412 Prof. S. R. Milner : Does an Accelerated 



the conservation o£ energy. The electromagnetic system 

 must, in fact, be a conservative one, capable of existing 

 without the introduction of energy by imaginary processes 

 from outside. (Radiation, if present, is of course part of 

 the electromagnetic system.) We may, perhaps, further 

 expect that the energy and momentum of each part of the 

 system which is in motion should be related to each other in 

 accordance with the laws of mechanics. 



From this point of view the solution of electronic motion 

 which has been discussed is a fictitious one, as it does not 

 obey the conservation of energy. As the electrons are 

 approaching each other the energy of the system gets 

 smaller, and it increases as they move apart. By a simple 

 modification, however, the system can be made conservative. 



Superpose on the field (4), which forms one solution of 

 Maxwell's equations, another solution, in the form of a 

 uniform electric field X of infinite extent and parallel to the 

 axis of x. Let X<? = F, so that X is the field which is 

 required, on the basis of Lorentz' s equation, to produce the 

 actual motion which the electrons possess. Then, since for 



2 e 2 

 a Lorentz electron we have m = " -y-, we have by (2) 



d c a 



x =fs • • < l2 > 



The resulting field is now given by 



E* = Ecos0 + X, E y = Esin0, 



H = /9sin^.E, 



where E and X are given by (4)' and (12), and it forms of 

 course a solution valid over all space and time. 



These equations represent a system of two electrons of 

 opposite sign and Lorentz mass moving in a prescribed way 

 in a uniform electric field. We shall, however, as before 

 consider only the case in which a is infinitely small, when 

 the electrons become point-charges, still of Lorentz mass, 

 and the field X, though of infinite strength, is the limit of 

 the definite value required to cause in the point-charges a 

 finite acceleration. 



Although the total energy of the system is infinite, the 

 question of its .conservation can be examined by considering 

 the flux of energy in the field. For mathematical purposes 

 we can regard such a field as a solution of Maxwell's 

 equations which extends over all space and time, except 



