The Stability Theory of Charge Distributions on Conductors

Stability of solution is considered for the charge-density problemof several rigid charged conductors, stability being with respectto the geometrical configuration of the system. Under the analyticalrestrictions usual in potential theory for the surfaces on whichthe charges reside, two theorems are proved establishing thestability: the first for when the potentials of the conductorsare fixed, and the second for when their total charges are fixed.Subsequently, it is shown how the rigidity condition can berelaxed, but a counterexample to stability is described fora situation where the surface analytical conditions are notfulfilled. Finally, an application of the theory is given, inwhich the second of the said theorems is used to close a gapin the theory by the energy method of the electrostatic forceon a conductor.