Second Derivative of Locus of Cycloid

Theorem
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian coordinate plane.

Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.

Consider the cycloid traced out by the point $P$.

Let $\left({x, y}\right)$ be the coordinates of $P$ as it travels over the plane.

The second derivative of the locus of $P$ is given by:
 * $y'' = - \dfrac a {y^2}$

Proof
From Equation of Cycloid:
 * $x = a \left({\theta - \sin \theta}\right)$
 * $y = a \left({1 - \cos \theta}\right)$

From Slope of Tangent to Cycloid: