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

ADDENDA AND CORRIGENDA.
I. IN ~~ 27-28 the number of tangents which can be drawn to a
curve from a node or a cusp should be m- 4 and m- 3 respectively.
From any point 0 not on a curve the number of tangents is m.
When 0 lies on the curve, two of the tangents coalesce with the
tangent at 0, leaving m - 2. When 0 is a node, two pairs of tangents
coalesce with the two nodal tangents, leaving m -4. When 0 is a cusp,
three tangents coalesce with the cuspidal tangent, leaving m- 3.
II. The Cayleyan of a nodal cubic is a conic.
The investigation of ~ 118 is not applicable to nodal cubics, since
the canonical form has been used. We shall therefore prove that the
Cayleyan of the logocyclic curve is a conic, whence by projection the
theorem is true for any nodal cubic.
The equation of the curve is x (x2 + y2) = a (x2 - y2); whence writing
down the polar conic of any point (h, k) from (8) of ~ 130 it will be
found that the equation of the Hessian is
5hk2+ r h =-a (k2- h2)........................ (1);
also if $ and r be the reciprocals of the intercepts which the polar
conic cuts off from the axes
a k- a- 2
h = 3 2 t 3 k=............... (2).
3-22a4' (3 - 2a) -q. ()
Since the polar conic of every nodal cubic passes through the node,
it follows that if (h, k) lie on the Hessian, the polar conic consists of
the line xe + y I = 1 and a line through the origin. The envelope of the