582 BELL SYSTEM TECHNICAL JOURNAL 



I will arranpje the optics much as I have previously arranp^ed them 

 in the description of direct-current discharges: ^ first, the observations 

 on gases independently ionized and exposed to an oscillating voltage 

 which displaces the independently-formed ions; then the observations 

 on breakdown or transition, in which the oscillating voltage is raised 

 to such a value that it initiates and maintains a self-sustaining dis- 

 charge; and finally, the observations on the self-sustaining discharge 

 itself. Beforehand, though, it will be convenient to derive some 

 equations describing the presumable behavior of ions in gases exposed 

 to alternating fields. 



First, consider a charged particle free to move in a vacuum, and 

 exposed to an alternating electric field. One might expect it to 

 oscillate to and fro across a fixed point, like a simple harmonic vibrator. 

 This however occurs only in a special case. In general, the particle 

 will oscillate about a point which glides at uniform speed along the 

 direction of the field. For, let us write down the equation for the 

 acceleration of the particle, denoting by e its charge, by m its mass, 

 by £o the amplitude and by v the frequency of the field E, which last 

 we suppose directed along the axis of x: 



m{(Pxldt^) = eE = eEo sin l-rrvt. ■ (1) 



Integrating once, we obtain for the speed of the ion, 



dx/dt = :r^— (1 - cos 2TPt) 4- vo, (2) 



and integrating a second time, we obtain for its displacement from the 

 point which it occupied at / = 0: 



1 eE^ , \ , 1 e£o 



X = I ^ + t'o I / — . o o sin lirvt. (3) 



The first term on the right of equation (3) represents the steady 



drift of the centre of oscillation, the second term the oscillation to 



and fro across this centre. 



The speed of the steady drift is the coefficient of t in equation (3). 



Its value depends on that of the constant of integration I'o, which, as 



one sees from equation (2), is the speed (or rather, the .r-component 



of the velocity) of the ion at the instant t — when the electric field 



passes through zero. If this speed just happened to be equal to 



eEajlirvm, and directed in the sense opposite to that in which the 



increasing field was about to draw the ion, the particle would oscillate 



about a fixed point; the amplitude of its vibrations would amount to 



eEo/^ir'^v'^m, its maximum kinetic energy to e^Ei^l^ir'^v^m. But this is 



1 In a recent book, "Electrical Phenomena in Gases," abstracted in the July 1932 

 Bell Sys. Tech. Jour., and to which 1 refer at various points in this article. 



