THEORY OF THE PARTITION OF NUMBERS. 
635 
Art. 22. Again, the partitions of all unipartite numbers into s different parts, limited 
in magnitude to p and in number to q, are enumerated by the coefficient of a q li s x pq in 
the development of the product 
1 
abx 
I 1 - 4 - 
1 — x 1 — a 1 — ax 
1 + 
abx 3 
1 — ax 2 
• 1 + 
abxP 
1 — axP / ’ 
or by the coefficient of a q x pq in 
a s 
x 
/i'j + 7c 2 + Jc 3 + . . . + k's 
1 — x . 1 — a ^ (1 — ax kl ) (l — axf (1 — axff ... (1 — ax k ‘) ’ 
where k x , k 0 , Jc 3 ,.. . k„ are any s different numbers drawn from the natural series 
1, 2, 3, . ■ p ; 
and the summation is in respect of all such selections. This is the coefficient of 
ai~ s xPi- O 2 1 ) 
m 
x 
l'i + 1*2 + &3 + .. . 4 - ht — "2 ^ 
1 — x . 1 — a (1 — a®* 1 ) (1 — axff (1 — axff ... (1 — ax h ) 
Taking the former of these two expressions ; inasmuch as we know from the 
(p\ AT 
reticulation theory that the coefficient of a q x pq is p j y j, we find that the effective 
portion of the generating function is 
which is 
(A (■ axP ) s . 
\sj (1 — axP) s+1 
. ad inf., 
viz., we have succeeded in isolating that portion of the generating function which 
proceeds by powers of axP only. 
The latter expression of the generating function also, is seen to have an effective 
portion 
p \ xp s ~( 2 ) 
s / (1 — axP) s+1 
The isolations thus effected would, I believe, be difficult to accomplish algebraicall 3 T . 
4 m 2 
