OF THE COMPOSITIONS OF NUMBERS, 
861 
i. P\ li 
t 
'P + # 
P 
Inverse Bipartite Comp)Ositions. 
27. A line of route being marked out on a reticulation from A to B, the inverse 
line of route is obtained by rotating the reticulation through two right angles and 
interchanging the letters A and B. 
Consider a line of route from A to B having essential nodes a, h. On the inverse 
line of route a, h', c are the essential nodes. 
The princii^al compositions along these lines of route are— 
(21 11 32) from A to B. 
(02 3l 11 from B to A. 
* Consider also tlie points of the graph distant t segments from A. The number of such points is 
the coefficient of in the product 
and if 
(1 + a; + ic” -f- . . . x'P') (1 4" a; “h a;“ • P 
t < q -\- 1 is equal to ^ + 1, 
>2<P + 1 ,, g + 1, 
> p „ p + q ~ t + 1. 
These points lie on lines parallel to the line 01234 . . . s . . . and we obtain a graj^hical proof of the 
identities, 
t\(p-t q-t\ (t\(p + q-t 
A Ip + 1 ~ /p + 2 
p 
on ,j / + hA 3-1 / + ■■■ +V\ 0 
(P + q — _ jp + q 
P 
for ^ < g + 1; 
for p -\- \ > t > q', 
0 
where the total number of terms in all these identities is 
2 A + 1) + (P — 2) (2 + 1) + S {p -\- q — t + 1) = {p + 1) {q + 1), 
0 t = 'p + l 
which is the total number of points in the graph (including A and R), and is therefore right. 
