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

AXISYMMETRIC DETERMINANTS (SYLVESTER, 1852) 123
Sylvester arrives at it by squaring
x1 1 0 1
X2 Y2 0 1
X3 Y3 0 1
X4 4 0 1 1
in a special fashion, and then performing certain transformations
on the result.
SYLVESTER, J. J. (1852, Nov.).
[Sur une propriete nouvelle de l'equation qui sert a determiner
les inegalites seculaires des planetes. Nouv. Annales de
Math., xi. pp. 434-440; or Collected Math. Papers, i. pp.
364-366.1
This paper of composite authorship probably originated in a
letter from Sylvester giving his theorem and demonstration,
with a remark or two additional. To these, which were made
~~ 7, 7', 8, the editor prefixed an introduction (~~ 1-6) on
determinants and determinant-multiplication.*
The theorem is an extension of one which is the basis of his
paper in the Philosophical Magazine of the same year, and may
be shortly enunciated as follows: If (11) (22)... (nn) be
axisygmmetric and have 1[11] [22]... [nn]l for its pth power,
then the roots of the equation
[11]-x [12].... [In]
[21] [22]-x.... [2n] =
[nl] [n2].... [nn]- x
are the p"t powers of the roots of the equation
(ll)-x (12).... (In)
(21) (22) -.... (2n) 0.
(nl) (n2).... (nn)*In the Coll. Math. Papers ~~ 1-6 are omitted, and ~~ 7, 7', 8 are numbered
~~ 6, 7, 8. The theorem of the original ~ 6 is incorrect.