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

346 HISTORY OF THE THEORY OF DETERMINANTS

whence,
for r even Al, = (2 4i i) Ar
and
AjA3... A —2
for tr odd Ar (A2A4.
-a result in agreement, as far as it goes, with Sturm's of 1842,
Sturm's non-determinant p,. being the equivalent of Brioschi's A,.
The obtaining of A1. in terms of the coefficients of f(x) is next
illustrated by changing A, into the form *
so 81 82 84
s o81 82 83 84
so S1 82 83 84 85
S1 82 83 84 85 86
and performing operations which we may denote by
col6 + a~col3 + a2col +-.... + a~coll,
col5 + acol4 +.... ~ a4col1,
col2 + a. Col.,
* Brioschi (and afterwards Baltzer, 1 P2, 9) would have done better to change
into the similar form of the seventh order, for then the result would have been
1 al a2 a3 a4 a3 a6
1 a, a2 a3 a4 a.
1 a, a2 a3 a4
A4 it (n.-a(-l)a, (it-2)a2 (n-3)a3
*. (n-l)a, (n-2)a2 (n-3)a3 (n2-4)a4
a (n - )-1)a, (n-2)a2 (n-3)a3 (le-4)a4 (n-5)a.
It (m1- )a, (n-2)a2 (it-3)a3 (nL- 4)a4 (it- 5)a- (n-6)a a,
in which the determinant on the right has a simpler law of formation than
Brioschi's.and yet is readily reducible to the latter, and which, as we see on
putting n1=4, has the further merit of showing that A,, equals the dialytic
eliminant of f(x) = 0, f'(x) = 0.