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VII. Researches on the Partition of Numbers. By Arthur Cayley, Esq. 
Received April 14, — Read May 3 and 10, 1855. 
I PROPOSE to discuss the following problem ; “ To find in how many ways a 
number q can be made up of the elements a, b,c,.. each element being repeatable an 
indefinite number of times.” The required number of partitions is represented by 
the notation F(a,b,c,..)q, 
and we have, as is well known, 
P(a, b, c, ..) 9 = coefficient x'^ in 
where the expansion is to be effected in ascending powers of x. 
It may be as well to remark that each element is to be considered as a separate 
and distinct element, notwithstanding any equalities which may exist between the 
numbers a, b, c, .. ; thus, although a=b, yet a+a+a+ &c. and + &c. are to 
be considered as two different partitions of the number q, and so in all similar cases. 
The solution of the problem is thus seen to depend upon the theory, to which I now 
proceed, of the expansion of algebraical fractions. 
Consider an algebraical fraction 
where the denominator is the product of any number of factors (the same or different) 
of the form \ —x'^. Suppose in general that [l— j-”*] denotes the irreducible factor 
of 1 — x™, i. e. the factor which, equated to zero, gives the prime roots of the equa- 
tion 1— a:'“=0. We have 
1— a7“=n[i— x“'], 
where m' denotes any divisor whatever of m (unity and the number m itself not 
excluded). Hence, if a represent a divisor of one or more of the indices m, and h 
be the number of the indices of which « is a divisor, we have 
fx=X\.\\—x^~f. 
Now considering apart from the others one of the multiple factors [_\~x’'f, we 
may write fx= [l —x'^ffx. 
Suppose that the fraction^ is decomposed into simpler fractions, in the form 
<px 
fa 
= 1 ( 1 ) 
+ &C., 
[1 — a?“] 
k-2 
[1— a?®] [1— a;®] 
s 2 
