Anglin — On some Theorems' in Determinants. 



645 



XL. — On some Theoeems in Deteeminaxts. By A. H. Akglin, 

 M.A., P.E.S. (Edin.), &c. 



[Read, April 12, 1886.] 



[Absieact.] 



1 . The method employed to establish the general results being that 

 of Mathematical Induction, the corresponding results in one or two 

 particular cases are first noticed. 

 By definition 



where ^„ is the sum of the homogeneous products of a, I, c and their 

 powers, all of n dimensions. 



But also it may be shown that 



B 



C 



{\~ax){\-hx)(l-cx) I -ax 1-hx \-cx 



where A = r- -, 



{a-b){a-c) 



similar expressions holding for B and C; and thus 



a' 



5-7 r. (1 - a.r)-H &c. = \+'hiX-^'hiX^ + . ..^-hnX'^-if . .. 



{a-o){a-c) 



Expanding, reducing to the common denominator (c - h){a - c){h-a), 

 which = a^ (i - c) + P {c - a) + c^ {a - 3), and equating coefficients of 

 like powers of :c in both sides of this equation, we shall obtain a series 

 of similar results, the general one of which is — 



a" {l-c)\ I" (c - a) + c" (fl - 5) = (c - l){a - c){l) - a) A„.j ; 



or, expressing in the form of determinants, ' 



a, h, c = a, b, c A„-j = («ic) ^J' 0) 



1, I, 1 1, 1, 1 



where (abc) denotes the second determinant, or the product of the dif- 

 ferences of a, b, c taken two at a time. 



R.I. A. PROC, SER. II., VOL. IV. — SCIENCE. 3 1 



