On some inverse problems in potential theory

It is a well known result that the inverse square law has the property that the attraction, at an external point, due to a sphere of uniform density is the same as if the sphere was concentrated at its center. The most general law of attraction having this property for all spheres among the class of central force laws is known to be the inverse square plus linear. For the corresponding internal problem, it is well known that there is no attractive force acting on a particle inside a uniform spherical shell and, inversely, this property characterizes the inverse square law. We establish the first inverse result under the weaker condition that the desired property holds for just two thin spherical shells of any radii. For the corresponding internal problem, we need three incommensurable radii.