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 \( \angle SPS_{1}=\alpha \), 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 \( SP_{1} \), and \( \beta=S_{1}P \). Then the greatest integer less than or equal to \( \dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2} \) is ________________.
