62 BELL SYSTEM TECHNICAL JOURNAL 



and, in fact, the means of extending his theory exists primarily in the 

 selection of much more general boundary conditions than were as- 

 sumed by him. It will, therefore, result that some repetition of 

 Benham's work will appear in the following pages. However, in view 

 of the new state of the theory and the importance of accurate founda- 

 tions for it, this repetition is advantageous rather than otherwise. 



With these preliminary remarks in mind, the next step is the selec- 

 tion of the idealized starting point for a mathematical analysis. Ex- 

 actly as was done by Benham we take two parallel planes of infinite 

 extent, one of which is held at a positive potential V with respect to 

 the other, and between the two electrons are free to move under the 

 influence of the existing fields. The next step in the idealization con- 

 stitutes the separation of alternating- and direct-current components 

 not only of current and potential, but also of electron velocity, charge 

 density, and electric intensity. With this separation, the restriction 

 that the direct-current component of the electron velocity and acceler- 

 ation is zero at the negative plane may be made while leaving us free 

 to select much more general boundary conditions for the alternating- 

 current component. It is true that the more general conditions now 

 proposed will not fit the original physical picture where the negative 

 plane consists of a thermionic emitter. Nevertheless the extension is 

 of importance since it allows application to be made to the wide 

 number of physical cases where "virtual cathodes" are formed. One 

 such example is the convergence of electrons toward a plate maintained 

 at cathode potential while a grid operating at a high positive potential 

 with respect to both is interposed between them. In a stricter mathe- 

 matical sense, the broader boundary conditions come about because 

 of the fact that the general equations containing all components are 

 separable into a system of equations, one for each component, and 

 that the boundary conditions for the different equations of the system 

 are independent of each other. 



The concept of an alternating-current velocity component requires 

 a few words of explanation. In the absence of all alternating-current 

 components, electrons leave the cathode with zero velocity and acceler- 

 ation and move across to the anode with constantly increasing velocity 

 under the well-known classical laws. This velocity constitutes the 

 direct-current velocity component. When the alternating-current 

 components are introduced, there will be a fluctuation in velocity 

 superposed on the direct-current value, and the alternating-current 

 component need not be zero at a virtual cathode. This separation of 

 components will come about naturally in the course of the mathe- 

 matical analysis which follows, but since the interpretation of the 



