836 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



For any individual electron, (2) and (5) may be written in the form of 

 total differential equations (7) and (8) 



where for that individual electron 



at 



and X is the coordinate of the electron. This group of total differential equa' 

 tions describes U and E only in the immediate vicinity of the one moving 

 electron, and it is therefore a restricted picture in comparison to the one 

 given by the original partial differential equations. It should, however, be 

 clearly understood that we are not seeking the solution of this group of total 

 differential equations; we are merely using them as aids for solving the 

 original equations. 



Equation (8) may be written in the form 



f^[e£+//i.] = (10) 



the bracketed term is regarded as a new variable 5, that is, 



S = eE + j I dt (11) 



and (10) says that S is an invariant for any individual electron, it remains 

 constant as the electron moves along. The solution of (10) is 



S = Ci (12) 



where Ci is any arbitrary constant. 

 Turning now to (7) it may be written in the form 



§--![-/'-] 



(13) 



and its solution for any particular electron — remembering that S is an in- 

 variant for that electron — is 



U = C2 --Isi - jj I dtl 



(14) 



