Anglin — On some Theorems in Determinants. 



657 



Generally, to establish the Extensions in the case of «z - 2 general 

 indices (which are necessary to prove the final case of m - 1 general 

 indices in the main proposition), we assume the Extensions in the case 

 of w - 3 general indices. 



We have 



aTy 



&c. 



a, 

 1, 



= B 



X - 3, X - 4, 



X - m + 1 



q - 3, q - 4, . . . q - m + 1 

 p - 3, p - 4, . . . p - m + 1 



n - 3, n - 4, . . . n - m + 1 

 which, according to the previous notation, we write 



[n, p, q, . • ■ x^ = D (345 . . . m-1) ; 

 the Extensions of which are m - 4 in number, and are as follows : — 

 :^ln+l, p, q, ...x'] =D {245 ...m-1), (a') 



:S,[n+l, p + 1, q, ,..x] =I){235 ...m-l), (y8') 



^ln + 1, p + l, q+1, r, ... x'] = D{2346 ...m-l), (/ 



^[n + l, p + l, .. . u+ 1, x'] = D{234 . .. m-3, m-1), (/t-4)' 



the S's consisting respectively of m - 3, .^-aCa, m-aC's, . . . , and „,_sC„j_4 

 terms, where ,„C^ denotes the number of combinations of m things 

 taken r together ; and the law of formation of the coefficients of D 

 being obvious on inspection. 



To deduce the Extensions for m-2 general indices n, p, q, ... x, y, 

 we have, by equation (m - 1), 



Now 



[w, p, q, . . . y']= D (234 . . . m - 1). 

 %\ji+l, p, q, . . . y] = DS, suppose, 



where S denotes the sum of the m-2 corresponding determinants ; and 

 it is easily shown that S = (134 ... m-l) + A, where by (a') it is 

 shown that A = (2245 ... m - 1), that is, 0. Thus we have 



2[n+l, 2), q, ...y] = i)(134 ... m-1). 



(") 



