222 Prof. J. J. Thomson on the Electrical 



Thus a may be regarded as the time during which the 

 collision lasts : 



*« A f735$* ! 



and if X x is large compared with a, this may without 

 appreciable error be written 



. f cos ^X A _ ?a 



^ T_« a 2 + X 2 a 



~+x*' 



If this acceleration reduces a particle moving with a velocity w 

 to rest 



i+QO 



X C dX ttA 



J-oo^ + X 2 a 



thus 0i==Me-* a , (7) 



and the energy radiated is equal to 



I^A-^sin 2 ^.^. . . . (8) 



By taking other expressions for / we can get other forms 

 of <j), but it will be found that fa is always of the form 

 w>fr(qa), where a is the time occupied by a collision, and that 

 ^(qcc) vanishes when qa is infinite and is equal to unity when 

 q is zero. 



In the preceding investigation we have only considered 

 one collision ; in considering the effects of a large number 

 of collisions we must distinguish between the case when there 

 is any regularity in the occurrence of the collisions and when 

 these occur entirely at random. If there are fixed phase 

 relations between the collisions, the energy radiated in a 

 number of collisions will not be the sum of the amounts of 

 energy radiated when the collisions occur separately. Thus, 

 take the case of two collisions occurring simultaneously and 

 in close proximity. If the two collisions were of the same 

 character, i. e. if the accelerations were the same in the two 

 cases, the energy radiated by the two collisions occurring 

 simultaneously would be twice as great as that radiated when 

 the collisions were separated by a long interval. If, on the other 

 hand, the accelerations in the two collisions were equal and 

 opposite, the energy radiated by the two collisions would be 

 infinitesimal in comparison with the energy radiated when the 

 collisions did not coincide. If, however, the collisions occur 

 quite at random, we may take the energy radiated in these 



