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McCullough's Formula
According to Equations (903) and (908), if the Earth
is modeled as spheroid of uniform density then its ellipticity
is given by
where the integral is over the whole volume of the Earth, and
would be the Earth's moment of inertia were it
exactly spherical. Now, the Earth's moment of inertia about its
axis of rotation is given by

(923) 
Here, use has been made of Equations (869)(871). Likewise,
the Earth's moment of inertia about an axis perpendicular to its
axis of rotation (and passing through the Earth's center) is
since the average of is for an axisymmetric mass distribution. It follows from the above three equations that

(925) 
This result, which is known as McCullough's formula, demonstrates
that the Earth's ellipticity is directly related to the difference between
its principle moments of inertia. It turns out that McCullough's formula
holds for any axially symmetric mass distribution, and not
just a spheroidal distribution with uniform density. Finally, McCullough's
formula can be combined with Equation (909) to give

(926) 
This is the general expression for the gravitational potential generated outside
an axially symmetric mass distribution. The first term on the righthand
side is the monopole gravitational field which would be generated
if all of the mass in the distribution were concentrated at its center of mass,
whereas the second term is the quadrupole field generated by any deviation from spherical symmetry in the distribution.
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Richard Fitzpatrick
20110331