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

THE CISSOID.

87

OM O _-OP b
and — AM PQ AQ'
A Q2 b
whence Q
or AQ3 = a2b.
144. Since a cissoid is a curve of the third class, three
tangents can be drawn from any point not on the curve. We
shall now explain a geometrical construction by means of which
this may be done*.
The polar conic of any point (h, k) is
h (3x + y) + 2y (-b)=by2...............(3).
Multiply (1) by 3h and (3) by x and subtract and we shall
obtain
2kx (x - b) = (b + 2h) xy - 3bhy.............. (4).
Multiply (3) by 3h and (4) by 2k and add and we get
(9h2 + 4k2) x2 + 3h (h - b) y2 + 2k (h - b) xy - 4bl2x =0.
In this write (X2 + y2)/b for y2/x and we get
3h (h - b) (x2 + y2) + (9h2 + 4k2) bx + 2kb (h - b) y - 4b2k2 = 0.........(5),
which is the equation of the circle passing through the three
points of contact of the tangents from (h, k).
A circle and a cissoid intersect in six points, two of which are
the circular points at infinity; and we shall now find the fourth
point of intersection R.
Transform (5) to polar coordinates, eliminate r by means of
(2) and we shall obtain the equation
(3h tan 0 + 2k) {(h - b) tan3 0 + 3h tan 0 - 2kc = O...(6).
Putting a= 0 in (9) of ~ 130, it follows that the second factor
gives the vectorial angles of the points of contact of the tangents
drawn from (h, k); whence the equation
3h tan + 2k =........................(7)
determines the fourth point R in which the circle cuts the
cissoid.

* J. J. Walker, " On tangents to the cissoid," Proc. Lond. Math. Soc. Vol. ii.
p. 161.