loith Electron Currents in Different Gases. 433 



IE we confine oar attention at present to the discharge 

 between two parallel plane electrodes, (2) reduces to 



y=^p ( 2> ) 



Eliminating p and v from (1), (3), and (2'), 



aV ~3* cu- 2 ' ' • ' / • • ^> 



where 



8tt /2^\ 



fl= lV7 l < 5 > 



A first integral of (4) is 



fcVM+O-g^ (6) 



dV 

 If the currents are a long way from saturation, -=— =0, 



when V and z = Q; so that C = 0, and dx 



(2a)**+C=2V* (7) 



Since V = when #=0, C' = 0, and if V, (V^Vj) is the 

 potential difference between the electrodes, whose distance 

 apart is /, 



2a =( 2 f) W 



Thus 



l =\/kh^> ■ ■ ■ ■ ^ 



and the current varies as the potential difference raised to the 

 power 1*5. 



So far we have only considered parallel plane electrodes. 

 Other cases lead to differential equations which, up to the 

 present, have proved intractable. The relation i oo Y 3/2 can, 

 however, be shown to be independent of the shape, size, 

 and relative position of the electrodes by a general argument. 

 In general, the relations between v, V, /?, and the coordinates 

 are governed by equations (1), (2), and (3). Consider any, 

 the same, geometrical system under two different potential 

 differences. Let v, V, and p be the variables at any given 

 point in the one case, and v', V, and p' in the other. Then 

 the dashed and undashed variables respectively will sepa- 

 rately satisfy equations (1), (2), and (3). If possible let 



Phil. Mag. S. 6. Vol. 32. No. 190. Oct. 1916. 2 G 



