840 
MAJOR P. A. MACMAHON OR THE THEORY 
s black nodes and m — s white nodes; to the left there are similarly placed s white 
nodes and m — s black nodes. 
Of the graph of the number there are m points to the right of the centre at which 
white and black nodes can be placed in 2”^ distinct ways. Hence it at once follows 
that the number 2m + 1 possesses in all 2'" inversely conjugate compositions. 
Otherwise, we may say that the number of inversely conjugate compositions of 
2711 + 1 is equal to the number of compositions of m + 1- 
It will be observed that the number 2^7^ + 1 has precisely the same number, 2“, of 
self-inverse compositions. 
There is, in fact, a one-to-one correspondence between the compositions of 2m -b 1, 
'which are inversely conjugate, and those which are self-inverse. 
To exjDlain this, take the graph last represented. Read according to the black 
nodes we obtain the inversely conjugate composition 
(23121) of the number 9. 
To proceed to the corresponding self-inverse composition obliterate the black nodes 
to the right of the centre, and also the white nodes to the left of the centre 
Substituting white nodes for the black nodes then remaining we have the graph 
---e—-----—---©--- 
of the self-inverse composition 
(252). 
Again, reading the original graph according to white nodes, we have the inversely- 
conjugate composition 
(12132), 
and proceeding, as before, with the exception that black and white nodes are 
obliterated on the left and right of the centre respectively, we obtain the graph 
- © --- © - ©-©-©---©-• 
of the self-inverse composition 
( 1211121 ). 
The process is a general one, and shows that we can always pass from a composition 
which is inversely conjugate to one which is self-inverse. 
E.g., of the number 7 we have the correspondence 
