text: For an ellipsoid with equatorial semi-principal axes  and polar semi-principal axis  , the angular velocity  about  is given by
  where  is the density and  is the gravitational constant, subject to the condition
  For fixed values of  and , the above condition has solution for  such that
  The integrals can be expressed in terms of incomplete elliptic integrals. In terms of the Carlson symmetric form elliptic integral , the formula for the angular velocity becomes
  and the condition on the relative size of the semi-principal axes  is
  The angular momentum  of the Jacobi ellipsoid is given by
  where  is the mass of the ellipsoid and  is the mean radius, the radius of a sphere of the same volume as the ellipsoid.