664 BELL SYSTEM TECHNICAL JOURNAL 



i\ . Now if /i is the time a particle passes Xi and h the time the particle 

 passes .V2 , the convection current at .vj will be 



h = (-/o +h)~ + /o. (hi) 



a/2 



This merely states that the charge which passes .\\ in the time interval dti 

 will pass .r2 in the time interval dt^ . Suppose the electrons are accelerated 

 at .Ti by a voltage 



If Vo is the voltage specifying the average speed of the electrons, the veloc- 

 ity will be 



V = (2,,Fo)'(l + {V/Vo)e"'''f (h2) 



We then have 



/2 = /i + t(1 + (F/Fo)e''"'0"* 



(h3) 



r = x{2-nV,r 



dh 2(1 + (F/Foy-'O" ^ ^ 



Now assume (F/Fo) <3C 1. If we neglect higher powers than the first 

 we can replace /i by 



h = ti — T 



and obtain 



and from (hi), neglecting products of two a-c quantities 



i, = i,-J^Ve^-'^^-\ (h6) 



Suppose we consider an electron stream travelling through a longitudinal 

 field of potential fluctuating as <?"'" and of magnitude F(.v), where V{x) 

 may be complex. Then the current at .T2 due to the action of the field at 

 Xi is, omitting for convenience the factor e^" , 



dn = •' — — - — 7(.T2 - Xi)e dxi (h7) 



7 = co/wo (h8) 



uo = (2r,Fo)*. (h9) 



