246 
RESEARCHES ON THE PARTITION OF NUMBERS. 
[140 
where fx is a product of factors of the form 1 — x a , and cf>x is a rational and integral 
function of x. And it is clear that the fraction will be a proper one when each 
of the fractions in the original expression is a proper fraction, i.e. in the case of 
P(0, 1, 2 ... k) m \(km — cl), when for k even, a<\k(k+2.), and for k odd, a<i(& + l)(/c+3); 
and in the case of P'(0> 1, 2 ... k) m \{km-cl), when for k even, a+1 < %k (k+ 2), and 
for k odd, cl + 1 < | (k + 1) (k + 3). 
We see, therefore, that 
and 
are each of them of the form 
P (0, 1, 2 ... k) m \{km — a), 
P' (0, 1, 2 ... k) m km — cl), 
coefficient x m in 
(px 
f*' 
where fx is the product of factors of the form 1 — x a , and up to certain limiting values 
of a the fraction is a proper fraction. When the fraction ^ is known, we may there- 
jx 
fore obtain by the method employed in the former part of this Memoir, analytical 
expressions (involving prime circulators) for the functions P and P'. 
As an example, take 
which is equal to 
coefficient x 3m in 
— coefficient x m in 
P(0, 1, 2, 3) m \m, 
1 
(1 - X 2 ) (1 - X i ) (1 - X 6 ) 
1 
(1 - X 2 ) (1 - X 2 ) (1 -~P)' 
The multiplier for the first fraction is 
which is equal to 
(1 - x 6 ) (1 - a; 12 ) 
(1 — X 2 ) (1 — £C 4 ) ’ 
1 + x 2 + 2x * i + ocf + 2 ¿c 8 + x 10 + x 12 . 
Hence, rejecting in the numerator the terms the indices of which are not divisible 
by 3, the first term becomes 
coefficient x 3m in 
1 + of + x 12 
(1 - a?) (1 - x 12 ) (1 - x 6 ) ’ 
or what is the same thing, the first term is 
coefficient x m in 
1 + X 2 + X* 
(l—a 2 ) 2 (i—O’