INTERFERENCE OF POLARIZED LIGHT. 223 



sin a. Each of these pencils will be again subdivided into 

 two by the analyzing prism, which we shall designate by 

 the symbols 4 0', + E', E + O x , E + E x ; and it is readily 

 seen that their respective amplitudes of vibration are 



for the ray + O x , cos a cos (a - /3) ; 



+ E', cos a sin (a - /3) ; 



E+0', sin asin(a-jS); 



E + E x , - sin a COS (a - )3 j. 



The first and third of these compound the ordinary emergent 

 ray, and the second and fourth the extraordinary. 



The amplitudes of the vibrations in these two resultant 

 pencils are given by the formula 



A 2 = 2 + a' 2 + 2aa' cos 2ir 



\ A J 



in which a and a' are the amplitudes of the component vi- 

 brations above given, and o - e the difference of the lengths 

 of the paths traversed by them. Hence the square of the 

 amplitude of the resulting ordinary ray, or the intensity of 

 the light, will be 



cos 2 a cos 2 (a - ]3) + sin 2 a sin 2 (a - P) 

 + 2 cos a sin a cos (a - /3) sin (a - j3) cos 2?r ( -y- j. 



Adding and subtracting 2 cos a sin a cos (a - |3) sin (a-/3), 



this is reduced to 



cos 2 ]3 - sin 2a sin 2 (a - /3) sin 2 TT i r \ 



Compounding in like manner the second and fourth pen- 



