On integral equations of axisymmetric potential theory

It is shown that two integral equations of the first kind, muchused in, respectively, axisymmetric electrostatics and hydrodynamics,are wrong in the sense that they do not in general possess solutions.A theorem is established giving the precise conditions necessaryfor solutions to exist, but perhaps more important practicallyis the fact, brought out by examples, that the necessary conditionsare far from sufficient. An alternative integral equation inthe electrostatic case is proposed and justified, one havinga similar form and the same computational advantages, but freefrom existence difficulties. The apparent paradox that ‘solutions’are found to problems when the governing equations may not possesssolutions is explained by the fact that these purported solutionsare obtained by numerical or asymptotic analysis, when an approximatingequation possesses a solution, but one which cannot be saidto approximate to the solution of the problem if the equationto which, formally, it approximates cannot, through being meaningless,represent the problem. The arguments are given mainly in theelectrostatic context, but it is shown how they are modifiedto carry over to the hydrodynamical one.