Small-amplitude swimming of a sphere

We study small-amplitude swimming of a deformable sphere in a viscous incompressible fluid. We assue that the surface displacement varies harmonically in time and calculate the resulting swimming velocity in perturbation theory to second order in the diplacement. The optimum efficiency of swimming may be evaluated from the maximum eigenvalue of a Hermitian matrix. We find the optimum efficiency for the class of swimming motions for which the first order flow velocity is irrotational. A closed form expression is constructed for the corresponding surface displacement.