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

148 HISTORY OF THE THEORY OF DETERMINANTS
The basis of his procedure is Cayley's result that
x... -1
W(X> S(Y~lf(Y-X) 1
I(x) q(y) + (y-x) = 1 x.,-.W(x) (y) Y
D
yn-l
where D is the array of Bezout's condensed eliminant I D.
Denoting the left-hand member of this by F(x, y) or F(y, x),
we see that
1 X1...I... 1 F(,y) F(x,y... F((x,y,)
1 2 X..2 D1 ID. ' Y2 *Y F(x2, yl) F(.2, Y2) *.. F(=2,.n)
1 C 2,, -1 - 1 y -... 1 F(x,, y,) F(x,,y2)... F(x,,y,)
1 n-1
and.. D = [F(al, a,) F(a2, a2)... F(a,,an,) [[a a2... an, 12
which, were it not for the illusory elements F((a, ar) in the
diagonal, might be viewed as constituting a solution of the
problem.
To obtain unobjectionable expressions for these, the interpolational forms of p((x), +(x) are necessarily taken, namely
i=n
(ao) f(x) '. + ((an).f() or /f(l(- ai)- ( )
f'(a0). (-a) f'(a,,) (X - ) fa -ai)
i=O
(ao)(.f(x) + (a /f(x)/(x - a. )
4a.)- ( f() + or V( '(a ),
f'(a0). ( - 0) / f'(a,) (x - a,,) f(a<)
i=0
where f(x) stands for (x- a0)(x - a)... (x- a). The resulting
expression for F(x, y),
^V ^-T^ ^ - ___i)(
( ) i f(y)1(y - ai) 3 (a) ) b(a ) N -(i) I fY a))
f! (WOi) ( i f f (a-) f/ (ai(
YIX
is then developed with a view to the actual performance of
the division by y-x. Each of the two multiplications in the
numerator gives (n+1)2 terms; but, as the product of any term