ELECTRON STREAMS IN A DIODE 837 



where C2 is an arbitrary constant. [In repeated integrations with respect to 

 time, the increment dt is written only once, it being understood that dt 

 is repeated in each integration.] Now any arbitrary function Fi of 5 is a 

 constant for the particular electron under consideration, so we may replace 

 C2 by Fi{S) and write 



U = Fr{S) -}\^^ - jl I ^M U5) 



, (9) may now be written in th 



For the same electron, (9) may now be written in the form 



dx 

 dt 



and its solution is 



x = C, + F^{S)I - ; [§ - llfldtj (17) 



where Cz is an arbitrary constant that may again be replaced by an arbi- 

 trary function F2 of 5, and 



X = F, (5) + Fi {S)t - ^ [f - /// I dl'j 



(18) 



By considering one individual electron we have thus arrived at two gen- 

 eral relations (15) and (18) which, taken together, describe U and E as 

 functions of x and t. Now the reader will probably be much surprised, as 

 was the writer, to learn that these two equations when standing alone are 

 not solutions of the group of total differential equations (7), (8) and (9). 

 The solution of that group is (12), (15) and (18). In other words, the two 

 general relations are solutions of the total differential equations only in the 

 very special case of S equal to a constant. But this constant may have any 

 value, and the general relations therefore apply to all electrons in the diode 

 space. 



We are therefore practically forced to the conclusion that (15) and (18) 

 are the solution of the broader group of partial differential equations (2) 

 and (5), and this turns out to be true. This solution, which is here rewritten, 



U = F,{S) --\s^- 11 ^ ^^1 (15) 



X = F,(S) + F,{S)t - - [f - /// I dt'] 



(18) 

 S = eE + j Idi 



