Transformation of Spherical Harmonics Under Change of Reference Frame

The problem of transforming spherical harmonics under a linear change of reference frame is solved analytically and numerically. In contrast to other methods both translation and rotation are treated with the same background theory which is comparatively straightforward and is formulated in geophysical terms; the directions of translation and rotation are arbitrary; and generally, combinations of harmonics of the same order can be dealt with equally as readily as individual harmonics. It is found that the expansion coefficients of a translated spherical harmonic are themselves spherical harmonics in the co-ordinates of the translation vector. As a special case, the potential of an eccentric dipole is briefly discussed. Very useful byproducts of this paper are the ways in which the given numerical algorithms can be used to evaluate any linear combination of harmonics.