102 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
Thus 
A 2 = 1+ (2) + (2 2 ) + (2 3 )+ ...ad inf. 
A 3 = 1+ (3) + (3 2 ) + (3 3 ) + ... „ „ 
A, — 1 + (i) + (i 2 ) + (i 3 ) + ... ,, „ 
. ?? )) 
We proceed in this manner because we desire the development of the generating 
symmetric function 
_ 1 _ , _ 
(l— a) (1—a.o) (l— /3a) ... (l— ol[3o) (l— ayd) ... (l— a/3ya) ... (l— a/3ySa) ... 
there being a denominator factor for every a, /3, y, ... product in which no letter is 
repeated/ The expansion of this fraction involves the whole of the homogeneous 
product-sums of such a, (3, y, ... products; and we form these product-sums through 
the medium of the sums of the powers of the products which are, in fact, 
A 1} A.j, ... A,, .... The development is 
1 T A | 
+ f,(AWA a ) 
where 
+~j (A, 3 + 3 AjA 2 + 2A s ) 
+ . 
+a k F* (A) 
+ ... 
__ A A ^‘2 A ^ 
= ^D‘2a!. yU£ t ! 
precisely similar to the Q, development with A written for Q. 
It may at this point be worth stating that the two developments may be written 
exp («Q, l + |-« 2 Q 2 + Ii« 3 Q 3 +...), exp («A, +1-« 2 A 3 + ^-a 3 A 3 + ...) 
respectively. 
We now examine the effect of the Hammond operators upon this infinite set of 
A functions. It is clear from the well-known fundamental property of the operators 
that 
DA = A;, 
and 
D m A t = 0 when 
results'of great simplicity. 
