NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 411 
(29.) We have therefore now only to consider the remaining portions, which we 
shall call Z, viz. 
Z=Q^_i+Q^_,_i +....?/ terms, 
may represent any circulating function. Suppose it to be such that 
Qx-l = %o(-^) + 
and let any term of this, as (which for brevity we will write simply 
putting z=x—i), be separately considered. Let R be the portion of Z which origi- 
nates from this term. Then 
\ 
B.=-)c{x) .m^+-)C(^—s) •'>^.- 2 . iy terms). 
Let ts=:n be the first multiple of s, which is also a multiple of in. Then after t terms 
the value of &c. will recur, and therefore R resolves itself into t separate 
series, as follows : 
R 
1 
Now we have 
whence 
and similarly. 
+%(^— n)-\-x{x—2n) + .... + 1 ) terms j 
s)-\- (== + 1^ terms 
+ — 2^) + %(x — w — 25) + . . . . + 1 ) te r 
+ &c. {t series). 
t t'y ' t 
y-\ y-\ 
t 
t 4- -i-f . 
y-2 y-2 ^ 
t 
y-3 
^„_ 3 + .. . 
t t 
y-t-l^ 
~l t "y-^~ t t -y-^ t 
and so on. But by equation (25.), since st—n, we have 
u oc \ 
If therefore we put 
Px—-\ 0-w^+ 1 1 )— 1 
we shall have 
= — 1 .... 1 
+ — n^-s+i 
%-2s+l 
&c.. 
■'==^-ip.+g'.) i+==l-(?*+7i^) ; &c., 
3 G 2 
