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

HESSIANS (TRUDI, 1862: BALTZER, 1864, -70, -75) 3365

1 xI x2... Xn
XI XI2 x1X2 X.Xn
A,6+12
fX2 XflX1 X2.. X.Xn
Irl n;X1 XnX2~~ Xn"
which, as it ought, manifestly vanishes, whatever A may be.*
STUDNICKA, F. J. (1868).
[U'eber die Anwendung der Hesse'schen Determinanten in der
Theorie der Maxima und Minima von Functionen mehrerer
unabhingigen Variablen. Sitzungsb.... Ges. d. Wiss. (Prag),
Jahrg. 1868, pp. 67-72.]
As is known, from Lagrange (1797),t the expression whose sign
decides whether an extreme value of
AXI1, X2,., Xn)
is a maximum or minimum is
~fnl *z 241Ih2f12 +. +. +1 h2fn,n,
where the h's are infinitesimal increments of the x's, and
is denoted by fq.
The origin of Studni'ka's so-called application lies in the fact that
the form of this expression is that of a quadric in the h's with the
Hessian
Ifnl 122. fnrtl
for its discriminant. There is nothing fresh in his paper save the
use of the determinant notation and of the name 'Hessian.'
* It may also be noted that an identity, which in this connection we may appropriately write in the form
Xi d2. 2 m X,,
H(u) 2
is attributed by Baltzer to Lacroix, the reference being Gaic. diff., ~ 91 ' at first,
and then later to 'OCalc. diff., ~ 292.'
t LAGRANGE, J. L. Thgorie des Fonctions Analytiques, 2e Partie, chap. iie;
or (iEuvres, ix. pp. 280-295.