240 ON THE REFLEXION AND REFRACTION OP SOUND. 



regards the co-ordinate x an exponential must take the place of 

 the circular function. In fact the equation, 



may be satisfied by 



< y = e- 



(where, to abridge, ty is put for by + ct) 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 </>. We must 

 therefore take, instead of (8), the formula, 



$ = a sin (ax 4 by + ct + e) + ft sin ( ax + by + ct + e t ) .... (9). 



Hence when x = 0, we get 



-^- = aa cos (ijr + e) aft cos (^ + e^), 



U& 



- = ca cos (^r + e) c cos (^r + e,), 



these substituted in the conditions (A), give 



a cos ty + e)ft cos (-^ + e,) = ^ ^ sin ^, 



a cos fy + e)+ft cos (<\fr + e t ) = - B cos ^ ; 



these expanded, give 



a cos e /3 cos e t = 0, 



a sin e, + ft sin e t = -- - B, 

 a 



a. cose-}- ft cos e t -r-' J5, 

 a sin e + ft sin e / = 0. 



