Spherical approximation of the boundary condition

Before we make use of Equation (2.265), let us transform the boundary condition (2.264) in the following way. With an accuracy of small quantities, which have the same order as the square of the geoid heights, a differentiation along the normal can be replaced by differentiation along the radius vector, and correspondingly the condition (2.264) becomes 

In essence, in place of the surface of the geoid we use a spherical surface. Now let us make one more approximation, which has the same order as the flattening, and, therefore, causes a very small error in determining the height N. We assume that the normal field is inversely proportional to the square of r. This gives 

To facilitate derivations and express the field T in terms of the anomaly of gravity let us multiply both sides of this equality by r. Then, in place of Equation (2.268) we have 

It is proper to notice that the function E is harmonic outside the geoid. Now, multiplying both sides of this equation by rdr and integrating within the range  

we perform some transformations of the integrand in the integral by r. By definition  

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