140] RESEARCHES ON THE PARTITION OF NUMBERS. 
Now putting for a moment x — x^ 6 , we have 
249 
1 1 = 1 - 
Xj^ — xe* ^ (1 — e 8+t ) x 1 (l — e d ) 6 x x {\ — e 6 )^ ' 
1 
and de = xd x , whence 
—-— t = —- 1-1 xd x — h ~ (xd x y 
Xi — x& x 1 — x 1 x 1 — x 1.2 
x, — x 
+ 
the general term of which is 
n(s-l) 
Hence representing the general term of 
09*)* 
X-, — X 
by x x J~ s > so that 
X x (f) (x^ 1 ) 
/OiO 
%Xi = coefficient - in t s 
1 «-i ^iOi e_i ) 
< ” ' /Oie _t ) ’ 
we find, writing down only the general term, 
= • • • + fT7~—T\ O^)® -1 
/«J*. n (s — 1) v 2:7 
¿»i — x 
+..., 
where the value of x x i depends upon that of s, and where s extends from s = 1 to s = k. 
Suppose now that the denominator is made up of factors (the same or different) 
of the form 1 — x m . And let a be any divisor of one or more of the indices m, 
and let k be the number of the indices of which a is a divisor. The denominator 
contains the divisor [1 — x ai y, and consequently if p be any root of the equation 
[1 — x a ~\ = 0, the denominator contains the factor (p — x) k . Hence writing p for x x and 
taking the sum with respect to all the roots of the equation [1 — = 0, we find 
№\ = , i 
Ml i-*“] n(s-i) 
=... + 
W- 1 s 
p — X 
n( S -l) 
(xd x ) s 1 
6x 
[1 — ¿c a ] 
+ 
where 
XP = coefficient ^ in t s ~ x 
t /(pO 
and as before s extends from s = 1 to s = k. We have thus the actual value of the 
function Qx made use of in the memoir. 
A preceding formula gives 
^-coefficient lin 1 * ( ° 
f X . 
t 1 - x* f(e- 1 ) ’ 
<f)X 
which is a very simple expression for the non-circulating part of the fraction ^ . 
This is, in fact, Mr Sylvester’s theorem above referred to. 
C. II. 
32