The general conditions are determined for stable multiple orbits of non-relativistic charged particles in a race-track-shaped optical system. These conditions are then considered in the context of an electrostatic storage ring consisting of two 180° hemispherical deflector analysers connected by two separate sets of cylindrical lenses. The race-track configuration of this type has already been constructed and demonstrated to achieve storage of low-energy (tens of electronvolts) electrons (Tessier et al 2007 Phys. Rev. Lett. 99 253201). Incorporating the aberrations of the energy-dispersive electrostatic prisms is found to modify and restrict the general stability conditions. This modified formal matrix theory and the results of charged particle simulations described in this study are in excellent agreement with the observed experimental operating conditions for this electron recycling spectrometer.