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

164

SPECIAL QUARTICS.

real or all imaginary, or may coalesce into two real points of
undulation.
247. The nodal tangents at the circular points are
2 = C2I2, 2 = c212;
or, in Cartesian coordinates,
(x - ty)2 = c2, (x + y)2 = c2,
which intersect at the points x = + c, y =0; also, since both
tangents are stationary tangents, their points of intersection are
triple foci.
Since every binodal quartic must have eight real foci, of
which two or more may unite into multiple foci, it follows
that the Cassinian must have two single foci. Their positions may be found by determining the conditions that the line
x - a + t (y - /) = 0 should be a tangent to the curve.
Writing p = a + tU3, and eliminating y from (2), we shall obtain
(p2 - c2) (4x2 - 4px + p2 - 2) = (a2 - 2)2,
which will have equal roots if
(p2 - C2) (p2C2 + a4 - 2a2C2) = 0;
the first factor gives a= + c, 3 = 0, which are the coordinates of
the triple foci S and H. As regards the other factor, we observe
that when c /2 > a, in which case the Cassinian is bipartite, we
obtain pc = + a (2C2 - a2)2, which gives a = a (2c2 - a2)2/c, fa = 0;
hence in this case there is a pair of real single foci on the
axis of x. But when c /2 <a, in which case the Cassinian is
unipartite, we obtain pc= + a (a2- 2c2)2, which gives a= 0,
8 = + a(a2 - 2c2)2/c; hence in this case there is a pair of real
foci on the axis of y. When a = c /2, the curve becomes a
lemniscate, and the origin is a double focus formed by the
union of the foregoing pair of single foci. It can also be
shown that the two single foci lie inside or outside the curve,
according as the Cassinian is bipartite or unipartite.
We shall now give some properties of the curve.
248. If P be a point on the curve, straight lines drawn frnom
the foci perpendicular to SP, HP will meet the tangent at P in
points equidistant from P.