16 BELL SYSTEM TECHNICAL JOURNAL 



lens. The differential equation for A necessarily has a particular 

 solution equal to zero at Z\. This solution is designated as F, and it 

 corresponds to an electron ray entering the lens parallel to the axis. 

 At Z2 this solution is equal to — ■\[2v2lf2, where /2 is the second principal 

 focal distance measured from Z2. The path of such a ray is illustrated 

 in Fig. 9. This particular solution obeys the same differential equation 

 as A. By subtracting the differential equation of A from that of Y 

 and making a slight transformation, we obtain 



^,„,(^_K)=-^, (49) 



and it should be noted that 



^/V2^=-^ = ^logr. (50) 



r^2v dz 



An integration from 2i to Z2 and a transformation to focal distances 

 then gives the relation 



(/2-^2Mi^^ Jl^ls (51) 



fidi \ V2 ^2 



where k2 is a constant of the lens, given by 



1/^2 = exp. r^. (52) 



By proceeding in the same manner with a particular solution X for a 

 ray leaving the lens parallel to the axis, we obtain a second relation 



(f^-ddd.^j^ jv^rj^ (53) 



fidi \viri 



where 



The differential equations of X and Y may also be subtracted and 

 integrated, and this gives a third relation 



f If - -^"^ S. (55) 



A multiplication of the first two relations 51 and 53 gives 



(/2 - ^2)(/i - d,) = -^1-^2/1/2, (56) 



