304 BELL SYSTEM TECHNICAL JOURNAL 



assumed small compared to their spacing was discussed in detail only for the 

 case of large distances between the grating and each of the conducting planes. 

 Figure 1 shows the assumed geometry of the grid, anode, and cathode. 

 End effects are neglected. The origin is taken at the center of one of the 

 grid wires which have radius c, and the X-axis is along the grid plane. The 

 spacing of the wires between centers is a, the distance from grid to anode is 

 do, and that from grid to cathode is di. No restrictions are placed on the 

 sizes of a, d^, and di. Above the anode and below the cathode is shown a 

 doubly infinite set of images which may be inserted to replace the conducting 

 planes of the anode and cathode. By symmetry the potential from the 



-t-qo o o o 

 -q o o o o 



d2 fy 



r^*l ' G G G X 



>IU-2C d| 



-q o o o o 



hq o 



-q o o o o 

 Fig. 1 — Array of images for production of equipotential surfaces in planar triode. 



array of charges there shown must be constant for all x when y = d^ and 

 also for all x when y = —di. The double periodicity of the array suggests 

 immediately an application of elliptic functions. The solution of the sym- 

 metrical case was actually stated in terms of the elliptic function sn z by 

 F. Noether". The extension to the non-symmetrical case shown in Fig. 2 

 is fairly obvious. One of the authors worked out such a solution in terms 

 of Jacobi's Theta functions in 1935, but abandoned any plans for publishing 

 his analysis in view of the excellent treatment appearing shortly after that 

 time in the Proceedings of the Royal Society by Rosenhead and Daymond®, 

 who appUed Theta functions to both tetrodes and triodes, and both cylindri- 

 cal and planar tube structures for the case of fine grid wires. Some of their 

 formulas were later included in a book by Strutt''. Methods of calculating 



