Transformation properties of spheroidal multipole moments and potentials

Introducing definitions of solid spheroidal harmonics which contain those of solid spherical harmonics as special cases for vanishing ellipticity it is shown that the formalism of the multipole expansion of a 1/R-potential can he consistently extended to incorporate prolate and oblate spheroidal multipole moments. For finite ellipticity one can transform between regular solid spheroidal and spherical harmonics and multipole moments through simple relations given before and independently proven here. Corresponding relations between irregular solid spheroidal and spherical harmonics are presented for the first time, together with an investigation of the convergence properties of the resulting series expansions. Explicit formulae are derived for the transformations between spheroidal multipoles calculated in coordinate systems of different ellipticity, origin and orientation. These fromulae can be utilized to calculate the energy of interaction between two arbitrarily oriented spheroidal charge or mass distributions of different ellipticity. The performance of spheroidal multipole expansions is illustrated with some numerical examples.