DIELECTRIC PROPERTIES OF INSULATING MATERIALS 533 



In this equation the following abbreviations are used: r' = rjf, 

 Aki = po, Ak = pi. The last two abbreviations are introduced to 

 facilitate comparison with an analogous derivation given in "Polar 

 Molecules," p. 86. D — DoC"'^ is a solution of the homogeneous differ- 

 ential equation obtained by letting Ei = 0, provided that the following 

 equation is satisfied : 



, , r / 1 - ^0 \ 1 I (1 + 2^0 + 2p{) 1 ] 



As we are interested here only in the special case where CR <3C r', the 

 1/t' term in the coefficient of a can be neglected in comparison with the 

 l/CR term, and the roots are 



I - po - pi\ i / 1 - po\ 1 



' or a = — ' ' 



^ - po / r \\ + 2po) CR 



to a degree of approximation which improves the larger the ratio 

 TJCR. The general solution of the non-homogeneous equation (6) is 



r> = z),r(So)^ + z),e-(^w)^'+ /A-+^^^±i^\£, = 0. (8) 



\ 1 — po — pi I 



For the special case of a random or cubic distribution of molecules 

 (in which case A = 1/3), Eq. (8) can be simplified by means of the 

 following relationships: 



(1 + 2p,)Kl - Po) = 6., (9) 



(1 + 2p, + 2pi)/{l - Po- pi) = 60, (9') 



(e. + 2)/(6o + 2) = (1 - />o - ^i)/(l - ^o), (10) 



Equation (8) then becomes 



D = Djg-'/'xCfi + D^e-^'^ + €o£i (11) 



and introducing the initial conditions, / = 0, D = 0; / = 0, —r 



» at 



= Ei/CR, we obtain, 



D = €o£i - 6.£ig~'/^-^« - (eo - 6^)£le-'/^ (12) 



