THJS 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



*fc.. 



[SIXTH SERIES.] 



G o ' 



}/ 9AUQ UST 1919. 

 O^FlCt— 



XYII. Notes on Electric and Magnetic Field Constants and 

 their Expression in terms of Bessel Functions and Elliptic 

 Integrals. By Professor A. Gray, F.R.S* 



1. fT^HE following notes were for the most part written out 

 JL several years ago, and were suggested by points 

 which arose in the course of a revision of part of my book on 

 6 Electrical Measurements ' on which I was engaged. I have 

 to-day noticed that in the idea of calculating the exhaustion of 

 potential energy involved in the formation of a uniform disk 

 of attracting matter, I have been anticipated by Clerk Maxwell 

 and by Lord Rayleigh f. Some of the other processes in like 

 manner may not be novel, but on the whole the notes may 

 perhaps be found to be not altogether devoid of interest. 



As a preliminary we take some expressions for potentials 

 and forces in terms of Bessel Functions. Some of these, if 

 not all, are quite well known. 



Case (1). A thin disk of matter of uniform surface density a 

 and radius a is situated with its centre at 0, fig. 1, and its 

 axis along OC : it is required to find an expression in 

 terms of Bessel functions for the potential produced at 

 any point P. 



Let the axial distance of P from be z, and its distance 

 from the axis be y, as shown in the diagram. Consider a 

 ring of the disk with as centre, and of radius p and 



* Communicated by the Author. 



f "Theory of Resonance," Rayleigh, Collected Papers, vol. i. 



Phil. Mag. S. 6. Vol. 38. No. *224. A ug. \ 919. P 



