Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 1741

HESSIAN CURVE

329

If xl, X2, X3 is a set of solutions, not all zero, of
(5) fO0, - fO, - '0,
axl aX2 3ax
and hence by (4) off=0, the point (xl, X2, X3) is called a singular
point of the curve f=0. At this point,
Nf aE f ax, = (j = 1, 2, 3).
yj ai=l x ay
Hence the definition of a singular point is independent of the
special triangle of reference chosen. It is readily proved,
but not presupposed in what follows, that two or more branches
of the curve pass through any singular point, which is therefore called a double or multiple point.
174. Hessian Curve. The Hessian of f is
a2f 32f a f
aXl2 al ax 2 aXl EX3
h=32f a2. a2f.
ax2 x1 aX22 x2 ax3
a2" a2f a32
aX3 ax ax3 X2 EX32
Let transformation (3), of determinant A, replace f by ((yi,
y2, y3). The product hA is a determinant of the third order, in
which the element in the ith row and jth column is the sum
of the products of the elements of the ith row of h by the corresponding elements of the jth column of A, and hence is
a2f a2f a C2f
-- lj - C2j + C3j.
ax axi C+ ax at2 ai aax3
The latter is the partial derivative with respect to xi of
af aXl af a2 af axa3 _
axi ayj ax2 ayj ax3 ayj ayj
Let A' be the determinant obtained from A by interchanging
its rows and columns. By the same rule of multiplication,