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 Demonstrate that the volume of an ellipsoid whose bounding surface satisfies
is
.
 Demonstrate that the moments of inertia about the three Cartesian axes of a homogeneous ellipsoidal body of mass
,
whose bounding surface satisfies
,
are
 According to MacCullagh's formula (Fitzpatrick 2012), the gravitational potential a relatively long
way from a body of mass
whose center of mass coincides with the origin, and whose principal moments of inertial are
,
, and
(assuming that
the principal axes coincide with the Cartesian axes), takes the form
where
. Demonstrate that if the body in question is a homogeneous
ellipsoid whose bounding surface satisfies
then
 Show that the gravitational potential external to a homogeneous ellipsoidal body of mass
,
whose outer boundary satisfies
, takes the form

(D.38) 
where
and
.
Here,
is the positive root of
Demonstrate that, at large
, Equation (D.38) reduces to Equation (D.37).
Next: Calculus of Variations
Up: Ellipsoidal Potential Theory
Previous: Analysis
Richard Fitzpatrick
20160122