Anglin— 0/i some Theorems in Determinants. 655 



+ ...+(-l)'"-^2 



«*+S &c. 

 a'-', 

 a, 

 1, 



/V„,+2+(-l)^ 



&c. 



a, 

 1, 



■svliich by the equation (w- 1) and its Extensions is equal to («5c . . .1) 

 nniltiplied by 



(234 ... m - 1) ^3_i - (134 . . . w - 1) ^,.3 



+ (124 ... w - 1) A,_3 - (1235 . . . w - 1) A,_4 



+ (- 1)"-^ (123 ...m-Z,m- 1) h^^. + (- l)"'-l (123 ... m - 2)^,.„+i ; 

 and tbus we have 



fl", 



5», . 



. . i^ 



«^ 



^^ . 



. . P 



«', 



K ■ 



. . z* 



rt=, 



K . 



. . p 



1, 



1, . 



. . 1 



=: {ale ...I) 



z-\, 



s-2, 



2-3,. 



. . %-m + \ 



y-1, 



2/-2, 



y-3, . 



. . y-wi + 1 



?-i. 



?-2, 



S'-3, . 



. . q-m + l 



i^-i, 



^-2, 



i>-3, . 



. .^-w+1 



w- 1, 



w-2, 



w-3, . 



. . n-m-\-\ 



{m) 



and hence the proposition is completely established. 



5. In the foregoing investigation reference has been made to Ex- 

 tensions of various results, which it is necessary to state and prove 

 (not only because they are necessary to establish the main proposition, 

 but also on account of their own individual importance and interest). 

 Strictly speaking, these extensions should be incorporated with the 

 foregoing work; but to avoid confusion, and to allow all sme7«r results 

 to follow one another without any digression, it is desirable that the 

 extensions should be treated separately, which we now propose to do. 



Denoting, for convenience, the determinant involving the general 

 indices n, p, q, ... ^ by the symbol [n, p, q, . . . x'], we have, by (3), 



\ji, p'] = (ahe . . . I) 



p-m + 1, p~m+l 

 n-m + 2, n-m-j- 1 



2) (21), suppose. 



