AND REFRACTION OF LIGHT. 15 



the value of w r We thus see the analytical reason for what is called 

 the change of phase which takes place when the reflexion of light 

 becomes total. 



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 + fi sin e t , 



— B = a sin e — 8 sin e , 

 a 



B = a cos e + /3 cos e t . 



Hence there results a = 0, and e t = — e, and 



tan e = — = -j- •£ t = -f tan 9. 

 a b b b 



But by (11), 



5 

 b 



if-^-^-^Fii-y^-OT)* 



by introducing ^ the index of refraction, and 6 the angle of incidence. 

 Thus, 



to n,- *^' rin ''- 1 > ; 



M 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 vibrations 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 re- 

 fracted 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 normal; and therefore, besides the 



