1888 .] 
Professor Anglin on Alternants. 
469 
A^^a^-dtly^-anh 
= %a% - - c^%a + 2a^ ; 
and thus K=% 
And, in the latter, 
A = a^%a% - a‘^%a^ - a^%a + 2a^ , 
when K also = 2.] 
In general, A consists of the sum of a series of terms of the form 
h^c^dn . . . , which we may denote by 
. . . Ihn^ • • • ; 
and we shall consider separately the cases where the indices in % 
are (1) all different, (2) all alike, and (3) of the most general 
character. In all cases the value of K is independent of the order 
of the alternant or of the values of the indices in 2, and depends 
only on the number of indices or factors in a term of 
(1) When the indices are all different. 
If the terms in A consist of one factor only, so that A = we 
have 
A = '^b^ = %a^-a^. Thus K= -1. 
If the terms in A consist of two factors, we have 
A = - a^%b^ - ; 
and A=0 + l + l = [2, 
In like manner we deduce that, when A = ^b^c^dP., A = - | 3 . 
]N"ow assume this law for any number /x of indices, so that if 
A=^%b^c^d^ . . .W, A=(-lf . 1^; 
then, taking /x + 1 indices, we have 
A = V)^c^d^ . . . TcH^ 
. . . ¥-or%b^c^ ... If 
- ... A:" - ... - d^b^c^ . . , ; 
thus 
^=0-(AC+l)(-lf. Ij. =(-ir+'. |_^ . . . (1). 
(2) When the indices are all alike, A = + 1 according as ' the 
number of factors in a term of is even or odd. 
Thus when A = , A = - 1 ; 
when A = , that is, , A = 1 ; 
when A = , A = - 1 ; and so on. 
