Discharge in a Transverse Magnetic Field. 63 



But i = i' + i = i' + ne(q + q') ; 



i' ? + ?V 



IV >TA/ X 



Vo-^V=/;£?X\ 



*o z + ne{q + q ) 



As, moreover, during this stage V is small compared 

 with V , and i' should be small compared with i , we get the 

 simple equation 



P 



q' + q 



V 



or — = const, nearly. 

 P J 



q + q' 



This, as we have seen, is the case in air (curve VII.) ; 

 when, however, the pressure is sufficiently reduced, a, y are 

 no longer zero. In fact, the terms in E become sufficiently 

 effective in making V large, as is found to be the case 

 (art. 24), since a, 7, k, k' are all proportional to pressure, 

 and it is reasonable to suppose «<7, remembering that 



1 i 1 



a cc — , and 7 cc - . 



A, A 



26. Although it is not possible to work out completely the 

 theory of this variation of pressure without a knowledge of 

 a, 7, Jc, k', we may get some insight into its nature in 

 special cases by proceeding as follows : — 



From (8), we have 



(a^,q/ + yn 1 q 1 )d«-» s =(*N 2 q 2 ' + yn 2 q 2 ) . . (9) 

 if N = N 2 ', n = n 2 at the anode, ' q = qi and q = q'/q 2 ' ; 

 but e(N 2 q 2 '-\-n 2 q 2 )=i = e(N 1 q l / +n 1 q 1 ). . . (10) 



If, now, n 2 = 0j 



a— 7 a — 7 



We have also 



i V -i , Y=:Xe'N 1 q l '[\'-ocP], since n 2 = 0; 



and since - = small, 



h 



we get X as an exponential function of p. 



It is obvious, however, that the above investigation is not 

 capable of giving a complete account of the variation of the 

 potential difference, for we have assumed (1) that the 

 potential varies uniformly from cathode to anode, and (2) that 

 a, 7 are constant. As neither of these suppositions can be 

 true always, it is not surprising that the curves obtained are 

 more complicated than those given by theory. 



