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

240

THEORY OF PROJECTION.

If ABC be the triangle of reference, the equation of the cubic
is
a2 (/ + vy) + MnS/2 = 0,
and the nodal tangents at B are a2 + my = 0, which together
with BA, BC form a harmonic pencil.
It might however have happened, in the case of a nodal cubic,
that the four lines in question did not form a harmonic pencil, in
which case it would not be possible to project every nodal cubic
into the logocyclic curve. We shall have examples of this in the
case of quartic curves; and it is necessary to warn the reader that
counting the constants is not always a safe process, since the
condition thereby furnished, although a necessary one, is not
always a sufficient one.
It can be shown in a similar manner that every cuspidal cubic
can be projected into a cissoid; also since the reciprocal curve is a
cubic of the third class, properties of one cuspidal cubic can be
deduced from those of another by reciprocation.
362. We shall now give some examples of the projective
properties of nodal cubics. From the theorems of ~ 131 and
127 we obtain:(i) From the point of contact A of the tangent from any point
of inflexion I of a nodal cubic, draw a chord APP'. Join IP'
cutting the cubic in p. Then the tangents at P and p intersect on
the curve.
(ii) If from a point of infiexion of a nodal cubic a tangent be
drawn, and through the point of contact any chord be drawn, the
locus of the point of intersection of the tangents at the other two
points where the chord cuts the curve is a cuspidal cubic, whose cusp
coincides with the node of the cubic.
The reciprocal theorem is the following:(iii) Let any cuspidal tangent of a tricuspidal quartic cut the
curve at 0; from any point on the tangent at 0 draw a pair of
tangents to the quartic. Then the envelope of their chord of contact
is a cuspidal cubic.
363. Every anautotomic cubic can be projected into the circular
cubic
x (X2 + y2 + a2) = b (x2 - y2).