8G2 
MAJOR P. A. MACMAHON ON THE THEORY 
All the contacts in these comj^ositions are necessarily positive-positive; hence the 
leading and ending parts are the only ones that can involve a zero element; in the 
leading part the leading element may he zero, and in the ending part the ending 
element. 
Every part, of a principal composition, which does not possess a zero .element 
necessitates an essential node on the graph of the principal composition on the 
inverse line of route. Thus, if the one principal composition has s nodes, s -f- 1 parts 
and t j)arts without a zero element, the other principal composition has t nodes, 
^ -f- 1 jDarts and s parts without a zero element. 
These may be called “ inverse princi|)al compositions.” 
The number of pairs of inverse principal compositions is equal to half the number 
of distinct lines of route through the reticulation. The number of pairs is thus— 
Otherwise, we may say that in regard to compositions, all of whose contacts are 
positive-positive, there is a one-to-one correspondence between those having s + 1 
parts and t parts without a zero element and those having ^ + 1 parts and s parts 
without a zero element. 
The number s + 1 — i is either 0, 1, or 2. 
E.g., for the bipartite 22 the correspondence is 
s = 0, t = I 
( 22 ) 
s = I, t = i 
(01 2T) 
5=1, t = 2 
(IT U) 
5 = 1 , ^ = 0 
(02 20 ) 
5=1, ^ = 1 
(12 lb) 
5=2, f = I 
(01 Ti lb). 
28. This particular case of inversion leads easily to the general idea. Suppose a 
line route traced on a reticulation, and suppose marked the s essential nodes on the 
line drawn from A to B, and also the t essential nodes of the inverse line of route. 
These 5 + i nodes are distinct. Along the line we may place an additional node at 
an unoccupied point and interpret the new compositions as read before and after 
rotation of the reticulation. The new compositions have each acquired an additional 
part; they each have the same number of parts without zero elements as before; the 
added node has either introduced a zero-positive contact into each or a positive-zero 
contact into each, according as the node is on an a or on a /3 line between adjacent 
nodes. 
