Transformation properties of spheroidal multipole moments and potentials

Introducing definitions of solid spheroidal harmonics which contain those o f solid spherical harmonics as special cases for vanishing ellipticity it i s shown that the formalism of the multipole expansion of a 1/R-potential ca n he consistently extended to incorporate prolate and oblate spheroidal mul tipole moments. For finite ellipticity one can transform between regular so lid spheroidal and spherical harmonics and multipole moments through simple relations given before and independently proven here. Corresponding relati ons between irregular solid spheroidal and spherical harmonics are presente d for the first time, together with an investigation of the convergence pro perties of the resulting series expansions. Explicit formulae are derived f or the transformations between spheroidal multipoles calculated in coordina te systems of different ellipticity, origin and orientation. These fromulae can be utilized to calculate the energy of interaction between two arbitra rily oriented spheroidal charge or mass distributions of different elliptic ity. The performance of spheroidal multipole expansions is illustrated with some numerical examples.