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BELL SYSTEM TECHNICAL JOURNAL 



where co is equal to 1,32/i?. The closeness of this approximation is 

 shown in Fig. 17. Its introduction gives 



V = Vo -\- (v2 — vo) tanh cos. 



(4) 



A similar approximation is found for the potential on the axis to 

 the left of the division plane, 



V = I'o — (vi — Vo) tanh coz, 



(5) 



and it turns out that the potential on the axis of both tubes can be 

 expressed by the single equation 



V = f [(^2 + Vi) -\- {v2 — Vi) tanh wz]. 



(6) 



3 0.6 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 



UJZ 



Fig. 17 — An approximation for the exponential series. 



With the potential on the axis expressed in terms of a known 

 function of z, various series methods may be used for locating the 

 principal planes and calculating the principal focal distances. They 

 are, however, complicated and it may be preferable to use the approxi- 

 mate lens equation obtained by treating the structure as a thin lens. 



When treated in this manner, the expression 33 for the inverse focal 

 term can be exactly integrated, and the lens equation is 



>V2 



^1 3(-yjv2 -\- -yfvi) 



(Vz^ — V^'l)^ 



= \,32IR. (7) 



Division by either ^2v2 or V2i^— as desired — reduces this equation 

 to one that involves the voltages only in the form of a ratio. The 

 error in a focal distance d calculated from this equation is of the 

 order of R, when the focal distance is measured from the division 

 plane of the tubes. 



