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

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

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