AND REFRACTION OF LIGHT. 19 



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 



\// = f {ax + by + ct) + F{— ax + by + ct), 

 and (18). 



<P — X ( ~ a 'x + by + ct) ; 



and for the lower one 



^,=f, («> + h + ct), 



(19). 



The coefficients b and c being the same for all the functions to 

 simplify the results, since the indeterminate coefficients a' a t a' 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 t d, as, however, it evidently = — a, it was 

 for the sake of simplicity introduced immediately into our formulae. 



By substituting the values just given in the general equations (14) 

 and (16), there results 



(f = ( a * + b 2 ) 7 2 = (a* + ¥) 7/ 2 = (a' 8 + ¥)g 2 = («/ 2 + b*)g% 



we have thus the position of the fronts of the reflected and refracted 

 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 



" § " . 



x = -p x, ; 



-a' x ' + b(f' + F') = a; x ; + bf;, 

 b x ' - a(f - F') = b x ; - aj; ■; 



where to abridge, the characteristics only of the functions are written. 



C2 



