The elliptic integral in electromagnetic theory

Abstract

The expression of the integral by the theta function would imply then a comples argument, and a table of the theta function would not be of complete utility unless made a triple entry table. When required in a problem of electromagnetism it is the complete third elliptic integral which usually is sufficient for a solutionz and this, as shown by JEGENDRE, can be expressed by elliptic integrals of the first and second kind, complete and incomplete; and for these the Table IX of LEGENDRE provides the material for a numerica} evaluation. As an important application we may cite the calculation of the mutual induction of two coaxial helices, employed in the ampere balance designed for weighing their electromagnetic attraction by the late Viriamu JONES and Professor AYRTON, and so arriving at an independent determination of the electrical units. The present investigation was undertaken in the lifetime of Professor Viriamu JONES, with the object of exhibiting the result of his complete third elliptic integral in its simplest form, suitable for immediate numerical computation, and also to reconcile the conflicting notation of different writers on the subject by adopting MAXWELL'S Electrictty and ]agnetism as the standard. Incidentally the quadric transformation of LANDEN is required here so frequently that a digression has been made on the theory, and an elucidation submitted of its essential geometrical interpretation. The references in the course of the work will be chiefly toMAXWELL, Electrictty and Magnetism. WEBSTER, Electricity and Magnettsm. GRAY, Absol?hte Measurentents in Electrbeity and Magnetisnz. CAYLEY,Proceedings of the London Mathematical Society,vol.6.