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

CHAPTER V.
CUBIC CURVES.
90. THE general equation of a cubic curve contains nine
independent constants, that is one less than the number of terms
in a ternary cubic; hence a cubic curve may be made to satisfy
nine independent conditions. It also follows from ~ 24 that
not more than six tangents can be drawn from any external point
to the cubic; nor more than four from a point on the curve; nor
more than three from a point of inflexion. Also since a straight
line cannot intersect a cubic in more than three points, a cubic
cannot have more than one double point unless it breaks up into
a conic and a straight line or into three straight lines. Moreover
every tangent cuts the cubic at one other point; and since the
asymptotes are tangents at infinity, every asymptote cuts the
curve at one other point, which may be at a finite or infinite
distance from the origin. Also by ~ 40 a cubic has three
asymptotes, one of which must be real.
Cubic curves are divided into the following three species, viz.:
(i) Anautotomic Cubics, which have no double point; (ii)
Nodal Cubics, in which the double point is a crunode or an
acnode; (iii) Cuspidal Cubics, in which the double point is a cusp.
Since n = 3, Plicker's numbers for the three species are found by
successively putting in equations (4) to (8) of ~ 89, K = =0;
C = 0, 8=1; c = 1, 8=0, which lead to the following table:
n K8 m T D
3 0 0 6 0 9 1
3 1 0 4 0 3 0
3 0 1 3 0 1 0