126 ON THE LIGHT REFLECTED 



And the intensity of the reflected light will be the sum of the in- 

 tensities of the two component portions. 



When the obliquity of the incident pencil is so small, that 

 the squares and products of v, v w, iff*, may be neglected in com- 

 parison with unity, the intensity of the reflected light will be, 

 nearly, 



(p 2 + 2m, cos a + 2 2 ) cos 2 7 + (tf + 2ww, cos y + w?} sin 2 7. 



This expression is independent of the phase, and therefore the 

 intensity is constant for a given incidence, when 



W Z 



w, cos 2 7 + vcw-i sin 2 7 = 0, or tan 2 7 = --- . 



It will be easily seen, on substituting for v, v z , w, iff z , their well- 

 known values, that the value of tan 7 will be real, and therefore 

 the disappearance of the rings possible, only when the angles of 

 incidence at the two surfaces of the plate are, in the one case 

 greater, and in the other less, than the respective polarizing 

 angles.* This is the explanation of the phenomenon observed by 

 Sir David Brewster. 



Again, since the values of v and v z are, in general, different 

 from those of w and w z , it follows that the phases of the two com- 

 ponent vibrations are unequal, and consequently that the result- 

 ing reflected light is elliptically-polarized. This consequence of 

 the wave-theory does not appear to have been noticed by observers. 



8. We may now proceed to consider more particularly the 

 case in which the media are the same on the two sides of the 

 plate. 



In this case, v z , = - v, w z = - w, and the general expressions for 

 the phases of the two polarized pencils are reduced to 



The difference of phase is given by the formula 



tan (,/, - v) = ^ ~ sin 



1 - 



p* + w 2 ) cos a + vW 

 It follows from this, that i// - x varies with a, and therefore with 



* See the paper above referred to, Proceedings, vol. ii., p. 268. 



