16 Ma GREEN, ON THE REFLEXION 



incident wave, there will, in general, be an accompanying reflected and 

 refracted wave, in which the vibrations are transverse, and another pair 

 of accompanying reflected and refracted waves, in which the directions 

 of the vibrations are normal to the fronts of the waves. In fact, unless 

 the consideration of the two latter waves is also introduced, it is im- 

 possible to satisfy all the conditions at the surface of junction ; and these 

 are as essential to the complete solution of the problem, as the general 

 equations of motion. 



The direction of the disturbance being in plane (xy) w = 0, and as 

 the disturbance of every particle in the same front of a wave is the 

 same, u and v are independent of * Hence, the general equations (4) 

 for the first medium become 



d 2 u _ 2 d idu dv\ , d idu dv\ 

 df ~ dx \dx dy) ' dy \dy dx) ' 



d*® _ 2 d (du_ dv\ 2 d fdv du\ 

 ~d? =S d~y \dx + dy) +y dx[dx~dy)' 



where g* = — , and y 2 = — • 



These equations might be immediately employed in their present 

 form; but they will take a rather more simple form, by making 



d<p dty 

 dx dy ' 



(13). 

 d(p d\f/ 1 

 dy dx ' 



<p and ^ being two functions of x, y and t, to be determined. 



By substitution, we readily see that the two preceding equations are 

 equivalent to the system, 



d? ~* Ux 2 + dy 2 )' 



(14). 



df " 7 \dx* + dy* I ' 



