ON THE REFLEXION AND REFRACTION OF SOUND. 239 



Hence 



tan 6 = 2 

 7, 



which is the precise angle at which light polarized perpendicular 

 to the plane of reflexion is wholly transmitted. 



But it is not only at this particular angle that the reflexion 

 of sound agrees in intensity with light polarized perpendicular to 

 the plane of reflexion. For the same holds true for every angle 

 of incidence. In fact, since 



2A A, 7 2 sin 2 6 



- = 



and the formulae (7) give 



sin 2 6 tanfl 



J?_ sin 2 6, ~ tan 0, _ tan (0-0) 

 f " sin 2 tan 0" -tan (0+0,)' 



sin 2 0, tan / 



which is the same ratio as that given for light polarized perpen- 

 dicular to the plane of incidence. (Vide Airy's Tracts, p. 356)*. 

 What precedes is applicable to all waves of which the front 

 is plane. In what follows we shall consider more particularly 

 the case in which the vibrations follow the law of the cycloidal 

 pendulum, and therefore in the upper medium we shall have, 

 (j> = a sin (ax + by + ct) + ft sin ( ax + by + ct) ......... (8). 



Also, in the lower one, 



<, = a, sin (a t x + ly + ct) : 



and as this is only a particular case of the more general one, be- 

 fore considered, the equation (7) will give 



If y / > 7, or the velocity of transmission of a wave, be 

 greater in the lower than in the upper medium, we may by de- 

 creasing a render a y imaginary. This last result merely indicates 

 that the form of our integral must be changed, and that as far as 



* [Airy on The Undulatory Theory of Optics, p. Ill, Art. 129.] 



