the Cathode in the Electric Discharge in Air. 613 



carried by negative ions produced close to its surface, it is 

 easy to calculate what proportion of the air-molecules which 

 strike the cathode must be ionized to give a current-density 

 of 0'4j» milliamperes per sq. cm. 



If e is the charge carried by one ion, and n the number of 

 molecules striking the surface of the cathode per sq. cm., 

 then if every molecule striking the surface were ionized there 

 the current would be ne. Now n=^NG where N is the 

 number of molecules in a cubic centimetre of the gas, and G 

 the square root of the mean square of their velocities. For 



air at the ordinary temperature Gr=5xl0 4 ^. Also for a 



gas at 760 mms. pressure 2Ne=^^ coulombs. 

 Hence 



p x 5 x 10 4 

 ne ~ 760x6x2x0-115 



= 47*7 p amperes per sq. cm. 



The actual current observed is 0'\p milliamperes per sq. cm. 

 so that the fraction of the molecules striking the cathode 

 which would have to be ionized to carry the current is 



0-40 1 



47-7 x 1000 120,000 



It is therefore clear that the amount of gas present in the 

 tube is amply sufficient to account for the observed current- 

 density at the cathode. J. Stark {Physikalische Zeitschrift, 

 2 Jahrgang, No. 5) has given a formula which represents the 

 variation of the cathode fall of potential with the current- 



k 

 density. This formula is K = K n +-- - (C— #p/)i where K 



is the cathode drop, K n the " normal " drop when the cathode 

 is only partly covered by the glow, p the gas pressure, / the 

 area of the cathode, C the current, and k and x constants. 

 When the cathode is only partly covered, as in the present 

 experiments, K = K Wj , so that Stark's formula becomes C = xpf. 

 This equation becomes identical with that which my experi- 





 ments have led to, viz., t = 0'4:p on putting ~-.=l. Stark 



does not give the value of his constant x and his experiments 

 do not appear to be well adapted for its determination since 

 he was mainly concerned with the variation of the cathode 

 drop with the current-density when the cathode is en- 

 tirely covered by the glow. 



