Tlieory of Matter and on Radiation. 193 



owing to radiation, fall into the centre of attraction and 

 correspondingly reduce its strength. In every case we may 

 treat the system as one of which all the electrons are arranged 

 in groups. It remains to examine the radiation from a group 

 of electrons ; it is of two types : (1) that due to the perma- 

 nent motion; (2) that due to disturbances produced by causes 

 external to the group. 



§ 5. J. J. Thomson* has shown that each of n equidistant 

 electrons, moving uniformly in a circle, radiates energy at a 

 very much slower rate than a single electron does for the 

 same orbit and same velocity. He finds 



E= 



2Cg*ff 2 W 3 (tt-H) (lift) 2 ' 1 



p' z '2n-\-l \2n 



This expression is only true for small values of vft. We 

 require one which holds for all values of ft less than unity. 



Strictly speaking, in all that follows ft must be less than 

 unity by a small amount depending on the ratio of the radius 

 of the electron to that of the orbit. For the negative electron 

 this ratio is of the order 10~~ 5 ; if /3 = *99 the results would 

 be in error to about one part in one thousand, if /3 = *999 to 

 one part in one hundred. 



Let us consider the case of n equidistant electrons moving 

 uniformlv in a circle. Take the centre as origin, the axis 



as Oz. The azimuth of the Oth electron E may be taken to be 

 cot + 8, where a is the angular velocity, so that ft = cop/C. 



o * 



The azimuth of the tth electron will be at + o -\ . Let 



n 



the polar coordinates, referred to Oz as axis and xOz as initial 



plane, of a point P be (r, #, <£>) ; and let the components 



along the radius, meridian, and parallel at P be, of the electric 



* Phil. Mag. [6] vol. vi. p. 681. 



