108 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
We deduce that the multipartite number 
666 ... s times 
has only one partition of the nature we consider, and this of course is quite obvious. 
That the multipartite numbers 
have each of them 
% 
the multipartite numbers 
have each of them 
and the number 
555 ... s times, 111 ... s times 
-g- (3 s + 3) partitions ; 
444 ... s times, 222 ... s times, 
g (6 s + 3 . 2 s ) partitions ; 
333 ... s times 
-g (7 S + 5) partitions. 
lias 
Also the multipartite number 
333 ... s times, 222 ...t times 
i(7 s . 6‘ + 3. 2 s ) 
partitions, and various other results. 
The symmetry shown above in the case of multipartite numbers is of general 
application in the subject and is very remarkable. I do not see any other a priori 
proof of it at the moment of writing. 
Art. 18. In general, we consider the case in which no constituent of the multipartite 
parts is to exceed the integer j. We Strike out from the functions Q, all partitions 
which involve numbers exceeding j and reach the infinite series of functions 
j . j.!. 
These functions are operated upon in the manner 
D t J t - D 2i J, = D :k J t = • • • = DjiJi = ; 
while every other Hammond operator causes J, to vanish. By the same reasoning as 
was used in the special cases we find that 
where 
... J/“ = F ) (m ; 1*'2*’... i ki ) . W ... J/’, 
Fj(m; l /,:i 2 fe ... A) 
is equal to the coefficient of x m in the expansion of the function 
(l + x +... + x’f 1 (l + x 2 +... + x 2j Y 2 ... (l +x'+ ... +x ij ) kt , 
