Motion of a Perfectly Conducting Electrified Sphere. 643 



With this form of f (x being now interpreted generally as 

 ct — r + a) we have fulfilled all the conditions, satisfied by the 

 field, to the first order. The field is, therefore, completely 

 determined under the restrictions imposed. 



The density of the charge on the sphere is given to the 

 first order by 



47ro- = X at r = a-f|cos# 



e 2 cos , j,,, > , j.. . . N 



= - 2 4 —3— 0/ ( ct ) + / M - e^) ; 



Ut Co 



or considering the equation satisfied by/, 



47TCT = -= ! COS 0. 



a' 1 a 



TVe now find the resultant force on the sphere, which is 

 obviously along the polar axes. The component of the 

 electrodynamic force in this direction is 



P'=Xcos0-Ysin0, 



and at the surface of the sphere this reduces to 

 p , (e 2f"cos6\ r 



Thus the total force on the sphere in the direction of motion 



is 



■-iff F 



oo? sin 6 dd df\ 



3 a 



This is the effective electromagnetic force on the si)here. 



Now ° * 



/"=2A(l-^cos^)_ 



x*/'d\ 2Ae k . x^'6 



sin 



Vo 2a 



and since 



A _ ^ Se 



A ~ 2 ? 



we have 



p _2« 2 5[7 * ctv/SX <?-£ r-V'31 



JL = « — II — e ** cos — % — — ef n ,J 



3ac-LV 2a 7 v /^ m 2a J' 



