Self-inductance of the elliptical loop

The paper states briefly how Maxwell´s geometric mean distance is used to reduce the self-inductance of a thin conductor to that of a pair of equivalent filaments. It then develops a method for splitting the standard Neumann double integral, used for finding the inductance of such filaments, into two sets of infinite series. The idea is first used for the known case of the circular loop, taking only the first terms of the sets of series to demonstrate their rapid convergence, and is shown to give the standard result. It is then applied to the hitherto unsolved case of the elliptical loop to derive two formulae, each valid for ellipses of different proportions. The accuracy of the second, which is semi-empirical, is examined, and finally the results are summarized. As the formulae appear in terms of the elliptical integral of the second kind, a simple formula for calculating the numerical value of this function over the range of interest is given in the Appendix. The formula expresses this in terms of the parameter of shape appearing in the inductance formulae, and may thus be used in preference to performing an additional calculation for the modulus that would be required to find the figure in a table of elliptical integrals.