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

DETERMINANTS IN GENERAL (HESSE, 1868)

29

We are thus led up to the important fact that when in addition
A is axisymmetric, the two factors on the right are identical. The
theorem may consequently be viewed as a generalisation of Salmon's
of the year 1859.
It may be noted, however, in passing that Hesse's notation rather
obscures the true character of his result, which is nothing more
nor less than the simplest case of Jacobi's theorem regarding a minor
of the adjugate. For dispensing with differentiation by using
A,. for the cofactor of a,.s in A, we readily see that B is A,,: that
the first factor of P is the determinant got from B by changing
the suffix /3 into q, that is to say, is Ap: that similarly the second
factor is Aaq that aB/3a a is the cofactor of I acpq | in A: that
aB V 2 B
DB Clt_ — TC I a Lqpj is Ad:
K.A
and that therefore Hesse's result is
| aja22.. a |. cofactor of I aapapq | = ApqAa - A A
= I A.Ap.
STUDNICKA, F. J. (1869, 1870): ZELEWSKI, A. v. (1870).
[0 determinantech. 64 pp. V Praze.]
[Einleitung in die Theorie der Determinanten. Fir Studirende
an Mittelschulen und technischen Anstalten. vi+66 pp.
Prag.]
[The Elements of the Theory of Determinants (in Russian). Prag.]
[Ein Beitrag zur Theorie der Determinanten. 31 pp. Breslau.]
The German title of Studnicka's booklet accurately describes its
character. It consists of an introduction followed by ten sections
(~~ 1-10) under descriptive headings, each of the last four sections
dealing shortly with a special determinant by name. The edition
in Russian, published the same year, I have not seen.
Zelewski's so-called 'Beitrag ' is a very simple but very formally
arranged introductory sketch.