﻿50 Prof. G. A. Schott on the 



motion, the expression (2) for the electromagnetic momentum 

 follows as a matter of course, but the expression (3) for the 

 radiation pressure requires a special hypothesis to justify its 

 introduction. It must, however, be borne in mind that the 

 deduction of the Lorentz momentum, as for instance by 

 Planck *, also implies the existence of a kinetic potential, 

 •and that this has only been defined for reversible changes, 

 whilst accelerated motions of an electron involve radiation 

 and therefore are irreversible. If, on the other hand, we 

 adopt the usual equations of the Electron Theory of Larmor 

 and Lorentz together with the hypothesis that the electron 

 occupies a finite though small region of space, whether sur- 

 face or volume, then the terms on the left of (1) represent 

 merely the first two terms of an infinite series. If a be a 

 length of the same order of magnitude as the linear dimen- 

 sions of the electron, and I a second length of the order of 

 the radii of curvature and of torsion of its path and of the 

 distance within which its speed is doubled, this series pro- 

 ceeds according to ascending powers of a/l, and converges 

 with sufficient rapidity only when a/l is small compared with 

 1 — /3 2 . When the acceleration of the electron becomes very 

 large, or its velocity nearly equal to that of light, the series 

 fails entirely ; indeed it is probable that under these con- 

 ditions the usual definition of the electromagnetic mass, im- 

 plied in (2) , can no longer be upheld. For the rigid spherical 

 electron of Abraham this has been proved definitely by 

 Sommerfeldf; he shows, that when the velocity of a uni- 

 formly accelerated electron is equal to that of light, the 

 largest term in the mechanical force on it due to its own 

 charge is proportional to the square root of the acceleration 

 when the latter is small. Unfortunately Sommerfeld's 

 method cannot easily be extended to the case of the Lorentz 

 electron, so that it is impossible to be quite sure of what 

 happens here, but it does not seem likely that the result 

 would be very different. However that may be, it is clear 

 that the expressions (1), (2), and (3) must be used with 

 •caution in cases where the velocity may be expected to 

 approach that of light, or in very strong electric or magnetic 

 fields, where the acceleration and curvature of the path of 

 the electron may reach large values. Thus we must be 

 careful in using them for an electron which approaches very 

 closely to the nucleus of Rutherford's model atom, and in all 

 problems of a similar kind. May not the failure of the 



* Planck, Sitzungsberichte der Preussischen Akademie der Wissen- 

 schaften, 1907, p. 8. 



t " Zur Elektronentheorie," Gottinger Nachric7ite?i, 1904, p. 411. 



