1899-1900.] Dr Muir on the Theory of Alternants. 131 
difference-product of a, b, c, . . . It is thus seen that the given 
alternating aggregate 
vr ± 1 l=k. VV ' 
(x — a){x - b)(x - c) . . .J Y ’ 
where P,P',V are known, and k has still to be found. An easy 
step further is made by inquiring as to the degree of k, it being 
noted in this connection that the degree on the one side is — n , 
and that on the other side the degree of P = — 1 ), the degree 
of P' likewise =-- \n(n - 1 ), and the degree of Y = n 2 . The resultant 
degree of PP'/Y on the right is therefore inferred to be 
= - 1 ) + \n{n - 1 ) - n 2 , 
= -n; 
and as a consequence the degree of k must be zero. In other 
words, k must be constant in regard to x, y, z, . . ., «, b t c, , . . . : 
so that for its full determination the best thing to do is to select 
as easy a special case as possible. Cauchy’s choice falls on the 
case where x = a, y = b, z — c , . . and preparatory for this 
substitution he transforms the above result, 
— {x - a)(y - b)(z - c) . . . — ^ ' Y ’ 
into 
k . pp' = v. y.[±, w J, V 1, 
(x - a)(y- b)(z - c) . . . J 
~^>L ± (x-a){y-b)(z-c) ]" 
As for the right side of this, it has to be noted that, since Y 
contains each of the binomials x- a, y — b, z — c, . . . once and 
once only, any one of the 1 . 2 . 3 .... n terms under 3 will 
vanish when the substitution 
x, y, z, . . . — a, b, e, . . . . 
is made, unless the denominator of the term also contains all 
the said binomials. But by reason of the interchanges which 
produce the other denominators, the first term is the only one 
of this kind : and the value of it after the substitution has 
been made is 
