of Inertia = M
^| (2a)
n+4fl+4
^
458. Centroids,
The distance
2
'
v
.
Polars.
etc.
of the centroid of an area
whose boundary is
by a polar equation, from any straight line in
the plane of the area and passing through the pole, may
be found, as before (Art. 454).
Take the line proposed
as the #-axis and a perpendicular through the pole as the
Then the distance of the centroid from the o>axis is
?/-axis.
obtained by forming the sum of the moments of the masses
of the polar elements of area about that line and dividing by
defined
the
sum
^/m.ii
of masses
i.e.
;
by the use of the formula y
-^~
.
Let cr be the surface density. Then o-rSOSr being the
element of mass and rcos$, rsmO being its abscissa and
ordinate respectively, its moments about the axes of y and x
are respectively
through
r cos
THUS
.
<rr
TC S
-
x=
If
-
and
SO Sr
(Trd
r sin
dr
y-
-fr
\\
<rr
.
r sin
SO
Sr.
"
'
r
d dr
r?
>
crrdfldr
||<rr<20<fr
the limits to be assigned so that the summations for
elements are thereby effected.
459. Ex.
1.
Find the centroid
the surface density
is /xr
of the circular
all
lamina of Art. 457 when
n
.
Obviously the centroid
lies
on the diameter through
Hence y = 0.
0.
tr
ft /"2aco0
To
find
divide
x we have
to integrate
2|
r cos 6
/
by M, which has been found before (Art.
r
This integral
=^
=
E.I.C.
n
pr r dQ
457, Ex.
dr,
and then
to
1).
IT
($a cos 0)"+
3
cos 6 dQ =
22'
+ 3 2ar n + 4n + 2 ...,
v
7i
.
-2n
^
(2a)
neven.
w+4 l9
w+3
^cos
dB
CHAPTER
482
_
+2
# = --
Hence
and
Ti
7i
.
2a
+3
XIV.
Ti
+-2
.
2a
=0.
y
upper half only of the lamina had been required,
we should have had the same value of x but for y we shall have to
If the centroid of the
evaluate the additional integral
rsf
rsctcosO
r sin
\
.
ur n r dO dr
Jo Jo
and divide by
|J/,
where
M is the
mass found for the whole lamina.
K
This integral =
n
r~*z
cos 0) n+3 sin 6 dO
-^
+ OJQ (2a
/
- ^ --TT.. Ti
or
7
,
Ex.
2.
3
7i
7
Find the centroid
+ 2r.
7i+l
of a
n
22
r...T'
n-lr...T'i
1
TT
TT'
n even.
lamina in the form of the cardioide
uniform surface density.
is an axis of symmetry, y is evidently
find the abscissa we have
in the case of
As
To
the initial line
r cos
d r dO
.
frr dQ
=0 (see Fig.
82).
dr,
the limits for r being
from
and
for
r
from
6,
=
=
to r
to Q
= a(l+cos
= TT
#),
(and double to include the lower
ra(I+cos0)
r cos
o
e.rdOdr =
2JQ
cos B
= | a3
o
/
Jo
[1
'(cos
=^a
o Jo
/
(3cos
r_
4 ,37T
2
3
1 TT
5
dO
+ 3 cos 2 + 3 cos 3 + cos 4 0) dO
IT
3
4
half).
i- r 3 -, a (l+cos0)
C*
5
483
CENTROIDS, ETC.
pr
Th e denominator = 2
r- r 2-|(l+cos0)
dd
/
Jo
= 2a?
L2_I
I
Jo
3?ra2
r37ra
-f-f
Ex.
2
5a
Calculate the surface integral of
3.
2"
fir
taken over one loop of a
Bernoulli's Lemniscate.
The curve is r 2 = a 2 cos2(9 (Diff.
The surface integral is plainly
naV^s2
Gale., Art. 458).
iu**.
where
^.=.20,
2
2n+T
If the
moment
of inertia be required
about an axis perpendicular to
the plane through the pole,
Mom.
In.
=2
(2)
where
If
M
density
and
is
the mass.
weput?i =
/A
^
=!
in (1),
we get the mass 3/of
viz.
gives the area, viz.
x
A=~
.
the loop for uniform surface
_
CHAPTER
484
XIV.
Putting w = l in (1), we have the moment of inertia for a uniform
lamina about a perpendicular through the pole to the plane (or the
mass for a superficial distribution /x/- 2 ), viz.
Similarly n = 2 in (1) gives the moment of inertia for a superficial distribution fjir2 or the mass for a superficial distribution /xr4 etc.
,
EXAMPLES.
Find the centroid
1.
when
when
(a)
(fi)
of a sector of a circle
the surface density is uniform
the surface density varies as the
;
th
7i
power
of the direct
distance from the centre.
Find the centroid of a circular lamina whose surface density varies
n th power of the distance from a point
on the circumference.
2.
as the
Find
also its
3.
moment
of inertia
(2)
about the tangent at 0;
about the diameter through
(3)
about a perpendicular to the plane through
(1)
(a)
Show
that the
surface density,
moment
;
about the y-axis,
the triangle of uniform
of inertia of
bounded by the y-axis and the
0.
lines
is
-
Mi
6
where
(6)
M
the mass of the triangle.
is
Find the moments
bounded by the
density,
of inertia
of
the triangle of uniform surface
lines
about the coordinate axes; and show that
if
M be
triangle, they are the same as those of equal masses
points of the sides.
the mass of the
placed at the mid-
4. Find the centre of gravity and the moments of inertia about the
coordinate axes of the rectangle x = a^ x = a 2 y = &i, #=&2> tne surface
,
density being
5.
If A,
a-
B be
=
the
moments
about an axis through
its
at right angles to the plane, prove that
C
for
any law
any plane area about a pair of
C the moment of inertia
and
plane,
of inertia of
perpendicular axes Ox, Oy in
of surface density.
TRILINEARS AND AREALS.
Show
6.
that the
moments
of inertia of a
485
uniform
ellipse
bounded by
O
T/7
x 2 la*+y'2 /b'2 = I about the major and minor axes are respectively
and
-, and about a line
plane,
7.
M
,
M being
-
through the centre and perpendicular to
its
the mass of the ellipse.
Find the area remote from the pole between the
circles
r = 2acos0;
r = a,
and assuming a surface density varying inversely
as the distance
from
the pole, find
(1) the centroid;
(2) the
moment
of inertia about a line through the pole perpen-
dicular to the plane.
8.
Find for the area included between the curves
(i)
the
(ii)
the
moment
of inertia
moment
about the z-axis
of inertia about an axis
;
through the origin and
at right angles to the plane of the area.
9.
Find the coordinates of the centroid of the area bounded by the
catenary y
10.
= c cosh -, an
If the density at
ordinate,
any point
and the coordinate
of a circular disc
axes.
whose radius
is
a vary
directly as the distance from the centre and a circle described on a radius
as diameter be cut out, prove that the centroid of the remainder will be
at a distance
^j~ from
the centre.
[MATH TEn>
}
?
187g
-,
Trilinears and Areals.
These coordinates are not well adapted for metrical purposes.
Their special role is the discussion of descriptive properties
460.
of curves.
With the usual notation
Conies,
as
of the trilinear system [Smith's
Chapter XIII.], we have
an identical relation between the three coordinates
and in the areal system this is replaced by
a, ft,
y
of a point,
The transformation formulae from the one system
other are
x
-?
*~2A'
7
v-^A
2/
~2A'
~-y
~2A'
to the
CHAPTER
486
Variations da,
or dx, dy, dz of the coordinates are
the
by
equations
dy
d/3,
therefore connected
XIV.
respectively.
.
dx+ dy+
dz = 0j
The evaluation
of an area for such coordinates is best done
back
the homogeneous equation given into a
by throwing
Cartesian form, taking two sides of the triangle of reference as
Fig. 87.
coordinate axes.
CB
Thus taking
and CA, sides of the
be the Cartesian
reference triangle, as axes of
and ^, if
Y\
coordinates of the point a, /3, y, we obviously have
y = (2A
and
c
\
T
do/
/'
and then the evaluation of the area
A=
or
any
8mC\rjdg
of the
or
will be obtained
sinCl^^
or
by
smtmcT^^
methods customary for Cartesians.
461. Formulae can, however, be exhibited expressing the
area directly in terms of areal or trilinear coordinates for use
if
necessary.
In the Case of Areals, since x,
a point, are linear functions of
y,
z
y,
}
the areal coordinates of
the Cartesian coordinates
TRILINEARS AND AREALS.
487
with reference to any chosen rectangular axes and
we have
= xl \dxdy
I
|<i<i>7
where
xcts
or
x+y+z= 1,
or
are determinate constants depending upon the
To determine X we shall apply
triangle of reference alone.
the first of these formulae to the triangle of reference itself.
If
X,
A
/UL,
v
be the area of the triangle of reference,
where the integration
Now
The
i.e.
to
us evaluate
let
limits of y,
y= 1
Thus
I
x,
conducted over the triangle.
is
\dxdy
I
keeping x constant, are from y =
to x = 1.
\dx dy for the triangle =
if
= 0.
I
dxdy
X = 2A.
be the equation of a closed curve in
f(x, y, z)
area
areals, its
to 2
and for x from x
.'.
Hence
for the triangle.
is
the limits of integration being obtained from
f(x, y,
The corresponding
I-x-y) = Q.
result for trilinears will
be
-^-ri {(dad/3,
smCJJ
where the
/(a,
/3,
y)
limits are to be
=
found from
being the curve to be considered.
462. Illustrative Cases.
Ex.
1.
As
a test let us apply this method to find the area of the
circum-circle of the triangle of reference,
areals).
viz.
a 2yz + b 2 zx + c 2 xy = Q
(in
CHAPTER
488
The
result,
XIV.
from elementary considerations, should be
TrIP = ir(
\-x-y
Substituting
R
,
7-^ J
for
being the radius of the
circle.
we have
2,
a?f + (2ab cos Cx - a?)y = b' x 2
a
ear
be
I
==
+ C*^2 COS ^
I
ft
b
<37
4
2
.
~^
S111
C^
t*
_1 + Ic cos ^
_lcc
S111 Zfl f
~4 4^2 sm2 C~^
26
\
2
2
=i cosec 2 A
fe
l"l
2
.
- 4 sin 2 B sin 2
-i
(?(.r
The
limits for
y
are therefore
1
and
for
^^^V]
/26 cos
C
r--
.r,
The area = 2A
j ldxdy
= 4&(
= 4A.
2^
=
9A
^[sin-U-sin-H-l)]
A A
= 2;rA._.
L;UCW;~^I
4 sm-B
TrA
~
2 sin
.
sm 6
A sin ^ sm C
_~
2
the result to be expected.
Ex.
2.
More
generally consider the area! equation of an ellipse
ux2 + vy* +
To obtain the
We
where
obtain
m + 2u'yz + Zv'zx + 2w xy =
'
2
integration limits put
z=l-xy.
ax2 + Zhxy + by2 + tyx + 2/y + c = 0,
a = w+u-2v',
g=-w + v',
h~w + w'
u'
v'j
f= - w + u',
0.
TRILINEARS AND AREALS.
Solving for
bt/=
y,
- (hx+f)
-A =
v
A
where
and
#=the Hessian, viz.
The
489
limits for
-(hx+f}
dU
?>H
-dH
a,
*,
g
U,
W,
V
k,
b,
f
w',
v,
u'
9,
f,
c
v'
11'
w
y are
G
and for
.r,
Writing the radical
area = 2A
f
I dx
Now q=*JC =
vvhereA'=
dy=
2
^(>Jp*-q
(x-r)* dx
>Jab-
f
w,
w',
1,
1,
v
r
,
,
the "bordered Hessian," and
G2 - AC= - bff.
1,
-H
Hence
IT
Therefore the area sought
2:rA
is
~
taken, where
(
A = area of
H=the
j,
the positive value to be
^)
triangle of reference,
Hessian,
viz.
',
',
K=t}\e bordered Hessian,
v,
u'
%',
viz.
w',
v,
u',
v',
it',
?/:,
1,
1,
1,
1
1
CHAPTER
490
XIV.
Corresponding Points and Areas.
463.
Let /(a?, y) be any closed curve.
Its
-
I
area
(A^
is
by taking the
expressed
line
-
integral
y dx or the line-integral x dy round the complete contour.
I
If the coordinates of the current point x, y be connected
the relations
(
rj) by
with those of a second point
this second point will trace out the curve
/(mg
whose area (4 2 )
?i>;)
= 0,
expressed by the line-integral
is
\tjdg or
the line-integral \^drj taken round the contour.
And we have
l
or
=
\y dx =
1=
or, if
we
la?cfa/
=
mn\g dtj = mnA 2
,
use surface integrals,
I
l
whence
mn\t] dg =
\rngn d*] =
A = \dxdy = Jmn
is
d=
\ntjm
dfdif^=mn\ \dgdt) = mnA 2
,
appears that the area of any closed curve f(x,
of the closed curve f(mx, ny)
0.
it
2/)
=
mn times that
464. Ex.
1.
Thus, in the ellipse
#2
y
2
?+! =
The corresponding point
= ~ x area
ab
the ellipse
Ex.
Put
.
,
P ut
!.
x
ST>
y = rj
b
r-
traces out the circle J2 +
.
,
of circle
=
ab
2
-y7rr
2
7/
= r2
,
and area
= irao.
Find the area of the curve (m 2 ^2 + n V) 2 = a 2 ^2 + b*y 2
mx=, ny = r). Then the corresponding curve is
2.
or in polars
r2
= a s2
m
cos 2 6
.
+ Ti- sin 2 0,
the central pedal of an ellipse, symmetrical about both coordinate axes.
of
CORRESPONDING AREAS.
Hence the area
491
of the given curve
=
inn
x area
of derived curve
dd
mn
It will be noted that it is often possible by a selection of such a
change of the variables to arrange that the derived curve is of a much
more convenient form, and its area readily obtainable when expressed
in polars.
Ex.
where
Find the area of the curve
3.
c is less
ay
Let
^
.
T=
than both a and
-=
bx
r>,
b.
.
Then the derived curve
is
2
2
2
or in polars,
(c
sin 0\
/cos
+ r2N)~~~ + -2~ =
,
obviously symmetry about both axes, and though there is a
conjugate point in the original curve at the origin, the curve does not
pass through the origin, and the derived curve is one which could be
There
is
obtained from an ellipse by writing r2 + c 2 for r 1
Let r2 + c 2 =
ellipse is irab.
.'.
r' 2
.
Then
The area
^ /2
2
-
&2
465.
=7r(6-c
of
thi
is
M^also
and the area
of our first derived curve is therefore
the area of the original curve
which
.
2
).
2
).
In connection with the last example, it is worth noting
if the area if any portion from
that in any curve r=f(0)
= a to = /3 be found as
IflfWT
- Ja
and
=A,
CHAPTER
492
XIV.
then the sectorial area of the curve r 2 = [/(0)] 2
the same limits
c2
between
is
1
2
and
if
both be closed and the origin within both, then the
new curve differs from the area of the original
area of the
curve by the area of a circle of radius c, supposing c to be such
that r is real throughout the range of integration in each case.
EXAMPLES.
1.
Find the whole area
of a loop of each of the curves
(i)
(ii)
[ST. JOHN'S, 1887.]
2.
Trace the shape of the following curves, and find their areas
(tf
(i)
2
(z
(ii)
+ 2?/ 2 ) 3 =
[BARNES SCHOLARSHIPS,
3.
:
1887.]
Prove that the area of
-rf
,2
ls
4.
Prove that the area in the positive quadrant of the curve
is
5.
Prove that the area of the curve
is
fab
+ (6 2 -
ft
2
)
tan" 1
.
[ST. JOHN'S, 1883.]
6.
Show
that the area of the loop of the curve
5
x5
7.
Find the area
8.
Show
y
K
x2 y*
5
,
of the curve
that the area bounded by
2
(x
+ y*~
c2
2
)
(z
+ y2 ) = 4a 2 T2
is
(2a
2
+
c
2
)
TT.
PROBLEMS.
493
Find the area included within the curve whose equation
9.
W
10.
,a
area
its
\a
,0/
half as great again as that of the ellipse
is
b2
a2
.
[COLLEGES, 1885.]
Trace the curve
and show that
11
is
= ayx + y
+y
[MATH. TRIPOS, 1884.]
of the curve
Prove that the area
x
~
s
[ST. JOHN'S, 1889.]
12.
of the curve
Prove that the area
2 5
(aW + 6V ) = 8a*b*xy (aW + b*f)
is
a2
b2 .
+
[ST. JOHN'S, 1889.]
13.
Show
that the area in the
25
x2
14.
the area of a loop
is
47r(2
quadrant of the curve
32
/
+ 2y2 - 2ay) 2 = x2 (x2 + 2y2
Trace the curve 4 (x2
between the loops
first
&
2
-\/3)
Find the whole area
and that the area included
/\/3,
[TRINITY, 1896.]
of the curve
^a
and
proving that
is
8a 2 (2;r - 3s/3)/3x/3.
15.
),
+
2
b2 ) ~~ab'
[OXFORD
I.
P.
,
1890. ]
of a loop of the curve
#4
a
16.
Show
4
ly
4
%xy
64
'
ab
[OXFORD
II. P., 1900.]
that the area of either oval of
x2 {x 2 /a 2
+ y 2 /b 2
-
1
}
+ c2 =
is
^b(a -
2c).
[ST. JOHN'S, 1890.]
17.
If /(*, y)
=
be a closed curve, show that its area is mn
= 0. Trace the curve
times the area of the closed curve f(mx, ny)
2
(4a;
+ 9# 2 ) 4 = axy*, and
18.
find its area.
Trace the curve
a
loop
is
s ab.
+ ^3 =
3
b
ab
[OXFORD
II. P.
and show that the area
,
1890.]
of its
CHAPTER
494
19.
A
curve
is
XIV.
defined by the equations
= 6a sin 2 <,
x
2
y = 6a sin (f> tan
<f>,
a variable parameter.
Show that the centroid of the
portion enclosed between the infinite branches and the asymptote
is situated on the z-axis at a distance 5a from the
origin.
where
<
is
[OXFORD
II. P., 1889.]
20. (i) In an involute of a circle, show that the area swept out
by the radius vector drawn from the centre of the circle to a point
on the curve varies as the cube of the central perpendicular upon
the tangent, the initial line being the radius to the point where the
involute meets the
(ii)
a
< b,
circle.
= a sec
In the Conchoid of Nicomedes r
show that the area
6 - b in the case
when
of the loop is
a2 (a sec 2 a - 2 sec a cosh~ 1 sec a + tan
a),
and that the distance of the centroid of the loop from the node
2
3a sec a - 3 cosh~ 1 sec a - sin a tan 2 a
a sec a - 2 cosh~ 1 sec a
3
21.
Prove that the area contained by the curve
x2 is
+ a4 =
+ 2x2f + 4ax2y + 2a2 (y 2 -
Find
also the distance
s
Show
2ay)
from the axis
of that portion of the area
22.
+ sin a
a = cos^a/k
where
x*
is
which
Tra 2 (4
-
5/>/2).
of y of the centre of gravity
lies in
the
first
quadrant.
[COLLEGES
j3,
1890.]
that the area included between the curve
= a tan ^,
its
2
\a tan
is
=
and its tangent at ^ =
tangent at
2
2
a
a
tan
+
<j>
<j>
log (sec <f> + tan <).
\//
<f>
[TRINITY, 1892.]
23.
Show
coordinates
that an expression for the element of area in trilinear
is
cosec
Show
a~ l /3y
is
C da d/3.
that the area of the conic whose trilinear equation
is
+ b~ l ya + c~ l a/3 =
to that of the triangle of reference as
4?r
24.
Show
3\/3.
[OXFORD
II. P., 1890.]
that the coordinates of the centroid of the area bounded
half the cycloid z
by
and the
:
= a(fl + sin
y-axis are given
0),
y = a(\
- cos
0),
the line of cusps
by
3y = a
T
2
'
[
W ALLIS.
]
PROBLEMS.
25.
OB and OC are any two semi-diameters
to each other
B and
495
C,
;
of
an
and show that the area
conjugate
normals at
ellipse
find the locus of the intersection of the
of the curve is
lab
26.
[R. p.]
Tangents to a system of similar and similarly situated condrawn such that the distance of each from the
centric ellipses are
centre
is
the same.
Find the area of the curve formed by the points
of contact.
27.
Show
[TRINITY, 1885.]
that the
moment
of inertia of the portion of a
uniform
parabolic lamina cut off by the latus rectum about the tangent at an
extremity of the latus rectum,
rectum and
is
equal to
=-^-,
M the mass of the lamina.
40 being the latus
[Oxr.
I.
P., 1914.]
28. Prove by
integration that the moment of inertia of a uniform
of mass Jf about a perpendicular axis at
is
triangular lamina
ABC
A
iV M (3b
2
+ 3c2 - a 2
).
[Ox.
I.
P., 1915.]
CHAPTER
XV.
QUADRATURE
(IV).-
MISCELLANEOUS THEOREMS, CONNEXION OF A LINEINTEGRAL AND A SURFACE-INTEGRAL, MECHANICAL
INTEGRATION, ETC.
466.
A THEOREM
DUE TO STOKES.
Let u and v be two functions of x and y, finite, single-valued and
continuous at every point within and along the boundary of a given
region bounded by any given contour line in the plane of x, y
,
having no multiple points,
~-
,
be
also functions
continuous at
all
and
let
which are
points of the region
u-r
ds
Kdx
;
dy\
dsJ
the
differential
finite,
coefficients
single-valued,
and
then the line-integral
j
-\-v-+-}ds
taken round the perimeter of the contour is equal
to the surface-
dx
taken over the region bounded by the contour.
sider u and v to be real functions of x and
We
shall first con-
y.
Let the region referred to be indicated, as shown in the
accompanying figure, with an inner boundary and an outer
boundary, the inner boundary enclosing a region within which
the integration is not to be performed.
Divide the whole contour into two systems of strips of
Two
infinitesimal breadth parallel to the coordinate axes.
one
in
the
the
shown
are
parallel to
figure,
typical strips
the $-axis being bounded
by
lines
496
with ordinates y and y + Sy,
A THEOREM DUE TO STOKES.
497
and that parallel to the y-axis bounded by lines with abscissae
x and x + Sx, The first intercepts elementary arcs
P l Q l = os l P2 Q 2 = (5s2 P^QB = SsB) etc.. an even number,
,
,
and the second intercepts
P 'Q = 88
P/Q/ = (5s/,
'
2
2
P 'Q = $s
'
2 ',
3
3
3 ',
etc.,
an even number.
Fig. 88.
The
direction of integration is indicated in the figure the
region to be integrated over being on the left hand as a
person travels along either boundary, following the direction
;
P P
The signs of Sy at the several points 1% 2
s.
are respectively -8y, +Sy,
Sy,
Sy, ..., and
the signs of 8x at the points
... are
',
',
',
1
2
3
4
respecof increase of
P P
3,
4,
P
tively
Let
>
+
...
/
,
P
P P
+ 8x,
ur
,
-Sx, +8x,
8x, etc.
vr be the respective values of u, v at
P
'
r,
'
and u r v r
t
those at P/.
And let the abscissae and ordinates of the points r
Qg be x, y with the corresponding accents and suffixes.
P Q P/
If
we
PiQfPtQt,
,
s,
integrate ox Sy with regard to x along the strip
...
,
we have
[v 8y],
taken between proper
limits, viz.
- ^ 2n_! Sy^-i)
2n 8y 2n
= 2v Sy,
K.I.C.
say, for the strip.
CHAPTER
498
If then
we sum
to the a;-axis
tion
by
XV.
the result for the whole set of strips parallel
integration,
we have
j
v dy, where the integra-
taken for the whole perimeter of the contour.
is
for the strips parallel to the 7/-axis,
if
we
Similarly
Sx
integrate
u
with regard to y along the strip PiQt-P&Qi,
[u Sx], taken between proper limits, viz.
- - -
,
we
obtain
=&e + u z 'Sx + u^'Sx ...)
= ItU Sx, say
;
and,
summing
integration
for the strips,
we
obtain --lucfo, where the
taken for the whole perimeter of the contour.
is
~
Hence
\j
Qi dp dx dy=\( udx + v d y)-
A
line-integral taken round a closed plane contour
therefore be represented by a surface-integral taken over
the surface bounded by the contour, and vice versa.
467.
may
we may say
Or,
that
if u,
v be the components parallel to
the axes of x and y of any vector quantity, then ox
oy
may
be regarded as another vector quantity at right angles to the
plane of xy, and such that the line-integral of u, v round a
contour in the plane of
x,
the vector quantity
J ^
is
is
-
3#
theorem
y
equal to the surface-integral of
taken over the surface.
This
cty
part of a more general three-dimension theorem due
to Professor Stokes.*
468.
Extension to Complex Functions.
If the functions
u and
separated into their real
v be not entirely
and imaginary
real, let
them be
parts, viz.
where u v u 2 v lt v z are single -valued finite and continuous
functions of x and y for all points within and upon the contour,
,
,
as also their first differential coefficients.
*
Smith's Prize, 1854
;
Maxwell,
Elect,
and Mag.,
vol.
i.,
p. 25.
A THEOREM DUE TO STOKES.
499
Then we have
"^ dx
dx dy
2
Therefore, multiplying the second line
dv
by
i
and adding to the
du
the integrations to be taken as before. Hence the theorem
true whether the functions u, v be real or complex.
=
In any case in which
it
is
will follow that
(udx+vdy) =0,
the integration being taken round the perimeter of the contour.
The theorem has many very important
An
469.
We may
Interpretation.
interpret the theorem thus
du
dv
rr= ^
dx
Let
Then
that
is
applications.
1
1
:
dx
-_. -j
.
;r
,
ds
dy
o-
dx dy
1
p ds
.
i
dy
y
--j-.
ds
;
the mass of a plane lamina bounded
tour for surface density
<r =
~;
by any
- is
equal to the
closed con-
mass of the
perimeter with a line density
dx
dx
470.
Ex.
1.
we have
Illustrations.
u=-y
Taking
at once
I
I
established (Arts. 409
Ex.
2.
and 452) as measures
w = e*sin y
Let
Then
v=x,'
dxdy = -\(xdy-ydx\
a?/,
which expressions have been
of the area.
v = e x cosy
a.
f[(**my-ay) ^+(^cosy-a)g]
ds
taken round the perimeter of the contour
=
/
I
[e*cosy
= ax area
- (ex
cosy- a)] dxdy =
of the figure enclosed
I
I
a dx dy
by the
contour,
CHAPTER XV.
500
Ex.
Consider the
3.
effect of integrating
1=1 [(cos x cosh y
Ay) dx + (sin x sinh y Ex) dy\
round any closed contour.
u = cos x cosh y
Here
Ay and
Therefore
v = sin x sinh
yBx.
/-j\
s-^
;~-
= cos # sinh y-Z?
and
~-=cosxsiiihy-A.
Hence
=
1=1 \(A-B)dxdy (A-B)x area
Ex.
i.e.
4.
V
If (7t
enclosed by the contour.
be any single-valued conjugate functions of x and y
x and y, such that U+t V=f(x + ty\ and if
real functions of
---B+A
then
and
I
[(
F- J^) cfo? + ( U
= A-B
[see Diff. Cal. t Art. 190],
Bx) dy\ round a closed contour
= \(A-B)dxdy=(A-B)x area bounded by the contour.
many different forms of U and Fmay lead to the same result is
/
That
obvious from the consideration that the mass of the area bounded by the
contour for a given distribution of surface density may be equal to the
mass of the perimeter for many distributions
471.
Two
of line density.
Resulting Theorems.
U
be any three functions of x and y, finite and
continuous throughout and along the boundary of a given
If P, Q,
contour, as also their first differential coefficients,
we have
the double integrals being understood to be taken over the
whole area bounded by the contour, and the single integral
being taken round the perimeter in the positive direction, i.e.
leaving the area bounded to the left in travelling in the
direction in
which
s is
measured.
MOTION OF A ROD IN A PLANE.
U
472. If R, S, T,
with
their
first
continuous and
a given
501
be any four functions of x, y which,
second differential coefficients, are
and
finite
throughout and along the boundary of
have, supposing suffixes to denote
we
contour,
partial differential coefficients,
3
fi
iff,
J7,
u.
= (RX UX +RU -RXX U-RX U X +SX UV +SU IJ
X:C
TT
fwi
,
TT
TT ffi
(Uyly + Ulyy- Uyyl
/
i
FT\
~
TT
O
TT
O
TT
)
U yl y+bxyU + S X U y)
fjl
i
\
= (RUxx +SUxy +TUyy)-U(Rxx +Sxy +Tyy
\
).
Hence
-
tt
a( R,
=
Jjfc\ u,
-J[{]?: ?:
U(Rxx +Sxy +Tyy )]dxdy
Rx
ux
u,
T,
^)$+{
R,
u,
uy
Ty
Rx
ux
the double integral being taken over the area bounded
contour and the single integral round the perimeter.
by the
Thus
+
-.
These results will be useful later (Chapter XXXIV.).
473.
Let
MOTION OF A ROD
IN A PLANE.
be the origin and Ox, Oy any fixed rectangular axes
in the plane.
Let a rod move in any manner in the plane.
Let Pr P2 P 3 be points attached to it, their coordinates being
,
Let
so that
P2 P 3 =
i,
a 1 +a 2 +a 3 = 0.
Let ^ be the angle the rod makes at any instant with
the x-axis.
CHAPTER XV.
502
Then
= x2 - a 3 cos 9,
x 3 = x^a-^ cos
= 2/2-^3 sin 9,
= 2/2+% sin
2/3
= dx2 -\-a 3 sin # <#, d#3 dx2 a sin # d9,
efo^
=
= dy2 -{-a^ cos # <^^
ty\ dy2 a 3 cos d9,
dy%
y^dxl = (x2 a 3 cos 0) (dyz as cos # (Z0)
x
6>,
2/1
.*.
;
6>
.'.
x1dy1
;
sin
~(2/2~ a 3
= x2 dy2
R = dy
where
and
y2 dx2 +a
2
B
d0a
2 -\-x2
% 3 dy 3
y 3 dx 3
x2 dy2
9)(dx2 -}-a B sin
3 (R
cos
/S
6>
sin
6>),
d9
2
y2 dx2 -\-a1 d9-\-a1 (R cos 9
S sin
0).
Fig. 89.
Hence, eliminating
i.e.
.R
cos 9
S sin 0,
+a1a 3 (a1 +a 3)d9,
^(x^-y^dx^+a^dy^-y^dx^a^dy^-y^dx^
-^-a^a^O = 0.
be the origin and dA l} dA 2 dA 3 the elementary
sectorial areas described by OP1} OP2 OP 3 respectively,
If,
then,
,
,
if
Hence,
,
a^dA^a^dA^a^dA^^a^dO = 0.
the points Pv P P describe closed
2,
3
curves,
and
Av A2 A3
be the areas of these curves, and if the rod
returns to its original position after making one complete
revolution, then
,
= 0.
474. Various Cases.
If the rod returns to its original position without completing a revolution, rotating in one direction during part
HOLDITCH'S THEOREM.
of
its
503
motion and in the opposite direction during another
part, then \d9
=
;
and
and A s be such that the rod
cannot complete a rotation, but must oscillate as in the case
of the connecting rod in a steam engine, we have
475. If then the contours of
A
./la
A
l
+a3A s
_al A ^^^
a +a3
1
t
1
476. If
m
times,
it
makes
and
several complete rotations forwards, say
backwards n times, whilst the several points
Pv P9 P5 describe closed curves once, then
,
477. If
two of the points, say
P
l
move on fixed curves and the rod
and
P
\dO = (m
3
,
n)%7r
',
and
are constrained to
rotates once round, as, for
Fig. 90.
instance, if the ends
were one on each of a pair of confocal
ellipses, or on a pair of circles, as in Fig. 90,
478. If P l and P3 move on the same curve
and the
theorem reduces to A 2 = A l Tra^.
This last result is known as HOLDITCH'S THEOREM.
A^A^
479. It should be noticed that in the above results,
if
any
of the contours are described in a sense opposite to others,
such areas are to be reckoned of opposite sign to the others.
CHAPTER XV.
504
480.
Leudesdorf s Theorem.
As an
application of this theorem, consider the motion of a
P are fixed points, the lamina being
lamina on which A, B, C,
move
constrained to
so that A, B,
C and P
describe closed
Fig. 91.
curves of areas
coordinates of
Let
AP
cut
[.4],
P
BC
referred to
at
Let
[B], [C], [P].
X
ABC
z be the areal
x, y,
as triangle of reference.
and the circumcircle at R.
Let
X
describe a curve of area [X].
Then
->(_/
Hence, eliminating the area [X],
PX _
~
v
AX
X>
'~ y
AXBC
AP BX _
~
AP XC _
'
AX' BC
~.
and
= rectangle
of segments of
circumcircle through
/.
[P]
x [A ] + y [B] + z [C]
any chord
P
mrX rectangle
chord.
of the
;
of
segments of
LEUDESDORFS THEOREM.
If
P
lies
outside the circle, instead of the rectangle of seg-
we may put
ments,
505
2
(tangent)
,
and the theorem may be
written
[P]
t
= x[A] + y[B] + z [C] + mrP,
being the tangent from P to the circumcircle.
This theorem is due to Leudesdorf.*
Motion of a Plane Lamina sliding in any Manner upon a
481.
Fixed Plane.
Two
Theorems.
When
a plane lamina moves in any manner upon a fixed
plane, so that in the end it again takes up its original position,
it is clear that
every point in the lamina will take up its
original position, that is that the several points in their motion
have travelled along paths back to the same points from
which they started, and may therefore be regarded as having
This will be supposed to intravelled along closed curves.
clude paths which are retraced, which may be regarded as
closed curves of infinitesimal distance between the outgoing
and returning paths. For instance, a finite straight line of
length 2a might be regarded as a closed oval say an ellipse
of semimajor axis a and infinitesimal minor axis.
Suppose two points on the lamina P l and P3 to trace out
known closed curves on the fixed plane. This will define the
motion of the lamina, and PJP$ may be regarded as a straight
rod whose ends are describing the given closed curves. Let
P be any other carried point on the lamina and PP2 a perpendicular from P to PjPg.
Let a fixed point O in the plane be taken as origin, and let
P P2 = a3
= 0.
04 + a 2 + a 3
PJP^O!, P3P =a2
]
so that
We
shall continue to
the area swept out
1
,
and
PP
2
=^,
adopt the convenient notation [P] for
by the radius vector
OP
to
any moving
point P.
Let
E
be the point of contact of PjP 3 with its envelope.
Through P draw a parallel PE to PJP lt and let the outward
f
normal to the
* See
1878.
E
Williamson,
locus
meet PE' at
Int. Calc., p.
E'.
Then EE'=p, and the
220 Leudesdorf, Messenger of Mathematics,
;
CHAPTER XV.
506
E
is a parallel to the
locus, the area between them
n
in
the
case
of
revolutions
complete
being
mrfP+pS, where S
is the perimeter of the envelope of the line
Pf^ (Art. 435),
E' locus
[E }-\E} = n-7rp
f
i.e.
i
+pS
or
7rp*+pS
if
there
be but one
revolution of the lamina.
Fig. 92.
Let E'P = EP2 = r. Then P l E
P1 P3 make an angle x//- with any fixed
and
let
line.
P-
Now
/.
,
multiplying by a lt a 2 a3 and adding,
,
(cf.
Art. 473);
and if the lamina reoccupies its original position after n positive
revolutions, or if n be the excess of the number of positive
revolutions over the
number
of negative ones, the right-hand
side is
(A)
Also
it
has been shown that
HOLDITCH'S THEOREM.
.-.
507
eliminating [P2 ],
which
may
be written as
482. Remarks.
assumed that
It is
"sense"
If in
all
the areas are described in the same
any case one
of
them be described by
its
tracing point in the clockwise direction, then in this equation
the corresponding quantity [ ] is to be interpreted as the area
and if one of the paths cuts itself so as to
the
interpretation of [ ] is the same as that
loops,
counted negatively
form several
;
in Art. 399, viz. the difference of the odd and even portions.
The sign of p is positive when in the same sense measured
from
P
2
as the
outward drawn normal of the envelope of P-f$.
483. Deductions.
P
P
and
is at
I.
When p = the tracing point
2
of
lamina
revolution
the
one
there
to
be
complete
supposing
we get the case already considered in Art. 477, viz.
Corollary
which
is
,
Woolhouse's Extension of Holditch's Theorem.*
484. Cor.
curve, then
II.
If in addition
[PJ^PJ
and
P
l
and
P
3
[P 2 ] = [P
are tracing the
a
1 ]-7ra 1 3
same
(Art 478),
Fig. 93.
and therefore a point upon any chord of constant length
inscribed in an oval curve, and which divides the chord into
two portions 04, a s> traces out another curve whose area is less
* See Williamson's Integral Calculus,
p. 206.
CHAPTER
508
XV.
than that of the original oval by the area of an ellipse whose
semiaxes are a v a y This is Holditch's original theorem.*
would not be affected
in this case.
If the tracing point be on the chord produced,
one of the letters a v a% is negative and the traced oval is
greater than the original oval by the same amount.
If
a lt a3 were interchanged the
485. Cor. III.
If the line
PP
X
3
result
oscillates
back to
its original
position without
performing a complete revolution, or if the
number of forward revolutions is equal to the number of
backward revolutions, n = 0, and
rp-i
aJPJ + aJPJ
fl^
This
is
the case
when
.
+ as
s
the contours are two ovals each lying
X
3 cannot revolve
entirely outside the other and the line
completely, but oscillates.
It is
PP
moreover assumed that the
line flj+G^ is sufficiently long to allow of the full description
If not, the particular oval which is not fully
of both ovals.
described contributes nothing.
For instance, if P3 travel along an arc of a circle ACB from A to B via
C and back along the same arc, it has described what we may regard as
a contour of zero area.
\
Fig. 95.
486.
(i.e.
Cor. IV.
PP = P P'),
and [P]
2
If P' be the
image of
P
in the line
Pj
2
[P'J
= 2pS, which is independent of the position of P2
*See Bertrand, Calc. Integ., p. 365; Williamson,
Lady's and Gentleman's Diary, 1858.
Integ.
Gale.,
p.
206
.
;
KEMPE'S THEOREM.
487.
Cor. V.
If
P
P
and
l
3
lie
509
upon the same curve,
2
[P] = [PJ mra^ + n7rp +pS.
In case a x =
0,
we have
and
488.
origin,
Let 0, the mid-point of
Cor. VI.
OP3
as z-axis,
and
let
P^,
be taken as
0P2 = z, P2 P = p = y.
Let the
length of the rod be 2a.
Fig. 96.
a^a
Then
and
x,
as
2a
^.e.
Hence the locus of point
P
contours [P] are
a circle whose centre
all
equal
is
on the lamina for which the
is
at
4
These coordinates are independent of
[P].
Hence, for specific
values of [P], the loci of the P-points are concentric circles on
the lamina.
This theorem
is
due to Mr. A. B. Kempe.*
We
489.
note that if [PJ and [P3 ] be the same contour, the
centre of this circle lies on the perpendicular bisector of the
line P^Pg.
*
Messenger of Mathematics, 1878, cited by Williamson, Integ. Gale., p. 210,
it is deduced from Holditch's form of the theorem geometrically.
where
CHAPTER
510
490.
If the closed
straight lines
or
when p =
"
contours
[PJ = [P3 ] = 0,
"
XV,
two
are merely portions of
and taking n = 1,
[P] =
also,
which is the case of a rod of given length sliding with its ends
on the coordinate axes, which are drawn in Fig. 97 as long
closed ovals to indicate the direction of rotation.
Fig. 97.
Note that
in the case
shown
in Fig. 97 the elliptic area
is
traced clockwise, the ovals, which are in the limit the axes,
are traced one counter-clockwise, one clockwise, and that
the areas of the two ovals traced
by
P
l
and
P3
are both
ultimately zero.
well-known theorem that in this case the locus of
ellipse of which the product of the semiaxes is the
of
the segments of the moving line, whether the axes
product
It is a
P
is
2
an
be rectangular or oblique.
491.
Cor. VII.
as diameter,
If
P
lie
anywhere on the
we have p 2 = a l a3 and
,
+pS,
or
if
[PJ and [P3 ] be the same
circle
on
P-fz
the theorem reduces to
contour,
A GENERAL THEOREM.
A
492.
GENERAL THEOREM on
System of Moving
511
the Motion of the Centroid of a
Particles, connected or otherwise.
If
mp m
2
,
2/ 2 >
m
...
3,
m
"
rt
,
n
be five groups of
2/3.
quantities each
'
2
it
X^y
,
readily be proved
may
by induction that
2m# 2my = 2m Sm^y 2m rm (#r
2m?/ 2m# = 2m 2myx 2mrm (?/ r
#g ) (y r
s
and
^
s
)
ys)
~ ^)>
(^r
and therefore that
m^ = 2m 2m
,
Let there be
and
(ojj, 2/i),
(^ 2
differentials of
The centroid
[(ar
-
05.) (y r
yx)
(a?2/
- y,) - (y - y
r
s)
(* r
~
particles of masses in the ratios
TI
y, viz. c&e,
of the
2m
coordinates
e ^ c -? their
2/2)'
x and
.
system
is
2^i
2m
.
.
and
let x,
y be the
cfa/.
given by
x = 2m#,
whence
;
2m y = 2m?/
dx = 2m
.
;
c?a?
dy
2m. dy.
Let each particle describe continuously a closed contour in
m^ describing a contour of area A lt 2 describing
a contour of area A 2 and so on, and let x, y in consequence
describe a closed contour of area A.
Also let the area of the
m
the plane,
,
contour which
m
2
describes relatively to m^ be called $12 and
Then the above equation may be written
,
so on for other pairs.
2
[2?)i]
dy y dx] = 2m 2??i(# dy y dx)
- -2m.w.[(av a;,) (dy r dy s ) (y r y s ) (dx r dx s )],
[cc
and therefore integrating round the contours
2
[2?>i] xf
= 2m 2mA
2m m
r
g /S rs ,
an equation which expresses the area of the contour described
by the centroid of the system in terms of the areas of the n
CHAPTER XV.
512
contours described by the several particles and of the
^^~
relative contours.
It will be noticed that the particles are in
no wise rigidly
connected, but are capable of independent motion also that
the result obtained is necessarily homogeneous as regards
;
the masses.
493.
If the revolutions of
any
particles of the
system be
not complete, the various integrals
^(xdy-ydx),
2
[(xr
-x
s)
j
(dy r -dy s )
-^(xdy-ydx),
- (y r -y
s)
(dx r -dx s )],
refer to the sectorial portions of the several contours which have
been actually described during the several displacements of the
particles, and represent sectorial areas swept out by the several
radii vectores from the origin to the centroid, or from the
origin to x, y in the first two cases, or the relative area by a
radius vector from xr, y r to xS) y s in the third class of integral.
494. When the several particles are rigidly connected, the
several relative contours are circles, with radii the distances
between the several
pairs,
and traced as many times over as
the whole system revolves before re-attaining
its
original
position and in case of no rigid connection,
the mutual distances returns to its original position without
making a complete relative revolution, in such case the
if
;
corresponding relative area
495.
S
one or more of
vanishes.
In the case where there are two particles only,
r
we have
_m^
l
by MR. ELLIOTT, and reproduced
a result established
Williamson's Integral
Calculus, p.
209, with
in Dr.
Mr. Elliott's
Enunciation of this Theorem.
496.
If in this case there be a rigid connection between
the points
A^ and
as the distances of
A
2
A
,
we may take a lf a2
= ^
centroid, and
say a connecting rod,
2,
A^ from the
-
1-
^lYL-t
//I/.?
ELLIOTT'S THEOREM.
513
Also the relative contour has area Tr^-h a 2 ) 2
J^m^.+m^,
Hence
+
-1
Aj = tt,^!
becomes
2
mg^
^1 2
!^2
^-rs2
K4-a
c^+a,
.
\2
/
-TT
(a,
-fa 2 )
2)
Fig. 98.
therefore deduced as a particular case
particle motion, there being a rigid connection.
Holditch's theorem
two
of the
is
theorem takes the form
497.
If there be three particles the
498.
Let us apply this result to find the area described by any
ABC
P
which moves in its own
attached to a triangle
plane and after one revolution re-occupies its original position.
If x, y, z, be the areal coordinates of P with reference to the
point
ABC,
triangle
m v ra m
2,
.
is
at A, B,
3,
the several
P
"
the centroid of masses proportional to
C
respectively,
relative areas
"
are ?ra 2 irb 2
,
=-^-
where
m^
,
?rc
in 2
t
97i
and
3
2
;
ip]=2
2
zx+c 2 xy)
"
[P]=x[A]+y[B]+z[C]-7r(a yz+b
whence
2
the square of the tangent from x, y, z to the
the point be without, zero if upon, or
the rectcircumcircle
angle of the segments of a chord through x, y, z if the
where
t
is
if
point be within the circumcircle which gives Mr. Leudesdorf's
result of Art. 480 already established in a different manner.
;
E.I.C.
2K
CHAPTER
514
499.
It is
XV.
worth observing that the locus of points
P
which
give equal areas [P]
a2 yz+b 2 zx+c 2 xy+lme&r terms
is
or
making
it
=0,.
i.e.
a
circle,
homogeneous,
7T
and the centre
7T
7T
of this circle is given
z}(x+y+z)
/
by
7T
7T
2fPl
- (x-\-y-\-z)=two
similar expressions,
i.e.
7T
which
is
independent of [P], and therefore indicates that such
values of [P] form a set of concentric
Mr. Kempe's Theorem of Art. 488 (Cor. VJ).
loci for different
which
is
500.
It is also
worth notice that the area described by
centroid of the triangle
revolution by
and for
is
the
given for the case of one complete
the orthocentre 0,
tan
where
circles,
A tan B tan C
R is the radius
of the circumcircle.
Fig. 99.
501. In
a,
b,
c,
the case of four particles in rigid connection if
d be the sides and e, f the internal diagonals of
MECHANICAL INTEGRATORS.
the
quadrilateral
formed,
we
have,
515
the
in
one-revolution
case,
(m l + m2 -f w3 -f wj
and similarly
if
there be a greater
502. In a case
where there
is
2
number
no
of points.
rotation,
i.e.
where the
line
joining each pair of particles remains parallel to its original
position, or if there be rotation of any of these joins and an
opposite equal rotation of the same join,
"
relative contours will disappear and
it is
clear that all the
"
503.
The same
result will also hold in the case
when
the
"relative contours/' though not individually vanishing, are
such as in the aggregate to destroy each other, some being
positive
and others negative, for
in such case 2,mr
mS
s
r8
=Q.
504. If the several particles be.w rigid connection and the
figure describe n revolutions before re-occupying its origi-
nal position,
by Lagrange's "Second Theorem." (Routh,
vol. i., Art. 437)
and in that case
Anal.
Statics,
;
r
ri
M = 2w and K the radius of gyration about the centroid G.
where
505.
MECHANICAL INTEGRATORS OR PLANIMETERS.
Consider the case of two rods OP, PQ of lengths a and a 2
freely hinged together at P and the first one OP hinged to
a fixed point
in a plane in which both rods can otherwise
l
move
Let
,
freely.
y be the coordinates of Q relative to a pair of
rectangular axes through 0, let the rods make angles P # 2
respectively with the x-axis, and let 2
l =\l^.
x,
CHAPTER XV
516
Then
xa
cos 9 1 -}-a 2 cos
l
dx=
a l sin O l dO l
2
a 2 sin
=a z d0
1
,
2
l
-{-a2
2
d0 2
,
y= a
\
dy=a
l
i
n #1+^2
cos
s ^ n $2>
^rf^+ag
cos
d0 2 -\-a l a.2 cos
Fig. TOO.
be a point on PQ at distance b from P, and let
be the positions taken up by P, Q, R after disP', Q\
placements dO l d0 2 of the rods.
Let
J
R
,
Then
R
has.
a l d9 l cos
\Is-\-b
d0 2 =ds, say, to the
Then xdyydx = a^ dO l +a 2
If
Q
PQ
advanced perpendicularly to
2
dO^a^ cos
a distance
first order.
\//-
d\]s
+ 2a (^5
2
6 d^).
be made to travel round the contour of any closed
is to be found, in the positive direction, on
curve whose area
to be outside
completion of the circuit, supposing the point
the contour and OP and OQ to have oscillated back to their
original positions,
and we have
Area bounded by the contour =
where S is the total distance travelled over by a point R
on the rod PQ, in a direction at right angles to the rod.
And
it
is
further to be noticed that
this result does
not
MECHANICAL INTEGRATORS.
517
depend upon b, the term involving b disappearing upon
Hence the particular position
integration round the contour.
of the attachment of the point R to the rod is immaterial.
be within the contour considered,
506. But if the point
and both rods make a complete revolution before regaining
their original position,
=2
id0 2
*->
= 27r,
f
A=
and therefore
2a 2 b)-\-a 2 S.
Q.
Fig. 101.
.Now a^-j-ag2
2a 2 b
the value of
is
OQZ when
clamped at the joint P in such a position that
2
2
pendicular to PQ. Call this value of OQ r
circle
and radius
with centre
When
the system
r
OR
is
per-
.
,
A
the rods are
called
is
the zero
clamped
position the
in
the direction
angles to OR, i.e.
of PQ, and R has no motion at all at right angles to the
rod PQ on which it lies.
Hence when
lies within the
contour the area of the zero circle, viz. 7rr 2 must be added
circle.
motion of
R
is
in this
is
at right
,
to a zS to give the area of the contour.
Again, if one rod, say OP 1? oscillates back to its original
position whilst the other PQ makes a complete turn, then
-O,
A = 7r(a2
and
Similarly,
if
PQ
oscillates
fdfl^Sw,
and
fd6> 2
2
2a.Jb)-\-aJS.
but
(d0 2 =0,
A=
fc
=27r,
OP
f
revolves,
CHAPTER
518
The general result
the
pointer is
by
507.
is
XV.
therefore that the area traced
S
(1)
a.2
or
(2)
or
(3)
Trfat+at-Za^+OyS
2
7r(a 2 -2a 2 &)+a 2S
or
(4)
Tra^+a^S,
according as
(1) neither a^ nor
(2)
a.2
complete a revolution,
both complete a revolution,
completes a revolution but a x does not,
c^ completes a revolution but a.2 does not,
(3) a.2
(4)
in each case the
arms of the instrument occupying the same
position as they did at the beginning of the tracing.
508. This principle
.
is
made use
of in the construction of
a Mechanical Integrator known as AMSLER'S PLANIMETER,
which is used for the practical measurement of an area. The
PQ
R
with a small graduated wheel with
axis parallel to the rod, which is allowed to rest on the paper
and to turn by friction with the paper. It can then only register
rod
is
provided at
the amount of travel of R at right angles to the rod, the
amount of travel in the direction of the rod being necessarily
unregistered as it is due to slide along the surface of the
reading of the
paper and not to the rolling of the wheel.
wheel gives the value of S. Then
A
area of contour
according as the point
aJS + Trr*,
outside or within the contour.
Or
=ag
is
509. Several forms of Mechanical Integrators are in use,
but for the most part they are modifications of Professor
Amsler's form and based upon the general principle described
above.
Description of the Instrument.
shown (Fig. 102) is an illustration of a form of
the instrument made by Messrs. John J. Griffin & Sons,
The
Scientific Instrument Makers, Kingsway, London.
The
figure
lettering corresponds to the preceding general explanation
is the fixed point,
the contour of
of the principle.
ABC
the area required,
Q
the tracing point which
is
being made
AMSLER'S PLANIMETER.
519
P is the joint connecting the two
beams of the instrument, R the graduated wheel or roller
whose axis is parallel to PQ and which rolls upon the paper
when there is any motion at right angles to PQ. Its position upon the beam PQ being immaterial, it is placed in
D is a dial
this form of the instrument on QP produced.
whose axis is perpendicular to the axis of the wheel and
turned by a worm on the axis of the roller. There is a
to traverse the contour,
pointer attached to the beam PQ,
amount of rotation of the dial plate.
ing to read small
a pointer at
followed.
amounts of rotation
Q by means
of
1, 2, 3, 4, ...0,
The
dial
D
is
There
is
of the wheel are such that the
divided into 10 equal segments indicated by
and each segment into 10 further subdivisions.
is
such as to rotate once for 10 revolutions of the
roller,
and
again
subdivided,
is
V
of the wheel.
which the contour can be carefully
The graduations on the rim
circumference
serving to mark the
is a vernier assist-
itself
divided
into
10 segments, which are
an advance of a
segment of the
indicating one complete revolution of the wheel.
dial
The read-
ings of the dial therefore indicate the number of complete
revolutions of the wheel.
In the vernier a length equal to
9 subdivisions of the wheel is divided into 10 equal portions
on the vernier.
If the figures on the dial be taken as units, the
figured graduations on the wheel will represent 10 th8 and the subdivisions
CHAPTER XV.
520
100 ths the difference between the distance of two consecutive
,
divisions of the vernier
and two consecutive subdivisions
of
the wheel, being ( T i^ rV X TITO ) f ^ ne circumference of the
wheel, is T irW f ^ ne circumference of the wheel. Hence, by
means of the vernier, readings may be made to three places
of decimals.
The area
to be found has been
shown
to be
Fig. 103.
proportional to the number registered by the roll of the
wheel, the component of motion parallel to the axis, i.e. slide,
being unregistered. Let S be the number registered by the
wheel, then
where C
is
some constant
called the constant of the instrument.
Apply the instrument first to any figure of known area A
say a square or a circle, as may be most convenient let the
,
;
and final readings of the instrument be
which determines G. If now we apply
it to the contour whose quadrature is required and S be the
difference of the initial and final readings of the instrument,
difference of initial
S
,
then
A = CS
,
A-- A
A
8
-
AMSLER'S PLANIMETER.
521
has been taken
It has been assumed that the fixed point
If inside, we have still
outside the perimeter of the contour.
"
"
to add the area of the zero circle, and
o
o
The area
the zero circle
of
is
usually
marked on the
instrument.
Mode of Procedure.
The procedure is then
(1)
(2)
(3)
(4)
(5)
(6)
as follows
:
Fix the point
to the drawing board on which the
area to be found has been previously pinned.
Bring the pointer Q to some point of the perimeter
of the contour and mark the starting point.
Read the instrument by means of the dial, the wheel
and the vernier, and note the initial reading.
w hole
r
Trace carefully the
with the pointer Q.
Read the instrument
perimeter of the contour
again.
Subtract the two readings. The difference is S.
Then the constant of the instrument being known,
or having been found previously in like manner,
rr
-=00OQ
according as
it
or
Q
has been convenient to take
outside
or within the contour.
EXAMPLES.
1.
Oy being perpendicular
any closed region
show
that the integral
quadrant,
and
Ox,
AMBA
{>'
taken round the curve from
/>'(#)
OB = b.
being
finite
A
to B,
is
B
A,
axes,
S
of area
(y)
fixed points on Oy
lying in the positive
e*-m
equal to
and continuous,
[J.
m
a constant and
OA=a,
MATH. SCHOL. OXFORD,
1904.]
CHAPTER XV.
522
2.
Pj,
P2
are points on a closed oval of area A, such that
P P2
lt
subtends a right angle at a fixed point 0. Show that the area of
the curve traced out by the middle point of P^P^ is equal to
where
OP = r i>
and
*2=#i*f
^P.2
i
= r2
.
[COLLEGES
3.
A fixed
is
point
taken on a central oval which
is
jS,
1889.!
such that
other than the centre one and only one
through any
chord can be drawn which is bisected at that point ; prove that the
point inside
it
locus of the middle point of the chord PQ for a constant sum 2o- of
the arcs OP, OQ cuts at right angles the same locus for a constant
difference
where
I
2ur'
of these arcs
;
and deduce that the area
the length of the oval, and 6
is
tangents at
P and
is
of the oval is
the angle between the
[MATH. TRIPOS,
Q.
1889.]
A bar AB carries at a point of its
4.
length a small wheel having
for axis and which turns about AB: the end
is constrained
AB
to
A
move
in a given straight line
;
show that
if
the end
B
is
carried
round any closed curve without singular points and which does not
cut the straight line on which A moves, the area of the curve is
measured by the product of AB into the whole length registered by
the revolving wheel.
[This
is
the
principle
[COLLEGES, 1892.]
of
construction of Coffin's Planimeter.
A
full
description will be found on p. 159, Practical Electrical Engineering, by
Briggs and others. It is the case when the rod OP of Fig. 102 is of infinite
length, so that
P describes
a straight line instead of a circle.]
A
5.
straight line of given length moves with its extremities on
the arcs of two closed curves of given areas, and a point is attached
to the moving line.
Prove that when the area traced by
this attached point has a
value for different positions of the point on the line,
the difference of the areas of the circles whose radii are the segments
into which the point divides the line is equal to the difference of
minimum
the areas of the given curves.
6.
Show
length
2e,
[ST.
JOHN'S, 1882.]
that the path of the mid-point of a rod of constant
2
lie upon an ellipse, is an oval of area Tr(ab - c ).
whose ends
PROBLEMS.
523
instead of both ends being on the ellipse, one end lies on the
ellipse and the other on the major axis, or if one end lies on the
If,
ellipse
and the other on the auxiliary
the paths described
7.
A
rigid cyclic quadrilateral
return to
angles.
triangles
circle,
find
the
areas
of
of the rod in both cases.
ABCD
moves
in its plane so as to
turning through four right
denote the areas of the curves
original position after
its
Show
that
described by A,
Find
by the centre
if
etc.,
BCD, CD A,
(A),
and
etc.,
etc.,
if
S19 S2
,
etc.,
denote the areas of the
then
the equation connecting the areas described by any
by the centre of the circumcircle
also
three vertices with that described
of the triangle.
[I.
C. S., 1909.]
8. Two bars OP, RPQ, of lengths OP = c, RPQ = b + a, respectively
and a joint at P. dSl dSz denote the
turn round a fixed pin at
of
of the curves traced by P and Q
area
about
elements
polar
,
respectively
;
prove that
dS - dS
2
where PQ = a, RP = b, p
l
= ad+a(a + b) dO -\adp,
is the perpendicular from
on RPQ, d is
is the inclination
the displacement of 72 perpendicular to RPQ and
of fiPQ to a fixed line OA.
[MATH. TRIP., PT. I., 1914.]
CHAPTER
RECTIFICATION
(I.).
XVI.
ELEMENTARY.
510. In the following five chapters we propose to illustrate
further the methods and processes of integration by showing
their application to finding the length of a curved line whose
equation is given by one of the ordinary modes of description,
and
Cartesian, Polar, Pedal Equation, Tangential Polar, etc.
further to discuss some subsidiary matters which arise in
;
connection with such problems.
The process of finding the length of an arc of a curve, i.e. of
finding a straight line whose length is the same as that of a
Curves, the lengths of
specified arc, is called Rectification.
whose arcs can be found, are said to be Rectifiable.
Any
formula which
may have
been
established
in
the
Differential Calculus expressing the differential coefficient of
"
the arc " s with regard to any independent variable, in terms
of that variable, gives rise at once by integration to a formula
in the Integral Calculus for the finding of s.
In each case the limits of integration to be assigned are the
values of the independent variable corresponding to the two
points which terminate the arc whose length is sought.
511.
THE WORKING FORMULAE.
Below are added a list of the most common of these formulae.
The references are to the articles in the author's Treatise on the
Differential Calculus where they are established.
524
RECTIFICATION
Formula in the
Differential Calculus.
(I.).
ELEMENTARY.
525
CHAPTER
526
XVI.
order than the second and proceeding to the limit
dx dy
(dx\*
-r- ) +2 -j- -f- cos
\ds/
,ds/
ds ds
(
7) +2^^cos + (^)
\ds/
=1,
and accordingly we should write
or
MM*
O
Fig. 104.
according as
we take #
or
?/
for the independent variable.
The formulae may be remembered
513.
manner
in a less formal
as
or
s
=
I
where the dx or the
may
be brought outside the radical as
circumstances demand.
514.
Further,
when
the curve
is
given by expressing x and y
separately in terms of a single variable
t,
as
i
we have
or
s
according as the coordinate axes are rectangular or oblique.
The coordinate axes will be always assumed to be rectangular
unless the contrary
the context.
is
expressly stated, or to be inferred from
WILLIAM NEIL'S PROBLEM.
The
515.
527
Rectification, therefore, of a curve
depends upon
the possibility of integration of the radical which occurs in
these formulae.
ILLUSTRATIVE EXAMPLES.
516. The
Earliest
William
Rectification.
Neil's
Problem
(1637-1670).*
Ex.
1.
Rectification of the Seraicubical Parabola.
The equation
of this curve is
ay
dy
Here
Taken between
quadrant,
#=0 (the
^^3
.
--
3 x^
cusp) and
x=x^
for the branch in the first
i
.
by Gregory and Walton to have been the first curve to
The priority is ascribed to Neil by Wallis, but the rectificathe curve was also independently accomplished by Van Huraet.t
This
be
2
is
stated
rectified.
tion of
517.
Ex.
2.
The Parabola.
Consider the arc of the ordinary parabola y 1
4cu;.
Here
To
effect this integration, let
Then
x = a tan 2 \/r.
G?<
i/r
= 2a lsQV3
= a [sec
If
^j/
\js
tan
\jr
d\fs
tan
>^
+ log (sec \p + tan
taken between any two limits, ^ K and x 2 corresponding to any two
/*, Q on the arc, which lie on the same side of the axis,
,
points
arc
PQ =
(
^v^f^ - V^v^+^) + a log
* Wallisii
Opera, T.
1,
551
;
Gregory and Walton,
fCajori's History of Mathematics, p. 190.
p. 420.
CHAPTER
528
XVI.
For example, if we require the length from the vertex
end of the latus rectum,
^ = 0, #9 = a,
to the
upper
= *Ja*Ja -f a + a log \fa 4- a 4- Va
and
arc required
Thus the length
of an arc of a parabola
= 2-2956... x a.
to the other
is
from one end
of the latus
rectum
1'1478... times the latus rectum.
i
'Q
M
P(*#)
a Q
Fig. 105.
It is worth considering the angle ^ which has been used as a subsidiary
variable to facilitate integration.
It is the angle which the tangent at the current point
makes with
P
PM
be the perthe tangent at the vertex. For if
the focus, SY the
the tangent,
pendicular upon the #-axis,
perpendicular upon the tangent, and if we call MFP, fa we have
the y-axis, viz.
PY
cosy
The
intrinsic equation of this curve
s
= a sec
i/r
tan
is
therefore
^ + a log (sec ^ + tan
or
the tangent at the vertex being the initial tangent.
Let us call PY, t. Then t=-asec^ tan fa
Hence
s
- 1 = a log (sec ^ + tan
^).
RECTIFICATION OF THE PARABOLA.
529
Hence the logarithmic portion of s, viz. a log (sec ^r + tan \js) denotes the
excess of the arcual distance of
from A over the " tail," i.e. the portion
of the tangent measured from
to the foot of the perpendicular upon
P
P
the tangent from the focus.
It will be seen later that in
cases this excess "arc -tail" plays
many
an important part.
In the case under consideration
be measured along the tangent.
viz.
the parabola
OY=s
Then
let
a length
P0 = s
is the
The point
which the vertex A would arrive if we regard the
t.
point on the tangent at
tangent as a fixed line, and the parabola to roll upon it without sliding.
Consider it in this way.
is then a fixed
Take the tangent OP
point.
as the
as the r;-axis. Then, if
-axis, and a perpendicular through
be the coordinates of the focus,
ij
,
77
To
have
= Y=s - 1 = a log (sec ^- + tan
= YS
=. a sec
i/r),
i^-.
S
find the path of
to eliminate ^.
as the parabola rolls
upon
its
fixed tagent,
we
*
1
sec^ + tani/r = e.
t
Hence
a
sec-- tan-=
f Therefore sec
i/r
= cosh *
Therefore the path of the focus of the rolling parabola
ri
i.e.
.
a
I
.
= a cosh -
a
is
,
the ordinary catenary or chainette.
We
also have, putting
=
?4,
tan i/r= sinh w,
sinYr
SP=a sec2 = a cosh%,
i/r
t
= SP sin
8
= a sinh u cosh u + a log (sinh u + cosh u)
i/r
= a sinh u cosh u = - sinh 2w,
= s sinh2-{-aM,
a
s-t = au,
S Y= asec\js=a cosh u,
Incidentally,
we may
= a log (sec + tan
\f>
may be used
is
etc.
note that the equation
to indicate the
"march"
\f/)
= a gd"
1
\j/
of the function,
gd~Y
= tan ^, ^
-77
the abscissa of a point on a catenary curve, and since
is
the slope of the tangent to the catenary. Hence a good idea of the graph
of y = agd~ 1 a; can be formed by first
plotting the catenary itself and then
E.I.C.
2L
CHAPTER
530
plotting a
new
XVI.
curve, taking as abscissae the circular measures of the
angles which the tangent to the catenary makes with
its directrix, and
for ordinates the corresponding abscissae of the catenary.
If PP' be a focal chord of the parabola, the arc
has been shown
AP
AP = asec\^ tan
if/
+ a log (sec
if/
-f
tan \^\
it by
writing 90- ^ for
P'A = a cosec ^ cot ^ + a log(cosec ^ + cot \j/).
and the arc P'A can be obtained from
i.e.
Hence, by addition, the whole arc P'AP cut
which makes an angle 2^ with the axis is
+ log(1 + 8CC
The evaluation
of the arc miglit
M
off
by a
^,
focal
chord
,,
1
+ C086C W
have been conducted by taking y as
the independent variable.
x
Then
which reduces to the same form as already obtained.
518.
Wren's Problem (1632-1723).
Sir Christopher
Rectification
of the Cycloid.
Ex. 3. The equations of the curve are
Here
dx = a(\+ cos 6)dO,
di/
Hence
ds 2 = 2<*. 2 ( 1
s
s
= a8\n 6 dO.
+ cos 6) d6 2 = 4a? cos 2 - d6 2
= 4asin-,
,
......................................................... (1)
= 0, i.e. the vertex.
being measured from the point at which
Again, with the same description of the figure as in Diff. Calc., Art. 394,
r\
chord
CQ = 2asin--
Therefore
arc
CP = 2
chord
CQ ..................................... (2)
n
Substituting for 6 from
2
y = 2asin -,
(3)
SIR CHRISTOPHER WREN'S PROBLEM.
P
If the tangent at
is
vertex,
inclined at an angle
V
fy = s n 9 = tan Y = -rr-7
1 -f cos 6
2
eta
531
to the tangent at the
j
tan
/
1
;
and
This
(4)
is
the intrinsic equation of the curve.
Y
T
Fig. 106.
The w hole length
T
of the curve
from cusp
to
cusp
is
7T
n7
r
= 8a
L 4asini/' Jo
which ^ = 30 gives s = 2a, and
2
The point at
arcual distance from vertex to cusp.
(5)
therefore bisects the
Fig. 107.
be drawn with any radius, and OA, OB be a pair of radii
at right angles, and OB divided into n equal parts so that
being, say,
the rth point of division, and
be then drawn parallel to OA to meet
If a circle
M
MR
the circle at
ft.
then
smAOR--n
If then in the cycloid a
so that the angle
chord
CQ
of the circle
XCQ = angle A OR, in Fig.
and cutting the cycloid at P, will cut
arc
CP = 4a sin
off
CQD
an arc -CP-- of the arc CA, for
- Y = 4a n = )i arc CM.
\L>
be drawn (Fig 106)
QP parallel to X,
107, the line
.
CHAPTER
532
Hence an arc
of
any proposed
XVI.
ratio to the
whole arc can be cut
off.
Many of the geometers of the seventeenth century devoted considerable
attention to the cycloid.* Wren, the architect of St. Paul's Cathedral,
discovered the rectification of the curve and determined the centroid
;
Fermat, the area bounded by an arc Huygens invented the cycloidal
pendulum Pascal and Wallis also greatly advanced a knowledge of the
;
;
curve.f
CENTROID OF AN ARC OF ANY LINE DENSITY.
519.
If
p be the
line density, the
mass of any element
\pyds
f
y=I
j
pds
p be constant,
\xds
r*\
**
\y ds
~/Tt
ds
that
8s,
8s is infinitesimally small,
\pxds
^__J
p ds
If
p
Hence, taking the limit
give the position of the centroid.
when
8s is
is,
s=|#cs, sy=\pdy,
\
s
*
v
ds
being the length of the arc
whose centroid is required, and the integration being taken
from one extremity of the arc to the other. (See Art. 446.)
And if x be the independent variable,
'
with corresponding formulae if it be desirable to express the
in the table
integral with other independent variables as shown
of Art. 511.
*See Di/. CWc., Art. 390.
tCajori's Hist, of Math., pp. 177, etc.
RECTIFICATION
ELEMENTARY.
(I.).
533
EXAMPLES.
1
y (2a x) = x?, the
of the arc of the curve
Find the length
1.
Of Diodes.
Find the curve for which the length
2.
cissoid
[HUYGENS, 1625-1695.]
measured from the
of the arc
origin varies as the square root of the ordinate.
The major
3.
axis of an ellipse
Prove that
is Aj.
its
is 1
circumference
is
foot in length, and its eccentricity
3'1337 feet nearly.
[TRINITY, 1883.]
4.
Find the length of any arc of the curve
x* -y% = 0?.
5.
Show
that in the "catenary of equal strength/
and that the
6.
Show
intrinsic equation of the curve is s
common
that in the
s
= \fy'2 -c2
s
,
5
?/
= a gd
catenary, or chainette,
= cta,n\^,
s-
= a log sec X-,
= c(p-c\
s
?/
= ecosh-,
= csinh-.
The area bounded by the curve, the directrix, the
is A = cs.
The centroid of the arc has coordinates
#-axis
and an
ordi-
nate
# cot ^
The centroid
bounded by the curve, the
given by
of the area
and an ordinate
is
.c
and that both centroids
lie
1
directrix, the ?/-axis
cx
/
on the ordinate through the intersection of
the terminal tangents.
7.
(.',,
8.
Show
//,)
that the length
to the point (#2 ,
Show
?/ 2 )
the curve
of
is
= logcothx
from the point
1<
that in the epi- or hypo-cycloid
x = (a + b)
?y
s
?/
a
cos 6
= (a + 6) sin
b cos
-
6 sin
being measured from the point where 6 =
+b
^
0,
-,
1
r
,
i.e.
a vertex.
CHAPTER
534
9.
For the four-cusped hypocycloid
show
that
(i)
s=
cos2^,
s
being measured from a vertex;
the whole length of the curve
(ii)
^oc x 1
(iii)
10.
XVI.
,
s
is
6a
;
being measured from the cusp which
lies
on the ^-axis.
In the tractrix
r
show that
s
.
c
,
c
vV
y2
2
= c log *y
Show
that the distance from the vertex of the centroid of a wire
form of portion of a cycloid, of which the vertex is the middle
point, is J of the greatest ordinate of the arc.
11.
in the
12.
Show
that the arc of a parabola of latus rectum 4a measured from
the vertex, and the radius vector from the focus, are expressible in terms
of a parameter t in the respective forms
s
_
t
1.
a~r^ + 2
1
+t
g r^?
r^_
1
a~i=7*'
[MATH. TRIP. PT.
Prove also that
520.
1
s = \/r (r - a) + a tanh^-
II., 1915.]
--
Polar Formula.
In the Differential Calculus (Art 201)
it
is
shown from
consideration of the small infinitesimal right-angled triangle
formed by the increments of arc,radius vector and perpendicular
on the radius vector from one extremity of the infinitesimal
arc,
that to the second order
This gives
rise at once,
on proceeding to the
limit, to the
formulae,
s
or
s
according as
we wish
and, as in Art. 513,
manner
as
=
to use 9 or r as the independent variable,
it in the less formal
we may remember
RECTIFICATION
POLARS.
(I.).
535
be given in
Further, as in the case of Cartesians, if r and
terms of some third variable t (though this is very unusual)
by <r=f(t), 6 F(t), we may say
=MI
ILLUSTRATIVE EXAMPLES.
521.
Ex.
1.
In the case of the Archimedean Spiral r
s
= a /V02T T dd = ~
[0
x/^TT + log (0 +
being measured from the vertex, where
As this may be written
we
see,
aO,
=
0.
on comparison with the result of Art. 517, that this is the same
2 =
2a.r, measured from the vertex of the
3/
as the arc of the parabola
parabola and expressed in terms of the ordinate.
Fig. 108.
Hence it will follow that when an Archimedean spiral r = ad rolls
without sliding on the concave side of a parabola ?/ 2 = 2a# so that their
come into contact, the roulette of the pole of the spiral is the
the parabola. In this case the r of the spiral is the y of the
is always at right angles to
parabola, and the motion of the pole
the line PO, and arcs AP, OT are equal.
vertices
;>xis of
For many examples
of this class, see
Qhapter XIX.
CHAPTER
536
522. Ex.
2.
The curve
TT
The Cardioide
r
=
XVI.
a(l -cos
symmetrical about the
for the upper half.
dr
is
(See Art. 424, Diff. Calc.)
and
varies from
to
6).
initial line,
dd
Hence *=
dd
Fig. 109.
OA P.
This gives the length of any arc
For the upper half the length
The whole length
523.
The
n,
of
arc=
4a(
is
1
-cos 5-1=
4a.
8a.
9 Formula.
The equation
of a curve is
u-.
where
u=f(9),
The appropriate formula
sometimes given in the form
for rectification in this case is
r^-
snce
s=
giving rise to
(
J
^m
1
/du\*
(
)
l
,
+-,
d9,
I
or
according as 9 or
2
fdO\ d,
u 52 (-?\du/
)
u
be taken as the independent variable.
CENTROIDS AND MOMENTS OF INERTIA.
524.
CENTROID OF AN ARC OF ANY LINE DENSITY
537
;
POLARS.
Again, exactly as in the case of the curve whose equation
given in Cartesian coordinates, if p be the lino density, the
centroid of the arc of a curve is given by
is
I
-}
px ds
dr
525.
dO
dO
Centroid of Arc of a Circle.
Ex. In the case of a uniform circular arc of radius a and terminated
by the
radii vectores
0=
a,
the line density being uniform, taking the
medial line as x - axis,
1
a cos
.
a dQ
CHAPTER
538
Thus,
Moment
of inertia about a?-axis
Moment
of inertia about i/-axis
Moment
and
XVI.
is
2
l/o^/
^,
px* ds,
of inertia about a perpendicular to the plane
through the pole =
and for ds
=
pr
2
ds,
from the table of Art. 511,
to be substituted
the appropriate expression according to the system of coordinates used in any particular case.
The Product
axes
is
of Inertia for such a wire
with regard to the
defined as
EXAMPLES.
1.
Find the length
of
for the following cases
r=a cos 6
(i)
,
2.
r=a
N
v>
Show
which
lies
equal to
3.
:
(circle).
sin
2
,
^o!T0'
.
.
r = aem
(ii)
r=asin 2 -(cardioide).
(iii)
(
any arc of the curve from the formula
(iv)
,.
cls
^
(
(equiang. spiral).
= 1 + cos 6
(parabola).
r=
V1 )
that the length of the arc of that part of the cardioide
r = a(l+cos0),
on the side of the line 4r = 3asec0 remote from the pole,
4.
Show
is
[OXFORD.]
that the whole length of the
limaon
r = acos6
+b
is
equal
to that of an ellipse whose semiaxes are equal in length to the maximum
and minimum radii vectores of the limacon. Hence show how to divide
the arc of the limacon into four equal parts.
4.
Prove that the length
of the
r
th
?i
m = am
[COLLEGES
a, 1888.]
pedal of a loop of the curve
sin
mO
K
rm
a(mn + l)l
is
5.
Show
sin
md,dd,
where m(k
n+
!)
= !.
that the length of a loop of the curve
^ T JOHN'S, 1881.]
-
RECTIFICATION
6.
Show
(I.).
ELEMENTARY.
that the rectification of the curve rn
a n sinnd
539
is
given by
the integral
a
=
.
7.
Two
radii vectores
OP,
OQ
^
/"*
J
\/T^f
[MATH. TRIP.,
1896.]
of the curve
drawn equally inclined to the initial line prove that the length of
the intercepted arc is aa, where a is the circular measure of the angle
[ASPARAGUS, Educ. Tii
POQ.
are
;
8.
Show
that the centroid of a wire bent into the form of a cardioide
t
r = a(l
+ cos#)j
an d with a line density &sec-, k being a constant,
the axis of the cardioide at distance
527. The Converse Problem.
The converse problem,
--
is
on
from the cusp.
Given
s,
find the Curve.
viz given s in terms of one of the
quantities x, y, r or 9, to find the equation of the curve,
leads in the first three cases shown below to an application
of the same formulae, but in the fourth case there is more
difficulty (Art. 529).
(1)
If
s=f(x),
we have
2
(S)
(2)
If
e=f(y),
'
(3) If
1.
s=f(r),
Find the curve
for
which
s
= ^2
.
Here
Say = x*lo*
- a- - a 2 cosh~ 1 - + constant.
CHAPTER
540
2.
Find the curve
in
Here
which
r-
s
XVI.
= rseca.
=sec 2 a-
(-77)
= tan 2 a,
I
y=rf6>cota,
logr =
r
3.
Find the curve
in
+
cot a
= rte
0cota
+ const.,
(Equiangular
.
spirals.)
s = \/8?.
which
ds
Let
6),}
= a(l-cos#) J[a
3/
529. (4)
cycloid.
But the case when s=f(6)
and the variables
r
and
are not
now
leads at once to
in general "separable
"
as in the former cases
(see Integral Calculus for Beginners,
Art. 175); nor does this differential equation fall under any
of the standard forms.
Nevertheless, in some cases useful
information
may
be derived from
its
consideration.
For example,
Is the circle r = a the onl
1.
Here we have
if
r
is
(
-^
not equal to
a,
)
curve for which
+ r2 = a 2 which
,
is
s
= a#?
of course satisfied
by r=a.
But
we have
dr
sin" 1 - = a
r = a sin (a
Hence
i.e.
0,
where a
is
a constant.
0),
a circle of radius - and passing through the pole will also give the
same
result, viz. s
= a@,
than r = a or r = a sin (a
as
is
geometrically obvious.
#) will
do
so.
But no curve other
RECTIFICATION
2.
ELEMENTARY.
(I.).
Is the equiangular spiral r = oe*
cota
541
the only curve for which
a
e cot
ae
cos a
2
Let r = ay/ cota where v
,
Thus
some function
is
/
_f.
of 6 to be determined.
vco a
fc
j
which
of
is
course obviously
But we have
y
if
= l, which
leads back to
in addition to this the general solution of
-j-f.
+ v cot a =
fJ y cot
where
To
satisfied
COta
r==a6
(3 is
2
\/(cosec'
v
a T v cosec^a
a - v-)
-
some constant.
=
integrate this, let v cosec a sin <.
r
.e.
I
a cot (< T a) ^ si n
{cos
a}
d<$>
fi
= ^-
j
sin
s
f)
,
ot
_n
or
furnishes a set of curves whose arcs are
which upon elimination of
same length as the corresponding arcs of the equiangular spiral
<
of the
EXAMPLES.
1.
Find the curves
(i) s
(v) s
(vii) s
2.
Show
in
= a sin"
1
which
-.
(vi) 5
<x r.
oc *JHc.
= 2 v/2ar.
that the equation
nx -v2-
-r
^-
=
leads to a cycloid or a four-cusped hypocyloid according as ?i=2 or n
= 3.
CHAPTER
512
530.
XVI.
Tangential Polar Equations.
,
-c,
formulae
t
dp
= -f-
ds
Legendre's Formulae.
== r,
,
d2
p
+ -rf-
9
2
d\[s
d\fs
.
d\/s
These results were proved in Article 221 of the Differential
Calculus, but are now established in a different manner.
Let PF, P'Y be the tangents at two contiguous points P, P'
of the curve, OF,
Y' the perpendiculars upon them from the
pole 0.
Fig. 111.
Let
t
be the projection of the radius vector upon the tangent
OY=p,
wcPP'=Ss,
OY'=p+Sp,
and
the angle YOY'.
<Sx/r
Then, projecting the broken line
upon
OYPP' upon OF' and
F'P',
(1)
p+p=p cos
(2)
t+St = Ss-\-tcosS\lspsmS\ls,
S\fs-{-t
sin c^/r-f second order quantities,
= SspS\ls,)1 to the first order.
d p
dt
ds
dp
And ultimately = -f
r,
TT =p-\- -j-r=p-\r~r9l
*
.
,
St
2
-
^
,
L
d\js
d\fr
2
d\fs
d\fs
531. It is to be noted that since
t
=
-j~
=r cos 0,
projection of the radius vector upon the tangent,
is acute or obtuse.
or negative according as
t is
i.e.
the
positive
<
The above figure (Fig. Ill) exhibits the standard case. In this
case t = -r is -f-PF, and is in a direction from P opposite to that
LEGENDRE'S FORMULAE.
of the direction of increase of s
is
as
p
is
increasing with
In cases where
therefore positive.
\//-
and
,
increases or decreases
p
=
PY.
The student should examine the formulae carefully
in all
\/r
decreases or increases,
four cases
(2)
(3)
(4)
Curve
Curve
Curve
Curve
concave to 0,
will
be seen that
= PY according as
The
negative,
<
<f>
convex to 0,
concave to 0,
convex to 0,
ft
It
(i.e. t) is
-r
and
:
(1)
t
;
543
^
ft
-r-r
= y>+-rr
a\fs
d\fs
is
acute.
acute.
<f>
obtuse.
<f>
obtuse.
^'/")
2
in all cases
and that
acute or obtuse.
measured from a point on the arc on the same
side of the radius vector as that on which
is measured
\fs
or
increase
decrease
the
with
increase
of
s.
may
arc s
is
<
;
The value
of the radius of curvature
and
positive;
p=^-r
is,
as
according
of course, essentially
and
s
^
increase
together, or the one increases as the other decreases.
Accordingly
we have
The formulae
cases.
532.
By
p=
in these
(p-\-^-j-^\ respectively
established are due to
integration of
d2p
ds _
~
we have
s
=
-
-
+
1
p d\lr,
st =\p d\]s
i.e.
where
t
is
LEGENDRE.
the
"
"
tail
;
referred to in Art. 517,
In the case of a closed oval of continuous curvature, the
"tail" t returns to its original value when the integration is
conducted round the whole contour.
the origin be within the curve and is only enclosed once
it, the length of the contour is given by
If
by
CHAPTER
544
n
If the origin is enclosed
XVI.
times (Fig. 112), so that the
as its point of contact
tangent makes n complete revolutions
travels continuously
round the curve, the length
will be
pd\}s.
i:
Further modifications may have to be made, for instance,
round a loop of a curve (Fig. 113) it may
in integrating
;
happen that the
initial
same, and
the
that
ds
and
values of
final
tangent
does
-rj-
are not the
make a complete
not
Fig. 112.
Fig. 113.
revolution, but the student should have no difficulty in such
cases in assigning the proper limits.
533.
Show
Ex.
that the perimeter of an ellipse of small eccentricity
3e4
exceeds by
^j
having the same
of its length that of a circle
e
axis.
[y, 1889.]
2
2
2
2
2
p = a cos + b' sm \^
Here
a 2 (l
i//
where $
is
the angle
p makes with
e
2
sin 2 ^),
the major axis.
Therefore
4
4
2
2
p = a(l --e sin' i/'--e sin ^ -..A
Hence
s
= 4a /7T
,
(z
=
The radius
and
its
is
2
1
~8
2- -a. -7T
r of a circle of the
circumference
1 7T
1
~^2
,
2
3
.
3
4 2
.
7r ae
same area
4
is
-
\
1 7T
e
2~ '")
...
.
given by
ARC OF AN EVOLUTE*
.'.
/
circumf ellipse - circuraf circle =
.
.
545
o
o
\
-
(
vrae 4
J
4
= 3e
^j {circ.
as far as terms involving
circ. ell.
3e 4
circ. circle
'
circ. circle
circ. ell.
terms involving
e
'
64
3e
circ. circle
4
3e4 \
//-
3e 4
64/64
~64/\
circ. ellipse
to
of circle}
e4,
4
.
534. Length of the Arc of an Evolute.
It was shown in the Differential Calculus (Art. 343) that
the difference between the radii of curvature at two points
of a curve of continuous curvature is equal to the length of
H:
Fig. 114.
the corresponding arc of the evolute; i.e. if ah be the arc
of the evolute of the portion
of. the
original figure, then
AH
(Fig. 114)
And
arc ah = Aa
Hh,
i.e.
p
(at
A
p
)
(at
H).
the evolute be regarded as a rigid curve, and an inelastic
be
unwound from it, being kept tight, then the points of
string
the unwinding string describe a system of parallel curves,
if
each of the parallels being an involute of the curve ha, one
of these being the original curve
itself.
HA
535. Ex. Find the length of the evolute of an ellipse.
If a, a', ft, /3'
be the centres of curvature corresponding to the extremities of the axes,
viz. A, A', B, B'
respectively, the arc a/3 of the evolute corresponds to
the arc
AB of
the ellipse, and
arc
E.I.C.
=
a/3 p
we have
2
(at
B)-p
2M
(at
A) = ~
7*2
-,
CHAPTER
546
the
for
radius
of
curvature at any point
2
(
the pedal
XVI,
equation being
62
= a? + b2 -r2
^-
length of the entire perimeter
symmetrical about the axes, is
of
the
P
of
and
evolute,
/>
the ellipse
T dr\
= -T
which
is
^-3-
^
Thus the
)
is
obviously
In the application of this rule care is needed, not to pass a point of
or minimum curvature on the original curve, for on travelling
maximum
Fig. 115.
round the original curve the difference of
successive radii of curvature changes sign at such points and the evolute
has a cusp as in the figure for the ellipse (Fig. 115). In that case, as
in a continuous direction
P
from A to B and through
the arc a/3 and upon the arc /?a'.
travels
continuously
maximum and of
and the positive
to A', the string PQ is wound of
And therefore the arcs af3 and (3a
signs, viz. -=--
would appear with opposite
in
B
and -
-j-,
if
P
travels
The intervals between the points of
minimum curvature must therefore be treated separately
one direction.
results
added together.
EXAMPLES.
that in the parabola 3/ 2 = 4a#, the length of the arc of the
e solute intercepted within the parabola is
1.
Show
2.
Find the whole length
4a(3\/3-l).
of the evolute of the cardioide
r
a(l +cos
0),
INTRINSIC EQUATIONS.
3.
Show
547
that the length of the evolute of the portion of the Folium of
Descartes # 8 +.y 3 = 3flw?y, which corresponds to the loop,
536. INTRINSIC
Let
s
is
(4
-
</2).
EQUATION OF A CURVE.
be the length of the arc of a curve measured from
to the current tracing point P
a fixed point
;
O
*
T
Fig. 116.
the angle of contingence at P, i.e. the angle between
i/r
the tangent at P and any fixed line in the plane, say the
tangent at
\
p the radius of curvature at P, or K
its
reciprocal,
viz.
the curvature.
Then any given
tities
the
s,
\IT,
curve,
relation
between two of these three quan-
p (or K) will suffice to determine the shape of
and may in many cases very conveniently
replace an extraneous specification of the curve by mean&
of coordinates, Cartesian or Polar.
These quantities s, i/r, p
depend upon no external system of coordinates and leave
the position of the curve undefined. The nature of the curve
itself is specified by the relation existing between two of
the three s, \//-, p, which has been very aptly styled by Dr.
Whewell the
Intrinsic Equation of the curve.
Some
notice
has been already taken of Intrinsic Equations in Arts. 346-349
of the Differential Calculus.
But the subject is more closely
allied to Integral Calculus, and it is convenient to develop
the matter more fully here, though at the risk of some
repetition.
shall
We
adopt the notation used in the Differential
Calculus as to the meanings of the letters involved for the
following work.
When the relation
is
between
say
s
and
s=f(\]s),
\//-,
CHAPTER
548
that between p and
\}r
XVI.
is
ds
The sign
to be taken
when
when
-f-
s is
s
increasing with
x//-,
increases or decreases as
x//-
decreases or increases,
and
if
K be used (viz. the curvature,
=-
instead of the
),
PJ
radius of curvature,
=
is
the relation between K and
1
with, of course, the same
\fs,
rule as to choice of sign.
Conversely,
if
the connection given be between p and
\js,
p=
say
then
and
8=\f(\ls)d^+C,
G being a constant which may be chosen to correspond to
the measurement of s from any arbitrarily chosen point of
the curve, and the sign selected as before.
When
the relation
is
given between K and
the same thing, except that
and
Finally,
s
when
=
y^r +
J/W
I
the relation
say
we have
\//-,
it is,
we have
is
const.
between p and
/>=/()
77 ~f( 8
)>
Jds
ffi
s,
of course,
INTRINSIC EQUATIONS.
549
Hence, these three systems of description of a curve, by
of a specified relation,
means
between s and \//-,
(6) between p (or K) and \]s,
(c) between p (or K) and s,
and
either forms a mode of specification which
are equivalent,
is intrinsically a property of the curve itself, and in no way
defining its position upon the plane upon which it may happen
to be drawn.
The s-\js description is the one which is usually under"
Intrinsic Equation," and it is the system used
stood as the
in
his memoirs on the subject (Camb. Phil.
Whewell
by
Trans., viii., p. 659; and ix., p. 150) and discussed in Boole's
(a)
Differential Equations, pages 264-269.
The p-\Ir specification was used by Euler.
537. To
obtain the
Intrinsic
Equation from
the
Cartesian
Equation.
When
the Cartesian Equation
supposing the initial tangent to
we have
given as y=f(x), then,
be parallel to the #-axis,
is
tan ^=f'(x),
.................................
(1)
and
8=lJi + [f'( x )]*dx
And
(1)
if
and
after integration x be eliminated
the required relation between s
(2),
say
s
(2)
between equations
and
x//-,
= F(\]s),
will be obtained.
Conversely, if the equation s = F(\fs) be given, and the
Cartesian equation be desired, we have
dx
whence
di
=
=
cos\lsF'(\ls)d\ls,
........................
(1)
ain
......................
(2)
Jai
A
and
B
^,F'(^)d^,
being arbitrary constants.
CHAPTER
550
And
(1)
and
if
XVI.
be eliminated from equations
\js
the Cartesian Equation of the curve will result.
after integration
(2),
538. Illustrative Examples.
Intrinsic Equation of a circle.
Ex.
1.
If
be the angle between the
\p
P, the centre being
therefore s
a^.
tangent at A and the tangent at
initial
A
and the radius
a,
A
we have POA - PTx=
Ex. 2. Intrinsic Equation of a catenary.
In this case the equation of the curve referred to
tangent at the vertex as coordinate axes is
its
axis
^,
and
and the
x
c
Hence
and
,x
smh -,
dy
W-T
dx
,
tan Y
.
c
~ = A/1 + sinh
dx
2
*
c
= cosh-;
c
the constant of integration being zero if we measure s from the vertex
where # = 0; therefore 5 = ctan^ is the intrinsic equation sought.
539. Case
when the Coordinates
are expressed in terms of a
Parameter.
If the equations of the curve be given as
we have
dy =
dx /
tan \r = -j2
-
'
Also
and
s
=
(t)
INTRINSIC EQUATIONS.
If s be
found in terms of
t
by integration from equation
then between this result and equation
The required
540. Ex.
1.
relation
551
between
s
and
(1)
\/r
we may
(2),
eliminate
t.
will result.
In the cycloid
= a(l-coat).
Hence
Also
whence
s
Hence
= 4asin-,
s
= 4asin ^
being measured from the vertex, where
s
is
the equation required.
397.
Ex.
2.
In the epi- or hypo-cycloid
x = (a 4- b)
y=
-b
cos
cos
j
(a + b) sin 6
- b sin
*=
Fig. 118.
T
0,
0,
See
Diff. Cede.,
= 0.
Arts. 395,
CHAPTER XVI
552
Also with the description of figure in Art. 405,
6=6
=
If s
46(0 + 6) cog
a
r
a
be measured from the cusp, the tangent at the cusp being the
initial line,
AP = s
arc
If
-F
fl+
and
Diff. Calc.,
=
=
we measure
46 (
+ &)/,1
cos
1
a
\
the arc from the vertex
-
7n - =
VP
s'
,
arc
.
a
where
F,
1
S
a
a+i
=
,
a
C os
OA being retained as the initial line for the measurement of
measure $ from the tangent at the vertex, we must write
7T
irb
_
+ ._ - +
,
,
i
for *,
CL
Hence the general
=A
,
a
intrinsic equation of such curves
s
d
a
.
5'
,. f
we
If
2
,
and
In the
measured
In the
measured
7T
M.
^.
sin Zty
or
s
is
= A cos B\j/.
case s = A sin B^, s is measured from a vertex and
from the tangent at that vertex.
case s = A cos .Z?^, s is measured from a vertex and ^
from the tangent at the next cusp.
is
\js
is
541. To obtain the Intrinsic Equation from the Polar.
Suppose the
initial line parallel to
the tangent at the point
A
Fig. 119.
from which the arc
notation, we have
r
= f(0)
,
is
measured.
Then, with the usual
the equation to the curve,
(1)
(2)
INTRINSIC EQUATIONS.
553
and therefore
If s be
found by integration from
by means
of
between
and
s
and
equations
will be found.
\//(2)
(4),
and
9,
equation be not that from which \js
equation (2) will need modification accordingly.
polar
542.
Ex.
1.
Find the
eliminated
is
measured,
intrinsic equation of the cardioide
cos
r=a(l
^=
Here
<f>
the required relation
If the initial line of the
(3),
6),
+ <,
a(I-cos0)
and
ds
_
,
2
r
~273-
'
dO
?
2'
r\
s=
4a cos -Q-l-C.
Fig. 120.
If
we determine
so that 5 =
when
= 0, we
have (7=4a
;
\
\,
/
the intrinsic equation sought.
If
A
If
we measure
be the vertex, the arc
write for ^,
\f/
AP=
from the tangent at the vertex
3:r
(Fig. 121),
we must
CHAPTER XVI
554
and
if
arc
AP=s'
= 4acos(
t
?--o
= 4asin ~.
Fig. 121.
Ex. 2. Find the intrinsic equation
Archimedean spiral r = aO.
of the first negative pedal of the
Fig. 122.
If
be the pole,
dicular to
P
OP touching
PT
a point on the spiral, and
be
the first negative pedal in T, then
drawn perpen-
INTRINSIC EQUATIONS.
Hence the normal TQ
and radius a.
centre
to the first negative pedal envelopes a circle with
It is therefore an involute of the circle.
If TQ
touches this circle at $, then
i.e.
involute,
p=ai/s, for
ds
dj
:
AQ
'd,Yc
y
where
A
is
the cusp of the
;
= a\l>
= a * and
2
ir-
Art. 455.)
Diff. Calc.,
Otherwise
pTQ =
\//=QOA
"
(See
555
If r
= aO
be the locus of P,
6 being the polar coordinates
r,
from the pole upon a tangent
of the foot of the perpendicular
negative pedal, the tangential polar equation of the pedal
ds
'
d}
d'2p
=P + W' = atf"'
.
'
a\l/
S=
is
to the first
p = a\js
;
2
'
543. To obtain the Polar Equation from the Intrinsic.
=
When the intrinsic equation s F(\js) is given, and it is
desired to get the equivalent polar equation, it is usually
best to obtain the Cartesian coordinates of a point on the
curve first, as above, from
and then, after integration,
to
form
and
as functions
of
\Js,
and
finally
0-tan- 1 !
eliminate
to
resulting equation will be the relation between r
If
we
when
and
the
0.
attack the problem directly without the intervention
of Cartesians,
we have
tan-'
which
\fs,
9
a troublesome second order differential equation
but one which, of course, theoretically furnishes the required
is
;
relation
between
r
and
9.
544. Illustrative Examples.
Ex.
Find the
1.
Here
<
=
a,
^=
5,
i/'
relation for the equiangular spiral
9cota
r = ae
:
6
+ a,
s
= a cosec a
e cot
(e
J
*dB = -
cos a
e
e
cota
CHAPTER
556
the constant being determined so that
- oo i.e. from the
pole
at which
XVI.
shall
s
be measured from the point
;
,
~
Ct,
(\J/- a)
Cot a
cos a
Ex.
2.
Conversely, find the polar equation corresponding to
(>/f-a)cota
,,.
,
We have
x
sma
a
a
-=
a
sin
y
C
,
lcosu/e
,,
0/,-a)cota
r
dy =
J
f
.
.
sin
/
\L e
M-a)cota
'
,
7
d\fr
J
the constants vanishing
if
r
*:**f*
cosec
= c*~
a
acota
,,
cosd/'-a),
/f
sin (\L
.
cosec a
- a),
we make x - a and y - 0, when ^ =
_
a;
-dr-aCOta.
V a~
and
tan
= tan(^ - a)
;
/.
^ ==
+a
;
545. Intrinsic Equation deduced from the Tangential Polar.
When
the tangential polar equation of the curve
p=F(^),
we have
is
given,
say,
at once
s=
and
the intrinsic equation required.
546. Tangential Polar form deduced from the Intrinsic Equation.
To
get back to the tangential polar form from the intrinsic
equation
we
have, of course,
we may
either
say
(See Integral
perform the operation indicated.
thus
or
we
XVI.),
may
proceed
Chap.
Beginners,
for
To solve
at once
this
differential
equation
p = A sin \\, +B cos
and
:
INTRINSIC EQUATIONS.
multiply
(a)
by cos
(1)
557
and then by
\/r
sin
\//-,
giving
respectively,
and
sin
(6)
integrating
where
and
J.
sin\//-
^
W
sin
we have
j?cos\//"
r
I
=
f'(\js)
sin \Isd\Is-B,
are arbitrary constants
eliminating jy
(c)
cos
^~p cos * = ^
=
jj-
=
+ sn
cos
;
,
p=sm\l/\ f'ty) cos \^ d\ls
cos \/r
I
/'
Jo
Jo
sn ir -f
cos ir
;
and the tangential polar result is obtained.
The result may obviously be written as
J*
/
(a))
sin (^
o>) dco.
Moreover, if we choose our origin of measurement of p to be
such that A and B both vanish, and suppose s to have been
measured from a point where \fs = Q, so that/(0) = 0, we may
by parts and further reduce
integrate
p=
cos
(\fs
this equation to
w) da).
Jo
547. Intrinsic Equation deduced from the Pedal Equation.
When
the pedal equation
(p, r) is
given, say p=f(r),
dr
Then
s
can be found in terms of
f_
r
by integrating
_rrfr_
Vr*-r
A
Again,
^5
~= p= rdr =
^
-
r
'
/ON
............................... (2)
CHAPTER
558
XVI.
If r be eliminated between equations
a differential equation between 6 and
and (2) we get
whose solution
(1)
\js,
furnishes the intrinsic equation sought.
548. Ex. Consider
p=r sin a
(equiangular
rdr
f
_
~
_
~J <Jr*~p z
r dr
ds
-n
T~
ds
s
log
s
=
dp
d\//
r
J ?*cosa~cosa'
r
-
sin
spiral).
rdr
C
a
= scota,
= cot a dy,
= ^ cot a + constant,
549. Pedal Equation from the Intrinsic.
if it
Conversely,
from the
p = sm\js
Upon
= f(\Js), we
have
ds
dr
and
be required to derive the pedal equation
intrinsic equation s
r*
r*
I
f (\js)cos\Isd\Is
elimination of
dr
between j-,
and
r
p,
\js
cosx/^
we have
f'(\ls)sm\js
d\js. ...(2)
a differential equation
which when solved gives the required
p-r equation.
550.
and
Ex. Starting with
p = sin
i/J
s
= Ce* cota
,
C cot a e^ cot a cos $d^- cos
j/'
(Ccot a e* cot a sin
= (7 cot a e^ cot a sin 'A cos A- a )- cos
(
l
/sin
'/
(
;
/
/
-a
)
cosec a
r dr
i.e.
r2
= ?2
sin
i.e.
a
,
if
p and r are taken to vanish together,
p = rsma.
551. Variations on these
modes
be adopted to suit special cases.
of procedure
may
of course
INTRINSIC EQUATIONS.
552.
WELL-KNOWN
INTRINSIC EQUATIONS.
The following are the most common
the "well-known" curves:
For the
(1)
(3)
= a^,
s = c tan
s = 4a sin
For the
Diff. Calc., p. 273.
\//-,
=
(4)
intrinsic equations of
a
circle,
For the catenary,
For the cycloid,
(2)
559
x//-,
273.
p.
340.
46
epi- or
or,
p.
p. 345.
generally,
hypo-cycloid,
= A sin
s = A cos
s
or
(5) Involute of a
circle,
(6)
Parabola
(7)
Evolute of a
J S=_a\Js
2
\
p.
275.
Int. Calc., Art. 517.
Diff. Calc., p. 275.
parabola
Semicubical
Int. Calc.
parabola
(8)
Equiangular
Beginners,
for
p.
151,
L=
spiral,
(9)
Diff. Calc., p. 358.
Tractory,
(10) Cardioide
r
included as a case of
the epi-cycloid,
= a(l-cos6>),
s
(11)
= 4a(
Catenary of
y
x
a log sec-,
i.e.
1-cos^J,
s
= a gd ~ l
s
=
equal strength,
Int. Calc., Art. 542.
\js,
Int. Calc.,
Ex.8,
Art. 517.
CHAPTER
560
XVI.
553. Intrinsic Equation of the Evolute.
Let s=/(^) be the equation of the given curve. Let s'
be the length of the arc of the evolute measured from some
Let
fixed point A to any other point Q on the evolute.
and
P
be the points on the original curve corresponding to
Q on the evolute p p the radii of curvature
the points A,
and P;
at
produced,
PT
or,
,
;
the angle the tangent QP makes with OA
which is the same thing, the angle the tangent
\//-
makes with the tangent
Then
S'=Q
at 0.
On^r-.
On
Fig. 123.
554. Intrinsic Equation of an Involute.
With the same
figure, if the
curve
given by the equation s'=f(\Js),
ds^
d^
AQ
we have
be the original curve
INVOLUTES.
p
is
now an
The
561
arbitrary constant, and
intrinsic equation of
an involute
is p
the particular involute whose radius of curvature at
For any of the other involutes, the whole set of which form
a family of parallel curves, replace p Q by a where a is the
is
.
radius of curvatnre of the parallel, corresponding to p
the particular involute considered.
for
f
The
OP, O'P' of these parallel curves
if the involutes form closed ovals,
and
(PQO)^',
the tangent making one complete revolution, the difference
is
difference of the arcs
therefore
of their perimeters
555. In the
is
27r(/o
case of an
a).
involute of a circle, already discussed in
Fig. 124.
Art. 542, if a be the radius,
the centre, A the cusp of the involute,
and Q the point of contact of the radius of curvature at P,
and
E.I.C.
2N
CHAPTER
562
For a
parallel traced
by a point
- of 'A
s'
=
of
measurement
+
s'
,
= s,
changing the origin
i.e.
of s suitably,
s
s
so that
.fii/',
= A B cos B^' - /o
dropping the accent, and writing p
or,
s'
P
and hypo-cycloids
A sin
s
s'
is
we measure
if
from
circle.
556. In the case of the epi-
the evolute
at distance b
when ^ = -,
a'
another involute of the
i.e.
;
)
a/
2\
XVI.
=
AB cos B\fr,
being measured from the point where ^-^jy> or
s
= ABsinB\j/
if
we
choose a suitable initial tangent, viz. that at the point from which s is
measured.
Hence the evolute of an epi- or hypo-cycloid is a similar epi- or hypocycloid.
is
Putting B = I we have a case which shows that the evolute of a cycloid
an equal cycloid.
Supplying the values of A and B (Art. 540), the equations of the curve
of the evolute may be written
and
s
rx
_ (a + 6)cos
= 4b,
a
__^
.
,
.,
,
a+b
=46^-^
w^;
a
cos
.,
^
with a different origin of measurement for s' and a different initial
tangent, and we can compare the linear dimensions of the two curves, viz.
a
linear dimensions of evolute
linear dimensions of original curve
a
+ 26'
in the case of a cardioide, for which a
6, the evolute
cardioide of one-third the linear dimensions of the former.
e.g.
is
another
557. Whewell's Theorem.
An interesting theorem is quoted by Boole from Whewell's
Memoir (above referred to) with regard to the ultimate form
to which the successive involutes of a given curve tend, the
"
involutes being such as have equal tails."
Whewell takes as his original curve s=F(\js), which he
WHEWELL'S THEOREM.
supposes
capable
vanishing with
x//-,
of
expansion
in
563
powers
of
\]s,
and
s
so that
and he further supposes the successive involutes to be defined
"
same " rectilinear tail at starting.
Let P P be the original curve, and Q Q, R R, S Q S,... the
"
"
tails
successive involutes, and the several
Q P R Q G>
as having the
,
Fig. 125.
S R
,
...
all
cessive arcs.
equal, say
The
\]s's
= a,
and take
are all equal
s lt
if
s2 ,
53
,
...
the suc-
measured from the
respective initial tangents.
Then for the arc Q Q, viz. the first involute,
no constant being required, as each arc vanishes with
Similarly,
sB
=a\lf-\-a^^-A l -^-{-A z -^
+
,
\Js.
CHAPTER XVI V
564
Proceeding thus,
And when n
is very large the terms in the first bracket
are
unaffected
(which
by the form of the original curve)
approximate to e*I.
And
ai*e
those in the second bracket have coefficients which
ultimately infinitesimally small.
to the limiting form
Hence the involutes tend
i.e.
an equiangular
spiral of angle
we
In a similar manner
1),
is
if we start off with a
an algebraic expression
,= a
say
since
F
= a(e*
.
note that
curve in which s=F(<j>), where
of the w th degree,
then,
-7
s
the radii of
curvature of
the curve and
its
successive evolutes are
ds
P
~W
_ dp
-
_
d 2s
pi
f
*-ty'***tRp'*
it
follows that r
a
a.
n-l
th
e
Hence the (n l)
volute
Therefore
s
=a
is
a
circle.
^y +& (n-l\\ +
'
one of the (n l)th involutes of a circle of radius a, or
parallels to such involutes, the "tails" being the successive
is
coefficients k,
558.
j, etc.
Involute of a Catenary.
Ex. The intrinsic equation of the catenary
Hence the
and pQ
is
is
s=ctan
^.
intrinsic equation of its evolute is
the radius of curvature at the vertex = c
7
p.
for
Hence the evolute
is
= c, when ^ =
2
s = c(sec
^- l) =
/o
=
j
2
^-r=csec
/'
J.
CURVE TRACING.
The
and
intrinsic equation of
if s
an involute
565
is
= c log sec ^ 4- A ip + constant,
s=
when ^=0, we have
s = c log sec ^ + A
be so measured that
\f/.
559. Tracing of a Curve from the Intrinsic Equation s=f(\js).
is
it
Generally
(1)
best to obtain the Cartesian or polar
possible by the methods of Arts. 537,
form of equation
and to trace the curve therefrom by the usual rules
if
543,
Chap. XII.).
be not possible by reason of the failure to intethe
grate
expressions occurring in the articles cited, find the
(Diff. Calc.,
(2) If this
curvature
\]s.
Note
ds
as
-jy
and examine how the curvature changes with
-^,
also concavity or convexity to the origin according
+
is
values of
or
.
Note whether
s
becomes unreal for any
and whether p changes sign for any values
>//*,
fj Q
of
where
^=
Also the inflexions where ^-r
\IT.
ds
= 0.
.
-j
oo
,
and the cusps
d\lr
Tabulate corresponding values of \js, s and p.
Observe whether a change of sign in \fs would alter the
value of
If
s.
from which
\Js
not there
is
is
symmetry about the
initial line
measured.
Examine whether
C'/'
x=\Jo
even though not (as in the case considered) integrable in
general terms, can be evaluated as definite integrals for any
particular values of \//-. Approximate values of these integrals
may lead to important information as to the position of some
For accurate plotting
points through which the curve passes.
the tabulated values of these integrals for various values of
\Js
in general
becomes necessary.
shape of the curve
when
For a general idea of the
close accuracy of plotting is not
CHAPTER
566
XVI.
necessary, an examination of the integrals and the behaviour
of the integrand may furnish sufficient information.
Ex. Trace the curve
560.
ks*
= ^,
(k
p = ^i =
Here
+ ve
).
Cornu's Spiral.*
JL
The curvature continuously increases with s. Hence, as s increases,
the osculating circle at any point will contain the whole of the remainder
of the curve ; and p diminishes more and more slowly as * increases.
Fig. 126.
Negative values of ^ would give unreal values of s. Each value of ^
It is to be inferred that
gives two values of s, one positive, one negative.
the origin of measurement of s is a point of symmetry.
We have
x=\
y=\
r/
cos
sin
^ ds
j
^ ds =
I
cosks2 ds,
sin ks 2 ds.
These integrals are not integrable in general terms.
cosh2 ds =
/oo
/
sin Jcs-ds has the
j=
same
is
a
known
value.
result (Art. 1163, Ch.
XXV1IL), and
These are known as Fresnel's
Jo
*
Journal de Physique,
t.
Hi., 1874,
M. A. Cornu.
integrals.
CORNU'S SPIRAL.
567
Hence, when s becomes very large the curve* dwindles down to a point
after an infinite number of convolutions about the point.
on the line
yx
And
is
the point
infinite
is
at a distance
=
from the origin
and changes sign when
5
= 0.
There
r7f=
The value
of p
therefore a point of
is
inflexion there.
dx =
Also
-j-
which show that the tangent
,
is
when
and perpendicular
dv
9
cos ks*,
-79
= sin
ks*,
-jj-
parallel to the initial line
&5 2
= 0,
2?r ...,
TT,
to it
,
when
3?r
TT
7
*5 2 =-,
5;r
...,
,
which, indeed, is obvious from the equation
Taking k as unity for convenience,
lA
give
= 0,
1,
0-500
We are now
1-414,
1732,
0-354,
0-289,
in a position to
2
=
f
\f
.
2'236,
2,
0'224,
0'250,
oo,
6,
5,
4,
3,
2,
1,
= 0,
p = oo,
5
h
2'449
oo
...
0'204...
form an idea of the curve which
is
,
0.
shown
in the figure.
This spiral is of considerable importance in the theory of light, the
length and direction of the radius vector at any point giving a graphical
representation of the amplitude of the resultant of a system of superposed
vibrations.^
The values
of Fresnel's Integrals
n
C7=
v
f
I
JO
2
^
cosr-dv,
4
o
S=
7ry2
^
sm-GW,
Z
f"
I
JO
have been calculated for values of v from
to QO
by
Gilbert. t
The tabu-
lated values are necessary for accurate plotting.
The general methods of evaluating these integrals are discussed
Verdet
((Euvres, vol.
undid, des Lichts),
v.),
Fresnel (CEuvres, torn,
Cauchy (Comptes Rendus,
t.
by
Knockenhauer (Die
See
xv.) and others.
i.),
Preston, Theory of Light, page 220 onwards.
Incidentally the spiral exhibits graphically
the march of these
and the ordinate representing the integrals and s
being the independent variable, showing their oscillatory character.
integrals, the abscissa
Thus
x=
cos
ds increases from
5
=
I
5
= v/l
to
5
= \^2, increases
from
s
= \/2
to 5=\^3,
to 5 = \/I, decreases
and
so on.
And
fory.
These integrals will be discussed more fully
later.
*
Preston, Theory of Light, Art. 141, onwards.
t
Mem.
couronnfa de V Acad. de Bruxelles,
t.
xxxi., 1863.
from
similarly
CHAPTER
568
XVI.
561. Length of Arc of First Positive Pedal Curve.
Let
from the origin upon the
be the perpendicular
p
tangent to
any curve, and x
makes with the
initial line.
^ ne
angle this perpendicular
then regard p, ^ as
We may
the polar coordinates of a current point on the pedal curve.
Fig. 127.
Hence the length of an arc
by the formula
s'
of the pedal curve
may
be
calculated
562. Ex. Apply the above method to find the length of any arc of the
pedal of a circle with regard to a point on the circumference (i.e. a
cardioide).
Here,
if
2a be the diameter, we have from the figure,
Fig. 128.
Hence,
arc of pedal =
PEDAL ARCS.
569
The limits for the upper half of this curve are x =
Hence the whole perimeter of the pedal
an d X = 7r
-
= Sa.
563. Arc of the Pedal Curve.
Again, the tangent to the locus of F, the foot of the perthe same angle that the radius
pendicular, makes with
makes with the tangent at the corresponding point
vector
of the original curve.
OF
OP
P
dr
-r-,=^r\
ds
ds
dp
mi
Thus
which again expresses the arc of the pedal in terms of
elements of the original curve.
The
result
may
be presented in various forms.
Thus
which
Also
8 '=\~ds={rd\ls,
is
equivalent to
s'=
=
or
(1)
I
f
=
2
J
-=--5
^
2+
2
-?2
-dQ,
for pedal equations (6)
dr,
(from equation
2).
Arc of a First Negative Pedal.
curve be r = f(6), then
If the original
,
,
1K
(5)
forpolars
(S
rf
1
v/r
564.
for Cartesians (4)
dx,
:
J
or
for
,
l-i
(3)
are the polar co?-,
ordinates of the foot of the perpendicular from the pole upon
CHAPTER
570
XVI.
the tangent to the first negative pedal, whose tangential polar
=
equation may therefore be written p f(\)> X being the angle
the perpendicular to the tangent makes with the initial line
= x'-
vz.
ds
Also
s
.*.
=
-
-\-
I
-jj
/\
/(x)
d-% "+"
constant.
v
Fig. 129.
Ex. Find the intrinsic equation of the
565.
ellipse
-=l
+ ecos#,
with regard to the
d
Here
pole.
I
I
* + const.
-i
(1+ecosx)
If
we choose
to
measure
negative pedal of an
first
s
+ e cos x
2
from the point where x
the constant =
--
-
v 1 - e2
;
cos-1
1=0.
PEOBLEMS.
1.
Show
that the whole length of the curve
2
2/
2.
2
(a
-
y
Find the whole length
2
)
=8
A
2
TraJ%. [OXFORD
is
I.
P., 1890.]
of the loop of the curve
Say
2
= x(x- a) 2
.
[OXFORD
I.
P., 1889.]
PROBLEMS.
571
Show
that the arcs of an equiangular spiral, measured from
the pole to the different points of its intersection with another
equiangular spiral having the same pole but a different angle,
3.
will
form a
Show
4.
found
series in geometrical progression.
[TRINITY, 1884.]
that the length of an arc of the curve yn
= xm+n
in finite
terms in the cases when
^r
2m
or -
2m
h-
is
2
can be
an integer.
Evaluate the expressions,
5.
dx
f
v
(i)
'
I
y
J
,
.
(n)'
-j- ds,
ds
f
1
dii
.
-.
/ ds,
x
(in)'
ds
J
f /
(
x
-,2
~ - y dx\
d'li
J V' ds
4, -j-
)
r2 ds)
7
ds.
wherein the line-integrals are taken round the perimeter of a closed
curve.
6.
[ST. JOHN'S, 1890.]
If s
be the length of the curve
n
?
and 6 = 2-n-, and
between the origin
points,
7.
show that
Show
that
= a tanh
^
A
be the area between the same
A=a(s- air).
[OXFORD
I.,
1888.]
the arc of the curve
if
a
r
= a tanh n
measured from the
^-
(n being integral),
origin, be called
s,
and
if
A
be the correspond-
ing area swept out by the radius vector from the origin,
if
n be odd and
if
8.
Show
2A
^ giving
= a (s - r) = a (2s - a6).
that the length of an arc of the Cissoid of Diocles
?*
= a sin
2
<9
COS0
is
"^
a\J3(z- tanh 2) taken between limits O l and
2
where
2,
n be even,
n
the results for r = a tanh
>
CHAPTER
572
9.
Show
that the intrinsic equation of the semicubical parabola
3af = 2x*
10.
XVI.
3
is
95=4a(sec ^-l).
In a certain curve
Show
that
s
= A/2 +
Also that for the curve
Name these
11.
is
Show
curves.
that the length of an arc of the curve
xcosd-ysm0=f"(Q),
+ C.
given by s=/(0) +/"(0)
(F
12.
Trace the curve y 2 *= -=- (a - x) 2 and find the length of that
od
,
part of the evolute which corresponds to the loop.
[ST. JOHN'S, 1881
13.
Show
that the curve whose pedal equation isp
for its intrinsic equation s
What
14.
curve
is this
has
= a~.
on a plane curve are given by the
cos 6 + 20 sin 6 -
of a point
2
)
1],
2
# = a[(l -0 ) sin 6>-2<9 cos 0];
3sa* = (2a + p)(p- ofi,
p being the radius of curvature at the point and
from the
1891.]
= r2 -a 2
^2
x = a[(l-
prove that
AND
?
The coordinates
relations
2
s
the arcual distance
[OXFORD
origin.
II. P., 1888.]
A
The
meets the
evolute of a parabola whose vertex is
Find the perimeter of the
and the parabola in Q.
figure bounded by AC, the parabolic arc AQ, and the arc of
the evolute CQ.
[OXFORD I. P., 1889.]
15.
axis
in
(7,
16. Prove that the length of the first negative pedal, taken with
respect to the origin, of the loop of the folium of Descartes
x3
is
equal to
17.
3axy =
6a -a{ir -s/2log(\/2
Find the length
the curve
+f-
of the arc
C2
(
_a
-f
1)}.
between two consecutive cusps
= C 2(r2 _ a ^
2)^2
t
[OXFORD
I.
of
P., 1889.]
PROBLEMS.
573
2
that the length of the arc of the hyperbola xy = a
=
=
arc
of
the
curve
c
to
the
x
is
x
b
and
limits
between the
equal
= a 4 2 between the limits r = b, r = c.
[OXFORD I. P., 1888.]
18.
Show
/-
,
19.
By means
curve r =
e
a sin 2 -
formula
of the
s
T
f
=
I
di'
.
2
,
find the length of the
Js/r--/
.
[COLLEGES
20. If s
of the
be the arc of an
+
ellipse -^
a*
.
=
1
measured from the end
o^
major axis to a point whose eccentric angle
s
+ ae 2 cos
<fr
a, 1887.]
is
prove that
</>,
= T >/a 2 cos 2 d + 6 2"5ii^ d0,
sin
Jo
= tan"
where
Show
21.
1
\ tan
\0
(
V
</> ).
[COLLEGES
a,
1883.]
that the circumference of an ellipse can be expressed
either as
"
or as
4a(l
-
e2
)
f
(1
-
'
6
2
sin 2
<9)
*d0,
Jo
where a
is
the semi-major axis,
Show
22.
the eccentricity.
[TRINITY, 1887.]
that the three-cusped hypocycloid has equations of the
forms
(i)
(ii)
Show
e
p = b cos 3^,
r4 + Sfo-scos 3(9+1 86V = 276 4
.
that the length of an arc of the inverse of this curve with
respect to the centre
is
proportional to
tan' 1 (2>/2 sin 3^).
Prove that the
23.
1
rt = at cos =
D
where
s
intrinsic equation of the curve
n
is
s
-
a
5
5
5
2^
= THR
r+4
7 sin ^ + ^ sm Q
o
o2
J
16
.
\l-
.
,
and ^ are measured from the point
that point.
[ST. JOHN'S, 1887.]
i)
,
0,
>
and the tangent at
[ST. JOHN'S, 1889.]
perimeter S and area A rolls externally in its
an oval curve of perimeter S and area B.
round
plane entirely
Prove that its centre describes an oval of perimeter 2S and area
24.
3
A
4-
A
B.
circle of
[OXFORD
I.
P., 1918.]
CHAPTER
574
XVI.
25. Find the centroid of a sector bounded by two
and an arc of the curve whose polar equation is
r*
= a 2 (l -sin
and show that an arc
20)(1
+ sin 20)- 1
radii vectores,
,
of this curve is expressible as
5a
c
f
"2J
(lW5sin
[MATZ, Educ.
Times.']
26. A rod moves always to pass through a fixed point and have
one extremity on a straight line distant h from the point. Show
that the arc of the curve traced out by its centre of instantaneous
rotation, as the rod
inclined at 45
is
J{log(\/5
27.
On
+ 2) + 2j5}h.
P
the tangent at any point
to the radius vector of
P
;
show how
[MATH. TRIPOS,
of a curve,
PT is
verify the result geometrically.
Find the arc
of the curve
x cos
</>
+ y sin
</>
Find the whole area
any arc
and
spiral
[ST. JOHN'S, 1884.]
enveloped by the line
= (a cos 2 + b sin 2 <)~ 3
</>
between the points corresponding to
29.
1883.]
taken equal
to find the length of
For example, take the equiangular
of the locus of T.
28.
to one
moves from the perpendicular position
to the line,
<
= 0,
-=
-
<j>
.
[T JoHN s>
,
1891
-,
of the curve
x = asmO-bsm20,
y = a cos B
b cos 26,
and show that the whole length of its perimeter
an ellipse whose semiaxes are a + 26, a-2b.
30.
Prove that
if s
equal to that of
[COLLEGES
a,
1885.]
be the arc of the curve
r
= a sec
a,
jca,
a -a,
where a
is
~\
J
a variable parameter, measured from the initial line to a
on the curve, and if A be the area bounded by the curve,
is
point P
the initial
1
line,
and the radius vector to P, then
Find the area swept out in any portion of its progress by the
intercept of the tangent to the curve between the curve and the
first
positive pedal with regard to the origin.
[TRINITY, 1890.]
PROBLEMS.
31. If a curve be given
radius vector and
</>
by
the angle
m tan -
r2
it
575
= sin 2 + m 2 cos 2 <, where
'l
r is the
<
makes with the tangent, show that
(<
nO)
= tan <,
6 being the angle the radius vector makes with the
(which is to be appropriately chosen).
initial line
Obtain also a formula for the rectification of the curve.
result
32.
is
not obtainable infinite terms.)
33.
(The
c. S., 1898.]
Consider the nature of the curves
(i)
when
[I.
m< 1
5^
2
= a,
^ = sin-,a
(ii)
(Hi) s
m>
and when
i.
Given a closed oval
of continuous curvature without
A
a series of parallel curves
denote the area of any one of them and
is
the same for
singularities
= /sin?n^,
2tir
:
is
I
drawn.
its
Prove
any
-that
if
perimeter, then
all.
[I.
C. S., 1895.]
34. In the equation of the curve r = a + eu, a and c are constants,
the latter being small and u is a function of
finite for all values
of 6 and periodic, with a period 2ir.
denote the
Show that if
:
A
area of the curve, then
small quantities of the
35.
The area
of
an
its
first
length
order inclusive.
ellipse differs
by 10 per cent, of the area of the
of the ellipse differs
cent,
36.
Assuming that
from that of
Show
latter.
for the catenary
a parabola
C
when
common
Jc
by 4 '93 per
[I.
elastic
\ku + sh u + ^k sh 2u,
catenary
when
k=
2/
and approximating to
[B.A. HON. LOND., 1899.]
= x-.
Show
ott
finite integral
that the only curve for which
functions of
s is
a straight
[OxF.
38.
c. S., 1910.]
formed by a hanging
is large.
In the cycloid
x and y are
latter.
/
-=
reducing to the
C. S., 1896.]
its auxiliary circle
that the perimeter
of the auxiliary circle
v
prove that
both
from that
[I.
approximately of the perimeter of the
wire
37.
2*JirA accurately as far as
is
line.
I.
P., 1913.]
Find the Cartesian equation (choosing convenient axes
of
coordinates) of the curve in which
p>
2
= (dp/ds)* +
1
.
[Oxr.
I.
P., 1917.]
CHAPTER
576
39.
2
Find the intrinsic equation of the curve
that the involutes of the curve 27'ay
x = a tan 2 ^
y=
c
XVI.
27aj/
= 4# 3 are given by
2
+ c cos ^ - 2a
- 2a tan + c sin
^
Show
Prove
.
the equations
y
i// t
being an arbitrary constant.
What happens when c = 0?
40.
= 4z 3
[Oxi\
I.
that the length of a quadrant of the curve x*
P., 1915.]
+ y* = a?
3a
is
equal to -^, and find the length of one quadrant of the curve
K. TRIPOS,
41.
x(l+t*)
as
increases
t
PART
I.,
1910.]
I.,
1910.]
Trace the form of the curve
Find
from -
oo
= l-t\
to oo
also the length of
,
any
y(l+P) =
and show that
'2t,
its
area
is
TT.
arc of the curve in terms of L
[MATH. TRIP., PART
42. Show that the length of the arc of the parabola y = lax which
intercepted between the point of intersection of the parabola and
2
is
3y = 8xis
a(log 2
43.
Prove that the perimeter
and semiaxes
neglecting
44.
e7
a, b is
of
+ ^).
an
[MATH. TRIP. L, 1908.]
ellipse of small eccentricity e
equal to
and higher powers.
Prove that the length
of
'dS
- taken over the
area, where
[MATH. TRIP. L, 1917.]
an
dS
ellipse
is
may
be expressed by
an element of the area of the
P
ellipse
and p the radius of curvature of the similar, similarly
and concentric ellipse passing through the element dS.
situated
[COLLEGES, 1892.]
Find the intrinsic equations
= a< 3
cycloid, and trace the curves s
of a circle, a catenary,
45.
At any
,
S(f>
= a and
s
2
=
and a
a'-'c/>.
P
of a cycloid the tangent is produced to a length
point
to
the
arc
measured from the vertex, and at T a perpenequal
dicular is drawn equal to the radius of curvature at P.
Prove that
PT
the locus of the extremity of this perpendicular is the same cycloid
parallel to its axis through a distance equal to twice the
moved
diameter of the generating
circle.
[ST.
JOHN'S COLLEGE, 1882.]
CHAPTER
XVII.
RECTIFICATION
(II).
CENTRAL CONIC, LIMAQON, LEMNISCATE, TROCHOIDS,
ETC.
APPLICATION OF ELLIPTIC FUNCTIONS.
We
566.
have reserved for a separate chapter the considerwhose rectification needs the employment
ation of those curves
of Elliptic Integrals.
If
Arc measured from the End
Rectification of the Ellipse.
56*7.
of the
MINOR Axis.
be the eccentric angle of a point
we have
a cos
x
dx =
x,
y on the
y = 6 sin
,
a sin 6 dO
dy
,
ellipse
0,
b cos 6 dO.
y
Fig. 130.
Hence
ds*
and
= (a 2 sin 2 + 6
I
cos 2 0) dO 2
-e* cos 2
BP
from the end
gives the arc
P
on
the
curve.
point
E.I.C.
2
577
B
6)**
,
d0
of the
minor axis to any
2o
CHAPTER XVIL
578
e=
Putting
g- X
,
N/l
Jx
- e2 sin2 x dx = aE(x>
e).
(See Chapter XI.)
568. This integral is
Legendre's elliptic integral of the
not expressible in terms of the ordinary
circular or inverse circular functions.
But its value can be
second kind, and
found
for
for
E
e
and x
E.
Thus,
values of
specific
calculated
tables for
is
the
function
corresponding to e
fr
m
for
the
tables
instance,
the
\ give
#(10) = 17431
#(20) = -34733
=
#(40) =
(30)
-51788
-68506
Values extracted from
-84832
tables given in Bertrand,
(50)=
#(60) = 1-00756
#(70) = 116318
#(80) = 1-31606
#(90) = 1-46746
Calc. Integ., p. 7 17.
Hence, taking an ellipse with a 20-inch major axis and
eccentricity J, the arcs for eccentric angles 80, 70, 60, ... 0,
measured from B, the end of the minor
6-85,
5-18,
8-48,
10-08,
11-63,
13-16,
axis, are:
14*67
T74, 3*47,
inches to
two
places of decimals.
The student should
construct a quadrant of such an ellipse
on squared paper, and by careful stepping with dividers round
the perimeter verify this calculation approximately.
The
where
case
total perimeter
E
4x
of
the ellipse in any case
The circumference
i.e.
is
4>aEv
the complete elliptic integral. And in the present
1
14'6746 = 58'7 inches very approximately.
is
of the auxiliary circle
4-1 inches longer than that of the ellipse.
= 20?r = 62*8318,
569. Approximation.
an approximate value be required, we may expand the
e 2 sin 2 x, and in cases where the eccentricity is
radical \/l
If
small the series
is
rapidly convergent.
ELLIPTIC ARCS.
We
579
then have
and
For a quadrant the limits are
and the arc of the
-^,
quadrant
7T_1 6 9
2
2
of
1
7T_1
1
2'4
^'2'2
e4
3
1
The first three terms give
587 approximately.
570. Other
7T_1
'4'2'2
3
1
'
2
e
'
4
3
5
6
'
'
6
.6
1
'
4
7T
'
2
_
"
"
2
for the above ellipse a perimeter
modes of procedure
may
be adopted.
Cartesians.
Keeping x for the independent variable,
we have
dy__^x.
dx~
a2
'
Hence
a2
o
If
we now put
x*
= asinx, where x
* s>
as before, the
com
plement of the eccentric angle, this reduces at once to
.
v/1
571.
we
e 2 si
Jx
ueiure.
Taking the
.
central pedal equation
rdr
get
Putting
= a2 sin2 x + b2 cos2 x
r dr = (a2
b 2 sin x cos
a 2 + b2 r2 = a2 cos 2 x + & 2 sin2 x = a 2 (1
2
2
2 - 2 =
2- 2 2
b
b
sin2 x cos 2 x
(a -r (r
(a
r2
>
)
and
)
)
)
{xo
vl
;
2
e sin
2
x ),
CHAPTER
580
572.
Taking the
Putting
r
XVII.
focal p-r equation
= a(l -fesinx)
_
this reduces at once to
e2 sin2
\/l
Jxo
x c? v
,
as before.
573. It appears then that aE(x,
a\
e), i.e.
e 2 si
-s/1
represents the length of the arc of an ellipse measured from
the end of the minor axis to a point, on the curve, whose
eccentric angle
eccentricity
This
may
_
is -5
x>
the semi-major axis being a and the
(See Art. 567.)
be written as
e.
x
2
2
*Ja cos
where
l-\-m
=a
and
lm =
x+b
574.
March
2
^
sin 2
And
b.
recognise these forms at once,
senting an arc of an ellipse.
rectifications.
,
it is
useful to be able to
when they appear, as
They occur in many
_
repre-
other
of the Second Elliptic Function.
The form
s
=
rx
aj ^
</i
-& s in
2
x dx
an ellipse gives a very clear idea of the "march" of
the "second elliptic function" corresponding to any given
modulus e, and it is easy to construct a graph of the relation
between x an(l s by measuring off ordinates equal to the
arc of the ellipse and abscissae proportional to the comfor
plement of the eccentric angle.
Taking a=l, the
figure (Fig. 131)
shows the march of the
581
ELLIPTIC ARCS.
function for the values e = 0, which gives a straight line, viz.
J
and
e
= 1,
which gives
s
= sin x
,
the curve of sines.
s
1-7
1-6
0-2
0-1
10
C
20
30
40
50
70 C
60
90 X
80
Fig. 131.
It will be seen that for the first 15
so small that there is
ordinates
is
between
ordinates
$=26180;
for e
= J,
in
the
the difference of the
no appreciable
drawings,
s='26106; and for
in
fact
e=l
f
difference
for
e=0,
s='25882, for
X = 15, which only gives a difference of ordinate of '0030
between the greatest and least, and the curve s = aE(x) lies
between these extremes. There is much more rapid deviation
of
s=aE(x, sin^J from the curve s = sin^
after
X = T-
575. Arc measured from the End of the MAJOR Axis.
FAGNANO'S THEOREM.
Another method of proceeding gives the length of the arc
AQ measured from the end of the major axis, and incidentally
CHAPTER
582
XVII.
a comparison of the two methods establishes a remarkable
result with regard to the difference of two arcs, one
measured from A, the other from B.
known
This
theorem
is
Fagnano's theorem, being discovered by Giulio,
It shows that two arcs
Count de Fagnano (1682-1760).*
as
an ellipse can be found in an infinite number of ways,
whose difference can be expressed by a certain straight line,
and really establishes in a particular case the addition formula
of
for elliptic integrals of the second kind.
Take the
central tangential polar equation
2
z
2
2
2
p = a cos i/r + 6 sin \fs
'
,
being the angle between the perpendicular
tangent and the major axis we have
\fs
upon
the
;
s
i.e.
Let
Q
of contact, whose
of
the equation,
comparison
be the
obviously by
=
with the equation
point
/
a2
or cos
2
b sin
=-
P
Also
~-=QY,
this case
and p
is
Y
is
coordinates are
\lr
P
the negative sign occurring, because in
on the "forward drawn" tangent from Q,
diminishing as ^r
is
increasing.
*Cajori, History of Mathematics, p. 241.
FAGNANO'S THEOREM.
583
Also
\p
d\f,
= Va 2 cos 2 \/s + b 2 sin 2 d^ = a
\[s
J
I
-s/1
-e 2 sin 2 i/r <Z\/r,
the same integral as obtained in Art. 567 for the
arc BP, \Js being in that case a different angle, viz. the
complement of the eccentric angle of P.
which
is
Hence,
if
these angles be taken the same in magnitude,
\/l-e 2 si
1^
and
= a\|>J\ _ e
BP
arc
arc BP - arc AQ
Thus,
This is Fagnano's result.
2
Si
= tangent Q Y.
-
P and
576. Algebraic Relation between the Abscissae of
^ Tr
OY=
Ar
Now
dp
(a
vv = ^
Also the coordinates of
z2 =
and those of
P
arc -BP
This result
is
sm\/rcos^
62
y 2 = -smVA
.
\//-,
arc
,
y\
= b cos
e2
(or
AQ-- x^,
p ^^J
for e 2
symmetrical as regards x l
BQ
arc
AP =
,
\//-
a;
1
,
iz;
.
p
x2
,
and therefore
2,
of course, immediately obvious otherwise.
e2
-2^2= tangent P7', if 07' be the
tangent at P from 0. Hence QY = PY'.
Also
the
Q.
being
-cos^,
= a sin
arc
is,
Q
a2
Q Y = e*x2 ^
we have
as
I
.
being
ojj
Hence
a 2e 2
6 2 )sin\/r cosx//21=
2
perpendicular on
a2 -
CHAPTER
584
577.
that
The corresponding
between y l and y2
relation
is
is
where
e'
XVII.
being the
578.
=1
?+?a
"
"
imaginary
THE FAGNANO
eccentricity.
POINTS.
It will be noticed also that
"
a3
Hence, at the point
coincide
when
is
F
'
63
AB
on the arc
which
at
P
and
suitably chosen,
x
*
22
"
t
a 2 "6 2
1
and the coordinates of the point are therefore
and
this is called the
"
Fagnano
Point,"
* for the first
quadrant.
579. Properties.
At
this point F,
arc
n
BFa,rc
t~t
n
AF=
A
=
"
-
^
a3
a
= the
-
u
=-
a +6
=a
7
6
difference of the semiaxes.
And the length of the projection of the radius vector
the tangent at
is also =a
b.
2 2
O, e SIH \JS COS \lr
rorv mi
e
^Tr
y
r
580. The expression for OF, viz.
F
OF
x/'
be written as
^,
on
may
'
2
2
\/a cosec
^ +6
and therefore
a
QY
6.
QY
2
sec 2 \/r'
V(a +6)
zero to a
maximum
is
value.
b in travelling
+ (a cot
maximum when
attains its
The Fagnano point
has a
2
from
x//-
6 tan
2
\/r)
V1Z
tan i^ <=*/=-, vi
6'
'
therefore the point for which
varies continuously from
QY
B
or
A
to F.
* Greenhill's
Elliptic Functions, p. 178
onward.
FAGNANO'S THEOREM
we
581. If
seek for a point
585
the quadrantal arc
AB
OQ upon
the
Q upon
of an ellipse such that QY, the projection of
tangent at Q, is of given length I, where
0<<a
two
will be
solutions, viz. the points
P
and
6,
there
whose positions
Q,
by the equations
are given
an(j
p
being the radius vector to either of the required points,
OP or OQ.
viz.
r
y
Fig. 133.
Eliminating
p we have
(1)
with roots
rx 2
,
r2 2
,
such that
(2)
and equal roots when
we
If
l
we
r
viz.
Z
call
j-
2
.
5P,
s 1?
^ri + r 2 ^r 2 = ldl*
and
= projection
J3Q, s 2
,
and remember that
of radius vector
on the tangent,
........................ (3)
is
a constant.
Taking the case when
= a 2 +Z2 and therefore
,
so that
/.
ab-\-b
in both cases,
where C
r22
b and r 2 = a 2
differentiate equation (2),
ri
If
=a
Q
is
arc
at
A
;
then
r1
r2
^ = 0,
= b,
s2
is
=a
P
at B,
and
l
= arc ^4 J3, 1 =
5P+arc BQ = Z-f arc J5^4,
*See Bertrand,
that
must
t.e.
arc
5P
CoZc. 7n^y., p. 380.
Q,
we have
for r2
^> a,
simultaneously
arc
^Q =
I,
;
CHAPTER
586
XVII.
Fagnano's result, and the points P, Q, in which
must be divided to give a definite value I for' Q Y
which
is
the arc
AP
',
are determined
by equation
(1).
EXAMPLES.
1. Show that if coaxial ellipses be drawn with a given centre such that
the areas enclosed between them and their respective director circles is
constant, the locus of the Fagnano points is a circle of the same area.
Show
2.
Show
3.
Fagnano points for similar and similarly
a pair of straight lines.
that the locus of the
situated concentric ellipses
is
that the locus of the Fagnano points which
lie
on confocal
ellipses is
2c being the distance
Show
4.
that
if
between the
F
foci.
be the Fagnano point on an ellipse of semiaxes
OA = a, OB=b,
2a,rcAF=aEl -a + bJ
E
where
5.
l
is
Show
point
is
the complete elliptic integral of the second kind
that the central perpendicular upon the tangent at a Fagnano
mean between the semiaxes, and equal to the semi-
a geometric
diameter conjugate to the radius to the Fagnano point. Further, that
the radius of curvature at this point is also equal to the perpendicular, and
that the normals at the corresponding point on the evolute pass through
Finally, that the arc of the evolute is at such a point
divided in the ratio
jfa$
the centre.
.
LM
6. Show that if a straight rod
of length a + b slides with its ends
on two axes Ox, Oy at right angles and carries a point
whose distance
from L and
are respectively a and 6, which thus describes an ellipse,
F
M
then at the instant when
LM
is
Fagnano point on the described
F is a
LM for
tangential to the path of F,
ellipse, and the circle on
diameter passes through the point on the normal at
touches the evolute.
7.
Show
ellipse
that the tangents at the points
+s
Ji
F where that normal
P(xlt y^
Q(x%,
which are related to each other so that
intersect on a confocal hyperbola
3/ 2 )
on an
^='r
2
which passes through the Fagnano
points.
[Many
properties of these points will be found in Greenhill's Elliptic
Functions, pages 182, 183.]
TAUT CORD ENCIRCLING AN OVAL.
587
582. Properties of the Locus traced by a Pointer which pulls
taut an Inextensible String passing round a given Oval.
Taking the case of any oval curve, let A be the point from
which s is measured; PQ, P'Q, the tangents at contiguous
*,
J9L
Fig. 134.
points
(s,
>//)
(s+Ss,
\Is-\-S\js)
of the oval;
and
let
a length
PQ = be measured upon the forward drawn tangent at P,
Let the tangent to the
P'Q' = t+St upon the tangent at P'.
with
the
an
locus of Q make
tangent at P to the oval.
angle
Draw QN perpendicular to P 'Q', and let the arc QQ' = So:
t
<j>
Then, to the
first order,
t+St+Ss = * cos S\!s+NQ'
and
= -\-Sar cos
$+& = cos0&r ............................... (1)
t
/.
<
;
t'+&' be the other tangents from
QR, Q'R' of lengths
which
can
be
drawn
to
the oval, and s', s'+Ss' be the
Q, Q'
arcs APR, APR' respectively, and if
be the angle which QR
makes with the tangent QQ' to the Q-locus and S\}s' the
difference of the angles of contingence at R, R', we have in
the same way, Q'N' being the perpendicular upon QR,
If
',
<$>'
Q'N'=t'W
t'+Ss'=
to the first order
,
So-
QN'= 8cr cos 0',
cos <f>'+t'+8t'
t
;
/.
&'-&'=- cos 0' &r ...................... (2)
CHAPTER
588
xvii.
(Hocus be such that the tangent at Q always
the exterior angle between the tangents from Q to
If the
bisects
the oval,
<
=
<j>'
QN -=Q'N' = Sa- sin
$+&+$' &'=(),
and
Therefore
and
to the first order.
<
\
tS\ls=t'S\ls')
These equations give
= P_
t_.
tdt'd'~t
P
f
t'
e
and
also
s'=constant ...................... (4)
t+t'+s
Equation
(4)
expresses that in such case
QP+QR-wc PR= constant,
<?P+G#+arc PAR= constant.
i.e.
In this case the Q-locus is an oval traced by a pencil at Q
which draws taut a loop of string placed round the original
oval.
583. DR. GRAVES'S
The
case
when
THEOREM.
the original oval
is
an
ellipse
and the
Q-locus is a confocal, when the necessary property holds,
viz. that the tangent to the Q-locus bisects the exterior angle
between QP, QR, gives the well-known theorem due to Dr.
Graves,
If
viz.
two tangents be drawn
a confocal
ellipse,
to
an
the excess of the
over the intercepted arc
is
ellipse
sum
from any point of
of these
two tangents
constant.*
Incidentally, we have a
confocal to a given one.
method
of
drawing an
ellipse
584. If the C-locus be such that its tangent bisects the
interior angle between the tangents QP, QR, as it would
do in the case of an ellipse and a confocal hyperbola, and
we measure
if
*
s
and
Salmon's Conic Sections,
p.
s'
357
in
;
opposite
directions
from the
Graves's Translation of Chasles's Memoirs.
THEOREMS OF GRAVES AND MACCULLAGH.
point A, where the Q-locus meets the oval,
we
589
have, in the
same way,
QN= So- smcj>=td^,
QN'= So- sin <j>'= t' <fy/,
NQ'=
N'Q'= So- cos 0'
and
So-
cos
,
t+8s+S(rcos<l>=t+St,
;
),
fto the first order
,
;
t'+Ss+So-cos <j>'=t'+St' j
and when $=$' we have dtdt'=dsds', and
}
,
dlot
sothat
and
also
^
dlot'
s=^
s'+const.
x
also, as
Z,
i
,
QP
arc
;
vanish at
8 =t'
f
tangent
all
s'
5,
s
^4,
,
AP= tangent QR
arc
AR.
Fig. 135.
MACCULLAGH'S THEOREM.
For the case of the ellipse and the confocal hyperbola,
where the condition $=0' is necessarily satisfied, we have
the following result.
If tangents
QP,
QR be drawn
from a point
Q on
a hyperbola
to a confocal ellipse cutting the hyperbola at A, the difference
of the tangents is equal to the difference of the arcs AP, AR.
This theorem
is
due to MacCullagh.*
* Salmon's Conic
Section*, p. 358
;
Chasles, Comptes Rendus,
Tom.
xvii.
CHAPTER
590
XVII.
585. Deductions.
we draw tangents to the ellipse at the extremities of
the axes, the particular confocal to the ellipse which passes
through the corners of the rectangle formed cuts the ellipse
If
in the
Fagnano
points,
and
if
Q
be the intersection of tangents
Q
Fig. 136.
F
A and J5, and the point in the first quadrant
confocals cut, MacCullagh's theorem gives
at
where the
QB- QA = a,rc FB-a.rc FA,
and
if
b, we have
a,rcFBa rcFA = ab,
the semiaxes be a and
>
which
586.
Qlt Q2
is
Fagnano's
From
result.
the theorem of Dr. Graves
it
appears that
be any two points on the confocal and
Q 1P
1
,
if
CA
J
Fig. 137.
Q2P2
,
Q2R2
are the corresponding pairs of tangents to the
original ellipse,
-arc
P& = Q P +Q R -wc P R
2
2
2
2
2
2
;
THEOREMS OF GRAVES AND MACCULLAGH.
and therefore that the
difference of the arcs
591
P^!, P^R^
^s
and is therefore rectifiable in terms of known lines.
The particular value of the constant to which
QP+QR-arcPR
is
be found by taking Q at a specified point on the
it cuts the conjugate axis.
where
e.g.
And a similar result follows also from MacCullagh's theorem.
may
equal
confocal,
587. Exactly in the same way, if Q be a point on the ellipse
and QP, QP' be tangents to the same branch of the hyperbola,
it
will be clear that
QP- wcAP=QP'-
arc^P',
for the tangent at Q still satisfies the requisite condition, namely
f
that the internal bisector of the angle
is a tangent
PQP
Fig. 138.
to the ellipse.
And the difference of the arcs AP, AP' is
therefore expressible as the difference of two straight lines
and
is
ellipse,
rectifiable.
Moreover,
such that tangents
QP
1
1
be another point on the
Q-fi can be drawn to the
Q
if
,
1
same branch of the confocal hyperbola, the
the arcs
PPlt
difference
of
In order that the point Q
P'P/
should be such that tangents can be drawn to the same
branch of the hyperbola, such point must obviously lie in
one of the regions between the asymptotes in which the
hyperbola
asymptote,
lies.
the
is rectifiable.
In the limiting case in which QP is an
difference of the infinite portion of the
CHAPTER
592
QP and
asymptote
QP
f
is
finite
of
at
the
intersection
the
of
and equal
Q being now
and the arc AP',
the difference
of
AP
the infinite arc
to
point
XVII.
asymptote with the
ellipse.
588. Rectification of the hyperbola
Let
G
an arc
be the centre,
AP
GA
measured from
perpendicular
-|Ar|rj=l.
the semimajor axis, s the length of
A in the first quadrant, CY the
p upon the tangent
at P.
Fig. 139.
=
Then
touches the curve
p*=a? cos \!/b sm \Is=a
2
2
2
z
if
e sin 2
2
(l
\]s).
In the case of the hyperbola, when P lies in the first
quadrant, \//~ is the angle xCY and is negative, and as s
increases from
to oo whilst P travels along the arc from A,
Y
from
travels
curve
r2
=a2 cos2
Bernoulli
A
towards
C
6 a sin 2 0,
which becomes a Lemniscate of
when b=a,
The angle
^
i.e.
when
along the
first positive
the hyperbola
therefore remains negative,
pedal
is
and
rectangular.
as its actual
magnitude is increasing \/r is algebraically decreasing and
an increment d\ls is negative.
When P has travelled to
oo
of
the
curve
this
the limiting position
branch
along
The tangents at the node of the
of YP is an asymptote.
pedal are therefore the perpendiculars to the asymptotes of
THE HYPERBOLA.
the
hyperbola, coinciding with them in the
rectangular hyperbola and
its
pedal
which \^=
We
x
have
=a2 cos 20.
AP from A
case
of
the
r2
Let us find the length of the arc
for
593
to a point
P
.
an d
TT'P-\-TT<>
f
2
d\}s
tjr\
d^
d\js
therefore, integrating,
f^
o
Now = -jj
t
the tangent
pd\Js.
the projection of the radius vector
is
=PY, and
is
_
f-x
\Js
cos \Is_ae 2 sin
__
Jo
o
arc
.'.
4P=Pro
2
sin v cos v
f
= ae
~TT=^ ^~ a
Vl
e2
==
2
sm x
x
\
x cos x
rx
e2 siu 2 \Isd\Is=
vl
pd^fapal
f^
upon
positive.
2
ae sin
OP
>
-
\/l-e 2
Jo
a\
Jo
sm 2 v^v,
PF-arc AP=a[*Jl -e 2 sin 2 x dx .................. (1)
Jo
"V,
not of the Legendrian form at present, e being
essentially greater than unity.
If P be allowed to travel to oo
x ultimately becomes
This integral
is
,
- ,a
1
^
o
Hence the excess
infinite arc ^4oo
TT
-
(.
(
i.e.
2
\
1 tan" J)\
)
of the infinite
.
a/
asymptote Goo over the
is
n
r
_1 a
6
<\/l
e 2 sin 2
<
It is easy to reduce the
x ^X-
integral in equation (1) to
of
integrals
Legendre's standard form.
Let esin x =sina>.
E.I.C,
2r
two
CHAPTER XVH.
=e(
cos 2 a
/T
Jovl
.
si
6
where
cota=-,
a
and a
is
.
e2
*.e.
^a
2
2
+6
= =cosec 2
a,
aj
the complement of the half angle between the
asymptotes.
Hence,
Arc AP=PY+ae[cos* a F(t,
E(to> sin a)],
sin a)
and E being the Legendrian standard integrals of the first
and second species, whose values are tabulated for particular
F
values of the modulus sin a,
a>
1
being sin-
upper limit and PY, written in terms of
ma
sm
where
i.e.
tan a)
^/i
sin 2 a sin 2
= A/1
V
Arc=ae{tanft>A+cos
w,
2
^siu
2
a-F(ft>,
589. In a rectangular hyperbola
sm
in the
)
a/
being
w = ae tan CD A
6
(x
(
\
Mod. - )
,
e;
co,
sin a)
=T>
,
E(w,sma)} ....... (2)
e=\/2, and
we have
CENTRAL
CONICS.
595
EXAMPLES.
1.
2
2
=
=
^=1, put a 6tana, A \/l -sin asin <, and
x=b tan a sec A, y = b cos a tan <, and that
In the hyperbola -|
show that we may take
<
b cos a
ds
and
2.
5
= 6 sec a tan <A + 6 cos a /^(<,
From
the polar equation r
rectangular hyperbola,
3.
If
PQ be
6
=
2
=a
2
sin a)
- 6 sec a
^(^
sec2# deduce the
sin a).
rectification of the
viz.
a chord of one branch of a hyperbola, touching a confocal
and the confocal cutting that branch of the hyperbola at A
and B, and if PR, QS be the other tangents from P and Q to the ellipse,
show that the elliptic arcs AR, BS exceed the elliptic arc AFB by the
excess of the tangents PR, QS over the chord PQ, i.e. that
ellipse at F,
arc Jfl + arc
is rectifiable
in
terms of known
BS-wc AFB
lines.
In particular, examine what happens
(1)
(2)
(3)
When Jf^is the vertex
When F is at B.
When PR and QS are
:
of the confocal ellipse.
at right angles to
PQ
and
F the
vertex
of the ellipse.
590. Another
Method
of Treatment for the
Use of Hyperbolic Functions.
In the case of the central conies
it is
Central Conies.
instructive to consider
another mode of treatment of the rectification.
The
x + iy
relation
gives
x=c sin u cosh v
Then v= const,
is
u= const,
is
y=ccosusmh v
,
the equation to the ellipse
c2
and
c sin (u -f- 1 v)
cosh 2 v
c2
sinh 2 v
the equation to the hyperbola
c 2 sin 2
u
c 2 cos 2
and different constant values of
and hyperbolae,
v
u
and u give confocal
ellipses
CHAPTER
596
dx
Now
c
XVII.
= cos u cosh v du + sin u sinh v dv,
=
u sinh v du+cos u cosh v dv.
sin
c
Hence
c
2
2
2
2
2
2
=(cos u cosh v-J-sin u sinh v) (du +dv )
= {(1
sin 2 ?/) cosh 2
v+sm z u (cosh 2 vl)}(du 2 +dv 2 )
sin 2 u) (du 2 -{-J.v 2 ).
2
=(cosh v
Hence, for any of the family of the ellipses
_ =Vcosh
2
sin 2
v
u du
(i
c
=const.)
and for any of the family of hyperbolae
c
sin 2
=N/cosh 2 v
where
and
e is the eccentricity,
And
ds
2
e sin
a\/l
2
u= const.,
x 2/a 2 +y 2 /b 2 =l,
b=c sinh v
,
;
u dv (w=const.).
591. In the case of the ellipse
a =c cosh v
v= const.,
.:.
a2
c2
,
b2
=a2e2
,
e=sechv.
u du,
Cu
s
=a
I
.-/ 1
e 2 sin 2
udu=aE (u,
e).
Jo
In the case of the hyperbola
a=csmu,
\Js
c2
and
b=ccosu,
With the notation
x 2 /a 2,-y 2 lb 2 =I,
=a 2 +6 2 =a 2e 2
of Art. 589, in
=
x,
sin
x
,
e
cosecw.
which
sin
u sin
o>,
we have
cos
The
x = Vl
line
- sin 2 u
sin 2 o>
= A and
a?cos\^+i/sm\^=p
A2 -
is
2
p
c sin
= PY=c tanw
A.
tangential, provided that
c c
/.
t
u cos w.
BERNOULLI'S LEMNISCATE.
P is given by
--6 sin
= c sin u A sec
y
The point of contact
a 2 cos\!s
2
.
x
597
x//-
CD,
and, as these are to be c sin
u cosh v,
c cos
=;c cos2 u taxi
u sinh
v,
we have
sinh v = cos u tan w.
=
cosh v dv cos u sec 2
cosh v =
It follows that
A sec
o>,
u day
cos
,
i.e.
A cos a)
Again,
v A 2 sec2 co
sin 2 u
sin 2 u
= cos u sec on
C
Hence
-
=
c
*
2
- sin 2 it.
|\/cosh v
dv
J
2
= COS92 U fsec-r w aft)
,
I
= A tan w -J-cos2 uF E
by Legendre's fourth formula,
(mod. sin u)
399
p.
;
1
,
sinu)aeE((0, sinw),
the same result as before.
592. The Lemniscate.
The equation
we have
r2
is
at once
-
= a 2 cos 2#
;
W
;
^=
tan
T (Hj
ds
^
whence
a
.
dO
ox/cos 20'
Put
cos 20 = cos 2
sin
,
<A
:
^ cos
ocos^\/l
.
f*
d?0
=
Jo\/2
^A
fo
^Jov/l-isin 2
or
n = sin
----Y
sin
c?0
cos 4 <^
f*
f*
/.
v/2
= am- 1
-s/2
d<f>
si
co,
CHAPTER
598
XVII,
=
am^
a
Hence
= cos = -
en
Hence
s
a
= -7=
,1
r
en" 1 -
^
mod.
,
x/2
Here
.
a
a
.
x/2
measured from the vertex.
from the beginning
might have expressed
and then
2
5 is
We
of
in terms
-
r,
a2
ds_
~
_
the work proceeds as before.
then putting r = a cos
For the whole length of the arc, we have
</)
JT
4a
f^
(Z0
I
/
1
-j=
The
=^=
1
= 2av2 ^j,
tables for F^ (Bertrand, C.L
p.
whole arc = 2o>/2 X 1-85407
whence
We
.
mod-
-j~
716) give
^ = 1-85407,
= a X 5-2441.
might, however, proceed as follows
:
dO
\/cos
Putting 20=o),
26
we have
o
It will be
shown
later (Art. 872) that
sm mr
where w
is
less
present purposes,
than unity.
Borrowing
this
theorem for
TT
Sin
4
=*,
say.
BERNOULLI'S LEMNISCATE.
The values
599
of the T- functions are calculated.
Tables of
these values are given in Bertrand's Calcul Integral, pages
285, 286, to seven places of decimals from Log T(l) to
LogT(2).
all
As the values
fractional,
10
is
of T(x) from T(l) to T(2) are
added to their ordinary logarithms for
convenience of tabulation, as is usual in tables of logarithms
of sines and cosines.
(See Chambers's Mathematical Tables.)
NOW
r(t)=iim
and
Lr(i)=Lr(f)+log4;
where
L
denotes the tabular logarithm,
9-9573211 from the tables of
=
+
L T(x).
-6020600
log 2
=F "3010300
10-5593811
lQg*"
=' 4971499
21ogr(J)= 1-1187622
- -7981799
3990899
log <s/2^= -3990899
\ogk=
-7196723
log 5-2441=
-7196710
13
Difference for
1
=
8
50
50
Hence &=5'244116.
Hence the whole perimeter
593. Incidentally,
it
r*=a2 cos 2$ is, as before,
5-244116 x a.
may be remarked that the equation
of
Fig. 140.
for a lemniscate gives a very
good idea of the graph of the
functions en and cn -1 for the case mod.
-j=
\/2
t
and we can readily
CHAPTER
600
draw a graph,
XVII.
= on
taking, for instance, as unit length
the X-axis, and any convenient unit on the ?/-axis, say
constructing the curve with abscissa s and ordinate r.
a,
and
Fig. 141.
The ordinate shows the march
march of cn~ l x.
of the function cnx, the
abscissa the
EXAMPLES.
1.
Find the length
of
the arc
a lemniscate r2 =a 2 cos20 from
of
e.oto-|.
Here
CL
and from the
IT
dtp
i
tables for F(<}>,
-j=),
VI -
sin 2 <
*
o
^J"
i
i
i
(Bertrand, Calcul Integral,
TT
p. 716.)
.= -82602;
-41301
= -5841a.
2.
Find the area of the curve y2 = _
for the portion in the first
.
quadrant. What connection is there between this problem and the
evaluation of the perimeter of the lemniscate ?
3. Draw a careful polar graph of the lemniscate r =25cos2$, taking
one inch as unit of length, and deduce a Cartesian graph of
2
?/
4.
Show
= 5cn W2
that the difference between the lengths of the asymptote and
^2 /a2 - # 2 /6 2 = 1 in the first quadrant is
the infinite arc of the hyperbola
_irar\
^~ S ~
2 L2*
1
l.l 2
1.1 2 .3 2
1
1.1 2 .3 2 .5 2
1
1
+ 2 2 .4-e3 + 2 2 .4^6e6 + 2 2 .42.6 2 .8V+-"
e
,
,
PASCAL'S LIMACON.
594 The Limacon
Here
/.
^--6 sin
601
r=a+bcos9.
(9
2
and
(^=a +2a&cos0+&
2
;
s= Va2 +2a&cos#+62 d0
Jo
(Let 0=20.)
4a&
f
where k*=,
=2(0+6)1
An
obvious modification will be necessary
if
v
&
2
>
a and 6 be of
opposite sign.
This curve very well illustrates the march of the second
The arc AP measured from the vertex
elliptic integral E.
142.
is
For the case a
proportional to E, whilst
</>
is
>
6.
half the angle
AOP.
See
also Art. 574.
The
to
result
shows that the arc AP of the limacon is equal
an ellipse of semi-major axis 2(a-f&) an d
the arc of
eccentricity
-^, measured
from the end of the semi-minor
a+b
axis to a point on the ellipse for
which the complement of
/a
the eccentric angle
is
= (compare Art. 573).
The semiaxes
2i
the ellipse in question are then 2(a-{-6) and 2(a
6).
of
CHAPTER
602
XVII.
This would also be evident upon writing
f
Jo
z
Ja?+2abcos0+b dO
a ~ 62 sin2
2
as
2
!a-f-6)
595.
cos 2
d6
2
2
6) sin
0+(2a
where
dty,
Ex. Consider the case of the lirnagon in which =
^
=
portion from #
to
=
?!* = -?=,
Here
0=20.
for the
-j=
.
"d
^
*2
*
^"
'
'
7T
= 8a(2-\/3)x-51788,
= 1-11012 x a.
The limacon
the cardioide
is
is
from the tables for
of course the focal inverse of a conic,
E(<f>,
\\
and when a = b
the inverse of a parabola.
596. Trochoidal Curves.
(See Diff. Calc.,
344)
p.
If a be the radius of the fixed circle, b that of the rolling
circle
and the carried point
P
be at a distance
mb from
centre of the rolling circle,
x=(a+b)
cos
y(a-\-b) sin
Hence
9mb cos
j
9mb sin -^o
-^= - (a-\-b) sin 9-\-m(a-\-b) sin
ctu
^L=
Let
(a+b) cos 9
ad
m(a -\-b) cos
x
26=2+^
0,
9.
=
o
j
0,
9
;
the
THE TROCHOIDS.
s=
Then
where
i.e.
s is
603
ex
07,
2
a
>/l-Jk sin x^X where
^ (a+b)(l+m)\Jn
measured from the point at which
x~>
i-
*=
e
-
@
T+m'
=
from a vertex V, as in the case of the epicycloid (Art.
>
540).
Fig. 143.
Hence again we can find the length of any desired portion
by means of the tables for Legendre's elliptic integrals of the
second form or, which comes to the same thing, such length
;
can be expressed as being equal to the corresponding arc of an
ellipse, measured from the end of the minor axis, the semiaxis
major
being
.
a=
--
,
-
a (a+6)(l+m),
the
eccentricity
and x being the complement of the eccentric angle
at the end of the elliptic arc.
For a
circle,
when
wi
= 0,
-
For the
being
epicycloid,
t
=
when
wi
-const.
= l.
(a + b) sin
x = -^
which agrees with the result of Art. 540.
(
+ 6) cos |g + const.
CHAPTER
604
We
might use
XVII.
this curve, like the ellipse
showing the march of
to construct a graph
and the
li
"
P
1
Jo
for
any modulus
I+m
597. The Cassinian Oval.
The
bipolar equation of this curve
Art.
Gale.,
458.)
2
is
r^r^b
if
the line of foci be
.
(See Diff.
Fig. 144.
If
Slf S2
be the
foci,
taken as z-axis and
polar equation
its
2
=2a, and
as origin, the equivalent
centre
is
Three cases arise
(1)
^
a>6, two
:
separate twin ovals with vertices distant
\/a
2
-& 2
from
0.
(2)
a=b, reducing to Bernoulli's lemniscate.
(3)
a<6, one
single oval lying outside
which may or may not possess
The equation may be written
r*
Take 'an auxiliary angle
M
6'
such that
^=26
2
cos20'.
the lemniscate,
inflexions.
OVALS OF CASSINI.
=a2 cos 20 +6 2 cos 20',
r2
Then
^^^a
2
cos 20 -6 2 cos 20';
a*-6*=a4 cos 8 20-6* cos
.-.
a4 sin 2 20=& 4 sin2
or
i.e.
605
the auxiliary angle
0' is
20',
such that
a 2 sin20=& 2 sin20'.
we have
Differentiating the original equation,
r
rdO_
dr~
\
2
_V
~
2
-a
2
cos 20
a 2 sin 20
1
1
2
2
4
a sin 20
C
-
&2J5HP
_
~a2 J
as_b
.
7
_^
dr
a
IZv
We
a
w=cos20,
shall
>
or
Let
A
is
<
adopt the
than 6.
tt
g
,
first
^=00820'.
or the second forms according as
(<&);
=cos2a, where
~2=sin2a;
a
where
-sin2^.
A4_|
<
b
u=cos 20=
,
A
I
,4
2 2
+ A-=:V2^+X,
I_ X ^V2V^\;
a
/;
F&+* dr
r
Vr=?'
b
,
In the case a
20
(
OY*
Jr
b~~ai r
where
sin
sin 20
r^tf+v
;
r
,
CHAPTER
606
'
6
1
b
r-
rr
if
2v
2UW(l-
6T
./sing
sn" 1 \cosa
/
=~
XVII.
!
2L
where sin2a=^
\
,
cos a
./sinfl
+sn-1 (
)
Vsma
.
sin a
\~|
/J
(Art. 388,4).
In the case a > 6
v
,
= cos 2$'=
f4
4_ a4
I
and the work proceeds precisely as before, interchanging
and 0', X and p a and /3 on the rightb, u and v
a and
,
hand
,
,
fl
side of the values of
o
-=-
.
b
a
a
/
e =4sin- 1 (sin2^sin2^)
where
The arc
in both cases
is
and
sin2/3=--p.
measured from the vertex, where
598. In the case of the Lemniscate,
a=b
then
= 0',
r2
,
=2a 2 cos 20=c 2 cos 2(9,
say
;
and either case gives
-^^/^
=_L en599. It
first
is
1
f-
,
72
4=Y
as in Arfc 592
-
-
a very instructive process to perform the
have
in terms of r.
We
expressing
,.
2
[(a
+6
2 2
)
-r
4
4
][r
- (a - 6
2
2
_
same
rectification
OVALS OF CASSINI.
/a 2
Let
+ 62 u and
^^ 2
607
,
&2
the positive value to be taken.
sin
2(9-(a
and
2
+ 6 2 )\/(l - u*}(u* - A 4 )/2a%2
,
dr
/.
s
=
Again,
[(1
+ A 2 ) ?* 2 - (u* + A 2 )] [(1 + A> 2 + (u* + A 2 )]
where
This transformation gives
1
Now
au integral of form
/=
2
,
can be converted at
once into the standard Legendrian form as follows (Art. 388, 4)
Put
+A
:
CHAPTER
608
XVII.
Then
2 (1
=
_ 3 /-*
_
<ft
cos
(ft
d<f>
^
N/2-(l+c)sin
Jo
+ c) sin
2
4>
d<f)
f*
V^
and
as in our case
c=,
1
is
positive
Hence,
2A
c-o,
+ A.*
and
less
cos<
= cn(//<s/2)
finally,
it is
than unity
numerically less than unity and
;
and
/=N/2cn~ 1
'V^-
we have
2;
1 en
~
l
the respective moduli being
V2 V(a 2 + 6 2 ) + (a 2 - 6 2 )
V2 V(a 2 + 6 2 ) + (a 2 -
For the twin-loop curve a > 6,
+ en
OVALS OF CASSINI.
609
with respective moduli
2a
2a
For the single-loop curve a<6,
r
en"
1
{
,
x/6
2
+a + V6 -a 2
2
2
_
x/F^
+ ^5Z?
+ en""
1
'
,
V6 2 + a 2 -
4
I
I
VF^J
,
with respective moduli
J
26
26
600. The expressions written in this rectification are less simple than
when written in terms of 6, as in Art. 597, but can readily be reduced.
62
In the case a>6,
Also
cos2a =
let
sin2a = ~2; then r 4 -2a 2 r 2 cos2# + a 4 cos 2 2a=0.
a
.1'
--4r,
Vl
'
sma =
/
a
cosa
4
-6
4
(V
,/a' +
2a
:
2a
\
t'Wa'-J
<7
=
Cn "'
2
cos2a
2
cos a
= en"
vcos 2a + cos 2^
1
,
f=
V2 cos a
2
2
= cn _, \i/cos a-sm
= sn _./sm0\
5
2
l
l
V
cos
).
a
Vcos a/
Similarly,
cn"
1
Hence a>6,
s
fsn- !- -, cosaj + sn- )^
2aL
\srna'
\cosa'
/
1
1
,
sina)
/J*
as before.
Also for the case
a<6,
since
M_
4
'
(Art. 597),
E.I.C.
2Q
,
CHAPTER
610
=4
r+
2
2
(cos
sin 2 6'
Similarly,
cos
= sn~
en
2r6 cos
601.
5
Sm
1
~, cos
(
\cos
/3
/3'
0'= Jsin~
Serret's
'
;
)
/sin 0'
o, an
cos
1
8
sin
-M"'^^/ ^"''(ili^'
2
where
2
/2-sin 0');
b 2 cos 2/3
26 cos/3
;.
XVII.
1
\sm
&
(sin 2/3 sin 20), the result of Art. 597.
Method
of Rectification of a Cassinian.
A
different method of rectification of a Cassinian Oval
given by Serret* connecting two arcs measured from
different vertices of the curve, and expressing these arcs
is
directly in terms of
0.
In the twin-oval case
one of the ovals, and
a>b,
let
let
A
B
and
be the vertices of
a radius vector
OQP
be drawn
Fig. 145.
cutting that oval in Q and P.
furthest from the centre 0.
.!
,
s 2 respectively.
Then
Solving,
Let b 2 = a 2 sin
r4 -2a 2 r2 cos
r2 = a 2 cos 20
2a.
20+a4 = 64
.
aVcos 2 20-cos2 2a,
the upper sign giving OP'2 the lower
,
A be the one
BQ be called
Let the vertex
Let arcs AP,
OQ2
* Calcul
Integral, p. 265.
.
OVALS OF CASSINI.
611
-
AT
Now,
T.
= --
\
-r-1
as before,
and
a- sin
-(&* cos 20
cfej
*
'
_6
2
+ Vcos
Vcos
dO~a
Similarly
/dsi
\dO
20 -cos 2 2a
2
Vcos- 0.-cos 2 2a
the positive sign being taken as
,
.
:,
ar
a
increases with
sx
2
v/cos
20-cos 2 2a
6
^2\ _ 6 2(cos20 + cos2a)_
~~
a 2 cos 2 20 cos2 2a
a2
dO/
2
4
4
4
2
0.
2 (cos 20
cos 2a)
cos 2 20 -cos 2 2a
/c^ _ (is 2 \ _ 6
\dO~do) ~a*
26
_
1
cos 20
4
""a 2
cos 2a
1
cos 20
+ cos 2a'
Hence
_
dO
o
In these integrals put
2
=
?>
= am
a
..
s1
f*
2
,
l-sin 2 asin 2
cos 2 asin 2 \/r'
Jo \/l
^
(
rg
(!
si
+s
2 ),
mod. sin
2 ),
mod. cos a
a
sin
,
+s =
62
a
/sin 9
sn-M-
\sina
~---
- =I
sin 2^
respectively.
.
= am
sin
'
v/cos a
)
a
^
\^
dO
2
J
= sin a sin
T v
= cos a sin ^ J
sin
sin ^
Then
e
= b*t
xcos2# + cos2a~
and
vsin 2 a-sin 2
a
cos20-cos2a
o
2
.
/sin
a,
,
\
,
smaj,
/
;
1
CHAPTER
612
2
.'.
sl
6 f
= ~sn" 1
s2
6
= o~
2aL
2
T
/sin
(
2al_
-
,
sin a
/sin
(
r-s
\
.
.
\sina
sn- 1t
XVII.
\
.
,
+ sn,
)
sin a
\sina'
,
1
\cosa
.
sn^ 1
)
-
/sin
I
cos a)
,
cos a
,
/J
/sin
-
Vcosa
/
\~|
,
\~]
,
/J
the former of these being the result previously obtained.
Reducing in the case of Bernoulli's Lemniscate, we have
= a cn -1 \/cos 2#,
= acn -1
602.
The
mod.
~r^
,
(**
T=,
a N/2
as in Art. 598.
Single-loop Case.
In the one-loop case a<6, the same method cannot be
adopted, and M. Serret considers the arcs traversed by a pair
of perpendicular radii vectores OP, OQ, starting
from the ends
Fig. 146.
A,
B
of the
two perpendicular axes. Let the arcs A P. BQ
s and a-, and let a 2 = b 2 sin 2/3.
Then, solving
be respectively
as before,
r4 -2a 2 r 2 cos 2(9+a 4 cos 2
r 2 = a 2 cos 20
and
2(9
= a 4 (cos2 20 + cot 2 20)
a 2 Vcos 2 20 + cot 2 20,
and the positive sign must now be taken.
Also, as before,
ds
b2
-a 2 cos 20'
ds
5 2 x/cos
a
W
4- v/cos
2
20 + cot 2 20
T
N/cos 20Tcot
2
20
OVALS OF CASSINI.
Writing 6 + 1 for
613
6,
2
20 + v/cos 2 20 + cot 2 2/3
d<r_b X/-CQS
"
dS~ a
Vcos 2 20 + cot 2 2/3
* *
+ cot 2/3 + cot 2/3
cos 20 + cot 2/3
a
d<r\* _ 26* Vcos 20 + cot 2/3 -cot 2/3
~
~
a
cos 20 + cot 2/3
do)
(ds + d<r\
\dO 30J
2
_ 2fr
4
2
2
x/cos 20
2
2
2
2
/ da
H
.
2
2
Vd0
2
2
In each of these change the variable to
0',
sin 20'
where
cos
,.
2^ (W
Then
W + MO
sin 2 2^
90/3
sin 2 2)8
2
i, cot.20/3
2
cos 20^,
co t 2 2/3 = 1
28
-h
.
2
2(9'
= cos2200
.
sm
.
2/3
Then
~
cos 2 20'
a
2
2
4
= 2629
9ft
r>
sin 2 2/3
-+
cos 20'
1
sin 8 20'
sin 2/3
cos 20'
cos 2/3
90/V
sin 2
a
sin 2 2/3
a1
cos 20'-cos 2/3
.
S111
20
a2
0/0
^P
sin 2 ft
sin 2 0'*
Similarly
W
(ds
d(T\ 2 _2b*
~~
dO'J
sin 2/3
_
fr
4
sin 2/3
a 2 cos 20' + cos 2/3" a 2 cos 2 /3^sm2 0''
7>
2
1-=.
*e.
,
-<T =
VS^8
-sin 2 0'
r
In these integrals put respectively
sin 0'
= sin ft sin
and remembering that
sin0' = cos/3sini/r,
and
sin 2/3
=a
2
,
r
Jo
vl
sin 2 /3 sin 2
Vl
cos 2 /3sin 2 >j>
>
o
^^/r
CHAPTER
614
= am
sin 0'
.
oli
= am S
S-\-<T
s
_
+
sin 0'
o-
7
J
6
sm^S
XVII.
or
s
Oil
7~i
;
s -er
=
<r.
7
)
b
cos/3
7
,/8in0'
&Bn-'(g,
,A
cos/3j,
whence
S
=
'
2
where
The
603.
0'
first of
these
= $ sin- 1 (sin 2/5 sin 20).
was
The Elastica or
This curve
of Physics.
C S
established in Art. 597.
Lintearia.
of considerable importance in various branches
is
form assumed by a uniform originally
bow by a bow-string, or by equal
it may take the form ABC or
i.e.
extremities,
It is (1) the
straight elastic
thrusts at its
rod bent into a
Fig. 147.
ABCDE,
E,
etc.
ing
etc.,
This
is slight,
is tied at A and C, A and
an undulating elastica. When the bendthe form is approximately the curve of cosines
according as the string
is
called
Routh, Anal. Statics, vol. ii. p. 281, "Bending of Rods").
(2) It is the form assumed by a flexible thin rectangular
(E. J.
sheet,
two
of
whose opposite edges are fixed horizontally at
Fig. 148.
the same height, the flexible rectangular sheet forming the
base of a rectangular
box with vertical sides into which water
O
is poured, the material being supposed impermeable for water
ELASTICA OR LINTEARIA.
and the base
fitting the sides so closely as to
able escape of water.
by
The
prevent appreci-
this property the second
name
= made
of linen).
curve also occurs in the case of water
arises (lintearius
(3)
From
615
drawn up
capillary action against a partially immersed vertical plate.
Fig. 149.
The curve may assume various shapes according to the
physical circumstances occurring. It may undulate, or there
may be any number of complete convolutions forming loops and
nodes.
Such cases are exhibited
in the
accompanying
Fig. 150.
Fig. 151.
Fig. 152.
Fig. 153.
Fig. 155.
Fig. 154.
Fig. 156.
figures.
CHAPTER
616
XVII.
The determination of the nature of this curve is due
James Bernoulli (1654-1705).
For much detailed information as to the curve and its
604.
to
physical properties, the student
Hydromechanics,
Minchin,
Statics,
W. H.
consult
may
Besant,
pages 168-171, p. 194, p. 201, etc.;
vol. ii. p. 204
E. J. Routh, Analytical
;
G. M.
Statics
}
"
"
Sir A. G. Greening
Bending of Rods
and the article on Capillarity in the
Elliptic Functions, p. 87
Encyclopaedia Britannica, by the late Sir J. Clerk- Maxwell.
vol.
ii.
p.
283,
etc.,
;
;
T7
605.
The
stress couple at
radius of curvature and
rigidity,
we have
any point being
,
where p
is
the
K a certain constant called the flexural
as the geometrical property of the curve,
where y is the ordinate from any point to the line of thrust
and T the thrust, or string tension if the bow is bent as in the
ordinary case by a bow-string.
Hence the equation to be considered
constant, and two cases
(1)
606.
is
py = c
2
,
c
being a
arise accordingly as the curve is
(2) nodal.
undulating,
Bow.
Rectification of the
as origin,
Taking the bow-string as a?-axis, its mid-point
and a perpendicular through
as the i/-axis, let y be the
1>*^
O
B
x
A
N
Fig. 157.
any point P, and let
makes
with the tangent
tangent
ordinate of
and
let arc
Then
VP =
s.
Let
\Is
\!/
= a when P
=c
2
be the acute angle the
at the vertex
is
at A,
V
and
of the arc,
let
OF = 2a
UNDULATING ELASTICA, OR BENT BOW.
TV.O-
A-
-= -=
c*
A-
Differentiating,
c
.*.
\ls
= a when
P
=
^
J
= 2 (cos \/r
and p =
Hence
sin
oo
i.e.
,
c
= -^=
s
Let
-
,
ds\
a),
^f.
^
;
x
X = am
\!s
at
sin
d\
/
sin 2 -sin 2 x
A/1
v
Jo
and
cos
i
i
~
5
.as
sm -|- = sin H sn - mod. sm
.
And
=
f*
= sin
s=c r*
.*.
,
dp = sin \/r d\Js,
c2
t/
/
.
2
-3
and integrating,
for
dy
dp
617
T
;
the intrinsic equation of the curve
is
.
a
~
.
therefore
.'
The student should note the analogous
(i)
result in Kinetics
in Art. 389, viz. the case of the oscillating motion of a
simple
circular pendulum.
For a comparison of the two results, see
Greenhill, Elliptic Functions, p. 87.
CHAPTER
618
The ordinate y
is
XVII.
given by
.
y=
2c
^ sm -sin
2
2
-^~
= o2c sin ^a cos x= o2c sin ^a
.
Z
2!
en -
;
//
.(2)
To
find the abscissa x,
we have
dx
-j-^cosx/^;
ds
dx
ds
.
2 sin 2 ^ sin 2
1
dx_
~
and
,
ds
,
dx
dx
,,.
rx
ds
and adding ^-
,
.
-=2c
.
2
sin 2
1
g
sm2, x
~
i.e.
We
;
s.
.(3)
thus have for the bow, or undulatory
s=c sn" 1
BULf
A
!
8m
a
.
,
sm
a
>
^
|
2
sin- 1
\.
a
,
sin-
8in^
a
elastica,
(8
.
a
J
s,
py=<
NODAL ELASTICA.
619
607. Rectification of the Elastica in the case
when
there are
several Convolutions, viz. the Nodal Elastica.
Taking the ?/-axis to pass through a vertex V as before and
the line of terminal thrusts as the x-axis and \js the angle
ON
Fig. 158.
P
which the tangent at
V
to P,
we have again
has turned through in passing from
=y.
2
c2
p
and integrating
c2
^
,
3
2cos\//"+a constant =2 cos \//- +^4, say.
We
have not, however, in this case, as we had before, any point
at which p is infinite.
Let 2a be the ordinate of the vertex.
Then
at F,
/o=|-.
c2
,,.
.*.
-
putting
being
>
1,
p=?r-,
2
when
as p cannot be oo
>//=(),
by
^= 4ay
supposition,
and
2;
CHAPTER
620
XVII.
or
c
"^
o
C2
P
a
Jo
2
.
8111
2
putting
x
2
<*X
/7~^
"
1
81
and
Hence the
intrinsic equation is
(1)
Also
Again,
==
THE CAPILLARY CURVE.
Hence, in the nodal case of
- am -1 V
~
\j
?
Compare with
,
this case the result
an
infinitely long rod,
touch the line of thrust at
= oo when
and
^2
ds
s
being
still
and process of Art. 390
pendulum.
608. In the case of
elastica to
,
i
a
for a revolving
c2
/oy
621
\fs
oo
,
imagining the
we have
= Tr,
= 2(l + cos^)
= c SeC \[r
and
s
=
measured from the vertex.
called the Capillary curve (see
Besant, Hydromechanics, p. 201), the shaded portion in
Fig. 159 representing the water raised above the normal level
This species of elastica
by
capillary action due to the presence of a partially
vertical plate
c
is
= a,
PQRS.
In this case
^
p=
^
immersed
at the vertex,
and
the modulus of the elliptic functions occurring in the
second case becoming unity.
CHAPTER
622
XVII.
609. Cotes' Spirals.
These Spirals are defined by the pedal equation
~2
There are
(1)
(2)
=
^+E
'
(
five varieties
.5 = 0, an
A = 1, in
the
See Diff' CalC
'>
Arfc
'
454 ')
:
Equiangular Spiral.
which case B is essentially positive
curve
is
(as
r
> p)
;
the
Reciprocal Spiral (Diff. Calc.,
and the other three are reducible to the
Art. 452)
polar forms
;
u = a sin nO, u = asmhn6 and u = acoshnO.
(1)
The
rectification of
an equiangular
spiral
has been effected
in Art. 449, Diff. Gale.
(2)
In the reciprocal spiral r
- we have r =
,
^
,
and
s_a
dO
The remaining
For instance, take the
functions.
the case
three are rectifiable
n>
/.
1.
as =
by the
first, viz.
aid of elliptic
u = asmn9
for
BIPOLAR CURVES.
measuring
s
from the vertex at 9 =
~
623
(See figure of curve in
.
Art. 387, Di/. Gale.)
nO = (b:
Let
,
^
2
A = \/l
where
2
2
K sin
and
y
2
=
lu
r
.-.
--K
-i
as= -Acot0
2
2
;
sin0cos0
--^
f*
where
7l
Bi-Polar Curves
610.
;
Plane Elliptic Coordinates.
H
be fixed points, and let the distances of a moving
P
from
be rx and r2 respectively. Let SH=2c
S and
point
the mid-point of SH, PN a perpendicular from P upon $#
ON = x,NP=y; also let r1 + r2 =2 rx r2 =2>;.
Let S,
H
;
;
Then
^
= const,
and
may
>/
be called the
const.,
elliptic coordinates of
hyperbolae.
Let
A
be the area of the triangle SPH.
Then
.e.
A 2 Hf-c2 )(c2 -V),
where f is necessarily
Hence
c
<
P;
for
give families of confocal ellipses and
c
and
CHAPTER
624
Also,
if
XVII.
m be the length of the median OP,
ON
S
H
Fig. 160.
Thus the Cartesian coordinates
of
P
are given
by
(1)
C2
_
2
V 2Z^And therefore,
curve traced by
If
if cZs
P
for
be an element of the arc of the Bi-Polar
any
relation
= c cosh f
we put
we have
,
s=c Vcosh 2 v
I
between
>/=c sin
and r2
r^
,
it,
sin 2 u \/du 2
+ dv
2
(3)
x = c cosh vsinu,
Moreover,
y=c sinh v cos u,
and
e-H2/=csin
the transformation used in Art. 590 for the rectification of the
central conies.
.
BIPOLAR CURVES.
The
(u, v)
625
17)
system and the (
system are therefore
"
"
may be regarded as elliptic coordinates.
have a definite interpretation of u, v as used in
connected, and either
Moreover,
we
Art. 590, viz.
+ r,
.^r.
l
v=cosh
-^
u=sm ,^-r,
-^
l
^,
-,
2iC
<'
and they are
thus expressed
determination of a point.
Ex. Employ Formula (3) in the case
sin
cosh
um
To what curve does
terms
in
this equation refer
of
the
bi-polar
v.
?
611. If we wish to express the result of Art. 610 in terms of
the original radii vec tores rv r2 we have
,
nd
4c 2
4c?
- (r t
(2c+r1 +r2 )(-2c+r1 +r2 )(2c-r 1 -fr2 )(2c+r1 -r2 )
^^(
,
where 2c
a and 2o-=
-a)(o--r1 )(a-r2 )
LIST OF
612.
WELL-KNOWN BI-POLAR EQUATIONS.
The
principal bi-polar cases of well-known curves are
Name.
Bi-Polar Equation. Form of Equation in Elliptic
Coordinates.
1.
Ellipse
2.
Hyperbola
3.
Cartesian oval
4.
Circle
<
5.
Circle
*,
6.
Straight line
7.
Cassinian oval
E.I.C.
i
r l r2 =K2
2R
CHAPTER
626
Ex.
613.
1.
=
Rectify the ellipse r 1 + r2 2a.
=a
Here
5
=
n
/
l
'V
a i _ -2
L
ofy
2.
Here
dg
>
(17
where
Ex.
XVII.
= 0.
increasing)
77
= c sin
(77
(cf
.
< c < a)
Art. 567).
Rectify the hyperbola r 1 -r2 =2a.
= 0.
r\ a
<&/
">
(> c>o)
(cf.
Art
388, Case 6),
= tan o> v/c2 -a 2 si
4
where
Ex.
3.
-=
^
sin 2 w
(cf.
Art. 588).
2
Consider the case of the Bernoulli's Lemniscate r1 r2 =c
=c
-
Here
and
(cf.
=ccn-
1
f^-,
i)
( cf -
Art. 388, Case
Dif- Calc
.,
614 Use
.
->
2),
Art 458 and
-
>
Art. 592).
of Bi -Angular Coordinates.
sometimes desirable to express an element of arc of
a bi-polar curve in terms of the bi-angular coordinates 6 lt 2
which r2 rt respectively make with the line joining the poles.
Let f(rlt r2 )= const, be the bi-polar equation of a curve, c
It is
,
the distance between the poles S, H. Let the angles of the
6 lt 3 so that rly 2 are the polar
2
triangle SHP be
of
P
with
SH for initial line, r2 9l the polar
coordinates
,
',
,
coordinates with
cut the line
and
let
Then
SH
at
HS
6r
for initial line.
Let the normal
and the circumcircle of
SHP at Q.
PG
Let
BIANGULAR COORDINATES.
Hence multiplying by p^ p 2
respectively,
627
and then adding
and subtracting,
*i
T7T
""
P2 r2 77^
c
J(\
^2^- =
tt't/T1
Now PSQH
being
PI
(0
2
^
On
2
,*.^
()
cyclic,
^>.
P
Fig. 161.
Hence these
results
may
be respectively written
(Ncp l rl )dO l ............ (iii)
cds=(Ncp 2 r2 )d0 2
= Pl r
Pz^Pi
and
dBi-pfa dOt-NcWi-dOi),
ds=p l rl dO l + p z r2 d0 2 +Ncd0 3i
......... (iv)
^ + ^2+^3=0.
for
The
l
equation (iv) is due to Mr. Roberts (vide Professor
Williamson's Integral Calculus, p. 501, for a somewhat
last
different proof).
Again, in travelling along the curve f(r lt
fn drt -\-fr^ dr 2 =Q
i.e.
Hence
(
r2 )=const.,
where ffl stands for
^-
,
etc.
\
/ri8inxi-/ra 8inx 2 =0.
(a)
W?^^
EG
r2
sm X2 r2 f
ri
(see Diff. Calc., p. 181, Ex. 32)
(M
2
Pi
=
i
sin X2
=
A'
;
CHAPTER
628
XVII.
In cases in which f(r lt r2 ) is homogeneous in r t and r2 and
of degree ?i, and if for convenience we write the constant as
QU-\
c
we
have,
n
/(r,,
9 )=c
J\ i> r2t
so that
,
by the theorems
a"- 1
n
,
Ptolemy and Euler,
of
= P* = riPi + r*P* - Nc _ N
fn fr rj^+r^ nf a-*
n~
Pi=fr^
=A"> N=a
Pi
.
2
Then
/
The
and
v
quantities
as follows
2V
1
v.
2
can be obtained in terms of rlt
:
(Hobson's Trigonometry,
/.
and
v is
And
Pi> P2>
V
;
2=
as
N
P\
are a^ so
=vfr^
k n wn
r i*
A1
1
203)
therefore found in terms of rlt r2 and the constant a.
Also, since
and
p,
TZ
/>/
l
v,
in terms of r lt r2 .
-r-fr
sin ^ 2
.
sin 6 l
N=a n ~
Pz~ vfr^
=
'
.
C
/n
+
sin (^j
*
/( r ij r2/ =c
TT-\
2)
n-l
n
LC
'
we have theoretically the means of expressing rlf r2 p lt p 2
and N either in terms of 9 l or in terms of 2 as required.
Hence the rectification of the curve depends upon the in,
,
tegration of either of the formulae
or
or
ANGELO GENOCCHI'S THEOREM.
615.
629
Genocchi's Result.
Rectification of a Cartesian Oval.
The last form was used by Mr. Roberts in a proof of Prof.
Angelo Genocchi's Theorem, that the arc of a Cartesian oval
can be expressed in terms of three elliptic arcs.
Thus, for this oval, viz. Zr 1 -M 2 r 2 =cZ 3
,
we have
f=
2
and
2
t
2
-2Np 2 cos0 = v
1
z
l
Nc N
=v
j-=r
a
3
3
-2NPl cos
2
>
say>
z
(l
=y2 (Z32
-
cos O l
1
cos
(9
2
+
+^
1
Z
3 ),
2
),
Hence
and
And
these are the integrations required in the rectification
This is Genocchi's result.
of ellipses.
For a
full description of
the elements of these ellipses and
for
many other important properties of the Cartesian Ovals,
the student should consult Professor Williamson's Differential
Calculus, pp. 375-382, and Integral Calculus, pp. 239-243.
Fig. 162.
616.
In a similar manner,
the circumcircle
of
if
the tangent to the curve cut
the triangle
SPH
at a point Q'
whose
CHAPTER
630
XVII.
and T be the length of the
o2
which
makes
tangent PQ',
angles Xl X2 with r l and r 2 we have
bi-polar coordinates are o^,
,
,
^=-cos Xl
,
=-
ds
j
j
/
.
ds
ds
'ds
=
cs
1
(T!
c?r2
|
and
^2
617.
A
a
<frV
General Theorem.
Let there be two given curves
and
let
OP P
2
be a radius vector from the origin cutting
1
these curves at
P
2
and Pr
Fig. 163.
Let a point
P
be taken on
i.e.
r = A!
and
r = \{
OP2 P
l
so that
\ v \ 2 being constants and dots denoting differentiation with
regard to
6.
Hence
r 2 +f 2 = A 1 2 (r 1 2 + r 1 2 ) + A 2 2 (r 2 2 + f2 2 )-f2A 1 A 2 (r 1 r 2 + r1 r2 )
Let s v
N ow
be corresponding arcs of the three curves.
2
r2r2 ) 2 = (r1 2 + r2 2 ) (r 2 2 + r1 2 )
(ryrg + r^) + (r^
s2 sp
,
(1)
A GENERAL THEOREM.
Hence there are two
(A) when
cases of simplication, viz.
r2r2 =
r^
631
when
(B)
;
r^ r^ = 0.
Case (A) arises when the given curves are so related that
r\ ~ rz = const. = a 2
.
/y*
Case (B) arises when
/v
i.e.
rz
= constant
/Y*
=
r2
^
,
and the original curves similar and similarly
situated with regard to 0.
In case (A)
+ fyr = (r2 + a + r2
and
If
2
2
2
y-2
2
2)
(r*
)
-a +f
2
2
1 )
= A; ^ 2 + X 22522 + 2X X 2v/(s 2 -a2 )(s22 +a2 >
we take
X = X 2 = X, say,
s>
2
2
1
1
x
and
If
another point
be taken on the same radius vector such
Q
\=
that
X 2 = X,
say,
then
The
radicals are placed in this order because
s2 2
as
may
be seen as follows
+ a 2 >6- 2 -a
r
If
is
,
:
r'l 2
and
2
1
2
^
r i 2_j,
M2
i
r/
2
positive.
we take
and
then the P-curve
the Q-curve
is
is the locus of the mid-points of
such that OQ = P 2 P = PP1 and,
P
so that the
P and Q
.
Q
loci are inverse to
each other.
CHAPTER
632
For such derived
loci
we
XVII.
therefore have
and when these integrals can be found, s p and
SQ
can be found.
p'
Fig. 164.
Again, the
P
and Q
loci
constant of inversion being
-=
being inverse to each other, the
,
ds p = -i-
-dsf
;
-
618.
whence
and
In Case (B),
r:
1
^
1
+X s
2 2
but as the curves are then similar this
part of the investigation
information.
this
is
an obvious
fact,
does not render any
and
new
A GENERAL THEOREM.
A
619.
633
Useful Case.
In Case (A), it may happen that the derived curves are
different branches of the same curve locus,
r*-bF(6)r + ^=0,
and
rp r Q
whose roots are
,
r1 2 -r22 =4rpr<2 =a 2
and therefore
say,
a2
.
In this case the two branches of the curve are
_
2
which are inverse to each other with regard to the
constant of inversion being ~
2i
And
the
And
if
"
"
given
Sj_
and
s2
.
curves from which this curve
Ex.
1.
is
derived are
be the differential coefficients of the arcs of
these curves, the arcs of the derived
620.
pole, the
P and Q curves are given by
Consider the rectification of the curve
4(.x
Putting this into Polars,
r2
+
-ar sec
_ a sec
The
original curves
from which
and
the
a tan
this
is
derived are obviously
r2
first
being a straight line and incidentally an asymptote of the curve
we wish to rectify.
The P and Q curves are branches
each other.
If
N be the node on
where the asymptote
x:La
of the
same curve and inverse to
and A the point
this curve (see Fig. 165)
cuts
the
.r-axis,
the
several
arcs
are
CHAPTER
634
Now
!
= a sec 2
s2
(9,
2
XVII.
= a2 (tan2 + sec* 0),
2
+1
(sec
i* -a* = a z tan 2
/T-r2
.'.
vs,
-
-a* = a
),
Sin 9 /
^-r.vl
cos 2 ^
J
+ cosTTT
0,
Fig. 165.
Now
cos2
0'
>s2
^+
/x/r+^0
= sec Vl + cos2 - sinn" (cos 0)
= \/sec2 0+1 -sinh- (cos 0),
1
1
and
2
[sec
N/l+cos2
^0 = tan
/
(
^ \
/7/1
l-Lsi
Hence
arc
^P - arc #Q = a [%/seWTl - sinh-
1
cos 0].
EXAMPLES OF THE THEOREM.
635
Thus arc NP and arc NQ are found by addition and subtraction.
It
is
NQ
to be noted in this case, that although each separate arc NP,
requires
for its expression the elliptic integrals of the first and second kinds, their
difference is free from these functions, and expressible in terms of
trigonometric and logarithmic functions.
Ex. 2. As a further example, consider the "derived" curves to be the
branches of the Cartesian oval
The
roots being rp
and
r
q,
we have
= A + B cos 0,
and these are the "original" curves from which the Cartesian ovals are
derived, the first being a
J
Hence the
^ 2 + B2 - a* + 2AB cos d dO.
difference
(See Art. 573.)
between corresponding portions of the inner and
outer loops of the curve
r2
- (4 + J3 cos 0)r
=0
+^-
can be expressed as the corresponding arc of a certain ellipse.
[This polar equation to the Cartesian oval is an ordinary conversion to
polars, retaining one of the poles as origin, of lr-\-mr'n^ writing
r2
+ c2 -2rccos#
for
r'
2
and performing the
rationalization.]
Fig. 166.
We may
remind the student that any arc of this curve has already
been expressed in terms of three elliptic arcs (Art. 615).
CHAPTER
636
The
arcs sP
s
AP^
q
= A'Q
XVII.
which the integration refers are shown
to
in the figure.
We may
rl
Having drawn the Hilton
construct the ovals as follows.
= A + Bcosd
explained in Art. 424, Diff. Cole., take any radius
vector OPj, and on OP1 for diameter construct a circle. Take centre P l
and radius a and draw a second circle cutting the first at JR. Then with
as
and radius
centre
Then
OR draw a
OP
OP
circle cutting
OP
l
at
P2
.
l
P^a at P and make
on
the Cartesian oval.
points
Bisect
OQ = PP
then the points
1 ;
P
and
Q
are
MISCELLANEOUS PKOBLEMS.
1.
Prove that the three equations
x=
c
log sec
y = c(tan ^-^),
i/',
s
= e(sec^--
represent one and the same curve.
2.
Find the area
considering
3.
cases
[I.
C. S., 1893.]
p denoting
the central
curve
of the
^=
all
1),
which may
Prove that the value
62,
arise.
of the integral
taken round the ellipse x2 /a 2
+ y2/b 2 =
1
,
is
-=
,
*j
perpendicular on the tangent at
(x,
y)
and ds an element
[I.
4.
the point
If
x,
y
lies
y = x + 2px + q,
dx
= dy = dx-\-dy
y
x+p x + y+p'
,,
prove that
2
,
and hence obtain the integral
of
-~=
dx
however, the point (x, y) lie on the
that the corresponding relation is
If,
dx
y
s
is
C. S., 1912.]
on the curve
2
where
of arc.
the length
of
Deduce the known formula
dy
x
~
circle
x2
+ y 2 = a2 show
,
ds
a
'
the arc measured to the point
(a;,
y).
dx
for the integral of
^- x
=..
V
2
2
[I.
C. S., 1908.]
637
PROBLEMS.
5.
Show
6
lf
that
if
IdR
'
*
8 stands for T-,
fx + g
-os' + te + e'
and if b 2 - 4ac be positive and the roots of ax2 + bx + c =
A and /*, prove that R = (x- \) p (x - /x) ? where
be
,
2
q
a=
And
if
If b 2
- 4ac be
-1,
6
= 0, c=l,
negative,
/
2ag-bf
i*"
If & 2
/2a
k
\
/
-ia
ROUTH, Proc. L.M.S.,
[E. J.
Show
+1
1
_
where J/ = (A; + Z)(^ + / -
1)
. . .
is
being an integer, though k
//v.
/
r2
1 \
I
i-\*
*;
to 2k
[Cf.
ABC
vol. xvi., p. 250.]
that
there being 2& -f 1 integrations, 2k
may be a fraction, is equal to
8.
Zgx+b
^^^
-4ac = 0,
^
7.
"1
1
+
1 factors.
ROUTH, Proc. L.M.S.,
A
a triangle with the corner
fixed
vol. xvi.. p. 249.]
and with
sides
AC, CB respectively Jn and >/w + l, given lengths.
The side ^4-B ( = r) makes an angle = nA-(n + l)B with a
fixed straight line
Show
^Z.
(1) that the
path of
B
is rectifiable
by the formula
mod.
(2)
AVhen n =
l
the rectification
Bernoulli's Lemniscate,
is
m-r
the same as that of a
CHAPTER
638
(3)
The
inclination of the normal
the
to
radius vector
A+B.
is
(4)
XVII.
The area
of the triangle is equal to the area of a sector
of the curve starting
AX.
from the axis
[M. SERRET'S PROBLEM, Cede.
C is
9.
A
and
maximum curvature on the Lima^on
b > a
r = a cos 6 + b,
A are the two vertices
the arcs
10.
a point of
AC,
If
AC
y=x
5
Prove that the difference between
is 4a.
- 3a
2
[ST. JOHN'S, 1891.]
prove that
dx
aj,
dy
~~
*Jtf~^ltf
and by integration express x
Apply
this
Int., p. 269.]
method
'
3V*/
2
-4a6
explicitly in terms of
y.
[OXFORD
I.
P., 1916.]
[OXFORD
I.
P., 1916.]
to solve the cubic
z3-3z 2 -45z- 473 = 0.
Prove that
11.
12.
Prove that
if
n be an odd
8 _2 w
-3
positive integer greater than
7i-2
2- 5<<[OXFORD
13.
The parameters
t
lt
t
2
of
3,
two points A,
I.
P., 1916.]
B of the unicursal curve
x/(l-P) = y/(t-t*) = a/(l + P)
are equal to tan
tan ft where
a,
-J7r<a< - JTT,
Prove that the area
the double point,
2
|
J7r</3<j7r.
of the curvilinear triangle
A OB,
where
2-^ + /2-a-sec/?secasin(/2-a)+-tantanasm2(/3-a)
[OXFORD
14. If
n be a
is
is
positive integer,
x *m'i
*
J"
show that
COMCZ dx = n^.
I.
I.
P., 1916.]
PROBLEMS.
639
By assuming
15.
x3
a + bx + cx 2 + dx* + ex*
form -
,
^dx
to be oi the
obtain the integration by differentiation and equating coefficients,
also obtain the result directly by putting z 4 = z.
Given a rational integral relation between x and y
1 6.
1
2
.
2
where A 19
2
,
form
of the
n~ +
+ A n = = F(x, y\ say,
y + A^- + A y
A
A n are rational functions of x, prove that
n
.
.
...
\ydx can be expressed algebraically in terms
of x,
when
then
Jy<fo-j
B B^
where
17.
,
-Z?
X
Assuming
and that
I
y dx
are rational functions of
...
2
x.
[ABEL.]
to be a rational function of
x,
and ym = X,-
integrable in algebraic form and expressible as
is
= PQ + P^y + P$2 + + Pm -iym ~\
fydx
... P
m ^ are rational functions of x, show
_ p
p
p3 _~
Jp
^>
m-*-l
-*2
.
.
.
I
where
P Plt
,
that
rv
'
that
-
'
that the integration must contain one term only, and that
is
is
\ydx
'
a rational algebraic function of
x.
[LIOUVILLE.]
9J
18.
If
f^r=dx
M
(1)
be two rational polynomials, then, provided
can be integrated in algebraic form at
the integral
Show
T
and
is
=, where &
a function of
is
all,
the form of
x.
also that
B
is
a rational function of
x.
T
(2)
ThatMr=2'^-i(9^
ax m ax
(3)
That 6
is
.
an integral polynomial expression and not
such form as
where
-p.,
U and V are
V contains x.
nomials, i.e. not such that
(4) That the degree of the polynomial 6
the degree of M.
Use these facts
to
show that
I
J
-=
v
A
-f*
is
of
complete poly-
greater by unity than
= is not expressible algebraically.
x^
[BERTRAND,
C.
L,
p. 94.]
CHAPTER
640
19.
P,
Q,
R
XVII.
being any rational algebraic polynomials, and
dx
-7= is integrable by means of the ordinary
Jp
-fi
elementary functions, the integral must be of the form
where
77,
av
6,
/3 lt
a2
etc.,
,
are rational functions of
x,
reduced to one term, the general type of the result
algebraic of form 0/>/5 or may be written as
^^^^
In the latter case show that
(1)
a*-p*R=Q.
(3)
2a'e-e'a =
where accents denote
Show
20.
^
8,
/
differentiation with regard to
x.
that
~
21.
,
Prove that
XdX
f+ 2x-5)(
)^
22.
O
,,v
Prove that
C5x 2
I
'
;
+ 3x+l
2x+l
dx
oL
/a4
9
2
=tanh~ 1
-
3tan 2 (9 +
f
(
a result
by Abel,* show that when the integration can be
established
} 2 tan^
_
~
* (Euvres.
See Bertrand, Cede. Intfy.
,
chap. v.
is
either
PROBLEMS.
Show
23.
%
<
fi
)
that
2n+l
-
-a
(i)
O
2
-
J
Ia 2n
f
'^\JV
1
641
-l
+ x2n - 1
dx
dx
1
1
+ sm 2
'1
I
-1
Integrate the following
f
+ x2il+l
T^
/.
,
sin
\
24.
la? n+1
"2n-lC
+ sin
--
l
0A/1
.
\l+sm
:
(2,+ Deb
;
[ABEL.]
25.
Show
that the whole perimeter and area of a single loop of
(n> 1) are respectively equal to the whole
the curve r = 2acos7i#
2
perimeter and area of the ellipse x
+ n z y2 = a 2
[Oxr.
.
I.
P., 1911.]
26. If an element ds of a curve lie at distance r from the origin,
and subtends an angle dO there, it is known that unit electric current
flowing along ds produces a magnetic force at the origin at right
angles to the plane of the curve proportional to
Show
that
if
.
unit current flows through a thin endless wire of
form of an ellipse, the magnetic force due to
given length in the
the current at the centre of the ellipse
the area of the ellipse.
E.I.C.
2s
is
inversely proportional to
[OXFORD
II. P., 1913.
CHAPTER XVII
642
27.
A
current
of
electricity
is
flowing
ABCD...KA
bent into a plane polygon.
the polygon, and perpendiculars OP, OQ, OR,
round
is
...
a
fine
wire
any point within
are drawn to the
respectively, and again perpendiculars
whose lengths are a, /?, y, ... from
upon the sides PQ, QR, .RS,
Show that the magnetic force
of the inscribed polygon PQRS...
is
on unit particle situated at
sides
KA, AB, BC,
etc.,
.
.
.
.
where
28.
t
is
the current strength.
Show
eccentricity
that the perimeter of an ellipse of axes 2a, 25 arid small
approximately equal to the perimeter of a circle of
e is
diameter a +
when
e is
b,
with an error which
as great as 0'2.
is
only about 0-0025 per cent.
[MATH. TRIP. PART
II., 1913.]
CHAPTER
XVIII.
RECTIFICATION (III). MISCELLANEOUS THEOREMS.
Arc of an Inverse Curve.
and s' be the corresponding arcs of a curve and of its
inverse with regard to a fixed point 0, the constant of inver621.
Let
s
sion being
k.
Q'
Q
Fig. 167.
Then
points,
And
if
P,
Q
we have
ultimately,
be points on the curve and P', Q' the inverse
PQ
r
when Q and
Q' are
made
respective paths to ultimate coincidence
to travel along their
with
P and
P',
Tods
(1)
giving the arc of the inverse in terms of elements of the
original curve.
622.
Modifications for Various Coordinate Systems.
This formula
may be modified as required for different
systems of coordinates, and with the usual notation, we have
for polars, the inversion
being with regard to the pole,
(3)
643
CHAPTER
644
Again,
i.e.
we may
XVIII.
write
as a formula suitable for tangential polars,
-
or for pedal equations,
lt)
dr
Clds, = 72 f
= k79 f
/^l-52
s =/c 2 |-s-r-^r
2
J
r
dr
dr
z
.
\
J r cos
2
J rx/r
,.
_ p--,z
.........
(6)
and for Cartesians,
the inversion being with regard to the origin
if
the inversion
623.
1.
is
with regard to the point
;
(a, 6).
Illustrative Examples.
Consider the arc of the inverse of the parabola
= ar
with regard to the focus
;
i.e.
=
for
a cardioide.
Here
'
r^
= k*l
."!_=- P
z
= 2k v 1 - au = 2k
a
a
2
,
sin
--
2
Rectification of the inverse with regard to the centre of the first
negative pedal of an ellipse with regard to the centre.
2.
The
ellipse
2
2
2
2
being # /a + ?/ /& =l, the
lo P eof
xcosys+ysmy/^p,
where
Hence the tangential polar equation
is
ab
P
/~>
o
i
first
1
%
=
negative pedal
cos 2
^
sinV
^H n^
1
is
the enve-
ARC OF AN INVERSE CURV&
Differentiating
dp ~
_
645
we have
(a
,
*
2
- b 2 sin cos
)
^
i/
whence
P+7to
d/\l/
and
'^\= aWHence
4
(a sin
Hence
if e
taken from
\j/
be the eccentricity of the
to
- and
if
cos
ellipse,
x be the complement
This curve therefore requires
all
and the integration be
of ^,
we have
three kinds of the Legendrian integrals
for its rectification.
Note for the
first
negative central pedal of an ellipse that
incidentally
(2)
(3)
-
=
(1)
2
>-
= a 262
^
6 2)
we have
CHAPTER
646
3.
XVIII.
Central inversion of epi- or hypo-cycloids.
Here
p = AainB\l/
9
where
A = a + 2b,
B=
*\
-
See
Diff. Cole.,
Art. 410
the inverses of epi-
}forcycloids,
for the inverses of hypo-
=
or
Is.
E.g. in the case of the inverse of the cardioide
of the fixed circle
a = b, A=3a, J5=g,
with regard to the centre
In the case of the inverse of the three-cusped hypocycloid
1
[tan-
(2 N/2 cos
Note that these inverses are such that their
logarithmically if derived from epicycloids, or
functions if derived from hypocycloids.
4.
Inverse of the parabola
y=o.
The general problem
for
2
?/
= 4a#
arcs are expressible
of circular
by means
with regard to the point
x=
-3a,
any point on the axis is discussed by Mr. R. A.
London Mathematical Society, vol. xviii.,
Roberts, in the Proceedings of the
p. 202.
JOHN BERNOULLI'S THEOREM.
Taking am 2am as the current coordinates
=4ax, an element of arc is given by
2
,
= 2a Vl
ds = \d3?~+~d
of a point
647
P on
the curve
+ m* dm.
Fig. 169.
OP2 = (am2 + 3a)2 + 4a2w2 = m4 +
2
Also
and the element
;.
s
=
ds' of
fc
the inverse
2
.
2^
a2( m2 +
ds'=
is
^m
2
dm
r
=
^-/JO 7-77-^-7^=7
2fc
10a 2 m2 + 9a 2
+ 9)
2
,,
(Itla
I /
cos
sin 2 <
a
= 2^
a
/
Jo
+ 9
cos 2 ^
2 r
rfsin
rfsin^)
2
sin </)~ 4a J g - sin 2
^^
9-8
Mr. Roberts shows in the article above cited that for
- co and - 3a on the #-axis the arc of the inverse curve
can be expressed as a pure logarithm. For
points from -3a to a such
arcs are partly logarithmic,
partly inverse circular. For points from a
Example.
points between
to
+ QC
Examine the truth
the arcs are inverse circular expressions.
of
this.
John Bernoulli's Theorem.
Let a number of points Pl (x l yj, P2 (x2 y 2 ), etc., be moving
in a plane, and let ds ds ds etc., be elements of the
lf
z
paths
s
described.
Let us impose upon their motion the condition
624.
,
,
that they are
all
moving
,
,
at every instant in parallel directions
CHAPTER
648
in the
same
respective
sense.
Let
\/r
XVIII.
be the angle the tangents to their
the z-axis.
paths make with
m m
Suppose heavy particles of masses
v 2
at P1? P2 etc., and let x, y be their centroid.
,
etc.,
to be placed
,
-~
Then
ds
Sm dx = 2m
~
Sm
2m cos ur,
,_
ax = =
"Zmds
Fig. 170.
-*L =
Hence
cos
centroid
particles
\js
^
sin
and
therefore the
motion of
the
\/
always parallel to the motion of the several
moreover, if ds be the corresponding element of the
is
;
path of the centroid,
^
,
ds--
and
-Sras
This result is ascribed by Mr. E. A. Roberts, in the
before
cited, as due to John Bernoulli, the intention
paper
a
method for the generation of new rectifiable
to
give
being
625.
curves from any system of curves whose rectification has
already been effected.
BERNOULLI'S THEOREM.
It is to
649
be remarked that the same theorem obviously holds
any system of particles moving in the manner prescribed
upon twisted or tortuous curves in space.
Again, several of the points may be moving on different
branches of the same curve.
for
m m m
... can
9
appears from Bernoulli's result that as
lt
3
be arranged at will, we can from any set of rectifiable curves
It
,
,
generate an infinite number of other curves which are rectifiable
in the same manner and in terms of the same functions.
Thus, for instance, taking any set of catenaries with parallel
directrices
or
any
or any
or any
or any
whose typical equation
is
s=a+ct&u\Js;
set of equal equiangular spirals, type s=a
set of circles,
type s=a
set of involutes of circles,
type s a
set of similar epi- or hypocycloids,
type
s=a+b sin (n^-fa)
or
any
set of semi-cubical parabolas
or, in fact,
any
pression of the
of the cases in
which
same form, the locus
5
with parallel axes, type
-i/
-=
reduces to an ex-
of the centroid
another curve of the same kind, and the length of any
portion of its arc is to be found from the formula
is
And
further,
when curves
of different nature are taken as
the original curves, though the derived locus be not of the
same nature as that of any one of the original curves, yet it is
still rectifiable in terms of the same functions as those in terms
of
which the original curves are
626.
Extension of Bernoulli's Theorem.
When
not
rectifiable.
the forward-drawn tangents at the several points are
the same sense, we may still apply the theorem, but
all in
with the precaution of reckoning
which are traversed
in the
remaining ones as negative.
all
those elementary arcs
positive, and the
same sense as
CHAPTER
650
XV11I.
Thus, if P l (x1 2/i), P2 ( iC 2> 2/2) b e a ^ opposite extremities of a
diameter of an ellipse, or centric oval, and if cos \fs, sin \(r be
,
the direction ratios of the tangent at PI}
sin\//- will
cos^,
be the direction ratios of the forward drawn tangent at P 2 and
,
,_
_m
x
dx l +m.2 dx 2
Fig. 171.
or centric oval, obviously ds1 =ds z
and s,=s2 and if we make 1 =m.2t 5=0, as it should be, since
all diameters are bisected at the centre, and the centroid locus
Moreover, for an
,
ellipse,
m
degenerates into a point.
In the case when one of the curves degenerates to a point
and one other point describes a given curve, Bernoulli's
and similarly situated centroidlocus is such that corresponding arcs on this locus and on the
original curve are proportional, which is a priori obvious.
Theorem
627.
states that the similar
Ovoid with One Axis of Symmetry.
Let us consider the case of any ovoid with one axis of
symmetry, and discuss the locus of the mid-points of chords
which are such that the tangents at their extremities are
Let PjP2 be such a chord and G its mid-point. If
parallel.
we take
the direction ratios at
P
l
as cos
\!s,
sin
\//",
then at
P
2,
where the forward- drawn tangent is parallel, but in the
If
sin \Js.
cos
opposite direction, they must be taken as
\//-,
SYMMETRICAL OVOID
651
be a question of applying the theorem to the locus of the
mid-point G of the chord P-f^ we have
it
,
-ds
_dSl
~~~
2
where ds v ds2 do- are the elementary arcs traced by P I} P.2 G
respectively, and as the inclination of all three tangents to the
,
,
#-axis
the same,
is
P=
p\p*
2""*'
where p lt p 2 p are the corresponding
radii of curvature.
,
172.
.
in integrating to find
Now,
a-
whole length of the
for the
necessary, for when the points
v 2 pass through positions at which the radii of curvature
become equal, ds l ds 2 in general changes sign. So that in
path of G, considerable care
is
P P
estimating
take
<r
=
\dar
I
^
for the
^
whole
some parts we must
6r-locus, for
*
and for others
I
'
2
1
^
;
i.e.
we must take
care that the difference of the elementary arcs at the ends of
the chord is reckoned positively.
Hence we
of
shall write the result
In such an ovoid there will in general be points A, B, C, D,
which the first and third are the extremities of the axis of
symmetry, where the
minimum,
radii of curvature are respectively
maximum,
minimum,
maximum
;
CHAPTER
652
XVIII.
D
be a pair of points, one between
and A and
and C, at which the tangents are parallel, and
such that the radii of curvature at those points are equal and
and there
may
one between
B
;
true of the portions AB, CD of the ovoid.
In such
case, on the 6r-locus there is therefore a point at which /o = 0,
with a change of sign of p. Hence there is at such a point a
the same
is
singularity on the 6r-locus, in general a cusp at which the
tangent is parallel to the tangents at the corresponding points
on the ovoid.
628.
Geometrical Examination.
Let us examine more
is
in general
closely, in
a geometrical manner, what
happening at such a point.
Let P<f lt P^, P2P3 P3P4 P4P5 ... be elements of the
ovoid, with equal increments d\fs in the angle of contingence,
,
,
,
and drawn in the neighbourhood of a point on the ovoid,
which has the peculiarity under consideration, viz. that the
radius of curvature at that point
opposite extremity of the chord.
And
is
equal to that at the
PoPi,pip z ,pdp s ,psP4>p4p 5 ...be the opposite parallel
the
elements,
angles between consecutive pairs of either system
and let P 3 P 4
therefore
d\fs,
being
let
,
SYMMETRICAL OVOID.
G G lt G 2
Let
,
,
6r 3
,
6r4
PiPi> PzPz'P-zP^
>
,
653
...be the mid-points of the chords
then it will be obvious
respectively
;
that
etc.
The points G3
2d\]s
,
6r 4
6r 4 6r 5 makes an angle
the direction of the tangent to
coincide, the element
with the element
G 2G3
,
G 3 am/G<
Fig. 174.
the path having turned through an angle 7r-f2dh/r. Ultimately then we have at G 3 two coincident tangents to the
6r-locus, i.e. there is a cusp on the 6r-locus at such a point, and
this
cusp
lies
upon the envelope of the chord, for G 3 is the
two consecutive positions of the chord.
point of intersection of
Again, at the points E, F on the double ordinate at the
widest part of the ovoid the radii of curvature are obviously
629.
equal,
and at the mid-point
Y
of
EF
there will be a cusp on
the (r-locus, whilst at X, the mid-point of the axis of symmetry
AC, the tangent to the G-locus will be perpendicular to AC.
Let
at /
IJ be
and
that chord of the ovoid for which the tangents
parallel and for which the radii of curvature at
J are
the ends are equal, and whose mid-point is situated at the
cusp L of the (/-locus, and let I'J' be the corresponding chord
through the cusp M, symmetrically situated with regard to
the axis of symmetry.
CHAPTER
654
XVIII.
Then, integrating along corresponding
, rirr
aucd'CI
Si
arc
MXL =
arc
LY =
arcFJf =
Thus the whole perimeter
arcs,
z
of the tricuspidal G-locus
= J(arc 7'<77-arc 7#+arc EJ'-a,rc /'J+arc JF-arc FL'\
i.e.
in short, half the difference of the
two sums
of alternate
arcs of the original ovoid, the points of division being those
D
Fig. 175.
at which, whilst the opposite tangents are parallel, the radii
of curvature are equal.
630.
Of
course, in the case of
any
closed oval symmetrical
about two perpendicular axes, such as an
ellipse,
the diameters
are all bisected at the intersection of the axes of symmetry,
and the tricusp is evanescent, the radii of curvature at all
opposite points being equal and the tangents parallel.
631.
Note
(i)
that
if lines
be drawn through the points
G
parallel to the tangents at the extremities of the chords through
G, then the points G are the points of contact of such lines
with their envelope
;
that the cuspidal tangents to the 6r-locus are parallel
to those parallel tangents to the ovoid at whose points oi
(ii)
contact the opposite radii of curvature are equal
;
SYMMETRICAL OVOID.
R
if
(iii)
described,
655
be a point on such a chord PjP 2 as has been
it in the ratio
2
v then the theorem
m m
and dividing
:
_
true for the whole perimeter s of the ovoid,
is
round the curve s 1 s 2 = s\ provided that
intermediate between a certain pair of points
on the chord, for which
can vanish,
2 /o 2
1 p1
=
(for in integrating
-
R does
Rv R
not
2
lie
m
m
1
X and X" be the greatest and least values of the ratio
Pj travels round the perimeter of the ovoid,
= \m2
the points R v R2 are the positions of R for which
l
and ?/i 2 = Xm1 respectively. Thus, for all points R on the chord
or the chord produced which do not lie between R l and R 2
i.e.
if
/i//2 attained as
m
,
the perimeter of the /2-locus
is
R
R
But for points between 1 and 2 thus defined, precautions
similar to those described for the mid-point must be taken.
Fig. 176.
632.
An
Instructive Problem.
Let us discuss the locus of the centroid of the triangle PQR when
these points lie upon a cardioide and are such that the tangents at
P, Q,
R are
always
parallel.
CHAPTER
656
The equation
= 2a
normal to the curve r=a(l-fcos#) at the point
of a
.3
is
XVIII.
_a
2
^tana
where
(Diff. Calc., p. 158).
Th3 three normals
will be parallel at points such that
'3t
t
.
3
jr-g
= &j
sav
>
i" e '
tan 3a = k.
Let
Then
Hence
2a,
2a + -^-, 2a +
o
o
are
points at which the normals, and
therefore also the tangents, are parallel.
Let these be called
If (#!, yj), (^2 ,
?/ 2 ),
2a, 2ft 2y.
(^73
,
^ 3 ) be the coordinates
of P. Q,
J?,
f\
x =2
l
3/!
cos - cos
2
= 2a cos 2
--
sin
= 2a cos 2 a cos 2a = - (1 + 2 cos 2a + cos 4a),
^= 2a cos 2 a sin 2a = ^ (2 sin 2a + sin 4a),
etc.
etc.,
;
Fig. 177.
(i)
/.
on the
= and
5=|, ^ 0,
axis.
the centroid
is
therefore at a fixed point
G
HOMOGENEOUS COORDINATES.
RG
cut the sides of the triangle at L, M, N. Then,
the points L, M, N, i.e. the mid-points of the sides
on another cardioide of half the linear dimensions of the former.
Let PG, QG,
(ii)
since
lie
657
GP=2GL,
etc.,
N
to this cardioide are parallel to the
M,
(iii) The tangents at
tangents to the original cardioide at P, Q, R.
(iv) The triangle PQR might have been described as one in which each
,
an angle 120 at the pole 0.
of the sides subtends
(v) All other points which divide the sides, or the medians, in a
constant ratio, or any points connected with the triangle PQR by the
formulae
J
,_^lx
~~2T'
7
_^
1
>~~Xr
l
m, n are either numerical or not dependent upon the magnitude,
and
position of the triangle, also trace cardioides and lines through
shape
where
t
;
such points parallel to the tangents at P, Q,
633.
It
jR,
envelope cardioides.
Areals and Trilinears.
has already been explained that such systems are not
well adapted for metrical purposes (Art. 460).
We can, however, readily obtain suitable formulae for such
cases
if
necessary.
Denoting the
(i> Pi>
7i)>
( Q 2>
given triangle
any two points by
reference being some
and area A, the distance
trilinear coordinates of
&>
72)* t ne triangle of
ABC
of sides a,
between these points
h, C)
is
or
(Ferrers' Trilinears p.
,
6).
Accordingly, the length of an elementary arc ds between
two points
may
(a, fry),
(a + da,
be written either as
or as
rfr
2
=
where
and therefore
(a cos
/3
+ dfr y + dy)
A da 2 + 6 cos BdfP + c cos C dy2
aa + b/3 + cy = 2 A,
ada + bd/3+c dy = 0.
),
CHAPTER
658
XVIII.
The corresponding expressions in Areals will obviously be
ds2 = - (a2 dy dz + b 2 dz dx + c 2 dx dy)
or
ds 2 = be cos A dx 2 + ca cos B dy 2 -f- ab cos C dz 2
,
with the identical relations
x + y + z=l,
The Areal
dx -\-dy-\-dz-0.
results are a little the simpler.
Unicursal Curves.
634.
In the case of a curve being unicursal, i.e. such that the
coordinates of a point upon it can be expressed as rational
functions of some parameter t, then if we have taken areal
coordinates x, y. z, so that their sum is unity, we may write
x
z
y
1
where
/(0=/i (*)+/() +/(*)
Let these functions be made homogeneous and
of the
same
degree, say the n
by the insertion of a proper power of
another letter r, where r=l.
th
,
The
1 '(t)-f(t)fl (t)
dx_f(t)f
~
.
2
dt
Now, by
{f(t)}
Euler's Theorem,
df
df_
dt
BT'
t
dj\^_dA
dt
frr
WW
where
and T
Jl
is
Thus
is
the Jacobian of /j and
to be put
=1
/ with
regard to
t
and
T,
i.e.
after the differentiations are performed,
dx
-
2
n f~^dt.
TJNICURSAL CURVES.
-
dy = n
Similarly
659
-J dt,
jr
n
where
J
2
J
and
Thus the
3
are respectively
areal formulae for rectification in the case of a
unicursal curve become
or
\ JV7I
^ l)C
C S
AJ + Ca C S BJ
*
These simplify a little further in the case where
possible to take the reference triangle equilateral.
635.
Ex.
1.
For example,
if it
t
1
dy_
dt
is
be required to apply this method to
rectify a circle referred to a pair of tangents inclined at 60
chord of contact, the equation is
and we may put
it
l+t + t*
dz
and the
CHAPTER
660
XVIII.
We
= 1,
take the negative sign, because we measure the arc from 0, where
the nearest point to J, and as the current point
moves from
towards
P
B (Fig.
t
178),
i.e.
decreases,
increases as
s
t
decreases,
i.e.
r 2a ^
.2* + IT
s=
j=L --ptan"
\/3
\/3
1
\
I
=
Clearly the radius -7=; hence
meaning
of the
parameter
*,
viz.
we can determine
the geometrical
=
t
Ex. 2. Take as triangle of reference any pair of tangents to a parabola
and the chord of contact. The equation of the curve then is
and we may write
'
^-
dx_oz 1-*
(i + if'
=
4
2
y (c
==
/i
yj
'
^
2
di~~(T+W'
:
di
-
fJ(bt
-
- c cos .4 ) 2 + c 2 sin 2 J
Put ^-ccos4=csin^4 tan ^
.
J [(b
;8A2
~2
<fy_
~dt~
(1
==
i
bdt
/.
;
,
+ c cos J ) cos
/(cosfltUi7W'
4- c
sin
A
sin
Where
<jf=csin A,
r
4A 2
tl
where
-<7 =
p
c sin
-,
^4
b + c cos
r
.4
/.
[^tan^-tan~
and
fe<
tan 6
- c cos /I
c sin
A
,
which, when taken between limits t lt t>, determines the length of the
intercepted arc in terms of ^, U and the elements of the triangle of
reference.
QUADRATURE AND RECTIFICATION.
661
Connexion between Quadrature and Rectification.
perhaps of historical rather than mathematical importance to point out the connexion between the problems of
636.
It is
rectification
and of quadrature.
y =f(x) be the Cartesian equation of the curve to be considered, we shall suppose a new curve to be constructed from
If
Fig. 179.
it,
\/s
taking the same abscissa and an ordinate *\ = a sec \//-, where
is the
slope of the tangent to the original curve and a is
any constant.
Then
cfe
= dx sec \]s = - doc
;
vt
Ldx
Hence the
rectification of the first
curve
the quadrature of the second.
Sec \]s may be interpreted in various
drawing of the graph of the new. curve
sec
Tangent
Subtangent
or
=
;
be regarded as
may
ways
to facilitate the
for example,
Normal
Ordinate
,
!
etc.
Accordingly, if the ordinates of the original curve be all
increased to a length tj so that
Tangent
Normal
Ordinate'
""Subtangent
a new curve will be found for which the area bounded by the
new
curve, the z-axis
and the terminal ordinates is equal to a
is a and the other side is the
rectangle, one side of which
corresponding arc of the given curve. -Also
choice, may be taken as unit length.
a,
being at our
CHAPTER
662
637.
Ex. If the ordinate of the semicubical
produced to a length
new
XVttl.
so that
tj
point thus found
17
=a
,.
parabola
2
ay =jt? be
show that the path
,
of the
the parabola
is
Find the area of a portion of
and deduce the result
bounded by two given
Art. 516, for the length of
this parabola
of Ex.
ordinates,
1,
the corresponding arc of the semicubical parabola.
Van Huraet's rectification of the semicubical parabola referred to in
Art. 516 was effected thus.
On a
638.
(Williamson, Int. Cak.,
Class of Rectifiable Curves.
If
=F(t)cosf(t)}
tiju
and
Hence
p. 249.)
_
.
/...
.
v
..
we have -^ = F(t).
Cut
i
in the curve
x=\F(t)cosf(t)dt]
J
\
y=F(t)smf(t)dt
we have s=
9
The functions F(t) and f(t) being
number of rectifiable curves arise.
at our choice
In constructing a rectifiable curve, a
make f(t) = n tan* 1 1 and make
cos(ntan-
l
<)
=
\F(t)dt.
a large
common method
is
to
use of the formulae
[
z(l-
sn
and either to choose an even value
one of the factors of F(),
639.
Ex.
1.
if
for
n be odd,
or to take (1
<
2 a
)
Thus, taking
rfrsf
x=t 2
= 2 and
\
,
^4
and
+
*ds
ji
= 1 + /2
t
;
as
to facilitate integration.
here
we have
7?,
whence
s
3
= + ^.
^(0=1 +< 2
,
A CLASS OF RECTIFIABLE CURVES.
The curve
in question is
then
-
a cubic,
2J
{X\2
1
and we have
Ex.
is
2.
,
in this curve
Let us take
the Cartesian equation of the curve.
Alsc
and the
*-*=^i
4a'
intrinsic equation is
s
Ex.3. Take
=
|
tan 2
1+
a log tan
^^(i-^
=.2^.
and
5
Then
-4o(l*-|),
and
Hence
s2
i2
Jt
= ^ y 2 and
t
Ex.
4.
,
the intrinsic equation
In the curve for which
is
663
CHAPTER
664
XVIII.
~
we have
"'++
#=*
where
y = W-t\
tan
i/>=^
*
= tan 4(9,
,
and the
if
<=tan
;
4
intrinsic equation is
the Cartesian equation being the J-elinrinant from the values of x and y.
Several examples of this class of curve will be found in Wolstenholme's
Problems (No. 1800 onwards).
640. Since
(m
2
-n2 2 + ( 2,mn)2 =(m2 +n2 } 2 we may
<
}
construct a curve
such that
and then we
shall obviously
have
where <(0> /i(0 /i(0 are a ^ a ^ our cn
form of the last method.
641.
i
ce
-
This
Ex. Let
Then
2
f2i
+
2^T2-2^
(^2;>+
artifice
amounts
to a
ROBERTS'S THEOREM.
A
642.
Theorem by Mr.
665
R. A. Roberts.
An
important transformation may be used in some cases to
derive one rectifiable curve from another, as follows
:
x + ty = u,\
Put
where
x-iy = v,)
J-
Then
2
clearly
ds'
,=v
1.
= dx + dy 2 = (dx + idy)(dxidy)
= du dv.
2
In cases where the equation of the original curve takes the
form
if
= const.,
say unity,
another curve be derived from this one by taking
it is
(p(u}(j)(v)
plain that
du' dv' = [0(^)] n
and therefore
ds'
2
=ds
du dv = du dv,
n
[</>
2
(v)]
and
ds'=ds,
and corresponding arcs will be equal.
The theorem is given by Mr. R. A. Roberts
[Proc. L.M.S.,
vol. xviii.].
Precautions.
643.
Some circumspection is necessary in the inference to be
made as to the whole perimeter of the derived curve. For
instance when the point P(x, y) of the curve, supposed closed,
traces out the complete path
P
f
0(w)0(v) = l, the corresponding
on the derived curve
point
the derived curve, or
it
may
may
not trace out the whole of
trace the derived curve several
This point must be examined in
times.
all
cases of applica-
tion of the theorem.
644.
In illustration
case, viz. that in
With
it
will
be instructive to consider the most elementary
which the primary curve
the proposed transformation,
is
the circle a?
viz. x-\-iy=-u,
uv = a 2
.
Taking the derived curve as
u
'
=
f u2
I
.
j
du.
/
v
=
J a*
we
get rf'=cfo,
N W
U
'
=
3
and corresponding arcs are
2
f^ dv.
j
/
J
r,
a2
equal.
x
+y 2 =a 2
.
iy=v, we have
CHAPTER
666
Therefore
And upon
XVIII.
3aW=#*-3zy*,
................................... (1)
3a2y = 3.^-/,
.................................... (2)
squaring and adding,
Hence the corresponding
locus
is
the circle
a2
viz.
one of radius -
.
o
The whole perimeters
But noticing that
if
are obviously not equal.
tan
and
^ = tan 6'
we put
6'
= tan 30,
= tan 0, we
and
or
6'
get
= 30,
appears that the derived circle is traced out at three times the angular
rate of the primary circle, and whilst the point P(x,y) traces the whole
of the primary circle, the derived point F(x',y') traces the derived circle
it
thrice,
and the circumference
ference of the second,
645.
this
As an
i.e.
of the
3x1
first,
viz. 2?ra, is thrice
-TT-)*
illustration of the derivation of a
new
rectifiable
method, take as primary curve the lenmiscate
?'
2
= a2 COS20,
i.e.
i.e.
or
Let us derive a new curve from this by putting
and therefore
whence
Now
ds'
du' dv' = -^Y u 2
= ds, and
-
I
j
v2
J
du dv = du dv
corresponding arcs are equal.
W=
the circum-
;
curve by
ROBERTS'S THEOREM.
667
which may be written as
50-2 sin
26* [sin
Hence
the
f
0],
parameter.
J
as arcs of a lemniscate can be expressed as elliptic integrals of
kind, the same is true of this derived curve.
elimination of u and v from the equations
first
The
in this
example may be performed as follows
:
Let
3' =
Then
(^-2),
,
Then
.'.
*
2
2)
t>(i
;
=.6, say.
say,
=-
+
3' =
=f
5
say;
A 3 B*-21A 2BZ + 93AB + 27 (^+5) + 8 =
is
the locus required, where
The desired curve is therefore one of the 12th degree, and its arcs are
same length as corresponding arcs on Bernoulli's lemniscate.
of the
646. Serret's
Mode
of Derivation of Rectifiable Curves.
M. Serret (Calcul Integral, p. 252) indicates a process by
means of which algebraic curves can be produced which are
rectifiable in terms of arcs of a circle, i.e. without the aid of
the elliptic functions.
Taking
i
and
i,
Let
*
a and
s v/
a,
b
1.
and
/3,
c
and
y, etc., to be
C any real
k pairs of conjugate constant complex quantities,
CHAPTER
668
constant quantity, and
etc.,
positive integers,
o>
xviii.
a real constant angle, and
??i,
n, p,
q,
and putting
he states that the proposed problem
is
answered by the formula
,
.................. (1)
k1
pairs of constants (a, a), (b, /3), etc., be
the result of integration algebraic. As
repeated factors in the denominator of the
provided the
chosen so as to
k
are
there
make
integrand, this will entail the satisfying of k 1 independent
conditions (Art. 149), for the degree of the denominator is
greater
To
by
2 than the degree of the numerator.
see the truth of M. Serret's assertion,
ds 2 = dx> + dy 2 =
Hence
7
and
as
s
giving
647. M.
Liouville's
Serret
observe that
fttti
=U
= Ct&u-
discusses
Journal,
C2
we
a
l
z ............................ (2)
slightly
different
form
in
vol. x.,* viz.
Here
whence
dz 2
ds2 =
'-a 2 )
and
a form readily
s
made
to
=
f
dz
depend upon an
elliptic integral.
*See also Lond. Math. Soc. Proc., vol. xviii.; Mr. R. A. Roberts; and
l(0
Cayley, Ell. Funct., Art. 448 (where the Ce is omitted).
SERRET'S
MODE OF DERIVATION.
669
In the equation (3), the denominator is still in degree higher
by 2 than the degree of the numerator, and there are two
repeated factors in the denominator hence one condition only
;
necessary that the resulting rectifiable curve should be
purely algebraic (Art. 149). The integral (3) is not in all
is
cases obtainable, but
integer and
if
one of the indices, say m, be a positive
be satisfied, the
if
the equation of condition
integration can be effected in terms of 0, involving complex
constants.
Then, equating real and imaginary parts, x and y
can be found, and when z has been eliminated the Cartesian
form of the equation of the derived curve will result.
648. The Equation of Condition.
The form
of the conditional equation
taking
<ft
_(q+a)
very remarkable,
is
viz.
2
4aa
m
itis
is discussed at length by Cayley, chap, xv., Ell. Fund..
which we must refer the advanced student for the work.
This
to
MISCELLANEOUS PEOBLEMS.
1.
Show
that any point on the Lemniscate
represented by
z
and hence obtain the
+z
z- z
%
?
%2
= a 2 cos 20 may
be
z
rectification of the curve.
[SERRET.]
Show
that the integral obtained for s reduces to the standard
Legendrian form by the further substitution
ws"
[CAYLEY,
2.
By
z
the transformation
ition
+
-
=i
t
Ell. Functions,
u,
c
show that the equation
I
takes the form
x
4-
/
=A
\
}
.
where
^
= (a -f i)Q(a
\
-
C)
(.
r(
-)'
C
d-u,
/a
A=-r-e tu>(
+ 1\ n+1 (a - iV
)
I
Art. 63.]
CHAPTER
670
XVIII.
Hence show that the condition that x +
algebraic
should be purely
iy
is
a and a being supposed conjugate, and m, n positive integers.
Discuss the roots of this equation.
[SERRET, Cak. Intfy., p.
3.
r2
In Bernoulli's Lemniscate
show that
x+ty = u
if
x-
and
iy
may
2
2
2 =
2
a4
(w -a )(v -a )
Further, expressing u
v,
be written
the equation of the curve
2
254.]
= 2a2 cos 20,
and
v
2
2
.
2
as a (l+t'
)
and a2
H + ^j
respec-
of the angle which the tangent
tively, show that the tangent
at any point makes with the #-axis is
Hence, putting the coordinates of two points at which the tangents
w 2 /z where o>3 = 1, show that the locus of the midw//,,
are parallel, as
points of chords joining such points
is
+ ^) + 9a],
i.e.
a curve of the eighth degree.
[R. A. ROBERTS, Proc.
L.M.
Soc.
y
vol. xviii.]
Obtain an integral for the rectification of the inverse of the
2 =
with regard to a point on the axis whose
4ft:r,
parabola ?/
4.
coordinates are
If
h= -
3ft,
(h, 0).
show that
s
=
3
1
-^
log
6a\/2
where
ft
tan 2 w,
2ft
tan
to
3
+
2 s/2 sin
o>
j=,
>
-2y 2 sin w
are taken as the current coordinates of a
is measured from
point on the parabola, and the arc of the inverse
the point corresponding to the vertex of the parabola.
Show
5.
that the semiperimeter
Show
{#
is
[MR. ROBERTS, loc. cit.}
bisected at the point w = sin- 1 f.
that the tangents to the parabola
points
where u
is
S i n h2
(uv)- a,
2a sinh (u
2
?/
+ ft)
at the
v)},
is
first,
then
TP + TP2 - arc P^2 = a (sinh 2v l
is
(x
a constant, intersect on a confocal
be a point on this second parabola, and
variable but v
parabola; and that if T
TP lt TP2 the tangents to the
and
=4
constant.
2*'),
[OXFORD
I.
P., 1911.]
PROBLEMS.
Show
6.
latus
where
that
taken over the area cut from a parabola of
,
rectum 4a by an ordinate distant c from the vertex (c<a),
r denotes the distance from the focus, is equal to
Show
7.
I
671
that
wv,-
.
-,
H
,
4
sin u
J
o
1
o
/
If
8.
r
-
u=e
where
c
r
dx C
</>
\
$>
c
v
2,
.
.
,
regard to
=a +
a v -..,am are (w +
greater than n,
-*-
2
+
...
,
dx
<p
\
+cnx n )fdx+Ce
J
2) arbitrary constants, arid
+ afi2 +
!
1)
a;
.
.
+ amxm
.
,
^e?i constants, show that
m be not
if
obtained by the direct differentiation of u with
,
contains only
x,
+
are (n
c n , (7
,
j
where a
+ c^ + c 2
(c
.
f
-
dx
j
|e
c
,
I
J
1) arbitrary constants.
(n+
[MATH. TRIPOS, 1878.]
xm
n
(cosh x)~ dx, where
m
and n are positive
integers, each greater than 2, prove that
(n
-
l)(n
10.
-
2)f(m, n)
Given that a and
and w->oo of
if-r+
w \a
is finite
1
f
(a
L
11.
= (n
V
cV +
+ -)
The
;
increase
1
/
+
\
and
dS
/(^, n 2)
2
2)
are positive,
c
(a
?t/
when r>l
-
1
2c\ 1>+ /
)
w/
(a-*-
\
-
x-a
dx.
-
l)/(m
[OXFORD
show that the
+->>
+
3cV
nj
Similar laws,
is
--.,
c
w
I.
S by an
P., 1914.]
increased
expressed by the law
*
z-c
,
hold for two other commodities, where X, //, i/, a, b,
Find how the man should expend a given sum
is
when
viz.
y-b
that his total satisfaction
2).
+ m?i-
[Oxr.
man's satisfaction
w-
limit
\
find this limit.
in a
2,
I. P., 1914.]
1
/
(rt
J
expenditure dx on a certain commodity,
dS =
m(m -
greatest.
c
are
all positive.
E (>a + b + c)
[OXFORD
I.
so
P., 1914.]
CHAPTER
672
Show
12.
that the
maximum
XVIII.
satisfaction
measured b
is
Evaluate
dx
J^-aLL +
i
vT^>
[OXFORD
1 3.
Show
II. P., 1914.]
that the tangent to the curve
at the point whose abscissa is h, cuts the curve again at the point
whose abscissa is - 2h, and that the area included between the curve
and the tangent
14. It
9A 4/4
2
[OxF.
.
I.
P., 1918.]
are both polynomials in x, show that the
with
/1 (#)//2 ()
respect to x can always be written in the
f^x)
integral of
is
and/2 (z)
form
^
(x)/<j> 2 (
x)
+ log
</>
3 (z)/< 4 (x),
where 1? 2 $3> ^4 a ^ so denote polynomials, not necessarily real.
Find the general form of the integral with respect to x of
[Oxp. I. P. 1918. ]
Jx*^l).
/i (a; + N/a^l )//2 (a;
<
</>
,
,
15.
Show
bounded by the curve
Sat
_ 3aP
that the area
=
asymptote x + y + # 0, and by two lines at right angles to
= -a, t = of the curve, is
asymptote through the points t
its real
this
4 {
and
find the
1+
(^ + tt + l)2}'
iy
whole area between the curve and
its real
asymptote.
[Oxr.
16.
If <f>(z)
^z^
range
ft
I
<
I.
P., 1917.]
be a rational function of z without singularities in the
1,
prove that
2
(sin 2x) cos x cos
2x dx =
rf
I
^ (sin
2
2
2x) cos .? cos 2x dx
1
Jo
Jo
=
I
</>(sin 2a;)
cos% cos 2x dx.
Jo
[OXFORD
17.
Integrate
(i)
I
J
(x-
(li)
1 C
b}\(x
- bY b -
i,
(x
- a} 2a |*
I.
P., 1907.
PROBLEMS.
18.
In the curve
(
\a
(i
s
being measured from the
Show
that the curve
is
~( a
\
]
/
673
?/-axis is
/
Too
(Q
loy/
)
19. If
2</>
be the eccentric angle of the point
= r(l -e cos 8),
K\
an axis
of
"2"
(Q a2
\o/
'
)
on the
r,
ellipse
prove that
+e)
{(1
Use the
,
origin.
a quintic of which the
\
c
= x 2 +ij?/ 2
s2
show that
i
)
/
2
- 4esin 2
</>}
f
-y-7
=4(1-
j
fact that
co.*0 <W =
and the above
ofVco
J
to obtain a value of a, such that
Jo
[OXFORD
20.
r
i>
l/i')
A
a>
uniform rod of mass
2>
3V
Show
respect to the axes
is
I.
P., 1917.]
M
has its extremities at the points
that the product of inertia of the rod with
given by
o
Hence show that the product
of inertia of the rod
is
the same as
that of three particles of masses
M M
6'
6'
2M
3
'
placed at the extremities and the middle point of the rod respectively.
[OXFORD
21.
Show
P., 1913.]
that the coordinates of any point on the curve whose
intrinsic equation is
where n
I.
s
= a(sec n i^ -
1),
an odd integer greater than unity, can be expressed
= Q the curve
rationally in terms of tan ^, and show that when x
is a cubic with a
[Oxr. I. P., 1911.]
cusp.
22.
is
Show how
to evaluate the integral
y
and f(x,
E.I.C.
I
f(x, y] dx,
= ax
y) is a rational function of x
2u
and
y.
where
CHAPTER
674
XVIII.
Prove that
a
dx
the positive sign being taken for the radical in each of the subjects
[MATH. TRIP., PART. II., 1913.]
of
integration.
23.
Show by means
and verify the
of the transformation y
+ $
-that
x ^-
= -(x
2
1
-}-
result in
1
an independent manner.
[MATH. TRIP., PART
^
24. Integrate
II., 1914.]
dx.
Evaluate
25.
Jx
+2
2
(a;+l) (x
dx
f
2
f
2
+ 4)'
J(x +l)
4
'
J (5
dx
- 3 cos 2
z)
'
and the corresponding definite integrals taken between the limits
oo ) and (0, TT) respectively.
[MATH. TRIP., PART II., 1914.]
(0, oo ), (0,
26.
,.,
Show
that
fsin4
\/3
j
JsTn^=T
M
,
tan
,/sm2x\
\
COS
6/
27T
sin 3x ,
fsi
n)' I -; =- dx
sin 5x
Jsi
\
27.
=
1
-=
5
sm v
r
r:
+ sm
V
Prove that
r
28.
log
de
Prove that
2 cos
I (1 + sin
+ sin
6 cos 0)%
2 sin 9
JQ
av
'
(1
+ sin
cos
0$
CHAPTER
XIX.
MOVING CURVES.
Quadrature and Rectification of Loci of Carried Points and
Envelopes of Carried Lines.
649.
"
Instantaneous Centre."
a very well-known geometrical theorem that if two
triangles ABC, abc are equal in all respects and lie in the
It is
same plane, the one can be superposed upon the other by a
rotation about some point in the plane.
Fig. 180.
7.
Let XI, YI, the perpendicular bisectors of Aa, Bb, meet at
Join I A, la', IB, Ib\ 1C, Ic\ and join / to the mid-point
Z of
Cc.
675
CHAPTER
676
Then I A, AB, BI being
XIX.
respectively equal to la, ab, bl,
A
A
the triangles IAB, lab are congruent, and angle IBA=Iba.
A
A
Hence IBC=Ibc, and having also IB, BC respectively equal
to Ib, be, the triangles IBC, Ibc are congruent, and 7(7 =7c;
whence IZ
bisects
dicular bisectors of
A
A
A
A
\
AIB
being equal to alb, and BIC being equal to
A
A
A
clear that
angle
it is
Cc perpendicularly, so that the perpenAa, Bb, Cc are concurrent. Moreover
AIa=BIb=CIc,
blc,
A
and therefore a rotation through the angle Ala about the
point / in the proper direction will accomplish the superposition of the one triangle upon the other.
Aa, Bb are parallel, 7 is at oc in the plane, and the
motion is one of translation without rotation.
If
Two of the three points A, B, C may be regarded as fixing
the position of the lamina upon which the triangle is drawn,
and the third point may be regarded as any point carried by
the lamina.
any shape in its own
plane may be regarded as brought about by a rotation about
a point in its plane, and any consistent motion of two points
Thus a displacement
of a lamina of
attached to the plane lamina will define the motion of the
lamina in
its
own
plane.
If the equal angles
650.
Ala, Bib be infinitesimal, Aa, Bb
may be regarded ultimately as the direction of the tangents
to the paths of A and B, and 7 is called the instantaneous
The position of this point is immediately discovered
the direction of motion of the two points A and B are
known, by drawing through A and B perpendiculars to the
centre.
when
direction of motion of these points these perpendiculars meet
"
"
If 7 be joined
7.
instantaneous centre of rotation
;
in the
to
any other point
path of
P
is
P
of the
moving lamina, the tangent
and PI is the normal
at right angles to PI,
to the
to the
path.
651.
For instance,
if
a hoop of any shape be in motion in a plane, and
the direction of motion of two points of the hoop be known, say, P7\ QT
then / is at the intersection of perpendiculars through
and Q to PT,
P
respectively,
and the motion
of
any other point of the hoop,
,
QT
/?,
is
at
INSTANTANEOUS CENTRE.
right angles to IE.
motion of
Hence
all particles
at
677
any instant the directions
on the hoop envelop the
first
the hoop with regard to the instantaneous centre.
of instantaneous
negative pedal of
When the hoop is
Fig. 181.
be an ellipse if / falls within the hoop, a hyperbola
without the hoop, and a point if / falls upon the hoop.
circular, this will
falls
652.
if
I
The instantaneous centre
fixed point.
If it
itself is not in general a
has a path upon the fixed plane, it has
another path relatively to the moving lamina.
When
a circular hoop rolls along the ground in a vertical plane, the
is the instantaneous centre, for at any instant the
point
point of contact
Fig. 182.
hoop in contact with the ground is not moving along the ground,
by supposition there is no slipping, and it has just ceased to approach
the ground, and is on the point of beginning to leave the ground, and
therefore for the instant it has no motion at right angles to the ground.
The path of the instantaneous centre on the fixed plane is evidently the
line on which the hoop rolls.
The path on the plane of the hoop is the
of the
for
hoop
itself.
CHAPTER XIX.
678
653. Exactly the same is true when any curve traced upon
a lamina is made to roll without sliding upon a fixed curve.
The point
of contact
is
the instantaneous centre.
7-loci are respectively the fixed curve and the
themselves.
The two
moving curve
654. When a rod AB, of given length, slips down between two perpendicular axes 0y, Ox, the instantaneous centre I is at the intersection of
the perpendiculars AI, BI to Oy and Ox, and its locus on the fixed plane
y
Fig. 183.
is
and radius equal
a circle with centre at
The path
to the rod.
relative
to the rod is a circle of radius half the rod, described on the rod for
diameter.
Any point P attached to the rod describes an ellipse, of which
to IP is the tangent.
the normal and a perpendicular through
IP
is
P
655.
General Motion of a Lamina reduced to a Case of Rolling.
Let us define the manner of motion of the lamina to be
angular velocity at every instant is some given
I
72 7 3 7 4 7 5 ..., being the corresponding sucquantity;
19
cessive positions at equal intervals of time dt of the instansuch that
its
,
,
,
taneous centre on the fixed plane upon which the lamina
moves.
Let d^, d\Js 2 d\]s s ,... be the infinitesimal angles turned
,
Then
I19 7 2 7 3
through
(a) Let there be a rotation d\}s2 about 72
Then a line on the moving lamina, which was originally
coincident with 7 2 7 1? will be brought by rotation about 7 2
in successive rotations about
,
.
into the position 72 / r
,
THE TWO
Now
(6)
let rotation
Then the
line I3 I2 ii
the position /3
(c)
i
commence about 73 through d\]s B
on the moving lamina is brought into
.
2 i'.
Let rotation
Then the
679
/-LOCI.
now commence about
line 7 4 73 f 2 // is
74 through
brought into the position
now commence about 75 through d\//- 5
l^l^i^' is brought to the position 75 4 3 2'V"
(d) Let rotation
Then the
and so
line
.
/
/
'/
on.
Hence
it is
clear that
when
the intervals of time are infini-
tesimally small, and the chords $/, 7 2 73 etc., indefinitely
diminished, the motion of the lamina may be constructed by
,
the rolling of the curve locus of the instantaneous centres
"/
u P n the curve locus
relative to the lamina, viz. 7 5 4 3 '< 2"f 1
of the instantaneous centres upon the fixed plane, viz.
f
/
Hence the general motion of a lamina in its own plane
be constructed by the rolling of one curve upon another.
may
It therefore
and
becomes important to study the motion of points
which roll.
lines attached to curves
656. The
The
Two
Loci of the Instantaneous Centre.
locus of I both on the lamina itself
and on the
plane upon which the lamina moves becomes important.
may be readily found.
fixed
Each
Let OX, OY be fixed rectangular axes upon the fixed plane.
Let O'x, O'y be rectangular axes attached to the moving
lamina.
CHAPTER XIX.
680
Let
rj
(-,
be the coordinates of 0' relatively to OX, OY x, y
P on the lamina relatively to
;
the coordinates of any point
O'x, O'y.
Let
be the inclination of O'x to OX.
The motion
of the lamina will then be fully defined by the
three coordinates
q, 0, and their differential coefficients with
regard to time, where f and
//
are definite
known
functions of
0.
Fig. 185.
The coordinates
of
P
relatively to
OX,
OY
will be
X=-\-x
y
Y=r)+x sin 6 + y cos 0..J
cos
sin 0,,
\
(1)
Differentiating,
dX=dg+(dx-y dO) coa6-(dy + x dff) sin 0,
d Y=dtj + (efo
To
at
y
eZ0)
any
instant,
-\-(dy
+
a;
6Z0)
it is
for the
for the
Hence
0.
moment
moment
is
turning
stationary in space,
stationary in the lamina.
for this point
dX=dY=0
Therefore
dg-y dO cos
drj
r\
cos
which the lamina
we must remember that
(6) it is
(a)
and
sin
find the position of / about
being
y de
known
and
dx=dy=0.
0x dO sin 0=0,1
smO + xdO cos 0=OJ
functions of
0,
at such a point,
x and y are found from
.(2)
and the 0-eliminant from these equations gives the locus
on the lamina.
of /
THE TWO
Next, substituting in equation
681
/-LOCI.
(1),
.(3)
and the $-eliminant from these equations gives the /-locus on
the fixed plane.
657.
Ex.
straight
1.
lines
Taking the case of a rod AB ( = 2a) sliding between two
OX, OY at right angles, making an angle & with the
Fig. 186.
latter,
and taking the centre
and the rod
itself as
of the rod 0' as origin for the
= a cos 6 cos d
a sin
sin
= aBin20,
Q = a cos 2#,
X = a 8 in 6 + a sin
J
moving axes
the y-axis,
= 2 sin 0,
= a cos # -f a cos = 2a cos #,
and the locus of / on the lamina
and on the fixed plane
1
tiona (2)
J
tions
is
<2
,i
+ y 2 = 4a 2
i
;
geometrically obvious (see Art. G54) as indeed are also all the
equations established, the point / being at the intersection of the peras is
;
OY
pendiculars at B and A to OX,
respectively.
All carried points which lie on the circle with
for diameter
describe two cusped hypo-cycloids, i.e. straight lines, and all points
AB
attached to the line itself describe ellipses (see Besant, Conic Sections,
Art. 245).
CHAPTER XIX.
682
Ex. 2. Taking the case of an involute of a circle of radius a, sliding
between two perpendicular lines OX, OY, let the radius of the circle
with the line OX. Then
through the cusp make an angle
*= a ( s
e ~ cos *>
y = a(cos0 + 8in0),
} from equations
(2)
;
J
X = a(
equations ( 3 )
Y=
Hence the
locus
of
/ on the lamina
the locus of / on the fixed plane
These
loci
are
shown
is
is
2
;
,
Y X = 2a - 7T
,
i.e.
i.e.
a circle
a straight
line.
in Fig. 187.
Fig. 187.
.
The first of the loci is geometrically obvious, as the tangents from / to
the generating circle of the involute are at right angles.
The motion is that of the rolling of a circle of radius av2 upon a
makes an angle - with the axes OX, OY and an
on the F-axis. The locus of the starting-point C of
straight line which
intercept (2
-
W
is plainly a trochoid, and the locus of the centre of the
generating circle a straight line. Points on the circular /-locus describe
cycloids, all other attached points describe trochoids.
the involute
The student will find this example done (in a
and Glisettes, p. 37. The object here
Roulettes
the general formulae of the preceding article.
different
is
way)
in Besant's
to illustrate the use of
THE TWO
/-LOCI
683
Ex. 3. Consider a case of motion of apparently different nature.
Let a lamina PQR rotating at a constant angular velocity <o be moving
so that an attached point C describes a straight line with uniform
velocity
v.
Take the path of C as the axis of X, and
the coordinates of the
77
centre, and 6 the angle turned through in time t, and suppose that
and
both vanish. Let accents denote differentiations with
initially
,
regard to 6
P
Fig. 188.
Then,
being the starting point for the point
The equations
of Art. 656 give
,
V
V
w
.'.
i.e.
1
t.'
=:
the /-loci are a straight
'
a circle whose centre
The motion
x=
to
line, I
7
w
(7,
sin
=-, on
the fixed plane, and
on the lamina.
is C.
therefore that of a circle rolling on a fixed straight
All carried points describe cyc/oids or trochoids.
658.
is
the point C be made to describe a circle of
lamina rotates with an angular
to, whilst the
we have, taking rectangular axes through the centre of the
and rectangular moving axes through the point C attached
In the same way,
if
radius a with angular velocity
velocity to',
fixed circle,
line,
to the lamina,
and supposing
77
and
to vanish together,
6 = <a't',
9)= a sin cut,
o
*
=a co--- .-cos (a
=--rk
b
a>
a>
aQ
d
=w + -=g = a
"n
to
.
to
i
,
dQ
'
y=
*J
Ct(0
*.\
r
.
sin
to'
to
to
to
-r
/./
n
a,
a>
.
sin
t6',
to
CHAPTER XIX.
684
and the motion
that of a circle of radius
is
a
of radius
,
and therefore
all
upon a fixed
rolling
circle
carried points on the lamina trace
epi- or hypo-cycloids or epi- or hypo-trochoids.
659. Ex.
angular
Suppose that a point 0' of a lamina PQR travel upon an equiwith pole 0, fixed upon a plane over which the lamina
spiral,
Suppose that the lamina rotates at -th of the rate of the radius
slides.
vector 00'.
It
required to reduce this motion to one of rolling.
is
Fig. 189.
Let 00' make an angle 8 1 with the
on the rotating lamina make an angle
Suppose
If
.
17
OX
be taken such that
be the coordinates of
n
0',
initial line,
Then
f
'=
T/'
:.
by Art.
^=
fixed in space.
l
MI-
spiral.
= ~( = 7me* lCota (cot a sin
:
t
att
- sin
0,),
+ cos 0,)
;
656,
-n cot a sin
ta
[(1
Putting
a line O'x fixed
OX
we have
^ (cot a cos
nae* 1
let
Then B = nO.
vanish together.
SM
with the usual notation as to the
and
with the axis
- n) sin
X
0J,
+ n cot a cos 0J
.
l-n = kcosj3, ncota = ksin/3,
I
i.e.
the locus of X,
nates,
?'.?.
Y is R = kae^ & ~^ coia
'
1
where 72,6 are current
and
the fixed /-locus
is
an equal equiangular
spiral.
coordi-
CURVATURES OF THE
x = 'sm Q- if cos
Again,
6,
?/
=
685
/-LOCI.
+ rj'sin
'cos
t),
and
and
7^, Bj be the polar coordinates of a point on the /-locus
if
laniina
R, cos
>
/?! sin
e = f sin - r;' cos 0,
= cos d + t]' sin #,
x
'
X
the polar equation of the
/.
upon the
'
(.,
?/)
locus
nCOtg
/
is
@i _ a _T\
sina
i.e.
another equiangular
by the straight
spiral,
r\
(3 1
line.
= TT^ + a, when
replaced
n = l.
is
upon another
The
same rate as the radius vector of the original
660.
is
therefore that of one equiangular spiral with angle a,
of different angle, or when n=l, upon a straight
case when n = 1 is that in which the lamina rotates at the
The motion
rolling
but of different angle, which
line
The Curvatures
It will
of the
two
Loci.
spiral.
Analytical Consideration.
be found in later articles that
we
frequently have
to find the difference of the curvatures of these
Arid for convenience
Arts. 665, 667,
concavities
of
and
the
opposite directions.
of
drawing
it
is
two
/-loci.
customary, as in
in Diff. Calc., Ch. XX., to consider the
fixed and rolling curves as being in
That
is,
the expressions
which
1
Pi
Pz
and Glisettes are the algebraic
measured in the same direction.
occur in theorems on Roulettes
differences of curvatures as
For the present we consider the concavities in the same
Both the /-loci have been found in the form
direction.
CHAPTER
686
XIX.
x = F(0), y=f(0), and therefore the curvatures can readily
be obtained from the formula
Representing by accents differentiations with regard to
0,
we have
For the /-locus on the fixed plane,
(a)
X =f - n
x'=f- n
Y =, +f,
r =,'+",
',
",
and
and
zT"-z"r=(r-,")(,"+n-(i'+n(r-i'");
pj be the radius of curvature of this fixed Z-locus,
if
1
tf'
- 1") (l"+
For the locus
(6)
x
='sin0
of /
'")
~
(
on the moving lamina,
/cos0,
^^
and
And
if
movin
p z be the radius of curvature of the /-locus on the
lamina estimated in the same direction as 7,
Hence
1
1
=
which gives the difference of the curvatures sought.
BESANT'S EQUATIONS FOR THE FIXED LOCUS.
687
Finally, x'*+y'*=(
and therefore
if
ds be the elementary arc of either curve,
--- = -,-.
whence
p.2
as
pl
dO
Geometrical Consideration.
661.
This last result
and
ds
may
ds
for Pz
their dif-
be seen at once geometrically
;
.
are the angles turned through
by p 2 and p lt and
Pi
ference
is
the angle turned through by the moving lamina,
-~-=da
Pz
662.
(1)
(See Fig. 190.)
Pi
Thus, in the case of the sliding rod of Art. 657, Ex.
=
acos#,
r}"=
a cos 8,
77
"
=
a
sin 0,
P2
1,
\ve
have
*
'
and
i.e.
Pi
which agrees with the previous result for which p l = 2a, p 2 =a.
(2) In the case of the sliding involute (Art. 657, Ex. 2),
and
P2
Pi
which agrees with the previous
gives
2#
in case (1) above,
\/a?
+ a? <W2
result, for
and a0\/2
which
^ = 00,
p2 = a*/2; and
in case (2).
663. Besant's Equations for the Fixed /-locus for sliding curves.
When
the motion of the lamina
is defined by two curves
making sliding contact with fixed
perpendicular axes OX, OY, the equations
attached to the lamina
*=<?-,',
F=,+f
CHAPTER
688
X' = g*i'
give
= Y-
F_ +
and
and show that
X' -
F=
XIX.
,.
(n
+ rT) =
"
,
eSPe<
_
p v \ by Legendrc's
= p*)
formula,
where p and p 2 are the
radii of curvature of the sliding curves
Y.
at the points of contact with the straight lines OX,
These equations are obtained by geometrical considerations
(Roulettes and Glisettes, Art. 51), and are the
he
uses
for the determination of the /-locus on the
equations
fixed plane in such cases of sliding contact.
They require
the integration of two simultaneous differential equations for
the determination of the locus.
by Mr. Besant
When
the intrinsic equations of the two curves are known,
s=/2 (\^-), Mr. Besant's equations are very con-
viz. 8=fi(\fr),
venient, and the fixed /-locus can be
the simultaneous equations
dX
,,
.
deduced by solving
dY
the constant being determined by the starting conditions.
664.
"
Roulettes and Glisettes."
The path of a point carried by a curve which
another curve is called the Roulette of the point.
rolls
upon
(See Diff.
Cak., Art. 561.)
The path of a point carried by a lamina which moves so
that a curve drawn upon it slides in such a manner as to
touch two given fixed curves is called a Glisette.
The latter name is due to Mr. W. IT. Besant.
The terms Roulette and
Glisette
have been extended
include the case of the envelope of a carried curve.
very full and interesting account of the
A
properties of Roulettes and Glisettes was given
in his Tract on Roulettes and Glisettes (1870).
to
principal
by Mr. Besant
The Curvature of the Roulettes described by a Carried
and
as the Envelope of a Carried Curve are worked out
Point,
The
in Articles 564 and 565 respectively of the Diff. Cole.
revise
student who has not access to Mr. Besant's tract, should
665.
ROULETTES.
689
these articles before reading the articles which follow, which
are mainly concerned with quadrature and rectification.
The formula
established for the radius of curvature of the
envelope of a curve carried by another curve
ithout sliding upon a fixed curve is shown to be
which
rolls
r
___
R-r
rolling
R
that
of
p2
radii of curvature of
Here p v p 2 are the
curves
r+p~ Pl
respectively,
that
p
whilst r
its
envelope,
the point of contact of the
carried curve with its envelope
from the point of contact of the
of
is
the
the fixed and
carried
the normal
curve,
distance
of
fixed
rolling arid
the angle
is
r
common normal
If
all
can
be
\j/,
the
curves, and
<
makes with the
of the latter.
these several quantities
expressed in terms of
angle
with any fixed
which
line,
r
makes
then IjRd^r'
gives the length of an arc of the
envelope,
i.e.
COS
-f
II
(j>
""!""
Pi
This
COS
'
Pz
^
'
P
the general result. It
includes the roulette of a carried
point,
is
viz.
when p=0,
or
of a
carried straight line (when p=oo),
or the case when the fixed curve
a straight line (p l =cc), or when
the rolling curve is a circle (p 2 =a), or
is
is
when the rolling curve
a straight line (^2=00), or any combination of sucli cases.
The standard figure is that shown above and described in
Diff. Cole.,
Art. 565.
If the concavity of any of the curves be
formula will require modification
in the
opposite direction, the
E.I.C.
2x
CHAPTER XIX.
690
of sign in the particular radius of curvature or
particular radii of curvature involved.
It must be remembered from Diff. Cole., Art. 565, that the
by the change
angle between
two consecutive
positions of p l is
,
Pi
ds
.
r is
ds cos
.
is
p
mi
J
Thus
ds COS
'
i
arc of envelope^
H r + cos
p
|
JL >!
COS0
since
,
ZR-r
-
/
Rr
2+D
^D
plainly
>
I
,
p2
Pl
\
r
,
\
is
11
COS0
R-r-~\ r + p
-^
d> Ids.
IR-r
1
and
+p
p2/
Again, the area swept out by r
666.
^
;
r(
I
r
'
<f>
tty/=-=
.'.
R-r
)cos0=-2cos^ + ^
r/
/I
\p l
1
cos0\
----H
p2
r
+ p/
1
i
.*.
r
667.
sweeps out an area
When
the carried curve reduces
cos^)=r-T-, where d9
is
a
to
point,
i.e.
vectores of the rolling curve.
Hence, for a carried point,
Arc of roulette =
I
J
and
p=Q,
the angle between consecutive radii
Area swept out by
r= 5
r(
Vi
Ir
J
+
}
2
(
\pi
ds,
/v
|
2
}ds + ~ \r d9.
p2/
AJ
ROULETTES.
Hence the area swept out by
r
691
exceeds the corresponding
portion of the sectorial area of the rolling curve,
viz.
by
Iffdfl,
Zj
f
lf
ZJ a(!+i)(fe
\p 1 p 2 /
the rolling curve be a straight line, p 2 =
expressions reduce further to
And
if
fr
Arc=|
1 (V 2
Area swept = -,
and
ds
}pi
ds
oo,
and these
1 f
+ ^\r 2 dO
2J
2Jjpi
respectively.
Important Cases.
668.
case, perhaps, is when a curve which
or
a
carries a point
straight line rolls upon a, fixed straight
The most important
line.
In this case p l
=^
.
If also the roulette be that of
r
^r
a carried point, p = 0,
r2
r.
7
/> 2
cos0
r
/o 2
cos9
be that enveloped by a carried straight
If the roulette
line,
P =oc,and
In these cases
usual
</>
is
makes with a
roulette
the angle which the normal to the
and in accordance with the
fixed line,
custom in dealing with intrinsic equations
written
be
may
\//.
Hence the
intrinsic equations of the roulette in the
two
cases will be respectively
s=
I
\r-\~fa
cos
~
\Js
-J
s= \(r + p2 cos\ls)d\[s,
669.
It is to
for a carried point,
d\Is,
be further noted that
for a carried straight
if
the concavity of
line.
any
of
the curves concerned be turned in the opposite direction to that
in which they are represented in
Fig. 190, the general formula
for
will need modification by the corresponding change of
R
the particular radii of curvature involved with a
corresponding modification in all the deduced results. To
sign of
avoid error
it is
therefore desirable to examine each case on
CHAPTER XIX
692
its
own
merits, rather than to deduce the formulae required
from the general result
cos
cos
R-r
r
</>
_
+p
1
1
Pl
p.2
'
Moreover, special cases have their own special geometrical
Hence, in succeeding articles, we adopt this
peculiarities.
course though it necessitates some repetition.
This will
also
have the advantage of exhibiting a somewhat different
treatment.
670.
line.
A
1.
circular wheel rolls in a vertical plane along a straight
find the intrinsic equation of the envelope of a given diameter.
Ex.
To
Fig. 191.
Here p 2 = the radius
r=acos(f>
of the wheel = a, say
;
;
i.e. the
s being measured
envelope is a cycloid with an axis of length
from the vertex of the cycloidal envelope.
For a parallel chord at a distance h from this diameter, we have
,
and
s
= h<f> -\-2ci sin. <,
a parallel to a cycloid.
Moreover, the cycloid which is the envelope
the diameter of the rolling circle, is itself an involute of another
Hence the parallels to the cycloid are involutes of a cycloid.
cycloid.
viz.
of
This then
is
the result for any carried
line.
Let the rolling curve be rn = a n cosnd, and suppose the initial
position be that in which the vertex of a foil of the curve is in contact
with the line.
Ex.
2.
First, let us find the roulette of the pole.
n+l
r
We
have
_r
i
.^w
'
r2
_ dr_
,/~,
an
1
i
1
'
/v.n
1
ROULETTES.
Let
P
693
^
the pole, A its initial position,
the
to the roulette, x' Cx the fixed
be the point of contact,
angle turned through by the tangent at
line.
tan OPx' =
Then
y- = -
n
cot
OPx' =
/.
;
-
and
<b
+nd\
=
\lf .
R = - COS
p2
ds
n+l
a cos'
(f>
1
l
n
dy
and
s
= 71+1 a
f
I
~
cosn
^ dif/
is
the intrinsic equation sought.
Fig. 192.
If w=l, we have the case of a rolling circle of diameter a, and the
intrinsic equation of the cycloid traced is s
2asini/'.
In the general case, if we refer the curve to tangential polar coordi-
=
nates,
we can perform one integration. For taking
l
dz p
ds
n+1
A
as pole.
.
n
a cos"
d/.
d\f>
Multiplying by sin ^,
.
sin
and integrating,
for
and
-
sin
vanish
-
a
i
^
if
-
p cos ^ = - a
cos
1
cos"
+
,
\f/
,
sin y,
^ + a,
and p be measured from the vertex A
=a
cosec
;
CHAPTER XIX.
694
Again, multiplying by cos
i
dp
ay
\p
and integrating
.
+p sin
cos \p-fj
,
\j/
=n+ l a
n
r$
cos
n+l
dp
-ff+P tan y = n a
ay
.
,
or
Eliminating -^,
we
+i
.
7
,
y dy,
sec if;
r
r^
cos
1
1
+
.
.
.
y aw.
J
obtain
n+1
[^
?ir
1
+-
+ 1
1
/
-
I
Jo
= n + l a sin
or
i
J
w
"?
r*
it*
cos
/
/
^
d\L
-a cos
it'l
1
1
-cos
n
\
if/ ),
V
Jo
as the tangential-polar equation of the roulette, the origin being at the
vertex of the roulette.
To
find the roulette enveloped
by the
axis of the rolling curve,
we
the angle between a parallel to CA and
<j>'
</>',
the perpendicular upon the axis of the curve, and r' is the perpendicular
from P upon the axis.
have
R=r'+ p 2 cos
Then
where x
where
is
<f>'
is
the angle the axis of the rolling curve makes with the line CA,
C(
JL
r
cos
nO
1
sin n
= a cos n nQ\p si
+l
(n+l)cos7i6U
X
and the
si
intrinsic equation of the envelope of the axis is therefore
= /""/sin-V
?i+l
J ^
\
Ex. Special Case of the Epi- and Hypo-cycloids.
Here
pi
= a,
rolling circle;
and
the radius of the fixed circle; p* =
b)
p = 0.
cos
-5
R
r = 1 hr1
c/)
r
a
b
cos
-
r
(A
a + 26
1
= ^1 + 777-=
a
26
26
,
;
the radius of the
ROULETTES.
ds
.
i
,
measured from the vertex increases as ^ diminishes)
(s
s
a+ b
a
= ~ 46 ^T^
sm ^T1>A y
.
7JJ
695
being measured from the vertex (Art. 412,
Diff. Cole.).
V,
Fig. 193.
671.
When
p1 =
x
,
the formula for the roulette of a carried
point,
R=r
viz.
7 .2
,
/Q 2
cos
</>
expressible otherwise.
For with the usual notation, taking the carried point as the
pole of the rolling curve,
is
p9 =
rdr
1
T:
Hence
and
-.
=p
cos
P dr>
?'
2
d
T
-
.
r
dp
-
dp \r/
which gives a convenient measure for
,
R
in this case.
General Theorems with regard to Rolling on a Fixed
Theorems of Jacob
Roulette of a Carried Point.
Straight Line.
672.
Steiner and
W. H.
Besant.
be any curve rolling along a straight line xz, P
the
the adjacent point on the curve
being
point of contact,
which will come into contact with the line at Q. Let
be a
Let
APE
P
CHAPTER XIX.
696
carried point and 0' the point at which
rolling of the curve has carried P' to Q.
arrives
it
when
the
Let OY, OY' be the perpendiculars from
upon the contiguous tangents at P and P'. Let 00'= dor, the elementary
arc traced by
as the point of contact travels from P to Q.
Let O'O cut xz at R. Then OY plays a double part.
X R
Q
Fig. 194.
of the roulette of 0.
(1) It is the ordinate of the point
(2) It is the radius vector of the pedal of the rolling curve
with regard to 0.
Let the elementary arc of the pedal curve,
dsp
viz.
77', be called
.
A
n
A
^Lt sin zRO=Lt cos RPO,
Then
da-
tor
OP
is
the normal to the roulette (Art. 562, Diff. Calc.)
A
dOY dt/
=Lt
cos
077=-,ds
p
That
is,
in the limit,
Hence corresponding
=*
.
,
dsp
da-=dsp ............................... ( 1 )
arcs of the roulette of
and
of the
w
:
are equal.
th regard to
pedal of the rolling curve
This theorem is due to JACOB STEINER (179G-1863) *
673, Again,
we have
if
OZ
be the perpendicular from
on YY',
ultimately
y j=Lt y cos
zRO=Lt y sin RPO=Lt y sin OY'Y
=LtysinOYZ=OZ',
.-.
i.e.
the element
OYNO' is
y dx=OZ d<r=OZ (h
l)t
ultimately double the element OYY'.
*Cajori's History of Mathematics, p. 295.
STEINER'S THEOREMS.
697
Hence
integrating, the area swept out by the ordinate of
the roulette during any portion of the rolling is double the
corresponding sectorial area of the pedal curve.
This theorem appears to be due to the late
(Art. 26, Roulettes
We
674.
and
W. H. BESANT
Glisettes).
consider next the area swept out by the normal
OP
to the roulette.
Draw
PM
Let
perpendicular to O'Q.
makes with the
be the angle
<
OP
tangent.
PF or PQ
YOY=^.
We have PM=Sssiii(j), Ss being the element
the rolling curve. Let OP=r, POP'=S9 and
Then to the first order,
Quadrilateral
(for
OYY'P
OPQO'=%OP
being ultimately cyclic,
.
00' + 0'Q
of
PM
.
YF=diam.xsin 70 F)
area swept out by normal in any portion of the rolling
.*.
= corresponding
the limits for
\Js
being
curve +
sectorial. area of
its initial
and
final values.
If the curve be a closed oval, every point of whose
perimeter comes into contact with the line in one revolution,
675.
and
if
to the line,
27r,
the rolling to start with OP at right angles
so that the limits for ty may be specified as
to
we suppose
we have
for a complete revolution
Area swept by normal = area of rolling curve -f ^
*J
I
=area
r 2 d\{s
1 f2* r2
of curve -f -.
ds.
'2J
But by
Art. 426
2 area of pedal
.-.
= area of
1
curve -f
area swept out by normal
j
in a complete revolution J
This theorem
*See Bertrand,
is
also
due to
Calc. Intey., p. 362
=2
^
2r
f'
r
I
2
ds
area of
;
dftl
Steiner.*
andBesant, Houleltes and
Glisettes, p. 19.
CHAPTER XIX.
698
676.
It is
worth noting
also that
1 f2 *
Area of oval =2 area of
pedal-2 J
I
(Besant, R. and
677.
1.
G., p. 19.)
Illustrative Examples.
When
of the focus
an
ellipse rolls
upon a straight
line,
any arc of the roulette
equal to the corresponding portion of the circumference of
the circle which is the first positive pedal of the ellipse with regard to
is
the focus,
The
i.e. the
auxiliary circle.
roulette of the centre is of the
same length as the corresponding
arc of its central pedal, viz. r2 =
\
a
Fig. 195.
And
in
both cases the areas swept by the ordinate are double of the
corresponding sectorial area of the pedal. In a complete revolution
these areas are 2:ra2 for the area swept by the ordinate of the focus in
a complete revolution of the ellipse and 7r(a 2 + 6 2 ) for the roulette of the
centre.
These paths are illustrated in the accompanying diagram.
2.
The arc of the
on a straight
line
first positive
pedal of the circle with regard
the point is on the circumference of the rolling
see that the arc of a cycloid is of the same length as the
of the limagon
to the point.
circle,
we
roulette of a point rigidly connected with a circle rolling
a Trozhoid) is equal to the corresponding portion
(i.e.
which
is
the
And when
corresponding arc of a cardioide.
3. If a rectangular hyperbola rolls along a straight line, any arc of the
roulette of the centre is equal to the corresponding arc of the lemniscate
which is the pedal of the hyperbola with regard to the centre,
therefore expressible as an elliptic integral (Art. 592).
When
and
is
rolls along a straight line, the arc of the roulette of
to
the arc of the cissoid which is the first positive
equal
pedal of the parabola with regard to the vertex.
4.
the vertex
a parabola
is
ROULETTES.
Many
699
may be cited and many curves may be discovered
whose arcs can be found this being so whenever the arc
other cases
as roulettes
;
of the pedal of the rolling curve can be found.
In each of these cases we also find that the area
ordinate
678.
is
swept out by the
double the corresponding sectorial area of the pedal.
General Theorems with regard to Rolling on a Curve.
Rectification of Roulette of a Carried Point
P.
P
We may
as follows,
prove the results for a carried point
without
deduction
from
and
the
directly
general formulae.
Let A be the point of contact,
B2
B
l
an adjacent point on the fixed curve,
the point on the rolling curve which will come into
contact with B 2
,
Fig. 196.
P,
P the two points on the roulette corresponding to the
points of contact
A
and
B
2
,
so that
PA, P'B 2 are con-
Let these meet in 0.
tiguous normals to the roulette.
Let C p C 2 be the centres of curvature of the rolling
A
and fixed curves respectively at A,
P I} p 2 the radii of curvature,
PAC =
l
(p,
CHAPTER XIX.
700
r=AP PY, P Y' perpendiculars
;
011
tangents at
A and B
I}
elementary arcs of the fixed and rolling
the
curves,
roulette, and the pedal of the rolling
i.e.
curve with regard to
8s, So; 8sp the
P
AB^AB^Ss,
Then when C l B l comes
into line with
B2 0.
;
YY'=Ssp
PP'=Scr,
into line with
B C. PB
2
2
,
l
will
come
Let
Then the angle turned through by the
A
A
J?<j
rolling curve
is
^o
AGA + AOA= PiJ+P2,
turns through the same angle, and B1 B2
1
order small quantity.
Hence, to the first order,
Also
PB
YY' = r,
Again,
is
a second
to the first order,
Pi
since
YY'APis
ultimately a cyclic quadrilateral, as in Art. 674
;
Pi
= 1+
and
l
+
i^
........................... (A)
(the formula of Art. 667 for p l dsp =rds).
679.
Also, as in Art. 674,
Area
PAB
2
P'
= ^r(PP'+Sssm 0),
to the first order,
ROULETTES.
And
roulette
701
integrating, the area swept out by the normal to the
between the roulette and the fixed curve
= -, \r*dO + \ ir2 (~ + -}ds
/
2J
2j
VyOi
(the formula of Art. 667).
(B)
/0 2
the rolling curve is closed, we have for the whole
area swept by the normal in one turn of the curve, such that
the original point of contact has again come into contact,
680.
When
Area swept = area of curve + ^
2
\r
J
(
|
\pi
)
ds,
PZ/
the limits of integration being from the initial to the final
value of s.
681. It should be noted that in the investigations above,
p l and p 2 are drawn in opposite directions. If the rolling
curve be on the concave side of the fixed curve, the formulae
will
become
Arc of roulette = <r =
andArea
S
wept|
|
by normal
J
682.
If
/3 1
=
/o 2J
upon an equal
as will
I (
= lf
1
If
2J
2
2J
/
1 _ i ^s
\ Pl
......
pz /
always happen when a curve
one, the
rolling
rolls
being started so that the
points of contact are initially and always corresponding points,
formula (A) shows that
a- = 2s
p,
the length of
i.e.
oj the roulette is double the corre-
any part
sponding part of the pedal.
683. In the case of an ellipse rolling upon an equal ellipse and placed
at starting with the ends of the major axes in contact, the paths of the
foci are obviously circles of twice the radius of the auxiliary circle, which
the pedal of the ellipse, which is a verification of the general theorem.
In the case of the epi- and hypo-cycloids and the epi- and hypotrochoids, p and p-> are the radii of the rolling and fixed circles and
is
l
constant.
Hence the
sponding arcs of the
arcs of such curves are proportional to the correpositive pedal of the rolling circle, i.e. to the
first
arc of a cardioide or of a liinac,on,
and are therefore
rectifiable in the
same
manner.
684.
Rolling along both sides of a Curve.
If the rolling curve be allowed to roll first on the convex
side of a fixed curve and then upon the concave side, starting
with the same pair of points
common and
rolling in the
same
CHAPTER XIX.
702
manner as before, so that corresponding points again come
into contact, formulae (A) and (B), (A') and (B') show that if
or, or' be the arcs of the roulette, and A, A' the areas described
by the normal
two
in the
cases,
and A p the corresponding area
of the pedal of the rolling curve, then
and
And
both results being independent of
p.2
,
are independent
in each case
and therefore
of the nature of the fixed curve,
double of the results for rolling along a straight
685.
In
the case of
itself slides in contact
line.
a curve carried by a second curve which
with two other curves, or moves in its
own plane in any
given manner, the same formulae as those
established for a roulette can be used for the curvature and
rectification of the envelope of the attached curve.
For the motion being a case of rolling of the locus of
the instantaneous centre /, traced on the moving lamina, upon
the locus of the instantaneous centre / traced on a fixed plane, it
is a matter in general of first determining these loci and their
radii of curvature
;
or,
what
is
equivalent,
if Ss
be the arc of
the angle which the normal to the
the fixed /-locus and
/-locus makes with the normal through I to the carried curve,
and if S% be the angle turned through by the moving curve
whilst I travels over Ss on its locus,
(j>
ds
j
dx=
Pi
,
and the formula
...
may
be written
--h COS
COS
^
R-r
,
ds
>
P2
1
1
Pl
p2
_ + cos0
_= 3dv-
cos0
]g
r+ P
+
,
having the same meanings as before, p v p z
as
referring to the two /-loci, the values being obtainable
the various letters
explained in Art. 660.
GLISETTES.
When
j
,
which
iiiv.ii
Y
is
10
cis
cos
,
|
i
pi
703
d>,
r
and p have been
ana
p%
expressed in terms of i/r, the angle which the normal to the
carried curve makes with a given line, the radius of curvature
of the envelope
is
cos_
cos
and 0-=
I
R d\}r
gives the intrinsic equation of the envelope of
the carried curve.
Also, as before, the case of a carried point is included as
that for which p = 0, and the case of a carried straight line is
included as that for which p=cc which respectively give
.
,
and
0-=
(Y
I f
,
^ds\
r+cos <p
j
r\ d\{r
.
as the intrinsic equations required.
686.
When
a Curve slides in such a manner as always to touch
a Given Straight Line at a Given Point, the glisette of any
carried point is obtainable at once.
Let the carried point be taken as a pole, and let p=f(^) be the
tangential polar equation of the curve with regard to this pole.
Fig.
Then
197.
the point of contact be taken as the origin and the
given straight line as the z-axis, we have
if
and the x/r-eliminant
is
the "glisette" required.
CHAPTER XIX.
704
687.
Ex.
1.
Illustrative Examples.
an equiangular spiral, we obviously have
,
and dp
If the curve be
p = rsma
:.
?/
= #tana
/,=rcosa;
dy
the path of the pole, as
is
is
geometrically obvious.
y
c
Fig. 198.
Ex.
2.
If the curve be
an
2
^>
p-fi
and the ^-eliminant gives
ellipse,
= a 2 cos 2 i/'4-^ 2 siii 2 ^',
-(a -6
2
"|
2
)sini/'cosj/',
j
for the glisette of the centre the quartic
O
x N
-V
Fig. 199.
Ex.
3.
In a parabola of latus rectum 4#, we have for the glisette of
the focus
y
[
W being the angle subtended at
from the vertex to the point
i.e.
being the angle
OB makes with
the y-axis.
the focus by the arc
of contact (Fig. 199)
;
(t,
y)
705
EQUATIONS.
y) Relations.
In many curves the relation between the ordinate y and the
angle i between the ordinate and the tangent takes a very
688.
(i,
simple form, and is, moreover, very useful (1) in the determination of the envelope of a straight line carried by a curve
which always touches a given straight line at a given point
and also (2) in the problem of Brachistochronism for a law of
force which is always in the same direction.
Fig. 200.
Let
(1)
be the fixed point at which the curve always
touches the fixed line Ox.
Let
AB
Then
1==
f( l )>
be the carried
line.
the equation of the curve has been expressed as
with
as the .x-axis, the tangential polar equation of
if
AB
AB
the envelope of
is
clearly
(2) The laws of force for the
tion of a curve,
(a)
for
p=f(\^\
y=p and
i=\fs.
BRACHISTOCHRONOUS
under a central force making
I
minimum and
a
- =k, a
constant, v being the velocity
(6)
;
under a force parallel to a given straight line which
Cfj
- a minimum
we
take as the
*?
may
and
y-axis
=u
,
making
a constant,
are respectively
2
2 dr
E.I.C.
descrip-
and
2Y
P=~
d
I
CHAPTER XIX
706
These will be found in books treating of kinetics of a
They are placed here for the convenience of the
particle.
and to illustrate further the use of the (i, y) equation
of a curve which is necessary for the glisette of a carried line
The central force formula
with motion described above.
student,
we
are not
now concerned
with, but
it
will serve for practice
in the use of (p, r) equations.
689.
To
find the
(/,
y) Equation.
Let the tangent at P meet the ic-axis at T.
The relation between i and y is easy to get, for
1
and
if
x be eliminated between this and the equation of the
curve the relation between
T
N
i
and y
will result.
x
Fig. 202.
Fig. 201.
LIST OF
COMMON
(<,
y} EQUATIONS.
-
Circle,
Catenary,
Tractrix,
Cycloid,
',=!-.
-
Evolute of a parabola,*
-
Directrix for #-axis, Lat. Rect. =4a/3.
(f,
E volute
y)
of a catenary,
707
EQUATIONS.
-
-
sin 2 j=l
--%.
n
Curves of the
dy = Ja
----- y
n
~-,
class
n
j
-f-=JL=
* ^ an_n
Curves of the class
,
sin 2 <=
Parabola,
.
The student should
sn*
establish each of these results.
It will
they are expressed as sm i=f(y).
form
the
convenient in discussing Brachisobviously
be noted that in
is
-
-
Rect. hyperbola,
This
sin 2 f=l
z
all cases
tochronism.
690.
Ox
line
Ex.
1.
If,
for instance, a catenary slides in contact with a
straight
we have for the envelope of the directrix the
at a fixed point 0,
tangential polar-equation
p=
-c-
c
,
for y
.
is
the
(t,
y] equation.
Fig. 203.
It
with
also,
is
obvious from this equation that the directrix touches a parabola
for focus and 4c for latus rectum.
This is clear geometrically
for the locus of the foot of the perpendicular upon the directrix
is
obviously a line at a distance c from the fixed line, and the envelope
negative pedal of a fixed line, i.e. a parabola.
of the directrix is the first
CHAPTER XIX.
708
-2
G
Since
-%,
..
.-i
,,2 fl
T-fctfcCv/.oV*
2
the equation
-P=-g -r-(sm t) gives
Hence, the catenary is Brachistochronous for a law of force which acts
perpendicularly towards the directrix arid varying inversely as the cube
The line of zero velocity in this case
of the distance from the directrix.
is
at infinity.
Ex.
2.
What
is
An
ellipse slides,
touching a straight line at a given
the envelope of the axis major
point.*
?
cot 2 1
Here
Fig. 204.
.'.
the tangential polar equation of the envelope of the carried axis
p2 (
by writing p
for y,
$
for
2
is
cos 2 ^ + 6 2 sin 2 $) = 64 sin 2 \f/ t
t,
and reducing.
Fig. 205.
Ex.
3.
A
fixed point.
cardioide slides in contact with a fixed straight line at
What is the envelope of the axis ?
GL1SETTES
709
Here
4asin3 -cos-,
_
2'
for y
ivelope of the axis
Putting
p
and
for
t,
the tangential polar equation of the
is
a
.
.
691.
Two Curves
in the
Lamina touching Fixed Straight
Lines.
Let two curves be drawn upon a lamina, and let the lamina
move so that the curves touch two given straight lines Ox, Oy
Fig. 206.
and
P
be a point carried by the
let
the
be
PM,
perpendiculars upon Ox, Oy,
and \js the angle they respectively make with two initial
lines PA, PB drawn upon the lamina, including an angle
inclined at
lamina.
TT
an angle
Let
o>,
PN
w, and initially at right angles to Ox and Oy
Then the path of P can be obtained at once.
Let
JP=JW, P =
respectively.
FW
P
be the tangential polar equations of the curves, with
for
of
of
and
measurement
PA,
origin
respectively as
p,
PB
initial lines.
CHAPTER XIX.
710
Let
P
y be the coordinates of
x,
Oy as coordinate axes.
x sin to =/(^)>
Then
with regard to the
lines
Ox,
yaina) = F(\Js),
an d
\//"-eliminant furnishes the path of P.
It is clear that instead of the two curves on the
and the
might have one single curve drawn,
be identical, except as regards the
is measured in the two cases.
\ls
The
rectification of the
path of
/(^) and
i.e.
initial
line
P follows
from
F(\]s) might
from which
dx = cosec (*)f(\!s) d^, dy = cosec w F'(^r)
2
2
cfe = dx +2dxdy cos w
and
whence
where
stands for
Two
692.
oS//-,
= cosec o> V/ 2 + 2f'F' cos w + F'2
[
8
f
lamina we
%&
and F' for
Straight Lines in the
Lamina touching Fixed
Curves.
ABC
are
When three straight lines forming a triangle
traced upon a lamina, and the lamina is made to move in
such a manner that two of the sides AB, AC, say, touch given
BC
motion envelop a
third curve, and there is a linear relation between the three
It has been shown
arcs described by the points of contact.
fixed curves, the third side
(Diff. Gale., Arts.
will in its
568-9) that the tangential-polar equation
of the envelope of the carried side
BC can
be found at once.
of any point 0, fixed
/3, y be the trilinear coordinates
in space, with regard to the triangle ABC taken as a triangle
If a,
of reference,
we have
the relation
aa + fy# + cy = 2A,
........................ (1)
are the lengths of the sides and A the area of
the triangle, with the usual trilinear convention that a, ft,
y are positive when drawn from a point within the triangle.
where
a, b, c
Hence
a
it
da
j~r
,d8
dy
+k-T7 + c ;rr=
,
d\js
d\Js
where
-v/r
follows that
is
d\js
dz
d z a , d?$
^-TT2z + &-^-T^+2
,
>
d\]s
,
ax//
the angle any line fixed in the lamina
a line fixed in space
;
makes with
GLISETTES.
And
same
711
the increment of the angle of contingence being the
for
693.
we have
all,
& /0g
api
Cps
= 2 A.
Caution.
as the origin of
measurement
for perpendiculars
for the tangential polar equations of the envelopes of BC,
CA, AB, it is to be noted carefully that we are in the presence
Regarding
two separate conventions with regard to the signs of the
perpendiculars, which may be antagonistic.
(1) The trilinear convention is that stated above, that the
perpendiculars are reckoned as positive when the point from
which they are drawn lies within the triangle of reference.
of
(2) In the general treatment of curves,
=p +
the formula
-^
i.e.
in establishing
and others involving
p, the per-
pendicular from the origin has always been reckoned positive.
A
C
B
Fig. 207.
p p 2 pz
If
l
,
,
be the positive perpendiculars from
/y
sides,
we have
o
in all cases --f
=
//
upon the
'TO
^i+'
etc
-
$8 i>
<$
S 2> $ ss
being
elements of the three arcs described by the points of contact.
Hence, so long as the origin from which the perpendiculars are
measured lies within the triangle, we have a =p 15 &=P2>
and
ds
ds
ds _
d2 3
d2 a
yP%>
~
If,
and
however, the origin
AC produced, a=
~~
d8
i
_
+
,
d*a
lie
Pi,
between
BC
and
AB
produced
/3=p 2 y~Ps>
ds.2
>
_
*This method is stated by Mr. Besant to have been suggested by the late
Master of Caius College, Dr. N. M. Ferrers.
CHAPTER
712
XIX.
with similar changes for other positions of the origin relative
to the triangle of reference.
In addition to this, when
ture, it will be
is
but
remembered that
+p
if 8
and
p
if s
and
=
we estimate the
ds
-r-r t
which
radius of curva-
is
always
\fs
are increasing together,
^
are such that
when one
increases, the
other decreases.
This point has been discussed in Art. 531.
Hence we have written
ap l
ap 2
ap3
2 A,
the signs to be determined in each particular case. But in
any case this equation is sufficient to prove that if two of
the three quantities p l} p 2 p 3 be constant, the third is also
constant, i.e. if two sides of the triangle envelop circles
,
or pass through fixed points, the third side also envelopes
circle, which is the theorem of Ex. 1, Art. 569, Differential
a
Calculus.
694. The ambiguity as regards sign necessitates careful
attention to the position of the origin relatively to the triangle.
Three straight lines on a plane divide the plane into seven
regions, and the signs of a, /3, y in these regions are indicated
in the figure.
GLISETTES.
Accordingly \ve have,
from points for which
x//-
\/r
if
we assume
= 0, and
s 1? s 2
,
s3 to
be measured
to be each increasing
when
increases,
as l + bs2
if,
713
and so long
as,
the origin
+ cs3 = 2 Ax//lies
within the triangle
;
asx -f- bs2 + css = 2 Ax//if,
and so long
of a,
/3,
y
as, the origin
are respectively
the region where the signs
-K and so on for the other
lies in
h
five regions.
moves the origin may pass from one
Hence care must be taken in integrating
Also, as the lamina
region to another.
between specified limits for
to observe the sweep of any
of the three lines BC, CA, AB through the origin, and to take
proper account thereof by using the appropriate case or cases of
x//-
as l
for the intervening
hs.2
cs3
= 2 A\/A
sweeps of the several
sides.
Thus, in integrating round an oval which the arms AB, AC
touch, the oval lying within the triangle (Fig. 209), we have,
taking the origin within the oval,
as 1 + bs.2 + CSB = 2 Ax//-,
and for a complete revolution
where
S is
and
S
l
the perimeter of the oval
that of the curve enveloped by BO.
CHAPTER
714
695.
In the same way,
if
XIX.
the oval be always external to the
triangle as in Fig. 210,
and similarly
in other cases.
Fig. 210.
696.
A Limiting
If the triangle
of a line
Case.
ABC
becomes evanescent, we have the case
B'C' (Fig. 211), carried by the pair
and
AB, AQ,
making constant angles B, C with
through A,
of tangents
viz.
B'
B,-
Fig. 211.
them
making a constant angle A
respectively, the tangents
The sides a,
with each other.
sin
A
:
sin
B
:
sin C,
b,
c
vanish
in
the
ratio
and the theorem becomes
Si sin
A = (sin B + sin C) $,
cos
i.e.
.
Sm
697.
shown
A
S.
2
If the carried line B'C' lie within the angle PAQ, as
it is the limit of the case in Fig. 213
in Fig. 212,
GLISETTES.
where the signs
tively
of
715
the perpendiculars
a, /3,
- + -,and a=-p fi=p 2 y= -p3
,
lt
y
are respec-
,
(and ultimately
A = 0),
Fig. 212.
EC does not sweep through the origin and if it
never does do so during the whole motion of the lamina
during a complete revolution,
so long as
;
- aS" + (b - c) S = 47r A
(and ultimately
A = 0),
c
Fig. 213.
giving S" the perimeter of the curve enveloped, or in the
limit, when the triangle is evanescent,
o0// =
sin
B
-.
sin
sin
A
A
C
sin
A
COS^-
698.
If,
however, the line
BC
does sweep through the origin
must be per-
in the course of the revolution, the
integration
formed separately for the several complete portions for which
the line BC moves without a
sweep through the origin, and
CHAPTER
716
XIX.
BC being found thus, the posiadded together, using the formula
the arcs of the envelope of
tive results
must be
finally
sin
A
for each portion.
Taking the case of any oval with two perpendicular
e.g. an ellipse, TP, TQ a pair of
tangents at right angles, and the carried line being the
699.
axes of symmetry AOA', BOB',
bisector of the angle
PTQ
(Fig. 214), this line will obviously
Fig. 214.
Fig. 215.
sweep through the centre every time the point T crosses one
of the axes of symmetry, and whilst T travels along its locus
first quadrant, the perimeter of the corresponding
portion of the envelope of the carried line is
over the
arc
sin
4
Off
P,P 9 ~
= v/2(arc BP
where
P v P Ps P4 are
make an
2
,
,
angle of
j-
2
sin
~ arc
-f-
4
AP
arc
P Po
9
2 ),
the points of contact of tangents which
with the z-axis (Fig. 215).
ISOPERIMETRIC COMPANIONSHIP.
be noted that the arc in question
It is to
717
is
described
in the opposite order to that of description of the ellipse
by the several points of contact.
The whole perimeter
curve
rectih'able in
is
is
then 4\/2(arc
BP
terms of arcs of an
2
arc
AP
2 ),
and the
ellipse if the oval
be
elliptic, or in terms of arcs of whatever curve the doubly
symmetric oval happens to be.
the point A is at oo we have the case of parallel
is a line parallel
tangents to the oval, and the carried line
to the tangents and dividing the chord of contact in the ratio
When
700.
,
AD
B and G are indefinitely
which we may assign, say p q,
sinJ5:sin(7 (see Fig. 213), where
in
definite ratio
any
and we then have
small,
i.e.
y
:
2
-g
8
P+V
for the perimeter of the envelope of
701.
A
AD replacing the
result
Case of Isoperimetric Companionship of Curves.
Let us consider the form of a curve O'PQ with pole N, which
will be such that, when it rolls upon the fixed curve OP whose
equation
is
known, y=f(x), the
pole'
N
will travel along n
straight line, say the x-axis.
and 0' be the points originally in contact, Ax, Ay the
the point of contact, PN,
ordinates, PT the tangent at P making an angle \js with Ax, O'N the radius vector
of the rolling curve from which
is measured and r the radius
Let
axes,
OM
P
vector
NP, $
curve and
its
the angle between the tangent to the rolling
radius vector.
CHAPTER
718
Then
r=y,
rdO
= tan = cot = dx
-y\ls
--j
dr
.
dy
rdO=dx
=dx\
Hence
and
0)
dr =dy.)
We
therefore have
0=j^==J-^L,
r
and
XIX.
if
(2)
=f( x \
=y
x be eliminated from equations
(2),
the polar equation of
the rolling curve will result.
y
Fig. 217.
Again,
if
the form of the rolling curve had been given, say
then
x=\F(8)de]
J
f
y=F(0),
and
}
be eliminated between these equations, the Cartesian
equation of the fixed curve will result.
It follows that, since there is pure rolling without slipif
ping,
the corresponding arcs
of the
two curves must be
equal.
This follows at once also from equation
a.rcs OP, O'P,
(1),
for if s
and
s'
be the respective
whence ds=ds' and s=s'
if
originally been in contact.
measured from such points as have
ISOPERIMETRIC COMPANIONSHIP.
719
It also follows that
the area swept over
i.e.
by the ordinate PN, that
is
MNPO,
is
double of the area swept out relatively to the rolling curve
by its radius vector, that is the sectorial area O'NP.
The polar subtangent of the rolling curve is the Cartesian
subtangent of the fixed curve, and the subnormals are the
same.
Hence, given
we
y=f(x),
by the transformation yr,dx=rd9, obtain another
curve r=F(0) for which
can,
(1)
(2)
corresponding arcs are equal;
the area travelled over by the ordinate of the one
is
double the sectorial area swept out by the radius
vector of the other
;
and
(3) if
the second be allowed to roll upon the first, having
been properly adjusted at the start, the locus of
the pole of the rolling curve
is
the ie-axis of the
other.
702.
Generalisation.
More
generally,
if
we take any
polar curve
r=F(f)),
and construct from
it
a Cartesian locus, such that for each
Fig. 218.
CHAPTER XIX.
720
point
(r,
0)
on the one there
which
is
dx=dr cos x
a point
(x, y)
rdO sin x
on the other for
,
dy=dr sin x -\-rdQ cos x.
where x i g an y angle whatever at our choice,
elimination of r and 0, a new curve in which
where
have, upon
are corresponding elements of arcs in the
It follows that
ds, ds'
curves.
ds=ds'
if
we
and
two
s=s,
the origins of measurement of arc are so chosen that
s
and
s'
vanish together.
The geometrical meaning of this is plain. We are
projecting dr, rdO upon a pair of perpendicular axes Ox, Oy
with an arbitrary origin, and such that the x-axis makes an
703.
angle x behind the radius vector of the polar curve, and
therefore makes an angle
x with the initial line of the
polar curve, or what is the same thing, with a fixed line
parallel to the initial line of the polar curve ; and
reserving choice of x we can make the new axes either
through
by
>
fixed axes or
If
moving
we make x~
in
any given manner.
^ we make the #-axis turn
i- e -
at the
same angular rate as the radius vector of the polar curve, we
have
dx=dr, dy=rd9,
the transformation discussed in the last
article, except that
the axes of x and y are interchanged.
If we make x=0 or 9-\- const., we have fixed axes.
1
f)
If
we make
0x=-,
of the radius vector,
we make our
and so
axes turn at -th the rate
on.
Moreover, either or both of the axes AX, Ox may be regarded
as a fixed axis, the matter being purely a relative one.
These transformations establish a remarkable connection
between many curves of common occurrence, and further will
furnish us with a method of deriving new rectifications.
70
4-.
Reverting to the more elementary case of
dx=rdO,\
y=r, f
ISOPERIMETRIC COMPANIONSHIP.
we
721
shall find that,
A
straight line
y=xcota
has for
its
analogue
an equi-
r=ae ecoia
A
A
straight line
parabola
angular spiral
has a companion in a catenary.
has as companion a spiral of
r=c cosec
-
.
Archimedes.
An
ellipse
-
-
has as companion one of the Rho-
-
-
has as companion a cycloid.
doneae. (Diff. Gale., Art. 385.)
A
cardioide
And when any
curve is rectifiable, a companion is also
same manner, and even when neither curve is
terms of arcs of a circle or an ellipse, arcs of
rcctifiable in the
rectifiable in
the one can be expressed in terms of arcs of the other.
And in addition the property as to the relative magnitude
of the area
swept out by the radius vector of the one and the
ordinate of the other holds good.
Such pairs of curves may perhaps be termed Isoperimetric
Companions.
As
illustrative
examples,
we
consider these examples in
detail.
705.
1.
Taking the straight
.'.
line
dr
dr
y=xcota
r
as the fixed "curve,"
dB cot a,
= a#cota,
7/1
Fig. 219.
Hence an equiangular spiral r = ae ScoitL and the straight line y = xcota
correspond in the manner described, corresponding arcs being equal, and
the Cartesian area bounded by the line, the .r-axis and two ordimites
CHAPTER XIX.
722
being equal to double the corresponding sectorial area of the
spiral.
(See Dif. Calc., Art. 449.)
2.
Take
as the rolling polar curve the straight line r
= r = csec@,
Then
/.
x = c log tan
.-.
cosh -
c
is
^
cos 6 cosh -
or
which
dx = rdO = csec Odd
4-
=1
,
~J
= c gd" 1
(A rt.
= csec 0.
;
;
69),
the catenary,
therefore the isoperiinetric companion to the straight line, and
has been seen (Art. 538). See also Dif. Calc.> Art. 444.
rectifiable as
We
note in addition to properties proved in Dif. Cole., Art. 444, that
Area
3.
Take
NO'P=%
area
ANPO.
as the rolling polar curve the cardioide
r=a(\ -cos 6).
Then, for the Cartesian curve,
r
a\
-cos 8
a cycloid with cusp at the origin and vertex upward. These curves
are therefore isoperimetric companions. When the cardioide is placed
with its vertex in contact with the vertex of the cycloid on the concave
i.e.
and .allowed to roll inside the cycloid, the roulette of the pole is
the line of cusps of the cycloid and the propositions of Art. 701 with
of
regard to equality of corresponding arcs and the relative magnitudes
side
PROBLEMS.
723
the areas swept by the ordinates of the cycloid and the radius vector of
the cardioide both hold good.
V
Fig. 221.
Take
4.
as fixed curve the ellipse
^+P
Then y=r,
dxrdd
=1
give
and
r=6cos-0,
i.e.
which
is
doneae
the isoperimetric companion of the ellipse. Hence the Rhoare rectifiable in terms of arcs of an ellipse.
r=A cosnQ
PROBLEMS.
A
round the circumference of an equal
Prove that the area of the epitrochoid, described by a point
carried with the rolling circle and distant c from its centre, is
1
.
circle of radius a rolls
circle.
2
(4ft
+ 2c 2 )7r.
[Oxr.
I.
P., 1918.]
2. If a circle roll on the convex side of a
parabola from one
extremity of the latus rectum to the other, and can just pass between
the vertex and the directrix, prove that four times the area traced
out by that radius of this
circle,
which always passes through the
CHAPTER
724
XIX.
point of contact, exceeds the area of the circle by half the rectangle
contained by the latus rectum and a line equal to the arc it cuts off.
[R. P-]
3.
An
equiangular
P2 of the spiral.
to a point
path Oj0 2
straight
4.
A
upon a straight line from a point 1\
the
0,
pole of the spiral, traces out the
spiral rolls
.
N
From
0^0^ are drawn perpendiculars O^N^ 2 2 on the
Find the area of O^NjO^.
[COLLEGES a, 1881.]
line.
closed oval curve rolls
upon a fixed curve. Find an exby any carried
pression for the area of the roulette traced out
point.
In a complete revolution of the closed oval curve, prove that the
of the areas of the envelopes of two carried lines at right angles
sum
to one another
is
constant.
which pass through a point fixed to the rolling curve
Prove also that this sum exceeds the area of the
roulette generated
by the
point,
by the area
of the rolling curve.
[COLLEGES
5.
line,
7, 1887.]
a closed oval curve roll with angular velocity o> on a straight
while a point moves along its evolute with relative velocity
If
prove that the area included in any portion of a revolution
between the straight line, the curve generated by the moving point,
w/>',
to the former drawn through the extremities
double the corresponding portion of the area between
and the perpendiculars
of the latter,
is
the curve and
its
evolute,
bounded by the
initial
and
final radii
of
curvature, provided the moving point is initially at the centre of
curvature of the point of contact ; p being the radius of curvature
of the evolute at the point corresponding to the point of the rolling
curve in contact with the straight
[COLLEGES
line.
5,
1883.]
6. The cardioide ? =
(!- cos 0) rolls on a straight line; prove
that the intrinsic equation of the roulette of the cusp is
measuring from the point of contact of the cusp.
Prove also that its Cartesian equation is
'la
that
its
area
is
-1
/ 7m2
>
an^ that the radius
of the cusp is three times its distance
of curvature of the roulette
from the point of contact.
[TRINITY, 1888.]
Find the evolute
of the roulette of the pole
equation of the envelope of the axis,
and the
intrinsic
PROBLEMS.
7.
A
Show
closed curve
is
moving
in
725
any manner
in its
own
plane.
p be the radius of curvature of the envelope of the
tangent at any point of the curve, then
that
if
\pds
is equal to twice the area of the curve, the integral being taken
round the curve, ds being an element of arc of the moving curve.
all
[COLLEGES, 1879
A
]
plane lamina moves in any given manner on a fixed plane
is a fixed point on the fixed plane, P a point attached to the
moving lamina and fixed upon it. If the area described by P about
8.
:
be given, show that the locus of all points (P) in the moving plane
which the area is the same, is a circle, and that for different
for
values of the area the corresponding circles are concentric.
[ST. JOHN'S, 1881.]
Examine the isoperimetric correspondence between the parabola
=
f ax and the Archimedean spiral r = 2a0, showing that the spiral
can be made to roll upon the parabola in such manner that the pole
9.
of the spiral travels
10.
Show
along the axis of the parabola.
that the reciprocal spiral rO = a and the exponential
X
curve y
fiable
are isoperimetric companions, both curves being rectiarcs equal, and interpret the result by
ae
and corresponding
reference to the locus of the pole of the spiral
when
suitably started
rolling.
11. Establish isoperimetric
-
y
a
and the
cissoid
r
companionship between the curve
= log tan
sin 2
cos
=a
(
^
+^j -
sin
4)
<f>
sin 2 B
a
cos u
-
.
Establish isoperimetric companionship between the semi-cubical
2 =
x3 and the spiral Sau + 3 = 0.
parabola ay
1 2.
13.
Show
that the curve
r = a log sec
is
rectifiable
and
of equal strength
in isoperimetric
t,
companionship with the catenary
y = a log sec
x
-
.
CHAPTER XIX.
726
Show
14.
that the curves
4# = a ( cos
ty = a
and
f*
are rectifiable
15.
Show
=
(
-9
<
3 sin
cos
-
sin
-5
^
J,
<
j
5
sin ^6
and isoperimetric companions.
ft
that the curve
/I
\/a 5 + r<
01
= 3log
1
0/3
20+6
is rectifiable
and
,
/j
x
Show
a
2
2
16.
companionship with
in isoperimetric
3
3
-y = a
3
that the curve
A
'
r = 4a sin
.
is
r
rectifiable
= a(l + cos
in isoperimetric
,
companionship with the cardioide
0), its
Show, by taking
x=
r
= aO and \ = nO
% g [nd
cos
y = ~2 [%# sin
is
cos d
pole travelling along the axis of the cardioide as
within the cardioide, the two poles being initially coincident.
it rolls
17.
and
t
.
nO + (n ?i#
(ft
in Art. 702, that
1) sin ?i#],
- l)(cos?&0-
1)]
an isoperimetric companion of the Archimedean spiral r = aO.
Hence show
2
(1) that x
=2ay
is
isoperimetric with r
--
( Jr*
(2) that r
= a cosec |
a
2
= a&;
}
I
is
rectifiable
and
in isoperi-
metric companionship with r = 4a6.
18.
Show
that an ellipse of semiminor axis b and eccentricity
roll upon the curve
can be made to
'
b
x
j
= dn
T
b
y
,
(mod.
x
e),
so that the path of the centre of the ellipse is the x axis.
e
PROBLEMS.
Show
that
if
the origin be taken at the point for which the end of
is in contact with the curve, this may be reduced to
the major axis a
the form
U
a
Kb-x
[Write
727
for x
,
JU
= dn T
.
b
and reduce,
see Ch.
XXXL,
Art. 1352.
See
also Greenhill, Elliptic Functions, p. 72.]
19.
x2
Show
that the perimeter of the ellipse
twice the perimeter pf one outer
and that the area of the ellipse
outer foil of the same curve.
Show
further, that
if
is
^-
is
equal to
bO
a
,
equal to four times the area of one
the vertex of the
with the inner side of the
v2
+ j-2 =
of the curve
foil
= b cos
r
2
ellipse at the
foil
end
be placed in contact
of the
minor
axis,
and
without sliding upon the
ellipse, the pole of the rolling
foil will traverse the major axis of the ellipse.
Deduce a well-known proposition as to a circle rolling in the
the
foil roll
interior of another circle of double its radius.
An
involute of a circle is made to slide, touching the rectShow that the locus of the instantaneous
axes
Ox, Oy.
angular
What is the locus of the
centre on the plane x, y is a straight line.
instantaneous centre relatively to the curve.
20.
Show
that
trochoids,
the
glisettes
of
and the envelopes
carried
points
are
cycloids and
are either
of carried straight lines
cycloids or involutes of cycloids.
[BESANT, Roulettes and Gluettes.]
A
Show that the intrinsic
cycloid rolls along a straight line.
the
of
of
the
axis, (2) the line of cusps, (3)
(1)
envelopes
equations
the tangent at the vertex are respectively
21.
measuring
s
(1) s
= a^ 2 + 3asin 2 ^
(2) s
= 30(^ + 1 sin 2^),
(3) *
= a(^ + $sin2^),
in each case
from the point on
its
locus for which
^ = 0.
Trace each of these curves, supposing the cycloid to be continued
both ways, and the rolling to continue with successive arches of the
cycloid,
and
find the positions of their cusps.
CHAPTER
728
Show
XIX.
that the whole perimeter of the last of these curves
Sa \/2
+ 8a sin"
-
1
v/f
is
2ira,
area = Jra 2
Show that the first evolutes of the second and third curves, and
the second evolute of the first are four-cusped hypocycloids.
and
its
22.
.
A
parabola rolls on a straight line
(1) the locus of the focus
(2) the
show that
;
a catenary (Art. 517),
is
envelope of the directrix
an equal catenary,
is
(3) the tangent at the vertex and the 'latus -rectum envelop
parallels to a catenary,
(4) the intrinsic
s
23. If the cardioide r
line
equation of the envelope of the axis
=a
2
$ + tan
(2 log sec
= a(l - cos
0)
1//).
move
so as to touch a straight
always at the same point, show that the locus of the pole
and that the
63
\
a
If
an
25.
of the centre
A
-
is
.
7sm 4
-
with a given straight line at a
ellipse slide in contact
given point, the glisette of the
and that
is
intrinsic equation of the envelope of the axis is
-s = 12 sin 2 ^
24.
is
foci is
2
2
a% = (a - f)(f - I 2
).
lamina moves in such manner that a certain point in
it
describes the path
= c sin ^ - c cos ^ log (sec + tan ^),
= c cos + c sin ^ log (sec ^ + tan ^) i/*
>7
\l/
referred to fixed axes
through
OY
OX,
]
e, J
in its plane, whilst a straight line
this point attached to the
lamina makes an angle ^ with
the F-axis.
Reduce
this
motion to
the curvatures of the
.
and on the fixed plane
Also show that the difference of
rolling.
loci of
the instantaneous centre on the lamina
cos 2 \b
is
c
Show
further that the intrinsic equation of the envelope of the
line attached to the lamina is
ds
--r
c
sec
^ tan ^ +
c
log (sec
^ + tan
!
PROBLEMS.
26.
A
lamina moves in
its
own
729
plane, so that a point 0'
upon
it
traces out a cissoid,
r\
= - 2acos 2
COS 3
^,
7:
upon a fixed plane with reference to a pair of fixed rectangular
axes OX, OY in that plane, whilst a straight line O'x attached to
with OX. Show that
the moving lamina rotates, making an angle
the motion
is
that of rolling of one parabola upon another equal
of Art. 660, for the differ-
parabola, and deduce from the formula
ence of curvature of the
the radius of curvature of a parabola.
/-loci,
A
catenary moves in its own plane so as always to touch a
Show that the tangential
given straight line at a given point.
is
of
axis
the
of
the
polar equation
envelope
27.
where
28.
c is
The
the parameter of the catenary.
centre of a circular disc of radius a travels along a
2 =
2ax, spinning at an angular velocity o> in
parabolic path y
a clockwise direction, the centre receding from the axis with a
Show that the motion thus produced is that of the
velocity rtw.
an involute of the circle upon the axis of the parabola,
and that the velocity of the point of contact is the same as the
velocity with which the centre of the circle recedes from the
rolling of
tangent at the vertex.
29.
A
Bernoulli's lemniscate
moves
so as to touch a fixed axis at
a given point.
Show that the tangential polar equation of the
envelope of the axis is
p
and that the
3
30.
A
tangent.
'
3
glisette of the pole is
r2
tangent.
C( S
1
circle rolls
Find the
= a 2 sin d.
on an equal
circle
and
carries with
it
a fixed
intrinsic equation of the envelope of the carried
[OXFORD
II. P., 1887.]
CHAPTER XIX.
730
31.
A
A
triangle of area
touch an oval of perimeter
moves
I
p, p ; prove that the radius of
the envelope of the third side c are
(2A
- ap bp)
of its sides (a,
b)
where the radii of curvature
curvature and the perimeter of
are
-
two
so that
at points
-
and
{47rA-
(a
+ b)l}.
[ST. JOHN'S, 1883.]
32.
An
will
on a fixed horizontal straight
ellipse rolls
Show
of x).
line (the axis
that the locus of the highest point of the ellipse
be
and reduce the
integral to the standard form.
JOHN'S COLL., 1881.]
[ST.
33.
Prove that the
intrinsic
equation of the envelope of the
directrix of a catenary of parameter c, rolling on a circle of radius
will be found by eliminating a between the equations
s
c
and
\j/
1 + sin a
= * tan a sec a + i log
&1
T
,
,
!
I
a.-\-
tan
-
c,
^
=
sin a
L
J
a.
[ST.
A
34.
given right-angled triangle
outside of a fixed oval curve with
is
made
JOHNS,
1886.]
to slide round the
the point
P
on the curve,
PR
touching it and the side PQ normal to it. If s be the
the oval, prove that the length of the curve enveloped
of
perimeter
the side
by
QR
is
equal to
(
s
+ 2vPQ) sin PQR.
[ST. JOHN'S, 1889.]
35. When a curve rolls on a straight line, show how to find the
locus of the centre of curvature at the point of contact, and prove
that, in the case of a cardioide, the locus is
an
ellipse.
[ST. JOHN'S, 1889.]
When
a curve rolls on a fixed curve, prove that the locus of
the centre of curvature is inclined to the common tangent at the
36.
tan- 1 {p dp/(p + p)
angle
where
/o,
67s},
p are the radii of curvature of the fixed and rolling curves
at the point of contact.
37.
A
cardioide
[ST. JOHN'S, 1889.]
r = &(l-cos0)
the vertices coinciding during the
the pole of the rolling curve is
r
= 4a sin 2
rolls
roll.
f
^
+^
upon an equal
Show
cardioide,
that the roulette of
PROBLEMS.
731
that the tangential polar equation of the envelope of the axis
p = 4a sin
and that the area
j/'
sin 3
is
JT -j. jA
^->
6
of the roulette of the pole is
38. A. cardioide of
perimeter Sa
on the outer side of a
rolls
cycloid of equal perimeter from cusp to cusp, the vertices coinciding
during the roll. Show that the area of the roulette of the cusp of
the cardioide between the roulette and the cycloid = I? 2
Show also that the arc of any portion of the roulette of the cusp
measured from the vertex of the curve is double the distance of the
.
point of contact of the two curves from the axis of the cycloid.
Show further that the tangential polar equation of the envelope
of the axis of the cardioide is
p = 2a sin
where
from
p
is
\f>
(^ +
drawn from the vertex
2 cos 3 ^),
of the cycloid
and ^
is
measured
its axis.
A
cycloid of length Sa rolls on the outside of a cardioide of
equal length, a cusp of the cycloid starting from the cusp of the
cardioide.
Show that the intrinsic equation of the envelope of the
39.
line joining the cusps of the cycloid is
2s
=
a
+
a
sn
i
being measured from the tangent at the vertex of the cardioide.
[Oxr. II. P., 1913.]
CHAPTER XX.
RECTIFICATION OF TWISTED CURVES.
706.
Let
PQ
be any elementary arc cs of the curve.
P and Q be respectively
Let
the coordinates of
(x, y, z)
and
+ Sx,y + Sy, z + Sz)
(x
with regard to any three fixed rectangular axes Ox, Oy, Oz.
Then
^
chord pgy> =
^+
Sy2
+
^
be made to travel along the curve so as
approach indefinitely near to P, the chord PQ and the arc
ultimately differ by an infinitesimal of higher order than tl
Now,
arc
PQ
if
Q
itself, i.e.
the chord
and the arc PQ ultimately
Hence we have to the seconc
PQ
vanish in a ratio of equality.*
order of small quantities,
Now
suppose the curve to be specified in one of the tw<
usual ways,
(a) as the line of intersection of two specified surfaces
or (b) the coordinates of any point x, y, z upon it expressed ii
terms of some fourth variable t, and defined by th<
equations
x =f l (t\
y=ft (t),
*=/,(<).
The First Case.
In Case (a) choice must be made of one of the three variables x, y, z to be considered as the independent variable,
x, and the equations /=0, vF=0 are then to be solved to fin<
*For a
discussion of this point see De Morgan, Differential and Intel
See also Diff. Gale., Art. 34, for a plane curve.
Calculus, p. 445.
732
TWISTED CURVES.
the other two, y and
and
we express dy
-fctx
,
We
then have
Then
in terms of x.
z,
dz
733
differentiating,
.
x
in terms oi
-ju/x
;
say
-J
And when the integration has been effected, the length of
the arc between the points specified by any particular limits
which may be assigned to x, will have been obtained.
707.
A more
Symmetrical Mode of Procedure.
We
might also proceed as follows
Along the line of intersection of
:
f=Q and
dx
+
-f
fy dy fz dz=
fx
Fx dx + Fy dy + Fz dz=0,
and
dx
dy
dz
ds
FQ we have
ds
J~^
Jj,
J J3
2
,
being the Jacobians
i.e.
Then
making use
of the one
which
is
most convenient
;
and which-
used, both the dependent variables occurring must be
expressed in terms of the independent one before integration.
ever
is
708.
The Second
In Case (b)
Case.
we have
and
whence
+ (//(O) + {//
2
CHAPTER XX.
734
and we obtain the arc by integration, as before, between
any two points corresponding to the limits assigned for th<
variable
709.
t.
If the equations of the
x
.
we have
rr-
=
/
2
dz
dy _
Wl/
and
/3
I
'
Similarly
where
z
y
_
curve be presented in the form
\JVV
/
J
have meanings corresponding to
dx
Hence
= ds
= dy = dz = dt
-^
^2
-^=,
where
Hence
710.
8
The
=
rectification of a curve therefore
possibility of
performing the integration
When /15 /2 /3 /
,
functions of
t,
,
we have
are rational
I
depends upon
-rf
integral
th(
dt
and
algebrai<
the case of a unicursal twisted curve
The advanced student is
memoir by Mr. R. A. Roberts,
referred to the very importani
"
On the Rectification of Certaii
Curves," in vol. xviii. of the Proceedings of the London Matht
matical Society, which has already been referred to in oth<
places.
711.
Ex.
1.
Find the length
of
an arc of the curve which
intersection of the parabolic cylinder y* = kax
Here we take x
dy_
la
-V'
is
and the cylinder
as the independent variable
and obtain
the line
TWISTED CURVES.
= \/2
I
735
dx =
where xv and &2 are the lower and upper limits of integration.
Hence,
upon the
in this curve
makes an angle
any portion
of the arc
of
- with the
Taking the same curve,
\/2 times its projection
let
.r-axis.
us put
i.e.
Then
we then have a
is
In other words, at every point of this curve the tangent
#-axis.
a cosh 2 -.
y
case such as that discussed in (6) of the preceding article,
x, y and z in terms of an auxiliary fourth variable /.
having expressed
^
a2
r s inh2 w + 4 sinh 2
.
|+(cosh
u - 1)2
]
CHAPTER XX.
736
whence
s
=
\
^ cosh u + C
v 2 L. a
The curve
of
|
= </2 (x.> - Xj)
-1
as before.
-J*i
intersection
of
the two cylinders
is
represented in
Fig. 222.
Ex.
2.
To
find
an expression in the form of an integral for the rectitwo right circular cylinders whose
fication of the line of intersection of
axes intersect at right angles.
If we take the axes of the cylinders as the axes of
we may write the equations
Let us take a
From
>
of the cylinders as
b.
the equations
T
dx
we have
yd
,
dz
tidy
J
and
Put y = b sin
0,
and
let
Then
dO.
z
and x
respectively,
THE HELIX.
When
the cylinders are of equal radius, k=l, and this becomes
s=b
[
the result of Art. 573, for an ellipse whose axes are in the ratio
i.e.
\/2
737
:
to
1,
which the curve of intersection then reduces.
It is interesting in this connexion to note more generally that when
the axes of two equal cylinders cut at right angles, and a sphere rolls
completely round in contact with both cylinders, the locus of its centre
is
two
ellipses.
In our case the rolling sphere has a zero radius.
In the "right circular Helix" or " Helicoidal curve," which
an ordinary thread on a screw, we have a curve traced on a right
circular cylinder and cutting all the generators of the cylinder at the
same angle.
712.
is
Fig. 224.
Let a be the angle the screw-thread makes with a circular section of
P any point on the curve, coordinates #, ?/, z referred to
the cylinder,
rectangular axes, the z-axis being the axis of the cylinder and the .r-axis
taken to cut the curve at a point A.
Let 8 be the angle the plane
through P and the axis makes with the plane of xz and let a be the
radius of the cylinder.
OPN
t
We
have
Hence
und
x=a
cos 0,
y = a sin
0,
z
= a6 tan a.
ds 2 = dx2 + dy* + dz* = a'J sec 2 a
d<9
2
s= ad sec a.
obvious from the fact that in this case the surface may be
becomes a right-angled
developed into a plane, and the triangle
triangle with sides ad, aO tan a and *, with P. ne of its acute angles a.
E.I.C.
3A
This
is
ANP
CHAPTER XX.
738
Since the curve develops into a straight line when the surface
developed into a plane, the surface itself being supposed entirely inextensible, the distance between any two points which it connects upon
the cylinder is a minimum distance on the cylinder between those two
713.
is
Such
points.
Geodesies
of
lines
minimum
length on any
surface are
termed
on a right circular cylinder are
helices.
(see Smith's Solid Geom., Art. 259).
Hence geodesic
lines
A Property of Geodesic Lines.
714.
an obvious property of such curves that if P, Q be any
points upon a geodesic line upon any surface, the path from
P to Q via this line being less than from P to Q via any contiguous supposititious paths from P to Q, viz. PBQ, or PCQ, on
opposite sides of it and of the same length, and the three
It is
c
m
P
Fig. 225.
lengths PAQ the geodesic, and PBQ, PCQ the supposititious
paths being unaltered in length by any deformation of the
surface on which they are drawn, supposed inextensible, the
deformed path to which PAQ is changed will still be in
length intermediate between the lengths of the contiguous
paths to which PBQ and PCQ are changed and which are
Hence, in the limit when PBQ and PCQ and their
equal.
deformed lengths are made to close up to ultimate coincidence
with PAQ and its deformed length, it will be clear that the
deformed PAQ is still a line of minimum length on the
deformed surface, being entrapped between two supposititious
paths which are both of greater length on opposite sides of it.
Thus geodesies on inextensible surfaces remain geodesies after
any deformation of the surface on which they are drawn.
715.
helix
if
It follows that a right circular helix remains a right circular
6
&
it
be the radii
helix
is
it is drawn be transferred from the cylinder
was wrapped to a cylinder of different radius. Let a and
of the first and second cylinders and J3 the angle the new
the paper on which
upon which
makes with the
the angle in the
circular section.
new
Then
s
=
0=
?>
6',
where
cos p
cos a
helix corresponding to 6 in the original one
;
CYLINDRICAL COORDINATES.
and the new coordinates
of
P can
739
be written down, the axes being placed
as described for the first helix.
716.
Cylindrical Coordinates.
For many
cylinders,
it
particularly for curves drawn upon
desirable to use cylindrical coordinates, viz.
cases,
is
the ordinary Cartesians are transformed to the polar
system as regards the x, y plane, and the z-coordinate is left
r,
0, z, i.e.
unaltered.
and r+6V, 6-\-S9, z-\-Sz as the coordinates of
contiguous points P, Q on a curve, we have, since Sr, r 80, Sz
Taking
r,
0, z
are mutually perpendicular elements,
Fig. 226.
N, N' be the feet of the perpendiculars from P,
the plane of x-y we have, to the second order,
For
if
Q upon
,
and plainly
PQ*=NN'*+Sz2
.
Hence, if the distance measured along the arc
have, to the second order,
whence
~ Udr
2
2j
+(rd9) rdz
2
,
PQ
be
Ss,
we
CHAPTER XX.
740
which we may write
in
any
of the forms
or
according as
it is
pendent variable
in case
r,
variable
6, z
;
convenient to take
or
we may
also write
it,
or z as the inde-
as in Cartesians, as
are expressed in terms of a fourth auxiliary
t.
The most common case
pendent
0, r
is
when
is
taken as the inde-
variable.
Curves on a Right Circular Cylinder.
are discussing a curve drawn upon the surface of
a right circular cylinder of radius a, we have
717.
When we
r=a
and the
718,
rectification
If
we apply
dr=0,
this to the case of the helix already considered, viz.
we have
r
s=
and
formula at once reduces to
I
= a, z=aQ tan a,
Wl + tan' a dd = a8 sec
2
a,
as before (Art. 712).
be at once remarked, however, that in all cases of
curves drawn upon a right circular cylinder, the length of the
It will
readily be considered by first developing the
surface
into a plane, and in fact the formula above
cylindrical
is merely the Cartesian formula
arc
may
as
for the developed surface, dx replacing a dO*
SPHERICAL-POLAR COORDINATES.
719. Ex.
Find the length of an arc
741
of the curve of intersection of the
cylinders
Putting #=acos0, we have
and
dz
Hence
whence
s
= agd
*0
or
a
f=alogsec^,
720.
in
= a log sec Q.
ds
,
and
-777=
=
alogtan(J
In this case the developed curve
viz.
z
is
+
-J.
the Catenary of Equal Strength,
which ^ = a\^ and
s
= agd~
l
\j/
(see
Ex.
5,
Art. 519).
General Polar Formulae.
The general polar formula
radius vector
r,
for rectification in terms of the
the co-latitude
9,
and the azimuthal angle, or
longitude, <, is easily obtained.
Fig. 227.
In passing from the point P(r, 0, 0) to a contiguous point
Q(r+Sr, 9+89, + S</>) along an elementary arc Ss of a curve,
the projections of the chord PQ in the three directions,
</>
(a) along the radius vector, increasing r
(c)
;
the meridian plane, increasing
perpendicular to the meridian plane, increasing 0,
(6) in
are respectively
;
oY,
r 39,
r sin
9
8<p
;
CHAPTER XX.
.742
and these being mutually perpendicular elements we have,
to
the second order,
and as either
r,
9,
<j>
or a fourth variable
t
can be regarded as
we have
the independent variable to suit circumstances,
or
or
or
721. Modification for Curves on the Sphere and the Cylinder.
There are two important cases to consider.
(1) If the curve under discussion lie on a sphere of radius a,
r=a,
dr=0,
and
s=
or
or
if it
be deemed desirable to use the latitude
the co-latitude 6
/
instead of
(^=|
s=
5=
or
(2) If
the curve under discussion
lie
on the surface of
a, and whose
a right circular cone whose semivertical angle is
axis is the 2-axis and vertex the origin, we have
0=a,
dr
or
dO=Q,
RHUMB
Ex.
722.
This
at the
is
1.
LINES OR LOXODROMES.
Line or " Loxodrome
"Rhumb"
"
743
on a sphere.
all the meridians
a curve on the surface of a sphere which cuts
same angle.
Fig. 228.
Let PQ be an element ds of such a line, zOP, zOQ meridian planes.
Let a small circle of the sphere parallel to the equatorial plane x-y pass
be the
through Q and cut the meridian plane of P in N. Let I and
latitude and longitude of P, a the radius of the sphere and a the
<
constant angle
Then
y^
cot a
or
whence
-
NPQ.
Tt^Q = Lt
Tt acosl8d>
tan a = Lt
*-,
</>
d(j>
cot
= sec I dl
a = gd" 1 ^
.
7
d<j>
i.e.
cosl-^~
= tan a,
;
i.e.
logtan(^ + -J,
which, with r=a, form the equations' of the curve.
Also
Hence
s
in this curve
r=a,
Ex.
2.
equation
we have
l
= gd(<f>cota) and
s=alseca.
In the case of a spiral traced on a sphere and defined by the
= <tana, where a is constant, we have
dl
=a
I
Vl + cot a cos 2Z dl
2
= al \/cosec 2 a - cot2 a sin 2Z dl
I
a cosec a
E(l, cos a),
,
=
cos 2 a sin 2 / dl
a cosec a
v'l
CHAPTER XX.
744
and the arc
of this spiral is therefore expressible as an arc of an
semi-major axis acoseca and eccentricity cos a (see Art. 567).
Ex.
ellipse of
In the case of a curve drawn upon a conical surface to cut all
a, we have, taking the origin
3.
the generators at the same constant angle
Fig. 229.
and the
at the vertex
axis of the cone as the 2-axis
and
j3
for the semi-
vertical angle of the cone,
8 dd>
^=
r sin
,"
as in
Example
(1),
tanoc,
and therefore
for the sphere,
= sin ft cot a d<j>
r=Ae
whence
where
A
is
the curve
;
<j>s
an arbitrary constant, determinable when some one point on
is specified.
The projection
of the curve
upon the x-y plane
1
angular spiral of angle cot" (sin
/5
is
therefore an equi-
cot a).
We also have
=
between limits r lt rz
a
through the origin, and
s
.
If the spiral passes
be measured from that
= rseca,
point,
which
\/l+tan 2 a dr =
r sec
/
is
from the consideration that if the curve be developed
become an equiangular spiral of angle a.
also obvious
upon a plane
it
will
PEDAL FORMULA.
745
The p, r Formula.
723.
=
The
p, r formula of Art. 547, viz. s
for curves of double curvature.
For, with the
same notation as
p
-=sm0
r
T
and
\-j====,
*
still
holds
before,
dr
-j-=cos(b
ds
}
being the angle which the tangent makes with the radius
<t>
vector from the origin
;
whence
and
-^
For cases
the formula
all
drawn upon a
of curves
useless.
is
For
sphere, the centre being at the origin,
in that case, the tangent being necessarily at
points at right angles to the radius vector,
^=
and p
r
throughout.
In the case of a curve drawn upon a right circular cone whose vertex
at the origin, we may use the formula with advantage but it is to be
remembered that we are doing no more than if we regarded the conical
is
;
surface as developed
Ex. For the case
upon a plane.
already
considered
of
generators of a cone at a constant angle a,
and s=
I
J cos
a
=rseca, as
a
curve cutting all the
at once p = rsina
we have
in the last article.
There are but few curves of double curvature, however, for
which the p, r relation is known, with the exception of course
of such as, having been originally plane curves, have been laid
upon a developable surface. For such cases the formula is
useful, as also of course whenever the relation can be readily
found.
724. Ex. Let BAA'B' be a strip of thin inextensible ribbon lying
upon a plane. Let OAA be a perpendicular from any point O of the
plane upon AB and A'B' and OPP' any other radius vector from 0.
Let OA = 1 OP=l, PA=s.
2
2
Then obviously
Z =
+Z 2
,
6-
Now
.
imagine this ribbon wrapped tightly without folding or creasing
with OAA' as a generator, the
upon a right circular cone of vertex
semivertical angle being a, the wrapping commencing with OA in con-
CHAPTER XX.
746
tact
When the wrapping has been completed, OP coming
and becoming a generator, let us unwrap the triangle from
keeping OP in contact and starting the unwrapping with
with the cone.
into contact
the cone,
O
B
A'
P'
B'
Fig. 230.
fixed at the vertex
when the
OA, keeping
unwrapping is just complete, the triangle has taken the position OYP,
A.
and is the same triangle as we started with, OYP being a right angle.
releasing the generator
;
O
Fig. 231.
It appears
AP
AP
2
2
the arc
upon the cone has a length \/ -Z
is a geodesic
the
cone
arc
that
the
upon
(2)
lies on a sphere of radius ?
(3) that the locus of Y in the unwrapping
and vertex at
on the cone is p = l for this
(4) that the p, r equation of this geodesic
(1) that
;
;
;
,
is
(5) the
so on the plane
formula
s=
from which
\-^
--=,
is
it
merely
was constructed
;
INVERSION.
F
747
an involute of the geodesic
taking a sphere of any radius with centre at O, cutting the axis OZ
at M, the generator OP at L and OY the perpendicular on the
is. a
right-angled spherical triangle, where
tangent at N,
the
(6)
(7)
locus
is
;
LMN
A
ML=a, LN=ta,n~
whence
cos
and
l
j-
MN = cos a cos LN
A
If
<
o
be the angle between the plane ZOY and the plane ZOA, and 8
ZOY, we have thus shown that
the angle
cos
= cos a cos LN. and
=
r
1
therefore
'
cos
Q
and radius
Now, if we take a circle on the plane OP 7 with centre
OP, and consider the arc bounded by OP and Y produced, this arc will
wrap upon the cone and will coincide with the corresponding arc of the
circular section of the cone through P
whence if x he the angle between
the plane ZOP and the plane ZOA,
;
Zsina.x = ZxanglePOF,
x sma = ^an
i
and
i
S
T'
& = \-LMN = -
Hence
sin
i.e.
is
<p
=-1
sin
,
a
tan" 1
Vcos 2 a
a
cos 2
tan-
V- tanIQ
tan- 1
cos
sin
?
1Q
r
sm a
\/cos 2 a
1
-,
1
a
cos
,
cos2
,
the equation of a cone which by its intersection with the sphere of
Z and centre O
gives the Y locus, which is also an involute of the
radius
geodesic on the cone.
725.
Inversion.
The process
of inversion
may
sometimes be employed with
advantage.
particularly the case when a twisted
curve lies on the surface of a sphere. By inverting with
This
is
regard to a point on the surface of the sphere, the spherical
is inverted into a
plane and the twisted curve into a
surface
plane curve, and vice versa.
Let
be the pole of inversion and
the diameter
OA
Jc
the constant, and
let
of the sphere meet the plane into which the
sphere inverts at C.
Then OA
OC
.
OC=k2
=(g=c,
say.
,
CHAPTER XX.
748
Let the element PQ,
viz.
spherical surface invert into
plane inverse curve.
PQ,
viz. &',
PQ
PQ-
Then
of a twisted curve on the
8s,
an element of the
'
OP'.OQ"
ds'
or ultimately
OP'*'
Let
CP'=r.
Then
and
if
S=!
this integral for the plane curve
rectification of the twisted curve
can be found, the
will have been
on the sphere
effected.
Fig. 232.
The method may
curves which
726.
lie
also be used to discover rectifiable twisted
on a spherical surface.
Extension of Art.
The angle between
inversion.
If
if
two
230, Diff. Calc., for
intersecting curves
Present Purposes.
is
unaffected
(Extension of Art. 230 of Diff. Calc.)
intersect in the line
planes QPP'Q',
RPPR
PQ, PQ' make the same angle with PP'
PR
by
PP' and
in opposite directions
and PR, then the angle QPR=Q'P'R. For, take
as also
distances PN and P'N' equal to each other in opposite directions
from P and
respectively on PP' produced, and let two
planes perpendicular to the line PP' be drawn through N and
N' to cut PQ and PR at Q and R, and to cut P'Q and PR' in
Q' and
respectively.
P
R
INVERSION.
749
PNQ and P'N'Q',
PNR and P'N'R respectively, we have NQ=N'Q' and
NR = N'R', whilst QNR=Q'N'R, and therefore the triangles
QNR, Q'N'R are congruent and QRQ'R: whence the angles
Then, from the congruent pairs of triangles
and
QPR, Q'P'R are
also equal.
PQ, P'Q' be the directions of
P' to inverse elements of curves in the
It follows therefore that if
the tangents at
P and
Q'
Fig. 233.
plane PP'Q'Q and PR, P'R be the directions of the tangents
f
at P and P to inverse elements of curves in the plane PP'R'R,
then, as in this case
PQ
and P'Q make equal angles with PP
do PR and P'R' (as proved in
in opposite directions, as also
curves in a plane), it will
follow 'that the angle between two curves meeting at P is
Hence
equal to the angle between the inverses meeting at P.
the result of Art. 230 of Diff. Calc. is now extended to any case
of inversion, the curves not being necessarily plane, and the
Differential Calculus, Art. 229, for
pole of inversion
727.
If
now
lying anywhere.
etc.
Stereographic Projection,
we take
as constant of inversion the diameter of the
on the sphere, the
sphere inverts into the tangent plane at the opposite end of
the diameter through the pole.
If the constant of inversion be taken as
sphere,
and the pole of inversion a point
diameter
=
.
,
i.e.
r=
v2
,.
.
radius,
the sphere inverts into the equatorial plane of which the origin
of inversion is a pole.
CHAPTER XX.
750
In
all
such cases the inversion amounts
to
a conical pro-
jection with the origin
as pole of projection.
the projection is upon an equatorial plane with
is called a Stereographic Projection.
When
pole, it
for
In any of these cases, the angles of intersection of any spherical curves project or invert into equal angles of intersection
of the projected or inverted curve.
Orthogonal intersection
in
the
intersection
remains orthogonal
projected curves curves
which touch on the sphere project or invert into curves which
;
invert
touch; circular arcs which pass through the pole
all other circles, great or small, into circles.
into straight lines
;
Ex. Consider the rectification of the line of intersection of the sphere
with the
elliptic
cone
Inverting with regard to the origin, and with c for constant of
=
sphere becomes the plane z c and the cone remains
inversion, the
t
unaltered, but cutting the plane
If
z
=c
in the ellipse
PQ, PQ' be corresponding elements
inverse curves,
fa'
ds,
7/
a
+Y^~1-
ds of the original and the
C2
*r$%i*'*?+4&?
where r
is
the central radius vector of the ellipse to the point P'.
STEREOGRAPHIC PROJECTION.
751
Hence, taking Q as the complement of the eccentric angle of
have for the ellipse,
2
2
2
tf=asin0, y = 6cos0, r = a sin + 1? cos? 0,
2
<fc'
=(a
2
/",
we
cos 2 0-f& 2 sin 2 0)d0 2 .
and
"
2
(c
+ 62) cos2 + (c2 + a 2 ) sin 2
A.
2
{(c
2
{(c
by the
e is
}
Va 2 cos 2 + 6 2 sin 2
_ |( C2 + ^2) cos 2 + (c2 + ^ 2 ) sin 2 0} + (a 2 + 6 2 + c2 )
2
where
+ b 2 ) cos 2 + (c 2 + a 2 ) sin 2
+ 6 2 )cos 2 + (c- + a 2) sin 2 0} \/a 2 cos 2 + 62 sin 2
the eccentricity.
And
thus the arc of this curve
elliptic integrals of the first
is
expressible
and third kinds.
Fig. 235.
728. Curves on Spherical Surfaces in particular.
Formulae for
the Rectification of Curves on a Spherical Surface, analogous to
the p, r and p,
APP'
\/r
Formulae for a Plane Curve.
be any curve drawn upon the surface of a sphere
of radius unity.
Let P, P' be contiguous points, and let
Let
CHAPTER XX.
752
arc-PP' = (Ss.
Let
PY, FT be the great
be any fixed pole on the sphere, and
F
P
let
T
and
the
07,
tangents at
a
and
fixed
OAx
great circle perpendiculars to them from 0,
great circle cutting the curve at A, the point from which s is
circle
;
measured.
YOT = S^
xOY = \lr,
Let
PY =
OY=p,
t
FT =
t,
t
+ &.
P
and F,
be the great circle perpendicular upon OP'.
Thus,
Let OP, OP' be the great
circle radii vectores of
Then, from the spherical triangle OYP,
cos r = cos p cos
PN
Let
we have
and sin p = sin r sin
t
as in plane geometry,
we have
dr
(
(p.
.
^=cos0(^viz.
Li
and
/.
OT
Let
triangle
i.e.
intersect
YOZ,
at Z, then,
from the right-angled
fAnOY = cot YQZ tan 7Z,
to the first order,
Also to the
PY
= ............................ (1)
.
YZ =
<S\/r
sin
^?.
first order,
.e.
-.
And
.
.,
T
.,
in the limit,
dt
= yrf-sm
,
7?,
s= t+ \8mpd\Js ......................... (2)
t.e.
Formulae
ds
^-r-
(1)
and
(2) are
^f-J^L:2
Jjr*-p
for plane curves.
analogous to
and
1^1
CURVES ON SPHERICAL SURFACES.
Convention of Sign of
729.
In regard to
regard to sign.
t
753
Closed Oval.
t.
necessary to make a convention with
It will be in agreement with the convention
it
is
for plane curves, Art. 531, if we fix that t is to be reckoned
is measured
positive when, as in the case of Fig. 185,
PY
from the point of contact in the direction opposite to that of
increase of the arc
As
s.
in plane curves,
it appears that if the curve considered
be a closed oval on the sphere, t returns to its original value
when integration is taken round the oval. Hence for a
closed curve surrounding the pole, encircling
it
once,
T27T
I
Jo
sin p
d\//\
If the radius of the sphere be a instead of
unity, which lias
been taken for convenience, the absolute length of the arc
will be changed in the ratio a 1, so that if s' and t' be
lengths,
:
p and
whilst
r
are measured
centre of the sphere,
by the angles subtended at the
formulae (1) and (2) become respectively
sin r dr
f
and
sm>
s
J
Loxodrome cutting meridians at a constant
6 be the co-latitude and azimuthal angle of any current
upon the curve.
Ex. In the case of a
730.
angle
a, let r,
point
P
Fig. 23G.
Then
.
=t +a\smpd\!s.
^>
Hence
= a and
= sinr sin
=
tt
f
\
a.
ninrdr
:
,
J^sinV-sin
being the radius of the sphere,
E.I.C.
i.e.
SB
= a
cos
2
/?
CHAPTER XX.
754
Arc
measured from the pole =
.
of curve
as in the case of the equiangular spiral
also have in this curve
arcual radius vector
OP
,
cos a
upon a plane.
....(a)
(See also Art. 548.)
We
o
i.e.
if
log
r
=r
when
,
= 0,
i.e.
=
I
I
tan ~ =
cot a,
cota
(6)
,
tan^|e'
= gd(0cota),
another form of the property
(c)
already established in Art. 722, a relation between the latitude and
which
is
longitude analogous to that between y and x in a Cartesian equation.
To
731.
find sin p.
The expression
for sin
in terms of
p
\js
which
is
required
Take
in the integration of Art. 729 may be found as follows.
the 2-axis through 0, the pole of the curve. Let C be the
O
Fig. 237.
(See also Fig. 235.)
centre of the sphere and F(x, y, z)=0 be the equation of the
2
in the given curve.
cone which cuts the sphere x 2 -\-y 2 + z 2
is a homogeneous function of x, y and z.
Then
=a
F
The tangent plane
curve
is
The equation
z-axis
is
to the cone at the point
x', y', z'
of the
x Fx + yFy + zF* = 0.
-
>
of a perpendicular plane
x F y - yFx >=0.
>
COY
through the
THE POLAR CURVE.
Hence
And
tan ^s
= j/-,
the perpendicular
t.e.
755
.(A)
.
cos
P (=ON,
sin
\[s
Fig. 237),
plane from the pole 0, whose coordinates are
From ^ =
and equations (A) and
upon the tangent
(0, 0, a), is
(B), the ratios
xiy'iz
are to be eliminated, and there will result a relation between
P and
^, say,
Again,
Hence the
732.
P = 0/0/0P=
a
relation required
sin p.
is
Relation with the Polar Curve.
Let any curve be drawn upon a sphere of centre
and
radius r. and let the cone with vertex 0, and passing through
238.
the curve, be drawn.
Let a plane through the centre of
the sphere, and therefore cutting the sphere in a great circle,
roll upon the surface of the cone.
The poles of this plane
then trace out two equal loci on the surface of the sphere,
Either of these equal and similar loci is called the polar curve
CHAPTER XX.
756
The great circle arcs which are the lines
of the given curve.
of intersection of the sphere and the plane touch the curve as
rolls, and are great circle tangents.
Let Q, Q' be two positions of one of the poles corresponding
to the great circles PT, P'T, intersecting at T and touching a
the plane
curve C^ drawn upon the sphere. Let the curve locus of Q
be referred to as the curve C 9
Drawing the great circles
.
PQ,
TQ
}
TQ',
PQ', we have
PQ = TQ,
TQ' = P'Q',
TQ = TQ',
and
both quadrants,
both quadrants,
both being quadrants.
Hence, in the limit when P' and
T
upon C 19 and
P
are indefinitely close,
the pole of a tangent plane
ultimately
which
cuts the sphere in C.2
at
vertex
to the cone with
0,
Hence the relation between the two curves is reciprocal.
lies
is
.
Each one
is
the locus of the poles of tangent planes of the
If QRQ' be the great circle arc
cone which defines the other.
Q and Q', T is
which pass through
joining
and the poles of all great circles
on QRQ' or QRQ' produced, that is
its pole,
T
lie
the great circle chord QRQ' of the arc QQ' of C 2
the poles of great circles through T.
The figure bounded by the arc QQ' of the
is
the path of
C2
locus and.
the great circle arc Q'RQ is thus the reciprocal of the figure
of the C 1 locus and the two great
bounded by the arc
Also the angle between two great
circle tangents TP, TP.
PP
circles
being the same as that subtended at the centre by
their poles,
we have
Angle
733.
A
PTP^TT-QOQ',
Theorem given by
ir-QRQ'.
i.e.
Schulz.
Let a circumscribed polygon consisting of an infinitely large
number of infinitesimal great circle tangents be drawn to the
one curve (7 lt and let the reciprocal inscribed polygon of great
circle chords be drawn in C.2
Then, if the angles of the one
.
be A, B, C, D,
sponding sides
and the angular measures of the
of the other be a, b', c, d',... we have
...,
,
A = TT
a
B=-7rb'
}
etc.
corre-
A THEOREM OF SCHULZ.
757
We
have Area of the polygon
(Todhunter and Leathern, Spherical Trigonometry, Art. 129)
= 2(7T-*>
if s'
2
,
be the angular semiperimeter of the polygon A'B'C'D'
B
A
Fig. 239.
is stated by Todhunter and Leathern
by Schulz, Sphdrik, ii., p. 241.* The author-
This remarkable relation
as "referred to"
ship does not appear to be clear.
Proceeding to the limit
when the sides are indefinitely small, if (C^), (PJ be the area
and linear perimeter of C lt and (C2 ), (P2 ) the area and linear
perimeter of (72 we have
,
2
(Cy-f r(P2 )=27rr :=half the surface of the sphere,
(C2 ) + r(Pl )=27rr
and similarly
that
is
27rr
2
-(C = r(P2
1)
)
and
2-7rr
;
2
2
>
-(C2 )=r(P
1 ).
Thus when the area of the one curve can be found, the
perimeter of the other can be found and vice versa.
*See
also Williamson's Integral Calculus, Art. 188.
CHAPTER XX.
758
It appears also that the area included
and any great
circle
which
it
between either curve
does not cut
is
equal to a
Fig. 240.
rectangle of length the perimeter of the other curve and
breadth the radius of the sphere.
734.
Formula analogous to that
for the
Area of a Plane Curve
in Polars.
It is
a well-known result in the mensuration of a spherical
any belt on a sphere is equal to the
surface that the area of
corresponding belt on the enveloping cylinder whose axis is
perpendicular to the bounding planes of the belt. Let A PA'
O
Fig. 241.
be any small
of the circle
circle of a
and
be the pole
sphere of radius a. Let
of
circle
radius
from
vector
great
OP any
length r, subtending an angle p at the centre. Then the area
of the spherical cap cut off by the small circle
2
cos p).
27ra(a a cos /o)=27ra (l
<
Let the azimuthal angle, of OP be 0.
OP and OP' for which
the area between
Area
OPP'=~ X27ra
2
Then we have
is
increased to
(l-cos p )
for
0+SO,
ANALOGY WITH A PLANE CURVE.
759
2
analogous to the result |r <5# for a plane (and indeed becoming
2
Jr o$
when we put
Hence, taking
any
p,
M
- for
p and the radius a becomes
ct
we have
as coordinates,
in the
same way as
If the curve be
A= ^
8=0^
\r 2 dO for
0=--0 2 ,
a plane area (Art. 407).
an oval encircling the pole
2
\
Jo
once,
*
"
A=a
for the area of
bounded by a curve on
portion of the spherical surface
the sphere, and the meridians
oo).
(l-cosp)dO=27ra
2
-a 2
\
Jo
cospdS.
Fig. 242.
The area therefore between the curve and the equatorial
plane of
is
fz n
a2
cos p d9,
\
Jo
or
if
we
use
I
for the latitude,
i.e.
the complement of
p,
and
for the longitude or azimuthal angle,
sin
I
d9.
(27T
o
If,
curve
then, this integral be evaluated for the polar or reciprocal
C2 the result will be aP lt i.e.
,
Perimeter =
(I,
0)
P^a]Jo
sinldO,
being the latitude and longitude of a point on the
reciprocal curve.
CHAPTER XX.
760
Illustrative Examples.
Ex.
1.
To
test this result in a
and
with pole at
known
of angular radius p.
27r<x
The polar curve
is
sin p.
another small
circle of
therefore the latitude of any point on
The formula gives
J\ =
rin
al
sin
case, take Ci as a small circle
Its perimeter is obviously
it is
angular radius --/>, and
p, in this
case a constant.
pd0 = '2,7ra sin p,
Jo
which
is
in
Ex.
2.
Find the length of the spiral, traced on a sphere, whose
is defined by the equation
4p = 9 corresponding to limits
to 2?r, p and Q having the meanings assigned to them in
agreement with the stated
result.
reciprocal
for 6 from
Art. 734.
The area between
IQ
is
4
Hence the perimeter required = 4a,
sphere.
and the equatorial plane
the reciprocal spiral
i.e.
twice the diameter of the
(Fig. 243.)
Fig. 243.
Fig. 244.
Ex. 3. To find the area bounded by any arc of a great circle and two
spherical radii vectores.
Let the plane of the great circle be at right angles to the plane of the
paper and cut the meridian in the plane of the paper at a point A whose
co-latitude
is a.
(Fig. 244.)
of the great circle
Then the equation
is
cos 8 = cotp tan
a,
from the spherical triangle OPA, right angled at A.
Then we have
SPHERO-CONICS.
761
and the integral
a?
f
J \/cot 2 acos 2 #
+l
v cosec 2 a
= tan a
= tan a
Hence the area between two
radii
meridian in the plane of the paper
a 2 (^ 2 -^i)-
2
1
[sin- (sin^ 2
sin 6
cot 2 a sin 2 #
dsiuO
.-
I
J \/sec 2 a-sin 2
sin -1 (sin B cos a).
making angles 8 and
1
0.2
with the
is
1
cosa)-sin- (sin^
1
cosa)].
(See Art. 781.)
735. The Case of a Sphero-conic.
DEF. A sphero-conic is the line of intersection of a cone of
the second degree with a sphere whose centre is at the vertex
of the cone.
Fig. 245.
Let the equation of the sphere be x 2 +y 2 +z 2 =d2 and that of
z
z
*
the cone
,
The
reciprocal cone has for equation
Putting p for the co-latitude and
for the azimuthal angle of
any point, we have x=dsinp cosO, y=dsinpsmO, z=dcosp,
and the equations of the sphero-conic and its reciprocal become
respectively
*0
c
in p,
+
coordinates.
M
and
c . cot
p=0 . C os3 + 6sin9
CHAPTER XX.
762
A
The area
bounded by the arc of the sphero-conic
l
cos 2 ^
c
2
a
and the meridians 6 = 0,
= d 2 (0-cl
l ),
=
sin 2 fl
2
is
given by
say;
and putting
a
sin
= b cos
b
an
^7' ^t ^=^tan x
"ihT7
,
,
whence
II
=
J
o
<*
2
- (<*2 - & 2
)
sin 2 x
V(a
2
+ c - (a2 - 6
2
2
)
a
)sii
55
and
A =d
2
1
-
.0
I
2
a\/c*< -f-c
and
is
2
Hi tan -My- tan 9)
\6
\/-T^t 6 2
,
/
Vc^-[-
'
'
therefore expressed in terms of a Legendrian integral
of the third species.
For the reciprocal sphero-conic
the area
=-
is
A 2 bounded by
C 2 cot 2 o
/
= a 2 cos 2 ^ + o 2 sin 2 ^
= and
the arc and the meridians
given by
(a
2
+ c 2 )cos $ + (b + c
2
2
2
)
sin 2 S
and putting
b sin $
cos
x
a cos #
sin
,%
x
i.e.
V.I-'.
tan^
^ =
jcotx,
L/CAIAJ.
.
i
SPHERO-CONICS.
763
we have
whence
c
and the area
where
of the
I
=
same curve from
a 2 -I) 2
to
=
is
same elliptic integral as occurs in the value of
II t is its complete value.
II is the
A and
736. Again, for the Rectification of
sin 2
cot 2 /o_cos 2
:
c*
~^~ ~F~'
the tangent plane to the cone
xz
at
any point P(x,
y',
y
2
22
z) of the sphero-conic
APE
(Fig. 245) is
f
yy _^_
a*^b*~c*>
xx_
and the perpendicular plane
tan \/r=
giving
where
OGY
\js
is
,/-,
b
x
*
Also the perpendicular
tangent plane at P, viz.
c
ON
GPY,
is
,
the azimuthal angle of the plane
a 2 cos
c
through the z-axis
OCY,
i.e.
y
\//"
b 2 sin
\fs'
(=P') from the pole
is
given by
upon the
CHAPTER XX.
764
Therefore, if p be the angle OCY subtended at
great circle arc OY, P'=eZsin_p, and we have
sin
v=
A
C by
the
a 2 cos 2 \/r -f- 6 2 sin 2 \/r
'
2
(a -f c
2
)
cos 2 ^+(6 2 +c 2 ) sin 2 x/A
'
(Art. 388, Ex. 7.)
and
if s
Hence,
conic from
be the lengths of the arcs of the sphero-
t
P to 5, and of the 'tail' PY respectively
(Fig. 245),
and
and
t
remains to be found.
Now
is
t
the arcual measure of the great circle arc PY.
of CY, CP (C being the centre of the sphere)
The equations
are,
from
(1)
and
(2),
x
y
?L
L
^Y'L
2
2
\a
4
,
/z
a
I
6
z
,
fsr
2\
2
z
i
and
x y z
,=-,=-.
4i
z
x y
'
*y
SY*
v' \
Hence
7
cos(
Vcos7C'P=.
7
/2
c2
/2
,
?/
2
/x'
^V^+i-zVa
2'
:
5
/
a4
T6
4
-^
4
,
y'*
,
BURSTALL'S THEOREM.
765
Also
acosr/r
2
-f c
2
(X
)cos
VVa 2)(a
+ c )cos
/
2
Hence
is
2
.
g\
2
found, viz.
i
-M
Ja 2 cos
/
PY
the negative sign being prefixed because
is measured from
P in the direction of the measurement of the arc increasing
from P to B. (See Art. 729.) Finally then we have
arcPB
,r
,
__
&GT*LJa* cos- ty + b'2 sin 2 ^
^
+a
737.
A
62
flT=IS5
Mr. Burstall's Theorem.
remarkable property of the curve
Burstall,
in vol.
xviii.
of
the
is
established
Proceedings
of
the
by Mr.
London
Mathematical Society, giving a result analogous to that of
Fagnano for the ellipse.
Fig. 246.
Let
AB
be the sphero-conic represented for convenience
a
upon
plane, and let A'B' be an arc of the reciprocal
CHAPTER XX.
766
A
sphero-conic,
being an end of the major axis of the one
A being the
corresponding point on the reciprocal curve.
be corresponding points of the sphero-conic and
and let ARP' be the great circle chord of the
its reciprocal
reciprocal sphero-conic; and AT, PT the great circle arcs
and
Let
P
and
P
f
;
tangential at
A
and P.
Then, since the areas
areas, we have
ATPMA
and
ALPRA are reciprocal
2
d(Arc ^P+tangentPT-f tangent 2M)=27rd -areaof ALPRA.
Now, putting A and A' for the spherical areas OALP' and
OARP'
respectively,
AOP'=&, and
= d9,
the same indefinite Legendrian integral that has occurred
both in the rectification above and in the quadrature of the
reciprocal curve with specified limits,
Arc AP + tang. PT+tang.
we have
TA = 27rd-(A- A')/d,
and
C
where
whilst A' can be found free from elliptic integrals (Art. 734,
Ex. 3).
JV"
.Q
Fig. 247.
Again, as in Art. 736, if Q be any point of the original
sphero-conic, QY' the great circle tangent at Q, and OY' the
A
great circle arc perpendicular to
Arc
and
Arc
it,
AOY'
being
AQ + tang. QY'=d.I*',
0",
BURSTALL'S THEOREM.
then
If
we take
the angles
AOQ(Q"} and A'OP(8') equal and
we have
eliminate the integral,
T^ + tang. TP+d.
Arc AP+tang.
767
0'^
= 27rd+ arc AQ -f tang. Q Y',
or
Arc
AQ- arc AP= tang. TA + tang. TP
tang.
QY'+d. fl'-A'/d
circumf. of a great circle,
giving the difference of two arcs in terms of certain arcs of
circles
and
A'.
P
,T
Fig. 248.
Hence we have the
difference of the arcs AP, AQ expressed
terms of elementary functions, free from elliptic integrals,
which is Mr. Burstall's result, and in its peculiarity resembles
in
Fagnano's result for a plane
738.
Artifices for the Construction of Rectifiable
Some
curves
1;
If
may
be noted.
we take
-Y,
v are
Hence
2
ju
dt,
y = */2luvdt,
any functions of
G|Y
*
Twisted Curves.
the construction of rectifiable twisted
artifices for
x=
where
ellipse.
t
jiPdt,-
at our choice,
= W* + 2tt2 + v
t,2
z=
4
we have
alld
jjf
s=
I(u
2
+ v*)dl = x+z + const*
For a very similar method,
y=
viz.
taking
lJ%fW dx,
see Williamson's Int. Cole., p. 244.
z
=
ff(x)
dx,
'
CHAPTER XX.
768
E.g. consider the line of intersection of the cylinders
Putting
di
we have
2.
= **>>
-$
dt
the case
u = t,
we
If
=^
v=
x=
take
s= #-1-2 + const.
and
l
j(u-v)(u-w)dt,
y=\(v-w)(v-u}dt,
z
where u,
v,
w
=
I
(w
are any functions of
^
=^
ds
we have
~dt
-
t
u)(w
- v) d^
at our choice, then, since
dx
^
^ vw ~~d~t
.,
du
dz
~ti
~dt'*
and
rj.o.
w = 0,
takino
1
dt
v
dt
whence we have
the equations of the curve.
And
for the rectification,
/3
s=^ -g- 4- + const. =x
<
and any
3.
may be
specified limits
Again,
if
taken.
we take
2
x=l(v- w)
dt,
y=f(w- uy*d>,
z
=
-v
f(u
we have
and
and the values
s
of w,
= v^
v, w;
2^ 2 ^ - 2^'* + const.
f
are at our choice, as before.
ARTIFICES FOR TWISTED CURVES.
In
all
these cases
functions of
if u,
w
v,
769
be chosen as rational integral algebraic
the equations of the curve can be found and
f,
between any specified
its
length
limits.
4. Similarly, other algebraic identities which express the sum of three
squares as a constant multiple of the square of a fourth expression may
be used in the same manner to construct rectifiable twisted curves.
Hence, putting
+
--/v2
dx
2
M> \2
with any arbitrary choice of
It will
be noted that
a
(ift)
dz
dy
w
u, v,
as functions of
,
we have
these methods proceed with a view to
all
making
square and avoiding the necessity
P er ^ ecfc
of
integrating an irrational expression.
5.
One type more may be given
illustrating the construction
of a
twisted curve upon the same plan, but of non-algebraic
character.
Taking u, v, w any arbitrary functions of put
rectifiable
,
x=
fdu
/
.
.
-y sm v sin
J at
V = Ifdu
-E sin v cos w dt.
,.
Then
ds
= du
~Tt
~di
dx
37
dt
=
sin
t
8# = 2< 2 -
dt
2
^=o)
dz
cos L
-j-
dt
=
dt
^=
Then
and
t
I
= w + const.
t
dy
-~ =
sin-%
-s
t
z
J
a
v=w =
E.g. taking-
the curve being
dt
J
= fdu
-y- cos v dt.
,
.
w dt,
=^ + C,
t,
2 sin 2t
- cos 2J,
z=
t
sin
t
+ cost.
Methods 1, 2, 3, 4 either give rise to unicursal twisted curves, viz.
those in which the coordinates x, ?/, z can be expressed as rational
algebraic functions of a single parameter t or may be made to give rise to
curves in which .r, y, z and s are irrational functions of f, this depending
upon the choice made for
K.I.C.
?/,
v,
u\
3c
CHAPTER XX.
770
Generalised Formulae.
739.
If the Cartesian coordinates of a point x, y, z be expressed
functions
as
any other three independent parameters
of
u, v, w, as
=fi( u v
>
dx
then
And
if
w \ y=M u
dv +
du-\-
we
,_dx dx
>
>
v
w
>
z =Mu> v
)>
dy = etc.,
dw,
>
w
)>
dz = etc.
write
_ dx
dz dz
dy dy
-,_dxdx
dx_
we have, for the element of distance ds between x, y, z and
x+dx, y + dy, z+dz,
ds2 = a du2 + b dv 2 + c div2 + 2/ dv dw -f 2g dw du + 2h du dv,
two assigned relations between u, v and w, defining a
linear path for x, y, z, we have the rectification formula
and
s
=
for
\[a
740.
du 2 + b dv 2 + c dw2 -f 2fdv dw + 2g dw du -f 2/i du dv$.
If
one relation only between u, v and w be assigned,
on an assigned surface. Let the relation be
x, y, z travels
x(u, v,w)
Then
= 0.
||^
and this being a linear relation between du, dv, dw, one of
the letters u, v, w, and one of the differentials du, dv, dw
may
may
be eliminated, and the square of the linear element ds
then be expressed as
where the forms of
The values
x, y, z
G
of E, F,
/
*^
/
r
I
are
now
derived from these equations are
*~\-
m
I
3
*"\
.
^
.
.
v
*-N
*"\
i
<^\
OA,'
\
GENERALISED FORMULAE.
741.
The quantity
EG F
2
is
EG F = ~,
2
For
essentially positive.
-f two
and
is
positive.
Eliminating du, dv from the equations
'dx
30
'dx
-,
we have Jl dx+Jz dy-}-Js dz = 0,
equation of the surface
743.
similar expressions
say,
,
742.
771
Dr.
Salmon
identically, viz. the differential
on which the curve
(Solid
Geom.,
lies.
252)
p.
shows that the
differential equation of the lines of curvature is
dx
t/1
dy
dz
Ja
Jn
= 0,
dJ
and obtains
in terms of
u and
v a formula for the evaluation
of the principal radii of curvature.
Now ds2
the square of the linear element connecting
the point u, v with the point u+8u, v+Sv, and lies on the
surface
=
x=
744.
is
^(u,
v),
y
3 (u, v).
v +bv
Fig. 249.
If
d(r1
we
travel along a line for
= \/Edu,
constant,
and
we have
if
we
which v
is
constant,
travel along a line for
d<rz = \fGdv
}
and ds
is
we have
which
u
is
the corresponding
CHAPTER XX.
772
diagonal of the infinitesimal parallelogram whose adjacent
Let o> be the angle between them.
edges are d<r 19 c?<r2
.
Then
whence
ds
it
2
= E du*2*jEG du dv cos
appears that
= F
,
and
and that the area of the elementary parallelogram
= d^ da-2 sin w = \iEG~F2 du dv.
We
therefore have also a formula for the quadrature of
the surface,
viz.
=
2
+J +J
2
2
[ [\/e/!
When the two families of curves on the
u = const., v = const., cut orthogonally, we have
coso>=0
and
8
viz.
^=0
and
=
surface,
8=
udv.
u=
This will necessarily be so, for instance, when
const.,
v= const, are the equations of the lines of curvature on the
surface.
PKOBLEMS.
1
r
.
may
2.
that the equations of a Ehumb line on a sphere of radius
be written as
x 2 + y2 + z2 = r 2,
Show
Show
that the curve of intersection of the cylinders
y
is
given by
3.
A
= 8ax, x = ae a
s = x + z + const.
2
,
K
touches the plane of an ellipse of
sphere of diameter
is the other end of the
a, b at its centre C.
A
principal axes
diameter of the sphere through
the sphere by lines through A.
C.
The
Show
ellipse is projected
on to
that the length of the curve
so described will be
K*Ja* sin 2
J
#
2
+
fl
2
cos
<
<ft
2
+ b 2 cos 2
,
.
JOHN'S, 1884,]
PROBLEMS.
4.
A curve
is
drawn upon the
</>
</>
773
surface of a sphere such that
sin 6
= const.,
being the longitude and 6 the co-latitude of any point.
= a\og(^ ^4^ ) is the length of the arc between
/
\ tan
points where 0= 6 l and 6= 2 and a is the radius of the sphere.
= 1 upon the
Give a sketch showing the nature of the curve
sin
=
sphere r a.
Show
that
s
-i
t/2/
,
<
5.
Show
that the line of intersection of the sphere
r=
and the cone
tan<9
=
c
cos
V
cota
c
is
rectifiable,
and that
s
= cO sec a.
Also show that the conical projection of this curve on the sphere
upon the tangent plane at the end of the diameter remote from the
origin, the origin being the pole of projection, is an equiangular
Hence deduce the same result by inversion.
spiral.
6.
Show
that the curve of intersection of the sphere
and the cone
(2z
2
+ 2f + zx)* =
2
(x
+ f)
projects conically from the origin into
plane
7.
z= 2a.
Show
Hence obtain the
Show
z,
z, is
?/,
that for the curve
60#
the arc measured from the origin,
s
9.
3fl
is
given by
+ x = \*fz +
const.
In the curve for which
J-O-Od-O. J=-'(i-0
show that
cardioide
upon
the
that the length of the arc of intersection of the cylinders
measured from the origin to any point
8.
a
rectification of the twisted curve.
s
2
,
J
x + z.
CHAPTER XX.
774:
10.
Show
that in the curve of intersection of
r
= a cos
and
cos 2 6
= tan 2
<
a5!L^\*^.where
x = \/2
sin
sin <.
Show
that the inverse of this curve with regard to the origin
lemniscate, the constant of inversion being a.
11.
is
Show
is
a
that the rectification of the line of intersection of
given by
c
s^-r-7
2*32
[
I
[V
sin0
,/
tan" 1 /
-(/COS
7T
B
\
"
/
V
\+
J
COS
I2/
V
and show that
this curve can
,
sin0
,
/tanh" 1
,
/
tan ^ = </-
where
A
1
\
.
oi
I
7r
Sln
l
l2\
.
9
I
*
'V Sm l2
,
be inverted into a parabola lying upon
a tangent plane to the sphere.
A
Loxodrome is drawn on a sphere to cut all the meridians at
12.
the same constant angle a ; show that the area of the surface of the
sphere, included between any arc of this curve and the two
meridians through
its
ends,
2
is
,
a tan a log
where
^ and
^ 2 are the
radius of the sphere.
1
r
1
+ sin ^,
-^
+ sin ^
2
:
,
latitudes of the ends of the arc
[Ox.
and a
is
the
II. P., 1900.]
CHAPTER XXI
VOLUMES AND SURFACES OF SOLIDS OF REVOLUTION,
745.
AND THEIR CENTROIDS.
Volumes.
Supposing the 0-axis to be the axis of revolution, the
typical equation of such a surface
is
Fig. 250.
formed by the revolution about the z-axis of the curve
y =f(z) which lies in the y-z plane.
It is
2
It
was shown
in Art. 24 that the solid contained
surface and the planes
z=z l9 2=z2
formula
775
,
is
by
this
to be obtained by the
CHAPTER XXI.
776
y being the perpendicular from any point of the revolving
curve upon the axis of revolution.
It is obvious that if we regard the surface as defined by its
three-dimension equation x 2 +y 2 =f(z), we must replace the
2
2
2
y and the dx of Aft. 12 by x +y and dz respectively. The
formula therefore will stand as
v=
i.e.
More generally, if the revolution be about any line
in the plane of the curve, and if
be any perpendicular
drawn from a point
of the curve upon the line AB, and
746.
PN
AB
P
P'N' be a contiguous perpendicular, the volume
or
if
is
expressed as
be a given point on the line AB,
the limits being the values of
formed.
planes of the solid
ON
which mark the terminal
VOLUMES AND SURFACES OF REVOLUTION
Illustrative Examples.
Find the volume formed by the revolution
777
747.
1.
curve y
z
=x
zt
-
a-x
the loop of the
(Art. 403, Ex. 3) about the #-axis,
bounded by the closed portion
Here volume = 7r
#2
/
Jo
Putting
of
a+x = u,
this
-
a+z
i.e.
the volume
of the surface
dx.
becomes
= TT
= TT
~
-
log
u - ba?u + 2aw 2 -
3
|~2a
^
= 2ra s [log2-].
2. Find the volume of the spindle formed by the revolution of a
of the
parabolic arc about the line joining the vertex to one extremity
latus rectum.
Let the parabola be y 2 =4ax.
Then the
axis of revolution
is ,y
= 2.r, and PN=^^--
Also
S
V5
x
N/5
and
Fig. 252.
Hence
4?r
6~
748.
75
Surfaces of Revolution.
Again, if S be the area of the curved surface of the solid
traced out by the revolution of any arc AB about a given line
in its plane, let PN,
be two adjacent perpendiculars
XY
from points P,
QM
Q
of the arc
upon the axis of revolution,
Ss
CHAPTER XXI
778
the elementary arc PQ, SS the area of the elementary zone or
belt traced out by the revolution of PQ about XY.
Fig. 253.
Let p l and p z be the greatest and the least of the perpendicular distances of points on the arc PQ from the axis of
Then we may take it as axiomatic that the area
revolution.
traced out by
PQ
is greater than it would
each point of PQ were at the distance p.2 from the axis,
and less than if each point were at a distance p 1 from the
be
in its revolution
if
axis,
i.e.
SS
lies
between
and
ZirpiSs
Also
first
p l and p 2
%7rp 2
Ss.
.
by a small quantity of at least the
Hence
Ss and 2-rp.., Ss differ by a
least the second order from %7rPN Ss.
differ
2^
order from PN.
small quantity of at
Therefore in the limit
we have
ds
S={27rPNds.
or
749.
Various Forms of the Formula.
If the axis of revolution be the #-axis, this
ds
^y m
ds
may
7
r,
etc.,
be written as
as
may happen
the values of
CENTROIDS.
779
to be convenient in
any particular example,
-,-,
-T-,
-^,
being obtained according to
etc.,
the rules of the Differential Calculus, viz.
ds
\*
Centroids.
750.
The
and volume of a
centroids, both of the surface
solid of
revolution bounded
by planes perpendicular to the axis of
are
revolution,
plainly upon the axis of revolution, supposing
the surface density and the volume density in the respective
cases to be either constant or some function of the distance
from a point on the axis of revolution, i.e. so that the
bution of density is symmetrical about the axis.
Take the &-axis
as the axis of revolution,
<j
distri-
the surface
density and p the volume density, both symmetrical as to the
axis, and functions of x alone, so that the elementary zones in
the one case and the elementary discs in the other case, into
which the surface or volume is divided, have their own
centroids upon the axis of x, and we have, on application of
the formula
(1)
==-
2m
,
_
For the Surface,
^ f(a-27ry
____
ds)x
I
-js
.
(2)
2
_ -J\(p7ry
dx)x
_
_ J\pxy
-5
\
be noted that in the
pendent variable
If
it
;
first
2
dx
z
py dx
case s
is left
as the inde-
in the second case, x.
be desirable to take x or
in the first case,
we must
as the case
be.
may
J
\a-yds
-J5
It is to
ds
-jj
\((r27ryds)
For the Volume,
rp
_a-xy
J
as the independent variable
replace ds by
CHAPTER
780
In cases where p or
from the formulae.
751. Ex.
1.
a-
XXI.
are constants, they of course disappear
Find the surface
of a zone of a sphere
planes z=z^, Z=ZI + /L
If a be the radius of the sphere, and
on the sphere, we have (Fig. 254)
bounded by
parallel
be the latitude of any point
S=
P
and
27ra 2 [sin0 2 -sin<9 1 ]
Fi<y
and therefore equal to the corresponding
belt intercepted upon the enveloping cylinder
by the same planes, the 2-axis being the axis
254
of the cylinder.
arrived at in a Newtonian
manner
This
is
the result usually
It has
in books on Mensuration.
already been used in Art. 734.
Ex. 2. Find the surface of a belt of the paraboloid formed by the
revolution of the curve v 2 = 4a.r about the #-axis.
Here
= 2?r
and
/
JXi
and
= 4ir^a
y~dx
CLX
Jx
a dx
/
l
since for the parabola the radius of curvature
we have
p
p
Va
c,
47ra,
~~
~
is
given by
where p lt p 2 are the
radii of curvature of the generating curve at the
points where
by the planes bounding the
Ex.
3.
it is
cut
The curve
belt.
r = a(l+cos#) revolves about the initial line.
Find
the volume and surface of the figure formed.
Here
V=
1
^-mj
dx =.77 f r2 sin 2 d d(r cos 0)
the limits being such that the radius vector sweeps over the upper half of
the cardioide.
ILLUSTRATIONS.
F=
Hence
T (1 + cos 6)
3
7r
781
+ 2 cos 0) sii
2
(1
.
= 2ira 3 "(1+5 cos 2 0) sin 3 ^ d0
/
87ra 3
Again,
r
= %TT fyd8=2ir[
= 2?ra 2
r
amen
['
(1
+ cos 0) sin 8
.
2 cos |
o
327TO 2 r
-
Ex. 4. Find the centroid of the
volume density being uniform.
The centroid obviously
The denominator has
The numerator
= TT
(r cos
= 7ra
.
r2 sin 2
3
4
J(l+cos^)
lies
=
cos5
solid
upon the
formed
axis.
To
in the last example, the
find its abscissa
just been calculated, viz.
=
7ra
x we have
3
.
d(r cos 0)
cos^sin 2 ^d(cos^ + cos 2 6>), the limits being
TT
and
CHAPTER
782
XXI.
"*
= Tra4
(1
/
+ cos Of cos 0(1 + 2 cos 0) sin 3
d0
Jo
= Tra4
+ 9 cos3 + 7 cos
2
(* (cos Q + 5 cos
+ 2 cos 5 0) sin 3
4
Jo
n-
= 2Tra 4 f
+ 7 cos4 0) sin3
2
(5 cos
dd
JQ
x=
Hence
Ex.
5.
Find the centroid
the cardioide, as in Ex.
=-=-
lo
= fa.
^
o
f
of the surface
formed by the revolution
of
the surface density being uniform.
3,
Here
The denominator was
The numerator
=
calculated in Ex.
[ r cos 0. r sin
2-rr
Jo
= %*[
f cos 6 - sin - (2 cos 2 Z
Jo
7
6.
-
1
)
dS
.\
COS 9 -
COS?-
= 64 ra 3
Ex.
-^ dQ
dB
a 2 (l + cos0) 2 .cos0.sin0.
= 32Tra3
Hence
3.
_-^-2_J^
#=^Tra3 /^Tra2 =|a.
As an example
of the case
when x and y
are given in terms of a
third variable, consider the case of the surface of the solid formed by the
revolution of a cycloid about the line of cusps.
n
Here #=a(0 + sin0), y = a(l -cos0), ds = 2acos-d0, and the perpendicular
from
Hence
a,
y upon the
=2
line of cusps
= a (1 + cos 0).
/**"
/
/-J
2rra (1
Jo
= 32Tra 2
r-s
/
.'o
647ra2
+ cos 0) 2a cos -* d8
cos 3 <f>rf<f),
where
<6
= Q^,
GULDIN'S THEOREMS.
752.
783
THE THEOREMS OF PAPPUS OR GULDIN.
When any
in its
own
plane closed curve revolves about a straight line
plane which does not cut the curve we have the
following theorems
:
The VOLUME of the ring formed is equal to that of a
cylinder whose base is the revolving curve and whose height
I.
path of the centroid of the AREA af the curve.
The SURFACE of the ring formed is equal to that of a
cylinder whose base is the revolving curve and whose height is
the length of the path of the centroid of the PERIMETER of the
is the length of the
II.
curve.
These theorems were given by PAPPUS, in his Mathematical
Collections, in the latter half of the fourth century,
and
redis-
covered by GULDIN, and published in his Centrobaryca early
in the seventeenth century. 1
753.
THEOREM
I.
Divide the area (A)
with sides parallel to
in the accompanying
Let the ic-axis be the axis of rotation.
into infinitesimal rectangular elements
the coordinate axes, such as
P^fJP^
figure,
each of area SA.
P12V 1 =y.
.
Let rotation take place through an infinitesimal angle
Let the ordinate
SO.
Fig. 256.
Then the elementary
height to
first
solid
formed
order infinitesimals,
infinitesimals of the third order its
1
is
is
on base SA, and
its
yS9, and therefore to
volume
Cajori, History of Mathematics, pages
is
SA y
.
59, 167.
SO,
CHAPTER XXI.
784
If the rotation be
through any
finite
we
obtain
by
If this be integrated over the whole area of the
curve,
have for the volume of the solid formed,
we
summation or
angle
a,
integration,
SA.y.a.
Now the formula for the ordinate of the centroid of a number
of masses
m
1
,m.2 ,...
)
with ordinates y^ y 2
Hence the ordinate
revolving curve
of the
y=
is
and therefore
\ydA
jydA=Ay.
the volume formed = A (ay).
Hence
A
..., is
centroid of the area of the
\ydA
But
,
is
the area of the revolving figure, and ay
length of the path of the centroid of the area.
theorem is established.
If the
solid ring,
curve perform a complete revolution and form a
we have
a=2?r
754.
THEOREM
and
V=
II.
Again, take the axis of revolution as the z-axis.
the perimeter s into infinitesimal elements, such as
length
the
is
Hence the
Divide
PP
1
2
,
of
Ss.
Let the ordinate P1 N l be called y.
Let rotation take place through an infinitesimal angle
80.
Then the elementary area formed, P^^Q^Q^ is ultimately a
rectangle with sides Ss and y SO, and to infinitesimals of the
second order
its
If the rotation
summation or
area
is Ss
.y
SO.
be through any
integration, Ss ya,
,
finite
angle a
we
obtain,
by
THE ANCHOR-RING.
If this
we have
785
be integrated over the whole perimeter of the curve,
for the curved surface of the solid formed,
i\yds.
If y be the ordinate of the centroid of the perimeter of the
curve in the plane of x-y, we have
[yds
y=^
f
I
(yds
O
7
\ds
Then
\yds=sy
the surface formed =s (ay).
and
But s is the perimeter of the revolving figure, and ay is the
length of the path of the centroid of the perimeter of the
revolving curve. Hence the theorem is established.
N.,
N,
Fig. 257.
If the curve
ring,
we have
perform a complete revolution and form a
a=2?r, and
Illustrative Example.
The volume and surface
of
its
surface
solid
is
an "Anchor-ring" or "Tore" formed by
the revolution of a circle of radius a about a line in the plane of the
circle at distance d from the centre (d>a) are
respectively,
Volume- 7m 2 x27rc?=27r 2 a 2
Surface =2?r
E.I.C.
x
3p
c?,
CHAPTER
786
XXI.
In this case the centroid of the perimeter and the centroid of the area
This of
are at the same point, viz. the centre of the revolving figure.
course would not generally be the case.
755.
Precautions.
In these theorems it has been stated that the axis of
If the curve consists of
revolution does not cut the curve.
more than one closed oval,
whole portion to which the
it is
to be further noted that the
rules apply
must
lie
on one side of
the axis of revolution.
When
the axis of revolution cuts the curve, or
bounded by the curve
when
regions
on opposite sides of the axis of
the
both
as to volume and surface, give
revolution,
theorems,
the difference of the volumes or surfaces traced by the portions
on opposite sides of the axis of revolution.
lie
Note by Mr. Eouth.
Again, it has been pointed out by Mr. E. J. Routh (Anal.
Statics, vol. i., p. 293) that during any elementary rotation
through an angle SO, the axis of revolution need only be an
756.
instantaneous axis of revolution.
Let G be the centre of gravity
of the revolving area A, G' a contiguous position of the centre
Fig. 258.
of gravity, Ss=GG', and let the plane of
angles to the tangent to the path of G.
A
be always at right
Let I be the centre
The rotation through SO may be
of curvature of G's path.
regarded as about a straight line through 7 perpendicular to
the plane GIG', and the volume generated
A X IG SO
or
A Ss.
is
ROUTH'S EXTENSION.
And
volume generated
integrating, the
Area x length of the path of the
And
further, for the
787
is
centroid of the area.
theorem with regard to the surface
;
if
the plane of the revolving curve be always at right angles
to the tangent to the path of the centroid of the perimeter,
the surface generated is the perimeter of the revolving
curve x length of the path of the centroid of
Ex.
A
circle of radius c
(<b) moves with
its
the perimeter.
centre on the ellipse
2=
the direction
1, the plane of the circle being perpendicular to
jt?/a?+y*/b
The volume and
of the tangent to the ellipse at the centre of the circle.
surface of the ring generated are
the perimeter of the ellipse,
i.e.
irc 2
P and
4#( |\
e\
2-rdP respectively, where
when
62 =
2
(1
P is
-e 2 ).
In this case the centroids of area and perimeter of the moving curve are
the same point,
757.
viz.
the centre of the
circle.
Axis not in the Plane of the Curve.
Theorem as
Extension for the
to the Volume.
Consider next the case
in the plane of the area,
when
but
the rotation
parallel to
is
about a line not
it.
Fig. 259.
Let Sx Sy be an element of the revolving plane, the #-axis
being taken parallel to the axis of revolution and the z-axis
on one side of the
cutting it at R, and the area lying entirely
x-axis.
CHAPTER
788
XXI.
be the element SxSy, and let P^, P2 N 2 be
the z-axis and P 1 l P2 2 perpendiculars
on
perpendiculars
on the axis of revolution, and let be the angle A 1 P 1 1
Let
P P2P P4
3
1
M
,
M
M
r
.
Let Pf^QsQi be the projection of SxSy on tlie plane
that is, the normal section of the elementary
2
l
lf
ring formed, and let a represent the angular extent of the
M M Pf
revolution.
Then, to the second order the volume traced out by the
revolution of Sx Sy about
RS
is
pN
/
\
[
3xSycos8x(a\ COS ^)
u/
.e.
and
or
8xSyXaP N
l
1
,
the same as that of SxSy about the x-axis.*
Hence, taking the limit when Sx, Sy are infinitesimally small,
and integrating over any area which lies on one side of the
is
ic-axis in
the x-y plane,
we have
the theorem that the volume
generated by the area revolving about a line parallel to the
plane of the area, but not in its own plane, is the same as
would be traced out
if
the revolution were about the projection
upon the plane of the area through
of the axis of revolution
the same angle.
Axis not parallel to the Plane of the Area.
Finally, suppose the axis of revolution not parallel to the
plane of the area. Let the area lie in the x-y plane and
758.
entirely on one side of the x-axis, and let the z-axis be the
projection of the axis of revolution upon the x y plane, and
AXIS NOT PARALLEL TO THE AREA.
789
the origin the point where that axis cuts the x-y plane and
its inclination to the plane.
Then the equations of the axis of
revolution are
The perpendicular upon
1\
M=
The equation
this
from
P
x
,
(x, y, 0), viz.
P^M,
is
s/
of the plane
OMP^
is
-Xsin0 + -si
y
The
direction ratios of the normal are
The
direction cosines of the normal to the element
u
i.e.
dy
Sz,
are
(0, 0,
1
1
2
3
,
).
The angle between these normals
The
P P 7J P4
projection of Sx 6y
is
upon the plane
OMP
The volume generated by the revolution
any angle a about OM is therefore
1
of
is
therefore
SxSy through
to the second order, and is therefore the same as if the rotation
took place about the projection of the axis of rotation upon
the plane of the area, the angle of rotation being a cos 9
instead of
a.
And
integrating over the whole area, we have the theorem
that the volume generated by the revolving area, the revolution
being through an angle a, is the same as the volume generated
by revolution about the projection of the axis on the plane of
the area through an angle a which is multiplied by the cosine
of the angle between the axis of revolution and the plane of
the area (see Ex. 1, p. 294, Anal. Statics, E. J. Routh); or,
CHAPTER
790
which
the same thing,
is
XXI.
the volume
may
be found by
revolution through an angle a about the projection and then
multiplied by cos$. This supposes the revolving area to be
entirely on one side of the projection of the axis on its plane.
759.
Ex.
semiminor
4
=-
1.
axis.
The area
The volume traced out
.
X1
A quadrant of the ellipse
is
2
i/
-7,+^
The
-^
4
=
]
revolves (1) about
abscissa of the centroid
in a complete revolution
4a
n-ab
its
is
is
2
(2) If the revolution were about a straight line outside the plane of
x-y but parallel to the minor axis, and which projects upon the minor
axis, the volume would still be
the revolution were about a straight line through the centre at
right angles to the major axis, and making an angle Q with the minor
axis, the volume would be
&ira 2 b cos Q.
(3) If
Ex.
2.
An
X
2
<L
ellipse -2
+^= 1
?/
revolves about
its
tangent
x cos a+y sin a =p,
Fig. 261.
The volume generated
is
7ra6x27rj?,
where
2
2
p =a cos
with this tangent,
If the revolution were about a line making an angle
and which projects upon the tangent, the volume generated would be
Trab X Z-rrp x cos #.
PROBLEMS.
1.
Prove that the volume of the
ellipse round its minor axis
solid generated
by the revolution
mean
proportional between
those generated by the revolution of the ellipse and of the auxiliary
circle round the major axis.
[I. C. S., 1881.]
of
an
is
a
Find the volume of the solid formed by the revolution
cycloid round a tangent at the vertex.
2.
of a
PROBLEMS.
3.
The loop
straight line
curve
the
of
y = d',
2ay
2
volume
find the
791
= x(x-a)~
revolves about the
of the solid generated.
[OXFORD
4.
Show
I.
P., 1890.]
that the volume of the solid formed by the revolution
= x 3 about its asymptote is equal to 2?r 2 a3
of the cissoid y 2 (2a-x)
.
[TRINITY, 1886.]
5.
Find the volume
of the solid
//
the loop of the curve
2
*/
._!_.
= z2 - -
2;r 2
(1
-
x.
[I.
e2
of
+
*
**
6
\
shr 1
A
/
[OXFORD
Prove also that of
of
an
all
C. S., 1882.]
the oblate and prolate spheroids
major axis 2a and eccentricity e
of
formed by rotating an ellipse
about its principal axes are
and
about the axis of
jC
(t
Prove that the areas
produced by the revolution of
7*
prolate spheroids formed
II. P., 1914.]
by the revolution
ellipse of given area, the sphere has the greatest surface.
C. S., 1891.]
[I.
Find the surface of any zone of an
by planes perpendicular to the axis of
ellipsoid of revolution cut off
revolution.
[COLLEGES
a, 1888.]
7. If the evolute of a catenary revolve about the directrix of
the catenary, show that the area of any portion of the surface
generated, cut off by two planes perpendicular to the directrix,
varies as the difference of the cubes of the radii of its bounding
circles.
[COLLEGES
Find the volume
a,
1892.]
formed by the revolution about
the prime radius of the loop of the curve
8.
of the solid
r3
between
9.
If
=
the
and
= ^,
cardioide
p = r cos (6 - y),
= a3 6 cos
[OXFOID
r
= a(l-cos6)
assuming that the
line does
round
revolve
prove that the volume generated
not cut the cardioide.
II. P., 1890.]
the
line
is
[ST.
JOHN'S, 1882.]
CHAPTER XXI.
792
10.
Prove that the area of the surface generated by the revolu-
tion of a portion of the arc of a cycloid about the normal at one
extremity is equal to the area of the cycloid multiplied by
--[(
-
7) sin
ft
cos y
+
| cos
(/3
+ y)
-
s cos (3/3
-f-
where y and /3are the angles of inclination
and of the normal at the other extremity
-
y)
-
cos 2y],
of the axis of revolution,
of the arc, to the axis of
the cycloid.
Deduce the areas of the surfaces generated by the revolution of
the whole cycloid about its axis and about its base.
[COLLEGES
Find the volume
e,
1884.]
formed by revolving a loop of a
lemniscate of Bernoulli about the straight line in. its plane which
passes through the pole and is perpendicular to the axis.
11.
of the solid
[OXFORD
12.
pole.
I.
P., 1901.]
The lemniscate r = cos20 revolves about a tangent at
Show that the volume and surface of the solid generated
2
respectively
2
7r'
2
a 3 /4 and 4?ra 2
the
are
.
13. A surface is the locus of points which have their distances
from a fixed plane inversely proportional to the fifth power of
in that plane.
Prove that its
their distances from a fixed point
volume equals twice that of the sphere which, with its centre at 0,
touches the surface.
[OXFORD II. P., 1880.]
14.
Find the volume
of the solid
curve (a - x)y* = a-x about
its
formed by the revolution
asymptote.
[I.
of the
C. S., 1883.]
Show that the rate of increase of the volume of an anchor
when the radius of the generating circle is increased while its
15.
ring
centre remains at a constant distance a from the axis of revolution
is
Wad,
the diameter of the generating circle being
rate.
d,
increasing at unit
[TRINITY COLL., 1881.]
A
=
loop of the curve r asinnO revolves about the initial line.
of the solid thus generated, and verify the result
by deducing the volume of the ring formed by the revolution of a
circle about a tangent.
[COLLEGES a, 1889.]
16.
Find the volume
17.
If
the curve r
= a + bcos
that the volume generated
provided a be<&,
9 revolve about the initial
line,
show
is
[COLLEGES
a,
1884.
PROBLEMS.
The curve
18.
= a(l -
r
ecos
<9),
793
when
e
is
very small, revolves
about a tangent parallel to the initial line ; prove that the
of the solid thus generated is approximately
27T 2 a 3
The curve
19.
r3
= u 3 cos30
(l+
2
[I.
).
revolves about
(9
= 0.
volume
C. S., 1892.]
Prove that the
loop in the third quadrant generates a volume
37TV
'
8
A
[OXFORD
I.
P., 1902.]
drawn from a point A on the earth's surface
be the longitude and co-latitude of B, and
lt
the
#,,,
</\>
corresponding quantities for A, show that the area contained between the meridians of A and B and the loxodrome is
20.
loxodrome
to a point B.
is
^
If
2 (O l
-
6>
2)
log (cos
log (tan
i</>!
1^
-f-
cos
tan i<
<
2)
2)
the radius of the earth being taken as unity.
#*
-.
JOHN'S, 1884.]
Prove that the whole area bounded by the curve
21.
is
[Si-.
r
Also show that
.
2V2
if
+ #* = 2axy2
the area revolves about the #-axis, either
loop generates a solid whose volume
is
f Tra
3
.
When
O
about the
y-axis, the
whole volume generated
is
j
the area revolves
Q
\/2.
Determine the curve which generates, by revolving about the
a volume proportional to the length cut off from the axis
the
terminal
[TRIN. HALL AND MAGD., 1886.]
by
bounding planes.
22.
axis of
,
23. The axes of two cylinders of radius a intersect at an
angle a
show that the whole volume common to the two is
-y-fl^coseca.
24.
Evaluate
I
,
[TRIN. H.
AND MAGD.,
;
1886.]
taken over the surface of a sphere of radius
a,
the perpendicular on the tangent plane from a fixed point
within the sphere at a distance b from the centre showing that
p being
;
f|%/
2?r
]]*">[OXFORD
II. P., 1892.]
CHAPTER
794
XXI.
25. Show that the volume traced out by the part of the area of
= a, when the
the curve r=f(6) which lies between 6 = /3 and
curve revolves about the line = y, taking a > ft > y, is
[OXFOBD L p
1902]
any portion of a surface revolving about an
volume
that
the
generated is the sum with the proper
prove
volumes
of
the
generated by the projections
corresponding
signs
26. In the case of
axis,
of the surface
on any two planes at right angles to one another
through the axis of rotation.
1900.]
[-y,
A
is taken on a diameter of a sphere (centre C,
27.
point
radius a) .so that 00 = c (c<&); the radius vector of length r drawn
with OC,
to any point P on the surface makes an angle
from
and the radius CP makes an angle 6' with OC produced, dS is an
element of area of the surface containing the point
f cos
integral
cos
0'
P
;
evaluate the
,
r*
J
taken over the larger of the portions into which the surface
divided by a plane, through 0, at right angles to OC.
[OXFORD
28.
Prove the formula
formed by the revolution
71-
3
1?*
sin
6dO
for the
volume
of a closed plane curve
I.
is
P., 1901.]
of the surface
about the
initial
line.
The outer loop
Show
of f*
= a? cos 10
i$
29.
Find the area
Show how
y
+
2"y*
[Oxr.
= h.
to find the
[OXF.
volume
the curve r=f(0) about the line
curve passes through the origin.
line.
is
i.
P., 1911.]
of the finite portion of the surface 2z
cut off by the plane z
30.
revolves about the initial
that the volume of the surface generated
I.
= x2 + y2
P., 1913.]
formed by revolving
being assumed that the
of the solid
= a,
it
Prove that the volume of the solid formed by revolving one loop
= a 2 cos2# about one of the inflexional tangents is
of the curve r2
|7rV.
Show
4a
origin
is
[Oxr.
I.
also that the distance of the centroid of this solid
^
O7T
P., 1915.]
from the
CHAPTER
XXII.
SURFACES AND VOLUMES IN GENERAL, AND THEIR
CENTROIDS, ETC. DOUBLE AND TRIPLE
INTEGRATION.
760.
Let the equation of a surface be
</>(#,
y,
z)=0
referred
to three mutually perpendicular coordinate axes Ox, Oy, Oz.
Let us discuss the volume contained between the boundaries
y,
3)=0;
a?=0,
x=a.
X=x,
Let planes
Y=y,
Z=z,
be drawn.
Fig. 262.
Planes
X=x,
X=
intercept between
or lamina of thickness Sx.
795
them a thin
slice
CHAPTER
796
XXli.
Planes Y=y, Y=y + Sy cut from this lamina a prism or
tube on rectangular base Sx Sy.
Planes Z=z, Z=z + Sz cut from this prism an elementary
"
"
cuboid of volume Sx Sy Sz, represented
rectangular box or
in the figure as
infinitesimals of the first order, the
P-^Q^^P^Q^S^
Regarding Sx,
volume of the
Sy, Sz as
slice is
a
order infinitesimal, the volume of the prism is a second
order infinitesimal, and the volume of the cuboid is a third
first
order infinitesimal. Let the prism intercept on the surface
a curvilineal quadrilateral figure PQRS, and on the plane x-y
the elementary rectangle pqrs, viz. SxSy. These areas are
both infinitesimals of the second order.
If we add up all the complete cuboids on base SxSy from
2=0
to
0=the
smallest
of
the values of z of the surface
within the quadrilateral PQRS, we get the volume of the
prism, less by a third order infinitesimal, viz. the portion of a
cuboid bounded by a base Sx Sy for its lower surface, by the
curvilinear quadrilateral PQRS for its upper surface, and by
four plane faces parallel to the y-z or z-x planes.
may
regard the infinitesimal Sz as having been taken not less than
We
r
the difference of the greatest and the least values of z for
This remnant of the
points on the quadrilateral PQRS.
prism
therefore less than one of the elementary cuboids
is therefore an infinitesimal of
is
forming the whole prism, and
not less than the third order.
Next
let
us add up
X=x
all
the prisms which
lie
between the
X=x
and
+ Sx, and bounded on its upper side
planes
surface
from the plane Y=0 to any definite
the
specified
by
value of Y. The sum of these second order complete prisms
from the volume of the lamina between the planes
and X=x + Sx by the sum of the third order infinitesimal remnants of the prisms, and by a second order tubular
element on a base less than Sx Sy at the end of the slice, that
differs
X=x
is
by a second order
infinitesimal, the
sum
of the complete
prisms being of the first order.
X=0
to
Finally, let us add up all the slices or laminae from
of
these
X.
sum
the
of
The
of
value
definite
portions
any
laminae made up of complete prisms is a finite quantity.
The sum
of the
remnants of the laminae
is
the
sum
of a set
I
TRIPLE INTEGRATION.
of second order infinitesimals,
797
and forms a
first
order infini-
Hence it appears that the sum of all the complete
tesimal.
cuboids within the figure bounded by the coordinate planes,
and the surface, differs from the
the planes
lt Y^y^ say,
X=x
whole volume of that figure by a first order infinitesimal at
most, and in the limit when Sx, Sy, Sz are diminished without
limit, we have the volume given by
The
from
limits for z are
z)=0
in
0=0
to
z=the value found from
terms of x and
y, say z=f(x, y).
y will be from y=0 to the value of y specified
in any particular manner, say y=F(x).
The limits for x will be such as to go from x=0 to x = a.
(j>(x,
y,
The
limits for
761.
Ex. Consider the volume of an octant of an ellipsoid
7
i
b1
Here the
prism, to
limits for z are z =
add up
all
to z =
c^l-^-^
for the elementary
the cuboids in the prism.
Fig. 2G3.
-'
V~x?
}
in the slice.
2
for the slice, to
add up
all
the prisms
CHAPTER
798
For x
from x = Q to
;
and taking
[z]
between
.r
=
its limits, this
c
__
~b
And
add up
to
<7,
.
TT J0
AJ
__ D
4
XXII.
/
xy
I
Obviously
the
_
3
_ a
c
_
\
1
,
TT
8
is
I9
/i^
.
3aV~6
V
slices.
integral
the volume of the whole ellipsoid
762.
all
4
V=
_
irafoc
*
6
^irabc.
where the volume
in cases
2a
_
~
3
of a slice can be
down
written
In the
distance
at once, the labour of computation may be saved.
case just considered, for instance, the section at
X=x
from the plane of yz
VX
I
and the area of the quarter
Hence the volume
2
--
is
I
,
an
ellipse, viz.
X2
1
c\ll
^',
ellipse in the first octant is
of the slice in the first octant
is
to the first order.
And
the
sum
of the slices
is
as before.
763.
may,
When
the volume contained
in general, start
V=
i.e.
we may
is all
that
is
required,
we
with
1
1
z
dx dy,
use the elementary prism on Sx Sy for base as our
This amounts of course to integrating with
element of volume.
VOLUMES.
regard to z in the triple integral formula
0=0 and
limits
2
799
1
1
1
dx dy dz between
the ordinate of the surface under con-
sideration.
upper surface of the region whose volume is required
z=f^(x y) and the lower surface be z~f2 (x, y\ instead of
If the
is
t
z=Q,
y
as taken in Art. 760,
we have
764. Illustrative Examples.
The curve z(az +aP)* = a* lying in the plane z-x revolves about the
Find the volume in the positive octant included between this
z.
surface and the planes #=0, x = a, # = 0, y = a.
[COLLEGES
188,3.]
The equation of the surface generated is
1.
axis of
,
and
Then
Hence
and we have to evaluate
Let
.r
Then
=
/=
CHAPTER
800
2.
XXII.
Express the volume contained between the surfaces whose equa#2 + ?/2 + 2 2 = a 2 #2 +y 2 = rt 2 z=a and the coordinate planes in
tions are
,
,
V=
the forms
\zdxdy,
I
V
I
\xdzdy; investigating the
integrations and determining the value of V.
(i) For the portion of the elementary prism on 8x8y
between the sphere and the plane z = a, the length is
limits of the
for base lying
Fig. 264.
This
is
.?
=
to
Then
by 8x8y and summed for values of ?/ from
*Ja*-x\ and afterwards the result is to be summed from
to be multiplied
?/=0 to
x=
_V=T f
a?
'
:
2
V<
~X
7r_7T
2
4
\a-*Jd*-x*-y*)dxdy
2a
'
3
MOMENT OF
If
(ii)
we
use the formula
V=
INERTIA, ETC.
801
\xdzdy, integrating with regard to y
I
we have for the length of the prism on base 8y8z intercepted
between the cylinder and the sphere \/a 2 - y 2 - v/a 2 - - y*, until the
2
2
prism ceases to cut the sphere, i.e. from y=0 to 3/ = \/a -;s and afterfirst,
,
wards the length
the limits for
z
of this
are from
\fa?-y 2 from y=*Ja*-z* to
is
prism
y=a, and
to a.
Hence
- * 2 - y 2 ) dz dy + f* {*
Jo
we
If
(iii)
we
before
use the formula
2
y
,
and
which, as betore,
765.
If
filled
we
\xdydz, integrating with regard to
I
z
we have the same peculiarity as
of length \fa'2 - y* - \/a2 -y*-z2 from z=0
integrate with regard to y,
before, viz. that the prism
to z = *Ja 2
V=
}</&
is
of length *Ja?
y
2
from z=*Ja?y'2 to z=a, and
=--
Mass, Moment, Centroid, etc.
regard the space bounded as described in Art. 760 to be
with matter of
specific density
p at each point, the Mass
of the elementary cuboid Sx Sy Sz is p Sx Sy Sz, where -p
be either a constant or a variable. And following the
argument as in finding the volume,
the body thus enclosed,
M=
\ 1 1
we have
p dx dy
for
may
same
the mass of
dz.
In the same way, if the Moment of this mass be
required about any line whose equations are known, say
x a_y b_z c
766.
E.I.C.
CHAPTER
802
m, n, being direction cosines
from x, y, z upon this line, viz.
moment
p
of the solid about this line
is
;
then,
\ppdxdy
dz.
To determine the coordinates
767.
be the perpendicular
if
I,
the
XXII.
of the Centroid,
we have
only to translate the expressions
_~
And
into the language of the Integral Calculus.
p Sx 8y
dz,
1 1 1
px dx dy dz
1
required,
line,
i.e.
Moment
If the
and
we have
if
1
\pydxdydz
\ \
\\pdxdydz
\\pdxdydz
7G8.
m
being
we have
of Inertia
\
pz dx dy dz
\\pdxdydz
about a straight line be
p be the perpendicular from
Moment of inertia = Smp2
(x, y, z)
upon the
,
in the language of the Calculus,
2
1
1
\pp dxdydz.
A, B, C be the moments of inertia about the
coordinate axes Ox, Oy, Oz respectively,
if
Thus,
Similarly for "Products of Inertia,"
769.
such as
i.e.
for quantities
D = 2myz, E = 2mzx, F = 2mxy,
we have
Z)=
\
\
\pyzdxdydz,
E= \\\pzxdxdydz, F=
\\pxydxdydz.
CENTROIDS, ETC.
803
770. The integration in all such cases takes the same course
as in the finding of a volume, first as regards the proper
assignment of limits, and second as regards the successive in-
tegrations (1) with regard to
regard to
z,
(2)
with regard to
y, (3)
with
x.
The order
of integration may be changed to suit circumlimits being suitably changed to ensure
several
the
stances,
that the elementary cuboids into which the specified region is
divided are thereby all added up.
As in the case of finding a volume, in some cases one, or
perhaps two, of the integrations may be avoided by taking
the elementary prism, or the elementary lamina described
above, as the primary element, as was done in Art. 762 in the
evaluation of the volume of the octant of an ellipsoid.
Ex. In the case of a sphere, viz. x2 + y 2 + z2 =a 2 let us find the
mass of an octant of the sphere, the density at any point being
771.
,
= kxyz.
Here
M = HI xyz dx dy
Tc
Fig. 265.
The
limits for z in the positive octant are
2
for y,
from
for x,
from
=
y=0
x=0
tO
Z=V
to
to
x=a.
dz.
CHAPTER
804
M=k Jo Jo
Hence
I
T.
ra
"l
/
XXII.
xyzdxdydz
/
Jo
rVtf^tf
-'
JferoW "
~8L
2a 2a4
2
~48'
If
D
be the density at a
of the octant,
i.e.
where
specific point, say the centre of the surface
x=y = z = -j=, we
have
a3
and
Jf=
EXAMPLES.
1. Establish the following moments of inertia for uniform density, J/
representing the mass in each case
:
(1)
For an
x
elliptic disc ~2
v'2
+p= 1
>
......
......
about the x
axis,
about the y
axis,
about a
through the centre perpendicular \
line
to the plane,
(2)
2
For a rectangle
;
^a
2
-f
&2
of sides 2a, 26,
26,
through the centre perpendicular
to the plane,
(3)
-
4
about a line through the mid-points of sides 2a,
line
;
-
about a line through the mid-points of sides
about a
Mb*
;
-^-
;
^a? +
b2
3
-
For a sphere about any diameter
\
-
,
a being the radius.
ROUTE'S MNEMONIC RULE.
For an
(4)
ellipsoid of semiaxes a, 6,
about the axis of length 2a,
about the axis of length
26,
about the axis of length
2c,
c, viz.
(1)
...
-
the positive octant of the sphere,
(2)
-
-
-
Obtain the position of the centroid of
the quadrant of an ellipse,
_
4a
2.
805
M
o
M
M
;
-
.
-__46
___.
3a
O
the positive octant of the ellipsoid,
_ 3a
(3)
_36
O
3.
Show
that in
all
3c
O
O
the above cases for the whole elliptic disc, rectangle,
sphere or ellipsoid, the products of inertia with regard to two axes of
symmetry are zero.
Routh gave the following
Dr.
moment
the
of inertia of
sphere or ellipsoid
Moment
;
or
elliptic
disc,
rule
for
rectangle
the
and
viz.
about an axis of symmetry
of inertia
_
mnemonic
useful
circular
sum of squares of
....
perpendicular semi-axes
4 or 5
3,
according as the body
772.
rectangular, elliptical or ellipsoidal.
is
Element of Surface.
In estimating the element of surface SS cut from the surface
S by the elementary prism on base Sx Sy, we may note that if
y be the angle the normal
SxSy=cosy SS
at
P
makes with
ttte z-axis,
to the second order of infinitesimals,
the projection of SS upon the x-y plane.
SxSy
The equations of the normal are
for
is
:
X-x_Y-y_Z-z
where
<p x
Hence
=^-,
etc.
cosy=-r=^
CHAPTER
806
XXII.
Then
when we proceed
the limit and
to
sum
the elements
by
integration.
Fig. 266.
If the equation of the surface be
thrown into the form
*=/(0 y\
and
if
we
use the ordinary notation
this equation
We may
S=
becomes
1
1
\/l
+>
2
-f q
2
dx dy.
note in passing that the equation Sx
Sy=SS cosy
also gives another expression for the volume, viz.
Y=
I
\z
dx
dy= \z cos y cZ$.
We have taken, as is ordinarily the case, x, y as the independent
variables.
If this
be inconvenient,
according as
y, z
or
z,
we should have
x be chosen as the independent
variables.
SURFACES.
807
773. We may note that the coordinates of P, Q, 8 and R,
"
the coordinates of the curvilinear " parallelogram bounding
SS are
:
for P,
x
y,
z;
for Q,
x + Sx,
y,
z+^Sx;
f or s,
x,
y+Sy,
z
f or
t
R, x+Sx, y+Sy,
to the first order
;
-\-^-$y,
z+
Sx+
Sy,
Fig. 267.
and the projections of
this curvilinear parallelogram upon the
coordinate planes are parallelograms of areas
(1) upon the x-y plane, Sx Sy
:
;
(2)
upon the y-z
plane,
1
*,
V*
z,
0,
|?
3z
(3)
upon the
z-x plane,
1
cc,
and the area SS
Sx,
1
f^&/,
1
0,
the square root of the sum of the squares of
projections upon any three mutually perpendicular planes
(C. Smith, Solid Geom., Art. 33).
its
is
CHAPTER
80S
XXII.
Hence
S=
giving
dtc dy, as before.
Element of Volume
774.
for Cylindrical Coordinates.
Instead of taking as our elementary volume one defined as
bounded by planes parallel to three coordinate planes, other
In some investigations it may be
desirable to employ cylindrical coordinates, viz. ordinary polar
choices
may
coordinates
be made.
r,
in the x-y
plane,
retaining the
Cartesian
Fig. 268.
An elementary prism, with this system, will be
on a base r 86 8r with a height z, and to the second order its
^-coordinate.
volume
is
rSOSrx z, and
between suitable
the volume will be
If for
limits.
any reason
\
\
it
rz dO dr, taken
be desirable to
subdivide this elementary prism by planes perpendicular to
the 0-axis, our expression for the volume will be
\rdOdr dz.
Ill-
Such a necessity would arise, for instance, if the mass of
the solid be required and the density be not a constant, but a
known function of r, 9, z, when the mass of the elementary
prism
is
rSOSr \pdz, r and
this integration, so as to
being regarded as constants during
add up
all
the elements of varying
summing the
density through the elementary prism before
VOLUME ELEMENTS. POLARS.
masses of the several prisms themselves,
We
809
should then
write the integral as
Mass=
775.
1
1
\pr d9 dr dz.
Spherical Polar Element of Volume.
Again, a spherical polar element of volume may be emthe co-latitude and (/> the
ployed, using r the radius vector,
azimuthal angle as coordinates.
Here the element of volume has three of
its
edges, mutually
at right angles, Sr, r SO and r sin 9 S<f>, and to the third order
of infinitesimals its volume is r2 sin SO S<}> Sr, the difference
Fig. 269.
between this and the actual volume being at
least of
the
fourth order of infinitesimals.
and <j> in
integrating successively with regard to r,
the
accumulated
difference after the three integraany order,
tions between the volume of any
space required and the sum
Upon
of these elements will be a first order infinitesimal at most,
and therefore vanishes when the limit is taken.
Hence we have
for the
volume required
CHAPTER
810
XXII.
be required to integrate any function of
(r, 0, 0) throughout the volume, say f(r, 0, 0), that is to
add up all such elements as f(r, 6, <j>) r 2 sin SO S(f> 6V, the exFurther,
if
it
pression for the result will be
f
#>
f f/0>
dO
0) r2 sin
d</>
dr,
the limits being such as to include in the summation
elements
which are included
Ex. If
776.
centre
in the region
we apply
this
all
the
under discussion, and no more.
formula to find the volume of a sphere whose
at the origin,
is
the limits for r are from
for 6 are
for
<j>
and
from
are from
=JQ j
^
3
to a, the radius of the sphere
to
TT
;
;
to 2?r
;
"fV&
/
BinOdO
Jo
[-~<I
Elements of Surface.
777.
Cylindrical System.
In the cylindrical system of coordinates the element of
surface 8S,
has for
rSOSr.
its
the curvilinear parallelogram PQRS, Fig. 270,
projection upon the x-y plane the polar element
viz.
Its projection
upon the meridian plane through
to the first order, an oblique parallelogram of area 6V
P
^
is
SO,
for one of its sides is the
change in z due to increase of SO in
the independent variable
0, i.e.
between
this side
and the
^30,
parallel side
and the perpendicular
is oV.
the projection upon a plane through P parallel to the
0-axis and at right angles to the meridian plane, is similarly
And
r SO
6V, for
r SO
is
the height of this parallelogram, and
SURFACE ELEMENTS. CYLINDRICALS.
- Sr
the change in z due to an increase Sr in
is
constant,
viz.
811
r,
keeping
the difference of the ordinates parallel to the
z-axis of the points
P
and
Q.
+
Sr
Hence
A8=dr
(r SO)*
and taking the square
.
root,
#* + (r 80 Sr)*
proceeding to the limit and
Fig. 270.
and
Similarly, "if it were found preferable to take the pair
r
we
should
and
for the independent variables, or the pair
z,
have in these respective cases,
To
by
an element is taken on the surface bounded
on the surface along which z is constant and const.,
z + Sz, 0, 6 + SO, and projected upon the same planes as
establish (2)
lines
viz. z,
in Case (1), the areas of the projections being
And
and
rSOSz,
Sz
to establish (3)
an element
bounded by
lines
is
Sz
taken on the surface
on the surface along which
r= const,
and
CHAPTER
812
,
viz. r,
r + <$r,
3
0,
the same planes as in Case
+ 6X
(1),
and projection
is
made upon
the areas of the projections being
and
r
SrSz,
XXII.
r
figures are, however, somewhat troublesome, and we
deduce these formulae from a more general result later.
The
shall
778.
In the spherical polar system of coordinates let the
meridian planes
(j>
and
<5
cut the surface in the curves
Fig. 271.
PQ, SR, and let the cones 0, + SO cut the surface in curves
PS, QR. Then PQRS is our element of surface. Let the
coordinates of the points P, Q, R,
for P,
r,
for Q,
r+^SO,
S be
respectively
0,
for
:
</>,
0,
for/?,
fy,
+ 80,
<j>
+
S</>.
O(p
The
projections of this elementary area
(1) a plane through
P
upon
at right angles to the radius vector
;
SURFACE ELEMENTS. POLARS.
(2) the meridian plane
(3) a plane
through
P
through
P
813
;
perpendicular to these two planes
are respectively, to the second order,
r
and
^-rfy
we have
to the fourth order
whence, extracting the
779.
If it be
and
root,
SS r the element of area
for
taking the limit and integrating,
more convenient
to take r
and 9 as the indemust be chosen
+ SO, and the
dependent, elements
pendent variables and
on the surface bounded by r, r -f Sr and 9,
<
resujtant expression for the elements will be
the areas of the projections on the same planes, as in Case
(1),
being
r
SO
.
Sr,
(r sin
.
<5r)
olj
r
SO
and
(r
^ 30)
sin
.
Sr,
and the formula for S being
+r*sin*0
And
in the
same way,
if
we wish
*
g)1*>
\0(7/ J
to regard r
......... (2)
and
<
as the
dependent, an element of surface
independent variables and
is to be chosen bounded by r, r-\-Sr, <j>, 04-<50 and its projections upon the same planes, as in Case (1), being
\
rSr),
/
(Ji
(r)/9
/
7)f)
\
(r^S^.Sr,
/
\ @(1)
(r sin
S<f>)
we have
S<j>*
and
2
S=Jj^[r*sm
<9(f^
Sr*
.
Sr,
CHAPTER
814
But the
XXII.
figures required are, as in the Cases (2)
and
(3), for
cylindrical coordinates somewhat troublesome, and we propose
to deduce these formulae from the more general result of
Art. 790.
780.
Areas on a Spherical Surface, the Origin being at the
Centre.
Then, putting r=a, the
Let a be the radius of the sphere.
general formula
S=a
reduces to
d
sin
d<t>
]J
=a?
[
c
by two
the
limits
6
some
meridian arcs and
=/($),
specified curve,
for 6 are from 0=0 to 0=/(0), and
If
we apply
the result to find the area bounded
the result of Art. 734.
COR. For the whole sphere /(<)=TT, and
781. Spherical Triangle.
Ex. Let us apply the formula obtained to the case of the area bounded
by a great circle and two meridian arcs, the radius of the sphere being a.
Take as the plane of xz that through the centre which cuts the great
circle perpendicularly, and let p be the spherical perpendicular from the
pole
upon the great
circle arc.
The equation
of the great circle
is
then
cotfl
cotp'
cosec 2 6d6
Then
cotp
/*
and
Area=a2
/
/
(1
- cos 6) d<j> = a
J
2
/i
I
J
Area^
,oot^
f*o^ /9^ r*o^po^ /9
=====: o?^
vcot 2 #-cot 2
n _1
M
si
is the angle a meridian makes with the great
azimuthal angle.
where ^
;
circle
and
<f>
is
the
SPHERICAL TRIANGLE.
If
we
take limits
<=a
to
<f>
= a+A,
the limits for
815
x
will be
ir-C
to
where ABC is the spherical triangle formed by the meridians AB,
and the arc BC.
B
AC
Fig. 272.
This area
where
E
is
is
2
therefore a [A
+ B - (TT - C)]
the spherical excess, a result readily established in an
(GIRARD'S THEOREM. See Todhunter and Leathern,
elementary manner.
Sph. Trig., Art. 127.)
Art. 734.)
782.
Other illustrations have been given
earlier.
(See
Case of a Solid of Revolution.
In the case of any solid of revolution about the z-axis
but r is independent of
and depends only upon the
curve
the
solid.
revolving
generating
The general formula
(ft
varies,
now
reduces to
sin
-27TJV
d
z
0^r + ( ~^
2
dO=2T sin
in conformity with the result of Art. 748.
ds
t
CHAPTER
816
783.
XXII.
In the case of
s-axis of circles
solids formed by the revolution about the
whose planes pass through the s-axis, centred at
the origin, but of varying radius, r
is
a function of
dO
The shape
of the surface
bling the hermit-crab
may
alone,
<p
and
d4>.
be pictured as somewhat resem-
shell.
Ex. Let the surface be r=ae^.
S=a 2
and
0,
<f>
f
[
are independent,
Let
x
Vl-sin 2 x d x
x ;
and
if
the area be taken from r=0,
784.
i.e.
mod.
$2 =
-
;
to
oo
any value
of
In the case of an area of a portion of a right
r,
circular
vertex at the origin, axis the z-axis and semivertical
angle a, the general formula
cone,
reduces to
I
r sin
JJ
a dd>
dr=
2
-
-
\
[r
J
the area in question to be bounded by some
upon the cone, say r=/(<), and two generators,
And supposing
curve drawn
we have
2
[> ]
= {/(0)}
2
,
the lower limit being r
= Q,
and
AREAS ON A CONE.
785.
The formula
is
817
obviously the same thing as
which is the area of the portion of the cone developed upon
a plane, the angle between two generators so developed and
and <j>-\-S<j> on the cone,
corresponding to azimuthal angles
being
S(j>
786.
sin
a.
Or again
i.e.
the same thing as
it is
2
I
(r sin a) d(j>
= S sin a,
the area of the projection upon the x-y plane,
of the cone
As a
making an angle
^
all
elements
a with the x-y plane.
and elementary case, the area cut off by a
plane perpendicular to the axis and intercepting generators of
lenth I is
particular
where a
the radius of the base =
is
sin a
and
I
the "slant
height," the ordinary mensuration formula.
In the case of any cone with vertex at the origin, the
equation is of the form
=/(#), r being absent from the
787.
equation.
Hence i? = 0. The general expression
in this case reduces to
i.e.
Hence,
viz.
if
^ =/(#),
a surface cut a cone whose vertex
the area of the cone between two of
and the curve
in
which
it
meets the surface
3F
is
is
the origin,
its
generators
CHAPTER
818
788.
XXII.
Ex. The equations of a cylinder and a cone are
r sin
= a and
cot 8
A 19 A 2 A 3
= sinh <.
$=
be the areas of the cone from
@+ a respectively, then will
If
,
Jj-f A 3 = 2A Z cosh
In
this case
Hence
r>
- cosec 2
+in
-\h
>
= cosh d> -^
= /? -a,
<
/3
and
[MATH. TRIPOS, 1875.]
a.
.
e(^\
>
9> d6=-^sm u A/
2
= - a2 v/2 cosh
<^> c?</>,
^=-
and
Hence
789.
to
Az
sinh
Generalised Results.
If f(x,y,z)
=\
(3
Orthogonal Coordinates.
be any surface,
it
is
required to find the
normal distance between the surface and the contiguous sur-
Fig. 273.
\ + S\ at the point (x,y,z). Let the normal at P to the
surface X cut the surface \-\-S\ at Q, whose coordinates are
face
x+Sx,
The
direction cosines of the normal are
-j?
t
fi
-A -A where
fi
/i
2
=
2
+ Xy + X
\x
suffixes represent partial differentiations and h
line
broken
the
6z
Then
Sx, Sy,
upon PQ,
projecting
2
2
z .
we have
ORTHOGONAL COORDINATES.
f (x,y,z) = \,
Let
f,(x, y, z)
1
= fi
t
819
fs (x,y,z) =
i>
be three mutually orthogonal surfaces. Consider the small
element of space whose faces are the three surfaces X, //, v
and the contiguous surfaces X + cSX,
Fig. 274.
P
be the point (X, ^ v), PP' the diagonal through P
of the element and X + \,
-f SJUL, v + Sv the coordinates of P'.
Let
/UL
Let the edges of this element be PA, PB, PC, PA', P'B', P'C'
This
etc., PA being an element of the normal to X, etc.
is
elementary space
parallelepiped or
Its
volume
Moreover,
'
ultimately an
cuboid.'
infinitesimal
Its edges are
rectangular
t
j-,
where
is
if
,
?i ),
3
be
the
direction cosines of the elements
Sv
fi\
T-
^=the
projection of
PA
"i
=the small change
,
hence
JUL
and
v
in
upon the #-axis
x due to increase of X to
remaining unaltered,
CHAPTER
820
8\
-=-
Similarly
m^ 3v
^
<^
y1
o>
S\,
u/\
fi^
Su.
XXII.
1-^
'dx
7
-^l=S/uL,
tl
OfJi
2
hence
^z
7^= C/A
etc.;
we have
7
'dx
,
'dy
mi= h^<
%=
,
i= ki^'
l
^
'dx
,
h=h
Thus J or ~;'
to X,
3(X, M>
the Jacobian* of
\,
")
x, y,
z
with regard
/i, v,
1
2
1
,
3,
m
m
2
,
3
,
7i2
%
(See C. Smith, Solid Geometry, Art. 46.)
Thus the volume of the elementary cuboid is
J8\$ju.Sv
and F, the volume of any region which is divided up into
elements by this system, is given by
f
of sign disappears when the limits have been
suitably assigned for the evaluation of the whole volume
under consideration.
The ambiguity
COR.
(1).
In the Cartesian system
\=x,
v=z,
jm=y,
7^=^2=^3=1,
and the formula reduces to
V=(\(dxdydz\
the formula of Art. 760.
= 0, v=z, x=rcos6,
\=r
and the elements are Sr, rSO, Sz,
(2) In the cylindrical system
y=rsmO, zz,
7^=1,
h 2 =-,
*SeeDif.
t
/u.
^ 3 =1,
Gale., Art, 534,
GENERALISED RESULTS.
and the formula reduces
821
to
the formula of Art. 774.
(3)
In the spherical polar system \ = r,
y=rsinO sin0,
x=rsm6cos(j>,
and the elements are
7
'^1
Sr,
- *)
r SO, r sin
7
'k>
1
- ~>
r
j,
ft*
S<j),
= 6, v =
im
(j>,
z=rcosO,
and
- - !:
?k>
rsm0
and the formula reduces to
V=\{\r*smOdOd<t>dr,
the formula established in Art. 775.
790.
Element of Surface.
Suppose the region bounded by any surface
divided up in the
manner described by
Q
C
S
to
have been
three families of ortho;
B
Fig. 275.
gonal surfaces whose distinctive parameters are
X,
/UL,
v\
any
say /m, v, with their contiguous surfaces /*+<fyz, v+Sv, form
a tubular region within S. Suppose this tube to cut the tangent
plane at P to the surface in the plane P'RPQ, which may in
pair,
the limit be regarded as an indefinitely small parallelogram
element of the surface. Its area is an infinitesimal of the
second order.
We may take
it
as axiomatic that the difference
between the area of the intercepted portion SS\ of the surface,
and the area of this parallelogram is at least of the third
order, on the supposition that the curvature is finite and
CHAPTER
822
XXII.
The area
continuous over the portion considered.
of
the
readily found from the fact that the
square of any plane area is the sum of the squares of its
projections upon any three mutually perpendicular planes
Let the cuboid element of
(C. Smith, Solid Geom., Art. 33).
parallelogram
the
JUL-V
P'B',
is
which PP'
tube, for
Art. 789, with
PA',
P'EPQ
PC
PA, PB,
PW
is a
diagonal, be constructed as in
for adjacent edges through
and
P
opposite edges through P' (Fig. 275).
be drawn at right angles to PA. Join C'N
for
and RM
and B'M. Thus the parallelograms PBA'C, PQB'M, PRC'N
are the projections of PRP'Q upon three mutually perpenThe areas of these figures are respectively
dicular planes.
Let
QN
PC.PM,
PB.PC,
and
it
will be observed that
PB.PN,
PN=RC'=MA,
i.e.
PM+PN=PA.
we have taken f^x, y, z)=X, fz (x, y, Z)=JUL and
Now,
fz(x, y, z)=v, we can express x, y, z in terms of X, /m, v, and the
equation of the surface S may be expressed in the form
as
by substituting for x, y and z these values. In
form a new system of coordinates and of these
and v as independent and X depending
we are regarding
and v change to /x+<V and v-\-Sv, the
When
them.
upon
F(\, M> v)=
fact X, p, v
;
//,
yu,
total
change of X
\=
is
Su + ^-Sv to the
Now,
first order.
OV
Oft
PM represents that part
in our Fig. 275,
depends upon S/UL, and MA, that
which depends upon Sv, i.e.
is,
PN
of
PA
represents that part of
PA
SfjL
4
and
OfJL
4
OV
PA,
i.e.
<\\
the two
We
making up the
total length of
-j.
"i
thus have, to the fourth order,
.
1
PM) 2 +(PB PN)*
.
3X,
which
/
1
3X
GENERALISED RESULTS.
823
Similarly, if we had taken v, X or X, /u. as the independent
pair of parameters and constructed the corresponding tubes,
we should have had
&SA SS^, SSV intercepted
tubes
respectively, may be
by
taken as an element of the surface for integration for the
and any
fj.-v
of the three surface elements
tubes, v-\ tubes or
,
X-/x
Whole.
Thus we obtain, when we proceed
and integrate,
791.
COR.
1.
If the Cartesian
\=x,
and the elements are
viz.
fj.=y,
system be taken,
v=z,
Sx, Sy, Sz,
to take the square root
h 1 =h 2 --=h3 =l
and
the formulae of Art. 772.
COR.
2.
If the cylindrical
\=r,
system be taken,
]UL=0,
v=z,
)
CHAPTER
824
and
XXII.
z form an orthogonal system, the elements being
r, 0,
Sr,
rS9,
and
Sz
A 2 =-,
7^=1,
^
& 3 =1;
+ (r SO)*
2
/
according as
formulae
or z
r,
is
\2
^/Q
r
g? fr)
,
the dependent variable, giving the
which are in agreement with those of Art. 777.
COR.
and
and
In the spherical polar system,
3.
ic
r, 0,
Sr,
y = r sin
= rsin^cos^,
sin $,
z = rcosO,
form an orthogonal system, the elements being
<t>
r SO,
r sin
S(j>
and
/^
= 1,
A-
2
= -,
whence
.
r 2 sin 2 <9 ^0 2 + r 2 sin 2 (9
r 8 sin 2 <9 S<p Sr*
.
f=Sr* r*SO*
2
(ty
+ Sr* (r
+r
2
M
2
sin
7i
3
=
;
-r
;
CHANGE OF THE ^VARIABLES,
825
giving the formulae
}
ad)
ar
P
is taken as the dependent variable,
or
according as r,
with
those of Arts. 778 and 779.
formulae in agreement
792.
CHANGE OF THE VARIABLES.
Form
of
Element of
Area.
Supposing the coordinates x, y of any point in the plane
of x-y to be expressed in terms of two new variables u, v, let
us consider the nature of the figure .bounded
purves obtained by assigned values of u,
u,
u-\~Su,
v-\-$v.
v,
Let the figure thus bounded be
by the four
v, viz.
PQRS,
Su being zero along PS,
Sv being zero along PQ.
y
v+Sv
O
x
Fig. 276.
The
several Cartesian coordinates of the four corners are,
to the first order,
for P,
x,
y
forS,
x+ OV
y+^Sv;
OV
forfi,
;.
CHAPTER
826
The
direction ratios of
and
of
PQ
XXII.
and
SR
are
PS and QR
~
~$u, 3u Su,
Vu
^-Sv,
^Sv.
^v
dv
Hence the chords joining the corresponding points are such
as, to the first order, to form the four sides of a parallelogram
whose area is
y
'du'
SuSv
or
'du
,
v)
:
dv'
This then
to the second order, the area of the elementary
curvilineal "parallelogram" PR, the difference between this
is,
area and that of the chordal parallelogram being at least of
the third order of infinitesimals. Hence, taking the limit and
integrating between any assigned limits, for
'
^
u and
v,
we have
^{dudv=\JJ\Jdudv,
3(u, v)
J is
x, y with regard to u and v.
remembered that if J' be the Jacobian of u, v
with regard to x, y, we have JJ'\ (Diff. Calc., Art. 540).
And in cases where u and v are already expressed in terms
of x and y, instead of x, y in terms of u and v, this rule will
where
the Jacobian of
It will be
often facilitate the calculation of J.
Similarly,
if
we wish
to integrate
any function
of x
and
y,
over the area considered, i.e. to find 2/(oj, y) 8A
where SA is an infinitesimal element of the area, it is only
necessary to express x and y in terms of u and v, and then to
say f(x,
y),
transform the function f(x, y) so as to express it as a function
of u and v, say F(u, v), then to multiply it by JSuSv, and
integrate, the result being
F(u, v)Jdudv.
793. Illustrative Examples.
1.
f
Find the area of the Garnot's cycle bounded by the isothermals
= a l5
,ry
= a.2,
and the adiabatics
sty?
*=
PU
%y* = Pz-
CHANGE OF THE VARIABLES.
= v,
Putting xy = u, xy
y
827
take an element of the area bounded by the
curves M v u + 8u, v + 8v.
Here
/"'
=
/.
J=
7-1
*'
Fig. 277.
1
|
n& 7-1
and
'
^
{1
7-1
"K'
-
v
(See page 63, Ex. 28.)
=
y* b*, which lie in the
The former intersects
positive quadrant, are drawn intersecting at B.
in A. If every
the asymptote of the latter in 6', and the latter meets
2.
The portions
of
the curves .ry=a 2 ,
xz
OX
from the origin
794.
0,
OABC
be multiplied by the square of its distance
the sum will be equal to a a & 2
[COLLEGES a, 1884.]
element of the area
.
CHANGE OF THE VARIABLES.
Form
of Element of
Volume.
the coordinates x, y, z of any point in space be
expressed in terms of three new independent variables u, v, w,
the surfaces
const., not necessarily as
const., v= const.,
Again,
let
u=
in Art. 789,
w=
forming an orthogonal system.
CHAPTER
828
XXII.
Let us consider the nature of the figure bounded by the
by assigned values of u, v, w, viz.
six surfaces obtained
u,
u-\-Su,
w,
v-\-8v,
v,
w-\-8w.
Let the figure thus bounded be PQS'RP'Q'SR',
Su being zero over the surface PRQ'S,
}
I
'
Sv being zero over the surface PQR'S,
Sw being zero over the surface PQS'R,
i
}
Q
Fig. 278.
,
The
first
several coordinates of these eight corners are, to the
order,
forP,
x,
y,
z,
for<2,
x + ^-to,
y+&*>
***&$*>
for R,
x+
y + ^-ov,
z
f
Cl
for
Q\
for Rf,
forF,
Sv }
^"^
x+
CT
Sv+
70
Sw,
etc.,
Su,
x+^-w-\-^
OW
OU
etc.,
+^
of,
?
CHANGE OF THE VARIABLES.
The
direction ratios of
'dx
829
PQ, RS', Q'P, SR' are
dy
9
'dz
_
.
3^> &*+ *i*;
those of P#,
R'P',
Q/S',
and those of PS, RQ',
S'P',
SQ' are
QR' are
Hence the chords joining the corresponding angular points
first order, to form the eight edges of an
oblique parallelepiped, whose volume is
are such as, to the
Ba?
'dy
Su Sv Sw
u, v,
w)
_
'dw*
This
'dw
volume of the elementary
between this volume and that of
to the third order, the
is,
solid PP', the difference
the oblique parallelepiped being at least of the fourth order
of infinitesimals.
Hence, taking the limit and integrating
between any assigned limits for
where J
is
the Jacobian of
as noted in Art. 792,
Jacobian of
u,
v,
w
it is
u, v, w,
x, y, z
to be
we have
with regard to
remembered that
with regard to
x, y, z,
u, v,
w
if J'
;
and,
be the
we have JJ'=1
And for cases where w, v, w are
(Diff. Cak.j Art. 540).
as
of
functions
x, y, z, instead of x, y, z, in terms
expressed
of u,
v,
w, this rule will facilitate the calculation of J.
795. Ex.
Find the volume enclosed by the
six hyperbolic cylinders
CHAPTER
830
XXII.
zx=v,
Putting
J'
=
V=
0,
*,
0,
x
y*
x,
o
ICCC
we wish to integrate the function
the
volume
bounded by surfaces specified
/(a?, 2/, z) throughout
of
values
two
two
u,
specific values of v and two
by
specific
It follows that- if
796.
specific values of
w,
f(x, y, z)
we have only
i.e.
to
add up
X an element
to express
x, y,
of
all
quantities of the form
volume at
x, y, z,
z in terms of u,
v,
w, and sub-
stitute these values for x, y, z in f(x, y y 0), obtaining, say
F(u, v, w), as the transformed function. Then taking, as
the same
before,
element of volume,
viz.
/ Su Sv Siv,
the
integral required will be
F(u,
iff
797.
Thus,
if
we wished
v,
w) J du dv dw.
to obtain the product of inertia with regard
to the y, z axes in the above example (of Art. 795), each element of mass
pJSuSv Sw is to be multiplied by yz, i.e. u, and assuming a uniform volume
density
p,
the product of inertia required
is
I
du dv dw
where
M
798.
If
is
/
/
pu J du dv dw, or
~
the mass of the solid in question.
we wish
for the .r-coordinate of the centroid of the solid,
>
I
I
C C
and
A
I
pj du dv dw
Cdu dv
dw
similarly for other integrals.
du dv dw
du dv dw
CHANGE OF THE VARIABLES.
We
799.
831
consider next the case in which the three co-
are expressed, or expressible, in terms of two
independent parameters u and v, and therefore the point travels
upon a definite surface. Consider the four points P, Q, S,
ordinates
x, y, z
R
on the surface defined~by the values
(u-\-Su,v),
(u,v),
i.e.
x,
+ 8v),
(u,
(u-\-8u,
dx
dx
du
dv
v+8v),
z;
y,
.
8
dv
Fig. 279.
The
direction ratios of
PQ
and
SR
are each
dx
32;
"dy .
^-Su,
^-Su, ^-Su,
du
du
du
and those of
PS and QR
Ba;
are each
.
r-^V,
dv
and to the
order
first
dy
dv
PQRS
^2
&,
is
;~SV,
dv
a parallelogram.
Let
its
area
be AS.
The coordinates
of the projections of P, Q, S,
of x-y are
and the area of
this projection is
dx
du'
dx
dv'
y>
dy
du
Vy_
dv
R
on the plane
CHAPTER
832
and similarly
its
are
projections
upon the other coordinate planes
~
^(y,
V(y, z) . _.
r SuSv'y
SuSv;
~,
(
d(u, v)
^p
whence
its
area SS
is
XXII.
9(z, x}
'
-ouov,
given by
Hence, proceeding to the limit and integrating,
SJ^+Jrf+J/du dv,
i.e.
where
J^^j-^1,
J =etc.
.7 =etc.,
2
3
the surface integral of any function f(x, y, z) be
required, f(x, y, z) is to be expressed in terms of u and v,
as 0(w, v), and the surface integral required is
Also
if
u
If
we
tf
write
=
~du)
we
+
"*
2
4-
\du)
"*"
\du
a&3x
'
"du
have, from the algebraic identity,
(mri
.*.
m'n)
2
+ (nl
2
ril)
the surface integral
(u,
as
dudv.
shown otherwise
800.
I'm) +(ll'-\- mm'
2
'+(lm'
may
be written
v)jEGF
2
dudv,
in Art. 744.
Results connecting
SV and
SS.
SS be an element of the area S of a surface, and P be the
perpendicular from the origin on the corresponding tangent
is at
plane, we have for the volume of the cone whose vertex
the origin and base SS,
ip SS,
If
SOME SPECIAL FORMS.
833
Hence the volume of any region bounded by a given surface
and a cone with vertex at t'he origin, and generators passing
through the perimeter of any closed curve drawn upon the
or, which is the same thing, if I, m, n be the direction cosines
of the normal to the element SS, so that
P = lx-\-my + nz,
is
the equation of the tangent plane,
we have
3.
801. If the equation of the surface be written as z=f(x,
the equation of the tangent plane at x, y, z fs
y),
Z-z=p(X-x)+q(Y-y),
*dz
'dz
p=
where
and the perpendicular
Hence the formula
P
,
q=
,
from the origin upon
for the volume, viz.
it is
IP^,
becomes
for
where cos a,
i.e.
cos/?,
V=
cosy are the direction cosines of the normal,
l>z + qy -f(x, y)] dx dy.
802. Let the inward drawn normal at a point P on a surface
make an angle ^ with the radius vector from the origin, and
let p be the perpendicular from the origin upon the tangent
plane at P, r the radius vector from the origin to P, and SS
an element of the surface about P
Then - = cosx, and the formula
for
an element of volume
forming an elementary cone with vertex
lp 8S, becomes IT cos^ SS.
E.I.C.
3G
and base
SS, viz.
CHAPTER
834
XXII.
Hence we have another expression for the volume bounded
by any curved surface and a cone' whose vertex is the origin
and passing through the perimeter of the region defined by a
given closed curve drawn upon the surface, viz.
g
Fig. 280.
or again, seeing that this element of volume
is
we have
r
and
803.
S=\\ -smOd9d<f>.
Ex. Find the surface and the volume of the solid formed by the
r=a(l+cos6) about the initial line.
revolution of the cardioide
Fig. 281.
TETRAHEDRALS.
=
Here
835
p = rcos x =
,
r**
,..3
S
= 16/ra 2 1~ - f cos
5
L
V=
Also
2
ffjr
the limits for r being
(f>
from
V=
to 2;r
to
TT.
6d8
^? (*
2
-
rf r _
^>
804.
^^ d<$> d
to
^ from
Hence
sin ^
L
Tetrahedral Volume.
An
expression for the evaluation of a volume for a surface
given by a tetrahedral equation may be obtained in the same
D
way
as that adopted for
For let
and let a,
F
/3,
an area in areal coordinates
(Art. 461).
be the volume of the tetrahedron of reference,
y, S be the tetrahedral coordinates of a point P,
CHAPTER
836
XXII.
and x, y, z be their Cartesian equivalents with reference to
some given rectangular system of axes then x, y and z are
linear functions of a, /3 and y, for we have a + /3-hy-{-$ = l.
;
F=
Hence
where
[Jf
^ dy dz=K (\\da
K is some determinate
dft dy,
constant (Art. 794).
To determine K, apply the formula to the fundamental
tetrahedron itself. If we integrate first with regard to a for
the tube bounded by two given planes ft and ft + 8/3, and
two planes y and y + <Sy, keeping /3 and y constant, 'the
limits for a will be
the plane a
a=0 to a 1
from the point at which this tube cuts
which it cuts 8=0, i.e. from
Then we have
to the point in
y.
ft
Next, integrating this with respect to /3, keeping y constant,
the limits for ft will be from ft=0 to the point where a=0
and 8=0,
i.e.
where ft=l
and
y,
77
Lastly, integrating from
Hence
805.
7=6F
;
y=0
to
y=l, F
therefore the formula
-^-.
is
Surface generated by the Revolution of a Tortuous Curve
about an Axis.
Let a curve of double curvature revolve round the 0-axis
;
required to find the surface generated.
Let PP' be the element ds of the curve.
it is
Let revolution about the z-axis be made through the angle
d9, and let the perpendiculars PN, P'N' turn into the positions
PJN, P^N'.
PP
Then
NP = \/x
to the first order,
and
ment PPjPj'P'
NPdO
is
.
ds sin
2
-f
^
y
2
,
and the area of the elewhere x
to the second order,
REVOLUTION OF A TORTUOUS CURVE.
is
the angle between
PP
and P-f^
1
t
i.e.
837
between directions
whose direction cosines are
Hence
dx
dy
dz
ds'
ds'
ds
cos
x
_
'
x = (x ds
/--.y ~] /v/z
2
Q
+f
and
Fig. 283.
Hence
Area
PP P
of element
dO v/(z dx + y dyY +
2
(x
+ y2
1
)
dz 2
1
T
/
.
Hence, for a complete revolution the area traced out
or in cylindrical,
(p, 0, z),
is
CHAPTER
838
XXII.
That is the area of the surface described is the same as
would be traced out by a rotation about the 2-axis through
the same angle, of a new plane curve constructed by first
swinging back each point of the tortuous curve from its
actual position without alteration of its distance from the
upon the
axis of rotation into a corresponding position
initial
plane.
And
if
ds be an elementary arc of this
curve,
Area = 2w p ds'.
and therefore
806.
new
I
Ex. Let us employ this formula to find the surface of a hyper-
boloid of revolution included between two planes perpendicular to the
Fig. 234.
axis, the surface
being regarded as generated by the revolution of a
the z-axis, the line making
straight line about the axis, which we take as
a constant angle with the 2-axis and not cutting it. The equations of the
line are
x=a case - z tan a sin 0,
y=a sin + 2 tan a cos 0.
Hence
and
x dx +y dy = z dz tan 2 a
S = 2 Uz* dz* tan
TT
4
= %TT l\fa + z2 tan
2
l
= STT tan a sec a
;
a + (a 2
2
+ s2 tan 2 a
a sec'2 a dz
,/v.
)
dz 2
ANNULAR ELEMENT OF SURFACE.
839
Hence
AS^TrtanasecaF.?
807.
cos 4 a
cos 4 a
2
sina
silia
.
,
.
sum" 1
al**
acos 5"aJ 2l
Case of an Annular Element of Surface.
Surface of the
Ellipsoid
Legendre's Formula.
The equations of the
and
its
normal &t
x, y,
z are
direction cosines are *~s,jjr>2'
a u c
wnere
perpendicular upon the tangent planes at
P
*s
^ ne
x, y, z, viz.
central
such that
Fig. 285.
Let a cone be drawn whose vertex
cutting the ellipsoid at
makes a constant angle
pz = cos
c
Let
S
2
z2
ft
is
at the origin 0,
and
those points at which the normal
with the 2-axis. Its equation is
all
or
.
4
c cos
2
5Ti
2
#
2
x
y
= ~r+7^rH
4
4
a
Z>
z2
rc4
be the area of the ellipsoidal cap cut off by this cone.
CHAPTER
840
XXII.
we
eliminate z between the equation of the cone and the
equation of the ellipsoid, we obtain the projection of this
If
curve of intersection upon the plane of xy,
sec 2 #/
^! 17 1
_^!_^!\
== ^!
4 + 4+ 2
2
2
,
c2
or
-7-7
a4 sin1d v(a
2
an
6
If
c
(6
cos^H,--^
6 sm #
4
2
v
2
_^_yl
W
a2
V
sin 2 0+c 2 cos 2 <9)
= 1,
ellipse of area
sin 2 $
00
to
,
a
6 /
sin 2 (9+c 2
5
viz.
a
\
viz.
we
increase 9 to
6+89, we increase $ and
Now
S+SS and A+SA.
areas of
two
And when
A
respectively
&4, the difference between the
the projection of SS upon the x-y plane.
indefinitely small, all elements of SS cut off
ellipses, is
SO
is
by contiguous meridian planes make the same angle
with
Hence
which are the corresponding elements of SA.
_
,
SA
SA^ 38a COB 6 and SS =
and taking the
limit
their projections,
.
and integrating
-r
Icostf
To
effect
We
the integration of
dA
^,
we shall change the variable.
have
Put cos
=
a
sin
_2 sin = sm'
.
d>
.
2
ain2/Q
=
A = ira2 b 2
*-
-.
,
,
where
c
= a cos y.
Then
a 2 ^2_ C 2
V ^^^z^
sin 2 y
sny
Trafe
sin
which
is
2
sin
0l
cos
-
a 2( & 2_ c 2)
sin 2 y-sin 2
'
y
81
cos0A
<1, and A 2 =l
~6 2 (a2 -c 2 )
SURFACE OF AN ELLIPSOID.
And
841
dS = siny
A
.A cos
,
~]
d - -H
rj
sm 2 ^d(h
[, sm<
.
<
^4
7
[
sm0 sm*y
y
A sm
\
_7rab r,
cos
/sin 2 y
cos0
<
.
Acos0
1
rJ
sin^-sin^
A sin 2
/
<
7
d(h
^
2
sm /
^
r
/si^y-sin^X
sin
sin 2
sin 2 y
Trafr
c2 -.
sin 2 ^>\
sin 2 y
7._^~|
Asin 2 ^> ^
A J'
"sinyL \Asin </>cos0/
/-.>,
'
A
Now
^( Acot,)=-^p- sl
4-AA
A
si]
Hence
sin
y
\
sm <p cos
= ^L \ d (8 i"V- 8i "V-A sin^y cot *
sin y L \ A sm
cos
cos 2 y
sin 2 y A d(j>
-^
sin
_^2s
_ 1 _ ps n
.
.
n
.
n
y
h
where
and the
1
fc
2
limits for
sm 2 y=l
are
19
9
6 2 (a 2
to
-
^.2
c )
rr-= y^,
for the
-
a 29
b2
upper half of the
and the consequent limits for <j> are
double to take in the lower half of the surface.
ellipsoid,
,
y
to 0,
and
CHAPTER
842
Thus
for the
XXII.
whole surface
m
.
-^
r-
smyL A
c 2 \n
e
2
T=)
y
0V J y
d<f>~
--- sm 2 y/ P Ac0-fcos 2 y P d<f>~]
sm
Jo A J
J
smyL
ZTrab V
*
7
,
-
xi
where cosy = -, a form due
CL
808.
to Legendre.*
Cases.
In the case of the oblate spheroid, a = 6,
7c
= l, and
the
elliptic functions degenerate,
E
becoming
^
F becoming
and
J^
S= 27rc2 +
giving
^
sin 2 0<i0=sin
\/l
I
3
[sin
=
y
logtan(|+g,
y + cos y log tan (J + ^ J J
2
smy
and for the prolate spheroid b=c,
k=Q,E=y and F=y, giving
smy
y cos y
sin
"^
or
"V*
sin
Another Method
809.
From
the formula
y
sin y cos
(y
v f -f
for the Surface of
$=
f
1
an
y
).
Ellipsoid.
,dA
Jcostf
of expression for the area of an
the
value
of dA, we have
Substituting
we may deduce another form
ellipsoid.
cos
Put cot (9=.
c
* See
Serret, Calcul Integral, pages 338-342
Integral, p. 193.
;
Legendre, Exercices du Calcul
SURFACE OF AN ELLIPSOID.
843
Then
dS
2
v/(a
2
+X)(6 +X)
11
i
d\
a 2 +X)(& 2 +X)(c 2 +X)
c2 /
1
~~2 Va
2
^
XdX
"
(
X,
and the
are
limits of integration for the
0=0
to
6=%,
2
X=oo
i.e.
upper half of the
to X
0,.
The
result
ellipsoid
must be
doubled to include the lower half of the surface.
r/-
Nov
Jo
W
2
v/(a
2
+X)(6 +X)(c
2
+X)o
(See Art. 3G3, Ex.
Hence
for the
whole area of the surface of the
5.)
ellipsoid.
We now
revert to the consideration of the generalised
system of orthogonal coordinates discussed in Art. 789.
It will be remembered that we there obtained expressions
810.
o
n
n-v
for the direction cosines of the elements 7-,
~,
^2
l
^- in
'h
terms of
partial differential coefficients of x, y, z with regard to X, JUL, v>
may also readily express the same direction cosines in
We
terms of partial differential coefficients of
to
x, y, z.
*
Mathematical Tripos, 1896.
X,
/x,
v
with regard
CHAPTER
844
Regard
,
ftj
j^
/6
2
t
-^~
f)
3
XXII.
as the directions of a
new
set of three
coordinate axes OA, OB, OC.
Referred to such axes the direction cosines of the original
axes are:
for Q x
Oy;
^ ^ ^
m m m
for Oz\
n lt n 2 n s
-
for
2
lf
3,
,
,
;
Fig. 286.
then
l
is
=a, small
x to
X
:=T-
Similarly
and we have the system
7
whence
it
OA
element on OA
the projection of 3x upon
1
x-\-Sx,
y and
UU.
^r-
and
,
of equations
1
c)A
follows that
.
ox
'd\
'
J',
>
isj,
mp
m
3'
3'
Zj,
^' v
i.e.
2,
)'
2/
??!
7?
2
3
z
due to an increase of
remaining unaltered,
GENERALISED COORDINATES.
845
which might have been anticipated from the theorem JJ'=1
(Diff. Calc., Art. 540).
We
thus have the following relations between the several
by comparing with
partial differential coefficients,
,
-'
x_t/
2
3
7,
~'
2/_^
2
"*
7,
2
/3
Art. 789, viz.
?_.
~'
l/
Similarly
811. It
plain that the areas of the three faces of the elelie on the surfaces X = const., /x = const.,
is
mentary cuboid which
i/
= const.,
are respectively
5l/
and that the
y-\-$y,
(5X
infinitesimal distance
z+8z,
viz.
S]UL
between
the diagonal through
^2
is
cuboid,
8\
-
<?
P
2
z
x, y,
and
x-\- Sx,
of the elementary
? 2
[See Todhunter, Functions of Laplace, Lame and Bessel, pages
210-233 also E. J. Routh, Anal Statics, vol. ii., Arts. 109, 110.]
;
812. Elliptic Coordinates.
The most remarkable
case of these orthogonal surfaces
is
that of the three confocal conicoids, (a
'
"I
an
9
i
-k
== 1>
19~,
I
I
== 1>
r~T
~0~^
=
a hyperboloid of one sheet and a hyperboloid
c 2 /x between
of two sheets respectively, so that X is <t
viz.
ellipsoid,
,
c2
and
62
,
and
v
between
62
and - a 2
.
CHAPTER
846
y and z in terms of the parameters
a well-known algebraical device, viz.
To express
resort to
XXII.
x,
X,
/u,
v,
we
Consider the equality
&
f
c2
a?+0* b*+0
where
tions.
x, y,
,
(\-Q)(p-Q)( v -e)
'
2
2
2
(A
.
(a +tf)(& H-#)(c +0)'
z have the values obtained from the above equais either an equation to find 0, or it is an identity
This
true for
If
#
=1
+9
all
values of
an equation,
0.
it is
of quadratic nature
;
for
3
disappears
upon multiplying up by (a + 0) (b + 0) (c -f -0). Hence it
than two. This
could not be satisfied by more values of
= X, = and
equality, however, is obviously satisfied by
= v, i.e. more than two values. Hence it is not an equation,
but an identity and true for all values of 0.
2
2
2
JUL
Multiply then by Q-\-a\
a 2 hence
In this identity put
;
2
(a
-6 2 )(a 2 -c2
)
iVd
Similarly
and
2
(c
Hence
2z
=
2
(a
that
is
2^
and similarly
-a2 )(c -6
2
-62 )(a 2 -c 2
)
2
)
a 2 +X'
=
2^ =
Again, if we differentiate the identity (A) with regard to
obtain another identity, viz.
we
x2
v2
z*
0IUL0
v0
a 2 +0
0,
ELLIPTIC COORDINATES.
and putting 6=\ in
r/3s\
this result,
/asVI
/3;/y
"HaA/ "HaA/ J
/
847
(X-AQ(X-F)
2
2
2
(a -j-A)(6 -|-A)(c
+A)'
(A2
where A\^
Hence
(a
2
+A)(6 -fA)(c
2
2
-|-A),
A^^etc.,
2
z-
r
A v = etc.
We
thus have for an expression for a volume divided up
into elementary cuboids defined by the faces of the three
confocals A,
/m,
v,
and the three contiguous confocals
X + <5X,
+ Sfi, v -f- $v,
IUL
813. In case of integration throughout the
by the ellipsoid,
x
the limits are: for X,
for
for
yu,
t/,
2
y
volume contained
z2
2
1*
^tfr^gr*
to X=
from X=0
c2
from jm=
from i/=
62
to /UL=
to i/--
c2
62
a2
;
;
.
be integrated through
any specific region bounded, say, by confocals X 1? X 2 yuj, jm 2
v^ v 2 we must convert F into a function of X, /*, y by sub814. If
any function
J(.^, y, z) is to
,
,
,
stituting for x, y, z their
values, obtaining, say,
will be
^(X,
and then the required summation
,
A, M, v)
815.
For instance,
v
-T
/\
j
(A,
-V/
x
M
,,
the function to be integrated be
if
\
^,V)=
.
v)(v
A) (A
/x)
/^2 /M2 /'a
we have
/==o
A^
A
a/x
a
JUL,
i/),
CHAPTER
848
we may gather from
816. In particular
of
an
ellipsoid,
XXII.
viz.
-J7ra&c,
the
known volume
that the value of the definite
integral
817.
The elements
of surface of the three confocals at a
point of intersection are respectively
.
,
818.
We may
^ !/>,A ~
=
\/(i/
x
X)(X
/*)
y)(i/
X)
thus, for instance, express the area of any
X=0, bounded by confocals JUL V /w 2
portion of the ellipsoid
i/u 1/9, as
819.
The
,
distance Ss from X,
given by
Ss
2
/x,
v
to X+(5X,
v+Sv
fjL+SfjL,
is
= &c 2 + Sy 2 + ^s 2
2
_^X_
V ^
"V V V
And
In the case where the line
And when
lies
on the
ellipsoid
X = 0,
the curve on the ellipsoid is further defined by
and v, further reduction may be effected.
/*
a relation between
For instance, along the
line of curvature
section of the intersection of
X=
with
/JL
which
is
= const. =
the inter/x
>
ELLIPTIC COORDINATES.
849
or writing
,
=
we have
^
^-^.^--^
.
d,
for the length of a specified arc of a specified line of curvature upon the ellipsoid.
If
820.
we write
X+a =X 2
X+6 =X 2 -6 2
X+ c 2 =X 2 -c 2
2
1
,
2
1
1
,
1
1
,
the conicoids become
vl
"I
"I
"I
and we have a certain amount
of
simplification
of
the
formulae, but with a loss of symmetry.*
Thus we obtain
x*=
and for the volume of the
>
A
the limits are
:
l
2~>A l 2
for
it
A
l
for X 1?
for
Hence
ellipsoid
//!
2T\A
2
l
~C 2
/,
1
from c^ to Xj
from & t to c x
to b r
from
;
,
i/j,
;
follows that the value of the definite integral
s
being an octant of the
* This
and
is
ellipsoid.
the notation adopted
by Todhunter, Functions of Laplace, Lame"
Besstl; Bertrand, Calc. Int.
E.I.C.
SH
CHAPTER
850
XXII.
The suffix has been retained to prevent misconception as to
the meanings of the several letters, but may now be dropped.
For this and the values of other definite integrals of similar
nature,
Bessel,
821.
see Todhunter, Functions of Laplace,
Chapter XXI.
Lame and
Solid Angle.
Let C be any closed curve, plane or twisted, bounding any
a fixed point, and S a sphere of
region upon a surface,
unit radius, with centre 0.
Let a cone with vertex
and
generators passing through the perimeter of C, isolate on the
unit sphere an area w. Then w is called the "solid angle"
subtended at
by the portion of surface bounded by C.
Fig. 287.
The area
of a sphere being 4?r
2
x
(radius) it follows that
closed surface at a point
,
the solid angle subtended by any
within it is 4?r at a point upon it which is not a singularity,
at a point outside, 0.
The solid angle subtended at a
2-7T
;
;
corner of a cube
by the
rest of the cube is
=
-.
At a
point on the line of intersection of two planes cutting at right
angles, each of the regions into which space is divided by the
two planes subtends a
solid angle
=
TT.
At the vertex
of
a right circular cone of semivertical angle a, the solid angle
is the area of the portion of unit sphere, centre at the vertex,
cut
off'
by the
cone,
i.e.
2?r
.
1
.
(1
cos a),
i.e.
2?r
vers
a.
SOLID ANGLES.
851
A circular disc of radius a subtends at a point
whose distance from the plane of the disc is h, a
on the axis
solid angle
JLA
Fig. 288.
822. In the spherical polar system of coordinates, the face
of the elementary cuboid r2 sin 66 S<f> Sr, which is at right
Fig. 289.
2
angles to the radius vector, is r sin SO S<j>, and
the
at
solid angle subtended
origin 0, we have
I2
if Set
be the
'
the area pqrs,- viz. Sco, intercepted upon unit sphere by
radii vectores to the boundary of the element whose face is
i.e.
PQRS,
viz.
r2 sin# SO
The element
2
of
written as r S<aSr,
S<j>,
is
given by
volume r 2 sin
and
In the case of the sphere r
is
SO
S(j>
constant,
Sr
may
and
therefore be
CHAPTER
852
XXII.
Let the inward drawn normal at any point of a
make an angle x with the radius vector r to
823.
closed surface
the point, and let SS be an element of the surface about the
point; then the projection of SS upon a plane cutting the
radius vector perpendicularly is <5>Scosx> a n(l i n ^ ne limit
.
when SS
is
infinitesimal,
r2
AS cos
to the second order
;
we have
whence
S=
Also,
p
if
the point
I
r 2 sec x dw.
be the perpendicular upon the tangent plane at
r, 0, $,
we have
3
p=r cos x
Obviously
and
If
if
it
S= fr
1
do*.
follows also that
the closed surface surrounds the pole 0, this gives
lies
singularity,
lies
on the surface where there
at a point
f
J
If
and
cos
_
r2
outside the closed surface,
f^S=0.
r*
J
If
lies
at a conical point of solid angle
o>,
is
no
GAUSS'S THEOREMS.
853
These theorems are of great importance in the theory of
attractions, and are due to Gauss.
(See E. J. Routh, Anal.
Statics, vol.
824.
Let
XY,
ii.,
Art. 106.)
Solid angle subtended by a triangle at a point not in its plane.
a triangle of sides a, b, c lying anywhere in a given plane
ABC be
let
tively p,
q,
be a point not in this plane, and let OA, OB, OC be respecr.
Let the planes OBC, OCA, OAB intercept on the unit
sphere, centre O, the spherical triangle A 'B'C' of sides a', b', c and let p'
be the great circle perpendicular from A' on B'C', and let w be the solid
f
,
angle subtended by
ABC at 0,
and
27'
the spherical excess of the triangle
A'B'C'.
o
c'
X'
Fig. 291.
Then
to is
Hence
it
sum
measured by the area of A'B'C',
i.e.
appears that triangles bounded by planes such that the
between them is constant subtend the same solid
of the angles
angle at 0.
Cagnoli's theorem gives
.
E'
Vsin s
sin
(s'
- a') sin
2 cos
cos
2
or,
which
is
(s'
32
- b') sin
(s'
- c')
cos
2
the same thing,
sin a' sin
4 cos
a
6'
sin C'
-cos -cos C'
b'
[Todhunter and Leathern, Spherical Trigonometry, Art. 132.]
Now let the volume of the tetrahedron OABC be called V
= V,
\qr sin a' p sin p'
pqr sin a' sin b' sin C' = constant = 6 V.
'
.
i.e.
.
;
then
CHAPTER
854
Again,
q
XXII.
+ r'2 - a 2 = 2qr cos a',
2
>a'
and
if
II 2 represent [(q +r) 2 -a 2][(r+^) 2
II 2 = 64p 2gV 2 cos 2
cos 2
and
cos 2
Hence
si
Also, if h be the distance of
of the triangle,
-& 2][(^ + g) 2 -c 2], we have
"-t=
O
12
IT = Spqr cos
cos
.
of
ABC
and
A
the area
*
sin|
=
4^^.
own plane in such manner
2
2 2 2
- c 3] = constant,
a
+
] [(p + qf
] [(r+p)'
[(q
r)
moves
^
n-
from the plane
F = PA and
If then the triangle
cos
in its
as to
make
fc
the solid angle at O will remain constant.
If the triangle
be a fixed non-conducting lamina uniformly
electrified, this equation will determine the lines of equal density of
ABC
electricity
induced upon an
infinite
parallel
plane
conducting
uninsulated.
ILLUSTRATIVE EXAMPLES.
825.
1.
To
find the
volume
of the portion of the paraboloid
b
cut off
by the plane
Ix -\-rny
+ nz=p.
Fig. 292.
The
difference of the 2-ordinates of the plane
.
p-lx- my
and the paraboloid
is
and
ILLUSTRATIONS.
The
projection of the curve of intersection
p
a
The problem
of this elliptic
=0;
855
upon the x-y plane
an
i.e.
is
ellipse.
of finding the volume required is that of finding the mass
lamina with a surface density f. We have to evaluate
over the area of the
ellipse.
Keeping x constant, we have
where y lt y 2 are the ordinates of the ellipse on the x-y plane for any
given value of x.
Now, the quadratic for y being
we have
y l + y ,=
Hence the subject
and the
To
2
limits for
effect
Then the
2
9
of integration
are
Zlx
&
is
-
c2 =a
where
^ = ,/#(_ + _
-
and
^
^
=
-c and +c.
the final integration, let ^ = csi
limits are
and double.
to
Hence
1
66"
We
plane
might
elect to
lx+my + nz=p.
5!
f
'
C 4?
15:
422
do the same thing by taking laminae
parallel to the
CHAPTER
856
The area
[C.
of such a section
Smith, Solid Geometry,
XXII.
is
p. 99.]
The thickness of a slice is 8p.
The slice of zero area is such that
PI being the corresponding value of p.
The
limits of integration with respect to
F=
Hence
^f
n
(al
2
p
are from p\ to p.
+ bn* + 2jp) dp
Jpi
'ab
7r\a
as before.
We
may note that frusta of finite thickness whose bases are parallel
to a given plane are such that their volumes vary as the squares of
their thicknesses ; also that frusta of given thickness are such that their
volumes vary as the squares of the secants of the angles which the
normals to their bases make with the axis of the paraboloid.
2.
To
calculate the value of
/
tions being conducted through the
Z,
/
/
(f>(lx
+ my + nz)dxdydz,
volume of the
the integra-
ellipsoid
m, n being such that
lx+my+nz = S.
Let
The area
of this section of the ellipsoid is
Trabc/
wh ere p 2 = a-Z 2 + bW + c% 2
S2
.
Consider the ellipsoid divided into thin slices parallel to this plane.
The volume of such a slice is Ad8 to the first order, d8 being the thickness of the slice, and <(S) is, to the first order, constant through the slice.
Hence
ILLUSTRATIONS.
3.
To
calculate the value of
I
+
The volume
and S + d8 is
and
</>(<$)
is
Hence
4.
I
</>(^
tions being conducted through the
Take
I
volume
+ p + ^Vztfydz,
the integra-
of the ellipsoid
+ ?=
bounded by the similar
of the ellipsoidal shell
d(7rac) =
constant throughout this
[ [(
857
rrac
ellipsoids 8
,
shell.
4>(^+^
Find the mass
of a thick focaloid,*
i.e.
a shell bounded by confocal
equal density being confocal surfaces, and the
each
at
density
point inversely proportional to the volume contained
by the confocal through the point.
ellipsoids, the layers of
Let
f
t
\+7-/
*
+
be the confocal through the point, and
x=l
be the outer and inner surfaces of the
The volume contained by the
let
shell.
ellipsoid
A
is
F = *W(a2 +A)(6 2 +A)(c 2 +A).
The volume
The law
of the layer
of density
between the surfaces A and A + o?A
is
p = kj^ir </(* + A) (6
Hence the mass
if
Z)
2
+ A)(c 2 + A),
k being a constant.
of the layer is
and the mass of the thick
and
is
shell is
be the density of the outer layer,
k
Hence
M
^-rrdbcD log
,,
doc
,
,
.
*Fjr this term see remarks by K. ,T. Routh, Anql.
Tait's Natural Philosophy.
Thomson and
Statics, vol.
ii.,
p. 97,
and
CHAPTER
858
5.
XXII.
Consider the region bounded by
(1) a
We
2
2=
2
a2
sphere # +# + 2
;
(2)
2
a right circular cylinder x2 +y = bx
(3)
the two planes
shall first find the
y=x tan
(a
<fc
6)
;
a.
volume enclosed by these surfaces
in the positive
octant of space.
Take
cylindrical coordinates
r, 6, z.
Fig. 293.
The elementary prism on base
and
r
80 8r has volume rzSd 6> to the second
order,
V=ffrzdddr
and the equation of the trace of the cylinder upon the x-y plane being
to 6cos#, whilst the limits for 6 are
r=6cos#, the limits for r are
from 6 = to 6 = a.
=\
Hence
j
Writing
=
-<f>
and
I
-'n
{a
3
-
a=~-ft
in the integral,
ILLUSTRATIONS.
A =
2
where
-^ sin <A
2
(l
and by Legendre's formula (No.
10, p. 399),
4
k-
f<>
/
and
if
^1}
where
-P\
O
5
JO
859
5
be the real quarter periods, we have
E =\
And for the wftoZe volume of the sphere included between the specified
boundaries, we have four times this quantity.
When the cylinder just touches the sphere, i.e. 6 = a, the elliptic
functions degenerate.
then have for the volume in the positive octant
We
O
JO
a
J^.
f
3
Jo
=
3ain0-sin30
(\
4
\
^ [4a
-
3(1
- cos
a)
'
+ 1 (1 - cos 3a)]
3
= a (12a-9versa + vers3a)
OO
and
in the case
where the planes ij=
=^, the whole volume
r=acos0 is
i.e.
To find
we have
x tan a coincide with they-* plane,
cut out of the sphere by the cylinder
the surface of the sphere thus bounded in the positive octant,
y being as usual the angle the normal
the 2-axis
;
;
that
is
cos
z
x/a 2"- r2
n.
a
7= a- =
'
to the sphere at
r, 0, z
makes with
CHAPTER
860
Hence
XXII.
S=([-^=dddr
J J \Ja- - r
2
6 cos
*
f*
2
2
{a- \/a - 6 cos*
=aJQ
and putting as before Q = --<$> and a =
^
-/:?,
and when 6= a, we have
=a
and
2
-versa)
(a
for the further particular case
;
when a=^,
->(!->)
And
this
in each case the whole of the surface of the sphere intercepted in
manner
is
four times the portion which has been found.
6. At every point of an
elliptic lamina a straight line is drawn
perpendicular to the plane. of the lamina and of such length that the
volume (/A, say) of the rectangular parallelepiped formed by this length
and the distances of the point from the foci of the elliptic boundary is
are the semiaxes of the elliptic boundary,
constant. Given that a and
show that the volume of the solid thus formed is
fc
TT/A.
a
4
Taking
x+iy=ccos(Q + i<j>\ we
<*>
+b
<*-&'
[COLLEGES, 1891.]
have
y= -csin# sinh <,
= constant are the confocal conies
#=ccos0cosh<,
and the
loci
</>
= constant,
2
?/
inh 2
^
and
1
CC
2
ifi
C 2 cos 2 6>~c*sin 2 #
=1
'
and the focal radii r l5 r2 are such that ri + r2 = 2ccosh <, r l - r2 = 2c cos 6.
Let the elliptic area be divided up into elements by confocals in this
<-}-<$</> as a type.
way, taking the element bounded by 0, 6+- 88,
</>,
Now
where
J^i is
II
F (x, y) dx dy =
the equivalent of
F
in
ff
F, (0,
terms of
<j>)J
0, <.
dd
ILLUSTRATIONS.
Also
861
- c sin 9 cosh
</>,
c cos
si n
h
</>
ccos 0sinh
c/>,
c sin
cosh
</>
= c 2 (sin 2 8 cosh 2 + cos 2
= c2 (cosh2 (-cos2 0).
sinh 2 ()
</>
and by the condition
Thus
and the
for
limits for
of the question
=
are
to
n=zr1r2
= -,
and
= &,
that
which ccosh<=a and csinh<
.
$ from
for
is
=
/>
<
inTii,-
=
to the value
1
Fig. 294.
Thus
7.
/=/--
In the evaluation of such integrals as
J
taken over the
p*
and the
surface of an ellipsoid of semi-axes a, b, c, where the surface is
volume I7 p being the central perpendicular upon any tangent plane,
,
consider three points P, Q, R on the surface, which are the extremities
of three semi-conjugate diameters.
Let 8$i, 8S2 ,-8S3 be any elements
of the surface about the three points and p lt p%, p3 the corresponding
perpendiculars.
rp,
Then
T
/=
fdS,1
/
J Pi
n
,
or
Cd$
J
^,
n
CdS*
dS*
or
I
J
P2
Ps
Now suppose these elements of area SS^ SS2
chosen that
go
go
g^o
,
8S3
to
have been so
//
CHAPTER
862
XXII.
J_, JL, JL = i ,1, I
Then, since
PI*
P*
<*
P*
b
<F
'*
we have
we
also
7_ : =
have
whence we can readily
I
p dS = 3 I
7
and
,
7
=
;
infer the values of I 19 7 2 , 7 3 , etc., viz.
^-^l-Ifi
+ I+iY
~
&
3K 3
5
n
V
2
2
^K
'
cV
.
&*.!
PROBLEMS.
1.
Find by integration the volume of a frustum
(1) a pyramid on a triangular base,
(2) a
pyramid on a square
of
base,
(3) a cone.
2.
Find the volume
of the portion of a sphere
bounded by planes
through the centre which cut the sphere in the sides
spherical triangle
3.
Show
ABC.
that the volume cut off from the paraboloid
by the plane
x+y+z=a
is
4.
Show
that the volume of the solid bounded
X2
y2
by
Z2n
is
5.
Show
that the volume bounded by the surface
and the planes
is
6.
Show
that the volume of a slice of the ellipsoid
x2
2
t/
z2
of a given
PROBLEMS.
bounded by the
863
parallel planes
+ my + nz = 8 lt
Ix + my + nz = 8 2
lx
,
where p
the central perpendicular upon a tangent plane parallel
is
to the faces of the
If
7.
A
slice.
be the area of a central section of an ellipsoid parallel to
the tangent plane at the elementary area $S, show that
the integration being taken over the surface of the ellipsoid.
Prove that over an
.8.
ellipsoid of semiaxes a,
^
p dS'= 4:7rabc,
- = 43
dS
p
dS
dS being an element
ca
ab\
--r-T-"\
--
/be
7r TT
\
\a
b
v'
]'
cj
++ 1\
1
= 8(1
of surface,
b, c,
and p the central perpendicular
upon the tangent plane.
Investigate also the value of
Apply the formula
9.
of
an
10.
I
ftsu>er~
3.
(Ix
t>
+ my + nz) dS
5o~7
*>
<<>-*$
to find the
volume
being the coordinates of any point on the
m, n the direction cosines of the normal there.
[COLLEGES a, 1881.}
ellipsoid, x, y, z
surface,
then
F= -
I
If
its
and
/,
the ellipsoid of semiaxes a, b, c be very nearly spherical,
is, to the first order (inclusive) of the small quantities,
area
represented by the difference of the axes
[TRINITY, 1891.]
^.
1
1.
Show
that a portion of a spherical surface (radius unity)
may
be bent into the surface of revolution defined by the equations
~,
K
=*\
J
and explain the geometrical theory, distinguishing the two cases
^
k
<
1
,
k>l.
[MATH. TRIPOS, 1887.]
CHAPTER
864'
12.
The curve z=f(x), y =
XXII.
revolves about the axis of
x,
and the
by the right cylinder ?/ = </> (x),
which is symmetrical with respect to the axis of x prove that the
cylinder cuts off from the first surface a portion the area of which
can be determined by the evaluation of the integral
surface thus formed
is
intersected
:
J
between proper
1 3.
Show
2
sphere x
limits.
[OXFORD
- c) 2 + y 2 = (a- c) 2
(x
that the cylinder
a portion of which the area
+ y2 + z 2 = a 2
Sa {a cos- 1 (c*or*) -c*(aa being supposed greater than c.
1 4.
Prove that the volume cut
jrabc
2
fa p
2
-4- V-^T
Show
2
{
(x
and the cylinder
is
+
II. P., 1888.]
from the paraboloid
b 2 q2
+
2r\ 2
^r 7/;
- 2)c 3
[OXFORD
to the evaluation of the
volume
of
an
[OXFORD
.
application of the formulae
(
[OXFORD
ii. P., 1902.]
+ y2 + c 2 ) 2 - 4cV } = cY
x 2 + y2 = c 2
(TT
By
c)*},
that the volume enclosed between the surface
z2
16.
cuts off from the
is
zpx + qy + r
by the plane
15.
off
II. P., 1888.]
F=^\pdS,
V=
II. P., 1886.]
I
zco&ydS
ellipsoid, establish the results
^
c
r
2)'
f
JJo
(See Art. 820 for the notation.)
[TODHUNTER, Functions of Laplace, Lam6 and
BERTRAND, Gale. Int., pages 424, 426.]
17.
Show
that the volume bounded by the surface
and the planes
is
Bessel,
[LAME.]
pages 216, 217
;
PROBLEMS.
865
A
18.
cavity is just large enough
revolution of a circular disc of radius
same radius
to
c,
allow of the complete
whose centre describes a
while the plane of the disc is constantly
to
a
fixed
and
plane,
parallel
perpendicular to that in which the
Show that the volume of the cavity is
centre moves.
circle of the
be a point without a sphere of radius a and centre
If
19.
c,
and r the distance
of
over the surface,
integrating
A
will be the results
lies
if
0,
show
C,
that,
n = 2.
.
e
if
27r-log&c+a
c
What
from
we have
c-a
.a,
and
of the sphere
any point
within the sphere
1
obtained by making the diameter 2a of a semiparallel to itself, the path of the centre being perpendicular to the initial plane of the semicircle, whilst the plane of the
20.
circle
surface
is
move
semicircle rotates
round the diameter; and when the plane has
the distance which the diameter has moved
moved through an angle
is c
sin
Prove that the volume
0.
3
jTi-a
21.
of the
2
Tr
+
to find the
volume
ff
[TRINITY, 1890.]
(dxdydz = (((jdudvdw
of the parallelepiped enclosed
r?
Prove that the area
is
= 0,
a<p
2
+ b 2y + c^z = 0,
-l)(.^
2
+
2
i/
)
= ^,
cut out by the surface
z
where a and
= a-
l
x2
+ lr
l
f
b are positive, is
|m(- ])(+J).
E.I.C.
by the planes
of that portion of the surface
(m
which
ca?.
is
Use the theorem
P=
22.
whole surface so generated
3
1
(03tTOEDn .p. >189(K]
CHAPTER
23.
is
Show
when
that
f(x)
XXII.
a slowly changing function,
is
approximately equal to
Prove that this formula may be used to calculate exactly the
volume cut from a hyperboloid of one sheet by parallel planes
[COLLEGES a, 1881.]
meeting it in elliptic sections.
24.
Prove that the volume included
the surface
z
z
2
(^ + yi + a v)2n+2( x +
x=
and the planes
Q,
a
Tra3
= oo
in the positive octant
2
)
y = a,
,
between
= a8w + 4/
y=
2n
2
<x>
1.3.5...(4n-3)
2+i W 2.4.6...(4w-2)
>
n being a positive integer.
25.
Show
that the area of that part of the sphere
by the cone tan
26.
Show
^coe*.
r=l, enclosed
is ,.
[COLLEOES
a>
18gL]
that the volume of the solid, the equation to the
surface of which
is
# + aX + 28xy + ftf =
2
4ff
^
3
Va^-82
is
2/*8?,
'
[COLLEGES, 1882.]
tangent plane at the vertex of a paraboloid two
whose axes are in the principal sections and
be
described
ellipses
27. If in the
proportional to their parameters, the cylinders whose bases are these
ellipses, and whose generators are parallel to the axis of the
paraboloid, will intercept on the surface a portion whose area is
proportional to the difference between the radii of curvature of
either of the principal sections at the points
where
bounding curve.
it
intersects the
[COLLEGES, 1892.]
th
any point vary as the w
the point from the faces of the
28. If the density of a tetrahedron at
power
of the
tetrahedron,
sum
of the distances of
show that the mass
_ 1:F
'
of the tetrahedron
^
1.2.3
p,
(r+l)(r + -2)(r + 3) ^(p, -p,)(Pl -p,)( Pl Pt )>
where Fis the volume p lt p 2 P 3 ,p 4 are the perpendiculars from the
;
,
corners upon the opposite faces, and
the volume.
Examine what happens
Jc
the density at the centroid of
in the case of a regular tetrahedron.
PROBLEMS.
29.
867
Find the volume contained between any two planes perpenand the surface whose equation is
dicular to the axis of x
(f + *
2 2
)
= (a 2 + px )f + (a' 2 + P'x)z\
[ST. JOHN'S, 1884.]
Show
that the mass contained between a paraboloid of revolution and a sphere, with centre at the vertex and diameter 2a, equal
30.
to the latus rectum of the paraboloid, where the density at any point
varies as the square of the latus rectum of the paraboloid containing
it
and having the same vertex and axis
where p
31.
is
as the
bounding paraboloid,
is
the density at the external surface of the paraboloid.
[COLLEGES 5, 1883.]
Find the volume between the surfaces
[COLLEGES
5,
1881.]
32. Prove that if a, b, c be any positive quantities in descending
order of magnitude, the solid angle of that part of the cone
axW + (If - cz2 )(x* +
which
lies
2
?/ )
=
on the positive side of the plane xy
4
i
snr 1
is
equal to
A\*
-4
A
[COLLEGES
jS,
1891.]
Prove that the volume common to a sphere and a circular
cylinder which touches it, and also passes through the centre, is
33.
1
9
- -
of the
volume
of the sphere.
[ST. JOHN'S, 1891.]
Also show that the sum of the two spherical caps cut off by the
cylinder forms ^
34.
A
of the area of the sphere.
sphere of radius a
is
cut by two diametral planes so as to
form a lune of angle a, which is itself cut in two by a plane inclined
at an angle ft to its edge and passing through one end of it, and
equally inclined to the two faces of the lune ; show that the volume
of the pointed part is
Q
asin
(
ft \
(2
+ cos 2 /?) tan" 1 (sin ft tan
~
)
+
-
sin/? cos'/? tan."
^
1+ sin*/?
ta.i2J*
[ST. JOHN'S, 1881.]
CHAPTER
868
Prove that the moment
35.
(bl
+ am + 2pnab)
2
about the axis of
2
2
{bl (a
is
+ 7b) + am
2
(7a +
the density being taken as unity.
If
36.
A+B+C=Q
z of the
cut off by the plane
+ my + nz =p,
Ix
2
of inertia
= ax2 + bif,
part of the paraboloid 2z
XXII.
b)
+ 2pnab(a + &)},
[MATH. TRIPOS, 1890.]
and the coordinate axes be rectangular,
prove that
,
B, C, D,
=
E,F$x,
2
y, z)
x (A\ B, C,
D,
E',
F\
2
x, y, z)
^ (A A' + BB' + CO + WD' + 2EE' + 2FF
}d*
),
where the integration extends over the whole surface of a sphere
unit radius whose centre is the origin of coordinates.
of
[COLLEGES, 1892.]
Also show that the unconditional result
37.
that
A flexible
e is
envelope
is
form
in the
of
is
an oblate spheroid, such
the part between
the eccentricity of a meridian section
:
two meridians, the planes of which are inclined to each other at the
angle 27r(l -e), is cut away, and the edges are then sewn together.
Prove that the meridian curve of the new surface is the "curve of
sines," and that the volume enclosed is changed in the ratio
Sire 2
A surface
:
8.
[ST.
JOHN'S, 1889.]
ABCD
being any rectangle in the plane
and AP, BQ, CR, DS being
drawn parallel to Oz to meet the surface in P, Q, R, S, the volume
of the solid ABCDPQRS is equal to the base ABCD, multiplied by
38.
of
x, y,
with
the arithmetic
is
is
such that
its sides parallel
mean
of
to Ox, Oy,
AP, BQ, CR, DS.
a hyperbolic paraboloid.
39.
Show
Prove that the surface
[MATH. TRIPOS, 1876.]
that the integral
2
e
I-
taken over the volume of the ellipsoid
n-abc
13
4-
.
(
6
+3e
)'
[COLLEGES, 1885.]
PROBLEMS.
869
Prove more generally that
rr
ix+my+nz
a*++c*dx dy dz
\\e
over the volume of the ellipsoid
tl
= ^.irabc (*
cosh k - smh &),
,
,
7 \
i
p
and
find the values of
((ex dxdydz;
]\(e x+ y +z dxdydz
x
\\\e +vdxdy dz-}
through the same space.
40.
On
curvature
everywhere
finite, rolls
of the envelope of the sphere is
envelope
V
a closed oval surface of volume
is
S'.
F+ a ^ + ^ _ y ^3
is
and surface
S,
whose
a sphere of radius a ; the surface
Prove that the volume of the
,
.
[MATH TRIPOS>
.
1886>]
41. Show that the volume of the pedal of an ellipsoid taken with
the centre as origin is less than that taken with regard to any other
and that the sum of the volumes of the pedals, taken with
origin
;
regard to the extremities of three semi-conjugate diameters,
times that taken with regard to the centre.
[MATH. TRIPOS,
42.
Show
moment
that the
ax
about the axis of x
2
43.
M
is
six
1887.]
of inertia of the ellipsoid
+ If + c$ +
2fyz + Igzx
+ 2hxy =
1
is
2
- 2$M(ca -g* + ab- A ) (abc + 2fgh a/
where
is
the mass of the ellipsoid.
Find the envelope
of the conies
bg*
-
cW)-\
[TRINITY, 1890.]
2
sec 3 0-//2 tan 3
= a 2 where
,
the variable parameter.
Show that in addition to certain lines
a.
it consists of a curve whose asymptotes are x=
Also, if the
the
and
area between the axis of
an asymptote,
corresponding
6
is
,
branch of the curve be A, and the volume generated by the revolution of this branch about the axis of x be V, prove that
[COLLEGES
44.
Show
that the value of
xyzdxdydz
0, 1890.]
CHAPTER
870
XXII.
taken throughout the positive octant of the ellipsoid
a-V + b- f + c~% =l
2
aWc
2
bc + ca + ab
+ c)(c + a)(a + b)'
ig
15
45.
2
(b
[OXFORD
Prove that the mass of a sphere of radius
any point
a,
Jc
P
where k
is
distant/ (>a) from the centre of the sphere,
47rfra
is
whose density at
A
a constant and
is
-j-p,
II. P., 1888.]
is
a fixed point
equal to
3
'
3
46.
[OXF.
/
Prove that the volume which
and the
lies
ellipsoid
where 0<a</?<^7r,
|a
P
1914.]
within the sphere
z2 sin 2 a cosec 2 /? + */ 2 cos 2 a sec 2 /? + z 2
47.
I. P.,
is
= a2
,
is
-
3
(7r
2/3
+ 2a sin 2/3 cosec 2a).
a point of abscissa x
x1
[Oxr.
I. P.,
1916.]
(>0) on the parabola
= 2ay,
z
= 0,
the area of the segment bounded by the arc OP and the
OP; the straight line PQ of length 2Sa is drawn
to
Oz.
The locus of Q being a curve which passes through
parallel
the origin, prove that
and So?
is
radius vector
OQ
(1) the length of the arc
is
z+
tf
3
2
/6a
(2) the cylindrical area bounded by the arcs OP,
PQ
straight line
48.
Show
that the
Show
two cylinders x
= k (where
+ a 2 )*/90a 3
z
/a?
+z
2
2
/c
=
[Oxr.
.
the cylinder y
= 2b(c- z)
l
49.
1916.]
+
2
2
/c
=
1
by
is
The sphere x + y + z2 = a?
z
x2
Prove that the ratio
I. P.,
& 2 <c 2 ), a rectangle of area
l**itojh.
2
the
and f = 2b(c-z)
that the volume cut off from the cylinder x^/a 2
2
OQ and
is
a 2 /45 + (3a 2 - 2a 2 ) (x2
intercept on the plane z
;
is
+y
2
[OXF.
intersected
P., 1917.]
by the cylinder
= az.
of the spherical area cut off
to the cylindrical area cut off
I.
by the sphere
7r-2:2.
by the cylinder
is
[OXF.
I. P.,
1915.]
PROBLEMS.
871
50. Integrate
51.
x,
y
;
52.
[OxF.
Find the value
p
of
I I
,-=
j j (a
.
+ x + y 2 )f
taken
all
being greater than unity.
+ y2 +
2 2
)
1915.]
over the plane
[Oxr.
Find the four points where any
intersects the surface (x2
I. P.,
I. P.,
1915.]
line parallel to the axis of z
= 4(ft 2 2 + a%2
).
Prove that the volume enclosed by that part of the surface which
3
is Y^
lies above the plane z =
[Oxr. II. P., 1915.]
.
53. If the coordinates of a point
x
a sin
on a certain surface be expressed as
z = a cos u + a cos v,
y = a sin v,
w,
prove that the area of the portion of the surface bounded by
is
1
~
C2r
=
x(
c
c
f~ i~i~f '}
L
'
~2r(2r-2)...2
[Oxr.
II. P., 1915.]
ANSWERS TO EXAMPLES AND PROBLEMS.
VOLUME
I.
CHAPTER
I
PAGE
..
ft
2
a
1.
3
,
g
5.
b -a
2
12.
3
2.
3
Gradient at # = 15, 36
20'
;
fa
2
7rA 3
4.
.
slope= 735.
Slope at 9'5
is
tan 2 a.
^,
oc
r\5
ydx=
I
17*4 square units.
PAGE
2.
1,
1,
15
x/2-1.
1,
PAGE
.,
2
3.
^7=
--5
>i
|rr6
6.
ths
Using paper ruled to 10 and
4.
As
Harmonic
m -f+3
.
Mass a2
.
.
this should
be
j,
we have
3'141592...
,
the
showing
per cent.
'05
PAGE
1.
e-l.
-,
-
= 3 '1400, the true value being
approximation
an error of about
2,
5 inches to represent unity on each of
the axes, the area =78500.
TT
|, |, log
25.
a.
1.
,
3.
oscillation.
2.
/
28.
y dx.
4.
2
%Tra
b.
5.
2;raA 2
.
7.
c'/t.
JXO
10.
Mean by
True
=
trapezoidal rule with unit increments 2378.
result
= 23*026...
.
(Unit increments
for a very exact result.)
872
are,
however, too large
ANSWERS TO EXAMPLES AND PROBLEMS.
13.
About 141,550
14.
(1)
In.
\
= J/^
~
M=
17.
About 213
if
'
J
= density at the
po
tons.
at
10
Mom.
edge.
20.
ordinates
Taking
# = %a, V where M= mass,
M m ^T***'
as length.
(2)
I
;
2
15.
25.
cubic yards.
x=%a,
Mom.
873
=
In.
'- Ma?.
13,863 foot-lbs., 10,574 foot-lbs.
and four
intervals
figure
trapezoidal rule gave '2501 TT, the true value being
tables,
the
.
29.
A= -^b-cU
where
= 25;r
B=26
J
2
-cn
100 + 257T.
C= -
30.
True values
35.
-r
36.
Binomial Expansion to 3 terms gives '1204, q.p.
linear inch, the trapezoidal rule gave
Graphically with
When this was corrected for curvature of the arcs by the
'1178.
a?c
+ (b
(1)
and
(2)
a)ac + j:c(b af
59
33.
c.c.,
q.p.
cubic inches, 3438'3 cubic inches.
^=1
approximate addition of small squares, the approximation was
1203.
40.
8465-7
42.
The true value
43.
When
44.
Q^at + ^-c^
t is
large
is
This will appear
^.
V
/ becomes
-^
q.p.
later.
and Q becomes
V - VL
t
-7
-^
V=aR + bL + (bR-2cL)t-cRt 2
.
.3
2i
45.
Perimeter = 30*1026 cm.,
41.
In the 'Otto Cycle' of operations there
revolutions.
About 16
46.
Weddle's rule gives - 1*08873
48.
5 tV miles.
;
CHAPTER
PAGE
x,
,
x
one explosion for two
true value -1*08878.
53.
,
is
H.P.
7
-
a
*821, q.p.
II.
51.
i
-
2
&
,
x
ANSWERS TO EXAMPLES AND PROBLEMS.
874
3.
ac
a-x
'
,
p-l
a+x
5.
f. 2^ = 1-894...,
7.
-(7 + log
9.
4001oge 2.
|log.
I (5* -3*),
4).
The integration
6.
832421J.
8.
In 5 seconds at a distance of 25
is
that of finding the
feet.
work done
in
allowing a gas to expand according to Boyle's law from #=10 to
If p. and v be in Ibs.-wt. per sq. foot and in cubic feet
v==20.
respectively, the result is in foot-lbs.
10.
The portions are
8^, -J, ^, -^g, 8J.
below the
12.
log(^ + ^), ilo g sin2^-,
13.
logtair^,
14.
log log x, log log log x,
1
1
-
log cosh
a?,
53.
^2 +a2 ), 1 log (x* + a 2 ) + tan'
2
.
o.
-.
4.
c
0.
6.
#-.
---n.r
,
.
,
,
,
x
,
1
~
,
X
,
C
2'
,
log tan X, log sin
i
X
1
2
2
1
-,
4
,
,
#
1
i
,
^
tan-'
sm-i-, smh-i-, cosh-'-,
3
3,
a sec
and
(sin~
PAGE
sin x
^7+ -
alternately above
.r-axis.
.
S
,
x
---1
X ~
i-s
a ** c
2
'
cosec x.
1
3+x
1
.
,
,
x
1
,
x-3
-
ANSWERS TO EXAMPLES AND PROBLEMS.
7.
- % cosec a #,
(i)
(ii)
log tan x,
(iii)
(v)
8.
(i)
logtan-^
9.
(i)
log,
11.
(i)
i(*-l),
(ii)
12.
(i)
1,
(ii)
(iii)
(i)
i,
(ii)x/2-l, (iii)^,
13.
14.
,
Tn
+
(i)
1
__
^y2
| ("*-!),
("i)
-
_+
vH
+
sinh # + sin #.
(iv)
^,
+
y+j+
(iii)
(iii)
2
-f,
(iv)
f^n
...
|.
1
M
+ ^_f
(iv)
+
+ a" log (*-
+
+
56.
(a 4- & + c)
kg*
(1)
r esult
)
PAGE
1.
n+1
A-
(")
Cl
7***
-j^jp
(ii)
875
abc
(5)
(4)
^
i
(6)
(9)
(12) tan
?
?
tan
3
.r
(13) sec
b cosec #.
+
(15)
~
c
-cosecjt' + logsina;.
(11)
x+ log sec x.
(14) a sec x
(16)
-c
(10)
-cot|.
(8)
lo S tana;i
#+ log sec #.
-2(cosec# + sec.r).
-
1
(17) tan" log .r
(18) sin log x.
2
f+^+i +|
18.
22.
?,
of a mile.
^=the
with
19.
ordinate
OK:
?/
&a?b
PQ
;
;
about 9
l).
(20)
^
feet.
20.
-= -ax, -^ = a.v-by.
~ = tangent
2
of angle the tangent at
Q makes
= asec2 -.
a
x-h
h
23.
y
24.
Approx. value given by formula '122422.
26.
-V-i 6 "** e
4
!
*
Jo
a
r-^^T-,e~
(n
1)
!
c?a.
27.
True value '122416.
True value
of integral
=
?r.
ANSWERS TO EXAMPLES AND PROBLEMS.
876
y
28.
29.
(l)/v>!log^
97-25 units.
,
where z=
30
sin
37.
38.
.
x
W
sin
3
l-x
2
/(o?)=l-^
42.
CHAPTER
III.
PAGE 75
1.
(i)
log(l-Kr>),
(ii )
tan-U,
\
6
(e*
~r"
Tra 2
_
/>N
2.
(0
4.
sm-
,...
(u)
-^i
(vi)
Jtan
1
(vii)
.r,
ira 2
(i)
-
(i)
isec- ^
9.
(i)
***,
(n)
I/
1
5tan^^.
2
(ii)
,
?njp.
....
^,
5.
=
7.
gtan^.r.
x)
8.
tanh
Wl
a3
,. x
3.
-3-
6\
+ -).
-(aa:
c
1
(iv) tan-'
,
3
,!/
1
\
6.
),
/
1
(iii)
->
-isech- 1 ^'2
Tra4
1 g-
_=
^
.
(iii)
,
\ 2 " fl
-i
i
(iv)7
( V11 )
sin^t^,
jc + ex
bc-ae
~
2(6
2
-a2
(v)
le
1
.
3
+ 2^
^a?+ |rin-> ??,
1.
I
i
'a
)
PAGE
3+.r
( vi )
(v) e-'
(iv) c*<0,
1,
tan-i*
'a,
98.
x- 2
1
.
3^-
-2
v/3^5 - 5 v/3 cosh- ',
ANSWERS TO EXAMPLES AND PROBLEMS.
2.
2 cosh- 1
,,
,
.
sinh" 1
2 sin- 1
^|,
(a?
-yf
i
2 sinli- 1
x+a VA-9 + 2cu? - a2
i
- ,^
1),
2
sin" 1
|,
,;r+a
x ----
,
.
^coslr
2
_
,
877
^J-,
.
u
4.
O
_
2
1
,
15
.
4
with a similar result
f,
\.^ + 4*+5 +
7.
-
8.
/-!
,#
v.^-a-^ + acosh- 1 -,
I
r,
,a?
a sin" 1
I log
tan
(*+^)
-
1-9
1
va
6!
f, llogta(* + )
log tan
-
sinh- 1 (#+2),
ft
9.
if
1
o
*,
+ 4ar+5 +
a2
.
,#
9
.
Trsin- 1
X-
CK
log
tan x.
| log
ia
log (c sin
* + rf cos
^
0(1- vl-sm ^)"}.
2
13.
log {cosec
2,
^sin- -
PAGE
1
1
,
Trft
2
.
3.
99.
x/e^^Ta? -f a log (Ve* + a +
.
sin" 1
3
'
ANSWERS TO EXAMPLES AND PROBLEMS.
878
4
,2.r
(l)sm
+3
.12-*
1
(U)COS
;
:
(iii)
(iv)
1.
.
7.
V^2 + 2. - 1 - 2 cosh-
9.
10.
Mass =
11.
3C
13.
15.
log
n+3
-ga
x* + 2# + 5
(v) 3
+ 1) Ve-^ + e-^ 1) - f sinh-
2
1 (2e-
7T
;
_
log
(1)
1
,
where density = krn and a
_
2
x
(i)
^a2 + ^ y3 + const
is
-
-1 v
.
,
.
in)
where
.--
cosh' 1
e
),
sin-7=
v c
,
i
1
vo-
^=
~1
ac
(c-
VPN
ve
),
.
,
^-asi
,, 2
9
1
-psinh"
ac
v'e
where
(iv)
'
a+c
be
+
ve
,
and a modification (Art.
77)
^=
if
(9
Te
>ac, a + ),
<9
a
where
(6
in
# = c tan 2 + 26ta
-\
upon BC.
ati
where
/\
^4
-
= sinh
-
the radius.
2
va<
if
1
a being 5(7 and jt? the perpendicular from
.
+ sinn" 1
+c
ANSWERS TO EXAMPLES AND PROBLEMS.
^ [3 sin- -a -
(i)
H
<*>
Vsrs?
provided
if
17.
o?
v/<^T2
,.,
(ii)
^^ ?
and
89, 17
-
>a-cj
a sin
1
log
2ava 2 -c 2
-
==
log
&
tan"1 A//c+# )v;
a>c, -
_
d
2
(
tan-2-,--^
do \^J a - c
-
- \/a2 - c2
<
;
+ ^)
--
/c
,
1
A/
J-.
*a-c)
,
where
-,
J
asin<) + Va 2 -c2
.
,
(a<c)
^^cos"
,c
1
-;
tan"
a<c,
(ij-^sinh-
1
*
I
- -
\
x
--
2
.
-^
5.
(Hi)
r\
^;
\
(iij-^sin/^k
^.
1
*
y
/vZ
1
CHAPTER
PAGE
(^
2
(-
30.
/v,2
smacosa
113.
4. 3
.2^+5.4.3.
2.1),
+ 2) cosh # - 2# sinh x.
--
-
...:\Ismisja
5.4^.5.4.3.2*\
--2^
H
.
^sm^+cosjr
t
IV.
.
x sinh x - cosh #,
2.
;
]
VHTW cot ') where
i,v/
-- (>c);
tan- 1 'V
f
^-
(i)
*
)
be positive, with a modification (Art.
.
da\*J a - c
26.
(
(ii)(3aC -2& 2 );
(i)48;
-
23.
tan "
2
negative.
oo
22.
ft0
(2*2 + 3a
1
16.
879
,
.,.
.
4 3 9 1\
5.4.3.2.1
.3^2
2
fi
.
cos 2#.
x
/cos
2^7
~8 V"n~~
3.
cos4A'
-^e*sin(2^-tan-
1
V5
1
-
1
cos_6\
3*
e
-=
sin (2# - tan"
-f
*
2),
-
1
fj
)
fa
^
2v5
<?*
"
/sin_2^
~2~ "^"^TeV"!
2
cos (2^
sin (4^
sin 4.r
sin
x\
"^~ "s "/
2
-tan- 1 2),
- tan" 1
f.
),
18)
ANSWERS TO EXAMPLES AND PROBLEMS.
880
#4
xn ^
tf>
,
71+1
sn
5.
&ax
-r 1
-======
-
-
4
[
;
fc-ftwo similar terms
- + etc. -etc. -etc.
6.
+ cos (p - q - 2r) x - 2 cos (^ +
Then apply
8 COS^JP cos
rule for
(jM7
e
I
2
cos (p
ax
-f q)
g')
^ - cos ( p + q + 2r) x - cos ( p + ? - 2r)
NX dx
cos
^7.
to each term.
x = 2 cos (p + q)x + 2 cos p - q) # + cos (p + q)x
x = ^A cos NX, say.
(
e*cos,p-taii-
=
Integral 2 A
Then
JOr* -8);
~.
7.
TT;
8.
^ sin"1 x -H Vl - ^2
8.r
4
-3
.
^
sin" 1 x
7
;
^2.^ + 3
-
1.
e*(x*
4
.
5.r
- 6
.
5
.
7
\x \
- .*-
1
;
**!"
/r-
PAGE
- 6^ 5 + 6
-
4^ + 6
114.
.
5
.
4 3^ 2
.
-6.5.4. 3.2^ + 6.5.4.3.2.1),
1
4^3 + 5
4 3 2#) cosh
2
23
V2
- (S^4 + 5
.
4
2
9
2.
3^2
i(TT
3.
7T
5
-
ftV
*);
l28
37r2
3
+ ^2-'
.
3.r2
+5
.
4 3
.
.
2
.
1)
sinh x,
25
cosh2.^5.r*
^
.
5.4.
5.4*?
_
12
.r
-
T4
2i
^4++
~-3
2
2
V 2
3
4'
,
5.4.3.r2
24
5.4.3.2.1N
2
/
ANSWERS TO EXAMPLES AND PROBLEMS.
5.
t
1.
PAGE 130.
(2^-sin2^)/4; (cos 3^ 9 cos a?)/ 12 or
Zx - 8 sin 2.r + sin 4#)/32
(1
881
;
/
Z* \
_ COS_tf +
O
^ _ 10 cQg
cog
O
\
_ cog ^
Qr
C0g3 ^
,
/
_
,5
O
4
cos
cos 9.r
/
1
or -cos x+
cos
7^7
5^
cos 3.r
COS 7 ^
4^o-- 6 o^+4----y
.
COS 3 ^?
_COS 5 ^7
.
COS 9 A'
;
/
or
COS3 ^7
-
6
cos9 .r
sin 9 .r
sin 7 ^?
sin 8^7
sin
6^
sm
.
-^
+s
1
4.
(7r-2)/8;
1
'
"2 cos
|"2
~4L
~
(157r
cos (a + 26) ^?
+ 44)/192.
cos (a
- 1 sin 4 ^7 +
sin 6 #
cos (n
(i)
(iii)
.r
cos" 1 x - V 1
.- - tan"
1
- x2
^^--
131.
#sec~ #-log(#+\;r 1
;
(ii)
;
(iv)
(v)
E.I.C.
- 2)
-7i
PAGE
2.
m
;
r2cos
~4L
- Zb) x~\
a-26
a + 26
sin 2 .r
1
43^2/120;
q^7
SK
x tan x + log cos x
i
;
1)
;
ANSWERS TO EXAMPLES AND PROBLEMS.
882
(vii)
-
(viii)
(ix)
+ v2 tan"
tan- 1 x + J (tan- 1 x)* -
+ #) tan-
(
sin -1 .r
x2
x tan" 1 \f 1
- <s/*
1
f=
J^ +
(x)
;
1
;
- 2a 2 cos" 1
)
(&*
grt
1
= cos
(sin- a?
(i)
- cot" 1 o)
)
;
-
x - V 1 - .r2 sin" 1
(ii)
;
+
(Hi);
8
_
(xi)
3.
(1+ a
log
where a?=si
2
J
where
/,
I>\
sn
>m bx - tan" - .
1.
I
I
a/
\
e
n
2.r
2
+2
ax
.
+ 6/92
,
(a?
+b
fer
1
- 2 tan" ,6\
,
a/
\
/
goa
-
A
sin (
l\
- sin bx - 3 tan" 1 (
\
2
}%
(iiOKk-i-^'
6.
4L2a + 6
(v) 3
X
(P sin 4.r
- ^ cos 4^), where
2
^=
.r
cos$
-7
2^cos2(^)
2.r
and
^ = tan-
sin 2^>
1
(4/log3),
_
-73
~72
r
2
2 sin 3<
"^
'
= 4 2 + (log 3)2
;
)
;
ANSWERS TO EXAMPLES AND PROBLEMS.
7.
~
(i)
cosh # tan
(iv)
8.
(i)
(ii)
;
tan
|
(iii)
;
;
(v)
- e* cot
-
log
(1
|
883
;
+e~ x )
;
.r
1
--tan-'
(in)
(iv)
.
(v)
2'
(ix)
'
where
Q
,.,
-^^-tan-1
^[".r
~
2r
^4 fLoTp
'
(ii)
.
sin f bx
\
- tan- 1
a
-
a/)
I
-
cos ((6
-^
+
a2
62
sin
-c)x- 2 tan-
ffta?
\
- 2 tan"
1
1
a
)
/
ANSWERS TO EXAMPLES AND PROBLEMS.
884
12.
13.
14.
-|.cot*0; -fcosfy.
15.
w",
v",
w"
1,'
1,'
1
...
-78343.
20.
+ (- 1)''-
22.
24
i
2?
-
33.
7
r wfe
^=^
29.
34.
2sm^-i0cos^0.
CHAPTER
PAGE
-tan-.
2
3.
^ log (#
5.
^-
6.
2x
7'
+ 4# +5) -tan-
ad-
1
(o?+2).
n^a*.
V.
143.
2.
4.
-
ad -be
tB
'
,
ad-bc
2(ad-bc)
I
10.
x tan"1
11.
-
PAGE
+ (cf - de) 2 + (eb - a/) 2
i
12-
8
z
bc)
tan
161.
1.
where 2
refers to a cyclic interchange of the letters
a
t
,
b lt
ANSWERS TO EXAMPLES AND PROBLEMS.
1V \
..
where 2
refers to a cyclic interchange of
riFlog{(*-5)3(#+15y>;
(viii)
1
24 (^ 2 -l
ax
3.
(i )
(iii)
(iv)
4.
(i)
(iii)
885
_L_
Q tan-i ^
V
5
2
5)
-= tan" ^
1
4&v/6 2 + 4ac
Ing
8
/.r-l\
5
a;
16^-1
32
g
U+l/
;
;
-
tan- 1
1
-
/MJ
2a 2^2 + 2ac + 6 2 + 6\/6 2 + 4ac
2-l
/f
,
(
l
or
-l)
- ~ tan-
2
-c2 )^-c(a2
~+(a
o
-
2
bi
,
x
a
tan~ l o; -
x
.
4^
3
a<i
(ix)
(6
m
+4ac>0).
1
_
2
-
tan- 1
-j=
-V
>
which
is
the same thing
;
ANSWERS TO EXAMPLES AND PROBLEMS.
886
1
.,
,
x^-ax+a?
tan-'-tan-'
(vii)'
\/3
N/3
or
6.
7
"!
tan
</3
"
(i)
-
(ii)
(iii)
-j
ita
-^x^-x^_a + b
L
8.
9.
W;
(H
10.
"**
1_
ANSWERS TO EXAMPLES AND PROBLEMS.
14.
887
(i)
(ii)
(iv)
x
-
[log (x
+ 1 - cos ^ log ( xz - %ax cos + a2
^
J
)
O
o
/
- cos
z 2ax cos
log ( x
o
\
.
asm
+
fcot
p
36v 5 L
JP
\
tan~
+ 2 sin ^
o
2
o
)
/
28
,
- a cos 7T
5
J
.
asm
7T
o
37T
5
-
- log cot |
I,
= sec"
where
^-1
5
1
x-.x
cosec ^
~+a
5
590,
5
1
5
5
.
1
^^
\/3
;
23
(iii)
19.
20.
- tan"
+ v/cot ^).
1
'a^-1};
where
ising
risng
.
'
[4 cos^
"
5 sin
sn
8
.r
3
3,
3
\/2 + l
16 10g 1-1
l+s
.
ioM + 16 10g ra
1
05
Ti-1) a (x- a)
n ~l
2
(7i-2) a (.r-a)"-
_
a""1
^-a
2
an
ANSWERS TO EXAMPLES AND PROBLEMS.
888
26.
= 2m,
n be even,
If
mC
m
m
(a-b)
m(x-a)
m-1
If
7i
30.
on
-
45.
Le
1
be odd,
=2m+l,
Q, Q'
R, R' similar expressions obtained by a cyclic inter-
and
change of
;
letters,
CHAPTER
PAGE
(i)
[(ac + be)6+(bc-
Q
(iii)
VI.
200.
ae)log(csin ^ + ecos(9)]/(
3?r
aK-j- log ( + 6 cos ^), where
or
,.
(v)
,
... N
(
viii)
,.
1
.
(iv)
.
(x)
,
.
sin
a
^=p
.
cosh"1
2
,
,1+cosacos^; = -: 2 tanh"1 /,tan a,
- tan ^\
);
2
cos a + cos x
sin a
2/
\
,
cosh" 1
1
log tan I (x+ tan1);
- --
p=
S-v/lOcos^-tan^S)
.
1
(vi)
;
,
[ax + 6 log (a cos x.+ b sin *)]/(
2
,
+ 62
).
log (cos
6^
+ sin 6^)
ANSWERS TO EXAMPLES AND PROBLEMS.
2.
(i)
3.
x cos a + sin a cosh"1
flog 2;
(a>c);
(ii)
.
,
4
1
-
(iii)
;
(iv)
+ cosacos.r
cos a -f cos
#
-
[cosh-
(ii)
.....
5.
1
,
*
(m> 8
1(
j tanh
^.
/-v
l)
>=
/2
/
8.
5
4
-27
dx
sin e cos 6 log (1
9.
(i)
IA
/-\
a/2 sin a
,
C03h
5
_,
+ 4 cos x
4
b
=etc " by Art 173
;
(ii)
tanh~'(
/--\
n^i
(n) 7T/12
2
(iv) 7r(a
+
3
2
)/4a
TT
^
(i)
2Vtan ^
|
\
+ ^\
/sin a.
/-"\
;
a
b
bc-ad
1
(m) i
(v)
;
2 + 3 e2
vr/4.
2
'
'
2
16.
tan
(e
3
TT
'
where
+ tan 0) - 1 + 5 log sin
i
/-i
^4
+ cos ^7)
sin ^(1
9
1
8-**-|i.*
1-cos^-sm^
'
**'
;
a
17.
889
(i)
^tan-iQtan|)-|8 tanh-i(2tan|)
;
(ii)
TT
;
;
ANSWERS TO EXAMPLES AND PROBLEMS.
890
10
1
...
18.
(!)
- cosh-' (cos + sing);
(ii)
(iii)
19.
20.
cosec-1 (l
22.
+ sin2^).
-2>/l-sin#;
(i)
cosh
sin
24.
cos
26.
.
#
#+
(i)
(iii)
_1,loa
2
-sin^-\/2logtan(|+|);
cot -
.27
cos x
.
sin
or
25.
#
2^ tan" 1 ^7- log (1+^2 )
i
i
log log tan
x.
3x tan- 1 o;-|log(l
(ii)
2
1
27
-2\/l
(ii)
+ sec 6).
sec" 1 ( cos
coB
(iii)
23.
21.
^-ilog(l+^
J.^tan-
-
).
--
1
l-\/2 sin0
" - -=
l-sin0
T^
where x = tan 0.
-^
log
,
l+sm0 V2
,
,
l+\/2sm6l
28.
1-sin.r
1,
\/2 sin
1
^
8
sin
sin(^-a)
29
'
30
-
S- sin a
+
8a
.
%
<">
...
1
}
-^
---6^
4a2 -2a6-6 2
a2
2p
in 4 rj_
pLi^S taN-
3
unless w = l, 2 or
the integrations.
32.
-
2
,
lo
a+6
-^r
(m)
^t__
n -3 \__8^.rj_
-2ta M 2
J
(a + 6)
3,
when
6-2a
.....
;
2a
^p- + p
,
_j__2 \
(a + 6)"- /
a logarithmic term occurs from one of
ANSWERS TO EXAMPLES AND PROBLEMS.
43.
(i)
ie*{.*;
891
sin #+(.- 1) cos #};
(ii)
(iii)
44.
-j|
(sin
-
-4 sin 20 -12 cos 4 0), where
40
-
cos a
v
~.^~.
rr
sm (a -6) sin (a
3
-.
7
j
i
r
lOg Sin
^ -^- (#-)-# 2-^
sin
/
sinacos 2 a
\
;
(a
c)
=2tan~4 = 0'918...,
First integral
45.
--
= (2.Z + 1 )/\/3-
tan
Second integral = loge 3
=1-0986...
CHAPTER
PAGE
-
6) sin (a
1
- o):.
,
*
,
VII.
221.
J
and
...
7
r\
[)
~
__
or
~
1
if
x
7 f
.r
11 f
,^7
3+ 12aL8a(a + 6.r4 2+ 8\4a( + 6jp4 +
.
12a(a + 6^)
where 7
,..
F7S = r
=
\"1
:t
)
/
r
^-j
of unlike sign
;
and
and
if a,
fc'
4
=
)
b be of like sign
and
j,
~ 136
3
-
if
s
be - ve=
-*"
J
J'
4
=?,
;
ANSWERS TO EXAMPLES AND PROBLEMS.
892
PAGE
4.
If
/w
,
denote
n
267.
given integral,
tjje
_
13
6.
With
a similar notation,
n~3
"-^-M 9 /T 2-
X
m/m = a?"-
(c)
1
(a?
~
+x
- (m -
1)
a 2 / wl _ 2 ;
- 2)/w_ 3
_
-l>
(27i- l)a
;
2
r
"
- (2w + 1) /jjn+i, where
74 =
e
z
.
r
24.42
H
12.
the integral = J2w+1
24.02"! cosjt'(acos^+2sin^')4-2.
(1)
~
/ = - xn cos .v 4- nxn l sin ^ - ;i(?i -
(2)
1.-**
i
1)
In _ 2
;
a sin j?sin ax + n cos,v cos ax
2
n(n
smh
2 rosh
2
m(m-l)(m-2)(77i-3)...3.2
'
2
2
2
m - 2)
2
}
. . .
(n
2
+3
'
2
)
w2
1
.
[
|.
ANSWERS TO EXAMPLES AND PROBLEMS.
18.
J5
3m + 3wi(3m
m(m
o>i
34.
,
4)
m(m
4ac
>
+
2)(3?/i
3>7i(3'/u
1)...2
v
Ifm 2 s +
T
m(m-l)
.+
2)
?i
,
a
v
=
^
1)...^
4a?
,
1
|~l
P^S^^
11
,
1
and
62
_.a?
1 A
4^6*
1.C-'
893
- cos
..
r-
.
) .
>4ac,
n'm^
-^
,
11
A-~l
mJ
;
2
or
=
and
cos
^[sec<Manh->
2<=
--p=,
where
Z>
2
where a = cfc4
;
< 4ac.
2 >/ac
If
^=4ao, the
2
fl^
r
a + bx
/a;
.
.>
2
.
integral
1
-j.-_ +
/
3=-7==(mtan~
2
+cx*
6
,^7\
.^7
wtan" 1 -),
1
-4ac\
tan- *
-
i
if
/
if
,
,
36.
,
(a) / n
i2=4oc
[BEBTBAND,
"- 1
= /7n_2
if
43.
44.
/n -2/n _1 + 7 _2 =-sin2(M-l)^,
49.
See Art. 202.
}l
*<8*-,r^cot^
CHAPTER
PAGE
VIII.
286.
42 < 4oc,
.
/. C., p.
36.]
ANSWERS TO EXAMPLES AND PROBLEMS
894
2.
u*-.,
(
-
(iii)
(iv)
3t
-cosech" 1 ^
(i)
(ii)
;
sinh~ l # + -psinh- 1 r
(iii)
- -1 sinh" 1 1 - #
;
;
1
-
.
\/2
(iv)
_
,
5
IQQ-
7.
\2cot6 + 3-l
sinh-'(-^
1
=
,
loar
sec
20).
8.
n
9.
...
(i)
.
.
am--1
,*Jx-b
1
2
^tatir
lr
o
6
v
(11)
/...,
(111)
1
/\/\
(iv) (a)
n
,
Ja-b
log
5
x
---smh"
i
,
1
-
;
'
(b<a),
,f
-
Jb-a
/7\
(o)
-
1
tan" 1
ANSWERS TO EXAMPLES AND PROBLEMS.
895
-
modification
p
< q.
2
-
13.
(i>
14.
(i)
15.
if
(i)
A2
,....
,
(m)
PAGE
1.
(i)
7
...
T
cosn -1
x
2.
(,)
c
6.
/-\
(i)
1
-V^2 -l - 2 cosh- # + v/3 cosh10
4
u
cosh"
^
-p
19
V3
i
1
(i)
1 j
^-
S1
3in
-i
r
2 cosec" 1 ( ^.7+
(viii)
;
-
1
^+ 2 j
{
- 9 smh.
19
S
with similar results for other
7
-sinh- 1 --
;
sinh-
-sinh- !^)
^
-
/
(iv)
1
(ii)
2tan~ 1 v
-i
-
(vii)
314.
(ii)
(111)
log
cases.
,
1
.
_
*
).
ANSWERS TO EXAMPLES AND PROBLEMS.
896
8.
(i)
1
cos
a
ft) (cos
cos
-
x cosh-
__
> cos/3
cos a
1
cos
cos a
P
cos
y
!
cos a
\
for the case cos a
1
cos tf+ cos a
/
,
y)
2
/
- cos
cos a
fi
- cos
y
or cos y, with modifications for other cases
1
N/sin(a-/?)sin(a-yj
x cosh"
,
1
tan x - cot a
J
cot
9.
-j-k
XI
*-.v
.
,
co*-
10
.
1
17
7
c
-a
w
*
ft
cot a
cot a
cot
cot a
y
,
.
10- 3^
1
cosh
F=
-
,3^-2
,
*
sinh
^-4
3\/10
2
,
.
S11
_!
\/
'
(c-b)(c-d) sj^a,
(V)
XO
*-T*SW
|
1
13
1
A'-l
--
I
-5# -
- --
,
3
~
A
_
/
x
(in)' -r- sinh
;
c~e
^
=r c
-
(i
1
13.
- cot /}
a
1
11.
cot a
1
(i)
o sin ^
-
1
^ tanh""
\/3
(ii)
1
/I
(
$\
-p tan
V^/3
tan~ 1 {^( v/l+^ + .^2 )^}
;
-
where
cos
),
(iii)
-7^ cosh"
.
if
62
>a 2
,
0=x2
;
*/
with other forms for other
cases.
- a*
1
tan 2.r
- cot y
/
;
ANSWERS TO EXAMPLES AND PEOBLEMS.
18.
20
g
-
sec- (cos x+ sec #)
-
1
21.
(i)
25.
!<!=
(ii)
;
.
-s3
30.
v/2
34.
35.
(ii)
(i)
sin $ -
-
log
-
/]
+ tan-
v/5
4
where ^?=cos 8
;
- tan
v/5
g
-i[tan0-21ogtan0+flog(tan0-l)+ilog(Un0 + l)
(ii)
41.
_ rt
8in-irin^;
(i)
..>
1
2
.
.
-=
(ii)
.
s,n- I
52.
^P
2
(ii)
1
-1
logtan^+|j
sin'
+
,
where
CHAPTER
PAGE
1.
(i)log,8;
3.
2;
4.
v/2/a;
5.
326.
l/s/2.
PAGE
.
353.
,.v
(1
3.
(i)
2(n-
E.I.C.
IX.
(Hi)|;
(ii)j5
^=^2 sin 0.
sin
SL
(
897
ANSWERS TO EXAMPLES AND PROBLEMS.
898
/\
/
mx sin4 .r dx =
.
/
(11)
'
cos
i
*
J
COS
TfLX d>
-5 -;
m^5
Sill 00
4?dx cos mx
d sin 2 ^
2
4^3
C
2
2
2
2
(m -4 )(m -2 )
lX
dx cosm^
+
4.3.2.1
2
2
2
2
(m -4 )(m -2 )
sin
mx
m
3^-10c + 9
*JpPn* + q*
-
/
.
\/P% 2 +
f
cos
2
(
V
P=a
where
z
yn +.
Q = j3-Sn*+.
and P', $' are the corresponding expressions, with Capitals instead
of Greek letters.
8.
6
9.
|,
A
1
12.
13.
2.
1.
6
15.
= i log
Principal Value
{-
tan
(|+|)}
32.
16.
2-log2-..
41.
=
Principal Value
^ log
^
.
= i log tan
(^ -|
^-
~.
[See Art. 347
(c).]
47.
sin
x
cos ^7 + ^7 sin
,
.
(v)
(vii)
-.
x
-\
v i i
(vi)
i
j
;
i
*
i v/ti
2
/Iosftane \
^
tan
\ log
'
I
-
where
CHAPTER
PAGE
12.
X.
377.
The integrand becomes oo at the limit @ = a, but remains real and
finite from 0=0 to 6 = a, and the rule of differentiation is not
established for this case.
culty disappears.
But putting
sin
-=sin| sin^,
the
diffi-
ANSWERS TO EXAMPLES AND PROBLEMS.
899
2X-1
14.
l
y = Ax ~*, where
16.
y = Ax*&-v>
17.
A straight
19.
The density
2-n
*2_
J
the height of the centroid being - of the height of the
segment.
through the origin.
line
at each point varies
inversely as the square of the
abscissa.
20.
y = (Ax + B}*, A,B,k being
38.
The
first
=
The second
|.
order of integration
becomes
tion
39.
The
21.
The
=-|.
not established
is
infinite at
_ yS.
jro dx
a 2 + x2
r(j2
constants.
any point
when
e-^co^3xdx = e-^
case reduces to
2.
3.
4.
XII.
415.
.
h
2
(a)c sinh-;
c
,.
e'^ dx.
Jo
PAGE
a2
the subject of integra-
of the range of integration.
is infinite.
CHAPTER
-
F(x} = AjJx.
rule for the reversal of the
Jo
1.
If
b
Trab
(b)e -l-,
2
/-=
(d}
~T~*t
()
(i)
a
(ii)
Flog^,
(l)^fa
^
b
ab
cos
~-2
ab.
(2)
;
A(logA-l) + l,
b
a
;
(f)\(e*-l\
Fsinwlog^;
l
2
,
(c)
Area bisected
in either case.
l
(2)1^ =
^
- A - A 2 + cd,
the four regions are
l
r
t-+A -A 2 -cd,
l
'
5.
42.
K.I.C.
6.
Sra*.
7.
3L2
(4-7r).
11.
a
For
ANSWERS TO EXAMPLES AND PROBLEMS.
900
13.
19.
(i)??;
2
(ii)
~
2
16.
.
(^+4 v/3-^J.
21.
PAGE
"
-
TTrt
2
^
+ ilog2-i.
17.
<*(J* + l*l\
428.
7TO 2
3.
5
2
.
4.
;
"
10.
-
5.
.
1
,
'
4
17
^-
+ 9^/3).
^(10^
o
a2
15.
n odd,
,,
g
Area
I
/I
+v
sin a 1
12.
-
sin
B
(fife TWln^
of
-
20.
19.
PAGE
429.
3.
8.
---
4.
18.
17.
19.
21.
- 26
(
a,
7-
ANSWERS TO EXAMPLES AND PROBLEMS.
-,
22.
+ 2)
23.
2
v/-i
Tr^-a^os-
30.
(7r
35.
A=*Jl&-lP-bGasr -, where
43.
^tan2
31.
.
ofj'
ab
tan"
2
52.
55.
Tra 2
+
,
a 2 sm a sm
Tr^v/a-v^)
.
c^
cos
,
^-,
^^3--.
^+
2
At
56.
.
Area
of loop of first
=
Area
of loop of second
=
54.
-a 2
2
where
1
^sinhc
[sinh 2c +
= 157
^r
^2 = 222
sq. cm., about,
about
sq. cm.,
2
(7T+1K
.
PAGE
XIII.
466.
Double the area swept out by the portion of the tangent intercepted
between the original curve and the first positive pedal.
-^*
3.
_t
a
16a6
7.
Tra(a
z
14.
.v
25.
|rt
31.
y
2
z
b).
|
I
-
\'
z
\-~{(h
-^
= (a z -y z )(y 2 -b
# + O^2
?rotan
The
13.
2
a)
20.
).
where
c is
,
c
and
is
least
if
h = a.
being the constant.
the diameter of the
34.
A
circle of radius
PAGE
(i)
2
},
circle.
-1 0i
vertex.
'.
o
+ a2
7TC
CHAPTER
1.
c].
the cusps.
2
CHAPTER
1.
2
^^--#
$1)
+ \b* cos
:
r
58.
^ cos $
#6sin($o
v
53.
.
= (p
sin((9 2
fl^
2
sin $! sin $ 2
6j cos
.
1
fi?
33.
-
^A
2
,:..
+ ^cosh-ip.
l
1
24.
901
Density = /wv/
;
(\\)
a
;
vra2
.
XIV.
478.
x=y = ^a\
(iii)
B=$Ma*.
ANSWERS TO EXAMPLES AND PROBLEMS.
902
2.
(i)
*=
(H)
_
4.
(i)
_
.
3 2
i
w
(ii)
;
_
157T-44
^=fa; y = a.
(iii)
6.
#=
.r=fa,
of Inertia about base =
Moment
(i)
from the vertex
to the base
2
AM* + AN 2
^(AL +
(ii)
),
where
A
^
,
h being
tlae
perpendicular
;
is
the angular point and L, M,
N
the mid-points of the sides.
1.
(a)
#= o
-
PAGE
-,
X
ct
484.
2a being the angle of the sector, and a the
,.
radms;
= --T, y=0,
2.
X'
3.
(6)
a being the diameter
;
If (p l5 ^), (p2 , g3 ), (p 3 , ?3 } be the coordinates of 4,
Co
6 29"^ 1
~ &i 9
-
"*"
x
W
Co
'
^>
2 C,
-
+3
7.
ir.
2(3>/3-ir)
,
C, viz.
ANSWERS TO EXAMPLES AND PROBLEMS.
C=-^J\
(ii)
c
,
PAGE
TT\
.,/,
(i)
.
903
7;ra 2/2
(i)
7.
Tr^^
a 2n2 -b2m 2
....
.an
^
ab
-^-^-tan-^+^W
(u)
2^(1--);
2.
492.
9
(ii)
;
_J
9>
a&.
15.
(i)
17.
Il7ra 2 /2 15 .3 12
25.
2(aV + 6V) 3 = (a 2 -6 2 ) 2 (a 2^2 -6) 2
(ii)
^=
21.
.
26.
.
irtf
CHAPTER XV.
PAGE
7.
.
.
4 sin
,
D~^
D
.
^1 sin 1> sin
7^
C
~
"n"-^
2
?
-^
521.
being the radius of the circumcircle.
CHAPTER
8a - 3.2;
XVI.
PAGE 533.
- Tan^2
^ J
3^:
1
.
2.
A
(cf.
Ex.
cycloid.
PAGE. 538.
.
(i)
(0 2
-0i)
f)
cos
(
v) a
(vi)
1
[V
(ii)
;
- cos
fi
+ 3 c os2 ^ os ^
\
;
^cosh-Hi
+ 6 cos 2 (9)]
1, p.
533)
;
ANSWERS TO EXAMPLES AND PROBLEMS.
904
PAGE
1.
(i)
(iii)
(v)
A
An
An
541.
(ii)
;
involute of a circle
equiangular spiral
+ 2 sin-
(vii)
2.
circle
(iv)
A cycloid
;
tractrix
;
;
= const.
+ 2-
1
catenary
(vi)
;
;
A
The
PAGE
546.
PAGE
570.
4a/ v/3.
5.
the area
(i)
the area
(ii)
;
;
2?r, according as the origin lies within or without the area,
there being one convolution about the pole or if there be n
convolutions, %mr.
or
(iii)
;
10.
Equiangular
15.
2 [3 v/3
17.
Epicycloid.
25.
i^Y<*+0)/4
1)],
5a.
L
y=<*^(B-C)IA,
i/>
2
-^
2
B = fsec + cos ^
Jl
,
i/>
,
J-Ai
rsin
circle.
4a.
19.
Ai
3
(7=1
Involute of a
13.
4a being the latus rectum.
^~~-
A = ftan
where
12.
spirals.
+ 3 ^2 + log (^2 +
3,
3sin^
^\~1*2 and
^ = ^-^r?-52 cos rr
-B2 lo g tan \4j + o
2
2/J^'
!/'
/TT
,
.
TT
,
)
(
LCOS^I/'
!/'
[^
a
29.
Area=/r(a
1
39.
s
2
+ 2ft
2
- mz C
30.
)'
^tra?.
sin
= 2a(sec3 i/'- 1).
If c
= 0,
the involute
CHAPTER
PAGE
5
A=
j
is
#
2
=
XVII.
600.
^F,, (mod. -7=) = 1*31102... square units.
v/2
\
s/2/
5T.]
ANSWERS TO EXAMPLES AND PROBLEMS.
PAGE
n
-;
905
636.
^ mod. ~.
2
in Diff. Gale.,
Art. 458,
4r
b<a,
in
j.u.
2*
?=11 or tanh" v
1
(ii)
v
-tanh- 1
where
^=
'
1
(v) cosh-
(vii)
(xi)
^
2tanh- 1 (jp+l) x/^;
tanh- 1
fU*
tanh
( vi )
;
/2
~X
(viii)
tanh- 1
(xii)
5
CHAPTER
XVIII.
PAGE 669
7/2
= 4^.27
being the parabola,
F the
const, of inversion,
and
(A,
0) the
pole.
12.
__^_^__^___
/=
V( 1+ cos v) (cosh
u - cos v)
.
S1T1
.
1
C S
2
cos -(cosh
17.
(i;
u
cos
v)''
1
/
%/
+ cos v
-
+ (i(XS h w
cosh u + 1
X^-^K
-
^-
2^7
cos v
+1
ANSWERS TO EXAMPLES AND PROBLEMS.
906
19.
a=
25.
(i)
- J cos"1
x cos a + sin a log sin (#- a).
24.
*?.
/^llog^+D-i^-^log
+ 10)/50;
[7] "=(7r+14log2
1
7
/=5
(n)
/sin 60
.
+
3 sin 40
+
,
- -- +
15 sin 20
CHAPTER
PAGE
,
10 #> where
,
= tan~1 #,
XIX.
723.
2
3.
i(r 2 -7^)
6.
Evolute of roulette of
the
cusp
a
is
four-cusped
Intrinsic equation of envelope of axis with notation
is
670,
s
See Art. 657.
25.
The
30.
s
rolling of a catenary
= ai/* - 3a sin
hypocycloid.
Ex. 2, Art.
= a sin 2 v / + 7 cos 2 v\
!
(5
20.
of
Q + |j +
!j.
upon a straight
line.
const.
CHAPTER XX.
PAGE
_
6.
a
A
Arc=-^
[
1.
z
2
772.
-Rz + J~2
Zz
-nl<
f)
where
R2 = 2(^/2+1), z = cos-,
and
is
the azimuthal angle of a
point on the curve.
CHAPTER XXI.
PAGE
2
a3
3.
2.
7T
6.
'
\y = &sin0/J'
.
.
v
.,
~^a
3
^f=7ra fsin B Ja 2 sin 2
5.
.
For surface from
the
790.
3/-axis,
+ 6 2^os 2
= 0!
to
7ra3 (3log2-2).
=
2,
revolution about
ANSWERS TO EXAMPLES AND PROBLEMS.
8.
10.
.
1L
14
Aboutaxis,
16<
'
- Ja^c)
27.
7ra 2 (37r- 4);
'
907
about base,
22
A
circular cylinder.
{a(c- 2a) Ja + c + (2a
29.
CHAPTER
PAGE
1.
In each case
A B
,
2.
8.
t\+
21.
^^
3
,
=
f
862.
where h = height of frustum and
the areas of the ends.
a being the radius of the sphere and
|brafo(-2 +r2
@V1 C*<)(X"*
/
+ B),
V=-(A+JAB
o
XXII.
T
,
where
+ -2j
X,
E the spherical excess.
9.
A
A
+ c 2 cosh Ja 2 + 6 2 + c2 - sinh v/
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