628 
MAJOR P. A. MACMAHON ON THE 
The )ine of route partitions the reticulation into two parts, each of which may be 
regarded as a partition of a unipart.ite. In fact, a line of route graphically 
represents a pair of partitions. 
These partitions can be equally depicted by the essential nodes only that occur 
along the line of route. It will be more convenient, for some purposes, to take the 
line of route, and not merely the essential nodes, to be the graphical representation. 
Attention for the present will be limited to the partition lying to the North-West of 
the line of route (i.e., towards the point -J). A line of route involves bends_|, termed 
“ left-bends,” and bends | , termed “ right-bends.” Essential nodes occur at the 
angular points of the latter. The North-West partition has as many different parts 
as there are left-bends on the line of route. The number of lines of route wdrich have 
s left-bends is equal to the number which have s right-bencls. This may be seen by 
rotating the reticulation through two right angles. 
This number is (loc. cit., Art. 22). 
Hence :— 
“ The number of partitions of all numbers into s different parts limited in magnitude 
to p and in number to q is 
Art. 1 2. The number of different partitions is equal to the number of lines of route, 
and this is 
s 
(p<$). 
Hence :— 
“ The number of partitions, of all numbers, into parts limited in magnitude to p, 
and in number to q is 
/p + q\ 
\ P 1 
Art. 13. A line of route, with s left-bends, has either s — 1, s or s + 1 right-bends. 
If it commences by tracing an a-segment and ends by tracing a /3-segment, the number 
is s — 1. If it commences by tracing an a-segment and ends by tracing an a-segment, 
the number is s. If a /3-segment begins and a /8-segment ends the line of route, the 
number is s, and if a /3-segment begins and an a-segment ends, the number is s -f- 1. 
If an a-segment begins the line, the North-West partition has exactly q parts. If 
an a-segment ends, the highest part is less than p. If a /8-segment begins, the number 
of parts is less than q, and if a /8-segment ends, the highest part is equal to p. 
Inspection of the Memoir cited shows that the enumeration of the lines of route 
possessing s left-bends and s — 1, s, s + 1 right-bends respectively are given by 
