82 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
been forged. To evaluate the coefficients we have to operate repeatedly with the 
appropriate operators until a numerical result is reached. In order to accomplish this 
with facility and to establish laws we have to put the generating functions in such a 
form that these operations are carried out in a regular and simple manner. To make 
my meaning clear, I will instance the case of the simple operation of differentiation 
d x and the exponential function e* 1 . We have 
0 2 e“ = ae“ 
the effect of the operation being, to merely multiply the operand by the numerical 
magnitude a. 
Thence 
3 2 n e“ = a n e ax 
and we arrive at the conclusion, that if a given operand, a function of x, could be 
expressed as a linear function of exponential functions of x, the r times repeated 
operation of 3 2 could take place with facility upon each tenn of the linear function, 
and a general law for the repeated operation of 3 X upon the operand would be 
obtainable. This reflection suggests the possibility of finding symmetric function 
operands in a form which will enable the repeated performance of Hammond’s 
operators in a practically effective manner. It is quite certain that any such operand 
must possess at least two properties in common with the exponential function : (i) its 
first term must be unity ; (ii) it must contain an infinite number of terms. The first 
step was to find a symmetric function of the elements a, (3, y, ... such that the 
effect of every Hammond operator upon it is to leave it unchanged ; or, as I prefer to 
say, to multiply it by unity. is, in fact, the sum of unity and the whole of the 
monomial symmetric functions £a. p [3 q y k ... EE (pqr ...). It is in the partition notation 
Qi = l + (l) + (2) + (l 2 ) + (3) + (2l) + (l 3 )+... ad inf. 
It was then found that the effect of any Hammond operator upon any power of Qi is 
merely to multiply it by a positive integer. It then appeared that, denoting Qj by 
F (a, /3, y, ...), the function 
Qi = F (a‘, j3 l , y\ ...) 
possesses properties of a character similar to those appertaining to Q x . The fact is 
that any Hammond operator when performed upon any power of Q t , say Q/‘‘ has the 
effect of merely multiplying it by an integer, which may exceptionally be zero. 
Finally the important fact emerged that the performance of any Hammond operator 
upon the product 
Qi*‘Q 2 * 5 ... Q/' 
where k i} k 2 , ... k iy ... may, each of them, be zero or any positive integer, is merely to 
multiply it by a positive integer, which may, exceptionally, be zero 
