ORTHOGONAL POLYNOMIALS
FOR CURVE FITTING
by
James F . Price
A THESIS
submitted to the
OREGON STATE COLLEGE
in partial fullfillment of
the requirements for the
degree of
MASTER OF ARTS
June , 19 40
APPROIIED:
Redacted for Privacy
Professon
of Mathematlos
In Ch.arge of MaJon
Redacted for Privacy
Heed
of
Depantment
of
!{athematlcs
Redacted for Privacy
Chalrnan
of
School
aduate Cormlttee
Redacted for Privacy
0h.a1nnan
of Stato
Iege Gnaduate Councll
TABLE OF CONTENTS
Introduction
1
Legendre's Equation and Its Analog in Finite
Calculus
4
Properties of the Polynomials
11
Curve Fitting
20
Examples
27
Tables
34
Acknowledgment
The writer wishes to express his thanks
to Dr. W. E. Milne, Head of the Department of
Mathematics, who suggested this topic and di­
rected the work done on it.
ORTHOGONAL POLYNOMIALS FOR CURVE FITTING
INTRODUCTION
One of the statistical problems which has received
a great deal of attention from mathematicians for a good
many years is the problem of fitting a theoretical curve
to a set of observed data.
The usual procedure is to use
the method of least squares to determine a theoretical
equation in powers of x.
But when the theoretical curve
desired is of higher than second or third degree, the work
of determining the coefficients becomes very great.
For
a polynomial of degree k, k+l equations must be solved
simultaneously for the
k~l
coefficients.
Theoretically,
of course, these can be solved, but in practice k does not
have to be very large to discourage any but the most pa­
tient and industrious persons from attempting to solve them.
To simplify the work of determining a theoretical
equation, various types of orthogonal functions have been
introduced.
The function may be represented by a series
of sines or cosines, by a series of Legendre's polynomials,
or by a series of polynomials made up of factorials.
Tchebycheff first worked in this last field, and more
recently some improved forms of the ortho gonal polynomials
have been suggested--among which are Fisher's
~-polynomials,
theY-polynomials used by Sasuly, and Jordan's polynomials.
This thesis deals with polynomials which differ from those
of Jordan only by a simple multiplier.
2
The most important case is the one in which the
observed independent variates form an arithmetic pro­
gression .
points .
Suppose observations have been made at n+l
By a simple transformation these n+-1 points may
be transformed into the points 0 , 1 , 2 , 3 , ••... , n , and
it is in this form that the data must be , if it is to be
fitted to a regression curve consistinb of a series of
these finite orthogonal polynomials .
lf the points are
not eaually spaced , the work of this thesis does not
apply, (unless the data may be grouped in some manner so
that the resulting points are equally spaced . )
To work with these polynomials , some knowledge of the
calculus of finite differences is needed .
Here, the fac ­
torial function x(n).x (x-l)(x-2) • • ••• (x-n+l) takes the
place of the power function xn in the infinltesimal case .
The following formulas will be needed .
Yl
L_ f, {X) L\ f (X)
2
X=O
t
S-:0
lll'ltl
~
[ f, (x) fz (xlJo - L f2 (X+l) L1 f, (x).
X=O
stm)
(Y1 + I ) (1'Yl+1)
Yn+l
(1)
3
The first three of these f ormulas are standard and
The
may be found in nearly any book on finite calculus .
Assume
fourth formula may be proved by induction .
t
5(-w!)
(~+I) (W1+1)
(1')
m+J
S=O
Then
t
seW!)
I
( R+ I
)
(WJ-t-f)
(YYl+J)
S=o
+(k+ I )
(WI).
This reduces to the equation
L
k+l
SCm)
5::::0
Equation (1 1 ) is obvi ously true if k• O.
Therefore
it must be true for any positive integral value of k .
4
1.
LEGENDRE'S EQUATION AND ITS ANALOG
IN FINITE CALCULUS
The dii'i'erential equation
~ [(l-x2)~}m(m+l)y:O
is known as Legendre's equation.
Since it is a second order equation, it will in gen­
eral have two linearly independent solutions.
But at the
points x:-1 and x=l the equation degenerates into a first
order equation, so there will be only one solution possible
at these points.
If the substitution x:2s-l is made, these
critical points are transformed into the points s=O and
s=l, and Legendre ' s equation becomes
(2)
In the calculus of finite differences the equation
analogous to (2) is the difference equation
·f1tS[S-{"Yl+1)] fl/A(S-J)}- Wl(YVl+l) [;{(s} ==0.
l3)
This difference equation may be expressed in a some­
what different form.
By actually taking the differences
in (3), (and combining terms in the result), we obtain the
equation
(5+ 1)(11-s)U(stJ)+[2s(s-n)-)1+WI(YVI+tTI i{(s)+S{YJ+ l-s)~(s-1)=0.
.
(3')
5
In the general discussion of difference e quations
most books on the subject consider the equation expressed
in the form of (3 1 )
.
One method of solving the difference equation is to
assume the solution
00
Uls)
== \ A s<b)
L o
o=O
(4)
·
Under this assumption
So the difference equation (3) becomes
L {['1) 'b+1)oO
'VYI
(Wl+i)]A~ s1'b)_(n+1-()~z A'bs(~-' 1]=0.
~=0
In order that this equation be true for all values of
s the coefficient of s{q) must e oual zero for all values of
q . Thus
[9J (~+I) -
W1 (W\ + I)]
'rherefore
)
A.'b+l
AD - (YJ- b)( b+ I) z A'b
+I
0.
6
If we substitute for Aq in equation ( 4) , the result
is
U(s)
== Ao[ 1- VYl (WJ+I) .s. + (m-l)mCm+t)("Wl+z.) 5
Yl
2]
2
12
) _ ..
Y/ (Z)
If m were not an integer , the fore going would be
an infinite factorial series , the convergence of which
would have to be investigated if the series were to be
used .
If , however , m is an integer , the series (S ) termin­
ates , and the solution u ( s) is a polynomial .
lf m is a
positive integer ,
The coefficient Ao is an arbitrary constant and may be
chosen at pleasure .
If Ao is chosen equal to 1 , the value
of the dependent variable will be 1 when s : O.
this solution Pm , n ( s) .
PYVJ )1 (s)
)
Let us call
Then
(b)
7
The values of the first ten polynomials as gi ven by
this formula are:
Pl , n( s )=1- 2~ ,
n
P~ n(s)=l-l2s+30s( 2 )-20s( 3 ) ,
v'
=niT =-rn
n
n
n
8
+8 , 084s( 6 )- 51 , 480s( 7 )+12,870s( 8 ) ,
(7)
( 8)
n
n
n
--rsr
If in equation ( 6 ) the substitution s=nx is made , we
have the result
The limit of this
~uantity
as n ap p roaches infinity is
"W1
[
o=O
Hl'blm+
&f. I z. (Wl - q I
v.
v.
!
x'h
•
This is the polynomial solution of the differential
equation ( 2) .
So one could say that the polynomial solu­
tior. )f the differential equation ( 2) is merely a special
case _f solution ( 6) of the finite difference equation-­
the interval in the factorial terms ( which in general may
be taken at pleasure) has been changed from unity and made
9
to approach zero , and the limits between which the solution
is desired are taken as 0 and l.
The formula for Pm , n (s) may be exhibited in a form
analo gous to riodrigues 1 formula for Pm(s) in the infinite ­
From formula (6),
sima l case.
By summing m tlme s between the limit s 0 and s-l we
obtain the result
-mo
s -
I m,'Vl( )-
..
I
s("WI)-
Yn!
=(-t)
YYl!
(111-1}!
s(wt+l)
)1
+
. . . . . ..
l-1) "Wl s (Zm)
+ YYl! Yl'~)
s <1'11) [EWJ -(YHI)J~1 (rn)+ YVl [Wl-(11+ 1\lu~~ -l)(s- m)
"WI
)1 fm)
I
+..... + m . IM - Cn + 1)]
k!
(111-~)
(YYI-k)~
+ ...... + (s -YYI)
(WI)] •
(5- Y¥1)
lk)
(A)
The quantity in brackets may be shown to be e qual to
[s - (n+l)
J (m).
Set
10
where the A's are constants to be determined.
If s is
set equal tom in equation (b), it is found that
Ao =[ ¥Yl- {Y1+ 1)]
("»1)
.
The coefficient A1 is determined by
same equation.
A,
s=m~l
in the
Thus it is found that
Yn [
:=:
~etting
vn -(YI + 1)]
(Yl'\-1)
.
The general term is
A R= ( ~ ) [ Wl - (Yl+ 1)]
(m-k)
)
which is the coefficient of the
(k~l)-st
term of (A).
The
m-th sum of Pm,n(s) becomes then
(-I)
Yv!
s
(m)
[s- f11+ 1)]
h1! Yl (-wt)
(WI)
.
Therefore
-p
I VYl
,
>
(s) =
(- I )
-wJ
m! .Yl c~)
b WI { 5 (WJ) [ S - (Yl + I)] ("»'~)} •
(7)
11
_g. PROPERTIES OF THE POLYNOMI ALS
The polynomials Pm,n(s) have the orthogonal property.
Consider two of these polynomials Pm,n(s) and Ph,n(s) .
Each satisfies the difference equation (3), so we may
start with the equations
and
Multiply the first of these equations by Ph,n(s), the
second by Pm,n(s), and subtract.
the limits s=O and
~
Then if the sum between
is taken, the result is
L
Ph,Y>(slLl{s [s-1-n+Jl]L'I Pm,Y>(s-1)]
S=O
)1
- L_ f:n,Y> (s) Ll{S [S-(1mll Ll fh,.nls-1)]
5=0
-[YY!(Yn+l) -hi h+ 1)]
t
5=0
Pm,Y>(s) fh,,,lsl=
0.
12
The first two of these sums may b e evaluated by summation
by parts .
The resulting e uu s tion is
J:[_
-[~."'{sl{S [s- (YI+ 11~ )s-/)}
1)]
0
~ ,-, (s)fi~,n (s)
(st ,)(s-n)11
S=O
== [WI (YVJ +I)- h{h+ 1)]
L Pm,,
(s)
fh,--.. (s) .
s =-0
Upon substitution of the limits 0 and n +l, this equation
becomes
)')
[Yi'l (T'VI+I)- h(h+I)] L_ P.,,...., (s) Ph,YI {s)
= 0.
S-=-0
Therefore , if mf h ,
L
Yl
P.,.,,.,{s) Ph,.,{s)
= 0.
(8')
.S=O
This is the most important property of these polynomials .
Indeed , t h e polynomials Pm, n(s) may themselves be derived
directly from this property--not making use of the differe nce
e quation at all .
13
Assume polynomials
(C)
which satisfy the equations
11
~,11 (s) Pm,,(s):::: 0
[_
.S:.O
1
(k = 0, I, 2,3,. "·, YYH).
(d J
Actua lly inste ad of u s ing th ese equations , it is found
ea sier to start with the e quations
u~)
")')
L\
5==0
(s+k)
(YJ+ k) <~)
.
~
-
m,'rl
(s) == 0.
(e)
This is permissable , since (s+k)(k) may be expanded
(n+k) ( k )
in a series of polynomials Pm n ( s) , the degree of none of
which is greater than k .
'
If we started with k =O, 1 , 2 , 3 ,
etc ., we vmul d a lready h a ve used t he fact that
t
Pk- i , n( s) Pm , n ( s l =o
( where i =l , 2 , 3 ••• ,. , k) , and
S:;O
the only new rela tion obt a ined from (e) would be equation (d) .
Upon substituting the v a lue of Pm , n(s) according to
e quation ( c) into e quation ( e) , we obtain the r e sult
14
If this sum is actually taken, the resulting equation
will be
This reduces to
+A, + Az +·······+ AyY! =0,
R+3
R+YVl+ I
R+ I
R+2
I
{f)
There will be m equations of this form, (k=O,l,2,3,
••••• , m-1), and these equations may be solved for them
unknovm A1 s.
Consider the left hand side of (f) as a
function of k.
Put all of these terms over the common
denominator (k•m•l) (m+l).
I
R+ I
+_AI_+
k+-2
Thus,
.. ,... . + R+AW\
111-+ I
where Q(k) is a polynomial in k of degree not higher than
m.
Equation (f) requires that Q(k) must vanish at
k:O,l,2,3, ••••••• , m-1.
Therefore,
15
where C is a c onstant to be determined .
Substitute this
value for Q(k) in e quation (g) and multiply through by
the common denomin a tor to obt a in the e qu a tion
In this equation set k = -1, which will give the result
m! =(-I ) -m m{c .
Therefore , C= (-l) m.
Then in e auation (h) set k = - q-1 (where q is a positive
integer not gr eater than m) , and obtain the result
Therefore ,
16
Then the polynomials we assumed are
This formula is exactly the same as (6) which was
obt&ined as a solution of the dif1erence equation (3).
It will now be necessary to evaluate
~ Pm2, n{ s) .
S=O
Substituting for Pm , n(s) its value a ccording to (7) , we
have
The quantity in brackets is e qual to zero when the
limits n+l and zero are substituted for s .
The process
of summation by parts is repe a ted m time s , a nd the
following result is obtained .
17
It is obvious that changing the upper limit of the sum on
the right from s=n to s • n- m would not affect the value of
the expression (j) .
After making this ch8nge , set s 1 =s+m.
Then
r
("Wl)
(SHVI)
[5+-Yri-(YJ-t-lll
("WI)
t
=
I ('W!)
I
[S-(Y!t-1)]
5
("WI)
.
S'="'Wl
S-= 0
Summin6 by parts leads to the equation
t
s
1
(
il (WI)
s - Yl+l )'jl (1'11'1) ::::: 5 1 (WI+/) Q.s11+1
1l-rn)[
I
(
s'= )'Y1
m+'
·
]
)1+1
-m
rhe first term on the right becomes zero upon substituting
the limits s =m and s =n+l.
t
s
,("l'lll) [
sI
(
·)
llWl)
Y1+1~
1
Repeating this process m times
(-I} t'J1 m/
== ( )i
2
2Yn,
S'=m
=(-I}
"WI
m!
(zm)!
Lr
I
s,.m
2
(s +rn
1
)(Zl'n)
18
Therefore equation (j) becomes
(YI±m±J)
("'Wit-1)
(9)
This formula will greatly simplify the determination
of the coefficients of a regression equation fitted to a
group of n+l points by the method of least squares .
It
is also useful as a check for the values of Pm, n(s) as
calculated in Table 1.
The sum of the squares of the
numbers in any given column must be equal to the value as
calculated by (9).
,
Another formula which is of use when tables of Pm n(s)
are calculated is the formula for Pm , n(l) .
Substituting
s =l in formula (6), all terms vanish except when q• O and
q=l , and after simplification the formula becomes
~)r1 (i)
,vhen calculating a column of fie;ures in the table of Pm, n( s),
one will find immediately by means of this formula if mistakes
have been made in reducing the general e quation for Pm n(s) to
'
a simplified form for the special value of m and n represented
by that column in the table .
19
In order to calculate Pm, n ( s ) for values of m and n
higher than those given in table 1 , use may be made of the
following recursion formulas :
(Yn+ 1Xn- YYJ) ~+r)n(s) + (z. rn+ 1)(zs- Y1) Pm> Yl {s)
+ Y¥1 (Yl-t- m + 1) ~-~}
11 (s)
==D)
(I 0)
20
3.
CURVE FITTING
In this chapter the problem of fitting a regression
or trend curve to a set of n+l equally- spaced points by
the method of least squares will be considered .
Suppose
the independent variable is t and that the n+l independent
variates are t 0 , t1 , t 2 , ••••• , tn .
If the transformation
si=ti _to n is made , the points t i will be transformed into
tn- to
the points s 0 =o , s 1 =1 , s 2 . 2 , •.••••••• , sn• n . In this form
the data may be utilized in the work that follows .
If it is decided to use ·a curve of degree k , the
fitted value will be given by
(12.)
where the A ' s are constants to be determined in such a
q
manner that
21
is a minimum .
To insure this , differentiate partially
with respect to the Aq ' s , and obtain k+l equations of the
form
Because of the orthogonal property , all of the terms drop
out except the one containing Am .
So each of the k+l
equations reduces to an equation of the form
Making use of equation ( 9 ), it follow s that
{
)
A.. -Zm+IY1
- ln + fn +J
(f1rl)
)'»!+')
I 'I
t_ Pl'n
s
s /1.1 Is).
\]
l1 { )
>
(13)
=0
This sum is not difficult to evaluate , although if n were
very large , it would of course involve c onsiderable
arithmetic computation .
Another of the advantages of these polynomials
is brought out in equation (13 ).
Supp ose one has cal­
culated the coefficients for & curve of degree k .
If it
22
is then decided that a curve of higher degree would be
better, only the coefficients Ai where i ) k would need to
be calculated, since the first
k~l
coefficients would be
the srume as they were in the k-th degree equation, because
(as equation (13) shows), the value of
Am
is entirely
independent of k.
Very often it would be useful to know how much vari­
ation there is between the observed data and the fitted
curve.
The mean square deviation of the observed from the
theoretical values is the measure usually considered.
square root of this quantity is called the "mean
and is denoted
by o-~
The
er~or 11
2
•
Algebraically CJh is given by the
equat~· == n~l cr~(s)- ~At~,)1(sf
This may be written in the
Uk
2
for~
=~~ ~'\s)-2 ~
+
Since
a-;
2.
J
= )1+1
L
s~o ~ (s)
2
A'bt ~
(s)
f1,,-.,(s)
t. t ~~'"(s)].
Ap2
t_
-~ Ao s~o ~+~,
R
2
2.
(s)
.
.
23
is denoted by R
q, n ,
If
(It)
T&ble 3 contains values of Rq , n to aid in the compu­
tation of
a-k ,
If it is known ahead of time what degree curve is
desired , the theoretical curve may be plotted without ever
actually computing the numerical values of the coefficients
Am·
Substituting the value of Am given by (13) into equa­
tion (12) leads to the result
If the value of y l at some point x is desired , we set
s =x in the final factor of the foregoing equation and
obtain for yl(x) the value
1
fld (X}
\J
~
~YI(s)fV1(sJ~
+rYf'Yl+l) L
' \1
==t{(2.lli±J)m
)1'[ =-0
(Yl +
{X),
n(m)
s ::.0
.
')1
24
Since Pm n ( x ) does not depend on s , and y ( s) does not
'
depend on m, the summation 'signs may be rearrange d as
follow s :
If we set
KL
(s X)=
K ))1
)
t
11'1:::.0
(Zm+ t) YltYlll)
(11+ )'yj+, rWI+I)
P-m Yl(s) Pvn
)
t1
)
(x)
7
(15)
.
then
~~(X)=
t
~ (s) Kk,
/I b)
11 (s,x).
Table 2 contains values of Kk n ( s , x) .
ln the actual
'
computation of these tables a formula analogous to that of
Christoffel in the infinitesimal case was used .
equation (10) through by Pm, n ( x )
(n+m+l )
Multiplying
N(m) leads to the result
(m+l)
25
Interchanging the variables s and x gives the equation
(
)
(Wi+l)
m± 1 Yl (
(¥1+ YVl +I)
"WI+ I
) ~+' 7 Y1 (x) F)'\11
+ (YlWl+YlYYll¥Y1)
Pm )
yWJ>
~ubtracting
11
(s)
J'1
(s)
~-I
)II
(s)
>
n (X) •
1
we ge t the e ouation
Z(X -sXz 'vYl+l)
n(rn)
Pm
(Yl+Yl'l+t)'"YY!i-1)
{Yn± 1) Yl
(YVJ+t)
(Yl + YYI+,y"Yrl+l)
- m Yl
(WI)
. (YJ+vn)I"WI)
"\11
'
(X)
Pm
==
1
[~+,)J'l(s) ~,Yllx} -~),(s) "Fm+,),(x)J
[Ph1
11 (s)
>
~-' Yl(x}- P'M-1 11(.s) PYvl1'1(x)].
'
)
>
26
Sum between the limits m=O and m=k to obtain the result
Equation (15) becomes
Of course this formula is of no use for cases when
s=x.
In this case formula (15) is used .
It becomes
(Is')
27
4.
EXAMPLES
The foregoing process may perhaps be clarified by some
actual problems.
Consider the data in Table A showing the
dielectric strength of varnished cloth insulators consist­
ing of various numbers of layers.
25°
The data was taken at
c ··*
No. of Point Puncture
Layers No . Voltage
s
y ( s)
1
2
3
4
5
6
0
1
2
3
4
5
6
7
8
9
7
8
9
10
Totals
Table A
P1 ,q(s)
times
y2 ( s)
y ( s)
P 2 ,q(s)
P3 ,q(s)
times
times
y (s)
y (s)
7.991
12.505
16.928
20.496
23 .638
26.718
29.890
32.452
35.136
37.515
63.856
156.375
286 .557
420.086
558.755
713.852
893.412
1053.132
1234.538
1407.375
7.991
9.727
9.404
6.832
2.626
- 2 .969
- 9.963
-18.029
-27.328
-37.515
7.991
4.168
-2.821
-10. 248
-15.759
-17.812
-14. 945
-5.409
11.712
37.515
7.991
-4.168
-14.105
-15.125
-6.754
7.634
22.062
27 .045
11.712
-37.515
243.268
6787.938
-59.224
-5.608
-1. 223
The column headed npoint numberu was inserted so that
the theory as developed in the last chapter could be
applied directly.
It would seem that the dielectric
strength should be approximately directly proportional
-lf-
Peek, F. W., Dielectric Phenomena in !!.!gh Voltage Engi­
neering, McGraw- Hill Company, New York, 1929, pp. 247.
28
tc the number of layers of the insulator , and that the
regression equations would be a straight line .
According
to equation (13)
~ (s)
Ao ==- n+-1
I
rL
5=0
and
A=
~
(
3 )1
)(2.)
YJ+2
rL
S-=-0
"F.1
>Y\
(s)
~(s)=_£2..f-54.2Zi/l::=
-Ji/.53b8.
II 0 L
j
The equation of a straight line fitte d to the data in
Table A by means of the me thod of least squares will be then
~
I
= 2'1.32&8 -/1.53b8
~,q (s).
According to equation (1 4 ) , · the square of the mean error is
CJ,z
= Jb-
f: ~z(s)
_,
Ai R'h,q
=b78. 7'13 8 -5'1 J. 7"132 - .t!?[
2/1.318&] = 0. '1078)
29
and
a-; =0.'153.
This value is rather large , and as a ma tter of fact , a
straight line does not adequately fit the data .
Due
probably to the non -homogeneous structure of the insula­
tor , the dielectric strength is not directly proportional
to the number of layers .
will be necessary .
A polynomial of higher degree
According to equation (13) the value of
A2 will be
t1
A2 =
5n <zl \
("Y1+3)(3)
k
Pz -n(s) 1/d (s) == 3bO(_s:bDff).::::- 1.52'15:
,
\J
13zo
I
The fitted parabolic curve is
By looking at equation (1 4 ) , it may be seen that
In the problem being considered this becomes
30
and
The value of A3 as given b y (13) will be
A3 = Z Yl 13l
(Y1+tf) £4)
~
~
~
)
11
(s)(l,j(s)=lfl(-l.i'.23)=0.2SI'-f.
\1
715
The fitted cubic curve is
~ =Ztf. 32.h8 -l'f.5"3b8 ~~{s)-1.5Z'J5" Pz.~ls)-0.25"!'/J;,~!s).
1
By using (1 4 ) again we find that
and the mean error is
It will be not ed that adding to the degree of the
equation always makes the fitted curve come closer to the
given points.
nomial
~ay
As a matter of fact, an n-th degree poly­
be made to fit n+l points perfe ctly .
is usually not desired .
But this
A curve that goes through every
point has been afrected by every unusual condition or
experimental error of each individual point .
The effect of
31
these errors is minimized by using a smoother curve (one
of lower degree).
In the problem above probably the pa­
rabola would be the best curve to use.
CJ3
is not a great deal smaller.
Cf2
is small, and
If we used the cubic
equation, the deviation of the cubic curve from the pa­
rabola would surely be found to be nnot significant. n-!fThe follo wing example will illustrate the use of
Table 2.
In this problem the independent variable is the
diameter of a 16-foot log, and the dependent variable is
the volume of finished lumber that can be obtained from
the log (measured in board feet).
Table B.
The data is given in
Since the volume of a log of given length is
proportional to the square of the diameter, it would seem
that a parabolic curve should fit the data, unless other
factors (such as methods of cutting, etc.) have an unusual
effect not evident on the surface of the problem.
*Rider, Paul, Statistical Methods, John Wiley and Sons,
New York, 1939, pp. 124-125.
32
Table B·l}
Volume of 16-foot Logs According to Scribner Log ·Rule
Diameter
in inches
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
Point
Number
Observed
Volume
Theoretical
Volume
s
y ( s)
y '(x)
0
18
32
54
79
114
159
213
280
334
404
500
582
657
736
800
923
1068
1204
12.0
30.3
54.8
85.7
122.9
166.4
216.2
272.3
334.7
403.4
478.5
559.8
647.4
741.4
841.6
948.2
1061.1
1180.2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
The theoretical curve will be plotted without ever
finding its equation.
Substituting n=l7 (for an 18-point
polynomial) and k:2, equation (16) becomes
~'txl
"
. .. Bruce and Schumacher, Forest Mensuration, McGraw Hill
Comnany, New York, 1935, pp. 179.
33
To find y 1 (0) look up the values of K2 , 17 (o,s) in table 2.
The result is the equation
YI (0):
54 ~ 64 [ 18 (21896) +32 (17136) +54 (12852) +79 ( 9044)
+114(5712)+159(2856)+213(476)-280(1428)
-334(2856)-404(3808)-500(4284)-582(4284)
-657(3808)-736(2856)-800(1428)+923(476)
+1068(2856)+1204(5712)] :12.0.
In a similar manner the other theoretical values y' (x)
are found.
The calculations are greatly simplified by
use of a calculating machine.
The graph on the following
page shows that a parabola fits the data very well .
5.
TABLES
Table 1 contains values of Pm , n(s) .
In order to
obtain the actual value of Pm , n(s) as given by equation
( 6) , it is necessary to divide the table value by the
table value of Pm n(O) .
'
Table 2 contains values of Kk , n(s , x) (see equation
(15 )).
In these tables when the number of points (n+l )
is greater than 10 , there are values of x indica t ed a t
both the top and bottom of the table .
Value s of x indi­
cated at the top of t he t able correspond to the v a lues of
s on the left .
Values of x given at the bottom correspond
to the values of s on the right .
In every case the table
values must be divided by the denominator common to the
whole table .
Table 3 contains values of Rq , n •
35
Table 1
Values of P 1 ,n(S)
n
1
2
3
4
5
6
7
8
9
10
0
1
1
3
2
5
3
7
4
9
5
1
-1
0
1
1
3
2
5
3
7
4
-1
-1
0
1
1
3
2
5
3
-3
-1
-1
0
1
1
3
2
-2
-3
-1
-1
0
1
1
-5
-2
-3
-1
-1
0
-3
-5
-2
-3
-1
-7
-3
-5
-2
-4
-7
-3
-9
-4
s
2
3
4
5
6
7
8
9
10
-5
36
Table 1
Values of P 1 n(S)
I
n
11
12
13
14
15
16
17
18
19
20
0
11
6
13
7
15
8
17
9
19
10
1
9
5
11
6
13
7
15
8
17
9
2
7
4
9
5
11
6
13
7
15
8
3
5
3
7
4
9
5
11
6
13
7
4
3
2
5
3
7
4
9
5
11
6
5
1
1
3
2
5
3
7
4
9
5
6
-1
0
1
1
3
2
5
3
7
4
7
-3
-1
-1
0
1
1
3
2
5
3
8
-5
-2
-3
-1
-1
0
1
1
3
2
9
-7
-3
-5
-2
-3
-1
-1
0
1
1
10
-9
-4
-7
-3
-5
-2
-3
-1
-1
0
11
-11
-5
-9
-4
-7
-3
-5
-2
-3
-1
-6
-11
-5
-9
-4
-7
-3
-5
-2
-13
-6
-11
-5
-9
-4
-7
-3
-7
-13
-6
-11
-5
-9
-4
-15
-7
-13
-6
-11
-5
-8
-15
-7
-13
-6
-17
-8
-15
-7
-9
-17
-8
-19
-9
s
12
13
14
15
16
17
18
19
20
-10
37
Table 1
Values of P 2 ,n(S)
n
2
3
4
5
6
7
8
9
10
11
0
1
1
2
5
5
7
28
6
15
55
1
-2
-1
-1
-1
0
1
7
2
6
25
2
1
-1
-2
-4
-3
-3
-8
-1
-1
1
1
-1
-4
-4
-5
-17
-3
-6
-17
2
-1
-3
-5
-20
-4
-9
-29
5
0
-3
-17
-4
-10
-35
5
1
-8
-3
-9
-35
7
7
-1
-6
-29
28
2
-1
-17
6
6
1
15
25
s
3
4
5
6
7
8
9
10
11
55
Table 1
38
Va lues of P 2 n(S)
I
n
12
13
14
15
16
17
18
19
20
0
22
13
91
35
40
68
51
57
190
1
11
7
52
21
25
44
34
39
133
2
2
2
19
9
12
23
19
23
82
3
-5
-2
-8
-1
1
5
6
9
37
4
-10
-5
-29
-9
-8
-10
-5
-3
-2
5
-13
-?
-44
-15
-15
-22
-14
-13
-35
6
-14
-8
-53
-19
-20
-31
-21
-21
-62
7
-13
-8
-56
-21
-23
-37
-26
-27
-83
8
-10
-7
-53
-21
-24
-40
-29
-31
-98
9
-5
-5
-44
-19
-23
-40
-30
-33
-107
10
2
-2
-29
-15
-20
-37
-29
-33
-110
11
11
2
-8
-9
-15
-31
-26
-31
-107
12
22
7
19
-1
-8
-22
-21
-27
-98
13
52
9
1
-10
-14
-21
-83
91
21
12
5
-5
-13
-62
35
25
23
6
-3
-35
40
44
19
9
-2
68
34
23
37
51
39
82
57
133
s
13
14
15
16
17
18
19
29
190
39
Table 1
Va lues of P 3 , 0 (S)
n
3
4
5
6
7
8
9
10
. 11
0
1
1
5
1
7
14
42
30
33
1
-3
-2
-7
-1
-5
-7
-14
-6
-3
2
3
0
-4
-1
-7
-13
-35
-22
-21
3
-1
2
4
0
-3
-9
-31
-23
-25
-1
7
1
3
0
-12
-14
-19
-5
1
7
9
12
0
-7
-1
5
13
31
14
7
-7
7
35
23
19
-14
14
22
25
-42
6
21
-30
3
s
4
5
6
7
8
9
10
11
-33
40
Values of P 3 ,n(S)
Table 1
n
12
13
14
15
16
17
18
19
20
0
11
143
91
455
28
68
204
969
570
1
0
11
13
91
7
20
68
357
228
2
-6
-66
-35
-143
-7
-13
-28
-85
-24
3
-8
-98
-58
-267
-15
-33
-89
-377
-196
4
-7
-95
-61
-301
-18
-42
-120
-539
-298
5
-4
-67
-49
-265
-17
-42
-126
-591
-340
6
0
-24
-27
-179
-13
-35
-112
-553
-332
7
4
24
0
-63
-7
-23
-83
-445
-284
8
7
67
27
63
0
-8
-44
-287
-206
9
8
95
49
179
7
8
0
-99
-108
10
6
98
61
265
13
23
44
99
0
11
0
66
58
301
17
35
83
287
108
12
-11
-11
35
267
18
42
112
445
206
-143
-13
143
15
42
126
553
284
-91
-91
7
33
120
591
332
-445
-7
13
89
539
340
-28
-20
28
377
298
-68
-68
85
196
-204
-357
24
-969
-228
s
13
14
15
16
17
18
19
20
-570
41
Table 1
Values of P 4 ,n(S)
n
4
5
6
7
8
9
10
11
12
0
1
1
3
7
14
18
6
33 .
99
1
-4
-3
-7
-13
-21
-22
-6
-27
-66
2
6
2
1
-3
-11
-17
-6
-33
-96
3
-4
2
6
9
9
3
-1
-13
-54
4
1
-3
1
9
18
18
4
12
11
1
-7
-3
9
18
6
28
64
3
-13
-11
3
4
28
84
7
-21
-17
-1
12
64
14
-22
-6
-13
11
18
-6
-33
-54
6
-27
-96
33
-66
s
5
6
7
8
9
10
11
12
99
42
Values of P 4 ,n(S)
Table 1
n
13
14
15
16
1?
18
19
20
0
143
1001
2?3
52
68
612
1938
969
1
-??
-429
-91
-13
-12
-68
-102
0
2
-132
-869
-221
-39
-4?
-388
-1122
-510
3
-92
-?04
-201
-39
-51
-453
-1402
-680
4
-13
-249
-101
-24
-36
-354
-118?
-615
5
63
251
23
-3
-12
-168
-68?
-406
6
108
621
129
1?
13
42
-??
-130
?
108
?56
189
31
33
22?
503
150
8
63
621
189
36
44
352
948
385
9
-13
251
129
31
44
396
1188
540
10
-92
-249
23
1?
33
352
1188
594
11
-132
-?04
-101
-3
13
22?
948
540
12
-??
-869
-201
-24
-12
42
503
385
13
143
-429
-221
-39
-36
-168
-??
150
1001
-91
-39
-51
-354
-68?
-130
2?3
-13
-4?
-453
-118?
-406
52
-12
-388
-1402
-615
68
-68
-1122
-680
612
-102
-510
1938
0
s
14
15
16
1?
18
19
20
969
43
Table 1
Values of P 5 ,n(S)
n
5
6
?
8
9
10
11
12
0
1
1
?
4
6
3
33
22
1
-5
-4
-23
-11
-14
-6
-57
-33
2
10
5
1?
4
1
-1
-21
-18
3
-10
0
15
9
11
4
29
11
4
5
-5
-15
0
6
4
44
26
5
-1
4
-1?
-9
-6
0
20
20
-1
23
-4
-11
-4
-20
0
-?
11
-1
-4
-44
-20
-4
14
1
-29
-26
-6
6
21
-11
-3
57
18
-33
33
8
6
?
8
9
10
11
12
-22
44
Values of P 5 ,n(S)
Table 1
n
13
14
15
16
1?
18
19
20
0
143
1001
143
104
884
102
1938
3876
1
-18?
-1144
-143
-91
-676
-68
-1122
-1938
s
I
2
-132
-9?9
-143
-104.
-871
-98
-1802
-3468
3
28
-44
-33
-39
-429
-58
-1222
-2618
4
139
?51
??
36
156
3
-18?
-'?88
5
145
1000
131
83
588
54
??1
1063
6
60
6?5
115
88
?33
79
1351
2354
?
-60
0
45
55
583
?4
1441
2819
8
-145
-6?5
-45
0
220
44
10?6
2444
9
-139
-1000
-115
-55
-220
0
396
J.404
10
-28
-?51
-131
-88
-583
-44
-396
0
11
132
44
-??
-83
-?33
-?4
-10?6
-1404
12
18?
9?9
33
-36
-588
-?9
-1441
-2444
13
-143
1144
143
39
-156
-54
-1351
-2819
-1001
143
104
429
-3
-??1
-2354
-143
91
8?1
58
18?
-1063
-104
6?6
98
1222
?88
-884
68
1802
2618
-102
1122
3468
-1938
1938
14
15
16
1?
18
19
20
-38?6
45
Table 1
Va lues of P6~n(S)
n
6
7
8
9
10
11
12
13
0
1
1
4
3
15
11
22
143
1
-6
-5
-17
-11
-48
-31
-55
-319
2
15
9
22
10
29
11
8
-11
3
-20
-5
1
6
36
25
43
227
4
15
-5
-20
-8
-12
4
22
185
5
-6
9
1
-8
-40
-20
-20
-25
6
1
-5
22
6
-12
-20
-40
-200
1
-17
10
36
4
-20
-200
4
-11
29
25
22
-- -25
3
-48
11
43
185
15
-31
8
227
11
-55
-11
22
-319
s
7
8
9
10
11
12
143
Table 1
46
Values of P 6 ,n(S)
n
14
15
16
17
18
19
20
0
143
65
104
442
1326
1938
6460
1
-286
-117
-169
-650
-1768
-2346
-7106
2
-55
-39
-78
-377
-1222
-1870
-6392
3
176
59
65
169
234
6
-918
4
197
87
128
481
1235
1497
3996
5
50
45
93
439
1352
1931
6075
6
-125
-25
2
145
729
1353
5088
7
-200
-75
-85
-209
-214
195
2001
8
-125
-75
-120
-440
-1012
-988
-1716
9
50
-25
-85
-440
-1320
-1716
-4628
10
197
45
2
-209
-1012
-1716
-5720
11
176
87
93
145
-214
-988
-4628
12
-55
59
128
439
729
195
-1716
13
-286
-39
65
481
1352
1353
2001
14
143
-117
-78
169
1235
1931
5088
65
-169
-377
234
1497
6075
104
-650
-1222
6
3996
442
-1768
-1870
-918
1326
-2346
-6392
1938
-7106
s
15
16
17
18
19
20
6460
47
Table 2
Va lues of K1 (S,X)
(3-point polynomial)
X
0
1
2
0
5
2
-1
i
2
2
2
2
-1
2
5
s
Denominator 6
Values of K (S,X)
1
(4-point polynomial)
X
0
1
2
3
0
?
4
1
-2
1
4
3
2
1
2
1
2
3
4
3
-2
1
4
7
s
Denomina tor 10
Value s of K2 (S,X)
(4-point polynomial)
X
0
1
2
3
·o
19
3
-3
1
1
3
11
9
-3
2
-3
9
11
3
3
1
-3
3
19
s
Denominator 20
Table 2
Values of K1 (S 1 X) (5-point polynomial)
2
4
X
3
0
1
s
0
6
4
2
0
-2
1
4
3
2
1
0
2
2
2
2
2
2
3
0
1
2
3
4
4
-2
0
2
4
6
Denominator 10
(5-point polynomial)
Values of K2( SIX )
X
0
1
2
3
4
0
31
9
-3
-5
3
1
9
13
12
6
-5
2
-3
12
17
12 .
-3
3
-5
6
J.2
13
9
4
3
-5
Denominator 35
-3
9
31
s
Values of K3 (S 1 X) (5-point polynomial)
2
4
X
0
1
3
s
0
69
4
-6
4
-1
1
4
54
24
-16
4
2
-6
24
34
24
-6
3
4
-16
24
54
4
4
-1
4
-6
4
69
Denominator 70
48
49
Table 2
Va lu es of K1 (S,X)
(6-point p olynomial )
X
0
1
2
3
4
5
0
55
40
25
10
-5
-20
1
40
31
22
13
4
-5
2
25
22
19
16
13
10
3
10
13
16
19
22
25
4
-5
4
13
22
31
40
5
-20
-5
10
25
40
55
s
Denominator 105
Va lues of K2(S,X)
(6-point polynomial)
X
0
1
2
3
4
5
0
115
45
0
-20
-15
15
1
45
43
36
24
7
-15
2
0
36
52
48
24
-20
3
-20
24
48
52
36
0
4
-15
7
24
36
43
45
5
15
-15
-20
0
45
115
s
Denominator 140
50
Table 2
X
Values of K3 (S,X)
0
1
(6-point polynomial)
2
3
4
5
s
0
121
16
-14
-4
11
-4
1
16
73
52
2
-28
11
2
-14
52
58
32
2
-4
3
-4
2
32
58
52
-14
4
11
-28
2
52
73
16
5
-4
11
-4
-14
16
121
Denominator 126
X
Values of K (S,X)
4
0
1
(6-point p olynomial
2
3
4
5
s
0
251
5
-10
10
-5
1
1
5
227
50
-50
25
-5
2
-10
50
152
100
-50
10
3
10
-50
100
152
50
-10
4
-5
25
-50
50
227
5
5
1
-5
10
-10
5
251
Denominator 252
51
Table 2
Values of K1 (S,X)
(7-point polynomial)
X
0
1
2
3
4
5
6
0
13
10
7
4
1
-2
-5
1
10
8
6
4
2
0
-2
2
7
6
5
4
3
2
1
3
4
4
4
4
4
4
4
4
1
2
3
4
5
6
7
5
-2
0
2
4
6
8
10
6
-5
-2
1
4
7
10
13
s
Denominator 28
Values of K2 (S,X)
(7-point polynomial)
X
0
1
2
3
4
5
6
0
32
15
3
-4
-6
-3
5
1
15
12
9
6
3
0
-3
2
3
9
12
12
9
3
-6
3
-4
6
12
14
12
6
-4
4
-6
3
9
12
12
9
3
5
-3
0
3
6
9
12
15
6
5
-3
-6
-4
3
15
32
s
Denominator 42
52
Table 2
Va lues of K3(S,X)
(7-point polynomial)
X
0
1
2
3
4
5
6
0
39
8
-4
-4
1
4
-2
1
8
19
16
6
-4
-?
4
2
-4
16
19
12
2
-4
1
3
-4
6
12
14
12
6
-4
4
1
-4
2
12
19
16
-4
5
4
-7
-4
6
16
19
8
6
-2
4
1
-4
-4
8
39
8
Denomina tor 42
X
Va lues of K (S,X)
4
2
0
1
6
456
25
1
25
2
(7-point polynomial)
3
4
5
6
-35
10
20
-19
5
356
155
-60
-65
70
-19
-35
155
212
. 150
25
-65
20
3
10
-60
150
262
150
-60
10
4
20
-65
25
150
212
155
-35
5
-19
70
-65
-60
155
356
25
6
5
-19
20
10
-35
25
456
s
Denominator 462
53
Table 2
Va lues of K5 (S,X)
(7-point polynomial)
X
0
1
2
3
4
5
6
0
923
6
-15
20
-15
6
-1
1
6
888
90
-120
90
-36
6
2
-15
90
699
300
-225
90
-15
3
20
-120
300
524
300
-120
20
4
-15
90
-225
300
699
90
-l5
5
6
-36
90
-1 20
90
888
6
6
-1
6
-15
20
-15
6
923
s
Denominator 924
54
Table 2
Va lues of K (S,X)
1
2
X
0
1
(8-point p olynomial)
3
4
5
6
7
s
0
35
28
21
14
7
0
-7
-14
1
28
23
18
13
8
3
-2
-7
2
21
18
15
12
9
6
3
0
3
14
13
12
11
10
9
8
7
4
7
8
9
10
11
12
13
14
5
0
3
6
9
12
15
18
21
6
-7
-2
3
8
13
18
23
28
7
-14
-7
0
7
14
21
28
35
Deno minator 84
Values of K2 (S,X)
(8-point polynomial)
4
5
6
7
-14
-42
-42
-14
42
66
42
22
6
-6
~14
66
78
78
66
42
6
-42
-14
42
78
94
90
66
22
-42
4
-42
22
66
90
94
78
42
-14
5
-42
6
42
66
78
78
66
42
6
-14
-6
6
22
42
66
94
126
7
42
-14
-42
-42
-1 4
42
126
238
X
0
1
2
0
238
126
42
1
126
94
2
42
3
3.
8
Denomina tor 336
55
Table 2
X
Values of K: (S,X)
3
2
0
1
0
413
112
1
112
2
(8-point p olynomial)
3
4
5
6
7
-28
-56
-21
28
42
-28
173
152
84
4
-53
-52
42
-28
152
193
1 44
54
-28
-53
28
3
-56
84
144
145
108
54
4
-21
4
-21
4
54
108
145
144
84
-56
5
28
-53
-28
54
144
193
152
-28
6
42
-52
-53
4
84
152
173
112
7
-28
42
28
-21
-56
-28
112
413
s
Deno minator 462
X
Values of K (S,X)
4
0
1
2
(8-point p olynomial)
3
4
5
6
7
s
0
1799
175
-175
-35
105
49
-105
35
1
175
ll99
725
-15
-335
-95
299
-105
2
-175
725
799
495
135
-85
-95
49
3
-35
-15
495
823
675
135
-335
105
4
105
-335
135
675
823
495
-15
-35
5
49
-95
-85
135
495
799
725
-175
6
-105
299
-95
-335
-15
725
1199
175
7
35
-105
49
105
-35
-175
175
1799
Deno mina tor 1848
56
Table 2
Values of K (S,X)
5
0
1
2
X
(8-point polynomial)
3
4
5
6
7
s
0
1709
36
-69
50
15
-48
29
-6
1
36
1529
366
-285
-40
219
-138
29
2
-69
366
969
660
-75
-306
219
-48
3
50
-285
660
941
450
-75
-40
15
4
15
-40
-75
450
941
660
-285
50
5
-48
219
-306
-75
660
969
366
-69
6
29
-138
219
-40
-285
366
1529
36
7--
-6
29
-48
15
50
-69
36
1709
Denomina tor 1716
Values of K1 (S,X
(9-point p olynomial)
X
0
1
2
3
4
5
6
7
0
68
56
44
32
20
8
-4
-16
-28
1
56
47
38
29
20
11
2
-7
-16
2
44
38
32·'
26
20
14
8
2
-4
3
32
29
26
23
20
17
ll
11
8
4
20
20
20
20
20
20
20
20
20
5
8
11
14
17
20
23
26
29
32
6
-4
2
8
14
20
26
32
38
44
7
-16
-7
2
11
20
29
38
47
56
8
-28
-16
-4
8
20
32
44
56
68
8
s
Denominat or 180
57
Table 2
Values of K2(S,X)
(9-point polynomial)
X
0
1
2
3
4
5
6
7
8
0
1526
882
378
14
-210
-294
-238
-42
294
1
882
644
441
273
140
42
-21
-49
-42
2
378
441
464
447
390
293
156
-21
-238
3
14
273
447
536
540
459
293
42
-294
4
-210
140
390
540
590
540
390
140
-210
5
-294
42
293
459
540
536
447
273
14
6
-238
-21
156
293
390
447
464
441
378
7
-42
-49
-21
42
140
273
441
644
882
8
294
-42
-238
-294
-210
14
378
882
1526
8
Denominator 2310
X
Values of K3 (s,X)
0
3
1
2
(9-point polynomia l)
4
5
6
7
8
8
0
1190
392
-28
-168
-126
0
112
112
-98
1
392
455
392
252
84
-63
-140
-98
112
2
-28
392
515
432
234
12
-143
-140
112
3
-168
252
432
435
324
162
12
-63
0
4
-126
84
234
324
354
324
234
84
-126
5
0
-63
12
162
324
435
432
252
-168
6
112
-140
-143
12
234
432
515
392
-28
7
112
-98
-140
-63
84
252
392
455
392
8
-98
112
112
0
-126
-168
-28
392
1190
Denomina tor 1386
58
Table 2
Values of K4(S,X)
(9-point p olynomial)
X
0
1
2
3
4
5
6
7
8
0
2462
350
-250
-150
90
162
10
-170
70
1
350
1412
1025
225
-330
-360
37
385
-170
2
-250
1025
1112
6?5
180
-105
-110
37
10
3
-150
225
6?5
912
810
405
-105
-360
162
4
90
-330
180
810
10?4
810
180
-330
90
5
162
-360
-105
405
810
912
675
225
-150
6
10
37
-110
-105
180
6?5
1112
1025
-250
7
-170
385
37
-360
-330
225
1025
1412
350
8
?0
-1?0
10
162
90
-150
-250
350
2462
8
Denominator 2574
Values of K5 (S,X)
(9-point polynomial)
X
0
1
2
3
4
5
6
7
8
0
1?00
?2
-108
32
60
-24
-52
48
-12
1
?2
1385
522
-213
-220
123
186
-187
48
2
-108
522
800
582
120
-202
-132
186
-52
3
32
-213
582
905
540
-2?
-202
123
-24
4
60
-220
120
540
716
540
120
-220
60
5
-24
123
-202
-2?
540
905
582
-213
32
6
-52
186
-132
-202
120
582
800
522
-108
7
48
-18?
186
123
-220
-213
522
13 35
?2
8
-12
48
-52
-24
60
32
-108
?2
1?00
s
Denomina tor 1716
59
Table 2
Values of K1 (s,X)
(10-point polynomial)
X
0
1
2
3
4
5
6
7
8
9
0
57
48
39
30
21
12
3
-6
-15
-24
1
48
41
34
27
20
13
6
-1
-8
-15
2
39
34
29
24
19
14
9
4
-1
-6
3
30
27
24
21
18
15
12
9
6
3
4
21
20
19
18
17
16
15
14
13
12
5
12
13
14
15
16
17
18
19
20
21
6
3
6
9
12
15
18
21
24
27
30
7
-6
-1
4
9
14
19
24
29
34
39
8
-15
-8
-1
6
13
20
27
34
41
48
9
-24
-15
-6
3
12
21
30
39
48
57"
s
Denomina tor 55
Table 2
X
60
Values of K (S,X)
2
2
0
1
3
(10-point polynomial)
4
5
6
7
8
9
s
0
408
252
126
30
-36
-72
-78
-54
0
84
1
252
184
126
78
40
12
-6
-14
-12
0
2
126
126
121
111
96
76
51
21
-14
-54
3
30
78
111
129
132
120
93
51
-6
-78
4
-36
40
96
132
148
144
120
76
12
-72
5
-72
12
76
120
144
148
132
96
40
-36
6
-78
-6
51
93
120
132
129
111
78
30
7
-54
-14
21
51
76
96
111
121
126
126
8
0
-12
-14
-6
12
40
78
126
184
252
9
84
0
-54
-78
-72
-36
30
126
252
408
Denomina tor 660
Values of K3 ( S,X)
(lO-point polynomial)
1-3
X
0
1
2
3
4
5
6
?
8
9
0
3534
1344
84
-456
-486
-216
144
384
294
-336
1
1344
1294
1064
?24
344
-6
-256
-336
-1?6
294
2
84
1064
1399
1264
834
284
-211
-4?6
-336
384
3
-456
?24
1264
1319
1044
594
124
-211
-256
144
4
-486
344
834
1044
1034
864
594
284
-6
-216
5
-216
-6
284
594
864
1034
1044
834
344
-486
6
144
-256
-211
124
594
1044
1319
1264
?24
-456
?
384
-336
-4?6
-211
284
834
1264
1399
1064
84
8
294
-1?6
-336
-256
-6
344
?24
1064
1294
1344
9
-336
294
384
144
-216
-486
-456
84
1344
3534
s
Denominator 4290
~
1-'
CD
[\)
Values of K4 (S,X)
X
0
1
2
3
(10-point polynomial)
4
5
6
?
8
sn
8
9
s
o'
1-J
(1)
ro
0
1608
300
-150
-150
0
108
90
-30
-120
60
1
300
808
650
250
-100
-240
-142
90
220
-120
2
-150
650
?33
4?5
150
-?0
-115
-1?
90
-30
3
-150
250
4?5
533
450
2?0
55
-115
-142
90
4
0
-100
150
450
608
540
2?0
-?0
-240
108
5
108
-240
-?0
2?0
540
608
450
150
-100
0
6
90
-142
-115
55
2?0
450
533
4?5
250
-150
?
-30
90
-1?
-115
-?0
150
4?5
?33
650
-150
8
-120
220
90
-142
-240
-100
250
650
808
300
9
60
-120
-30
90
108
0
-150
-150
300
1608
Denominator 1716
Values of K5 (S,X)
X
0
1
2
3
(10-point polynomial)
8
4
5
6
7
8
9
~
1-'
s
CD
0
2109
144
-171
-6
99
36
-69
-54
81
-24
1
144
1549
774
-111
-356
-69
246
151
-264
81
2
-171
774
919
624
204
-104
-174
-24
151
-54
3
-6
-111
624
999
744
156
-264
-174
246
-69
4
99
-356
204
744
859
576
156
-104
-69
36
5
36
-69
-104
156
576
859
744
204
-356
99
6
-69
246
-174
-264
156
744
999
624
-111
-6
7
-54
151
-24
-174
-104
204
624
919
774
-171
8
81
-264
151
246
-69
-356
-111
774
1549
144
9
-24
81
-54
-69
36
99
-6
-171
144
2109
Denominator 2145
t'\)
64
Table 2
X
Values of K1 (S,X) (11-point polynomial)
2
0
1
4
3
5
8
0
35
30
25
20
15
10
10
1
30
26
22
18
14
10
9
2
25
22
19
16
13
10
8
3
20
18
16
14
12
10
7
4
15
14
13
12
11
10
6
5
10
10
10
10
10
10
5
6
5
6
7
8
9
10
4
?
0
2
4
6
8
10
3
8
-5
-2
1
4
7
10
2
9
-10
-6
-2
2
6
10
1
10
-15
-10
-5
0
5
10
0
10
9
8
?
6
5
8
X
Denominator 110
65
Table 2
Values of K2 (SX)
(11-point polynomial)
X
0
1
2
3
4
5
0
1245
810
450
165
-45
-180
10
1
810
597
414
261
138
45
9
2
450
414
373
327
276
220
8
3
165
261
327
363
369
345
7
4
-45
138
276
369
417
420
6
5
-180
45
220
345
420
445
5
6
-240
-l8
159
291
378
420
4
7
-225
-51
93
207
291
345
3
8
-135
-54
22
93
159
220
2
9
30
-27
-54
-51
-18
45
1
10
270
10
30
9
-135
8
-225
7
-240
6
-180
5
0
s
s
X
Denominator 2145
66
Table 2
Values of K3 (S,X)
(11-point p olynomial)
X
0
l
2
3
4
5
0
678
288
48
-72
-102
-72
10
1
288
246
192
132
72
18
9
2
48
192
246
232
1?2
88
8
3
-72
132
232
251
212
138
?
4
-102
72
172
212
206
168
6
5
-72
18
88
138
168
1?8
5
6
-12
-24
2
52
112
168
4
?
48
-48
-64
-23
52
138
3
8
?8
-48
-88
-64
2
88
2
9
48
-18
-48
-48
-24
18
1
10
-?2
48
?8
48
-12
-?2
0
10
9
8
7
6
5
s
s
X
Denomina tor 858
67
Table 2
Va lues of K4 (S,X)
(11-point polynomia l)
X
0
1
2
3
4
5
0
393
90
-30
-45
-15
18
10
1
90
177
150
75
0
-45
9
2
-30
150
177
125
50
-10
8
3
-45
75
125
127
100
60
7
4
-15
0
50
100
127
120
6
5
18
-45
-10
60
120
143
5
6
30
-48
-35
20
80
120
4
7
15
-15
-23
-10
20
60
3
8
-15
30
10
-23
-35
-10
2
9
-30
45
30
-15
-48
-45
1
10
18
-30
-15
15
30
18
0
10
9
8
7
6
5
8
s
X
Denomina tor 429
68
Table 2
Values of K5 (S,X)
(11-point polynomial)
X
0
1
2
3
4
5
0
1671
162
-153
-48
72
72
10
1
162
1104
666
36
-264
-180
9
2
-153
666
719
456
156
-40
8
3
-48
36
456
684
576
240
7
4
72
-264
156
576
684
480
6
5
72
-180
-40
240
480
572
5
6
-12
72
-96
-96
144
480
4
7
-72
204
-48
-216
-96
240
3
8
-27
54
29
-48
-96
-40
2
9
78
-216
54
204
72
-180
1
10
-27
10
78
-27
0
8
-12
6
72
9
-72
?
s
s
5
X
Denominator 1?16
69
Table 2
X
Values of K1 (S,X) (12-point polynomial)
2
4
0
1
3
5
s
0
253
220
187
154
121
88
11
1
220
193
166
139
112
85
10
2
187
166
145
124
103
82
9
3
154
139
124
109
94
79
8
4
121
112
103
94
85
76
7
5
88
85
82
79
76
73
6
6
55
58
61
64
67
70
5
7
22
31
40
49
58
67
4
8
-11
4
19
34
49
64
3
9
-44
-23
-2
19
40
61
2
10
-77
-50
-23
4
31
58
1
11
-110
-77
-44
-11
22
55
0
11
10
9
8
7
6
s
X
Denominator 858
70
Table 2
X
Values of K (S,X)
2
1
0
2
(12-point polynomial)
3
4
5
s
0
2189
1485
891
407
33
-231
11
1
1485
1109
783
507
281
105
10
2
891
783
677
573
471
371
9
3
407
507
573
605
603
567
8
4
33
281
471
603
677
693
7
5
-231
105
371
567
693
749
6
6
-385
-21
273
497
651
735
5
7
-429
-97
177
393
551
651
4
8
-363
-123
83
255
393
497
3
9
-187
-99
-9
83
177
273
2
10
99
-25
-99
-123
-97
-21
1
11
495
11
99
10
-187
9
-363
8
-429
7
-385
6
0
s
X
Denominator 4004
71
Table 2
X
Values of K (S,X) (12-point polynomial)
3
4
2
0
1
3
5
s
0
6831
3168
792
-528
-1023
-924
11
1
3168
2511
1872
1272
732
273
10
2
792
1872
2295
2208
1758
1092
9
3
-528
1272
2208
2455
2188
1582
8
4
-1023
732
1758
2188
2155
1792
7
5
-924
273
1092
1582
1792
1771
6
6
-462
-84
357
812
1 232
1568
5
7
132
-318
-300
53
608
1232
4
8
627
-408
-732
-520
53
812
3
9
792
-333
-792
-732
-300
357
2
10
396
-72
-333
-408
-318
-84
1
11
-792
11
396
792
627
132
-462
0
10
9
8
7
6
s
X
Denominator 9009
72
Table 2
Values of K4 (S,X)
(1 2-point polynomial)
X
0
1
2
3
4
5
0
920?
24?5
-495
-1155
-660
132
11
1
24?5
380?
3285
1 905
420
-660
10
2
-495
3285
4023
30?5
1500
60
9
3
-1155
1905
30?5
3023
2300
1340
8
4
-660
420
1500
2300
2648
2480
?
5
132
-660
60
1340
2480
3032
6
6
660
-1068
-?80
460
1840
2800
5
?
660
-?80
-852
-140
880
1840
4
8
165
-15
-285
-3??
-140
460
3
9
-495
?65
495
-285
-852
-?80
2
10
-693
855
?65
-15
-?80
-1068
1
11
495
-693
10
-495
9
165
660
660
0
8
?
6
s
11
s
X
Denomina tor 10,296
73
Table 2
Values of K5 (S,X)
(12-point polynomial)
X
0
1
2
3
4
5
0
9361
1188
-891
-506
264
528
11
1
1188
5581
3834
789
-1136
-1320
10
2
-891
3834
4069
2532
852
-200
9
3
-506
'789
2532
3369
2952
1620
8
4
264
-1136
852
2952
3684
2880
'7
5
528
-1320
-200
1620
2880
3108
6
6
220
-312
-480
80
1200
2400
5
7
-264
796
-240
-912
-352
1200
4
8
-429
996
103
-870
-912
80
3
9
-44
-9
198
103
-240
-480
2
10
495
-1178
-9
996
796
-312
1
11
-198
495
-44
-429
-264
220
0
11
10
9
8
7
6
s
s
X
Denominator 9724
74
Table 2
X
Values of K1 (S 1 X)
1
2
0
(13-point polynomial)
3
4
5
6
s
0
50
44
38
32
26
20
14
12
1
44
39
34
29
24
19
14
11
2
38
34
30
26
22
18
14
10
3
32
29
26
23
20
17
14
9
4
26
24
22
20
18
16
14
8
5
20
19
18
17
16
15
14
7
6
14
14
14
14
14
14
14
6
7
8
9
10
11
12
13
14
5
8
2
4
6
8
10
12
14
4
9
-4
-1
2
5
8
11
14
3
10
-10
-6
-2
2
6
10
14
2
11
-16
-11
-6
-1
4
9
14
1
12
-22
12
-1.6
11
-10
10
-4
9
2
8
8
7
14
6
0
s
X
Denominator 182
'
75
Table 2
Values of K (s,X)
2
(13-point p olynomial)
X
0
1
2
3
4
5
6
0
517
363
231
121
33
-33
-7?
12
1
363
2?5
198
132
??
33
0
11
2
231
198
16?
138
111
86
63
10
3
121
132
138
139
135
126
112
9
4
33
??
111
135
149
153
147
8
5
-33
33
86
126
153
16?
168
?
6
-??
0
63
112
14?
168
1?5
6
7
-99
-22
42
93
l31
156
168
5
8
-99
-33
23
69
105
131
147
4
9
-7?
-33
6
40
69
93
112
3
10
-33
-22
-9
6
23
42
63
2
11
33
0
-22
-33
-33
-22
0
1
12
l21
12
33
11
-33
10
-??
9
-99
8
-99
7
-77
6
0
s
s
X
Denominator 1001
76
Table 2
Values of K3 (S,X)
(13-point polynomial)
0
1
2
3
4
5
6
0
2915
1452
462
-132
-407
-440
-308
12
1
1452
1100
792
528
~- 308
132
0
11
2
462
792
920
888
738
512
252
10
3
-132
528
888
1004
932
728
448
9
4
-407
308
738
.932
939
808
588
8
5
-440
132
512
728
808
?80
672
?
6
-308
0
252
448
588
672
?00
6
?
-88
-88
0
148
328
512
6?2
5
8
143
-132
-202
-116
77
328
588
4
9
308
-132
-312
-288
-116
148
448
3
10
330
-88
-288
-312
-202
0
252
2
11
132
0
-88
-132
-132
-88
0
1
12
-363
12
132
11
330
10
308
9
143
8
-88
?
-308
6
0
s
-:X
s
X
Denominator 4004
(13-point p olynomial)
Value of K4 (S,X)
X
0
1
2
3
4
5
8
ll'
6
o'
1-'
s
(I)
0
29,678
9075
-825
-3795
-2915
-572
1540
12
1
9075
11,528
9900
6270
2255
-990
-2772
11
2
-825
9900
12,428
10,140
5745
1280
-1890
10
3
-3795
6270
10,140
9992
7625
4460
1540
9
4
-2915
2255
5745
7625
8042
7220
5460
8
5
-572
-990
1280
4460
7220
8678
8400
7
6
1540
-2772
-1890
1540
5460
8400
9478
6
7
2420
-2860
-3072
-470
3140
6400
8400
5
8
1760
-1485
-2245
-1283
715
3140
5460
4
9
-55
660
-60
-990
-1283
-470
1540
3
10
-1947
2420
2160
-60
-2245
-3072
-1890
2
11
-2145
2178
2420
660
-1485
-2860
-2772
1
12
1815
-2145
-1947
-55
1760
2420
1540
0
12
11
10
9
8
7
6
ro
s
X
Denomina tor 34,034
....:1
....:1
78
Table 2
X
Values of K5 (S,X)
2
0
1
(13-point polynomial)
3
4
5·
6
s
0
9240 -
1452
-858
-704
66
528
440
12
1
1452
5005
3762
1221
-704
-1320
-792
11
2
-858
3762
4060
2586
906
-200
-540
10
3
-704
1221
2586
3045
2628
1620
440
9
4
66
-704
906
2628
3360
2880
1560
8
5
528
-1320
-200
1620
2880
3108
2400
7
6
440
-792
-540
440
1560
2400
2708
6
7
0
220
-312
-480
80
1200
2400
5
8
-396
924
94
-816
-858
80
1560
4
9
-396
759
294
-473
-816
-480
440
3
10
66
-242
108
294
94
-312
-540
2
11
528
-1089
-242
759
924
220
-792
1
12
-242
12
528
11
66
10
-396
9
-396
8
0
7
440
6
0
s
X
Denominator 9724
79
Table 2
X
Val es of K1 (S,X)
0
1
2
(14-point polynomial)
3
4
5
6
s
0
117
104
91
78
65
52
39
13
1
104
93
82
?1
60
49
38
12
2
91
82
73
64
55
46
37
11
3
78
71
64
57
50
43
36
10
4
65
60
55
50
45
40
35
9
5
52
49
46
43
40
37
34
8
6
39
38
37
36
35
34
33
7
7
26
27
28
29
30
31
32
6
8
13
16
19
22
25
28
31
5
9
0
5
10
15
20
25
30
4
10
-13
-6
1
8
15
22
29
3
11
-26
-17
-8
1
10
19
28
2
12
13
-39
-52
-28
-39
-17
-26
-6
-13
5
0
16
13
27
26
1
0
13
12
11
10
9
8
7
s
X
Denominator 455
80
Table 2
X
Values of K (S,X)
2
2
0
1
(14-point polynomial)
3
4
5
6
8
0
1781
1287
858
494
195
-39
-208
13
1
1287
989
726
498
305
147
24
12
2
858
726
604
492
390
298
216
11
3
494
498
492
476
450
414
368
10
4
195
305
390
450
485
495
480
9
5
-39
147
298
414
495
541
552
8
6
-208
24
216
368
480
552
584
7
7
-312
-64
144
312
440
528
576
6
8
-351
-117
82
246
375
469
528
5
9
-325
-135
30
170
285
375
440
4
10
-234
-118
-12
84
170
246
312
3
11
-78
-66
-44
-12
30
82
144
2
12
143
21
-66
- 118
-135
-117
-64
1
13
429
143
-78
-234
-325
-351
-312
0
13
12
11
10
9
8
7
8
X
Denominator 3640
81
Table 2
(15-point polynomial)
Values of K1 (S,X)
X
0
1
2
3
4
5
6
?
0
203
182
161
140
119
98
??
56
14
1
182
164
146
128
110
92
?4
56
13
2
1.61
146
131
116
101
86
?1
56
12
3
140
128
116
104
92
80
68
56
11
4
119
110
101
92
83
?4
65
56
10
5
98
92
86
80
?4
68
62
56
9
6
??
?4
?1
68
65
62
59
56
8
?
56
56
56
56
56
56
56
56
?
8
35
38
41
44
4?
50
53
56
6
9
14
20
26
32
38
44
50
56
5
10
-?
2
11
20
29
38
4?
56
4
11
-28
-16
-4
8
20
32
44
56
3
12
-49
-34
-19
-4
11
26
41
56
2
13
-70
-52
-34
-16
2
20
38
56
1
14
-91
-?0
-49
-28
-? .
14
35
56
0
14
13
12
11
9
8
?
s
10
s
X
Denominator 840
Values of K2 (S,X)
X
0
1
2
3
(15-point polynomial)
4
5
1-3
6
6­J-J
7
s
(1)
0
14,378
10,647
7371
4550
2184
273
-1183
-2184
14
1
10,647
8294
6201
4368
2795
1482
429
-364
13
2
7371
6201
5126
4146
3261
2471
1776
1176
12
3
4550
4368
4146
3884
3582
3240
2858
2436
11
4
2184
2795
3261
3582
3758
3789
3675
3416
10
5
273
1482
2471
3240
3789
4118
4227
4116
9
6
-1183
429
1776
2858
3675
4227
4514
4536
8
7
-2184
-364
1176
2436
3416
4116
4536
4676
7
8
-2730
-897
671
1974
3012
3785
4293
4536
6
9
-2821
-1170
261
1472
2463
3234
3785
4116
5
10
-2457
-1183
-54
930
1769
2463
3012
3416
4
11
-1638
-936
-274
348
930
1472
1974
2436
3
12
-364
-429
-399
-274
-54
261
671
1176
2
13
1365
338
-429
-936
-1183
-1170
-897
-364
1
14
3549
14
1365
13
-364
12
-1638
11
-2457
10
-2821
9
-2730
8
-2184
7
0
s
X
Denominator 30,940
ro
(X)
ro
83
Table 2
X
Values of K (S,X) (16-point polynomial)
1
2
4
0
1
5
6
7
3
s
0
155
140
125
110
95
80
65
50
15
1
140
127
114
101
88
75
62
49
14
2
125
114
103
92
81
70
59
48
13
3
110
101
92
83
74
65
56
47
12
4
95
88
81
74
67
60
53
46
11
5
80
75
70
65
60
55
50
45
10
6
65
62
59
56
53
50
47
44
9
7
50
49
48
47
46
45
44
43
8
8
35
36
37
38
39
40
41
42
7
9
20
23
26
29
32
35
38
41
6
10
5
10
15
20
25
30
35
40
5
11
-10
-3
4
11
18
25
32
39
4
12
-25
-16
-7
2
11
20
29
38
3
13
-40
-29
-18
-7
4
15
26
37
2
14
-55
-42
-29
-16
-3
10
23
36
1
15
-70
15
-55
14
-40
13
-25
12
-10
11
5
10
20
9
35
8
0
s
X
Denominator 680
X
Values of K (S,X)
2
3
1
2
0
(16-point p olynomi a l)
4
5
6
1-3
P>
o'
?
~
(1)
6
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
4445
413?
3819
3491
3153
2805
244?
20?9
1701
1313
915
50?
12,635
9555
6825
4 445
2415
?35
-595
-15?5
-2205
-2485
-2415
-1995
-1225
-105
1365
3185
9555
?539
5?33
413?
2?51
15?5
609
-14?
-693
-1029
-1155
-1071
-?7?
-273
441
1365
6825
5?33
4?31
3819
2997
2265
1623
10?1
609
237
-45
-23?
-339
-351
-273
-105
-339
-?7?
-1225
15
14
13
12
~ 89
2415
2?51
299?
3153
3219
3195
3081
28??
2583
2199
1?25
1161
50?
-23? .
-1071
-1995
11
?35
15?5
2265
2805
3195
3435
3525
3465
3255
2895
2385
1?25
915
-45
-1155
-2415
-595
609
1623
2447
3081
3525
3?79
3843
3717
3401
2895
2199
1313
23?
-1029
-2485
-15?5
-14?
10?1
20?9
287?
3465
3843
4011
3969
3?17
3255
2583
1701
609
-693
-2205
10
9
8
15
14
13
12
11
10
9
8
?
6
5
4
3
2
1
0
6
X
Denominator 28,560
ro
85
Table 2
Va lues of Kl(S,X)
(17-point p olynomial)
X
0
1
2
3
4
5
6
7
8
0
88
80
72
64
56
48
40
32
24
16
1
80
73
66
59
52
45
38
31
24
15
2
72
66
60
54
48
42
36
30
24
14
3
64
59
54
49
44
39
34
29
24
13
4
56
52
48
44
40
36
32
28
24
12
5
48
45
42
39
36
33
30
27
24
11
6
40
38
36
34
32
30
28
26
24
10
7
32
31
30
29
28
27
26
25
24
9
8
24
24
24
24
24
24
24
24
24
8
9
16
17
18
19
20
21
22
23
24
7
10
8
10
12
14
16
18
20
22
24
6
11
0
3
6
9
12
15
18
21
24
5
12
-8
-4
0
4
8
12
16
20
24
4
13
-16
-11
-6
-1
4
9
14
19
24
3
14
-24
-18
-12
-6
0
6
12
18
24
2
15
-32
-25
-18
-11
-4
3
10
17
24
1
16
-40
16
-32
15
-24
14
-16
13
-8
12
0
11
8
10
16
9
24
8
0
s
X
Denominator 408
Values of K2 (S,X)
X
0
1
2
3
(17-point polynomial)
4
5
6
8
7
~
8
......
s
{1)
ro
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1636
1260
924
628
372
156
-20
-156
-252
-308
-324
-300
-236
-132
12
196
420
1260
1006
777
573
394
240
111
7
-72
-126
-155
-159
-138
-92
-21
75
196
924
777
642
519
408
309
222
147
84
33
-6
-33
- 48
-51
-42
-21
12
628
573
519
466
414
363
313
264
216
169
123
78
34
-9
-51
-92
-132
372
394
408
414
412
402
384
358
324
282
232
174
108
34
-48
-138
-236
156
240
309
363
402
426
435
429
408
372
321
255
174
78
-33
-159
-300
-20
111
222
313
384
435
466
477
468
439
390
321
232
123
-6
-155
-324
-156
7
147
264
358
429
477
502
504
483
439
372
282
169
33
-126
-308
-252
-72
84
216
324
408
468
504
516
504
. 468
408
324
216
84
-72
-252
16
15
14
13
12
11
10
9
8
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
s
X
Denomina tor 3876
X
0
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
595
544
493
442
391
340
289
238
187
136
85
34
:..17
-68
-119
-170
-221
-272
17
544
499
454
409
364
319
274
229
184
139
94
49
4
-41
-86
-131
-176
-221
16
Values of K (S,X)
1
2
3
(18-point polynomial)
1-3
4
5
6
7
8
391
364
337
310
283
256
229
202
175
148
121
94
67
40
13
-14
-41
-68
13
340
319
298
277
256
235
214
193
172
151
130
109
88
67
46
25
4
-17
12
289
274
259
244
229
214
199
184
169
154
139
124
109
94
79
64
49
34
11
238
229
220
211
202
193
184
175
166
157
148
139
130
121
112
103
94
85
10
187
184
181
178
175
172
169
166
163
160
157
154
151
148
145
142
139
136
9
I»
o'
1-'
CD
s
tv
493
454
415
376
337
298
259
220
181
142
103
64
25
-14
-53
-92
-131
-170
15
442
409
376
343
310
277
244
211
178
145
112
79
46
13
-20
-53
-86
-119
14
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
s
X
Denomina tor 2907
X
1
0
Values of K2 (S,X)
2
3
(18-point polynomial)
8
4
5
6
?
p:l
8
o'
t-J
s
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
(l)
21,896
1?,136
12,852
9044
5712
2856
4?6
-1428
-2856
-3808
-4284
-4284
-3808
-2856
-1428
476
2856
5712
17
1?,136
13,832
10,836
8148
5?68
3696
1932
476
-6?2
-1512
-2044
-2268
-2184
-1?92
-1092
-84
1232
2856
12,852
10,836
8981
?287
5754
4382
3171
2121
1232
504
-63
-469
-?14
-798
-?21
-483
-84
476
16
15
9044
8148
728? .
6461
5670
4914
4193
3507
2856
2240
1659
1113
602
126
-315
-721
-1092
-1428
14
5?12
5?68
5?54
56?0
5516
5292
4998
4634
4200
3696
3122
24?8
1?64
980
126
-798
-1792
-2856
13
2856
3696
4382
4914
5292
5516
5586
5502
5264
4872
4326
3626
2772
1764
602
-714
-2184
-3808
12
4?6
1932
3171
4193
4998
5586
5957
6111
6048
5768
5271
4557
3626
2478
1113
-469
-2268
-4284
11
-1428
4?6
2121
3507
4634
5502
6111
6461
6552
6384
595?
5271
4326
3122
1659
-63
-2044
-4.284
10
-2856
-672
1232
2856
4200
5264
6048
6552
6776
6?20
6384
5?68
48?2
3696
2240
504
-1512
-3808
9
1?
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
ro
s
X
Denominator 54,264
(X)
(X)
X
0
1
Values of K1 (S,X) (19-point polynomial)
4
3
5
6
?
2
8
1-3
~
9
!-J
(])
s
0
1
2
3
4
5
6
?
8
9
10
11
12
13
14
15
16
1?
18
111
102
93
84
?5
66
57
48
39
30
21
12
3
-6
-15
-24
-33
-42
-51
18
102
94
86
?8
?0
62
54
46
38
30
22
14
6
-2
-10
-18
-26
-34
-42
1?
93
86
?9
?2
65
58
51
44
37
30
23
16
9
2
-5
-12
-19
-26
-33
16
84
?8
?2
66
60
54
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
15
?5
?0
65
60
55
50
45
40
35
30
25
20
15
10
5
0
-5
-10
-15
14
66
62
58
54
50
46
42
38
34
30
26
22
18
14
10
6
2
-2
-6
13
57
54
51
48
45
42
39
36
33
30
2?
24
21
18
15
J.2
9
6
3
12
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
11
39
38
3?
36
35
34
33
32
31
30
29
28
2?
26
25
24
23
22
21
10
ao
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
9
ro
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
s
X
Denominator 5?0
())
<0
Values of K2 (S,X)
X
0
1
2
3
4
(19-point polynomial)
5
6
7
1-3
8
~
9
I-'
(I)
6
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
13107 10404
10404
8483
7956
6732
5763
5151
3825
3740
2142
2499
714
1428
527
-459
-204
-1377
-2040
-76.5
-2448 .-1156
-2601 -1377
-2499 ~1428
-2142 -1309
-1530 -1020
-663
-561
459
68
1836
867
3468
1836
18
17
7956
6732
5603
4569
3630
2786
2037
1383
824
360
-9
-283
-462
.-546
-535
-429
-228
68
459
16
5763
5151
4569
4017
3495
3003
2541
2109
1707
1335
993
681
399
147
-75
-267
-429
-561
-663
15
3825
3740
3630
3495
3335
3150
2940
2705
2445
2160
1850
1515
1155
770
360
-75
-535
-1020
-1530
14
2142
2499
2786
3003
3150
3227
3234
3171
3038
2835
2562
2219
1806
1323
770
147
-546
-1309
-2142
13
714
1428
2037
2541
2940
3234
3423
3507
3486
3360
3129
2793
2352
1806
1155
399
-462
-1428
-2499
12
-459
527
1383
2109
2705
3171
3507
3713
3789
3735
3551
3237
2793
2219
1515
681
-283
-1377
-2601
11
-1377
-204
824
1707
2445
3038
3486
3789
3947
3960
3828
3551
3129
2562
1850
993
-9
-1156
-2448
10
-2040
-765
360
1335
2160
2835
3360
3735
3960
4035
3960
3735
3360
2835
2160
1335
360
-765
-2040
9
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
to
6
X
Denominator 33,915
<0
0
Values of KlS,X)
{20-point polynomial)
8
ll'
X
0
1
2
3
4
5
6
?
8
9
0
1
2
3
4
5
6
?
8
9
10
11
12
13
14
15
16
1?
18
19
24?
228
209
190
1?1
152
133
114
. 95
?6
5?
38
19
0
-19
-38
-5?
-?6
-95
-114
19
228
211
194
1??
160
143
126
109
92
?5
58
41
24
?
-10
-2?
-44
-61
-?8
-95
18
209
194
1?9
164
149
134
119
104
89
?4
59
44
29
14
-1
-16
-31
-46
-61
-?6
1?
190
1??
164
151
138
125
112
99
86
?3
60
4?
34
21
8
-5
-18
-31
-44
-5?
16
1?1
160
149
138
12?
116
105
94
83
?2
61
50
39
28
1?
6
-5
-16
-2?
-38
15
152
143
133
126
119
112
105
98
91
84
??
?0
63
56
49
42
35
28
21
14
?
0
13
114
109
104
99
94
89
84
?9
?4
69
64
59
54
49
44
39
34
29
24
19
12
95
92
89
86
83
80
??
?4
?1
68
65
62
59
56
53
50
4?
44
41
38
11
'Z6
o'
.....
('0
8
ro
134
125
116
10?
98
89
80
?1
62
53
44
35
26
1?
8
-1
-10
-19
14
?5
?4
?3
?2
?1
?0
69
68
6?
66
65
64
63
62
61
60
59
58
5?
10
19
18
1?
16
15
14
13
12
11
10
9
8
?
6
5
4
3
2
1
0
s
X
Denominator 1330
<0
.....
Values of K2 {S,X)
X
0
1
2
3
{20-polnt p olynomial)
4
5
6
7
8
t-3
ll'
9
o'
.....,
s
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
(l)
-3339
-3681
-3633
-3195
-2367
-1149
459
2457
4845
20,349
17,289
14,459
11,859
9489
7349
5439
3759
2309
1089
99
-661
-1191
-1491
-1561
-1401
-1011
-391
459
1539
15,105
13,437
11,859
10,371
8973
7665
6447
5319
4281
3333
2475
1707
1029
441
-57
-465
-783
-1011
-· -1149
-1197
10,431
9975
9489
8973
8427
7851
7245
6609
5943
5247
4521
3765
2979
2163
1317
441
-465
-1401
-2367
-3363
6327
6903
7349
7665
7851
7907
7833
7629
7295
6831
6237
5513
4659
3675
2561
1317
-57
-1561
-3195
-4959
2793
4221
5439
6447
7245
7833
8211
8379
8337
8085
7623
6951
6069
4977
3675
2163
441
-1491
-3633
-5985
-171
1929
3759
5319
6609
7629
8379
8859
9069
9009
8679
8079
7209
6069
4659
2979
1029
-1191
-3681
-6441
-2565
27
2309
4281
5943
7295
8337
9069
9491
9603
9405
8897
8079
6951
5513
3765
1707
-661
-3339
-6327
-4389
-1485
1089
3333
5247
6831
8085
9009
9603
9867
9801
9405
8679
7623
6237
4521
2475
99
-2607
-5643
18
17
16
15
14
13
12
11
10
32,547
26,163
20,349
15,105
10,431
6327
2793
-171
-2565
-4389
-5643
-6327
-6441
-5985
-4959
-3363
-1197
1539
4845
8721
26,163
21,531
17,289
13,437
9975
6903
4221
1929
27
-- -1485
19
~2607
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
ro
Denomina tor 87,780
<0
ro
Values of K1 (S,X)
X
0
1
2
3
4
(21-point polynomial)
5
6
7
8
1-3
9
ll:l
o'
10
1-..J
(I)
8
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
410
380
350
320
290
260
230
200
170
140
110
80
50
20
-10
-40
-70
-100
-130
-160
-190
20
380
353
326
299
272
245
218
191
164
137
110
83
56
29
2
-25
-52
-79
-106
-133
-160
19
350
326
302
278
254
230
206
182
158
134
110
86
62
38
14
-10
-34
-58
-82
-106
-130
18
320
299
278
257
236
215
194
173
152
131
110
89
68
47
26
5
-16
-37
-58
-79
-100
17
290
27.2
254
236
218
200
182
164
146
128
110
92
74
56
38
20
2
-16
-34
-52
-70
16
260
245
230
215
200
185
170
155
140
125
110
95
80
65
50
35
20
5
-10
-25
-40
15
230
218
206
194
182
170
158
146
134
122
110
98
86
74
62
50
38
26
14
2
-10
14
200
191
182
173
164
155
146
137
128
119
110
101
92
83
74
65
56
47
38
29
20
13
170
164
158
152
146
140
134
128
122
116
110
104
98
92
86
80
74
68
62
56
50
12
140
137
134
131
128
125
122
119
116
113
110
107
104
101
98
95
92
89
86
83
80
11
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
110
10
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
ro
.o
s
X
Denominator 2310
(!)
Vl
(21-point polynomial)
Values of K2 (S,X)
X
0
1
2
3
4
5
6
?
1-3
8
9
~
10
1-'
(!)
8
0
1
2
3
4
5
6
?
8
9
10
11
12
13
14
15
16
1?
18
19
20
59945
48?35
384?5
29165
20805
13395
6935
1425
-3135
-6745
-9405
-11115
-118?5
-11685
-10545
-8455
-5415
-1425
3515
9405
16245
20
48?35
40451
32832
25878
19589
13965
9006
4712
1083
-1881
-4180
-5814
-6783
-708?
-6726
-5?00
-4009
-1653
1368
5054
9405
19
384?5
32832
2?599
22?76
18363
14360
1076?
?584
4811
2448
495
-1048
-2181
-2904
-3217
-3120
-2613
-1696
-369
1368
3515
18
29165
258?8
227?6
19859
1712?
14580
12218
10041
8049
6242
4620
3183
1931
864
-18
-715
-122?
-1554
-1696
-1653
-1425
1?
20805
19589
18363
1712?
15881
14625
13359
12083
1079?
9501
8195
68?9
5553
4217
2871
1515
149
-122?
-2613
-4009
-5415
16
13395
13965
14360
14580
14625
14495
14190
13710
13055
12225
11220
10040
8685
7155
5450
35?0
1515
-?15
-3120
-5?00
-8455
15
6935
9006
1076?
12218
13359
14190
14711
14922
14823
14414
13695
12666
1132?
9678
?719
5450
28?1
-18
-321?
-6?26
-10545
14
1425
4?12
7584
10041
12083
13?10
14922
15?19
16101
16068
15620
1475?
13479
11786
96?8
7155
421?
864
-2904
-?08?
-11685
13
-3135
1083
4811
8049
10?9?
13055
14823
16101
16889
1718?
16995
16313
15141
134?9
11327
~ 8686
5553
1931
-2181
-6?83
-118'£5
12
-6?45
-1881
2448
6242
9501
12225
14414
16068
17187
1?7?1
17820
1733!
16313
14757
12666
10.040
68?9
3183
-1048
-5814
- 11115
11
-9405
-4180
495
4620
8195
11220
13695
15620
16995
17820
18095
1?820
16995
15620
13695
11220
8195
4620
495
-4190
-9405
10
20 t\')
19
18
1?
16
15
14
13
12
11
10
9
8
?
6
5
4
3
2
1
0
8
X
Denominator 168,245
<0
~
95
Table 3
Values of R
q
n
1
2
1
1
2
2/3
2
3
5/9
1
4
1/2
5
7/15
6
3
n
4
5
6
5
2
14
14/25
6/5
14/3
42
4/9
12/25
6/7
22/9
12
7
3/7
3/7
33/49
11/7
39/7
33
8
5/12
11/28
55/98
143/126
13/4
55/4
9 11/27
11/30
143/294
143/162
13/6
22/3
26/75
13/30
13/18
52/33
68/15
91/275
13/33
182/297
442/363
34/11
10
2/5
11 13/33
7/10
132
12
7/18
7/22
4/11
476/891
119/121
323/143
13
5/13
4/13
340/1001
68/143
1292/1573
3230/1859
14
8/21 136/455 204/637
15 17/45
51/175 969/3185
16
57/200
57/196
17 19/51
19/68
19/68
18 10/27
14/51
19
3/8
1292/3003 7752/11011 2584/1859
323/819
969/1573
969/845
19/52
57/104
1311/1352
209/612
437/884
4807/5746
55/204
1771/5508
23/51
6325/8619
77/285 253/969
1771/5814
805/1938
1265/1938
20 11/30 253/950 506/1995 2530/8721 1495/3876
759/1292
7/19
96
BIBLIOGRAPHY
Jordan , C. H., Calculus of Finite Differences , Budapest ,
1939 .
Milne-Thomsen , L . M., The Calculus of Finite Differences ,
MacMillan and Co . Ltd ., London , 1933 .
Rider , Paul , Statistical Methods , John
New York , 1939 .
v~iley
and Sons ,
Sasuly , ax , Trend Analysis of Statistics , The Brookings
Institution , vashington , D:c ., 1934 .