Striated Electrical Discharge. 249 

 Solving equations (1) and (4) for n x and n 2} we have 



1 f i . k 2 dX | ,r, 



' he= ^+kAx + ^fy\' ■ ■ ■ • (5) 



1 ( i k, dX.~) //M 



n2f =vr4{x-|i^j ; ■ • • • («) 



and the substitution of these values in either of the remaining 

 equations gives 



4tTre k x + k 2 dx\ dx J 



-* XV(* 1 + ^ + ^ X 5FJl f "& x lte> ' (7) 



dy 



1 ^#2 d 2 j/ 



Writing 2?/ for X 2 and p for -/ , this becomes 



4™ k, + k 2 dx* -^ 2ye\k l + k 2 )\ l + 5* ) V 4tt^ /' W 



and this is Thomson's equation for y. 

 The boundary conditions are taken to be 



n x = 

 at the anode, and 



n 2 = 

 at the cathode. 



For, considering any small element of volume dx, termi- 

 nated in one direction by the anode, it is clear that no positive 

 ions can occur inside this element except those which have 

 actually been produced by dissociation inside it. Now positive 

 ions are produced at a rate qdx, and the average time during 

 which each ion remains in this element after its production 



doo 

 is of the same order of quantities as -j-^%- Thus the total 



number of positive ions in a length dx measured from the 

 anode is of the order of (dx) 2 , so that at the anode n 1 = 0. 



Similarly, n 2 = at the cathode. 



§ 4. With a view to simplifying the discussion of the dif- 

 ferential equation, an assumption will now be made to the 

 effect that the quantities q and a depend only on the electric 

 force at the point at which they are measured. It will be 

 seen in a subsequent section that there is very little probability 

 that this assumption is a legitimate one ; but in the same 

 section I shall endeavour to show that the limitations caused 

 by its introduction can easily be removed. 



