ELECTROSTATIC ELECTRON-OPTICS 29 



We first consider the case of the plane preceding the tube. The 

 electric intensity at the plate is found by difTerentiating equation 4 

 with respect to z and then setting z equal to zero. A substitution of 

 this intensity in equation 41 of the text gives the lens equation of 

 the aperture. In addition to this lens there is an equivalent thin 

 lens located inside the tube at a distance .SR from the plate, and 

 having the inverse focal distance of equation 10. The system is 

 considered as a combination of the two lenses and the calculations 

 are carried through in the usual manner. When the plane follows 

 the tube, the constants of the two lenses are determined from equations 

 5 and 9, and the combination is treated in a similar manner. 



Appendix IV — The Velocity Function 

 The auxiliary functions u and w are a special case of the components 

 of a generalized vector function that is useful in developing series 

 solutions for electron motion. The equations of this function are 

 equivalent to the Hamilton -Jacobi equation ; they are briefly outlined 

 in the present system of units as follows. 



In a field that may comprise both an electric intensity E and a 

 magnetic intensity H, let v be any vector function of x, y, z that 

 satisfies the equations 



curl i; = HIc, (1) 



l/2|i;|2= + T7, (2) 



where PF is a constant equal to the energy of electron emission from 

 the source. Then u is a possible vector velocity for electron motion 

 in the field. 



If the magnetic intensity is zero, the vector function v has a po- 

 tential ^p, which may be any solution of the equation 



l/2lgrad^|2= c^-f TF (3) 



and grad \p is then a possible vector velocity for electron motion in 

 the field. 



The validity of these equations is established by transforming 

 them to the usual equations for electron acceleration. 



A List of the More Important Symbols and Equations 

 In the present theory of electron-optics, all distances along the axis 

 are measured in the direction of motion, as they are in the optics 

 of light. 



r, z — cylindrical coordinates 

 / — time 



