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

214 MISCELLANEOUS CURVES.
and that of the perpendicular tangent is
x cos *J + y sin r = (a + 2b) sin m (Tr + r),
whence
= (a + 2b) {sin m ( Tr + ') cos r + sin mrn sin k},
y = (a + 2b) {sin mn (-r + r) sin - sin m# cos}.
These equations may be written in the form
= (a + 2b) [sin { (1 + m) r cos (1 - ) (+ + 47r)
+ cos 1 (1 + m) 7 sin {(1 + m) ' - (1 - ) }],
y = (a + 2b) [sin 4 (1 + m) r sin (1 - m) ( + r)
- cos { (1 + m) 7r cos {(1 + m) - ~ (1 -m) 7r}].
Let (1 - ) (q + 47r) = 0,
and the equations may be written in the form
x= (a + 2b) {sin (1 + m) 7r cos - cos I (1 + m) 7r cos (1+ )
(ris \ln jEflLM? Cr;~ AAh4 1 - M }
which are the equations of an epitrochoid. Whence if A and B
be the radii of the fixed and rolling circles, and C the distance of
the fixed point,
A + B = (a + 2b) sin I (1 + m) 7r
(a + b) 7r
= (a + 2b) sin 2 (a + 2b)'
(a ~ b6) wT
C (a + 2b) cos (a + b)
2 (a + 2b)'
A+B l+m a+b
B 1-m- b
To verify this result in the case of a cardioid, put a = b, and we
get
A = B = 3V3a/4, C = 3a/2,
and the locus is the limanon
r=-a (A3 -2 cos 0),
which agrees with ~ 302.