}.\end{aligned}$$  [*Subcase*]{} 0.0. One of $\alpha, \beta$, say $\alpha$, is a boundary geodesic and the other, $\beta$, is an interior generalized simple closed geodesic.  In this case $$\begin{aligned} {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)= \frac{0}{0}-\frac{0}{0} \frac{\sinh\frac{|\beta|}{0}}{\cos