64 BELL SYSTEM TECHNICAL JOURNAL 



of X. This comes about because of the plane shape and parallel dis- 

 position of the electrodes, and the fact that current always flows in 

 closed paths. Thus, while the current between the two planes may 

 be a function of time, it is not a function of x. 



The separation of alternating- and direct-current components may 

 now be made. We write 



/ = /o + /l + /2 + • • • (4) 



with corresponding 



^ = Z7o + t/i + Z7, + • • • , I 

 V = Vo + Fi + F2 + • • • , I 



where the quantities with the zero subscript are dependent on x, only, 

 those with subscript 1 are dependent to first order of small quantities upon 

 time, those with subscript 2 are dependent to second order, and so forth. 

 As a result of this separation in accord with the order of dependents upon 

 time, (3) may be split up into a system of equations, the first of which 

 expresses the relation between Uo, Jo, and x and does not Involve time. 

 This is the relation governing the direct-current components. The 

 second equation of the system involves the relation between Ui, Ji, x, 

 and time, and contains Z7o which was determined by the first equation. 

 Likewise, the third equation contains U2, U\, J2, x, and t. Since the 

 series given by (4) and (5) are convergent so that, in general, the terms 

 with higher order subscripts are smaller than those with lower sub- 

 scripts, we may consider that, at least for small values of alternating- 

 current components, the total fundamental frequency component is 

 given by the terms with unity subscript. 



The first two equations of the system are as follows: 



l/.f(c/„^) = 4.i/., (6) 



dx \ OX J m 



d , jj d \/dUi . ,, dUi. ., dUo 



+ <.(^o^^«) = 4.^... (7) 



In the solution of (6), the boundary conditions are restricted so 

 that when x is zero, the velocity and acceleration both are zero. These 

 restrictions mean that initial velocities are neglected, and that com- 

 plete space charge is assumed. Thus the solution for Uo is 



Uo = ax"\ (8) 



