892 
MAJOR P. A. MACMAHOI^ OR THE THEORY 
for 
M - N = {k^ - k {k-l)]h.,- {F - /,• (Z; - 1 ) 2 ] 63 + . . . + {-Y - 1 )""^] 
and thence 
D b D A 1 - Sj (1 - S.) (1 - SJ . . . (1 - S.) 
75 . The investigation proceeds precisely as in the case of Z: = 2 with the following 
result. The fraction 
1___1__ 
{1 + «o + . . . -f «h)} {1 — ^2 (^“1 “t ka^ + . . . + ajj)} • ■ • {1 — (Zi«i + kci^ + . .. + Z;a„)} 
is equal to the product of the fraction 
1_1_ 
k 1 — + k (Jc — 1) 2s;^S2*l‘^3 — . . . + (— y^k [k — 1)"“^S2^S2 • • • . . . a„ 
and the series 
, , ^ k (A/^ + (Af^ + g/J ■ . ■ (A4 + x/J — + kx^y) (At^ + kxty) . . . (A/„ + 
(k- 1) (1 - S,^) (1 - S,,) ... (1 - SJ 
the summation having regard to every selection of it integers from the series 
1 , 2 , 3 , . . . n, and u takes all values from 1 to n — 1 . 
As in the former case the relation 
which becomes 
Ai + kat — A^+j^ + cLt+i, 
A/„ + kaf^ — k (Af^ + a/) 
where and are the highest and lowest sufBxes present, shows that the terms 
under the summation sign do not involve any products of 52 ^ 2 ? • ■ • 
and therefore as far as concerns the generating function may be put equal to zero. 
7 6. Hence the number of combinations of order k of the multipartite number 
Zh Ih- ■ 
is the coefficient of iii fke expansion of the generating 
function 
1 _ 1 __ 
k fl — S]^ (,k^\ H + . . . A “«)} {1 — % (^*^1 A kx^ A • • ■ A ‘^k)} ... {1 5„ (Z'a^ A ... A ^^m)} 
