  The swimming of a sphere immersed in a viscous incompressible fluid with
inertia is studied for surface modulations of small amplitude on the basis of
the Navier-Stokes equations. The mean swimming velocity and the mean rate of
dissipation are expressed as quadratic forms in term of the surface
displacements. With a choice of a basis set of modes the quadratic forms
correspond to two hermitian matrices. Optimization of the mean swimming
velocity for given rate of dissipation requires the solution of a generalized
eigenvalue problem involving the two matrices. It is found for surface
modulations of low multipole order that the optimal swimming efficiency depends
in intricate fashion on a dimensionless scale number involving the radius of
the sphere, the period of the cycle, and the kinematic viscosity of the fluid.
