Consider a spherical planet whose unperturbed surface corresponds to . Suppose that this planet is
subject to a small, externally generated, tidal potential
, where is a solid harmonic of degree 2.
The planet is modeled as a solid body of uniform mass density whose perturbed surface lies at

(C.8) 
where is a surface harmonic of degree 2. It is assumed that
.
The net gravitational acceleration in the vicinity of the planet takes the form

(C.9) 
where
is the gravitational potential.
According to standard gravitational theory,

(C.10) 
where
is the mass density distribution.
Thus, we can write

(C.11) 
where

(C.12) 
and

(C.13) 
[See Equations (E.2) and (E.3).]
Here,
is a Dirac delta function (Riley 1974b).
The physical boundary conditions are
as
.
As is well known, the solutions to Laplace's equation,
,
take the general form
and
. Moreover, the
axisymmetric solution to
that is well behaved at the origin is
.
(Riley 1974c.)
It follows that
and
where

(C.20) 
is the gravitational acceleration at the planet's surface. Note that
is a solid harmonic of degree 2 inside the planet (i.e., ).