REFLEXION AND REFRACTION OF POLARIZED LIGHT. 179 



and they compound a single vibration, inclined to the plane 

 of incidence at an angle whose tangent is the ratio of the 

 component vibrations. If, then, this angle be denoted by 

 a', we have 



tan a' = - tan a 



cos (i + r) 

 cos (i - r) 



The truth of this formula has been verified by the obser- 

 vations of Fresnel himself, and more fully since by those of 

 Arago and Brewster. 



When a = 0, a! = ; and when a = 90, a' = 90. Accord- 

 ingly, when the plane of polarization of the incident ray 

 coincides with, or is perpendicular to, the plane of incidence, 

 it is unchanged by reflexion. When i + r = 90, a' = 0, and 

 the plane of polarization of the reflected ray coincides with 

 the plane of incidence, whatever be the azimuth of the inci- 

 dent ray. 



(191) The plane of polarization of a polarized ray is 

 changed by refraction, as well as reflexion, but in an opposite 

 direction, the plane being removed farther from the plane 

 of incidence, instead of approaching it. This movement of 

 the plane of polarization increases with the incidence ; being 

 nothing when the ray falls perpendicularly upon the re- 

 fracting surface, and greatest when the incidence is most 

 oblique. The law of the change is expressed by the simple 

 formula, 



cot a' = cot a cos (i-r) ; 



which indicate that the former quantity is negative, and the latter positive (see 

 Professor Powell's paper " On the Demonstration of Fresnel's Formulas," Phil, 

 Mag. Aug. 1856). This is equivalent to saying, that one of the waves gains, or 

 loses, half an undulation in the act of reflexion. We shall see hereafter that the 

 complete theory of reflexion includes a progressive change of phase ; and that 

 the conclusions of Arts. (,184, 5) are only approximate. 



N2 



