196 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



We have now proved that the time factor of first order vibrations is 



where 



180/ 2 \ 390/' 



and higher powers of X J are neglected. 

 In a similar way condition (2) leads to 



Tlit- result shows that both the frequency and the damping coefficient of the first 

 order vibration diminish as the velocity of the sphere increases. The effect is clearly 

 not considerable until the velocity is nearly that of radiation, and a higher degree of 

 approximation for such a speed is necessary. 



We have assumed that the sphere is constrained to move uniformly. If the sphere 

 is uncharged no constraint is necessary, so that the solution applies directly to the 

 case of an uncharged sphere. If the sphere is charged and unconstrained the 

 equations are more complicated, so that 1 shall give only the result. 



The time factor of the vibrations is 



,(1 -A- 1 ) 1 '-TIM' 



i 



where 



t(>30 



X-- /'7(I-/- J ), and 



This approximation neglects squares ofm'/m and higher powers of X 2 , and if we also 

 agree to neglect products of in'/ in and X L> we get, when condition (l) is used, 



while condition (li) leads to 



So far as these calculations go they indicate the way in which the process of 

 attempting to establish a uniform acceleration, at speeds nearly that of radiation, 

 may fail. The damped harmonic train of waves may have such a small damping 



