OF THE COMPOSITIONS OF NUMBERS. 
865 
30. Self-inverse compositions can only occur upon self-inverse lines of route. For 
the existence of such a line, one at least of the elements of the bipartite number 
must be even. If both elements be even there is a central point in the reticulation, 
and the number of self-inverse lines of route for the bipartite 'Ip , 2q is equal to the 
number of lines of route of the bipartite p>'q that is to say, the number is 
y + Q' 
y 
If the bipartite be 2^^' + 1, 2q' it is easy to see that the number is also 
y + 9' 
y 
The number of self-inverse compositions in both cases is evidently equal to the 
total number of compositions of the bipartite qj'q. 
§ 4. The Graphical Representation of the Comjyositions of Tripartite and 
Multipartite Numhers. 
31. The graph of a tripartite number may be in either two or three dimensions. 
It may be derived from a bipartite graph in a manner similar to that in which the 
bipartite has been derived from the unipartite graph. In the tripartite number 
we take r -f- 1, exactly similar graphs of the bipartite p)q, and place them similarly 
with corresponding lines parallel, and like points lying on straight lines ; when 
these straight lines are drawn the graph is complete. The r 1 bipartite graphs 
may be in the same plane or in parallel planes according as the tripartite graph is 
required to be in two or three dimensions. 
B 
The figure depicts the graph of the tripartite 233. The points of the reticulation 
are identical with the points of the r-\- 1 reticulations of the bipartites pq. Observe 
that in two dimensions there are intersections of lines which are not points of the 
reticulation. On the other hand in three dimensions all intersections are also points. 
MDCCCXCIir.—A. 5 s 
