84 BELL SYSTEM TECHNICAL JOURNAL 



By equating corresponding terms, we find 



UaQ = Ubo + constant, 



Ual = Ubl, 



Ua2 = Ub2, etc. 



(40) 



The first of these equations is trivial when the boundary conditions 

 are inserted, for then it appears that Uao = — Ubo and the equation 

 merely states that at a given value of x the direct-current velocity 

 component is not a function of time. 



The second equation is much more enlightening and tells us that 

 although two values of the direct-current velocity may be present, 

 nevertheless there is only a single value for the alternating-current 

 component. The same conclusion holds for the higher order velocity 

 components. This conclusion supplies the key for the solution of the 

 general equations when applied to the stream of electrons moving in 

 both directions between the grid and plate of the tube. 



In general, the total current may be written 



1 f)F 



J = PMa + PbUb+^ ^' (41) 



47r at 



If 2 is the total area of each of the electrode planes and 



^ = a + b, 



where a and b are constants to be defined later, (41) may be written 

 as follows: 



In this expression, the two streams of current are clearly separated if 

 a and b are taken so that '^ 



Pa- = P and Pbj = P, (43) 



a 



where P is the total charge density, equal to the sum of P„ and Pb- 



The total current may now be expressed in terms of velocities, 

 only, giving similarly to the transition from (1) to (3), 



47r - 72 = a ( t/„|- + -^- )' U„ + b I Ub f + |: \Ub. (44) 



m \ dx dt / \ dx at / 



I 



" A more rigorous analysis, involving nu-an values of the motions of individual 

 electrons, leads to the same result. 



