882 
MAJOR P. A. MACMAHOX OX THE THEORY 
61 . The vinsymmetrical product may be written in the form 
i {«3 + . . . + {«! + «o + i(«3 + . . . + • • 
...{«! + aj + ... + a,,}/" 
and herein the coefficient of . . . a/- may be written 
2 C, (ly = tCs2^P-^-\ 
where C^, C,, Co, . . . are certain integers. 
Cbserve that is the coefficient of . . . a/" in tlie product 
+ A (a^ + . . . + {«! + ag + X (ag + . . . + a„)]P"~ . . . _j_ + . . . + a„]P' 
Considering the reticulation of the multipartite number Pj2:>2 . • • Pn, suppose there 
to be Dj lines of route which possess exactly s essential nodes. Upon these lines of 
route are represented 
distinct compositions, and the whole number of compositions is manifestly 
2D,2-^’-*-i = 2C/2-»’-^-’, 
Hence 
or = C^ presumably (see post, Sec. 5 , Art. 71 ), that is to say, the number of 
distinct lines of route through the reticulation which possess exactly s essential nodes 
is the coefficient of 
in the product 
a 
Pn 
; a 
1 + X (ag + . . . + a„)} {a^ + + X (ag + . . . + a,,)} 
2^2 
{«! 
+ tto + . . . + a„} 
62 . To interpret the product arithmetically we find that for a particular permutation 
of the letters in 
. . . a/" 
the factor X^ will occur if 
ag occurring times in the first p^ 
“3 >’ ’3 ” Pi + Pz 
places of the permutation 
) j 55 
a., 
r„. 
the sum r^, - h '^3 + • • • + 
P\ + Pc + • • • +P/'-i 
>3 
