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

CHAPTER VII.
CURVES OF THE THIRD CLASS.
154. WE have shown in ~ 54 that the class of a curve is
equal to the degree of its reciprocal polar, also that a node and a
cusp respectively correspond to a double tangent and a stationary
tangent on the reciprocal polar. Whence curves of the third class
are the reciprocal polars of cubics, and may be classified according
to the following scheme:
n 8 / m n r D
6 0 9 3 0 0 1
4 0 3 3 1 0 0
3 0 1 3 0 I 0
The first species, which are the reciprocal polars of anautotomic
cubics, include all sextic curves of the third class. They have
nine cusps, and no nodes, double tangents or points of infiexion;
also since six of the points of inflexion of an anautotomic cubic
must be imaginary, six of the cusps of the sextic must also be
imaginary.
The second species, which are the reciprocal polars of nodal
cubics, include all quartic curves of the third class. They have
three cusps, one double tangent and no nodes or points of
inflexion; also since two of the points of inflexion of a crunodal
cubic are imaginary, it follows that if the double tangent touches
the quartic in two real points, one of the cusps must be real and
the two others imaginary. If on the other hand the double
tangent touches the quartic in two imaginary points, all three
cusps must be real. Since an acnode is a real point, the corresponding double tangent must be a real straight line; but the
points of contact, which correspond to the tangents at the acnode,
will be imaginary.
B.. 7