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

THE GENERATING CIRCLE.

139

The pedal of a parabola is
(r2 + gy +fx) x = ay2
which is a circular cubic.
We also observe that in both these cases 8= 0.
205. The preceding methods are not the only ones by
which a bicircular quartic can be generated. We shall now
show that:
A bicircular quartic is the envelope of a variable circle whose;entre moves along a fixed conic, called the focal conic, and which
suts a fixed circle orthogonally.
In the figure to ~ 200, describe a circle whose centre is Q
and which passes through P and P'. Then, since (10) may be
written in the form OP. OP'= 82, it follows that the tangent
from 0 to this circle is constant and equal to the radius 8
of the fixed circle. Hence, if with 0 as a centre a circle of
radius 8 be described, this circle will cut the circle through
QPP' orthogonally.
Let Q' be a point on the conic near Q; then Q' may be
regarded as lying on the tangent at Q. Hence, if a circle be
described through Q'PP', PP' will be the radical axis of the
two circles, and both will be cut orthogonally by the fixed
circle. Hence P and P' will be the limiting positions of the
points of intersection of the two circles, and therefore the quartic
is the envelope of the moving circle.
The moving circle is called the generating circle; whilst by
~ 200 the fixed circle is the circle of inversion.
206. If through the centre of inversion 0 any chord be drawn
and P and P' be the two inverse points of intersection, the locus
of the points of intersection of the normals to the quartic at P
and P' is the focal conic.
Let OT =p, OP =r, q the angle which the tangent to the
quartic at P makes with OP; then
do r dp dob
'tan dr p dr pdp
pt OQT r dr
p dr QT dr'