Coefficient of friction, modulus of elasticity problem?

Two small rings, each of weight $\displaystyle W$, slide one upon each of two rods in a vertical plane, each inclined at an angle $\displaystyle \alpha$ to the vertical; the rings are connected by a fine elastic string of natural length $\displaystyle 2a$, and whose modulus of elasticity is $\displaystyle \lambda$; the coefficient of friction for each rod and ring is $\displaystyle \tan \beta$; show that, if string is horizontal, each ring will rest at any point of a segment of the rod whose length is:

$\displaystyle W\lambda^{-1}a\csc \alpha\{\cot (\alpha - \beta) - \cot (\alpha + \beta)\}$.