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

184 HISTORY OF THE THEORY OF DETERMINANTS
determinant-forms for P(a,, 2,...,,, ).P(31, /2,...,.)
passes thence to the persymmetric determinants in so, 81, s,...
and finally gives Cauchy's evaluation of the double alternant
I (ac - /3i) -1 (a2 - 2)-l ~. (a - /3,-) -1. The applications, which
come next, concern the solution of Lagrange's set of linear
equations, Sylvester's transformation of a binary quantic of odd
degree into canonical form, and the discussion of the equality of
two roots of the equation ax' + a. - 1-... +a0 = 0, or say
f(x)=0, viewed in connection with what, following Salmon,
he calls the " determinant" of the equation, although Sylvester's
use of the word " discriminant" is explained a page or two
later.
Under this last head an interesting transformation falls to be
noted. Calling the roots of the said equation a,, a2,...,,
and taking the determinant which is the square of their differenceproduct, namely,
S0 s8... 8_1 -81 89 '. '^O '
8I ~2 8 or Z say,
$n-1.'% 2n -2
he substitutes for it a determinant of the (2 - 2)th order
1 0 (.... 0 0 0.... 0
0 1 0....0 0.0....0
0 0 1....0 0 0.... 0
0...., 0,.... So *
0 0 0.... 8s s2 s3.... si
0 81 S1.... 8 3 8n- 2 -1.... 82-4
80 S 882.. -2 - S.... -3
81 82 83.. 8n-. 1 n 8n+l..... 2- 2
where the first n - 2 rows do not contain an s, and the rows
following contain all the s's in descending order from right to left,
beginning with s,i_i in the last place of the (n - 1)th1 row, with s