﻿168 Mr. G. H. Livens on Lorentz\ 



law, among these 



4,ir^wA^y^ 2 e' qu2 u 2 du 



have velocities between u and u + du ; the constant q is 

 related to the velocity u m already introduced above by the 

 formula 



3 



Thus the total value of the sum in (4) is given by 





2u0L 



— Oil 2 



l+' s 



or, using z = qu 2 , by 



4M A /l 



V 7TC 



{ 



6hc 2 



ze^dz 



sVlJq 



2 z 



This integral cannot be evaluated in definite terms, being 

 of the integral-logarithmic type, but we can obtain various 

 good approximations to its value. Iu fact a direct use of 

 the first theorem of mean values in the integral calculus 

 soon shows that we have 



[ 



e~ z dz 1 C" -zj 



ze ~dz 



. , sWq .. sWg 



1 + —¥~ 1+ ^ 2 . 

 1 



~ , sVlJq 



z Q denoting some mean value of z, which is ultimately, how- 



— ™ ■ J in the 



integral ; I find on trial that z is such a function of this 

 constant that its value lies between 1 and 2, the values it 

 assumes for small and large values respectively of the con- 

 stant. If, therefore, we define u by the relation 



qilQ 2 = Z , 



we shall know that u^, ultimately a function of I- — nf~^)j 

 must, however, lie between the limits 

 %u-J and 



3 "■m ? 



