48 



BELL SYSTEM TECHNICAL JOURNAL 



We easily see that 



where 



We = - Jo/(oUo 



(A-32) 

 (A-33) 



We obtain from (A-20), (A-28), (A-29) and (A-30) 

 raZ 



(2TraZ\ ( ry2irt 



,, (l-waZX r { rr 2TaZ\ 



= L 



(A-34) 



= 1 - 



W 



+ 



W^ 



.{Z-B^y {Z-B^)\ 

 Equation (A-31) is equivalent to Equation (12) of the text or to 



(A-35) 



2a 



_i_ 



■1 



II ^ N 



Fig. 6 — The diameter of the electron flow considered is 2a, and the length is /. 



Letting F = — Z and making use of the following well known relations 

 between the Bessel functions 



U{jx) = Jo{x) 



hijx) = jJxix) 

 Equation (A-34) becomes 



w 



(A-36) 



(A-37) 



W 



(Z - B,Y (Z - ^2)- 



Let the right-hand side of equation (A-37) be denoted by Fi{Z) and the 

 middle of Ft{Z). In order to find the real roots of equation (A-37) one can 

 plot F\ and F^ as functions of Z on the same chart. The abscissae of the 

 intersections of the two curves will then be the real roots. In Fig. 7, F\ 

 is plotted as a function of Z for B]. = 0.9 and B^ = 1.1. 



