﻿llieory of Long Wave Radiation. 165 



We now fix our attention on all the paths described by all 

 the electrons under consideration during the time 6, and we 

 use the symbol IS to denote a sum relating to all these paths. 

 We have then 



S7tt sir / r t\ 



We now want to determine the square of the sum S. This 

 may be done rather easily because the product of two terms 

 of the sum whether they correspond to different free paths 

 of one and the same electron, or to two paths described by 

 different electrons, will give if all taken together. Indeed 

 the velocities of two electrons are wholly independent of one 

 another, and the same may be said of the velocities of one 

 definite electron at two instants separated by at least one 

 encounter. Therefore positive and negative values of v x 

 being distributed quite indiscriminately between the terms 

 of the series S, positive and negative signs will be equally 

 probable for the products of two terms. We have therefore 

 only to calculate the sum of the squares of the terms in S or 

 simply 



S4:0 2 v x 2 . 2 sttt 9 S7r/ r , r\ /Q v 



7? 6m W ooar T\ t+ c + 2> ' • (3) 



Now since the irregular motion of the electrons takes 

 place with the same intensity in all directions, we may 

 replace v 2 by ^v 2 . Also in the immense number of terms 

 included in the sum (3) the quantities r and v are very 

 different, and in order to effect the summation we may 

 begin by considering only those terms for which the product 



/ 57rT\ 



( v sin -q-j- ) has a certain value. In these terms which are 



\ . w J sir / r' t\ 



still very numerous, the angle ^~(^-f _ + ^l nas values 



that are distributed at random over an interval ranging 

 from to sir. The square of the cosine may therefore be 

 replaced by its mean value J, so that 



sV /±v 2 6 2 

 a,"= ^. *, ■ ., S ( o » sm- 





240W o ^ ^2 «« 20 



or if we introduce, after Lorentz, the length of the path / 

 instead of the time in it, this may be written 



\ 2v0 



