It was earlier established that the lens field between the cathode and crossover can be represented by a simple functionV=Asinhkz, so that the main properties of the paraxial ``crossover'' (or minimum section of the beam) can be determined as a first approximation in terms of ``A'' and ``k.'' The distance of the crossover works out at 8&pgr;/3k√3; its potential isAexp(8&pgr;/3√3); and the angle of the beam is simply proportional to ``k.'' The radius of the crossover is then 2/kexp[−(2&pgr;/3√3)]·tan&thgr; for electrons of initial velocity ``A,'' where &thgr; is the angle of emission within the paraxial limit. The differential analyzer was also used to obtain the correct solution for the trajectory equation from the field plots for different systems, and shows that the simple theory gives an adequate approximation for design purposes.