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It has long been known that the potential distribution in axially symmetric electrostatic fields can be expressed in terms of a Fourier integral. This form of solution is not generally very useful because of the difficulties encountered in evaluating the integral. The present paper shows how this integral can be evaluated for a particular type of potential distribution, and shows how it may be applied to obtain the variation in potential along the axis when the potential variation along any cylinder is known. The result is then used to obtain an approximate solution for the particular case of two infinitely long cylinders of equal diameter separated by an amount comparable to their diameters.