OF THE COMPOSITIONS OF NUMBERS. 
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speak of a segments and of /3 segments indicating that the corresponding lines (on 
which lie the segments) are in the a and ^ directions. 
Suppose a traveller to proceed from ^ to ^ by successive steps. A step is per¬ 
formed by moving over a certain number of a segments and subsequently moving 
over a certain number of ^ segments. A step is thus made up of two figures—say 
an a figure and a /3 figure. The number of segments moved over may be zero in 
either, but not in both of these two figures of the step. 
A step may be taken from A to any point a of the graph ; a second step may be 
taken from a to any point of the graph aB which has a and B for its initial and final 
points ; subsequent steps are taken on a similar principle, and the last step termi¬ 
nates at the point B and completes the procession from the point A to the point B. 
A step which takes x, a segments followed by y, /3 segments, is taken to be the repre¬ 
sentation of a bipartite part, xy. A procession from the initial to the final point (from 
A to B) of the graph thus represents a sequence of bipartite parts which constitutes 
a composition of the bipartite number To every procession from A to B corre¬ 
sponds a composition of the bipartite pq, and the enumeration of the different 
processions is identical with the enumeration of the total number of different 
compositions. 
The .steps of a procession are marked out by nodes placed at the points of the 
reticulation, which terminate the first, second, third, &c., and penultimate steps 
When nodes are thus placed, we have the graph of a composition. If nodes be placed 
at the points a, h of the graph, we obtain the graph of the composition 
_ (l3 bl id) 
of the bipartite 54. 
The number of parts in a composition is always one greater than the number of 
nodes in its graph. 
Considering the number of compositions of pq, of two parts, it is clear that we may 
place the defining node at any one of [p A~ 1) (^fi" 1) ~2 points of the reticulation. 
Hence (and this may be verified by previous work) the number of two-part com¬ 
positions is 
(p+ 0 0+ 1) -2- 
18. The graph of a composition, traced from A to B, passes over certain segments, 
and may be said to follow a certain line of route through the reticulation. Other 
compositions in general follow the same line of route; they all have their defining 
nodes upon the line of route, and a certain number of defining nodes will, in general 
be common to all of them. 
Consider the path AcbB in the graph given below. All coinj)ositions whose gTaphs 
follow this line of route must have the node h ; h, in fact, is an essential node along 
this line of route. 
