968 Mr. S. Smith on Initial 



to such a position that the charges gathered on electrodes 

 1 and 3 were equal when the light had been turned on. 



In order to obtain an expression for the distribution of nega- 

 tive ions between the electrodes, let the geometrical centre of 

 the face of electrode 2 be taken as the origin of rectangular 

 coordinates, the normal to the face as the axis of z, and the 

 axis of y parallel to the length of the electrode. Let n be 

 the number of negative ions per cubic centimetre in the 

 neighbourhood of the point (as, y, z), Z the electric force at 

 that point in electrostatic units, and e the charge on an ion 

 in electrostatic units. The number of ions crossing per 

 second unit area normal to Z, may be represented as niv, and 

 is due to the processes of diffusion and motion under the 

 electric force, so that niv is given by the equation 



nw= — K-r — \-nw,2j. 



where w x is the mobility of the ions or velocity under unit 

 electric force*. This expression holds good as long as the 

 velocity due to the electric force is small compared with V, 

 the mean velocity of agitation. Reference to a paper by 

 Prof. Townsend and H. T. Tizardf will show that this 

 condition has not been violated throughout the experiments. 

 The velocity of the ions is therefore 



^ ' ' J \ n Qx n dy n dz J 



Taking a parallelopiped of sides Sat, By, and Bz about the 

 point (as, y, z) , it is clear that the excess of the number of 

 negative ions coming into the parallelopiped per second over 

 those going out per second is given by 



X \ nu By Bz — nu-{- <— (nu) Bx \8y Bz J , 



and this is easily seen to be 



|KV a W-w 1 z|?T8*.«y.&. . . . (1) 



Now the number of negative ions produced per second by 

 collisions in the parallelopiped is nwJL By . 8z . a Bos. In the 

 steady state the sum of these expressions must be zero, so 

 that 



K . V 2 (n) - WiZ |j + nuwyZ = 0. 



* Prof. Townsend, Proc. Roy. Soc. A. vol. lxxxvi. (1912). 

 t Proc. Roy. Soc. A. vol. lxxxviii. (1913). 



