OF THE COMPOSITIONS OF NUMBERS. 
893 
that is to say it is the coefficient of . . . «/“ in the product 
- (hai -j- • • • H" (^’*1 ”1“ ^’“3 “f~ “3 H“ • • • + • • • (^’<^1 + + • • • + 
which may be written 
liy,+ih+ + T (^3 + • • • + + 7 («3 + • • • + 
Tc 
h 
The coefficient of . . , a/'‘ is 
• (“l “b “l~ • • • “b ^nY“‘ 
where C. is the coefficient of 
in the product 
a/^a/^ . . . «/'■ 
{«! 4- ^ (“3 + • • • + (“1 + “3 + (“3 “b • • • “b 4“ “3 “b • • • + 
If in the reticulation of the multipartite number there be lines of route which 
possess exactly 5 essential nodes, 
combinations of order h may be represented upon these lines. Hence the whole 
number of combinations is 
Hence 
S D, = kCs 
a relation which is true for all positive integral values ofh. 
77. Hence 
D, = 0, 
the important relation temporarily assumed in the investigation concerning com¬ 
positions. (Art. 61.) 
78. The theorem thus established, viz., that the number of distinct lines of route 
through the reticulation of the multipartite 
PiPo . . .'Pn, 
which possess exactly s essential nodes, is given by the coefficient of 
. . . a/'‘ 
