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

GENERATION OF BICIRCULAR QUARTICS.

135

The equation of a cartesian may therefore be written in either
of the forms
S2+ u= 0...........................(8),
or S2+13UI = 0..........................(9),
where S is a circle and u is a straight line. The form of (9)
shows that the line u=0 is the only double tangent which the
curve can have; and also that the circle S has a contact of
the second order with the curve at each of the circular points
at infinity.
200. Bicircular quartic curves have formed the subject of an
exhaustive memoir by the late Dr Casey*, from which most of
the present chapter will be taken. He first of all shows that the
quartic may be generated in the following manner:If OT be the perpendicular from any fixed point 0 on to the
tangent at any point Q of a fixed conic; and if two points P, P'
be taken on OT such that TP = TP', and
OT2 PT2 =.... (10),
where 8 is a constant, the locus of P and P' is a bicircular
quartic.
When the fixed conic is an ellipse or hyperbola, the quartic
has two nodes at the circular points at infinity; when the conic
is a circle, the circular points are cusps and the quartic is a cartesian; and when the conic is a parabola, the curve degenerates
into a circular cubic.
Let EY be the perpendicular from the centre E of the conic
on to the tangent at any point Q. Let (f, g) be the coordinates
of 0 referred to E; (x, y) those of P referred to 0. Let OP = r,
EY=p, YEX =. Then

OT = p-f cos - g sin <,
PT=r-OT,
* Trans. R. I. A. Vol. xxIV. p. 457.