868 
MAJOR P. A. MACMAHOlSr 0'S THE THEORY 
33. I now enquire, in respect of the tripartite reticulation, into the number of 
lines of route which possess exactly y8a essential nodes, Sgg, y/B and Sgj, ya. essential 
nodes. The second method of investigation ado|)ted in the bipartite case will be 
employed. 
Consider the permutations of the symbols in aP/B'^y which possess 521, 
; ^32 > 
and 531 , ya contacts, and suppose 
^21 + ^32 + ^31 = 
It will be shown that the number of such permutations is equal to the number of 
permutations in which 
j3 occurs 521 times in the first p places, 
y }) ^31 ’» ” 
y ,, 533 „ in the q places succeeding the first p. 
A permutation of the former kind involves successions of letters 
^a, y/3, ya, y/Ba ; 
letters between consecutive successions being in alphabetical order. 
As regards these successions, and attending to them alone, the permutation 
exhibits some permutation of the terms in 
These permutations are 
{ylBaY (ya)*-. 
^21 "t ^32 “t 
^31 
r)! 
(Sgi - a )! (S32 — 0-) ! 0 -! S31! 
in number. 
Selecting any one of these there are, with reference to the terms, 
'^21 + '^32 +^31 ~ O' + 
different positions in which other letters may be jfiaced. It is clear that a cannot be 
placed after a term (y/3), or y before a term (ySa), for these placings would lead to 
additional {^a) and (y^S) contacts. The letter a may be placed before any of the 
terms {/Ba), {y/B), (y/Ba), {ya), and after any of the terms (/3a), (y/Ba), {ya). Hence 
out of the 531 + S 33 + 531 — a + 1 different positions (relative to the terms) 
^21 + '^31 + L positions may be occupied by a letter a. The letter fB may be placed 
in all the 531 + .933 + 531 — cr + 1 positions without the introduction of additional 
/Sa, yy 8 , ya terms. The letter y must not be in a place preceding (/3a), and hence it 
may occupy 533 + '^31 + 1 different positions. 
Besides the letters occurring in the terms 
{IBay--’^ (y^)*"-" {ylBaY (ya)*-, 
a occurs p — 5oj — 531 times, 
(B „ 2 ~ ^21 ~ % "1“ ^ times, 
y ,, P — hi — times. 
