﻿404 Dr. Norman Campbell on 



8. This equation is an accurate expression of the theory, 

 but, in order to solve it, it is necessary to depart from the 

 theory. For if n^ is finite, neither / (a?) or /' (a?) is con- 

 tinuous ; both functions obviously have a discontinuity at 

 a? = 0, and the form of (8) shows that they will also have a 

 discontinuity at x = mb, where m is any integer. If n 1 =0, 

 but n' is finite, /' {x), but not f(x), will be discontinuous at 

 t r = 0, but both functions will again be discontinuous at all 

 values of m other than 0. Now it is possible to calculate 

 accurately according to the theory what should be the current 

 between electrodes a distance I apart in a gas at pressure p, 

 when X has a given value, but the calculations are extremely 

 complex, and the resulting formula for the relation between 

 i, I, p, X is not of a form to which Townsend's measurements 

 can be applied. We will therefore assume that /(a?) and 

 f (a 1 ) are continuous, and examine the results we obtain. 



If /(a?) is continuous it must have, in order to satisfy (8), 

 the form 



/•(.!•) = A ^--°, (9) 



a 



where /3 = cce~P b (10) 



To determine A use must be made of a known value of/ (a;) 

 for some value of x. Now between a? = and x = b,f(x) 

 must have the form 



f(a:)=n 1 + n pa: (11) 



It is clear that no value of A will give this form to f (a?) for 

 all values of x within the range, and a choice must be made. 

 If we choose x = we get 



A=n 1+ ^°, (12) 



if we choose x = b we get 



A= jn 1 + tt (p&+-H*-#\ . • . (19') 



Now, since we are concerned experimentally with the value 

 of f(x) for x = l, where / is greater than b, we are likely to 

 get the more accurate results if we fit the curve to % the 

 larger value of x. Accordingly, taking (12') we get 



ile=f{l)=e^(n l + v ( r b + ^)\e-^--°. . (13) 

 L \ .«o/J «o 



It is advisable to transform (13) slightly in order to apply it 



