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

366 HISTORY OF THE THEORY OF DETERMINANTS

GUNDELFINGER, S. (1872, 1876).
[Ueber die Ausartungen einer Curve dritter Ordnung. Math.
Annalen, iv. pp. 559-572.]
[Intorno ad alcune formole della teoria delle curve di secondo e di
terzo ordine. Annali di Mat., (2) v. pp. 223-235.]
The subject here dealt with had already been examined by
Sylvester in 1852 (see Cambr. and Dubl. Math. Journ., vii. pp. 187 -188). Much of it does not concern us. The most relevant part
bears on Sylvester's fifth result, namely, that a ternary cubic when
it is proportional to its Hessian is resolvable into linear factors. This
Gundelfinger proves at length (pp. 227-230).
As an example we may note for ourselves the circulant
x3 + y3 + z3 - 3xyz,
the Hessian of which is
- 54(x3 + y3 + z3 - xyz).
CAYLEY, A. (1872).
[Theorem in regard to the Hessian of a quaternary function. Quart.
Journ. of Math., xii. pp. 193-197; or Collected Math. Papers,
ix. pp. 90-93.]
The function in question is
P7 + XQ1",
where P, Q are quaternary functions of the same variables, and X
is a constant. The result is lengthy and unattractive, and the
demonstration confessedly tedious.
CASORATI, F. (1874).
[Sui determinanti di funzioni. Mem. del R. Istituto Lombardo,
xiii. pp. 181-187.]
As already explained (chap. ix.), part of Casorati's object in this
paper was to ascertain the effect produced on the Hessian by multiplying or dividing the basic function by another function of the
same variables. In his quest (~ 2) no theorem analogous to those
which hold in the case of the Jacobian, prejacobian and Wronskian