1888 .] 
Professor Anglin on AlUrnants. 
475 
5, c, d, . . . ^ and (2) of the degrees 1, 2, 3, . . . , n- \ respec- 
tively. 
We now propose to determine the value of K for alternants of 
this form, the complete generalization following in a similar manner 
to the former case. 
Consider a particular case, as 
1 ^bc, Wc, m ' 
Since — a, we can replace '^b by a on multiplying the 
alternant by -1. Since ^bc = '^ah - a%a -\- we can replace 
%hc by a^. Since ^(6% = - a^%a - a%a^ 4- 2a^, we can replace 
'%b‘^c by on multiplying the alternant by 2 ; and by on 
multiplying by 2. Thus the alternant is equal to ( - 1)( -f 1)(2)(2)^^, 
that is, - 
Now these multipliers are respectively the values of K corre- 
sponding to the functions taken in order. Hence we see that the 
value of K in the case of alternants of the above general form is 
equal to the product of the values of K corresponding to the 
several functions. 
We may illustrate the rule further by taking a more general 
case, as 
1 %b, ^b\ Wc, ^b^c\ Wcde, . . . , %b^cH^eY-g%Hinnr 
We can replace the elements of the first row, beginning with 
^5, by a, . . . , respectively, on multiplying the alternant 
in succession by - 1, - 1, 2, 1, 4, . . . , - - — respec- 
|_2 |_3^ 1^ 
tively; and thus the value of K is the product of these multipliers. 
The cases which we have just considered involve only the simple 
symmetric functions of b,c,d,... But the most general forms of 
the several orders of symmetric functions have been already 
obtained; and the rule for finding K being obviously the same for 
any symmetric functions whatever, we conclude that the most 
general form of alternants (involving integral functions) which are 
constant multiples of the difference-product of the variables is 
