894 
MAJOR P. A. MACMAHON OR THE THEORY 
in the product 
{“i + ^ («3 + • • • + ^ («3 + • • • + • • • {^1 + ''^3 + • • ■ + Ci„Y‘ 
gives the theorem in compositions :— 
“ The number of compositions of the multipartite 
Ihlh’-Pn 
which jDOssess exactly 5 contacts of 
n — u zeros witli — 1 zeros, 
subject to the condition u > t, is 
Qm\+'P 2 + 
where is the coefficient of 
in the above-mentioned product.” 
Vn 
79. Also the further theorem :— 
“ The number of compositions of the multipartite 
P\V% • • • P'H 
which possess exactly 
contacts of n — u-^ zeros with — 1 zeros. 
'®«»4 ” ” ^3 ' ^ ” 
is the product of 
and the coefficient of 
in the product 
2lh+lh^r . . . [-5 _ 
X Sui^i\ '^“2^2 
^lhr^ Vi 
a„ 
(“1 + ^3l“3 + • • • H~ + <^3 "b ^33^3 + • • • ~b + “3 + • • • + 
80. The generating function which enumerates the combinations of order h of 
multipartite numbers may be written 
where 
i + JY 
k ^ k-I 
+ 
H'2 
(t-1)2 + 
H'3 H'“ 
- 4- -I- — -4- 
(k _ 1)3 ^ ^ (Z; - 1)'« “ • • 
a\ = {k — 1) a'o = {k — 1) a.,, See. 
h'^ is the homogeneous product sum of degree m of the quantities 
1 ) 35 • • • ^ n ) 
