648 



Proceedings of the Royal Irish Academy. 



and 



a% &c. 



«^ 

 1, 



= (ahcde) 



where {abode) denotes 



t-\, t-2, t-S, t-4: 



s-1, s-2, s-S, s-4 



r-1, r-2, r-3, r-4 



n-1, n-2, w-3, »-4 



a*, . . . e* 



a\ . . . e* 



a\ . . . ^ 



a, . . . e 



I, 1, 1, 1, 1 



. (9) 



or the product of the diifferences of a, h, c, d, e taken two at a time ; 

 and the same remark applying to the right-hand determinants as in the 

 preceding case. 



3. We now proceed to establish corresponding results in the 

 general case of any number (m) of letters a, b, c, . . . I; and, in order 

 to effect this, there will be a double application of the principle of 

 Mathematical Induction — the first, by reference to the number of 

 letters employed ; and the second, by reference to the number of gene- 

 ral indices in the case of any particular number of letters. 



By definition 



{l-ax){l-bx) ... (1 -to) 



where A„ denotes the sum of the homogeneous products of a, b, c, . . . I, 

 and their powers of n dimensions. 

 But it may also be shown that 



1 



+ 



B 



{\-ax){l-bx) . ..{\-lx) 1-ax 1-bx 

 where 



A = 



1-y 



I"' 



{a-b){a-c)...{a-ly 



and thus 



(a-b){a-c).. .{a-l) 



{1 -ax)-^ ■{■... + ■ 



{l-a)(l-b)...{l-ky 



{l.a){l-b)...{l-k) 

 = 1 + hix + hiX^+ . . . + hnX" + . . . 



.{l-lx)-^ 



