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

QUARTIC CURVES.

243

In ~ 301 we proved that:Tangents at the extremities of a chord drawn through the real
cusp of a cardioid intersect at right angles on a circle, whose centre
is the triple focus, and which touches the cardioid at the point where
the cuspidal tangent intersects it.
Whence by projection:(iii) If through any cusp of a tricuspidal quartic a chord be
drawn cutting the curve in P and Q, the tangents at these points
intersect on a conic which (a) touches the quartic at the point of
intersection of the cuspidal tangent with the quartic; (/3) passes
through the other two cusps of the quartic; (y) also the point of
intersection of the cuspidal tangents to the quartic is the pole of the
line joining the other two cusps with respect to the conic.
(iv) The tangents at P and Q, together with the lines joining
their point of intersection with the other two cusps, form a harmonic
pencil.
From the theorem that the evolute of a cardioid is another
cardioid, we obtain:(v) Through any point P on a tricuspidal quartic draw two
straight lines to a pair of cusps; draw the tangent at P and the
harmonic conjugate of these three straight lines. Then the envelope
of the harmonic conjugate is another tricuspidal quartic, two of
whose cusps coincide with the pair of cusps of the original quartic.
368. Any quartic having a node and a pair of cusps can be
projected into a limacon.
For the quartic can be projected into a curve having a node at
the origin and a pair of cusps at the circular points, and the
limacon is the only quartic having these singularities at the above
mentioned points.
The ninth species of quartics, of which the limaqon is one of
the simplest forms, is of great interest owing to the fact that such
curves reciprocate into quartics of the fourth class. We may
therefore deduce properties of quartics belonging to this species
by first projecting those of the lima9on, and then reciprocating the
projected curve with respect to any origin.
369. In the limacon, the nodal tangents, the line joining the
node with the triple focus and the line drawn through the node
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