  We derive new relationships expressing solid spherical harmonics as series of
toroidal harmonics and vice versa. The expansions include regular and irregular
spherical harmonics, ring and axial toroidal harmonics of even and odd parity
about the plane of the torus. The expansion coefficients are given in terms of
a recurrence relation. As an example application we apply one of the expansions
to express the potential of a charged conducting torus on a basis of spherical
harmonics.
