  The swimming of a spheroid immersed in a viscous fluid and performing surface
deformations periodically in time is studied on the basis of Stokes equations
of low Reynolds number hydrodynamics. The average over a period of time of the
swimming velocity and the rate of dissipation are given by integral expressions
of second order in the amplitude of surface deformations. The first order flow
velocity and pressure, as functions of spheroidal coordinates, are expressed as
sums of basic solutions of Stokes equations. Sets of superposition coefficients
of these solutions which optimize the mean swimming speed for given power are
derived from an eigenvalue problem. The maximum eigenvalue is a measure of the
efficiency of the optimal stroke within the chosen class of motions. The
maximum eigenvalue for sets of low order is found to be a strongly increasing
function of the aspect ratio of the spheroid.
