An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE HESSIAN.

21

35. The first polar of any point passes through every double
point on a curve.
By ~ 22, the coordinates of a double point satisfy the equations
dF/da = 0, dF/d/3 = 0, dF/dy = 0, which obviously satisfy the
equation AF = 0.
36. In ~ 9 we have defined the Hessian of a quantic; we
shall now proceed to investigate some of the properties of the
curve obtained by equating to zero the Hessian of a ternary
quantic, which we shall denote by H (a, 3, y)= 0.
The Hessian of a curve is the locus of the points whose polar
conics break up into two straight lines.
The equation of the polar conic is '2F' = O. Let A = d2F/df2,
F= d2F/dgdh &c. &c., then if the polar conic be written out at full
length it becomes
Aa2 + B/32 + Cy2 + 2F37y + 2Gya + 2Ha,3 = 0.
The condition that this should break up into two straight lines
is that its discriminant should vanish; and the discriminant of the
conic is obviously the Hessian of F (f, g, h). Hence
H(f,g, h)= 0,
and therefore the point (f, g, h) lies on the curve H(a, /3, y) = 0.
37. The Hessian passes through every double point.
The coordinates (f, g, h) of a double point satisfy the equations
dF/df= 0 &c.; and therefore by Euler's theorem
Af+Hg+ Gh = 0,
Hf + Bg +Fh=0,
Gf+ Fg+ Ch= 0,
which shows that the Hessian H (f, g, h) = 0, and therefore the
double point lies on the curve H (a, /, y) = 0.
38. If the first polar of a point A has a double point at B,
then the polar conic of B has a double point at A.
Let (f, g, h) and (,,, ) be the coordinates of A and B. The
condition that the first polar of A should have a double point is
that the differential coefficients of AF should vanish at B. Hence