The circular motion of two classical point charges interacting through their time-symmetric electromagnetic fields is determined with complete relativistic rigor. Expressions are obtained for the total energy and angular momentum of this two-body system, which include the contributions of the field. This work is a preliminary to a calculation, undertaken jointly with Schlosser and now in progress, of the relativistic electromagnetic interaction of two classical particles with charge, spin, and magnetic moment, and of the quantization of this classical two-body system. The purpose of the investigation is to check the possibility that some elementary particles may be atoms composed of other elementary particles in states where the principal interaction is due, not to the Coulomb force between point charges, but to forces where the intrinsic magnetic moments of the particles play an essential role and must not be considered as small perturbations. A very rough, but very simple, estimate by Corben indicates that this possibility is worth exploring.