650 



Proceedings of the Boyal Irish Academy. 



l\ &c. 



h 

 1, 



= {led ..'.I) 



r-m + 4, r-m+ S, r-m + 2 

 q-m + 4, q-m+S, q-m+2 

 p-m + 4, 2^~m+3, p-m + 2 



(4') 



And, generally, for m - 2 general indices p, q, r, . . . ^, %. 



1% &c. 



1, 



= {led ...I) 



z-1, s-2, . . . %-m^-2 

 y-l» y-2, ... y-m + 2 



q-1, q-2, ... q-'m-\-2 

 p-l, p-2, ...p-m + 2 



{m-\y, 



■uhere /?„' denotes the sum of th.e homogeneous products oil, c, d . . .1 

 and their powers of n dimensions ; and where in the determinants on 

 the right-hand side the elements are in reality A's, but for convenience, 

 and in order to better observe the law of formation, the suffixes only 

 are written. These m - 1 results are necessary and sufficient to ob- 

 tain the m corresponding results in the case of m letters. 



Expanding the terms in the left-hand side of {A), and equating 

 the absolute terms in both sides of the equation, we shall get by (1'), 



I""", . . . /" 



• 



• 





= {ale . 



.1) 



a\ 



l\ 



. . P 







a. 



I, 



. . I 







1, 



1, 



. . 1 







(1) 



Also equating coefficients of the several remaining like powers of 

 a? in both sides of {A), we shall get a series of similar results, the ge- 

 neral one of Avhich is, by (1'), 



