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

88

SPECIAL CUBICS.

The points where (5) cuts the generating circle are found by
transforming (5) to polar coordinat es and eliminating r by means
of the equation r = b cos 0. This leads to the equations
2k tan 0~ + 3h = 0)8
~2k n- =0....................
2ktanO2,-4h+b==O)
Equations (7) and (8) give the following geometrical construction for drawing three tangents to a
T cissoid from an external point.
G ^ K Let A be the centre of the geneAA-^ A -rating circle, OA the cuspidal tangent.
L O0 M From the point T (h, k) draw TM
-Q'-a_ R perpendicular to OA, and take K such
that KM =T TM. On the other side of
OM draw OR cutting the cissoid in R
such that angle ROM = MOK. Draw OQ perpendicular to OK
meeting the generating circle in Q. Produce AO to L so that
AL= OM; join LT, and draw OQ' cutting the generating circle in
Q', and making with OM an angle Q'OM = LTM. Let the circle
through QQ'R cut the cissoid in P1, P2, P3; then TP,, TP,, TPs
are the tangents from T.
We have tan ROM = tan MOK = 2k/3h,
whence by (7) R is the fourth point of intersection of the cissoid
with the circle through the points of contact. Also
tan QOM= cot MOK = 3h/2k,
tan Q'OM = tan MTL = (4h - b)/2k,
whence by (8) Q and Q' are the points where the circle through
the points of contact cuts the generating circle.
When the point T is on the curve, the tangent may be drawn
by the following simple construction.
Produce the ordinate TM to K such that KM= 2TM; join
OK and produce it to meet the curve in R, then TR is the
required tangent.
Putting a = 0 in (9) of ~ 130, the equation

(h - b)tan3 + 3htan 0-2 = 0...............(9)