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

FOCI OF THE OVAL OF DESCARTES.

183

Foci.
273. To prove that the three points F, F2, F2 satisfy Plicker's
definition of afocus.
The equation of an oval of Descartes referred to the point F is
r2 (1 n2) + 2nm2CX + a2 - Mn2c2}2 = 42r2....... (19)
and the points where the line x + ty = 0 intersects (19) are determined by the equation
(2M2cx + a2 - m2C2)2 = 0,
which shows that this line is a tangent to the curve. In the
same way it can be shown that the line - y =0 is also a
tangent; whence the point F satisfies Plicker's definition of a
focus.
Since the polar equation of the curve referred to F, and F2 is
of the same form as (19), it follows that these points are also
foci.
Since cartesians are bicuspidal quartics of the sixth class, it
follows that these curves have one triple and three single foci.
The latter have already been determined; the former may be
obtained by considering the bicuspidal quartic
{/y (1 - m2) + 2c (/3 + ) a + (a2 - 2c2) a2 = 4a2a2/3,
which reduces to a cartesian when B and C are the circular
points, and a the line at infinity. The two cuspidal tangents are
/ (1 - 2) + m2CA = 0,
y (1 -m2) + m2ca = 0,
which, when retransformed into Cartesian coordinates, become
(x + by) (1 -m2) + ^2c = 0,
which intersect at the real point
x=-mn2c/(1-m 2), y=)................(20),
which is the triple focus.
If the origin be transferred to the triple focus, it will be found
that the curve assumes the. form S2+L = 0; whence the triple
focus is the centre of the focal circle.