6 BELL SYSTEM TECHNICAL JOURNAL 



at points off its path. The equations may, however, be reduced to a 

 more convenient form by the introduction of two velocity functions ^ 

 defined as follows. 



Let u and w be any two functions of r and z that satisfy the equations 



f^-5^=0, (7) 



dz dr 



u^ -\-w^ = 2$. (8) 



Consider now an imaginary point moving with velocity components 



f = u, z = w. (9) 



The derivative of f with respect to time is 



.. du . , du . du , du ,._. 



r=^-r-\- — z=—u-\- — w (10) 



dr dz dr dz 



and from equations 7 and 8 



du , dw d^ /,.. 



r=— i^+-r-w=— (11) 



dr dr dr 



the component f thus satisfies differential equation (4) for electron 

 motion. In a similar manner it may be shown that the velocity 

 component z satisfies equation (5) . The motion of the imaginary point 

 is thus the same as the motion of an electron, and the velocity functions 

 u and w are therefore the velocity components of electron motion. 



The velocity functions are solutions of equations 7 and 8, one of 

 which is a simple algebraic equation and the other a partial differential 

 equation with respect to space alone. The inconvenient time deriva- 

 tives have been eliminated in these new equations for electron velocity. 



The existence of a velocity function is not confined to a single 

 electron path ; it exists over the electric field in general. Any pair of 

 particular solutions for u and w thus corresponds to an infinite number 

 of possible electron paths. In the converse manner, there are an 

 infinite number of particular solutions for any electric field, and there 

 is a pair of particular solutions corresponding to any given electron 

 path through the field. ^° 



^ These functions are the components of the generalized vector function described 

 in Appendix 4. 



1" The existence of such solutions is proved by the existence of the series solutions, 

 which are derived in the following pages. 



