84 Lord Rayleigh on the 



Equations (12) and (15) simplify considerably in their ap- 

 plication to a uniform medium, assuming the common form 



£ + |>+»VK(l-4™CK-)=0. • (16) 



To express the boundary conditions let us suppose that #=0 

 is the surface of transition between two uniform media. From 

 (12) we learn that the required conditions for case 1 are that 



h , 1 d (h\ 



must be continuous. 



In like manner, for case 2 we see from (15) that 



c , 1 d /c\ 



~IL and K(1-4tt«- 1 CK- 1 )^V/ 

 must be continuous. 



If the media are transparent, or but moderately opaque, we 

 have to put C = 0. The differential equation is of the form 



£+* + .VK-0 (17) 



In case 1 the boundary conditions are the continuity of the 



dependent variable and of — -=-, and in case 2 the continuity 



l± ax y d 

 of the dependent variable and of ^ — . Analytically, the 



results are thus of the same form in both cases. If 6 and S x 

 are respectively the angles of incidence and refraction, the ratio 

 of the reflected to the incident vibration is in case 1 



tan #i fi 



tanA £ ' ^ 



tan 6 fi 1 

 and in case 2 



tan flx K 



famtf, K' (1J) 



tan + Kj 



in which K, //, relate to the first, and K 1? fi x to the second 

 medium ; while the relation between 6 l and 6 is 



K lf jL x : K/*= sin 2 6 : sin 2 X (20) 



As Helmholtz has remarked, Fresnel's formula? may be 



