260 ON THE REFLEXION AND REFRACTION OF LIGHT. 



Substituting now (10) and (12), in the equations (9), and 

 proceeding precisely as for Sound, we get 



= a cos e /3 cos e t , 



= a sin e 4- /3 sin e t9 



B = a sin e sin e lt 



B=OL cos e + /S cos e t . 



Hence there results a = ft and e, = - e, and 



a' a' a a' ~ 

 tan e = - = - + r = -=* tan 0. 

 a b b b 



But by (11), 



by introducing /u, the index of refraction, and 6 the angle of 

 incidence. Thus, 



/i COS 



and as e represents half the alteration of phase in passing from 

 the incident to the reflected wave, we see that here also our 

 result agrees precisely with Fresnel's for light polarized in the 

 plane of incidence. (Vide Airy's Tracts, p. 362*.) 



Let us now conceive the direction of the transverse vibra- 

 tions in the incident wave to be perpendicular to the direction 

 in the case just considered ; and therefore that the actual motions 

 of the particles are all parallel to the intersection of the plane of 

 incidence (xy) with the front of the wave. Then, as the planes 

 of the incident and refracted waves do not coincide, it is easy to 

 perceive that at the surface of junction there will, in this case, 

 be a resolved part of the disturbance in the direction of the 



* [Airy, ubi sup. p. 114, Art. 133.} 







