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

PERSYMMETRIC DETERMINANTS (BRIOSCHI, 1854) 343
and therefore
S1 - S2 - S1X... Sn - Sn-1X
82 - six SR - 82X... Sn+1 SnX: / 0.
n i - 1X Sn, -SnX... S2n-1 - o2n-2.
Further, he points out that if the last determinant be denoted by
V,, and the cofactor of its last element by Vn-1, and so on, then
Y, being axisymmetric it follows from Cauchy's theorem of 1829
that n,) V,,6,1..., V1, 1 possess the characteristic property of
Sturm's remainders.
It is not noted that the set of n relations used gives each of
the a's in terms of the 8's, and that substitution in the original
equation then gives
~~~~~~S Si S S
so Si S. S.... sib~
81 82. 111s 11.. S-1
(X11+a xn-1+... +a,,) 1 82... 8z
fl-i Snf 592n-1
x... Xn 8n-1~S12- Sn, S211 - 2
as may be otherwise seen.
BRIOSCHI, F. (1854, February).
'[Sur les fonctions de Sturm. Nouv. Artnales de Math., xiii.
pp. 71-80; or Opere mat., v. pp. 89-97.]
Briosehi in effect here recalls that if f, ji, f2. be the series,of Sturm's functions originating in the consideration of the
'equation
Xn. + a~CL-1 + a 2x'-2 +.... + an = 0, or, say, f(x) = 0,
and q,, q21. -be the linear functions of x which are the,quotients obtained in the process of finding f2, f3,. then
(1) f= q1fi-f2, fA=q2f2-f3A......, fr-2=qr-i f;r-i fr,(2) fr is of the (ni - r)th degree in x;