112 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
Also 
DGDTMi (u,»+3U,U 3 + 2 U 3 ) = 4(6''. 3”. 9'U,*+3XT.tr,), 
where the term 311,11/ only appears if a- l and <r 2 are both zero. 
Hence in general, if a- x and a-., are not both zero, the multipartite number 
12 - 11 ^ 10 ^ 
possesses G 0- ' 1 . . 9 0-3 partitions of the [nature considered. In particular the multi¬ 
partite number 
12 12 11, for <T ! = 2, <r a = 1, <r 3 = 0, 
possesses the 18 partitions 
(775 553 003), 
(755 573 003), 
(573 705 053), 
(775 053 503), 
(575 753 003), 
(753 073 505), 
(773 555 003), 
(755 073 503), 
(573 703 055), 
(773 553 005), 
(575 703 053), 
(555 073 703), 
(773 505 053), 
(753 573 005), 
(553 705 073), 
(773 055 503), 
(753 075 503), 
(553 075 703). 
Art. 20. In considering the partitions of the multipartite number m x m 2 ...m s the 
partitions of the unipartite constituents of the number have been regarded as being 
subject to the same conditions and restrictions. This, however, is not necessary 
except in the case of the number of parts which has been denoted by Jc. 
We may for m 1 choose any of the restrictions that have been denoted by the 
symbols A, B, ... J ... Q, U. For m 2 similarly and so on. For instance, suppose the 
numbers rn } , m 2 , on 3 are subject to the restrictions denoted by B, Q, and U ; that is to 
say, the partitions of m x are such that no part exceeds 2 ; the partitions of m 2 are 
unrestricted ; the partitions of m 3 are such that the parts are drawn exclusively from 
specified integers u x , u 2 , ... u s . 
For the partitions of the multipartite number m 1 m 2 ...m s , subject to this 
combination of restrictions, we 
(i.) Take 
D w B/B/ ,..B/ 
l*'2 fa ... i ki . h ! k 2 \... L ! 
Z-.z-xZ-, •/ - \ 
B/ B/ 2 ... B/ 
with the result 
