Stability of solution is considered for the charge-density problem of several rigid charged conductors, stability being with respect to the geometrical configuration of the system. Under the analytical restrictions usual in potential theory for the surfaces on which the charges reside, two theorems are proved establishing the stability: the first for when the potentials of the conductors are fixed, and the second for when their total charges are fixed. Subsequently, it is shown how the rigidity condition can be relaxed, but a counterexample to stability is described for a situation where the surface analytical conditions are not fulfilled. Finally, an application of the theory is given, in which the second of the said theorems is used to close a gap in the theory by the energy method of the electrostatic force on a conductor.