The electrostatic force between charged spheres at constant potential from the Fredholm integral equation

The electrostatic problem of two spheres held at constant, not necessarily equal, potential is solved from considerations of action-at-a-distance (Coulomb force) by the way of using a solution to the Fredholm integral equation of the second kind. The solution, written in series form, is obtained from Gauss' generalization of the electrostatic force for distributed charges and yields the charge density distribution, important to liquid droplet stability, droplet deformation, and breakup. Evaluation of the electrostatic force is fast because the series expression for the charge distribution is rapidly converging, a characteristic trait for some Fredholm type equations.