304 



s—1 = 4jtP= -^ . . (14) 



e 4-2 



JLl__^_ ^(23 4- 2) as <Js 



«0—1 1 



4:Tr 

 where g = ~-- h~^ and denotes as before the proportion of the total 



space occupied by the material of dielectric constant f,. 



So far we have considered the tiehl only in the direction of one 

 of the axes of the lattice. If the lattice is rectangular and not cubical 

 the quantities ^, may be different for the three principal directions. 

 In the case of a cubical lattice however they are the same. Since 

 all the relations of the problem are linear the effective dielectric 

 constant in a cubical lattice is independent of the direction of the 

 field and may be thus justly compared with the effective dielectric 

 constant of a powder. 



If the first a[)proximation (13') is substituted for Ug into (14) the 

 approximate formula 



e - 1 = — -^ .... (14') 



•^ q- 2{2s-^'iy^,ös 



1 



is obtained. In the summation of this formula the density of packing 

 enters through the quantities ög and the intensity of polarization 

 comes in through {^g- The quantity e, — 1 occurs in these to the 

 first power. If the more accurate formula (13) were used higher 

 powers of s^ — 1 would come in. Hence if the density of packing is 

 kept constant and if deviations from the simple formula (1") just 

 become apparent due to an increase in f„ formula (14') is the proper 

 one to use. On using (11) it may be simplified to 



a - 1 = -i ^^- (14") 



s, + 2 =" 2s4- 1 -2 /eaNa"*"^ 



where Og is the value of Os for q = — which is the maximum possi- 

 ble q for the lattice. The quantities Og are rapidly diminishing as s 

 increases. Thus we find 



Writing 



3 (4ai)' = 0.0646, -j^^ (6a,)' = 0.00082. 



4s 



«-f47T^(^-^2,...(--^) .... (15) 



