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

THE CYCLOID.

207

Through P draw a line PM perpendicular to BY, and cutting.
the parabola in Q. Then
QM2 = 8a. BM = 16a2 sin2 #
by the second of (2); whence
QM= 4a sin = BP.
311. To find the tangential equation of the cycloid.
Let PC meet YB produced in R; then:-1 = BC 2a+,
tan =- -/7,
whence 1 + 2ac tan-1 / = 0....................(5).
Epicycloids and Hypocycloids.
312. The epicycloid is the curve traced out by a point on the
circumference of a circle which rolls outside another circle.
To find the equation of an epicycloid.
Let a and b be the radii of the fixed and rolling circles, so that
OQ=a, O'Q=b; also let* QOA =, PO'Q=, where P is the
point which initially coincided with A. Then since
arc AQ=arc PQ, a0=bq;
whence the coordinates of P are given by the equations
x = (a + b) cos 0 - b cos (a + b) O/b (6);
y=(a + b) sin 0-bsin (a + b) /b...
the elimination of 0 between these equations determines the
Cartesian equation of the curve.
The line EP is the tangent to the curve at P, whence if
OY=p, where O Y is perpendicular to EP, we have
OTE =r-*, p=(a+2b)sin 10;
also, = + ) = (a + 2b) /a,/
whence p = (a + 2b) sin a....................(7).
a+2b..................
* The point A (not marked in the figure), is the point between O and T where
the fixed circle cuts OT.