ELECTRON STREAMS IN A DIODE 85vS 



APPENDIX I— THE LAGRANGE SOLUTION 



Lagrange has shown that any partial differential equation of the first 

 order, linear in its derivatives, is equivalent to a group of total differential 

 equations. The Lagrange equations corresponding to (2) and (5) are 



(88) 

 (89) 



(90) 



Now we can find three independent solutions of this group. One solution is 



e£ + / Idt = ci (91) 



The first member of this solution is indicated as 5; then the other solutions 



are 



U -{- 



rjSt 



-1 jj Idt = C2 (92) 



-^'-f +!['//^*-///H=^' 



(93) 



Since each of these quantities is a constant, we may set any one of them 

 equal to an arbitrary function of another, and the resulting equation is also 

 a solution of (90). We can, however, obtain only two independent solutions 

 in this manner, and we naturally choose the two simplest combinations, that 

 is, 



U ■i-'^t -"^Ij Idt = FiiS) (94) 



Ut-I^f 



+ I [^ // ^"^^ ~ /// ^"^^1 ^ ^' ^^^ ^^^^ 



where Fi{S) and F^iS) are arbitrary functions of S. These equations contain 

 two arbitrary functions; they are solutions of the Lagrange equations (88) 

 and (89), and they therefore constitute the general and complete solution of 

 the partial differential equations (2) and (5). With the exception of a slight 



