(12.36) |

on the surface of a sphere (i.e., constant). Here, , , are standard spherical coordinates, and is the Laplacian operator. It follows that (Love 1927)

where the and are arbitrary coefficients, and the associated Legendre functions.

A *solid harmonic* of degree
(where
is a non-negative integer), denoted
, is defined as a well-behaved solution
to

in the interior of a sphere (i.e., the region constant). It follows that (Love 1927)

(12.39) |

In particular, the functions and , introduced in Section 12.3, are solid harmonics of degree . Note that the Cartesian coordinates (where runs from to ) are solid harmonics of degree . Moreover, is a solid harmonic of degree . Finally, , specified in Equation (12.35), is a solid harmonic of degree 2.

The following results regarding solid harmonics are useful (Love 1927):

Here, use has been made of the Einstein summation convention (Riley 1974).