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

CHAPTER XI.
PERSYMMETRIC DETERMINANTS, FROM 1841 TO 1860.
As has already been pointed out (History, i. pp. 485-487*), the
special form of determinant named "persymmetric" in 1853 by
Sylvester came first to light in 1835 in a paper of Jacobi's on the
elimination of the unknown from two equations of the nth degree,
the fact being that the adjugate of Bezout's condensed eliminant
-in other words, the adjugate of the determinant resulting from
Bezout's "abridged method" of elimination-is there shown to
be such that the elements of it whose place-numbers have the
same sum are equal.
The essentials of the proof are easily made clear if we accept
the fact that from the equations
a1x + acy + az = 0
b1x + b2y + b3z = 0
cdx + c2y + cz = 0
it can be shown for non-zero values of x, y, z that
x: y: z:: A A2: A:: B: B2: B3::1 C: C 3.
This is something more than what Jacobi had then occasion to
use, but in 1841 the portion of it which holds when there is one
equation fewer was stated by him in all its generality in ~ 7 of
* The 7th and 8th lines of p. 486 have unfortunately been transposed by the
printer. Also, in the first determinant of the footnote on the same page the
first b1 should be b0.