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

70

CUBIC CURVES.

The tangents at A and B to the Hessian are the coefficients
of a2 and /32 in this expression, and are easily seen to be B= 0,
A = 0. These equations obviously satisfy (29), which shows that
the tangents at A and B intersect on the Hessian.
For the purpose of simplifying the analysis, we shall take the
point C in which the tangents at A and B to the Hessian
intersect as the third vertex C of the triangle of reference, in
which case the tangents reduce to 8=0, a=0. This requires
that I +mn =, X + / = 0, and the cubic becomes
a - X2 + m3 - Xmn2+2 + V = 0..*...... (30),
whilst the Hessian is
a/3 (X2a + mn2/3 - Vy) + (mn4a + X43) y2 = 0......(31).
The polar conics of A and B will form a quadrilateral DEGF
as shown in the figure; and we shall now prove that:115. The diagonals DG and EF of this quadrilateral intersect
at C; and the polar conic of the cubic with respect to C consists of
the line AB, and another line passing through. the third point K
where AB cuts the Hessian.
B
G~~~ A
Since the lines BD, BE constitute the polar conic of A, whilst
AD, AF constitute that of B, the equation of
BD is a-Xy = 0, BE is a +X - 0,
AD is - nry = 0, AF is /3 +ny=0,
from which it follows that the equations of EF and DG are
na + X3=0 and na-/3 = 0,
which obviously intersect at C.
To prove the second part, we observe the polar conic of C is

- 2X2aXy - 2min2Iry + N^y2 = 0,