158 ON THE LIGHT REFLECTED 



-according to which the elliptic polarization varies, both with the 

 thickness of the plate and with the incidence. 



It is assumed in this investigation, that in the reflexion and 

 refraction of light at the surface of a transparent medium, the 

 phases of the incident, reflected, and refracted vibrations coincide 

 at the refracting surface. This assumption is that made by 

 Fresnel. Its theoretical truth is indeed now disproved by the 

 more complete analysis of Cauchy, and by the experimental labours 

 of M. Jamin; but its deviation from the phenomena is exceedingly 

 small, except within a small range of incidence in the neighbour- 

 hood of the angle of polarization, and the conclusions based upon 

 it are therefore sensibly in accordance with fact, except for the 

 same incidences. 



It is unnecessary to add, after what has been just stated, that 

 the elliptic polarization here considered is altogether dictinct, both 

 in its origin and in its laws, from that produced by reflexion at a 

 single surface. 



Let us suppose that the incident light is polarized either in 

 the plane of incidence, or in the perpendicular plane. Let u and 

 u denote the ratios of the reflected and refracted vibrations to the 

 incident vibration at the first surface of the plate, and for light 

 falling upon it from without; (u) and (u'} the corresponding 

 ratios for light proceeding in the opposite direction ; and u 2 the 

 'ratio of the reflected to the incident vibration at the second 

 surface. Then, the amplitude of the incident vibration being 

 unity, the amplitudes of the vibrations which emerge at the first 

 surface, after one reflexion at the second, = u f u 2 (u) ; after three 

 internal reflexions, = ' 2 (u} 2 (u'} = 1st portion x (u) i^ ; after 

 five internal reflexions, = 2nd portion x (u) w 2 , &c. But 



(u} = -u, u' (11} = 1 - u 2 * 



These amplitudes, accordingly, form a series in geometric pro- 

 gression, whose first term is u' u z (u') = w 2 (1 - w 2 ), and whose 

 common ratio is (u) u 2 = - u u z . 



But the interval of retardation, after one internal reflexion, 



* The truth of these relations is evident from the known formulae of Fresnel. It 

 has heen deduced independently, from very simple general principles, by Professor 

 Stokes, and has been shown by him to hold even in the case of change of phase. 

 See an interesting paper " On the perfect blackness of the central spot in Newton's 

 Hinge." Cambridge and Lublin Mathematical Journal, 1849. 



