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BELL SYSTEM TECHNICAL JOURNAL 



and the lens equation is derived by integrating it from a point Si to 

 a point 22, where the two points are taken at the substantial boundaries 

 of the non-uniform field. In carrying out this integration, the origin 

 of time is taken at the instant that the electron is at the maximum of 

 \v"\, as illustrated in Fig. 8, and for convenience the origin of z is 



Fig. 8 — Coordinates for a thick lens. 



also taken at that point. With this choice of the origin, the term 

 tIT in the second member of the equation is small compared to unity 

 in the region where v" is large and not very important in the regions 

 where v" is small. This term may therefore be neglected in lenses 

 when the time of transit is not too great a fraction of the focal times 

 involved. The integration of the equation then gives 



1 



1 



when the inverse focal term is again 



and 



1 _ r«» v" 



r'^ dz _ r 



Jo ^l2v Jo 



^ dz_ 



2v' 



(43) 



(44) 



(45) 



A transformation to space variables by means of equations 24 and 25 

 gives the lens equation in a form analogous to that for a thick optical 

 lens, 



■^2v2 _ V2yi 

 di — a2 d\ — oi] 



1 



where 



a2 = — V2»2^2 = — I ^^—dz, 

 ax = — ■42viti = — I -yA-^dz. 



(46) 



(47) 



