224 INTERFERENCE OF POLARIZED LIGHT. 



oils, + E' and E + E', we have, for the intensity of the ex- 

 traordinary pencil, 



sin 2 ]3 + sin 2a sin 2 (a - j3) sin 2 TT f -r- Y 



(232) The preceding formulas give the intensities of the 

 two resulting pencils, in terms of the angles a and /3, and of 

 the difference of the paths traversed by the two rays within 

 the crystal, for any given wave-length i. e. for any species 

 of homogeneous light. When white light is used, we must 

 take the sum of the expressions corresponding to the dif- 

 ferent values of A. Now, the angles a and /3 are independent 

 of A, and therefore the same for all the components of the 

 pencil ; and the same may be said of the difference of the 

 paths, o - e, which is independent of A, except in crystals of 

 extraordinary dispersive power. Hence, when the incident 

 light is compound, the intensities of the two rays will be 



cos 2 ]3 - sin 2a sin 2 (a - /3) S sin 2 TT (&* ] ; 



\ A J 



sin 2 /3 + sin 2a sin 2 (a - /3) S sin 2 * \- 



(233) The foregoing formulas contain the whole theory 

 of the colours of crystalline plates in polarized light. 



1. The sum of the two intensities is equal to 1, or to the 

 intensity of the incident light. Accordingly, when the 

 incident light is white, the colours of the two pencils are 

 complementary. The truth of this is seen by partially 

 superposing the two images, the portions superposed being 

 always white. 



2. The two pencils become white, whatever be the in- 

 terval of retardation within the crystal, when 



sin 2a sin 2 (a - 13) = 0. 



