303 



in virtue of (6). Equating coefficients of jPj we have 



(^^-~^A<l=\ + éAlS'i-}-6AlSl-\- (10) 



, I cos' 0, — -^ 

 where use is made of the fact that Sl= 2! -— — — = 0. In 



order to obtain Af^ we are thus in need of Al , A^ , etc. For these 

 it follows from (9) that 



[(2.+ l)8,-{-2s^2\A2s+i _ ^ (2.4 2p + 2)/^2s+ 2^+2 ^ 



6.-1 "^=0.1.. {2s)!{2p+l)f '^+' 



(5 = 1,2,3,...) 

 the upper subscript being dropped for the present. Writing 

 _ g.-l _4!5±L (2« + 2p + 2) .^ 



^'- 2,4-2' "'-- A ' ^''^^-(2.4- l)/(2p + l)./| 



we have 



or 



S2p-\-is-\-2 — ^p+s 



«, =^, :^M«./>)^/.+5«/., (« = 1,2,3,...). . . (12) 



p=0 



«* = ^s (s, o) as -{- ^s ^ {s,p) <yp-^s «;,.... (12') 



Substituting for ap on the right hand side the expression which 

 follows for it from (12') and proceeding in this manner indefinitely 

 we obtain purely symbolically on changing suffixes : 



00 

 as = ^s («,0) Os-]- ^ ^s ^51 («. \) («1. O) Os-\.SuSi + 

 Sl=l 



+ ^ ^s i?*, /?5,(», »a) («1. «,) («,. O) Os^s,, s,+s,.s, + 



+ ... 



+ 2 ^S^S,-' ^S (S,S,) (S,,»,) . . («/.-I. «;>) («/>. O) (Js^s,, 5,+53 ,»^_i+5^, s^ 



Sl,,.,«5^ = l 



where 



^^i.y.^,... =<7:rövÖ2 . (13) 



If the spacing of the lattice is large in comparison with the 

 diameter of each sphere this expansion may be expected to converge 

 rapidly. As a first approximation the first term will suffice giving 



«, = (2,4-2)^,(7, (13') 



Using (10) and (11) we have for the average polarization F=A^h~^ 

 and the effective dielectric constant 6 



