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

CHAPTER X.
SPECIAL QUARTICS.
243. HAVING discussed the general theory of quartic curves
and also that of bicircular quartics, we shall proceed to consider
the properties of a variety of well known curves of this degree.
It will further be shown in Chapter XII. that all the projective
properties of these curves may be generalized by projection; and
in particular that the theory of all quartics having three biflecnodes, a node and a pair of cusps, or three cusps may be deduced
from the properties of the lemniscate, the limanon and the cardioid
respectively.
The Cassinian.
244. The Cassinian, or the oval of Cassini as it is sometimes called, is the locus of a point
which moves so that the product of
( ( S\^ ( its distances from two fixed points is
constant. The two fixed points are, as
will hereafter be shown, triple foci.
To find the equation of the Cassinian.
Let S and H be the foci, 0 the middle point of SH; let
SH= 2c, and let a be a constant such that
SP. HP = a2 c2.....................(1),
then if (x, y) be the coordinates of P referred to 0,
SP2= (-c)2 + y2, HP2=( + )2+y2,

whence

(X2 + y2 + c2)2 4C2x2 = (a2- c2)2...............(2).