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The object of this paper is to make a contribution to the theory of attraction when the force is proportional to any given power of the distance. In the case of the law of nature, it is well known that when the attracting mass is a hollow shell of uniform density, whose exterior and interior bounding surfaces are both surfaces of revolution about the axis ofz , the potential at any point exterior to the exterior surface can be simply obtained from the potential at any point on the axis exterior to the exterior bounding surface, and a similar theory connects the potential at any point within the inner bounding surface with that at any point on the axis which is also within the inner surface. In the expression for the pointz on thez axis we have merely to writer forz , and intercalate the zonal harmonics P0 , P1 , P2 ,... in the well-known manner. We consider the analogous theory when the law is that of then th power of the distance instead of that of the inverse square.

Contribution to the theory of attraction when the force varies as any power of the distance