We present a theory of the resonances that appear in the scattering cross sections of electromagnetic waves from a perfectly conducting sphere coated with a homogeneous dielectric of uniform thickness. The individual normal mode scattering amplitudes are separated into an infinite series of resonance amplitudes interfering with a nonresonant background. The resonances are seen to fall into families that recur in successive partial wave amplitudes at successively higher frequencies and are shown to constitute the manifestation of circumferential (surface or creeping) waves that generate the resonances by matching their phases in the course of their repeated circumnavigations of the scattering object. Dispersion curves of these surface waves are obtained here from a knowledge of the resonance frequencies.