MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 159 



and <f> is supposed to be small of the same order as f, but there is no restriction to a 

 small value of k. 



In the particular case of a charged conducting sphere, we have to satisfy the 

 condition that the tangential component of electric force should vanish at r = a, and 



that the surface density of electricity is given by. - (normal component of electric 



4zr 



force). 



The appropriate form for i/> is 



where 



t - 



and e is the total charge on the sphere, which is not uniformly distributed. 

 The corresponding simplest form for < is 



where X = k/(lk 2 ) and ^ is an arbitrary function. 

 Thus the contributions to electric force are 



3 - 



Now, the finite terms due to the steady state are already chosen to secure the 

 vanishing of the tangential component of (X , Y , Z ), hence the tangential component 

 of (X, Y, Z) will vanish at r = a, provided 



*" 



when r = a, for all values of t. 



As in the simpler case in Section 3, damped harmonic vibrations will arise and 

 rapidly become negligible. We therefore proceed to consider the motion established 

 when the vibrations have subsided. 



Let 



f= ^f t 2 where f is constant, 

 then, if 



t P _ X 



the tangential condition is satisfied identically. 



