Potential and the Gravitational Field due to an Ellipsoid of Rotation

Now we will start to apply the theory of the potential U(p) and its field g(p) to study a gravitational field caused by masses of the earth. Earlier, it was pointed out that the behavior of the gravitational field on the earth s surface has mainly a regular character, while the irregular part is very small, less than 0.1%. Correspondingly, it is natural to divide the mass of the earth into two parts  

The regular part, which is symmetrical with respect to the axis of rotation and the equatorial plane of the earth. Also we assume that this mass is equal to the total mass M of the earth, and that the center of this mass and that of the earth coincide. 

we formulate the boundary problem for the potential of the attraction field, which has to satisfy the following conditions  

Here Uq is the potential of the gravitational field at the surface of the ellipsoid of rotation and r the distance between the axis of rotation and points of this surface. 

With an increase of the distance from the mass the potential tends to zero and far away it decreases inversely proportional to this distance 

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