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

206 MISCELLANEOUS CURVES.
with A. Let (x, y) be the coordinates of P referred to BX an
BY, as axes of x and y; and let GOP = -. Then
AG = arc GP =aO,
and therefore A Y= air.
Now x = a (7r - q) + a sin }.(1),
y = a + a cos..........
whence x = a cos-' (a - y)/a + (2ay -- y2).
Since PC is the direction of motion of P, PC is the tangen
and PG is the normal at P.
309. The evolute of a cycloid is an equal cycloid.
Let PCX =; then t = r - 2#, and (1) becomes
x = 2ak + a sin 2} (2),
y = a - a cosi 2q}..................... (2),
whence p = 4a cos.....................3),
and s = 4asin............................(4),
no constant being required, since s= 0 when * = 0.
Equation (3) shows that p = 2PG, whence if P' be the centre
of curvature of P, the evolute is another equal cycloid A'P'A,
whose vertex A coincides with the cusp of the original cycloid.
Equation (4) proves the isochronism of the cycloid; for the
equation of motion of a particle sliding down a cycloidal tube
under the action of gravity is
d2s
dt- +g sin = 0,
d's
or dts + (/wg/4a)s = 0,
whence the time of motion from any point P to B is 7r (a//tg)1.
Squaring and adding (3) and (4) we get
BP2 + PP'2 = CC2.
310. If a parabola be described which touches a cycloid at the
vertex B, and whose latus-rectum is the line joining the adjacent
pair of cusps, any double ordinate to the parabola drawn from a
point on the arc joining the extremities of the latus-rectum is equal
to the intercepted arc of the cycloid.