630 
MAJOR P. A. MACMAHON ON THE 
'p - A (q - A , (p - A h - 1 
k- 1 " r \k- 1 \ k 
If the bends be 2k \ in number we may have k right-bends and k + 1 left-bends 
or k + 1 right and k left.. The enumeration gives 
v - A fi - 1 
k 
k 
Hence the number of pairs of complementary partitions which have each k different 
parts is 
k ) U- - V U - 1/ \ k 
and the number of pairs in which the partitions have k and k + 1 different parts 
respectively, is 
Adding the number of lines of route with 2^—1 bends to the number with 2k 
bends we obtain the number 
Hence the identity, 
P + r i (p -Ah- 1\ 
k \k - 1/ \k - 1/ ‘ 
or 
0 
t 0 / 
pi ql 
which may be established independently. 
Art. 16. Consider the lines of route which pass through a particular point P of the 
graph, say the point distant from A, a a-segments and b /3-segments. In the 
reticulation AP we may draw any line of route, and any line also in the reticulation 
PB. In AB we can thus obtain 
a + b\/p+q — a — b 
« / \ P - a 
lines of route passing through P. 
In the associated North-West partitions, the highest part < a and the number of 
parts <£ q — b. 
Put q — b = c, and we obtain the theorem : — 
