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# Exercises

1. Demonstrate that the volume of an ellipsoid whose bounding surface satisfies

is .

2. Demonstrate that the moments of inertia about the three Cartesian axes of a homogeneous ellipsoidal body of mass , whose bounding surface satisfies , are

3. 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

4. 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 2016-03-31