OF THE COMPOSITIONS OF NUAIBERS. 
859 
thence the successions 
and the permutation 
in which /3 occurs twice in the first three places. 
Now, it happens that these transformed permutations are very easily enumerated. 
The number of permutations of the letters in which possess the property that 
the letter yS occurs exactly s times in the first p places is the coefficient of in 
the development of 
(a + ixl3)p {a + 
This is evident from a consideration of the actual multiplication process. 
Hence, in regard to the bipartite pq, 
{a + (a + (By 
is the generating function for the number of lines of route possessing a given number 
of essential nodes. The complete coefficient of aP^'i is found to be 
1 +(v Cl) ^ +(2) (2) +■ • ■+C) C)+• • •+w C'l 
generality not being lost by the supposition p^q. Hence the number of lines of route 
possessing 5 essential nodes is 
(P\ 
Observe that the known formula 
0 / VO 
1/ Vi 
s / \s 
+ ? 
supplies a verification of the result. 
It has also been proved that the number of compositions of s + 1 j^arts having all 
• • . • • fv \ f Q 
contacts positive-positive is ' ' 
s \s 
25. On each of the P j p j lines of route, which have s essential nodes, may be repre¬ 
sented 
2P+9-S-1 
5 R 2 
