Most references in a brief search under 'Apollonius' concern tangent circles. The problem I am interested in is the Apollonius pursuit problem. In the plane, the question concerns the point at which ships traveling in straight lines at constant (in general different) speeds will intercept.
According to S. Ogilvy (Tomorrow's Math), "To...characterize the locus of collision points on a sphere for constant speeds and great circle courses remains an unsolved problem [italics added]."
Is this still true, I wonder, and in either case is there a reasonably accessible discussion of the problem somewhere? Failing that, a brief summary of the more obvious obstacles would be appreciated.
So far, I see that we can without loss of generality orient a sphere of unit radius so that Ship 1 moves toward the north pole along a great circle at constant angular velocity $v_1.$ If Ship 2 moves at a faster constant angular velocity $v_2$ in the northern hemisphere we can ask at what angle we must point Ship 2 so it meets Ship 1.
With some loss of generality, we can launch both ships at a certain longitudinal distance d apart at points $p_1$ and $p_2$, respectively, at the same latitude and time. The faster ship cannot simply head for the starting point of the slower ship, which will make at least some progress as Ship 2 traverses d.
When/if the ships meet at some point $p_3,$ I think $p_1,p_2,p_3$ define a right triangle on the plane passing through them. Calculus (as in, related rates) via the Pythagorean theorem is no help because the relation between distance $\theta$ between two points on the sphere and the length d of the chord connecting them is $d = 2 \sin\theta$. For tiny triangles the surface of the sphere is approximately flat, but otherwise it's not so easy, and there is no obvious generalization of the solution in the plane.
I'll replace this with something hopefully less naive if no one answers before I have a chance to do some reading.