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

ORTHOPTIC LOCI.

99

From any two points T, t on a cuspidal tangent of a curve of
the third class draw two pairs of tangents TP, TQ and tp, tq to the
curve; and let P, p, Q, q be their points of intersection, then PQ
and pq pass through the harmonic point.
From (ii) it follows that in the case of a tricuspidal quartic,
the lines PQ and pq intersect on the double tangent; which may
be easily verified in the case of some simple curve such as the
cardioid or the three-cusped hypocycloid.
(v) If two tangents be drawn to a cubic from a point A on the
curve, the tangent at the third point where the chord of contact
intersects the cubic meets the tangent at A at a point on the curve:
whence,
Let a straight line touch a curve of the third class at D and
intersect it at B and C. Let the tangents at B and C intersect at A,
and let the tangent at A touch the curve at E; then DE touches the
curve.
156. The foregoing examples sufficiently illustrate the application of the method of reciprocal polars in the case of curves of a
higher degree than the second. It will, however, be shown in
Chapter XII. that any projective property of a nodal cubic may
be deduced from the corresponding property of the logocyclic
curve; and therefore instead of reciprocating the properties of
this curve, and thereby deriving properties of a special class of
tricuspidal quartics, the preferable course is first to generalize by
projection, and afterwards to reciprocate. But in the case of
properties which are not projective, the method of reciprocation
may be employed with advantage in the first instance.
Orthoptic Loci.
157. In ~ 68 we have explained a general method of finding
the orthoptic locus of a curve. We shall now apply this method
to examine the orthoptic loci of curves of the third class.
In dealing with this subject, the most convenient classification
to make is a fourfold one which is founded upon the position of
the origin of reciprocation.
(i) Let the origin not lie on the cubic. Then the reciprocal
polar consists of all sextic, quartic and cubic curves of the third
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