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

DETERMINANTS IN GENERAL (BELLAVITIS, 1858) 101
have two common roots, these roots are given by the equation*
a + bx + cx2 d
a - fx + yx2. e =0.
ax + bx2 c d
Another note (pp. 627-628), in view of Mainardi's 'regola'
above referred to, very properly draws attention to the rule
given in the appendix to Bellavitis' Sposizione of 1857.
ZEHFUSS, G. (1858): MAINARDI, G. (1855).
[Ueber die Zeichen der einzelnen Glieder einer Determinante.
Zeitschriftf. Math. u. Phys., iii. pp. 249-250.]
[Una regola per attribuire il segno proprio ad ogni parte di un
determinante numerico. Atti.. Istituto Lombardo (Milano),
i. pp. 105-106.]
Neither of these communications is of importance. Zehfuss,
using the recurrent law of formation and giving " drangement "
the very opposite of its original meaning, so that the principal
term of an n-line determinant has ~n(n-1) derangements, seeks
to show that the sign of any other term having / derangements
is (-l)2n(n-l)-.
Mainardi, employing Cauchy's "clefs algebriques," finds himself also face to face with derangements, and seriously advises
that in counting them we should say, not 1, 2, 3, 4, 5,..., but
1, 2, 1, 21, 1,..., the sign being - or + according as we end with
1 or 2.
GALLENKAMP, W. (1858).
[Die einfachsten Eigenschaften und Anwendungen der Determinanten. 12 pp. Sch. Progr. Duisburg.]
A workmanlike twelve-page exposition.
*No reference is made by either Sylvester or Bellavitis to the two other
similarly derivable quadratics
b + cx + dx2 a. c + dx a b
a + bx + x2. d =0, b + cx + dx2. a = 0.
f + yx + 62 a e 7 + y x + ex2 a p