Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

In A Treatise on Electricity and Magnetism, Maxwell determines the angles of intersection for which one may use Kelvin's inversion method to obtain the perturbed electric potential upon placing intersecting spherical conductors into a region with a known potential. There are numerous modern applications utilizing this geometric construction in potential theory and hydrodynamics, and generalized circle and sphere theorems play a foundational role in this area of mathematical physics. In his work, Maxwell gives an intuitive argument for obtaining the perturbed potential based on intersecting planar conductors and a spherical inversion, and in this paper we extend his ideas to a full proof using rotational transformations and reflections. In the process, we disprove results in [Proc Lond Math Soc., 1966:3(16)] and [Stud Appl Math., 2001:106(4); Z. Angew. Math. Mech., 2001:81(8)] on boundary value problems in hydrodynamics involving intersecting circles and spheres, and we detail the angles of intersection for which these theorems are viable. Moreover, our proof recovers a special case overlooked by Maxwell for which Kelvin's inversion method may be utilized to obtain full solutions.