294 Prof. Townsend on the Variation of the Potential 



the electric force. The effect of the positive charge in the 

 gas on the potential required to maintain a discharge can 

 easily be illustrated by finding the potential required to 

 maintain a current in a field made up of two portions in each 

 of which the force is constant, the value of the constant being 

 greater near the negative electrode. It is interesting at first 

 to investigate cases in which the discrepancy between the 

 forces in the two parts of the field is small compared with 

 the variations which are found when the ordinary cathode- 

 fall of potential is developed. In general, let a and /3 have 

 the constant values a x and ft from os—0 to x = b ; and let the 

 values be « 2 an d ft from x = b to % = a. 



Equation (1) may be expressed in general in the following 

 form : 



Xll-|«xe° dx) + e X^j^xe <fo?J = 



With the above constant values of a. and ft for the two 

 parts of the field, this equation reduces to 



"i— ft ft— «2 



= 1. 



As an example, let the gas between the electrodes be 

 hydrogen at 10 millimetres pressure, and let 5 = 2 millimetres. 

 Let the force be 80 volts per millimetre in the two millimetres 

 of gas near the negative electrode, and 50 volts per millimetre 



in the rest of the field. The values of — will then be 80 and 



P 

 50 in the two parts of the field, and the values of a lf ft, ot 2j 

 and ft are as follows : "* 



a 1 = 10*0, a 2 =:?f'§, 



ft= -081, ft= -021. 



The value of a may be found by substituting these values 

 in the foregoing equation. The distance between the 

 electrodes requisite for the maintenance of the discharge is 

 thus found to be 10*0 millimetres. Hence, if the electrodes 

 are a centimetre apart, a continuous current would be main- 

 tained in the gas if the force is 80 volts per millimetre in a 

 layer 2 millimetres thick near the cathode, and 50 volts per 

 millimetre in the remaining 8 millimetres of the gas. The 

 total fall of potential between the electrodes would be 560 

 volts, which is about twenty volts less than the sparking 



* J. S. Townsend, Phil. Mag. June 1902 and November 1903. 



