Show that the least force which applied to the surface of a heavy uniform sphere will just maintain it in equilibrium against a rough vertical wall is

$\displaystyle W\cos \epsilon$ or $\displaystyle W\tan \epsilon\left[\tan \epsilon - \sqrt{\tan^2 \epsilon - 1}\right]$

according as $\displaystyle \epsilon$ < or > $\displaystyle \arccos \frac{\sqrt{5} - 1}{2}$, where $\displaystyle W$ is the weight and $\displaystyle \epsilon$, the angle of friction.