required to maintain a Current in a Gas. 291 



ionized by a negative ion in moving through a centimetre of 

 the gas,??! the number of negative ions per cubic centimetre, 

 r, ft, and n 2 similar quantities for positive ions. Let X be the 

 electric force at any point in the gas, and V the difference of 

 potential between the electrodes. The current i per unit area 

 of the electrodes will be constant when the current is steady. 

 The quantities a, /3, u, and v (as well as ?i x and ?? 2 ) depend on 

 the distance x since the electric force between the plates is 

 not in general the same at different points. 



If e is the charge on an ion, then n l u-\-n 2 v = - is constant. 

 Let c denote the value of this quantity. 



When the current is steady there are no variations with 

 respect to the time in any part of the gas, so that 



and 



Hence 



chu d , , ~ ~ 



-T7 = — T~, \ n i u ) + an 1 w + /3??2i , = 0, 



~sr = j~ ( n 2 v ) "*" an i w "*" & n * v = ®" 



So that 



-j- (uiu) = oLn x u + /3n 2 v 



5 



n l u = Ae^~^ dx + e-*"- w *f £ X c X e~f^ )dx dx 



where A is a constant. Cx 



J (*-p)dx 



Let Z(x) denote the quantity e so that n x u may 



be expressed 



n lU =AZ(x) +cZ(x) ) [Z(x)Y l ftdx. 



Similarly 



fi 8 w=BZ(^)-cZ(o?)r[Z(a?)]- 1 («fdy. 



Since all the ions are produced in the gas, the following 

 conditions hold at the electrodes : — 



At the negative electrode (a? = 0), n 1 = and n 2 v = c. 



At the positive electrode (<r = a), ?? 2 = and ?? 1 m = c. 



The first two conditions give 



A = and B = c, 



U2 



