MOTION OF ELECTEIFIED SYSTEMS OF FINITE EXTENT, ETC. 193 



It is therefore desirable to discuss this point more fully, and it will be sufficient for 

 our present purpose to consider the sphere as constrained to move uniformly in a 

 straight line with velocity kC. We may state our problem as the determination of 

 the period equation for first order vibrations on the sphere, such as would be excited 

 by the prodiiction of a uniform field of electric force. 



When the sphere is at rest the vibrations depending on zonal harmonics of different 

 orders are quite independent of each other ; but when the sphere is moving this is no 

 longer the case. In particular, the vibrations excited in establishing a uniform field 

 no longer depend solely on a zonal harmonic of the first order, but involve an infinite 

 series of zonal harmonics, as might be expected from our examination of the steady 

 distribution produced by a uniform field. 



In general, the components of electric force are 



= __ v _ 



' 



3z 8 ' Si/ex' 



where 



Since the field is to be symmetrical about the axis of x we may write 



Y 1 3 9< v '/ &<!> v z 9 2 <i 



~ a~~ CT ~~ ' i i - = , z< = - i 



OT crs CCT is (JOT OX OT (JOT OX 



where 



', ox/ 

 If we make LORENTZ' transformation 



t' = t 



we get 



and 



We now assume condition (]), that the tangential component of X, Y, Z at r = a 

 should vanish. 

 Thus 



This may be written 



. _ 



* " 5 



VOL. OCX. A. 2 C 



