422 Scattering of Light by Small Dielectric Spheres. 



and the resultant electric force clue to the infinitesimal layer 

 is equal to 



(e s )r=^dz^2 ^r I sm-~ (ct-r) dr. 2tt 



or (e s)r = ^<fe__co 8 y (<*-*),. 

 while that of the original beam is equal to 



6! = ! .sm— (ct — z) 

 and the sum is equal to 



<? = *?i + (<?,),. = sin — (c£ — c) — N dz j— ? — ^- cos — (<tf — 2) 



... (7) 



27T 



or ^ = sin — - (ct —z — 8) 



2/7T ^7T ^7rS 



or e = sm-- r (ct — z) — cos— (ct — z)~-~ , (8) 



because the path difference 8 is a small quantity. Com- 

 paring (7) and (8) we obtain 



2tt8 Ic-ISttY 



— aJSlfc.j-j-j— - 



^H^t (*> 



^ On the other hand, Huygens' theory gives for the path 

 difference 



S=(7-l)«fc, 



hence y _i = N ^J-=l . . . (10) 



or 



v2N2 (UT=^- i ) 2 l- 



If we substitute this expression in (6) and assume Ej = l 

 we obtain 



an important relation first established by Kayleigh. 



