Velocity field and surface density in the problem of two fixed centers

Agekian (1972) has shown that in the case of a potential with rotational symmetry, the ratio of the velocity components in the meridional plane can be taken as the third integral of motion. The application of this proposed third integral to the problem of two fixed centers is examined. The potential of this problem is independent of the azimuthal coordinate and is amenable to integration by separation of variables in spheroidal coordinates. Expressions for the central orbit, the velocity field, and the surface density are derived.