646 



Proceedings of the Royal Irish Academy. 



The corresponding result, when the indices in the second row of 

 the first determinant have any general value r, is then obtained by the 

 application of the identity 



4-1 = Cl,K-2 + 



and it is shown that 



1, 1, 1 

 which, by (1), 



a«, 



5», 



c" 





a, 



h 



c 



hr-l - 



1, 



1, 



1 





0, h, c 



= (flic) 



1 



1, 1 



K-%, 



(2) 



2. Corresponding results are then obtained (by a process which 

 will be exhibited in the general case) in the cases of four and five 

 letters, making use of the results (1) and (2) ; and, for greater clear- 

 ness, and in order to discover the law by which the results in the 

 general case are formed, we will state the results in these two parti- 

 cular cases. 



"When four letters a, h, c, d are employed, it is shown that, h^ 

 denoting the sum of the homogeneous products of a, b, c, d and their 

 powers of n dimensions, 



fl", l'\ c", d'' 



a?, Ir, r, d'^ 



a, h, c, d 



1, 1, 1, 1 



{abed ) /i,i_ 



(3) 



and 



a", 













a% 

 a, 



&c. 



= {abed) 



r-2, 

 n-2, 



r-3 



n- 3 



y 



1, 













a\ 

 a% 

 1, 



&c. 



= {abed) 



s-1, 

 r - 1, 

 n- 1, 



s-2, 

 r-2, 



n- 2, 



5-3 



r-3 



n-3 



(4) 



(5) 



