220 Prof. J. J. Thomson on the Electrical 



Hence 



'+0O 



i 



\f(t)}*dt = ~f (\fa + cj> 2 cos qt 2 } 2 + fa 2 sir^ qt 2 ) dq 

 7r Jo 



if 00 



— — (</>i 2 + W + 2<^>i0 2 cos q t 2 ) dq. 

 t Jo 



Thus the energy of the waves with frequency between 



q and q + dq is equal to 



2 <? 9 



3 ^y (^>l 2 + ^ + ^Cjb^g COS ^ 2 ) ^. 



If the accelerations when the particle is stopped are equal 

 and opposite to those when it is started, cf> l =—^ and the 

 preceding expression becomes 



**ftffa*fdq . (3) 



We shall now proceed to find the value of fa in special 

 cases. If the acceleration is constant and equal to ft during 

 the collision, then 



J 2 *P &m ~2~ 

 (3 cos q\ dX = — » 



2 



where X x is the duration of the collision. If the collision 

 reduces to rest a particle which was moving previously to the 

 collision with the velocity u, then 



2 



ft d\ = u, 



A_i 



" 2 



or 





fl\i = u. 



Hence . tf\ x 



2w sm ^- 



fa = 



q\ ' 



and the energy radiated from the beginning to the end of the 

 free path is by (3) 



/ . qXi\ 2 



Instead of supposing that the acceleration abruptly changes 



