(D.2) |

where , , and is the constant mass density of the ellipsoid. Hence, we obtain

The net potential at is obtained by integrating over all solid angle, and dividing the result by two to adjust for double counting. This yields

From Figure D.1, the position vector of point , relative to the origin, , is

(D.5) |

where is the position vector of point , and a unit vector pointing from to . Likewise, the position vector of point is

(D.6) |

However, and both lie on the body's outer boundary. It follows, from Equation (D.1), that and are the two roots of

(D.7) |

which reduces to the quadratic

(D.8) |

where

(D.9) | ||

(D.10) | ||

(D.11) |

According to standard polynomial equation theory (Riley 1974), , and . Thus,

(D.12) |

and Equation (D.4) becomes

(D.13) |

The previous expression can also be written

(D.14) |

However, the cross terms (i.e., ) integrate to zero by symmetry, and we are left with

Let

It follows that

(D.17) |

Thus, Equation (D.15) can be written

where

At this stage, it is convenient to adopt the spherical angular coordinates, and (see Section C.4), in terms of which

(D.20) |

and . We find, from Equation (D.16), that

(D.21) |

Let . It follows that

(D.22) |

where

(D.23) | ||

(D.24) |

Hence, we obtain

(D.25) |

Let . It follows that

where

Now, from Equations (D.19), (D.26), and (D.27),

Thus, Equations (D.18), (D.26), and (D.28) yield

where

Here, and are the body's mass and volume, respectively.

The total gravitational potential energy of the body is written (Fitzpatrick 2012)

(D.32) |

where the integral is taken over all interior points. It follows from Equation (D.29) that

(D.33) |

In writing the previous expression, use has been made of the easily demonstrated result . Now,

(D.34) |

so

(D.35) |

Hence, we obtain

(D.36) |