668 
MAJOR P. A. MACMAHON ON THE 
To find the form of the graph between any two graph lines, say 2' 2 \, 2 i , we have 
merely to proceed forwards horizontally from 2 2 1 and backwards diagonally from 2 i 
till a junction is effected in the manner 
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In the A, B notation the specification is given by the position of the BA contacts. 
BAABABBAA. 
Every graph that can be selected has its own specification. To enumerate the 
graphs we must enumerate the specifications. 
We must discover the properties of the succession of graph lines which are able 
to constitute a specification. 
If 2 A H be a graph line in a specification, X must be greater than zero since the 
lower line of the scheme cannot be reached by diagonal succession. Also 2 A H may be 
followed by a graph line 2 A T' Xl so long as X 2 > X and X 2 + /r 1 > X + p. ; and may be 
preceded by a graph line 2 A °P° so long as X > X 0 , X -f- p > X 0 + p 0 . 
These laws may be gathered by simple inspection of the scheme. 
Hence we may form the graph lines of the specification into a graph of the specifica¬ 
tion, viz.:— 
2 A F 
2 a i]/i 
Q^2/ x 2 
which has the simple properties— 
(i) X > X 2 >\. 2 > . . . 
(ii) X -f- /r > X 2 -f- /r 2 > X ? -J- /r 3 > . . 
(iii) The lowest graph line contains, at least, one 2. 
Art. 61. Any selected graph has a certain specification graph. From the former 
we may suppose any graph lines, which do not also belong to the specification, to be 
absent, and thus obtain a number of graphs which all have the same specification, and 
may be considered to follow the same line of route through the scheme. Further, 
the graph lines, whether belonging to the specification or no, may occur any number 
of times, repeated without the graph of the specification being changed. 
Finally, we may associate any partition of a unipartite number with any specifica¬ 
tion graph whatever, so as to form a two-layer graph. The partition of the unipartite 
must be interpreted upon that line of route through the scheme which is associated 
with the given specification graph. 
