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

196

SPECIAL QUARTICS.

(iv) We have
QTQ' = - Q-r 7r QFQ'
by (ii).
The Trisectrix.
297. One of the most interesting applications of the hyperbolic limaeon is the trisection of an angle. This is effected by
means of the trisectrix, which is the name given to the curve
when b = 2a.
Let a be the angle which is to be trisected. Through the
vertex A of the loop draw a line PQR,
cutting the trisectrix in P, Q and R such
P that the angle PAF= a. Join FR, and
/ Q E draw AE parallel to FR. Let RFA = 0.
At F Then since b = 2a,
\FR = a (2 cos + 1).
But
R^ FR sin a
F --- sin (a -) == 2 cos 0 + 1,
a sin (t - 0)
2 (2 cos 0- 1)2 (1 + Cos 0)
whence cot2 a
(2 cos 0 + 1)2 (1 - cos 0)
1 + cos 30
1 - cos 30'
therefore cos 30 = cos 2a,
whence 30 = 2a.
Accordingly EAF 0 = = a = ~ PAF,
whence EA trisects the angle PAF.
The Cardioid.
298. When a= b, the lirnaon becomes a cardioid, and its,
equation may be written in either of the forms
r=a(1 -cos0)}
r = a/2. sin............
or (x2 + y2)2 + 2ax (2 + y2) = a2y2.