This paper presents Taylor expansions for the imaging and timing characteristics of spherical optical systems with infinitesimal mirror facets, sometimes referred to as "modified Davies-Cotton" telescopes. Such a system comprises a discontinuous spherical mirror surface whose curvature radius is different from its focal length, and whose mirrors are aligned to suppress spherical aberration. Configurations that range between two "optima" are, one of which minimises tangential comatic aberration and the other that minimises timing dispersion.