﻿Ionization by Collision, 403 



The number arriving at the negative electrode is / (I) ; so 

 that, if i is the current received by that electrode and e the 



i/ 6= (n 1+ ^W-!!o (7) 



\ «o/ a o 



6. It is immediately clear that this formula cannot be 

 completely accurate ; for, if V is less than V, there can be 

 no ionization by collision according to the theory, and con- 

 sequently f{l) for all values of Y less than V should be 

 constant and equal to ?i i -{- n pl; this condition is not fulfilled 

 by (7). A little consideration discloses the source of the 

 error. Townsend assumes that the proportion of the ions 

 passing through any given layer which have travelled freely 

 a distance sufficient to enable them to ionize at their next 

 collision is constant and independent of the position of the 

 layer. This assumption cannot be true, for, if the layer is 

 distant less than b from the positive electrode, none of the 

 ions in that layer can possibly have travelled a distance b ; 

 the frac ion is here zero, while elsewhere it is finite. 



In order to estimate rightly the number of ions passing 

 per second through the layer x which, having travelled freely 

 a distance not less than b, will make collisions and produce 

 fresh ions in that layer, we must consider, not the total 

 number of ions passing through the layer x, but the total 

 number passing through the layer (x — b). As will be seen 

 from § 4, otcLv is not the chance that an ion in the layer x 

 has travelled a distance b freely and makes a collision in 

 that layer, bat the chance that an ion in the layer x will 

 travel a distance b freely and make a collision in the layer 

 (.6*4-6); these two chances are not identical. The ions 

 passing through any given layer are not a random collection 

 in respect of the distance that they have travelled previously 

 to arriving at that layer; for some of them are known to 

 have been produced in the layer ; but the ions in any given 

 layer are a random collection in respect of the distance that 

 they will travel after leaving the layer, for this distance 

 is independent of the distance that they have travelled 

 previously. 



7. Accordingly, the number of ions passing per second 

 through the layer as which produce fresh ions by collisions 

 in it is the number passing through the layer (as — b) per 

 second multiplied by the chance that an ion in that Layer 

 travels a distance l> and then makes a collision. In place of 

 (5) we get 



f{x)=n Q p + *f(*-l) ( 8 ) 



