584 BELL SYSTEM TECHNICAL JOURNAL 



one oscillation. But for more than the order-of-magnitude we should 

 certainly not rely on it! 



Since allowance should be made for the collisions of the ions with 

 molecules of the gas, let us do what is customary in physics: rush to 

 the extreme, exaggerate the effect instead of neglecting it, and suppose 

 that collisions are so frequent that their net influence upon a wandering 

 ion amounts to a force of the type called "viscous" or "frictional" — 

 a force proportional and oppositely directed to the velocity of the 

 particle. Into the equation of motion we introduce a new term, so 

 that it takes the form: 



m{d^x/dt'^) + g{dx/dt) = eE = eEo sin nt (5) 



g standing for the new "coefficient of friction," and n being written 

 for lirv so as to avoid constant repetition of the factor 2x. One 

 integration is sufficient to bring out the important point; we obtain: 



dx _ eg „ me dE . . 



dt m^n^ + g^ m^n^ -\- ^ dt 



If now we multiply each member of the equation by Ne — N standing 

 for the number of ions per unit volume — each becomes equal to the 

 current-density borne by ions in the gas. 



Notice now an important thing about this expression for current- 

 density. It consists of two terms, one of which is proportional to the 

 fieldstrength E, while the other is proportional to the time-derivative 

 of the fieldstrength — two terms, therefore, of which one is in phase 

 with E and the other in quadrature with E. The first we may set 

 down in the following equation : 



Ne^g 

 rn^n^ + g^ 



and we may apply the name "conductivity" to the coefficient a. 

 The second must be treated differently. By itself it is not the whole 

 of the current-density in quadrature with the fieldstrength; there is 

 the displacement-current also to be taken into account, equal to 

 {l/4:ir){dEldt). Then we must write : 

 Current-density in quadrature with fieldstrength 



47r V m}n^ + g2 / dt 47r dt ^^ 



and we may apply the name "dielectric constant" to the coefficient e. 



Say now that we can ionize a gas abundantly, as for instance by 



maintaining an intense direct-current discharge across it between 



auxiliary electrodes; and that we can apply a small oscillating voltage 



Current-density in phase with fieldstrength = 2 2 f — 2 -^ ~ ^-^ (^) 



