128 
MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
where I(j?) denotes the integral part, and the &c. refers to the fractional terms 
depending upon the other multiple factors, such as [l —a?®]*. The functions 6x are 
to be considered as functions with indeterminate coefficients, the degree of each 
such function being inferior by unity to that of the corresponding denominator; and 
it is proper to remark that the number of the indeterminate coefficients in all the 
functions 0x together is equal to the degree of the denominatoi-yjr. 
The term {x'b^Y~'^ [i— reduced to the form 
4.&C 
[1 — [1 — ,2?“]*“' ' ’’ 
the functions gx being of the same degree as 6x, and the coefficients of these 
functions being linearly connected with those of the function 6x. The first of the 
foregoing terms is the only term on the riglit-hand side which contains the denomi- 
nator [l— .r®]*; hence, multiplying by this denominator and then writing [1— 
we find 
which is true when x is any root whatever of the equation [1— . 2 ;“]= 0 . Now by 
means of the equation [l — .r®]=0,^ may be expressed in the form of a rational and 
integral function Gj;, the degree of which is less by unity than that of [t — x“]. We 
have therefore an equation which is satisfied by each root of [l — x®]=0, 
and which is therefore an identical equation ; gx is thus determined, and the 
coefficients of ^x being linear functions of those of gx^ the function Qx may be con- 
sidered as determined. And this being so, the function 
fx [1-a;®] 
will be a fraction the denominator of which does not contain any power of [l — jc®] 
higher than [l— a;®]'""'; and therefore d^x can be found in the same way as 6x, and 
similarly d^x, and so on. And the fractional parts being determined, the integral part 
may be found by subtracting from ~ the sum of the fractional parts, so that the 
JX 
(bX 
fraction ^ can by a direct process be decomposed in the above-mentioned form. 
JX 
Particular terms in the decomposition of certain fractions may be obtained with 
great facility. Thus m being a prime number, assume 
\ o , . 
{\—x’^)[\—x^)..[\—x^) ’”^[1— 
then observing that (1 — = — x)[l — a?”*], M^e have for [l — x^~\—0, 
0x= 
\—x)[\—x‘^)..[i—x'" 
