Consider the hyperbola x ^ 2 100 - y ^ 2 64 = 1 with foci at S and...

Consider the hyperbola $\dfrac{{x}^{2}}{100} - \dfrac{{y}^{2}}{64} = 1$ 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 $∠ SP{S}_{1} = \alpha$, with $\alpha < \dfrac{\pi}{2}$. 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 $\dfrac{\beta\delta}{9}sin\dfrac{\alpha}{2}$ is _____ .