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

330 HISTORY OF THE THEORY OF DETERMINANTS

For ourselves we may add that the theorem becomes still more
interesting when it is pointed out that, by reason of the identity
-2y 1 n-2 - _ Y1 + Y2 Y
Iy... ', | I y, Y12.,,: I -, (+...+
where 0(y) = (YT-Y1)(Y-Y2) *. (Y-Yn), the R's like the w's
are all expressible as determinants of the order n + m+1, that
these determinants in both cases belong to the special type known
as alternants, and that R. differs from wp in the last column
only;-in fact, that
1 x1 x\... X^ 1 xl/(x1-x)
1 x3 X,... x,, -1 a'U8/(x3-x).;:
1 X 2... n-z-l 1 P UZ1 (2 - )
1 1...,n+m -' 1
2 n+m-1
' 2 2... $2 X2 U2(02 S) *X)
1 X3 X... X.+n-1 XPU3(X-x)
where f is the difference-product of x1, x2, * *, xn+.
1 2, — 1 X
BORCHARDT, C. W. (1847, February).
[Developpemnents sur l'equation a l'aide de laquelle on determine
les inegalites seculaires du mouvement des planetes. JoLrn.
(de Lioville) de 2Math., xii. pp. 50-67; Gesamrnelte Were,
The new section of this paper, which is an extension of
Borchardt's of 1845 (January), is the third (pp. 54-60), and
explains at length how, for the purpose of ascertaining the
total number of real roots of the equation of the neth degree
f(x) =, the coefficients of highest powers in the series of Sturm's
functions f(x), f,(x), f(x),.. may be replaced, according to
Sylvester, by
1, ",, — (x2 —xl)2n, (X2- xl)2(X3 — 1)2(X2-1) )2,...