244 Mr. S. H. Burbury on the 



because owing to the finite velocity of light, which shall be 

 denoted by v, r and 6 relate strictly, not to the present 

 position, S, of the sphere, but to the position, S x , in which 

 it was t seconds ago. And S' and t are determined by 

 the conditions S / P = fl£, and S / S = w* if u be constant, or 



S 7 S= I udt if u be variable. In like manner, if e be supposed 



variable with the time, the value of e in the expression 



— — g — is the value of e when the sphere was at S', but I shall 



not further consider the possible variation of e. If, as is 

 generally the case, u be very small in comparison with r, the 

 result given is a very near approximation, unless the rate of 

 time variation of u be very great. 



8. It is noteworthy, however, that if u be constant, being less 

 than v (which is the case considered by Lodge), the result 

 given is exact. For, S being the present position of the sphere, 

 let S^ be its position t seconds ago, S x its position t — dt seconds 

 ago, S 2 its position t—2di seconds ago, and so on. About 

 each of the points S , S 1? S 2 . &c. as centres describe spherical 

 surfaces having radii, vt for S*, v(t — dt) for S^ v(t— 2dt) for 

 S 2 , &c. Then, since u is less than v 9 no two of these surfaces 

 intersect each other. The sphere described about S* includes 

 that described about S : , and so on. Then the magnetic force 

 at P at this instant due to the motion of the charged sphere 



through S^ t seconds ago is K=ue — $- m The normal distance 



between the two successive spherical surfaces described about 

 S^ and S x is at the point (r, 6) is (?: — u cos 6)dt. And the 

 integral of H 2 /87T throughout the spherical shell between 

 these surfaces is 



that is 



, e 2 u 2 f-27Ti' 2 sin 3 (v - u cos 0) d6 



^ J„ Fi ' 



sin 3 6 (v — u cos 6) d6 



2 u? f n sh 

 4"Jo~ 



The term in cos 6 disappears in the integration, so when we 

 integrate for t the result is 



sin 3 6 



« 



d6 



vt 2 



which agrees with Lodge's integral. 



4. If, however, u be variable, we must put u 2 under the 



