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 110 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. "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. Are there any other laws of attraction which have this property?"2 The above problem was solved in the affirmative,2 i.e., the most general such central force law being C(r) = A /r2 + Br. For the problem concerning an internal point, Jeans3 notes that if there is no attractive force acting on a particle inside a uniform spherical shell, then the law of force must be inverse square. He also provides a neat proof and attributes the theorem to Laplace. In both cases above, the desired property was assumed to hold among the class of central force laws and for spheres or shells of all sizes. In this paper we establish the first result under the weaker condition that the desired property holds for just two thin spherical shells of radii 1 and 2. Our method has some features in common with that given by Jeans. If instead of using shells of radii 1 and 2 we use a pair of arbitrary radii, the problem becomes considerably more difficult and apparently requires a more sophisticated argument. Consequently, this solution is given secondly. Lastly, we extend the internal problem in a similar way. But here it will turn out that the radii of the shells cannot be commensurable. Let <J>(r) denote the unknown potential function. Since the desired property is to hold for shells of radii 1 and 2, $(r) must then satisfy f *(|R C|) dS( = <£(|R|) + Cl for |R| > 1 (1) 7T J|<|-1 and j[ *(|R <|) dS{ = S(|R|) + c2 for |R| > 2 (2) 47T J | {|n=2 where dS( denotes the surface area element of the spherical surface. •Received February 24, 1967. •And also Yeshiva University. 2Problem 4381, proposed by R. J. Walker, Amer. Math. Monthly, (1951), p. 51. This problem also appears in S. L. Loner, An elementary treatise on statics, Cambridge University Press, 1942, p. 323. zThe mathematical theory of electricity and magnetism, Cambridge University Press, Cambridge, 1948, pp. 38-40. 278 M. S. KLAMKIN AND D. J. NEWMAN [Vol. XXVI, No. 2 Since a spherical zone has area proportional to its thickness, we can reduce the above double integrals to the following single ones: | J $((r2 — 2rt + 1)1/2) dt = $(r) + c, for r > 1 (3)