OF THE COMPOSITIONS OP NUMBERS. 
857 
Of this theorem I give three proofs, because its thorough examination is necessary 
for the purpose of leading up to the more difficult theories connected with tripartite 
and higher multipartite numbers. 
First Proof. 
In each of the adjacent sides AD, AC of the graph of y)^', select any s “ points ” 
(see definition of “point”) a, h, c, . . . in order from the point A. The two 
points a, a are seen to determine an essential node a ; the two points h, h an 
essential node h', &c., and a line of route necessarily exists which possesses these 
essential nodes and no others. The points along AD and AC, from which s points 
may be selected, are in number p and q respectively ; along AD s points can be 
selected in ways, and along AC in ways; any selection on AD can be taken 
with any selection on AC. Hence the number of lines of route having exactly 
s essential nodes is 
f 
23. Second Proof. 
We determined the total number of lines of route by considering the permutations 
of p, a-segments, and q, /S-segments. Whenever in any such permutation there is a 
sequence j8a there must be an essential node upon the corresponding line route. We 
have simply to find the number of permutations of the p q symbols in olp^i which 
possess exactly s, ySa-contacts, 
Write down the s, ySa-sequences 
. . . ySa . , . fa . . . fa . . . fa , . . 
and the s + 1 intervals between them. In these intervals we have to distribute the 
letters in 
^p-S ^q-S 
in such manner as to introduce no fresh ySa-contacts. For each of these ^ + O' “ 2s 
letters there is a choice of s + 1 intervals. The p — s letters a may thus be 
distributed in 
ways, and the q — s letters f in 
and each distribution of the 
5 B, 
MDCCCXCIII.—A. 
