462 BELL SYSTEM TECHNICAL JOURNAL 



agreement could be obtained if the charge density over the black 

 squares was assumed to be uniform and equal to p + ju while over the 

 white squares it was assumed to be uniform and equal to p — 7i. In 

 terms of the previous notation, /3 = + 1 for all points of the black 

 squares while /3 = — 1 for all points of the white squares. 



If we use the coordinates indicated in Fig. IIA, such a charge 

 distribution can be represented by the following double F"ourier series 



, 16mv- (- 1)('V-i)/2 ttNx^ (- 1)(^'-i)/2 tKv 

 c= p + -^Z.v ^ cos-^Za- ^ cos-^, (/4) 



in which N takes on all values, 1, 3, 5, 7, etc., and for each TV, K takes 

 on all the values 1, 3, 5, 7, etc. If such a charge distribution is located 

 at a distance / above the surface while its image is located at a distance 

 / below the surface, then the potential energy of an electron due to 

 this double layer is given by 



PJe = — 300 X 47rp/ L-vLa ^^ 



/-ir(N^+K^y^ \ Ntx Ktv 

 X exp I ■ 7 2 1 cos — T— cos — j-^ • ( / 5) 



This formula is accurate provided //6 <C 1, which is always fulfilled in 

 any case in which one is likely to be interested. The electric field 

 normal to the surface due to the double layer is 



1 dP„ _ 300 X 64^/ (- iyN+K),2^^2 ^ x^y 



e dz b ^^'■^^ 7VX 



X exp I 7 z 1 cos —J- X cos —r- y. (76) 



Equations (74), (75) and (76) reduce to the corresponding equations 

 for the hill and valley distribution if IGjI/w^ is replaced by n and if only 

 the first term of the double series is used, i.e., if iV = 1 and K = 1. 

 See equations (70) and (71). 



Recently Mr. Albert Rose working with Professor L. P. Smith at 

 Cornell University has made calculations for a checkerboard with uni- 

 form charge distribution and has compared his computations with 

 experiment. The agreement is as good as we have found and his 

 computed values of b and m are about the same as ours. 



Linford '^ in an excellent review on the external photoelectric effect 

 has shown that a checkerboard distribution of charge or potential 

 satisfactorily accounts for a number of photoelectric phenomena 

 observed with composite surfaces. His equation (42) is almost 



