MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 



151 



accelerating force applied, the initial motion would be attended to a first order of 

 approximation by the production of a damped harmonic train of radiation depending 

 on a zonal harmonic of the first order, and that equations could be formed for the 

 motion of the sphere. This method proved successful ('Roy. Soc. Proc.,' A, vol. 77, 

 p. 260), and was extended to a second approximation. It was next found to be 

 applicable to a disturbance from a state of steady motion with any velocity in a 

 straight line. 



3. Initial Motion of a Charged Conducting Sphere. -This problem was solved in 

 the paper jxist referred to, but the accelerating force was supposed to arise from a 

 uniform electric field. In order to obtain the electrical effects of the motion free from 

 a superposed electric field, it is convenient to suppose that the accelerating force is of 

 a purely mechanical nature. As the procedure is almost identical with that given 

 (' Hoy. Soc. Proc.,' foe. fit.), it is sufficient to note that if f is the displacement in the 

 direction of ,r the primary equations are 



and 



~* <*-- 



a 



where F is the mechanical accelerating force. 

 The initial conditions are ^ = ^' = when r = 

 If we write m' for je 2 /aG a the solutions are 



and ==() when t = 0. 



X (Ct-r).= A 



sn 



eF 



'lam 





in 



, i 



where 



and 



A e 



A sin e = -- 



. 4m'\ 1 2 A 



3 + - - A cos e = 



m 



It may be observed that the initial displacement expressed by the damped 

 harmonic part is equal and opposite to that expressed by the non-periodic portion. 

 After one complete vibration the amplitude of the vibratory part falls to 



