412 Mr. W. J. M. Rankine's General View of an 



and the other a quarter of an undulation behind it. The maxi- 

 mum amplitudes of oscillation in these evanescent waves consti- 

 tute four unknown quantities ; the amplitudes in the two ordi- 

 nary reflected waves and the two ordinary refracted waves, dif- 

 fering by one quarter of an undulation, constitute four more 

 unknown quantities, making eight in all : four conditions having 

 been fulfilled by the waves polarized in the plane of incidence, 

 there remain to be fulfilled eight conditions ; viz. the identity, 

 as calculated by the formulae for the first and second substance 

 respectively, of the following eight functions at the bounding 

 surface ; the angular displacement, and the rotative forces, round 

 each of the two axes in the plane of incidence, for the incident 

 wave and the set of waves synchronous with it, and for the set 

 of waves retarded by one quarter of an undulation. These con- 

 ditions are sufficient to determine the unknown quantities, and 

 to complete the solution of the problem. 



The following is a general statement of the results of the 

 solution when the second medium is the denser. They agree 

 with the results of the experiments of M. Jamin, and are in every 

 respect analogous to those deduced from the hypothesis of vibra- 

 tions by M. Cauchy, Mr. Green, and Mr. Haughton. 



Light polarized in a plane perpendicular to the plane of inci- 

 dence, suffers by reflexion at a perpendicular incidence no altera- 

 tion of phase. 



At a grazing incidence (or when the angle of incidence differs 

 insensibly from 90°), the phase, like that of light polarized in 

 the plane of incidence, is retarded by half an undulation. 



The variation of phase with the angle of incidence is, in fact, 

 continuous; but it is, generally speaking, not appreciable by 

 observation, except in the immediate neighbourhood of an angle 

 called by M. Jamin the principal incidence , where the retardation 

 of phase is a quarter of an undulation. 



This angle differs by a very small amount, appreciable only in 

 certain substances, from the polarizing angle, at which the inten- 

 sity of light polarized in a plane at right angles to the plane of 

 incidence is a minimum. 



The " Law of Brewster/' that the tangent of the polarizing 

 angle is equal to the index of refraction, is, theoretically, only 

 approximately true ; but the error is quite inappreciable. 



When the second medium is the less dense, the phase of the 

 reflected light is half an undulation in advance of its value when 

 the second medium is the denser. 



In either case, light polarized in planes perpendicular to the 

 plane of incidence is less retarded, that is to say, is accelerated 

 in phase, as compared with light polarized in that plane, accord- 

 ing to the following table : — 



