MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
135 
and the series is a finite one, the last term being that corresponding to s=k, viz. 
Writing —x^ for z, and substituting the resulting value of 
(1 — 0 ?™+') ( 1 — 0 ?“+") . . ( 1 — 
in the formula for P(0, 1, 2, we have 
C ^m+is,s+l 1 
P(0, 1,2, coefficients" 
where the summation extends from .y=: 0 to s=k; but if for any value of s between 
these limits 5md-|^y(.y4-l) becomes greater than q, then it is clear that the summation 
need only be extended from s=0 to the last preceding value of s, or what is the same 
thing, from ^=0 to the greatest value of s, for which q — sm — ^ 5 ( 5 + 1 ) is positive or 
zero. 
It is obvious, that if q>km, then 
P(0, 1, 2..A:)’"^ = 0; 
and moreover, that if dA^\km, then 
P(0, 1, 2, ..^)’”^=P(0, 1, 2..kY.km—6, 
so that we may always suppose q^y^km. I write therefore q=\{km — Di) where a is 
zero or a positive integer not greater than km, and is even or odd according as km is 
even or odd. Substituting this value of q and making a slight change in the form of 
the result, we have 
I 1 
P(0, 1, 2..A-)-|(fa-.,) = 2,|(-)-coeffixa»-"--in-,— 
where the summation extends from s=-0 to the greatest value of s, for which 
{^k—s)m—\a—\s{s-^\) is positive or zero. But we may, if we please, consider the 
summation as extending, when k is even, from ^=0 to . 9 =^/::— 1 , and when k is odd, 
from . 9=0 to s—^{k — 1 ), the terms corresponding to values of s greater than the 
greatest value for which {\k — s)m—\a,—\s{s-\-\) is positive or zero, being of course 
equal to zero. It may be noticed, that the fraction will be a proper one if 
a<{k — 9 )(^ — A'+l); or substituting for s its greatest value, the fraction will be a 
proper one for all values of .9, if, when k is even, a<^k{k-{-2), and when k is odd, 
c<l{k^l){k-{-d). 
We have in a similar manner, 
P'( 0 , 1,2 ... /f)’"o= coefficient in — r- 7 ^ r\’ 
which leads to 
f „ia+i(s+l) 
P'( 0 , \,2..kY^{km — a) = ^A (— l^coeff. .r<**“''""in gr — j - — — ^ ^ ^ j- 
where the summation extends, as in the former case, from 5=0 to the greatest value 
of 5 , for which (^A:— 5)m— ^ 05 — ^ 5 ( 5 + 1) is positive or zero, or, if we please, when k is 
even, from 5=0 to 5=^/1:— 1, and when 5 is odd, from 5=0 to 5=^(A:— 1). The con- 
dition, in order that the fraction may be a proper one for all values of 5, is, when k is 
even, a-f-l <^k(^k-\-2), and when k is odd, os-j-l l)(A:-l-3). 
T 2 
