78 BELL SYSTEM TECHNICAL JOURNAL 



positive value of the constant in equation (7). They differ in nu- 

 merical values in that the current / must be replaced by the value 2/ 

 to allow for the reflected current. The correct equation is then 



(T = [(1 - 2/3i'2)(l + /3i/2)i/2 - (^1/2 _ 2i3i/2)(,^i/2 + ^i/2)i/2]2-i/2. (39) 

 The corresponding transit times are given by 



r = [(1 + /31/2)l/2 - (^1/2 + ^l/2)l/2]2-l/2. (40) 



To the right of ^ = the space is free of charge so that the potential 

 gradient is constant and equal to the value at (p = Q obtained by taking 

 the derivative of equation (39). This value is 



^^_4V2^_ (41) 



d(T 3 



Concerning Completeness 



We may now review our work and see that no possible space charge 

 distributions can have been omitted. Starting from the fundamental 

 equation (6) we obtain equation (7) with an undetermined integration 

 constant. Setting this constant equal to zero we could integrate once 

 more, obtaining a solution formally identical with Child's equation. 

 If we supposed that the cathode plane defined by the Child's solution 

 lay to the right of the initial plane, then the only freedom left in the 

 solution was represented by Z, the fraction of current passing through 

 the plane. All physically sensible values of Z, i.e., to 1, are included 

 in the solutions. If the cathode is assumed to lie to the left of the 

 plane, then Z must equal 1 and the solution which arises is given by 

 a = in equations (23) or (26), or jS = in equation (35) — that is, 

 a D solution. For a negative value of the constant, a further inte- 

 gration gave only two possibilities. Each of these was investigated 

 for all possible values of the constant. A similar statement is true 

 for positive values of the constant. 



As was stated in the text, space charge distributions corresponding 

 to injection from both bounding planes can be handled in formally 

 the same way as injection from one plane ; therefore we may conclude 

 that all solutions to the problem given by specifying the boundary 

 conditions on two planes, subject to the assumptions represented by 

 equation (6), have been determined. 



