636 
MAJOR P. A. MAC MAHON ON THE 
Art. 23. Again, regarding y> and q as constant and s as variable, we know that the 
coefficient of (ax p )' 1 in the product 
calling this expression B yi the effective portion of the generating function is written 
1 -f- B x ax p + B 3 ( axP'f + B 3 (ax p ) s + ... ad inf. 
Art. 24. I recall now the generating function which enumerates the partitions of 
all unipartite numbers into parts limited in magnitude by p and in number by q, 
viz. :— 
_ 1 __ 
(1 — ie) (1 — a) (1 — ax) (1 — ax ~)... (1 — ax p ) 
The coefficient of a q in this development is well known to be 
(1 — Xi +1 ) (1 — Xl + ~) ... (1 — Xi + P) 
(1 — a :) 3 (1 — £ 2 ) (1 — as 3 ) ... (1 — x ?) 
Hence the coefficient of a q x pq in the former is the same as the coefficient of x pq in 
the latter. This we know to have the value ft ^ 1 J • 
Art. 25. Numerous theorems of isolation, in the senses in which the word is 
employed in this paper, may be obtained from the reticulation theorems. I proceed 
to give some of those which present features of interest. 
A previous result was to the effect that the number of partitions of all numbers 
which have exactly q parts, a highest part equal to p and s different parts is 
enumerated by 
fP - r \ ft - 1' 
s- 1/ \s- 1. 
Hence, without specification of s, the number is 
s fr 7 v ) (NXVtH 2 
This enumeration is also given by the coefficient of a q 1 x p( - q ^ in the function 
(1 — x) (1 — ax )... (1 — oxp) 
