Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at S and $S_1$, where S lies on the positive X-axis. Let P be a point on the hyperbola, in the first quadrant. Let $\angle SP{S}_1=\alpha ,$ with  $\alpha <\frac{\pi }{2}$.The straight line passing through the point S  and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line $S_{1}P$ at $p_1$. Let $\delta$ be the distance of P from the straight line $S{p}_1$, and  $\beta =S_{1}P$. Then, the greatest integer less than or equal to $\frac{\beta \delta }{9}\sin \frac{\alpha }{2}$ is …………..