915 



introduce 7^^ e~''- as law of attraction and in the result take 6 = 0. 

 Then we can write 



U^o{e)=C{B)-\- 2:^- , 



.■ k 4jrr,jt 



where C(f) is the potential of the infinite space hoinogeneonslj' 

 filled, and thus is a constant, only dependent on e. 



If we substitute the values of U(e) in the double integral, the 

 term C{s) will consequently yield zero. In the other terni the sum 

 and the integration may be interchanged. The various integrals may 

 then be united into a single one over all rectangles. And so we 

 obtain : 



^ 2iim 2/in 



cos X cos 1/ 



JJ 



g—t^z*+y^ dx dy. 



By introducing pole-coordinates this integral may be reduced to 



— I dre~'*'' 1 COS (li sin a ) d(f> =z { I J ^(lr)e ~ ^' dr ■= — 



2nrJ J J 2l//'-f€» 



00 



and thus for 6 = 



ah __ 1 



The potential found can further easily be summed up for all the 

 layers of distance c in which the lattice may be divided by planes 

 perpendicularly to the z-axis. In a point for which <^ z <^ \c all 

 layers under the point yield 



(tc—zy , 2üxm 2.Tn 



^ -^ 2B,nn !«-'= + e-^^^^+') + «-^(-'-1-20 ^ . . . cos X cos^-^y = 



2abc a h 



{ic—zY e~'^ 2jrm 2jtn 



= — — h ^B„„, cos X cos — — y 



2abc 1— g— '« a b 



and all planes above it 



-'(c— a) 2nm 2nn 



2 Bmn Z r COS X COS — — y 



1 — e~«c a b 



so 



Uc-z)* 2 fi-lz ^ e-^i<--'^ 2nm 2nn 



V' = -\- 2 COS X cos —-y. . (1) 



2abc ^ abl 1— «-'-^ a 6 ' 



in which the sign 2' means, that we must take half of the terms 



for which 771 = or 71 = 0, whilst there is no term for 7??. = 7i^O. 



For the spherically limited lattice without space-charge we ulti- 



nately find 



