Angltn — On some Theorems in Determinants. 



653 



And in like manner it is deduced, in the case of four general in- 

 dices, that 



a", 

 a\ 

 a% &c. 



gm-o 



a, 

 1, 



= {ale . . .1) 



r-m+4, r-m+3, r-m+2, r-m+1 



q-m+4:, 



p-m+4, X 



n-m+4. 



, (5) 



^vhere X denotes the determinant (4) in the preceding case. 



The law of formation of the results in the cases of five, six, and 

 any number of general indices, is now obvious ; and in order to es- 

 tablish the most general case of 7n - 1 general indices n, p, q, ... x, y, %, 

 we assume, by the principle of Induction, the case oim - 2 general 

 indices n, p, q, ... z, y, viz. : — 



or, 



a% 



a, 

 h 



&c. 



= {ahc ...I) 



y-2, y-S, 



x-2, x-3, 



p-2, p-3, 

 n-2, n-S, 



y-m + 1 

 x-m+l 



p-m+ 1 

 n-m + 1 



(m-1) 



the right-hand side of which we will, for convenience, denote by 

 (abc ... ^)(234 ... w-1). 

 By the equation (m - 1 )' it is readily shown that 

 a", 



a", 



1, 



&c. 



fl» (led ...I) 



K-1, z-2, 



y-1, 2/-2, 



z-m + 2 

 y -m-\-2 



q-\, q-2, 

 p-\, p-2, 



q-m + 2 

 p-m + 2 



-&c., 



tlie (') referring to the homogeneous products of I, c, d, . . , I. 



