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

ROULETTES.

209

All cycloidal curves belong to the class of curves called
roulettes, for the complete discussion of which we must refer to
Dr Besant's Notes on Roulettes and Glissettes. We shall only give
one proposition on the subject.
314. A curve rolls on a straight line; it is required to find
the roulette traced by any point Q.
Let QA be a line fixed in the plane of the rolling curve AP,
Q
A<
0 N P
and let OP be the line on which it rolls. Let A initially coincide
with 0. Then if (x, y) be the coordinates of Q referred to 0,
ON=x, QN=y;
also if (r, 0) be the polar coordinates of P referred to QA, in the
plane of the rolling curve
AQP=, QN=p, QP=r, QPN= b,
whence y=p, tan = dx/dy = rdO/dr
y= r sin 4 = rdx/ds............
315. If the roulette is an ellipse, the rolling curve is an
epicycloid.
Let the roulette be
x2/a2 + y2/b2= 1,
ldx\2 a2y2
then -j- = T /., -9,dy) b2 (b2 - y2),
whence by (11)
a22 2, 2
a2p2 -tan2b- 2'
pr2 -b" (b2 _ p) = tan = 2 _ p '
or r2 b4/a2 + e2p2,
which is the p and r equation of an epicycloid. If therefore an
epicycloid roll on a straight line, the locus of the centre of the
fixed circle is an ellipse.

B. C.

14