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

THE CISSOID.

89

determines the vectorial angle of the points of contact of the three
tangents drawn from (h, k) to the cissoid. If (h, k) lies on the
curve, k2 (b - h) = h3, whence (9) becomes
h3 tan3 0 - 3hk2 tan 0 + 2k3 = 0.............. (10),
two of the roots of which are equal to k/h (as ought to be the
case), whilst the third root is equal to - 2k/h. This at once gives
the foregoing construction.
When T is on the asymptote, h =b, and we obtain from (9)
tan 0= =k/h. Hence if OK meet the curve in P, then P is the
required point of contact.
145. To find the tangential equation of the cissoid.
The equation of the tangent at (x, y) is
X(3x2 + y2)+ 2Yy(x -b)=by2............(11),
whence = (3x2 + y2)/by2,
= 2 (x - b)/by.
Eliminating x and y by means of (1) we obtain
27b2?2 = 4 (b - 1)3.....................(12).
By ~ 57, the reciprocal polar is obtained by writing xa/k2, y/k2
for I,; and is
27k2b2y2 = 4 (bx - k2)..................(13).
This curve is the evolute of a parabola.
The pedal of the cissoid with respect to the cusp is obtained
by inverting with respect to a circle of radius k, and is
27b2y (2 + y2) 4 (bx - y2)............(14),
and is therefore a sextic curve.
The orthoptic locus is a sextic curve which can be written
down by the method of ~ 68.
Foci.
146. The foci of circular cubics are best studied when the
curve is treated as a particular case of a bicircular quartic; we
shall therefore only make a few remarks on the subject.
Since nodal circular cubics are of the fourth class, it follows
that the curve has one real double focus and four real single