Electrostatic and Magnetic Fields. 4,17 



produced at any point internal to itself by a uniform 

 magnetic shell coinciding with the confocal surface. The 

 strength of this shell, though constant over the surface, 

 is proportional to the length of the perpendicular from 

 the centre on the tangent plane to the confocalat P, and 

 therefore varies with the position of P on the surface. 



The value of the integral \ cos / dS'/r 2 is of course 47r and 



so F the force at P due to the given elliptic homceoid, or its- 

 equivalent the magnetic potential within the confocal is 

 obtained at once *. 



4. Evaluation of the field intensities for the case of § 1. 



The values of the electric and magnetic field intensities 

 specified in § 1 in terms of elliptic integrals are of course 

 well known, (see for example a paper by Dr. Alexander 

 Russell, Phil. Mag. April 1870) . But a process of integration, 

 differing somewhat from the usual one, is of some interest 

 and seems possibly capable of some extensions, which may 

 be given in a subsequent note. 



The direct process of evaluation of the electric field 

 intensity F e from 



-^ a f " cos 0ds' 



v "^\-w-- • • • • < 13 > 



in elliptic integrals is as follows. By fig. 1 we see at once 

 that cos 0= (a' — a cos 0)/r so that we get 



_ a I v u ! — acos<f> , , ,.,,. 



F ' =2 vJ -- ~^ ds - ■ ■ ■ ( 14 > 



This leads to 



f --*K^ B( * )+ to k -<*>}' ■ • (15) 



if k = 2\/aa'l(a + a / ), and we use the relation 



i 



(16) 



chfr _ E 



! (l-Psin 2 ^/^)£ _ ^-I 2, ' * 



which can be verified by direct integration. This is an 

 elliptic integral of the third kind with parameter — P. 



* In the particular case m which the confocals are concentric 

 spherical surfaces, the attraction of a shell coincident with the inner 



sphere reduces at once to Airier [a" 2 , since \ cos0'fS'/?* 2 =47r. 



Similarly for the attraction at an internal point, the point A is 

 external to the sphere radius a', and the solid angle is zero. Hence the 

 internal attraction is zero. 



