INan * Find: The maximum distance between two points, one on the surface of a sphere with center at $(-2,-10,5)$ and radius 19, and the other on the surface of a sphere with center at $(12,8,-16)$ and radius 87.
Let $O$ and $O_{1}$ be the centers of two spheres, and $P$ and $P_{1}$ be the points of intersection of the extended line segment $O O_{1}$ with the surfaces of the two spheres, with $O$ inside $P O_{1}$ and $O_{1}$ inside $O P_{1}$. Clearly, the maximum distance between the two points is $P P_{1}=P O+$ 
$$
\begin{array}{l}
O O_{1}+O_{1} P_{1}=19+O O_{1}+87 \text {. Also, } O O_{1}=\sqrt{(-2-12)^{2}+(-10-8)^{2}+[5-(-16)]^{2}} \\
=31 \text {. Therefore, } P P_{1}=19+87+31=137 .
\end{array}
$$