421 
1909-10.] The Theory of Persymmetric Determinants, 
taneously vanish, or, say, that we shall have 
Brioschi, Fr. (1854, March). 
[La Teorica dei Determinant^ e le sue principali applicazioni ; 
viii + 116 pp., Pavia. French translation by Combescure, 
ix + 216 pp., Paris, 1856. German translation by Schellbach, 
vii + 102 pp., Berlin, 1856.] 
Denoting by s r the sum of the r th powers of the roots of the equation 
a n + a n _ X x+ . . . + cqaf _1 + = 0 
Brioschi recalls the n known relations 
a n S 0 
+ a M _ 1 s 1 + . . 
, . + 
1 + S n 
= 0 
a n S 1 
+ a ,i- l s 2+ • ■ 
, . + 
+ 
= 0 
a n S n- 
-1 + Mn-l S n+ • ■ 
, . + a 1 s 2n . 
-2 + S 2n-1 
- 0. 
and thus derives by elimination 
s o 
S 1 • 
S 1 
s 2 . 
• • s n+l 
S M-1 
s n ' . , 
• • ®a»-i 
1 
X 
. . x n 
and therefore 
h ~ *d* 
s 2 - S Y X 
• • *» “ «»-!» 
h~h x 
h - h x 
• • S »i+1 ” S n X 
s n - s n _ x x 
S w+1 — S n X • • 
• S 2n-1~ S 2n-2 X 
Further, he points out that if the last determinant be denoted by V n , 
and the cofactor of its last element by Y n _ x , and so on, then Y n being axi- 
symmetric it follows from Cauchy’s theorem of 1829 that Y n , Y n _ x , . . . , V x , 1 
possess the characteristic property of Sturm’s remainders. 
It is not noted that the set of n relations used gives each of the as in 
terms of the s’ s, and that substitution in the original equation then gives 
S 0 
*1 ‘ * * 
S 1 
s 2 . . . 
s w+l 
S n - 1 
S n ... 
s 2n-\ 
1 
X . . . 
x n 
s o s i • ■ 
' • S n - 1 
(x n + a 1 x n *+ . . 
• • + d n ) 
S 1 *2 • ' 
' • 
»«-l • ■ 
■ • S 2n-2 
.as may be otherwise seen. 
