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

ON THE HESSIAN AND THE CAYLEYAN OF A CUBIC. 69

On the Hessian and the Cayleyan of a Cubic.
113. We have proved in ~ 38 that if the first polar of any
point A has a double point at B, the polar conic of B has a
double point at A. In the case of a cubic, the first polar is the
polar conic; hence the theorem becomes,-If the polar conic of A
breaks up into two straight lines intersecting at B, the polar conic
of B breaks up into two straight lines intersecting at A. The
points A and B obviously lie on the Hessian of the cubic (which
is another cubic), and are called by Professor Cayley conjugate
poles*. The envelope of the line joining two conjugate poles was
called by Professor Cayley the Pippian; but it is now usually
known as the Cayleyan.
114. Tangents to the Hessian at two conjugate poles of a cubic
intersect on the Hessian.
Let the conjugate poles A and B be two of the vertices of the
triangle of reference; then the polar conies of A and B must be
of the form
dF/da = (a + xy) (a + ry) = 0,
dF/d/3 = (m/3 + Xy) ( +?ny) = 0,
and therefore the equation of the cubic is
i a3 + I (X + it) al2 + -Xtay2 + ] rn/33 - (1+ in)?32
+ ln3y2 + I Nrs3 = 0......(27).
Now if A = d2F/da2, A' = d2F/dlfdy &c.,
A = 2a + (X +)
B = 2m/3 + (I + mn) 7
C = 2Xpua 4+ 21n/3 + ~22y(
A' = (I + n13 + 21y.................. (28s),
A' = (1 + mn) i3 + 21ny
B' = (X +,uL) a + 2Xwy
C'=0
and the equation of the Hessian is
H = ABC- AA'2- BB'2.............. (29).
* Cayley, "A memoir on curves of the third order." Phil. Trans. 1857, p. 415;
Collected Papers, vol. ni. p. 381.
J. J. Walker, Phil. Trans. 1888, p. 170; Proc. Lond. Math. Soc. vol. xx. p. 382.