870 
MAJOR P. A. MACMAHOU OX THE THEORY 
Hence the number of permutations sought is 
/% + «3l\ / P \ / 2 \ / ? + S 31 \ / ’’ \ 
\ ^21 / \% "t Sgj/ XSgg/ ySji + 531/ \S32 + S31/ 
34. It can be shown by direct expansion that this number is the coefficient of 
development of 
(a + X^j/S + Xg^y)” (a + /3 + Xogy)^' (a -j- /3 + y)'", 
which expresses the number of permutations in which 
/3 occurs % times in the first places, 
y j) '^31 ” ” 
y „ 533 times in the q places succeeding the first p. 
This remarkable identity of numbers suggests a one-to-one correspondence between 
the permutations of the two kinds, but I have not as yet been able to determine the 
law of the transformation. It would appear to be a very difficult problem. The 
number of lines of route in the tripartite reticulation which possess 
/3a essential nodes, 
^32’ y^ >f >’ 
^3i> y^ >) >> 
is thus 
fhi + %i\ / P + \ 
\ / \% + ^31/ W 33 / W21 "t 31/ \%2 "t %l/ 
35. Hence the identity 
P + 1 + _ V A‘'21 + S3l\ / P \fq\/(l + %\ ( \ 
i’- (Z- \ % / vhi + %i/ V 32 / Wi + ^31/ \% + Soj/ 
the summation being for all positive integral, including zero, values of ^ 3 i> 
which yield positive terms. 
36. The generating function for the number of lines of route having s essential 
nodes ; that is 53 ^ + -f Sgj = 5 ; is 
(a -f X ;8 + Xy)^ (a + ^ 4 - Xy)? (a + + y)", 
the number in question being the coefficient of XW^S^y*’. 
