8G6 
MAJOR P. A. MACMAHOX OX THE THEORY 
Other than the initial and final points A and B there are {p + 1 )( 5 ' + !)(?’+ 1) — 2 
points. 
The graph involves lines in three different directions; say an a, a /3, and a y 
direction. These are parallel to AF, FQ, and QB respectively. 
Through each point of the graph pass lines in all three directions, and a segment 
joining two adjacent points is called an a segment when it is in the a direction. 
A line of route proceeds from A to B, from point to point of the reticulation. The 
iiumljer of lines of route is the number of permutations of the symbols in a^/3?y, and 
is, therefore, 
fp + q + ^ 
V IP q> I ’ 
A step along a line of route traverses in succession any number of a, /3, and y 
segments, and in any one step the segments must be taken in the order, a, y8, y. 
The number of segments traversed may be zero in one or two, but not in three of 
these directions. 
A step is represented by a tripartite number Piq^'i and a succession of steps, the 
first starting at A, and the last terminating at B, is represented by a succession of 
tripartite numbers constituting a comj)Osition of the tripartite p)qrA 
The graph of a composition is obtained by placing nodes at the points which 
terminate the first, second, &c., and penultimate steps. 
Essential nodes occur upon lines of route whenever the direction at a point changes 
from (i to a, from y to a, or from y to /3. These will be alluded to briefly as ySa, ya, 
or y/3 essential nodes. To these essential nodes on the graph of a composition 
correspond respectively zero-positive, positive-positive, and positive-zero contacts in 
the composition itself. It is convenient to call these collectively essential contacts. 
The theory of conjugate composition exists as in the bipartite theory. On every line 
of route there are p + O' + ■?• — 1 points, and if there be s essential nodes, + i-* 
distinct composition graphs can be delineated along the line of route. Each of these 
has s essential contacts, and, as in the bipartite case, we establish a one-to-one 
correspondence between the compositions having s essential contacts, and s fi- i -f 1 
parts, and those having 5 essential contacts, and p) A- q A- ^ i parts. The graph of 
a quadripartite number is derived from the tripartite graph, and generally the graphs 
of the multipartite numbers of order n from those of the multipartite numbers of order 
n — 1, on the same principle as the tripartite from the bipartite; and, moreover, in 
two dimensions. 
.32. For the multipartite number 
Pilh • • • P»-\Pn 
* Observe that the thirteen compositions of the tripartite ill are elegantly represented on the edges 
of a single cube, the sis lines of route lying between opposite corners. 
