62G 
MAJOR P. A. MACMAHON ON THE 
This method cannot be simply extended to the case of multipartite partitions, 
though some progress can be made in this direction as will be shown. 
Art. 10. Sylvester’s graphs naturally present themselves in the graphical repre¬ 
sentation of the compositions of bipartite numbers. 
As the graph of a bipartite ( pq ) we take p + 1 lines parallel and at equal distances 
apart and cut them by q -f- 1 other lines at equal distances apart and at right angles 
to the former ; (N.B. The right angle is not essential,) thus forming a reticulation 
or lattice. 
The figure represents the reticulation of the bipartite (76). 
c 
b 
+ 
+ 
+ 
a 
+ 
+ 
+ 
+ 
+ 
I recall that A, B are the initial and final points of the graph, and that the 
remaining intersections are termed the “ points ” of the graph. 
The lines of the graph have either the direction AK (called the a direction) or the 
direction AJ (called the /3 direction). 
Each line is made up of segments, and we speak of a-segments and of/3-segments, 
indicating that the lines, on which lie the segments, are in the a and /3 directions. 
If the bipartite (pq) have a composition 
(Pi9i M* 2Ms • • •)> 
the composition is delineated upon the graph, as follows :— 
Starting from the point A we pass over p x a-segments and then over q x /3-segments, 
and place a node at the point arrived at. Starting again from this node, we pass over 
P -2 “-segments and q 2 /3-segments, and place a second node at the point then reached ; 
we proceed similarly with the other parts of the composition until finally the point B 
is reached. At this point it is not necessary to place a node. In this manner the 
composition containing 0 parts is represented by 6 — I nodes placed at 6—1 
different “ points ” of the graph. 
The segments, passed over in tracing the composition, form a line of route through 
the reticulation. In general many compositions have the same line of route. Along 
every line of route there are p -f- q — 1 “ points,” which may be nodes. A certain 
