G 46 
MAJOR P. A. MACMAHON ON THE 
we proceed to the bipartite partition 
(<etc bd). 
The number is thus shown to be 6 X 6 = 36. 
Art. 36. In general, when the tripartite is pqr, the partitions are into r parts, zeros 
not excluded, the first and second figures of the biparts being limited to p and q 
respectively. 
The bipartite numbers partitioned extend from 
p~X~r, 'qx~r to 00. 
The partitions are doubly regularised and may be enumerated by observing that we 
have to combine every partition of p X r and lower unipartite numbers into r parts, 
zeros not excluded, and no part exceeding p in magnitude, with every partition of 
q x r and lower numbers into r parts, zeros not excluded, and no part exceeding q in 
magnitude. 
Hence (see ante , Art. 12) the number of partitions is 
p + r 
This expression also enumerates (1) the compositions which have only positive¬ 
positive and positive-zero contacts ; (2) the lines of route in the tripartite reticulation 
which are without (3a bends; (3) the permutations of a J ‘(3' ! y' which are without (3a 
contacts. 
Art. 37. The truth of the theorem may be seen also as follows :—Suppose a solid 
reticulation and take the directions a, (3, y as axes of x, y, and z meeting at the 
origin of the lines of route. The face of the solid in the plane xz is a bipartite reticu¬ 
lation in which 
p + r 
r 
(q + r\ 
lines of route may be drawn; similarly ^ j lines of route 
may be drawn in the bipartite reticulation which lies in the plane yz. One of the 
former lines of route is an orthogonal projection of a tripartite line of route on the 
plane xz ; one of the latter is an orthogonal projection on the plane yz ; any one of 
the former may be associated with any one of the latter, and such a pair uniquely 
determines a tripartite line of route which does not possess (3a bends. This may be 
clearly seen by considering the permutation 
ort / T ' 1 '/ 1 a n '(3"Y' . . . ; 
suppression alternately of the letters (3 and a yields two permutations, viz. : 
