292 Prof. Townsend on the Variation of the Potential 



and the second give 



l = Z(a)\ p[Zxy l dx and 1 = ( ' *\Z{x)]- 1 dx. 

 Jo Jo 



f"0B-«)[Z(«)]- 1 i.-[Z(j.)]^-l > 



Jo 



-0 



Since 



the two latter equations reduce to the same condition 



C a Jo ( P~ a)dx 

 1= ax« x dx, „ » . k (1) 



and the values of n x and n 2 become 



n l u = cxZ(x)x\ [Z(x)]~ l x /3 X dx, . . (2). 

 Jo 



n 2 v = cxZ(x) x \ [Z(«)] _1 xaX^. . . (3) 



Equation (1) represents the condition which must be 

 satisfied by the values of a and ft along the path from x = 

 to x = a in order that a continuous current should be main- 

 tained by collisions. Since the values of a and ft may be 

 found in terms of the pressure and electric force by inde- 

 pendent investigations, it would be possible to decide whether 

 any given field of force would maintain a steady current. If 

 this condition is satisfied by a field of force determined 

 experimentally, while a current is passing through the gas, 

 it is evident that the ionization must be produced principally 

 by collisions. 



The field of force obviously depends on the current, the 

 connexion between these quantities being obtained from the 

 equation 



dX 



which gives 



47r(n 2 — n L )e = 



dx 



_ [""I a TZ (x)]' 1 x a x dx ([Z(a)]- 1 xftxdxl 



4^ = 4*nZ(ff)N^ --J- - 



dx v /4 - v u -» 



The rate of change of the force should also satisfy this 

 equation, but since the velocities u and v are not known for 

 the large forces and small pressures w T hich are used in the 

 experiments, no very satisfactory verification of the theory 

 could be deduced from this condition. Equation (4) might 

 be used to determine the velocities at different parts of the 

 discharge. 



(4) 



