Reflection and Refraction of Light, 159 



conditions ; and so in what follows the compressibilities and 

 rigidities of the two media will be considered as unequal. 



III.* 



Taking, as before, the interface of the media as the plane 

 of yz, and the first medium on the side of positive x, let the 

 axis of z be parallel to the plane of the waves, so that the 

 plane of xy is the plane of incidence ; then, if f 77 f and f ' rj f 

 denote the components of the displacements in the first and 

 second medium respectively, £ n f, £' rf J ' will be independent, 

 of z. 



(1) Let the incident vibrations be perpendicular to the 

 plane of incidence. 



The general equations of motion are in this case 



P dt 2 " \da» + df} P dt*~ n \dx* + dy l J 1 



and the principle of continuity gives for the interfacial con- 

 ditions that for x = 0, 



t ,, az^dj 



b *' dx dx' 

 Assuming 



we get at once 



n a + a! n a — a' 



Of 



1 2a ' 

 sin(i-^r) 



a + a' sin(i + r)' 



since 



— =tanz, -/ = tan r ; 



a ' a 



where i, r are the angles of incidence and refraction. 



(2) Let the incident vibrations be in the plane of incidence. 

 The equations of motion in the first medium are 



< 



Pd7 =m TAd-z + Ty) +n {M + df)' 



P^ =m d^ + dy)" rn {d? + 'dJ'} 



* C. R. viii. p. 985 ; ix. pp. 1, 59, 91, 676, 726, 727 5 x. p. 347. Ex. 

 cTAn. et de Phys. i. pp. 133, 212. 



