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

30 HISTORY OF THE THEORY OF DETERMINANTS

TRZASKA, W. (1870).
[Krotkie wiadomosci o wyznacznikach. Appendix to Folkierski's
Zasady ruchunku vozniczkowego i calkowego.... Tom i.
xlvi+1087 pp. Paris.]
Trzaska's 'short account of determinants' extends to forty-nine
pages (pp. 1031-1079), and, as is appropriate enough in a book
dealing with the Differential Calculus, a considerable share of space
(pp. 1059-1075) is devoted to 'functional determinants.' It is
followed by a collection of seventeen exercises (pp. 1080-1087).
BALTZER, R. (1870).
[Theorie und Anwendung der Determinanten. 3te verbesserte
Auflage. vi+241 pp. Leipzig.]
In this edition the 'Theorie' is enlarged to the extent of four
or five pages. One interesting addition (p. 31) is a simple deduction
from the identity
a1 - a2 t3
bl b2-y b3
C1 C2 3- Z
- xyz+ acyz+ b2zx + c3xy
- 6 b3 [ - -'I cAl y- Ia.b2I +1 ab2c3
Putting x, y, z equal to the sums of the a's, b's, c's respectively,
we have
0 = - aZbEc+ al1b1c + b2Ec1Ea+ C3aZtEb
- I ab2 I Ec - Ib|c3 2ci a - c3a c Eb + |I acb2c,
a result which, notwithstanding Baltzer's different standpoint, we
may view as a generalisable relation between the coaxial minors
of the determinant [ alb2c3 I and the sums of its rows.
Another addition (~ 5. 8, pp. 46-49) concerns the result of the
substitution which is possible when we have given
x1, x2,..., x linear functions of x'x,,,,,
xl', X',..., x, linear functions of x", X2,..., xn
and so on. The subject, however, belongs more naturally to the
theory of matrices (see ~ 11 of Cayley's memoir of 1857).