914 



1 



density -j — , which with exception of a constant is equal to — 



aoc 



x' + f^-z' 

 6a he 

 We begin by calculating the potential U, which is caused by the 

 charges lying between the planes 2 = ± ^ c. Evidently U is a peri- 

 odical function of x and y with the periods a and b. So it may be 

 represented by a double series of Fourier: 



2jtm 2771 m=: 0, 1, 2 



U =z S Zmu cos X cos — — y ^ , « • 



a o n = 0, I, 2 . . . 



in which for the sake of symmetry only the cosinus appears. The 

 coefficients Zm,, are functions of z, which can be determined from 

 the equations: 



1 

 Li' = — for U :<ic AU=0 for ? > ic 



abc 



with the conditions of limit 



^==±« = na7).= (a7J. ••-- = -*'• 



Now the FouRiER-sei'ies for z ^ may be twice differentiated, so 

 that after substitution in these equations every term separately must 



1 

 fulfill the homogeneous equations, and Z^^ the equation LZ,, = — . 



abc 



From this we shall find 



•• {\c-\zW V a' b' 



2abc ' ^ ' 

 In order to determine still B,„n we can take z = and use the 

 ordinary form of coefficients: 



n b 



ab r r 2ftm 2jtn 



— Bnin = \ dx \ dy Uz=o COS X COS y 



^ J J a b 







ab 

 in which when m or n are zero we must have —. For 6^=0 we have 



o 



in which r,i- is the distance to the point (ia,kb)a.nó C)k the potential 

 of the parallelepipedum rt6c with that point as centre, homogeneously 

 filled. For the sake of convergence we shall here for a moment 



