146 ME. GEOEGE W. WALKER ON THE INITIAL . ACCELERATED 



where 



m' = fe 2 /aC 2 and k = %e s /C 3 , 



e being the charge and a the radius of the sphere. 



The distribution of the charge on the sphere is not specified, and LORENTZ does not 

 claim any great generality for the equation. We shall find, however, that more 

 refined calculation proves that LORENTZ' equation is an exceedingly good approxi- 

 mation for vibratory motion. 



The second result of importance is the rate of radiation from an accelerated charged 

 particle calculated by LARMOR (' yEther and Matter,' p. 227). 



The rate is found to be f e 2 (.V:) 2 /C :i . The result is based on the Poynting flux over 

 a surface surrounding the particle, and reference to the original calculation shows 

 that certain terms are neglected as small. If the motion is vibratory this is readily 

 seen to be correct, but it has been claimed that the result is true for a uniformly 

 accelerated linear motion. The substitution of the requisite -form for the displacement 

 of the particle in this case shows that the terms neglected are as important as those 

 retained, and the result must be modified. It is further important to note that 

 MAGDONAI-D (' Electric Waves,' p. 72) lias shown that a term (nil for periodic motion) 

 must be added to the Poynting flux in order to give the whole rate of radiation. 

 Since a uniform movement in a circle may be compounded of two vibratory motions 

 there is no reason why LARMOR'S result should fail in this case, although the 

 acceleration is uniform. 



LARMOR'S result has, I think, been applied to the motion of a charged particle, 

 without regard to the condition of validity. 



THOMSON gives the equation (' Conduction of Electricity Through Gases,' p. 543) 



where the term 



t if) 



3 c 3 .; 



is the reaction on account of the radiation 



Against such an equation two criticisms may be made. First, it does not appear 

 where the term m'x comes from, as it ought to arise from the total rate of radiation. 

 Second, for a given velocity it gives two values for (x), which may be real or 

 imaginary a conclusion which seems untenable. The difficulty here presented may 

 be partially removed by consideration of LORENTZ' equation 



