Problem source

In the figure, $W_{1}$ and $W_{2}$ are wheels, both of radius $r$.
Their centres $C_{1}$ and $C_{2}$ are fixed at the same height,
a distance $d$ apart, and each wheel is free to rotate, without friction,
about its centre. Both wheels are in the same vertical plane. Particles
of mass $m$ are suspended from $W_{1}$ and $W_{2}$ as shown, by
light inextensible strings would round the wheels. A light elastic
string of natural length $d$ and modulus elasticity $\lambda$ is
fixed to the rims of the wheels at the points $P_{1}$ and $P_{2}.$
The lines joining $C_{1}$ to $P_{1}$ and $C_{2}$ to $P_{2}$ both
make an angle $\theta$ with the vertical. The system is in equilibrium. 

\noindent \begin{center}
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\par\end{center}

\vspace{-0.5cm}
Show that
\[ \sin2\theta=\frac{mgd}{\lambda r}.
\]For what value or values of
$\lambda$ (in terms of $m,d,r$ and $g$) are there

\begin{itemize}
\setlength{\itemsep}{3mm}
\item[\bf (i)]  no equilibrium positions, 
\item[\bf (ii)] just one equilibrium position, 
\item[\bf (iii)] exactly two equilibrium positions, 
\item[\bf (iv)] more than two equilibrium positions?
\end{itemize}