424 
Proceedings of the Royal Society of Edinburgh. 
[Sess. 
whence, 
and 
for r even A r 1 = f ^ 2 ^ 4 2 ^) A,. 
. . . A r J 
for r odd A r , = [— i A « • ■ • ± r 
Va 2 A 4 . . . A r J 
— a result in agreement, as far as it goes, with Sturm’s of 1842, Sturm’s non- 
determinant p r being the equivalent of Brioschi’s A r . 
The obtaining of A r in terms of the coefficients of f(x) is next illustrated 
by changing A 4 into the form * 
1 . . • 
1 . . . . 
5 0 S 1 S 2 S 3 
5 1 S 2 S 3 S 4 
5 2 S 3 S 4 S 5 
and performing operations which we may denote by 
col 6 + cq col 5 + a 2 col 4 + . . . + a b cob » 
col 5 4- cq col 4 + . . . + a 4 cob , 
the result being 
i <q 
col 2 + <q 
2 a 2 
co h j 
3 a 3 
4a 4 
5 
<q 
a 2 
a 3 
a 4 
«5 
A, = 
1 
a i 
a 2 
a 3 
a 4 
4 
n 
( n - l)q 
(n - 2 )a 2 
( n - 3 )a 5 
n 
(n - l)q 
s 
i 
to 
V 
to 
(n - 3 )a 8 . 
(n - 4 )a t 
n 
(n - 1 )a 1 
(n-2)a 2 
(n - 3 )a s 
(n - 4)<q 
( n - 5 )a. 
To obtain the desired expression for D r _ 4 (^ Brioschi takes the set of 
* Brioschi (and afterwards Baltzer, § 12, a) would have done better to change into the 
similar form of the seventh order, for then the result would have been 
a 2 
CL% 
a 4 
a 5 
a 6 
a i 
a 2 
a 3 
a 5 
1 
«i 
a 2 
a 3 
a 4 
n 
( n - l)«j 
(n - 2 )a 2 
(n-3)a 8 
n 
( n - l)a 4 
(n ~ 2)a 2 
( n — 3 )cs 3 
( n - 4 )a 4 
(n - l)a 4 
(n - 2 )a 2 
(»-3)a s 
(n - 4)a 4 
(n-5)a s 
{n - 2 )a 2 
( n-s)a 3 
(n - 4 )a 4 
(n-5)a 5 
(n - 6)a 6 
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 n = 4, has the further 
merit of showing that A n equals the dialytic eliminant of f{x) = 0 , f(x) = 0 . 
