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

382 HISTORY OF THE THEORY OF DETERMINANTS

CAYLEY, A. (1847).
[Note sur les hyperde'terminants. Crelle's Journat, xxxiv.
pp. 148-152; or Collected Math. Papers, i. pp. 352-355.]
As already noted, the second section of this short paper
concerns what would, a few years later, have been called "the
Hessian of the discriminant
Gabed + 3b2C2 - a2d2 - 4ac3 - 4b3d
of the binary cubic." The result, which is rather inelegantly
verified, is that the said Hessian is a numerical multiple of the
square of the discriminant.
HESSE, 0. (1847, August).
[Ueber Curven dritter Classe und Curven dritter Ordnung.
Crelle's Journ., xxxviii. pp. 241-256; or Werke, pp. 193-210.]
Any homogeneous function of the Mzh degree in the variables
XI, x2, x, being denoted by m, its first differential-quotients by
'UzI U2 U.3, and its second differential-quotients by m1, 'a12,
there is obtained from Euler
a11x1 + '12X2 + 'a13x3 = (rn- 1)u'1,?%21X1 + '22X2 ~ 1%23X3 = (wn- 1)m3 j
'a31x1 +.'32a2 + 'a33X3 = (M -1)u,3
and thence, on solving,
r-l -x1 = U11 1+U12U2 + U13%,m - I
A
X2-" = U21ul + U22'c62 + U23
_ _ I
A 3 = U21UI +U222+U, + 333
m/ I
where, evidently, A is used for Hesse's determinant of 'a, and U,,
for the cofactor of a,1, in A. Using in connection with the latter