Attraction Field of the Spheroid

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is 

Substitution of these expressions into Equation (3.308) leads to a great simplification and we obtain 

This is an algebraic equation of second order with respect to ri and has the form  

Here is a positive number and C equal to zero for a point located on the spheroid surface, but in accordance with Equation (2.307), inside it is negative. For this reason, the difference B —AC is positive and greater than B. Thus, this equation has two real roots, one of which is positive. The latter represents the distance ri from the point p to any point q( i, rji, Ci) on the spheroid surface, but the negative root has to be discarded. Respectively, we have 

Introducing this expression into Equation (2.310) we obtain 

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