  Retarded or frequency-dependent hydrodynamic interactions are relevant for
velocity relaxation of colloidal particles immersed in a fluid, sufficiently
close that their flow patterns interfere. The interactions are also important
for periodic motions, such as occur in swimming. Analytic expressions are
derived for the set of scalar mobility functions of a pair of spheres. Mutual
hydrodynamic interactions are evaluated in one-propagator approximation,
characterized by a single Green function acting between the two spheres.
Self-mobility functions are evaluated in a two-propagator approximation,
characterized by a single reflection between the two spheres. The
approximations should yield accurate results for intermediate and long
distances between the spheres. Both translations and rotations are considered.
For motions perpendicular to the line of centers there is translation-rotation
coupling. Extensive use is made of Fax\'en theorems which yield the
hydrodynamic force and torque acting on a sphere in an incident oscillating
flow.
