A hoop stands in a vertical plane on a rough incline which the plane of the hoop cuts in a line of greatest slope. It is kept in equilibrium by a string fastened to a point in the circumference, wound round it, and fastened to a peg in the incline further up and in the same plane. If $\displaystyle \lambda$ is the angle of friction, $\displaystyle \theta$ the angle the hoop subtends at the peg, and $\displaystyle \alpha$ that of the incline, show that there is limiting equilibrium when $\displaystyle \theta = \alpha + \arccos \left[\frac{\sin (a - \lambda)}{\sin \lambda}\right]$. What will happen if $\displaystyle \theta$ has a greater value?