Field of Circular Currents. 421 



deduced from the expression for the magnetic potential at 

 the point expressed in a series of spherical harmonics. In 

 many cases the series converge very slowly. It seemed 

 desirable, therefore, to attempt to find formulae which can be 

 evaluated more readily. 



By means of Laplace's formula * for the magnetic force at 

 any point due to an element of current, formulae for the 

 component forces can be found very easily in terms of 

 elliptic integrals, the values of which are given in Mathe- 

 matical Tables f. The results found are also useful in 

 hydrodynamics in connexion with the theory of the circular 

 vortex filament. The formulae give directly the mutual 

 inductance between two coaxial circular currents and a close 

 approximation to the self-inductance of a thin circular 

 current. The author has not found it necessary to assume 

 Neumann's theorem in proving any formula for self or 

 mutual inductance. An expression is found for the axial 

 magnetic force at any point due to a cylindrical current 

 sheet, and particular cases are noticed. It is proved that the 

 mutual inductance between two cylindrical current sheets is 

 the same as that between one of them and a certain helical 

 current of the same diameter and axial length as the other 

 sheet. This theorem can also be immediately deduced from 

 a formula given by Viriamu Jones %. 



The exact formula for the mutual inductance between a 

 cylindrical current sheet and a coaxial helical filament is 

 expressed both in terms of elliptic integrals and in a series 

 which in general converges rapidly. Viriamu Jones left the 

 solution in the form of a definite integral. By the formulae 

 given in his paper § this can be expressed without difficulty 

 in terms of complete and incomplete elliptic integrals. By 

 utilising Jones's results, Professor Coffin in the ' Bulletin of 

 the Bureau of Standards 3 (p. 118, June 1906) has done this 

 when the axial length of the two helices is the same. The 

 author, however, starting from Laplace's formula, directly 

 deduces the complete solution in forms adapted for easy 

 computation. Lorenz's formula || for the self-inductance of 

 a helical filament is a particular case of the general formula 

 for mutual inductance given in this paper. It is shown also 

 that a well-known formula due to Lord Rayleigh is a 

 particular case of Lorenz's formula. 



* A. Russell, ' Alternating- Currents,' vol. i. p. 29. This formula is 

 sometimes attributed to Ampere in English text-books, 

 t For instance, Dale's ' Mathematical Tables,' p. 76. 

 X Proc. Roy. Soc, p. '203 (1898). § L. c. ante. 



II Wiedemann's Annalen, vii. p. 170 (1879). 



