90 
MAJOR, P. A. MacMAHON : SEVENTH MEMOIR ON 
which does not involve the elements a, /3, y, ... is unity. 
Hence the portion of 
FFQAQA • • • Q/" 
that is free from the elements is 
F q (m; 1*’2 ;<2 ... A), 
which is obtained directly from the result of the operation by putting 
Qi = Q 2 = ••• — Q, = l- 
We may represent this circumstance by the convenient notation 
* (IWV •.. Q*'*% = 1 = F ? (m ; 1*’2* 2 ... i ki ). 
The number of partitions, of the unipartite number m, into k or fewer parts, is by 
the present investigation 
DJMQ) q = 1 
= z (D.W... Q/'*)q = i 
l k '2 k > ... i ki . k x \k 2 \ ... k L ! 
F 9 (m ; l* l 2* 2 ... i ki ) 
1*‘2* 2 ... i ki . l\ \k,\ ... hi ! 
m • j. r m- v (1 — x)~ k ' (1 — x 2 ) * 2 ...(l— x { ) ki 
= coefficient of in 2 ^ . ki -yyf- 
h \ Jc 2 \ ... hi ! 
the summation being for every partition 
ki~\-2k.>-\- ... + 
of the number k. 
But we know, otherwise, that the number of partitions of m, into k or fewer parts, 
is given by the coefficients of x m in 
_1_ 
(l— x) (1 — X*) ... (l —x k ) 
Hence the identity 
y (t — x)~ kl (l — x 2 )~ kt ... (l — x')~ k ' _1_ 
H2* 2 ... i k % \k 2 \ ... /c,! — (1 — x) (1 — x 1 ) ... (l — x k ) ’ 
which being a known result supplies an interesting verification of our work. The 
present investigation in any case supplies one proof of it. 
Art. 9. There is now no difficulty in proceeding to the result 
DA 
D*. • Q/ QA 
H2* 2 
%■). F ? (m 2 ; B‘2* 2 ... i ki ) 
F (m,; H2* 2 ... i ki ). QrQA ... Q 
ki 
