The paper contains a direct attack on the electromagnetic two-body problem, based on the hypotheses (i) that the bodies are particles, (ii) that the fields are given by the retarded potential, (iii) that the force on a particle is the Lorentz ponderomotive force (without a radiation term). A method of successive approximation leading to an exact solution is outlined. General expressions are found for the rates of change of invariant quantities which are the constants of energy and angular momentum in the Kepler problem, and formulae are developed for the principal parts of these expressions in the case where the ratio of the masses of the two particles is small. This is applied in detail to the case where the orbit of the light particle is approximately circular. It is found that energy disappears from the motion, so that the orbital particle slowly spirals in, but the rate at which this occurs is much less than that given by the usual formula for radiation from an accelerated electron. Except in some final calculations, no assumption of small velocity is made, the sole basis of approximation being the smallness of the mass-ratio.