Electrostatic or gravitational interaction energy of coaxial ellipsoids

An expression, which is appropriate for high−precision calculations, is derived for the interaction energy of two coaxial ellipsoidal distributions of charge or mass. By starting from Dirichlet's expression for the potential it is shown that the (in general) sixfold integration can be considerably reduced by integrating over equipotential surfaces and introducing elliptic integrals. The evaluation of the final expression requires only a double numerical integration.