Geometry of four hard fused spheres in an arbitrary spatial configuration

The statistical mechanics of hard spheres and molecules consisting of hard fused spheres frequently requires the surface and volume of overlapping hard spheres. The problem considered is the surface and volume of four overlapping hard bodies. In the present paper these quantities are calculated analytically for four equally sized hard fused spheres which are arranged arbitrarily in space. To check the resulting formulae for the general four-fold overlapping body, two independent derivations are given.