Electron necessarily Radiate Eneryyf 411 



we find the ^-momentum associated with the positive 

 electron external to this surface to be 



H^ 1 -^-*- • • • • 'w 



This is precisely the same as the momentum of the Lorentz 

 electron of the same e and a in uniform motion with the 

 same velocity, 



It must be noted that this result has only been obtained 

 by making the surface (10), to which the integration in yfr 

 was extended, a slightly different one from that (5) used for 

 the determination of the energy. The difference, however, 

 disappears when, as is the case here, the surfaces are of 

 indefinitely small size, as in this case (10) as well as (5) 

 reduces to identity with the Lorentzian spheroid. 



There is a distinction between what we may call " ficti- 

 tious " and " real " cases of electronic motion, which is 

 brought into evidence by this example. Maxwell's equations, 

 as is well known, are of great generality, and can be satisfied 

 by cases of motion which cannot really exist. Theoretically 

 a solution for the field of a charged particle in an arbitrary 

 state of motion can be obtained. But from a physical 

 standpoint the purely arbitrary motion of a charge is a 

 condition impossible to produce. We know of no way in 

 which an electron can actually be set into motion except by 

 the application to it of an electromagnetic field (excluding 

 possible effects of gravitational fields). Not only is this not 

 entirely arbitrary, being subject to the fundamental equations, 

 but also its inclusion alters the problem in an important 

 respect. The problem becomes now, not that of finding the 

 field of an electron whose charge is imagined to be moved 

 about in a given way by external non-electromagnetic 

 means, but that of finding a solution of the fundamental 

 equations which will represent the real time-history of a 

 given electromagnetic field which contains an electron. But 

 for this problem Maxwell's equations are insufficient. Being 

 satisfied by the simple superposition of two electromagnetic 

 fields, they do not reveal the way in which, when an electron 

 is present in a given field, the two fields interact. An 

 additional theory is required for this, such as that of the 

 Lorentz equation. 



Leaving this aside for the moment, we can consider in a 

 general way the conditions which a system of two superposed 

 fields must be expected to satisfy for it to represent a real 

 case of electronic motion. There is undoubtedly the law o( 



