854 
MAJOH P. A. MACMAHON ON THE THEORY 
Essential nodes occur on a line of route at all poinfs where the course changes 
from the to the a. direction. 
We may regard a line of route as defined by these essential nodes since the line of 
route is cornpletely given by these nodes. 
Every composition-graph involves nodes of two kinds — 
( 1 ) Those which are essential to its line of route ; 
( 2 ) Those which, in this respect, are not essential. 
There are 'p q — \ points ” along every line of route. 
We have now to discuss the compositions whose graphs follow a given line of 
route, and we find that they naturally arrange themselves in pairs. 
19. Associated with any one graph is another obtained by obliterating those nodes 
which are not essential to its line of route, and by then placing nodes at those points 
on the line of route not previously occupied by nodes. 
These two graphs are said to be conjugate. 
The compositions, represented by these graphs, are likewise said to be conjugate. 
Conjugate graphs are shown above; AcbB is the line of route, and h is the essential 
node. 
The corresponding conjugate compositions of the bipartite number AI are 
(l3 01 40) and (Id 01 Ol 02 Id To Id Id). 
If the graph of a composition of pg', of r parts, possesses s essential nodes, it is clear 
that the conjugate composition has y) + 2 ' — ^'-fs-j-l parts. 
Of compositions, of the bipartite_p 5 ', whose graphs possess s essential nodes, there is 
a one-to-one correspondence between those of r parts and those of ^3 fi- j -f 5 -)- 1 
parts. 
Corresponding to an essential node in the graph of a composition there exist in 
the composition itself adjacent parts . . . pg^-^. . . possessing the property of 
and being both superior to zero. Thus, from inspection of a composition, we are 
enabled to determine the number of essential nodes in its graph. 
