THEORY OF THE PARTITION OF NUMBERS. 
627 
number of these points must be nodes. These occur at all points where there is a 
change from the /3 to the a direction. They are termed “ essential nodes. 1 ' 
In the line of route traced in the figure the points a, b , c are essential nodes. 
Along a line of route there is a composition which is depicted by the essential 
nodes alone. This is termed the “ principal composition along the line of route.” 
In the figure the principal composition is 
(4l 12 IT 12). 
I have shown ( loc . cit .) that the number of lines of route which possess s essential 
nodes is 
and that the total number of lines of route is 
SCX'HT) 
(p > q ). 
We remark that the line of route divides the reticulation into two portions, an 
upper portion AJBDC, and a lower portion CKD. 
Placing a Sylvester-node in each square of the lower portion, we recognise at once 
Sylvester’s regularised graph of the partition 
(32 2 1) 
of the unipartite number 8. 
Similarly, from the upper portion, we obtain Sylvester’s regularised graph of the 
partition 
(7 2 65 2 4), 
of the unipartite number 34 (thirty-four). 
Whatever be the line of route, we simultaneously exhibit two of Sylvester’s 
regularised graphs, one of a partition of the unipartite N, and one of a partition of 
the unipartite pq — N. 
The two partitions may be termed complementary in respect of the unipartite 
number pq. 
Art. 11. This interesting bond between the partitions of unipartites and the com¬ 
positions of bipartites, I propose to submit to a detailed examination. The partitions, 
with which we are concerned, are limited in magnitude of part to p, and in number 
of parts to q. 
Moreover (as in the bipartite), we may suppose the numbers p , q, to be inter¬ 
changed. This would simply amount to rotating the reticulation through a right 
angle. 
4 L 2 
