898 
MAJOR P. A. MACMAHON ON THE THEORY 
Q ~p ^p- 1)“"' + (fj ^p “ 2)""' - • • • + {-y^'^p} 
= A" 
in the notation of finite differences. 
We have now the well known identity 
^ P' ^ (y.-l) _|_ Q ^2 (0-1) + ■ ■ . + P) A' (0*-') + • ■ ■ + P) A' (0-1), 
and we have seen its interpretation in the theory of the combinations of order h of a 
given unipartite number m. 
Tables of Compositions of Multipartite Numbers. 
No. 
1 
part 
Total 
1 
No. 
2 
P 
parts 
1 
. 
i 
i 
1 
2 
T otal 
3 
No. 
] 
2 
3 
3 
1 
2 
1 
21 
1 
4 
3 
P 
1 
6 
6 
parts Total 
... 4 
... 8 
... 13 
No. 
L 
2 
3 
4 
4 
1 
3 
3 
1 
W[ 
1 
6 
9 
4 
2‘^ 
], 
7 
12 
6 
1 
10 
21 
12 
P 
1 
14 
36 
24 
. . 8 
. . 20 
. . 26 
. . 44 
. . 75 
