We start from the observation that the two-element cyclic group consisting of the identity and a sphere inversion can be viewed as a stereographic image of a one-mirror reflection group in 4D. This allows us to identify 19 three-parametric families of finite groups formed by four (or less) sphere inversions. Exactly as in the single sphere case, each member of each of the 19 families generates a solvable electrostatics problem of a charge inside a piecewise-spherical cavity with grounded conducting walls. We present a worked example of a member of the D4 family: a cavity formed by three mutually orthogonal planes (i.e. spheres of infinite radius) and a spherical segment at 60∘ to each of the flat walls. In this particular case, generating the induced potential requires 191 image charges.
9 pages, 3 figures