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

THE ELLIPTIC LIMACON.

189

curve at two imaginary points, and two imaginary points of inflexion. When 2b =a, the vertex A becomes a point of undulation, and the reciprocal singularity is a triple point composed of a
crunode and two real cusps. When a> 2b the points of inflexion
on the limanon are imaginary, and the double tangent touches the
curve at two imaginary points; hence the reciprocal curve has two
imaginary cusps and a conjugate point.
When a < b, the lima9on is hyperbolic, and has two imaginary
points of inflexion and a double tangent touching it at two real
points. Hence the reciprocal curve has a crunode, two imaginary
cusps, a double tangent touching the curve at two real points, and
two imaginary points of inflexion.
The reader will find it an instructive exercise to trace the form
of the reciprocal curve when the origin of reciprocation moves
along the axis of x from plus to minus infinity. When the lima9on
is elliptic and has two real points of inflexion, the form of the
reciprocal curve, when the origin lies between the vertex B and
the point of intersection of the two stationary tangents, resembles
that of figure 5 of ~ 159.
284. The limacon is the pedal of a circle with respect to any
point in its plane.
Let 0 be the point, C the centre of the circle; and draw OZ
perpendicular to the tangent at any point P on the circle. Let
CP = a, CO = b, PCO = 0. Then
O Y=a-b cos 0,
whence the locus of Y is an elliptic or hyperbolic lima;on according as 0 lies within or without the circle. When 0 lies on the
circle, a = b, and the pedal is a cardioid.
285. If T be the middle point of the arc PQ of the circle
circumscribing FPQ (see fig. to ~ 279), then TP, TQ are the
tangents at P and Q.
Let FPL = f, F1PL = qb; then differentiating (2) with respect
to s we obtain
a cos = b cos 0....................... (8),
which gives the relation between the angles which the tangent
makes with the two focal distances; whence the theorem can be
proved in the same manner as the corresponding property of the
oval of Descartes given in ~ 268.