THE DIELECTRIC 117 



rise above that of CD, or we shall again be able to get a positive 

 discharge from AB. 



It is easy to work out a more general theory wHere the dielectric 

 consists of any number of layers of different thicknesses with 

 different dielectric constants and different conductivities or rates of 

 decay of strain, if we assume that Ohm's law holds that is, that 

 the rate of decay is proportional to the intensity. If D is the 

 strain and E the intensity, we have 



_dD _ _ 47rcD 



dt TT 



_4irrt 



whence D = Y> e ~^~ 



where D is the initial strain and c is the specfic conductivity of 

 the material. If then at any instant the strains are D x D 2 . . . in 

 layers having conductivities c t c t . . . dielectric constants K x K 2 . . . 

 and thicknesses d^ d z . . . at any future time t they will be 



K, &c., 

 and the potential will alter from 



, 



But this general investigation has little value, for in the first 

 place the dielectric heterogeneity does not consist in a parallel 

 arrangement of layers, each homogeneous, but much more probably 

 in an irregular granular arrangement. That it is complex was shown 

 by Hopkinson (Original Papers, ii. p. 2). He found that the poten- 

 tial of a jar charged and then insulated could not be expressed as 

 a function of the time by two exponentials only. If it could be 

 expressed by a series of the above form, certainly more than two 

 terms would be required, or the heterogeneity is more than twofold. 



In the second place, even if Ohm's law holds, we cannot assume 

 that K and c are constant for each element of the structure while 

 it is breaking down. If the breakdown is, as we have supposed, 

 electrolytic, the products of decomposition may alter the values of 

 K and c. It is even possible that they may alter the values of d 

 if the heterogeneous portions are of molecular dimensions. 



Hopkinson (loc. cit. ii. pp. 10-43) investigated the rate of 

 fall of potential of a Leyden jar, and though he could not obtain 



a mathematical expression for the rate, he found that it was not 



/\ 

 very different in some cases from , where t is the time from 



