68 BELL SYSTEM TECHNICAL JOURNAL 



rises, finally reaching the limiting values indicated on the upper curve 

 of Fig. 11; the limiting conditions there are precisely the same as for 

 curve a of Fig. 10. Further increase of injected current produces a 

 transition to type B solutions. 



It seems questionable that this overlap type C condition can exist 

 in a practical case. An investigation of this question involving ex- 

 ternal circuit considerations is beyond the scope of this paper. 



Transition Between Distribution Types 



The physical choice between the different possible potential dis- 

 tributions which may exist with a given set of boundary conditions is 

 determined by the sequence in which the boundary conditions are 

 established. Extreme values of any parameter are seen from Figs. 7 

 and 8 to lie in regions for which only one solution is possible. If the 

 boundary conditions are varied slowly and continuously from these 

 values, the indicated type of distribution will persist until the limit of 

 this region is reached at which time a sudden transition must occur 

 to another indicated type of distribution. Inspection of Figs. 7 and 8 

 will show that at such transitions only one other type of distribution 

 is ever possible. The determination of the correct physical distribu- 

 tion can thus be made without ambiguity. 



Certain peculiarities are, however, to be noted. A survey of all 

 possible transitions in which 7 and v' are treated as independent 

 variables will indicate that, starting from extreme conditions and 

 changing conditions continuously in the same direction, distributions 

 of the overlap C type shown in Figs. 4 and 1 1 never occur. A second 

 peculiarity has to do with the unstable region of type B solutions 

 shown on Fig, 9. When this region is entered with insufficient 

 resistance in the external circuit, instability results with a sudden 

 transition to a corresponding stable type of distribution. 



The space model shown in Fig. 12 has been found to be of value 

 in visualizing problems involving transitions. The three coordinates 

 used in its construction are the second electrode potential <p (to the 

 right) the injected current 7 (to the left) and the transmitted current 

 Z7 (vertical) . Solutions corresponding to potential distributions of the 

 C and D type, for which Z7 = 7, appear as a celluloid plane inclined at 

 45 degrees, on which the values of the potential minima are indicated. 

 Solutions of the C overlap type have been omitted. Solutions of the B 

 type are represented by the concave surface of the model. Viewing the 

 model from a different angle as shown in Fig. 13 (where the celluloid 

 sheet is removed) the unstable B region appears as an over-hanging 

 cliff extending between 7 values of 0.5 and 4. Bounding curve c 



