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

202

SPECIAL QUARTICS.

Accordingly
p = - 2R cos ( + C)cos - 2R cos(20 + C- A)cos
+Rcos(3 + C- A)
= - 2R cos (A + C) cos 0 - R cos (4 + C - A)
=-2R cos(A + C)cos ( - C - A)- Rcos ( + 2 C-2A).........(1).
Let
K cos a = 2 cos (A + C) cos I (C - A) + cos 3 (C - A),
K sin a = 2 cos (A + C) sin I (C - A) - sin - (C - A),
then
2 = 1 - 8 cos A cos B cos C = IO2/R2.
Accordingly (1) becomes
p = -I0 cos (I 3 - a).
This is the tangential polar equation of a cardioid referred to
the centre of the circle, which is the locus of the points of intersection of tangents at the extremities of cuspidal chords, as
origin; and the radius of this circle is equal to IO.
304. A parabola is described touching a given circle and
having its focus at a given point on the circle; prove that the
envelope of its directrix is a cardioid.
Let C be the centre of the circle, S the focus of the parabola,
P the point of contact; draw SX perpendicular to the directrix
and meeting it in X. Let XSP = 0, XSC = r, SC = c; then
~ - 0 = CSP = CPS = 0,
whence 2# =30. Also
SX = SP (1 + cos 0) = 2c cos3 0
= 2c cos3 ) ~,
which is the tangential polar equation of a cardioid.
The Conchoid of Nicomedes.
305. This curve was invented by the Greek geometer Nicomedes for the purpose of trisecting an angle, and may be described
as follows. Let 0 be a fixed point and AB a fixed straight line;
let OA = a, and draw a straight line OQ cutting AB in Q, and