MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
137 
where fx is the product of factors of the form 1 — x", and up to certain limiting’ values 
of a the fraction is a proper fraction. When the fraction^ is known, we may there- 
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 P(0, 1, 2, 
which is equal to 
coefficient in 
-coefficient x” in 
The multiplier for the first fraction is 
(1 — a7®)(l — 
(1— a;^)(l — ’ 
which is equal to 1 2x^-1- 
Hence, rejecting in the numerator the terms the indices of which are not divisible 
by 3 , the first term becomes 
coemcient x* m 7; 12T71 gv? 
or what is the same thing, the first term is 
coefficient x” m ’ (i 5 
and the second term being 
x^ 
—coefficient x”* in 7-, 2^27; 47> 
(1 — xd (1 — x^) 
1 “ 1 “ 
we have P(0, 1 , 2, 3 )'"f /w = coefficient x” in 
And similarly it may be shown, that 
P(0, l, 2 , 3 )’"i( 3 m— l)=coefficient x” in 
As another example, take P'(0, 1, 2, 3 , 4 , 5 )f m, 
which is equal to 
coefficient in 
-coefficient in (Ti:si)(ii:^4^r^(rris, 
+coefficient a:" in 
The multiplier for the first fraction is 
(1-X2‘>)(l-X3«)(l-X^0) 
