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

330 INFLEXION POINTS OF A CUBIC CURVE [CH. XVIII

the element in the rth row and jth column of the determinant
equal to A' hA is
Ora (a& a (a& C6r a (a =a (a~
aXisie Y 2 cay isa3 p d es y
since cir is the partial derivative of xi with respect to yr. Hence
A2h a 2~ = Hessian H of.
yr ayj r, j= 1, 2, 3
In words, A2h becomes H under the transformation (3),
so that H=0 represents the same curve as h=0, but referred
to the new triangle of reference. Hence there is associated
with any curve f=0 a definite Hessian curve h=0 independent
of the choice of the triangle of reference.
175. Points of Inflexion of a Cubic Curve. Let f(xi, X2, X3)
be of the third degree. Choose a triangle of reference having
the vertex P= (0, 0, 1) at a point on the curve f= 0, not a singular point. Then there is no term involving x33, and the coefficients of the terms rX1x32 and SX2X32 are not both zero, since
otherwise the derivatives (5) would all vanish at P. Hence
we may take rxi+sx2 as a side of a new triangle of reference
with the same vertex P and obtain
X32X1 +X3(axl2 +bxlx2 +CX22) + ((XI, X2) =0
as the new equation of our curve. Replacing X3 by
X3 -(ax +bx2)/2,
we get
F1 =x32X +ex3x22+C(Xl, X2).
Denote the second derivative of the cubic function C with
respect to xi and Xj by Ci. Then the Hessian of F1 is
C11 C12 2X3
H1i= C21 C22 +2eX3 2ex2 = -8e33+..
2X3 2ex2 2xi
Hence P = (0, 0, 1) is on H1 =0 if and only if e =0.
If d is the coefficient of X23 in C, then xi =0 meets F =0 in
the points for which x22(ex3+-dx2)=0, and these three points