110 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
We have presented to us the generating function 
where 
1 +aU, 
+fi(u, 2 +u 3 ) 
+ |J(U 1 “+3U 1 U j + 2U 3 ) 
+ ... 
+ a k W (U) 
+ ... 
F*(U) = ^ 
U/'U/' 2 ... U/* 
l* 1 . 2* 2 ... i ki .k x \ k 2 \... k ( ! 
The effect of D m upon the product 
U/HJ/ 2 ... U/‘ 
is to multiply it by the coefficient of x m in the function 
(1 + x u ' + af* +... + x u, ) kl (1 + x 2u ‘ + x 2u ’ 2 + ... + x 2u ‘f 2 ... (1 + x iu ' + x iu2 + ... + x iu ‘) k \ 
which I write in the abbreviated notation 
V iiV *2 Y ki 
• 
If in the expression of F /t . (U) we write X Sii for U s we must reach the expression which 
enumerates the partitions of unipartite numbers into k or fewer parts, such parts 
being drawn from the series u x , u 2 , u s ... u s . That is to say we must arrive at the 
coefficient of a k in the expansion of 
_1_ 
(l— a) (l—ax u ') (l —ax 112 ) ... (l —ax 11 ') 
Hence this coefficient has the expression 
v tv a-2 y 
Y A 2 u • • • -W 
ki 
H.2 k *...i ki .k 1 \k 2 \...k i \ 
To enumerate the partitions we have 
