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

INVERSE OF A CONIC.

137

the inverse of which with respect to the origin is obviously
a curve of the same form. If, however, the origin is situated
on the curve, o0 = 0, in which case the inverse curve reduces to
a circular cubic.
202. Bicircular quartics and cartesians may be divided into
two classes according as the curve has two or three double points.
In the latter case the curve is the inverse and also the pedal of
a conic with respect to some point in its plane, which is the
third double point of the quartic. That a bicircular quartic
having three double points is the inverse of a conic, can be at
once shown by taking the third double point as the origin, in
which case (15) reduces to r4vo + r2v1 + u2 = 0, the inverse of which
is a conic. We shall now prove that:The inverse of a conic with respect to any point not on the
curve is a bicircular quartic having a third double point at the
centre of inversion; and this point will be a node, a cusp or a
conjugate point according as the conic is a hyperbola, a parabola
or an ellipse.
The equation of a central conic referred to any point (f, g) as
origin is
( +f)2 + (y + g)2
a2 b2
the inverse of which is
k Y' + 2 r +g- yv\ f2 + 92 1 74=0...(16),,a2+b (a2+ b2 ) + 2 b2
and the origin will therefore be a node or a conjugate point
according as the conic is a hyperbola or an ellipse. When the
conic is a parabola, the equation of the curve is
(y + g)2= 4a(x +f)
and the inverse curve is
4y2 + 2ki2r2 (gy - 2ax) + (g2- 4af) r4 = 0......(17),
and the origin is a cusp.
203. When the centre of inversion is the focus of the conic
the quartic becomes a cartesian, which is called a lima9on when
the conic is an ellipse or hyperbola, and a cardioid when the
conic is a parabola. When the centre of inversion lies on the
curve, the quartic degenerates into a circular cubic. We have
also shown in ~ 170 that the lemniscate of Bernoulli is the only