306 BELL SYSTEM BECHNICAL JOURNAL 



Fig. 1, the planar triode with fine grid wires. The potential of the cathode 

 is set equal to zero. Then in the space between anode and cathode, 



AV{x, y) = [2t d^{y - d^)/a + {d, + d,)f{x, y)]V, 



+ [B{y + ^0 - 27r d,y/a - d,f{x, y)]Vj, , 



(1) 



where 



f{x, y) = fn 



di [7r(a; -\- iy — 2i d2)/a] 



(2) 



t?i [ir{x + iy)/a\ 

 A = (di-\- d,)B - 27r dl/a (3) 



adiiliri di/a) 



B = (n 



TCt?i'(0) 



(4) 



Here we have used Jacobi's notation for the ??i-function, as explained by 

 Whittaker and Watson , rather than the Tannery-Molk notation used by 

 Rosenhead and Daymond. We write i?i(7rz) for their di{z). In our notation 



t?i(2) = 2 f; (_)V^"+i'==^*' sin {In + 1)2 (5) 



where the parameter r in the above formulas is given by: 



r = 2i{di + d2)/a (6) 



By t?i(z) is meant the derivative with respect to z: 



t?i'(s) = 2 E {-nin + l).«"+^/«*' cos (2« + 1) z (7) 



n— 



Verification of the solution is straightforward. The resulting V{x, y) is 

 seen to be the real part of a function which is analytic in the complex variable 

 X + iy except for logarithmic singularities at the points where the Theta 

 functions vanish. Hence V{x, y) satisfies Laplace's equation in two dimen- 

 sions in the region excluding the singular points. Since the zeros of t?i(z) 

 occur at z = mir + wttt, where m and n take on all positive and negative 

 values as well as zero, the singular points of the solution are at 



X -\- iy ■= ma + 2in{d\ -}- d^ — 2idi 1 



(8) 

 and X -\- iy = ma + 2in{dx + c?o) J 



which coincide with the centers of the image circles of Fig. 1. The logarith- 

 mic singularities represent fine charges with the first set arising from a i?i- 

 function in the numerator, yielding a positive charge, and the second set 

 from the «?i-function in the denominator giving a negative sign. The 

 equipotential curves are approximately circular in the neighborhood of the 



