Abstract The K = 500 MeV superconducting cyclotron being built here has an RF system containing three dees with spiral electric gaps. These gaps produce a significant force component transverse to the ion orbits. An analysis is presented which shows nevertheless that the effect of such gaps does not alter the relevant orbit period, in accordance with an unpublished theorem due to McMillan. The structure of the “Spiral Gap” orbit program is then described. This program calculates ion orbits in a given magnetic field and provides a systematic treatment of the transverse as well as the longitudinal electric force resulting from spiral electric gaps. Sample results are presented which show that there is no significant difference in the phase-energy history of ion orbits accelerated with spiral or non-spiral electric gaps. A discussion of central ray orbits and accelerated equilibrium orbits is also presented together with analytical and computer results.