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

CHAPTER VIII.

QUARTIC CURVES.
158. THE general equation of a quartic curve is of the form
u4 + U3 +2 U+ U + uo = 0, where u, is a binary quantic in x and y,
and therefore contains fourteen independent constants. A quartic
cannot have more than three double points; or it may have two,
one or no double points; also any double point may be a node or
a cusp. It therefore follows from Pluicker's formulae, ~ 89, that
quartic curves may be divided into the following ten species,
which are shown in the accompanying table.
n 8 K m T t D
I. 4 0 0 12 28 24 3
II. 4 1 0 10 16 18 2
III. 4 0 1 9 10 16 2
IV. 4 2 0 8 8 12 1
V. 4 1 1 7 4 10 1
VI. 4 0 2 6 1 8 1
VII. 4 3 0 6 4 6 0
VIII. 4 2 1 5 2 4 0
IX. 4 1 2 4 1 2 0
X. 4 0 3 3 1 0 0
From the preceding table it will be observed first that in the
last four cases the curve is unicursal; secondly, that the tenth
species is the only one in which the quartic is of the third class;
whence a variety of theorems relating to tricuspidal quartics can
be obtained by reciprocating the properties of nodal cubics.
Thirdly, the ninth species is the only one of the fourth class, and
is therefore the only species in which properties of one quartic
can be derived from another by reciprocation.