REFLEX OSCILLATORS 665 



Hence, the convection current at x^ is 



^' = ^ r y'(-^i)y(^^ - xi)^~'''''~"' dx, . (hio) 



The power flow from the electron stream to the circuit is 



P = 1 f V'{Xi)it dx2 (hll) 



P = {^ [ r V'{x,)V'*{x,)y{x^ - x,)^-'^'''-''^ dx, dx2 . (hl2) 



4 K "Loo J—oo 



We have in (hl2) an expression, based on the physics of the picture, for 

 power flow from the electron stream to the circuit. This expression is a 

 product of the electron convection current, due to bunching, and the electric 

 field, and the product is integrated from x — — oo tox= +oo, which is 

 merely a way of including all the a-c fields present. We could as well have 

 integrated between two points a and b between which all a-c fields lie. 



We will now go through some strictly mathematical manipulations of 

 (hi 2), designed to transform it to a more handy form. In the following 

 steps we may regard Xi and x^ as merely two different variables of integra- 

 tion, disregarding completely their physical significance. 



Suppose in the .Vi .Tj plane, Xi is measured + to the right and .V2 , + up- 

 wards. Then the plane is divided into two portions by the line x^ = X] , 

 and we are integrating over the upper left portion. If we reverse the order 

 of integration we obtain 



p = ill [ I V'{x^)V'*{x,)y{x2 - x,)e^''^"-''^ dx^dx,. (hl3) 



Let us (a) interchange the variables of integration x-i and Xi and (b) take 

 the conjugate of P. We obtain 



P* = T^l \v'{x,)V'*{x,)y{x,-x,)e'''''-'''Ux,dx,. (hl4) 



The real part of P is ^ (P -t- P*) ; hence 



Real = ^J^ r r V'{x^)V'*{x,h{x, - x,)e'''''-''' dx, dx, . (hl5) 



O VQ J— «o J— 00 



Let us consider the quantity 



A = [ V'ix)e''''dx (hl6) 



