86 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
and its cardinal property is 
D mt D ms ... D m . (m x m 2 ... m,) = 1; 
and this operation does not result in unity when it is performed upon any other 
symmetric function. 
In order to obtain the coefficient of 
a k {m 1 m 2 ... mj, 
in the expanded function, we first of all find the complete coefficient of a k and then 
operate upon it with the Hammond combination of operators 
D m ,D m ,... D,, 
The result is an integer followed by the sum of an infinite series of symmetric 
functions. The integer mentioned is the number we seek. 
Art. 5. We now expand the generating function. On well-known principles we 
can assert that the coefficient of a k in the expansion is the homogeneous product-sum 
of' order k of unity, and of the whole of the a., fi, y, ... products which occur in the 
denominator factors of the generating function. The elements, of which we must 
form homogeneous product sums are, in fact, 
a' 
1 
/T • • • •> 
a“, (3 , y , ... ct/3, cty , /3y, ... , 
/ 3 3 , y 3 , ... od/3, a 2 y, ...a/3y, a/3$, ... . 
We can form these product-sums from the sums of the powers of these elements, 
because we have before us the well-known symmetric function formula 
ft ^ln ^-2 
*1 *2 
S'- 
ki 
H2* 2 ... -T. kj ! k 2 \ ... k : ! 
The sum of the powers are readily formed ; for, calling them 
it is clear that Q,- is the sum of unity and the whole of the monomial (that is to say 
merely involving in the partition notation a single partition), symmetric functions 
of weights one to infinity. Hence 
Qi = l+(l) + (2) + (l 2 ) j-(3) + (2l) + (l 3 )+... ad inf. ; 
