of Light at Certain Metal-Liquid Surfaces. 91 



In the case of a metal this must be supposed complex and the 

 real part of VK is the index of refraction. If we substitute for % 

 in (1) from (2), replace the exponential by its equivalent trigo- 

 nometrical expression, and rationalize the denominator of the 

 left-hand side of (1), we obtain 



cos2i/< (14 ^sinAtg2i/') sin<£tg<£ 



1 — cos A sin 2\p VK— sin 3 <£ 



This may be simplified by making the following substitutions: 



sin A tg 2i[/ = tg Q ; cos A sin 2ifr = cos P ; 



(3) 



P 



cos 2t[/ = cos Q sin P; S = tg — sin cj> tg<£; 



which yield 



S 

 ofQ. — — (4) 



V K — sin 2 t£ 



In the case of the reflection in a transparent medium of 

 index of refraction n„ we have in place of (2) the relation 



sin <f> , 



», -^ = V K . 

 sin x 



Substituting from this in (1) and performing the same opera- 

 tions and making the same substitutions as before ; we get 



eiQo _ — . (5) 



VK-n,' sin 2 <j> V ' 



Dividing (4) by (5) and substituting for the radical in the 

 denominator its value from (4) we get 



n -£- &*-*>? - ^- V K-n/ sin 2 <f> . 



Using in this the value of K from (4), 



T7- ^ • 3 



K = — -=- + sin' (f>, 



we obtain 



n A.««Q-Q.) = /i- s _i^ (< _ 1)e «Q. 



