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

THE CISSOID.

85

y=,x; whence substituting and equating coefficients of x we
shall find that p = b/k, X = k"'/(k2- b2), and we obtain
(bx - ky) {(k2 - 2b2) - 3bcy + b (k2 - b2)} = 0.
The second factor equated to zero is the chord of contact, and
its envelope is the ellipse
9y2 + 4 (x + b) (2x + b) = 0.
Further information on this curve will be found in Booth's
Treatise on some New Geometrical Methods.
The Cissoid.
141. The cissoid is the inverse of a parabola with respect to
its vertex, and its equation is found by putting a= 0 in (3) of
~ 125, and is

rcos = b sin2 0.....................(2).
It is also the pedal of the parabola y2 + 4bx = 0 with respect
to its vertex.

It is, however, more usual to define the cissoid by the following
construction. Let OA be a diameter of
a circle, Q any point on its circumference;
draw QN perpendicular to OA. Let
Mbe a point on OA such that OM=An,
and let MP be drawn perpendicular to O- M - A
OA meeting OQ in P. Then the locus
of P is a cissoid.
Let POM =, OA =b; then
= OM = AN =b sin2 9,
y /(X2 + y2) = sin2 0,
whence the locus of P is the curve
x (x2 + y2) = by2.
The curve has one asymptote, viz. the line y = b also the origin
is a cusp; hence the curve is of the third class.

The circle OQA is called the generating circle.