ON THE REFLEXION AND REFRACTION OF LIGHT. 261 



normal ; and therefore, besides the incident wave, there will, in 

 general, be an accompanying reflected and refracted wave, in 

 which the vibrations are transverse, and another pair of accom- 

 panying 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 intro- 

 duced, it is impossible 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 z. Hence, the 

 general equations (4) for the first medium become 



d*u _ g d fdu dv\ 2 d fdu dv\ 

 d? ~ 9 SVd& 5/ +7 dy\dj/~~dx)> 



d*v _ 2 d fdu dv\ 2 d fdv du 

 ~dt* ~ 9 dy \dx + ~dy) +ri dx\dx~~dj 



, , A ' 2 B 

 where g , and 7 = . 



These equations might be immediately employed in their 

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

 making 



,(13); 



<H _ L. _1 __ L_ 



dx dy 



d6 d-Jr 



v = -T---T- 

 dy ax 



<f> and i/r being two functions of x, y, and t, to be determined. 



By substitution, we readily see that the two preceding equa- 

 tions are equivalent to the system 



de 



dy" 



.(U). 



