THEORY OF ELECTRIC DISCHARGES 65 



a method similar to that which was used to determine 

 the condition for sparking in a uniform field, but it is 

 interesting to investigate the problem on somewhat 

 different principles. 1 



26. Condition for the maintenance of a current by 

 effects of collisions in any field of force. 



Let the current pass between two parallel plate 

 electrodes at a distance Z apart, and let x be the 

 distance of any point in the gas from the negative 

 electrode. Let u be the velocity of the negative ions, 

 a the number of molecules ionized by a negative ion in 

 travelling through a centimetre of the gas, n\ the number 

 of negative ions per cubic centimetre, v, /3, and ?? 2 similar 

 quantities for positive ions. These quantities depend 

 on the force X, which is supposed to vary from point to 

 point. If e is the charge on an ion, then 

 represents the current in the gas, so that 

 is constant along the path of the discharge. 



When the current is steady there are no variations 

 with respect to the time at any point in the gas, so 

 that 



and C -~ = ^- 



Hence ^(niu)=aniU-\-8n^v=(a (3)niu-\-pc, 



so that 



f(a-0)efo f(a-3X*f -/(a-0)ete 



n lU =^ J +e J/3C6 ' dx 



1 See papers by the author, Philosophical Magazine, March, 

 1905, and June, 1906. 



I.G. F 



