AND TRANSMITTED BY THIN PLATES. 105 



sin 2 - cos 2 y u? sin 2 | sin 2 y 





1 - "2v~ cos a + 1 <4> ~ 1 - 2iv- cos a + ic 1 ' 



Let the ratio of the corresponding amplitudes be denoted by 

 tan y' ; then 



w Ifl - 2c 2 



- / -- 



*v\i - 1 



- c 2 cos a + 

 tan y = tan y - 



' 



The angle 7' will be the azimuth of the plane of polarization, when 

 the reflected light is plane-polarized. 



When the thickness of the plate is that corresponding to the 



maximum difference of phase, or cos a = ^ - - , the ratio of the 



1 + V 

 amplitudes becomes 



tc 



tan 7 = tan T,- 



The angles A and 7' being known, the character of the elliptic 

 polarization is completely determined. 



12. We may now proceed to examine the intensity and phase 

 of the transmitted light. 



It will be easily seen, by following the same reasoning as 

 before, that the transmitted light consists of an indefinite number 

 of portions, which emerge at the second surface of the plate after 

 0, 2, 4, &c., internal reflexions ; and that the amplitudes of their 

 vibrations form a series in geometric progression, whose first term 

 is wV 2 , and whose common ratio is - w 2 , in which u and u 2 denote, 

 as before, the ratios of the amplitudes of the reflected to the 

 incident vibrations, at the first and second surfaces of the plate 

 respectively, and u' and w' 2 the corresponding ratios for the re- 

 fracted vibrations. It is likewise evident that, if $ denote the 

 phase of the first portion, $ - a will be that of the second, <J> - 2u 

 that of the third, &e., in which 



4;r 



a = - T cos u , 

 A 



as before. Consequently the resulting vibration, which is the 

 sum of all these, is 



liu't !sin <p - mtn sin (/> - a) + u'u-i sin (^ - 2a) - &c.j 

 sin d> + HH-, sin (^ + a) 



" 



M X 2 



1 + Ktftfc cos a 



