to Maintain Currents between Coaxial Cylinders. 85 



Let i be the current per unit length of the wire, k the 

 velocity of the ions due to unit force, p the charge per cubic 

 centimetre of the gas, and <£ the potential at the distance r 

 from the axis. At the surface of the wire r=a, cf> = 0; and 

 at the surface of the cylinder r=b, <£ = V. 



If all the electrical quantities are expressed in electrostatic 

 units the following relations hold between cf> } p, and i : — 



i= — 2iTprk-^- 

 dr 



1 d ( d<t>\ . 



TdrVdr)=-^ 



when p is eliminated the relation between dfyjdr and r 

 becomes 



The constant of integration C is obtained from the condition 



jr =A 1 =- 1 jj- . when r — a. 



dr a log bja 



Hence 



(-£ )'-«.•-¥ ♦? 



When 2z/£ is negligible compared with Xj 2 this equation 

 becomes 



, . v A , 2ir 2 \i dr 

 which may be integrated by changing the variable r to 



The total potential fall V thus obtained is 



V=aX 1 {(l + 0)*-l + Iog2&-loga(l+(l-f0)*)}, 



2ib 2 

 where = ^-^ 2 , 2i being small compared with kX^. 



Hence V— -V 0l b „ , 9 



