For positive integers $n_i geq 2, i = 1, 2, 3, 4$, such that $Sigma frac{n_i}{1} < 2$, there exists a quadrilateral $P = P_1 P_2 P_3 P_4$ in the hyperbolic plane $H^2$ with the interior angle $frac{n_i}{pi}$ at $P_i$. Let $Gamma subset Isom(H^2)$ be the (discrete) group generated by reflections in each side of $P$. Then the quotient space $H^2/gamma$ is a differentiable orbifold of type $(^* n_1 n_2 n_3 n_4)$. It will be shown that the deformation space of $Rp^2$-structures on this orbifold can be mapped continuously and bijectively onto the cell of dimension 4 - left$mid$ {i$mid$n_i = 2} ight$mid$$.