Introduction

Let us adopt the right-handed Cartesian coordinate system $x_1$ , $x_2$ , $x_3$ . Consider a homogeneous ellipsoidal body whose outer boundary satisfies

$$\frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1,$$ (D.1)

where $a_1$ , $a_2$ , and $a_3$ are the principal radii along the $x_1$ -, $x_2$ -, and $x_3$ -axes, respectively. Let us calculate the gravitational potential (i.e., the potential energy of a unit test mass) at some point $P\equiv (x_1,\,x_2,\,x_3)$ lying within this body. More information on ellipsoidal potential theory can be found in Chandrasekhar 1969.