THE PARTITION OF NUMBERS. 
9. r ) 
Associated with another unipartite number m 3 we have the linear function 
\a 2 T f^b 2 T - i'C 2 T- ... . 
It has then been shown that the number of partitions, into k or fewer parts, of the 
multipartite number m 1 w 2 is 
A a x a 2 + mVa + vc x c 2 + ... , 
and in general the number of partitions, into k or fewer parts, of the multipartite 
number 
is 
m l m 2 ... m s 
A a x a 2 ... a s + fxbf 2 ... b s + ve l c 2 ... c s + 
The multipartite solution is thus essentially derived from the solutions which 
appertain to the separate unipartite numbers whose conjunction defines the multi¬ 
partite number. The numbers A,//, v, ... are those well known in connexion with the 
expression of the homogeneous product sum h k in terms of the sums of the powers 
s u s 2 , s 3 , ... s k , the whole question is therefore reduced to finding the numbers 
a,b, c, ... 
appertaining to the unipartite number m. 
This, as has been shown, depends upon finding the coefficient of x m in a function 
where 
(l— x) k, (l—x 2 ) k '...(l—x') ki 
ki H" k 2 +... + k{ — k. 
The possibility of the solution rests upon the remarkable circumstances that when 
the operator D,„ is performed upon the operand 
0'0 k 2 0 ki 
Hq H»2 * * * Hrt 
its effect is to merely multiply it by an integer. 
The Partitions of Multipartite Numbers into Three Parts. 
Art. 13. I will, in future, merely deal with the partitions into k or fewer parts, since 
the result for exactly k parts is at once derived by subtracting the result for k— 1 or 
fewer parts. 
The operand is 
(Qi 3 + 3QiQ 2 + 2Q 3 ), 
and since the result depends upon the divisibility of m by both 2 and 3 it will be 
necessary to consider the operations of 
1 Am> f-Am + lj ld(j m ^_2> +3? ^ bm + 4? 
6m+ 5* 
