﻿llieory of Long Wave Radiation. 167 



the length dl. This factor is known from the above to be 



o-'H 1 -!) 



so that 



f(i+dt).= (i-Q/it), 



or what is the same thing 



so that 



df 



dl' 



>-w. 



the arbitrary constant of the integration being determined 

 by the condition that/(0) = l. 



Thus the probability of the electron describing a free 

 path between I and l + dl is expressed as the product of the 

 probability that it has described a free path I, and that it 

 will collide in the next small distance dl, and is therefore 



e- l/l m cdl=~e-^ l mdl 



Thus of the total number of paths described by the electron 

 in the time 6 the number whose length lies between I and 

 l + dl is 



u6 



I, 



■Iflmdl, 



so that the part of the sum in (4) contributed by these 

 paths is 



u0_ 



Sm \Wu) 



\ 



sir 



e-Wmdl = ude- l - l n 



J 



m \2Su L) 



sirlm 

 20u 



/ 



dl 



On investigation of this expression from Z=0 to l=co , we 

 find the part of the sum in (4) due to one electron, which is 

 therefore 



?^u a l sm \2&u %J e 



■lllm 



dl 



2u6l m 



J- T /»n q 



Now the total number of electrons in the part of the 

 metallic plate under consideration is Nir_V and by Maxwell's 



