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

ON THE HESSIAN AND THE CAYLEYAN OF A CUBIC. 73

If we had eliminated X,,u, v and k we should have found that
X', /', v satisfy (41); hence we obtain the theorem:The two straight lines which constitute the polar conic of the
cubic with respect to any point on the Hessian are tangents to the
Cayleyan.
119. From the preceding theorem it appears that the four
straight lines AD, AF, BD, BE each touch the Cayleyan, and we
shall now prove that:-The points of contact of these straight lines
are collinear.
Let Q, q be the points of contact of AD, AF; and let 84,
r + A8q, $S be the coordinates of a point B' on the Hessian near B.
The polar conic of B' is
dlF dF dF E
- + (, + 8) = 0.
da d/3 dy
To find where this cuts AD, we must differentiate (30) and
put / = ny, and we obtain
dF/da = a2 - X272, dF/d3 = 0,
dF/d7 = - 2X27y - 2mn3y2 + Ny2.
Writing A, q, ' for a, /, y in (31), differentiating and putting:= '= 0 we obtain 8 =0; whence the points where the polar
conic of B' cuts AD are given by the equation
7 (2X2c + 2mn37 - N7) = 0,
and therefore the equation of BQ is
2X2a + 2mn3y- Ny = 0.
Putting / = - ny, it can be shown in the same manner that
the equation of Bq is
2X2a - 2mny - Nry = 0,
whence the points Q, q lie on the straight line
22a+ 2mn2 -Ny = O................(42).
By considering the points of intersection with BD, BE of the
polar conic of a neighbouring point A', it can be shown that the
points P, p lie on (42); whence the four points P, p, Q, q are
collinear.
Since equations (32) and (42) are identical, it appears that the
four points and also the point K lie on one of the lines which
constitutes the polar conic of C.