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

186

SPECIAL QUARTICS.

Two of the forms of the curve in the latter case are shown in
the figure, and further information will be found in Cayley's
Memoir on Caustics*.
The Limacon.
278. We have shown in ~ 266 that a lima9on is a particular
case of an oval of Descartes in which two of the foci coincide.
It is, however, more usual to define this curve as the inverse of a
conic with respect to its focus. The polar equation of a conic is
l/r = 1 - e cos 0,
whence the polar equation of a limaCon is
r= a - b cos 0........................(1),
where b/a = e. The curve is therefore the inverse of an ellipse or
a hyperbola according as a > or < b; in the former case it is called
an elliptic limacon and in the latter a hyperbolic limacon. This
curve appears to have been first studied by Pascal, who so named
it from a fancied resemblance to the form of a snail. When a = b,
the curve is the inverse of a parabola with respect to its focus and
is called a cardioid.
The Elliptic Limacon.
279. The form of the elliptic lima9on is shown in the figure.
The origin F is the point where the two internal foci of an oval of
Descartes unite, and is also a conjugate point; whilst the external
focus F1 is the inverse of the other focus of the ellipse. Since the
limacon has a node at the origin and a pair of imaginary cusps at
* Phil. Trans. 1857, p. 273; Collected Papers, vol. ii. p. 336.