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

CHAPTER XIII.
HESSIANS, UP TO 1860.
SPECIAL cases of the determinant
Dx2 axay?xaz
-a2 U 32u 3a2q
Dy x Dy2 )y z
3a20 32 a2 2t
Dzzx zzay Dz2
where u is a function of x, y, z,.., may well have appeared at
a very early date in the history of determinants. The case where
u=ax - 2bxy + cy2 may be viewed as traceable to Lagrange
(1773), and the case where =ax2 + by2 + cz2+ 2dyz + 2ezx + 2fxy
to Gauss (1801); but it is certain that in those cases the elements
of the determinants were not looked on as second differentialquotients of u. The general conception first occurred to Hesse
in the year 1843.
HESSE, 0. (1844, January).
[Ueber die Elimination der Variabeln aus drei algebraischen
Gleichungen vom zweiten Grade mit zwei Variabeln.
Crelle's Journal, xxviii. pp. 68-96; or Werke, pp. 89-122.]
In ~ 15 (p. 83) Hesse passes from the direct subject of his paper
to the special case in which the three functions fl, f2, f3 are the