254 Prof. G. A. Schott on Bolirs Hypothesis 



Corresponding conditions can be obtained for e and //, by 

 multiplying (2) by vrdivdz, integrating over the area swept 

 out by the electron, employing the usual expression for the 

 operator div in cylindrical coordinates, and bearing in mind 

 that C and K vanish at the surface of the electron by defini- 

 tion. Using (18) we find 



OG 



= cSft- ia z )k exp ik%, 



— CO 



with a similar equation for fi. In order that the integral 

 involving e, fi may not increase indefinitely with the time, 

 which is clearly physically impossible, we must have 



^30 = ^30 = ^30 = ^30 = for all integral values of k. (19) 



Also performing the integrations and indicating initial 

 values of e, fju by a zero suffix we obtain 



\\€.iffd'nrdz=z\\6Q'UFdivdz, \\fi^rd'Ssdz = \\fJL 'UTd^dz. (20) 



15. The conditions (18), (19), and (20) may be interpreted 

 as follows : — 



The radiation from an electron, which either moves uni- 

 formly or executes an oscillatory to and fro motion in a 

 circular path, or from a system of electrons, which move in 

 this manner in coaxal circular paths, can only vanish to a 

 first approximation when the following conditions are 

 satisfied : 



(1) The mean values of the electric and magnetic currents 

 for the whole area of a meridian plane swept out by the 

 electron, or electrons, must be independent of the time, but 

 may varj r from one meridian plane to another. (2) The com- 

 ponents of the currents perpendicular to the meridian plane 

 must vanish on the average for each meridian plane and at 

 each instant. (3) The mean values of the electric and mag- 

 netic densities for the whole area swept out by the electron, 

 or electrons, on any meridian plane must be independent of 

 the time, but may vary from one meridian plane to another. 



Obviously these conditions cannot possibly be satisfied for 

 any discontinuous distribution of charge in circular motion 

 about an axis common to the whole distribution, such as a 

 single electron moving in a circle of radius large compared 

 with its own, or a stream of electrons following each other 

 round such a circle in succession at distances apart large 



