3 Newton Theorem•The gravitational force should vanish everywhere inside an ellipsoidal homoeoid.Newton, I., 1686, Principia Mathematica, Book I, Prop. XCI,Cor. 3.
4 Xu and Szeto Theorem where • For a homogeneous ellipsoidal shell whose inner and outer surface flattening values are f1 and f2 respectively, the gravitational potential in any internal point of the shell is exactlywhere
5 Inferences (1)• When f1 = f2, Newton’s theorem is recovered as a special case.• Newton’s theorem predicts a constant potential inside a composite body consisting of concentric, aligned homoeoidalshells of varying density. If these homoeoidallayers are replaced by axially symmetric ellipsoidal shells of varying flattening, Newton’s theorem is incapable of predicting the potential inside such a body. The new theorem extends to this situation. We find a degree 2 harmonic potential whose magnitude depends upon the radial gradient of flattening.
6 Inferences (2)Parameter B, and hence the internal attraction, depends neither on the thickness of the shell nor the locations of the inner and outer boundaries. This is consistent with Newton’s theorem, since a homoeoidof arbitrary thickness can be added to the outside of a shell, followed by the removal ‘from the inside out’of another homoeoidwithout affecting B or the internal attraction.
7 Inferences (3)If an ellipsoid of revolution with density distribution lies inside the shell in such a way that its center of mass coincides with that of the shell and their symmetrical axes depart from each other by obliquityε, then the shell exerts a torque on the inner body given by
8 Inferences (4)If f1 and f2 are small, to first order in flattening, B is directly proportional to (f1-f2):
9 Inferences (5)When B < 0, the stable position of the inner body with respect to the shell lies in such a way that the symmetry axes of the inner body and the outer shell are coincident.
10 Inferences (6) • When B > 0, the corresponding stable position is one where the two symmetry axes are orthogonal.