134 MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
P(7, 8)?=i^|2j+43 
+ 7 (1,-1) per 2, 
+ 14 (1, -1, — 1, 1) pcr4, 
+ 16(3, 2,1, 0,-1, -2, -3) pcr7, 
+56 (0, - 1, - 1, 0, 0, 1. 1, 0) per 8,|, 
vyhich are, I think, worth preserving. 
Received April 14, — Read May 3 and 10, 1855. 
I proceed to discuss the following problem : “ To find in how many ways a number 
q can be made up as a sum of m terms with the elements 0, 1,2, ...h, each element 
being repeatable an indefinite number of times.” The required number of partitions 
is represented bv 
p(o, 1 , 2 , 
and the number of partitions of q less the number of partitions of ^—1 is repre- 
sented by 
P'(0, 1,2, ..A-)> 
We have, as is well known, 
1 
P(0, 1, 2, ..kYq= coefficient in 
x'^z) 
where the expansion is to be effected in ascending powers of 2 . Now 
(1— 2 ')(l — a72’)..(l — x^z) ' \—x ^ (1 — a7)(l — ' *■’ 
the general term being 
(l-^ft+l)(l-^fc + 2)_^(l— ^fc + . 
(1 — x) (1 — x ’^) . . (1 —x'^) 
or, what is the same thing. 
and consequently 
(1 — a:'«+q(l — 
(1 — ^)(1 — a .'^) ..(1 — a ;*) 
P(0, 1, 2, ..krq= coefficient in 
to transform this expression I make use of the equation 
(,+..)(! + A) .. ( 1 +.*z)= 1 
X 
,is{s+0 
( 1 — «) (1 — a?^) . . (1 — a?®) 
where the general term is 
