Problems relating to a thin plane annulus

The problems which arise in electrostatics and hydrodynamics, in regard to the thin annulus cut from a plane sheet of metal, are of some importance, more especially in regard to electrical instruments of precision, such as the electrometer. Their mathematical solution in an exact form is a matter of extreme difficulty, and only first approximations, which can be derived by simple methods, appear to have been used hitherto. In the present paper, higher approximations are obtained, to an order which appears to be effective for most of the applications which are of real importance. It is shown that the actual difference of radii of the circles bounding the annulus is of com­paratively small significance in such magnitudes as the electrical capacity of the annulus, a result which could not readily be foreseen. The whole investi­gation is only carried to the second order of significance, but by a method—treating the annulus as a special case of the elliptic anchor ring—which can readily be extended to any desired order. The convergence of such approxi­mate solutions is not discussed, but it is clearly analogous to the remarkable degree of convergence found by Lord Rayleigh in certain solutions of problems of vibration of discs in which eccentricity is taken into account.