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

72

CUBIC CURVES.

intersects the Hessian in one other point C, and from C three
other tangents exclusive of the tangent at A can be drawn to the
Hessian. The points of contact B, B1, B2 of these three tangents
are the three conjugate poles; also since the lines AB, ABE, A B2
are the only tangents that can be drawn from A to the Cayleyan,
this curve is of the third class.
118. To find the tangential equation of the Cayleyan.
We have already pointed out that every anautotomic cubic
curve may be expressed in the canonical form
X3 + y3 + + 61xyz = 0...................(37),
and that the Hessian is
H =-12 (x3 + + 3) + (1 + 213) yz = 0.........(38),
and is therefore a cubic of the same form as (37).
We have shown in ~ 115 that the polar conic of C is the line
AB and another line through the point K. Let the equations of
these lines be
Xx+Py+vz=0)..(39).
X'x + I' + V'z = 0.
Let X, Y, Z be the coordinates of C; then the polar conic
of C is
X (x2 + 21yz) + Y (y2 + 21zx) + Z (z2 + 21xy) = 0...(40).
In order that (40) may be identical with the product of (39)
we must have
XX'= kX, Bu'= kY, vv'= kZ,
- v' + 2'v = 2klX,
vX' - v'X= 2klY,
X/p' + X'/ = 2klZ,
where k is some constant. Eliminating X',,k', v' from the last
three by means of the first three, we obtain
- 2l1vX + v'Y +,2Z = 0,
zX-21vX Y + X2Z =0,,u2X + X2 Y- 21kylZ = 0,
whence eliminating X, Y, Z, we get
(X3 + + 3 )+(1-43) X = 0............ (41).
This is the tangential equation of the Cayleyan, and its form
shows that the curve is of the third class.