Relations between Laplace’s spheroidal harmonics associated with different spheroidal coordinates are derived. The transition matrices for the functions of the 1st kind are lower triangular and are related by inversion. The matrices for the functions of the 2nd kind are the transposed ones for the functions of the 1st kind. The series for the functions of the 1st kind are finite, and those for the 2nd kind are infinite. In the latter case the region of convergence is considered. Using the derived relations, the rigid solution to the electrostatic problem for the multi-layered scatterers with nonconfocal spheroidal boundaries of the layers is obtained and the Rayleigh approximation is constructed, as well as an approximate approach to a similar light scattering problem, which provides reliable results far beyond the range of applicability of the Rayleigh approximation, is suggested.