THE PARTITION OF NUMBERS. 
91 
and the number of partitions of the multipartite number 
m x m 2 ... m s 
into k or fewer parts, is 
D mi D n!j • • • Dm.F/c (Q)q = i 
_ y (D»nD„t, • • • • • • Q, % = i 
1 /ci 2 Aj ... i k '. % \k 2 \ ... k t ! 
_ y F„(m 1 ; \ k '2 k ' ... %') . F„ (m 2 ; 1*‘2 hl ...i ki ) .F„ (m,; 1*'2*’ ,, 
_ A l ki 2 k *... i k \ k x ! k. 2 \ ... k t ! 
This is the general solution of the problem of enumeration in the absence of any 
restriction upon the magnitudes of the constituents of the multipartite parts.* 
Art. 10. It will be convenient at this point to give a few results derived from the 
function 
(1 -x)~ kl (1 -x 2 )~ kt ... (1 - x')~ ki 
■ t*) 
which will be useful in the sequel. 
F s (m; l‘) = ( m ^F)’ 
F s (2m; 2*) = ( m ^- 1 ), 
F ,(m; 0 = ( OT ^7 1 
F ? (2 m; 12) = m+1, F ? (2m+1; 12) = m+1, 
F ? (2m ; 1 2 2) = (m + l) 2 , F ? (2m +1 ; 1 2 2) = (m + 1) (m + 2), 
F,(2m; IV) = (“l 2 ), F,(2m + 1 ; 12 s ) = ( m t 2 V 
* With regard to the algebraical identity met with above, the reader may compare ‘ Sylvester’s 
Mathematical Papers,’ vol. III., p. 598, where it is shown that for the roots of the equation 
yi --£-+-+ 
1 — c (1 - f) (1 - c' 2 ) (1 -c) (1 -c 2 ) (1 -c 3 )' 
+ • •. = 0, 
the general term being 
,(?) 
7 q~n 
(1 - c) (1 - C 2 ) ... (1 - C n ) 
the homogeneous product-sum of order n is 
1 
(1 — c) (1 -c 2 ) ... (1 -c n )’ 
and the sum of the n th powers of the roots is 
1 
1 - c n 
The expression of the homogeneous product-sum of order k, in terms of the sums of the powers, by the 
formula quoted early in this paper, gives the identity in question. 
o 2 
