264 ON THE REFLEXION AND REFRACTION OF LIFHT. 



i|r / are transverse or wholly in the front of the wave. If the co- 

 efficients A and B did not differ greatly in magnitude, waves 

 propagated by both kinds of vibrations must in general exist, 

 as was before observed. In this case, we should have in the 

 upper medium 



)"\ 



and </> = (-a'# + + <rt) J" 



=f(ax 

 = % y (-a 

 and for the lower one 



=* + + <* 



The coefficients Z> and c being the same for all the functions 

 to simplify the results, since the indeterminate coefficients a'aa 

 will allow the fronts of the waves to which they respectively 

 belong, to take any position that the nature of the problem may 

 require. The coefficient of x in F belonging to that reflected 

 wave, which, like the incident one, is propagated by transverse 

 vibrations would have been determined exactly like a'a^a', as, 

 however, it evidently = a, it was for the sake of simplicity 

 introduced immediately into our formulse. 



By substituting the values just given in the general equa- 

 tions (14) and (16), there results 



c 2 = (a 2 + V) 7 2 = (a,* + V) %' = (a 2 + V) f = (a/ 2 + 5 2 ) </*, 



we have thus the position of the fronts of the reflected and re- 

 fracted waves. 



It now remains to satisfy the conditions due to the surface 

 of junction of the two media. Substituting, therefore, the values 

 (18) and (19) in the equations (17), we get 



