The theory of determinants in the historical order of development, by Sir Thomas Muir.

AXISYMMETRIC DETERMINANTS (BORCHARDT, 1859) 149

of q((x) by the corresponding term of +l.(y) is cancelled by the
product of the corresponding terms of 0(y) and Vr(x), the
number of terms then remaining is
2(n+ 1)2 - 2(n+1), i.e. 2n(n+ ).
Next, taking the product of the (r+l)th term of ~ (x) by the
(s+l)th of +(y) and subtracting the product of the (r+l)th of
+(y) by the (s+1)tl' of VlI(x), we have
f(x)/(x-a) / ) f()y- ) (y )/((y - ~.) f/()/()lx- a,
fO(a.) ar) ~(-(. ) VI (a()).,) (ar) ( ),
i.e. f(yx) fy )(a) f (a f'( 1.e. f 'f() * '(as);(X - )(:y-as ) (y - a^)(X -as)
i (e ) f(x ) f (y)) (a,). (as) (y-x)(ae,. -s)
f'A) f(as) (x - a,)(X —as)(y -) (y -a)
in which y-x is a visible factor. From this, by the mere interchange of r and s, we secure the combination of another pair of
terms; and by adding the two results we see that F(x, y) or
F(y, x) can be expressed as a sum of n %(n + -) terms of the form
/f(x).f/( ) a,,) (a,)- (as) - (a,)}(a, -a.a-,
f(a) f(as) j r)- a/ ) / arr),)(y -,.)(y- a.)'
the individual terms of the sum being got by giving r, s the values
0,1; 0,2;...; 0,
1, 2;...;, n
n-l, n.
We thus arrive at
F(x, x) {f(x.)}2 F(a, a,) (a, - ct,)2
) f'(ar.) f'(as) * (X -.)2(X - a)2 '
r, s= 0, 1;...; n —, z
and so see that if x be put equal to one of the a's, say ai, all the:
terms under 2 which have r and s both different from i must
vanish because of the presence of the first factor of the numerator.
We have only therefore to consider the terms got by putting
r, s-z, 0; i, 1;...; i, 3-1;, i+1;...; 3, n,