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

364 HISTORY OF THE THEORY OF DETERMINANTS

support of the doubtful proposition regarding the vanishing of the
Hessian of a function; namely, that such evanescence necessitates
among the first differential-coefficients of the function the existence
of a linear relation with constant coefficients, and that therefore
through a linear transformation it is possible to express the function
in terms of one variable fewer.
In the second edition published in 1877 the account does not
appear, and in a note (no. 37) the real facts are shortly stated
and reference made to Gordan and Noether (see below, pp. 368-9).
TRUDI, N. (1862): BALTZER, R. (1864, -70, -75).
[TEORIA DEI DETERMINANTI,.... ix+268 pp. Napoli.]
[THEORIE UND ANWENDUNG DER DETERMINANTEN. 2te AufiL
vi+224 pp.; 3te Aufl. vi+241 pp.; 4te Aufl. viii+236 pp.
Leipzig.]
Trudi's treatment (p. 205, ~ 10; p. 208, ~ 3; pp. 218-228, ~~ 1-9) is
full, like Fiedler's, but is not in the same way faulty.
The successive editions of Baltzer show a number of changes,
but as a rule they are merely for the improvement of the
exposition.
In speaking formerly, under Baltzer (Hist., ii. p. 399), of the
vanishing determinant,
m
uI ZIt U2.. l
Ut1 t1 U2... Un
U2 q'tt2 tl s2n
Utn Utn Un2... tenn
we might have noted that because of evanescence and axisymmetry
every element of its adjugate will vanish if one of them vanishes,
and that in particular this is true when the latter is in the (1,l)th
place and is thus A-a fact which is naturally corroborated by
the adjugate being known from (a) and (/) to have A for a factor
of every one of its elements. It is there also seen that the said
adjugate