ON THE REFLEXION AND REFRACTION OF LIGHT. 259 



waves is exactly the same as that for light polarized in the 

 plane of incidence (vide Airy's Tracts, p. 356*), and which Fresnel 

 supposes to be propagated by vibrations perpendicular to the 

 plane of incidence, agreeably to what has been assumed in the 

 foregoing process. 



We will now limit the generality of the functions/, F and/, 

 by supposing the law of the motion to be similar to that of a 

 cycloidal pendulum; and if we farther suppose the angle of 

 incidence to be increased until the refracted wave ceases to be 

 transmitted in the regular way, as in our former paper on Sound, 

 the proper integral of the equation 



a w t _ 



~de~ ==J 



will be 



where Tjr = by + ct, and aj is determined by 



! ) (ii). 



But one of the conditions (9) will introduce sines and the 

 other cosines , in such a way that it will be impossible to satisfy 

 them unless we introduce both sines and cosines into the value 

 of Wj or, which amounts to the same, unless we make 



w a. sin (ax + by 4- ct + e) + /3 sin ( ax 

 in the first medium, instead of 



w = a sin (ax + by + d) -{- fi sin ( ax + by + ct}, 



which would have been done had the refracted wave been trans- 

 mitted in the usual way, and consequently no exponential been 

 introduced into the value of iv 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. 



* [Airy on the Undulatory Theory of Optics, p. 109, Art. 128.] 



172 



