THEORY OF THE PARTITION OF NUMBERS. 
637 
Hence we can isolate that portion of the generating function which contains only 
powers of ax?. It is 
i + (i) axP + ( l> 2 X ) ( axP f + 3 ~j ( axP f + •••••; 
or 
_ 1_ 
(1 — axP)P 
This fact leads as before to an expansion theorem. 
Putting; 
(1 ~ axP)P __ (1 _ azt-iy-i _ 
1 — x .1 — ax .... I — axP 1 — x . 1 — ax .... 1 — axP _1 p ’ 
then of course 
1 1 p = '-p 
- - - = - - - S V (r) 
1 — x . 1 — ax . ... 1 — axP (1 — axP)P v = i p ' 
and we find in succession 
y 2 (x) 
V 8 (») 
V* (X) 
ax 
ax’ 
ax - 
+ 
ax 1 (1 — ax 3 ) 
1 — ax 1 — ax . 1 — ax 3 
ax? (1 — ax?) ax? (1 — ax*) 
1 — ax . 1 — ax 3 1 — ax . 1 — ax? 1 
ax 3 (1 — ax 4 ) 3 
ax .1 — ax 3 . 1 — ax 
,3’ 
and in general 
y , . _ ax? 1 (1 — axP~ 1 y~ i (l — axP) 
11 ' ' 1 — ax . 1 — ax 3 .1 — ax? -3 1 — ax . 1 — ax 3 .... 1 — ax? -3 
ax? -1 (1 — axP)P~ 3 axP~ l (1 — ax?)?" 3 
+ 
axP 1 (1 — axP 1 )p 3 
Y~- 
+ 
+ 
1 — ax . 1 — ax 3 .... 1 — axP 3 1 — ax. 1 — ax 3 _1 — axP 3 . 1 — axP~ v 
The simplest cases, omitting the trivial one corresponding top = 1, are 
1 
1 
1 ax 
1 — x. 1 — ax. 1 — ax 3 (1 — ax 3 p [ 1 — x 1 — ax 
1 1 r 1 
1 — x. 1 — ax. 1 — ax ?. 1 — ax 3 (1 — ax 3 ) 3 11 — x 1 — ax 1 — ax 1 — ax. 1 — ax 
ax 
+ ;—-4 
ax 3 ax 3 (l — ax 3 ) 
