18 



Mr GREEN, ON THE REFLEXION 



dx dy 



dj, = d<f>, d^ y 

 dx dy 



d(p 

 dy 



d^f _ d(p / 

 dx dy 



dx 



d 2 d 2 ^ m £& j gjfc 

 d# 2 da; dTy 



d 2 



V (when a; = 0) ; 



rfo; 2 



dxdy 



d> _ d>, d>, 

 da; 2 d# dy 



da; dy da; 2 dx dy dx 2 



or since we may differentiate with respect to y, the first and fourth 

 equations give 



d 2 ^ , d 2 ^_d 2 f dS//, 



+ 



+ 



dx 2 dy 2 dx 2 T dy 2 ' 

 in like manner, the second and third give 



d 2 <p d 2 (p 

 dx 2 + dy 2 



d 2 0, + d'fr 



dx 2 ' dy 2 ' 

 which, in consequence of the general equations (14) and (16), become 



^NL-._^ flm l ** - d% *> 

 y 2 df ■ 7/ 2 df g 2 dt 2 ~ gfdf 



Hence, the equivalent of the four conditions relative to the surface 

 of junction, may be written 



d<p d^/ _ d(f> t d^ l 

 dx dy dx dy 



d(p d^> _ d(p t dy\r t 

 dy dx dy dx 



d 2 <p d 2 (p, 

 g*df- gfdf 



d> d>, 



7, 



> (when x = 0), 



7 



df 



2 df 



(17). 



If we examine the expressions (13) and (15), we shall see that the 

 disturbances due to <p and <f> / are normal to the front of the wave to 

 which they belong, whilst those which are due to ^ and ^ are trans- 

 verse or wholly in the front of the wave. If the coefficients A and B 



