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

CUBIC CURVES.

239

imaginary points respectively. Also by means of an imaginary
projection an elliptic lima9on with two imaginary points of
inflexion can be projected into a quadripartite quartic.
359. In order to apply the theory of projection with advantage,
the first step is to draw up a table of the simplest form or forms
which curves of any given species can assume. The next step is
to investigate the properties of these simple forms by any convenient method, and then to generalize those which are capable of
projection. We shall therefore proceed to examine some of the
simplest forms of cubic and quartic curves, and shall incidentally
show that a variety of results, some of which are known whilst
others are probably new, may be deduced from the properties of
various well known curves.
Cubic Curves.
360. Any nodal cubic can be projected into the logocyclic
curve; and every cuspidal cubic into a cissoid.
The equation of the logocyclic curve in its simplest form is
(x2 + y2) = a(x2 - y2), and therefore contains one constant.
Transfer the origin to any point in the plane (x, y) and two
more constants will be introduced, which make three. Project
on any plane passing through the new origin, and five more
constants will be introduced making eight, which is the number
of independent constants which the general equation of a nodal
cubic contains.
The logocyclic curve has one real point of inflexion I at infinity, and the asymptote is the inflexional tangent; hence the
tangent at the vertex A is the tangent drawn from the real
point of inflexion to the curve. Now since the nodal tangents
bisect the angles between 01 and OA (which are at right angles)
it follows that these lines together with the nodal tangents form
a harmonic pencil; hence the possibility of projecting any nodal
cubic into the logocyclic curve, depends upon the following theorem,
which we shall proceed to prove.
361. If from any point of inflexion A of a cubic whose node
is B, a tangent be drawn touching the curve at C, the lines BA,
BC together with the nodal tangents form a harmonic pencil.