Anglin — On some Theorems in Determinants. 



649 



E-educing the expression forming the left-hand member of this 

 equation to its common denominator (which consists of the product of 

 the differences of a, b, c, . . . I, taken two at a time), we haye 



aP'-^bcd.. . l){\-axy-b"'-\acd . . .)(\-bx)-'^-\-c'^\abd . . . l){l-ex)-^ 



= {abc .. .l){l + hix + h^x"^ + .. . + Kaf + ...), {A) 



where any symbol of the form (abc . , .) denotes the product of the 

 differences of the letters involved, taken two at a time. 



We now, in accordance with the principle of Mathematical Induc- 

 tion, assume all results for m - \ letters corresponding to those which 

 we propose to establish for m letters, viz., 





7m-J 

 Im-t 



b\ C\ 



b, c, 

 1, 1, 



72 



iP 

 Jm-i 



b\ C\ . 

 1, 1, . 



IP 

 Im-i 



b, C, 

 1, 1, 



. . . I 



{bed ...I). 



(1') 



= (bed ...I) h'p_^z. 



(2') 



^{bcd.,.l) 



j'-m + S, 2'-m + 2 

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



(80 



