Dynamical Motions of Charged Particles. 539 



particles are small, though not negligible, fractions of C, 

 the Aelocit}- of light — that is, we shall expand in inverse 

 powers of 0. That it should be necessary to approximate is 

 not surprising, as it is well known that according to the 

 classical electromagnetic theory an accelerating electron 

 will radiate, and the consequent dissipation of energy cannot 

 possibly be represented by a Lagrangian form. The radia- 

 tion of a single electron gives a reactive force on it of 



amount ~ — — , where v is the velocity vector. Hence 



we must not expect to be able to find a Lagrangian accurate 

 beyond the terms in C -2 . Though SommerfelcTs orbits are 

 worked out without approximation, the neglect of the radiation 

 terms implies that they are really only valid to this degree — 

 that is, it* they are regarded as based on the classical electro- 

 magnetic theory. 



Now this raises quite unanswered questions of funda- 

 mental physics, for there can be no doubt that radiation does 

 not really work in that way at all. But we have no right to 

 claim that the equations with these radiation terms omitted 

 will truly describe the motion simply because the radiation 

 does not in fact occur; for quite apart from radiation there 

 is something wrong with them. This is shown by the spiral 

 orbits* which an electron should describe about a heavy 

 nucleus, when its angular momentum is below a certain 

 value. These orbits involve an ultimate coalescence of the 

 electron with the nucleus, and if the theory were right they 

 should be of fairly frequent occurrence, because, whatever 

 the initial line of motion of the electron, a low angular 

 momentum can always be attained by a sufficiently low- 

 initial velocity. This result follows whether the accurate or 

 the approximate formula is used for the variation of mass. 

 As there can be little doubt that coalescence does not in fact 

 occur, it is necessary to invoke some modification of the laws 

 of motion to prevent it, and the quantum naturally suggests 

 itself. The question of these orbits in relation with the 

 quantum has been discussed by Sommerfeld f to a certain 

 extent, though the difficulties are not altogether removed 

 and as in most other applications of the quantum the actual 

 physical motion is quite incomprehensible. But, however 

 that may be, and whatever assumption is made, the type of 

 orbit must be quite altered and the validity of our present 

 methods destroyed. So it is safer to claim validity for our 



* C. G. Darwin, Phil. Mag-, vol. xxv. p. 201 (1913). 

 t Loc. cit. 



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