410 Mr green, on THE REFLEXION AND 



What precedes is applicable to all waves of which t\\e 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, 



(8) <p = a sin {a X + h y -\- c t) ■\- (i sin ( - a a; + i y + c /). 



Also, in the lower one, 



(p^ = a, sin {(iiX + b y + ct): 



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

 considered, the equation (7) will give 



. ^ VA a) ' 



If 7^ > 7, or the velocity of transmission of a wave, be greater 

 in the lower than in the upper medium, we may by decreasing a render 

 «, imaginary. This last result merely indicates that the form of our 

 integral must be changed, and that as far as regards the co-ordinate x 

 an exponential must take the place of the circular function. In fact 

 the equation, 



(IP ~ ^' \dx' df]" 

 may be satisfied by 



(where, to abridge, ■// is put for h y + c t) provided 



when this is done it will not be possible to satisfy the conditions {A) 

 due to the surface of separation, without adding constants to the quantities 

 under the circular functions in <p. We must therefore take, instead of (8), 

 the formula, 



(9) = n sin (rt iT -^ A y + c / + e) 4- /3 sin ( — « a; + i y + c / + e^. 



