292 Proceedings of the Royal IrisJi Academy. 



Explanation. 



By the " sum of tlie homogeneous products of aj, an, ... a,„ of 

 n dimensions" is meant the sum of all the products, each of n 

 dimensions, that can be formed of ai, a^, ... a„, and their powers. 



It is the coefficient of «" in the development of 



1— tti^ \ — anX 1- a„iX^ 



i.e., in the development of 



(1 + ttiiC + a^x- +...)(!+ ao^; + aJx^ 4- ...)... (1 + a,„a; + ajx'' + . . .) 



"We notice that it includes the poicers of a^, a^, . . . a,^, and it is 

 often expressed more fully thus : — 



" The sum of the homogeneous products of aj, a^, . . . a^ and 

 their powers, all of u dimensions." 



The number of homogeneous products of n dimensions that can be 

 formed out of aj, a,, . . . a,„ and their powers is found in the usual 

 way by putting %, 02, . . . «„» each - 1, and is • 



\m + n - \ 

 I m — 1 \ n 



