ELECTROSTATIC ELECTRON-OPTICS 7 



Solutions for the velocity functions are obtained by expressing them 

 as power series in r. 



u = Ar + Br^ + Cr^ -\- ■•-, (12) 



w = a -\- br^ -{- cr* -i- " • , (13) 



where the coefficients are functions of z alone. The above powers of 

 r are the ones required in a system symmetrical about the z-axis. 

 In such a system r reverses in sign with r and the w-series is odd ; 

 z does not reverse sign with r and the w-series is even. Aside from 

 such reasoning, the choice of the two series is justified provided they 

 lead to solutions of the differential equation in a form suitable for 

 the purposes of electron-optics. 



The potential $ obeys the equation 



_ a2<i> a2$ 1 a$ _ 



dz^ dr^ r dr ^ ^ 



and it may likewise be expressed as a power series in r. This well 



known series is 



v" v"" 



^ = ^-Y^r^+-(2^.''--'^ (15) 



where v is the potential on the axis of the system, and the primes 

 indicate differentiation with respect to z. 



On substituting the three series in equations 7 and 8 and equating 

 the coefficients of the various powers of r in each equation we obtain 

 a series of ordinary differential equations for the coefficients of the 

 w-series. 



^vA' + A' - -~y (16) 



^|YvB' + 4AB='^-^ 

 lo 2 



(17) 



^vC -h6AC= -^-j^- SB' - 3/4A'B', (18) 



and the coefficients of the w-series are 



a = V2z;, 



b = A' 12, (19) 



c = B'lA. 



