Uq 



On the Reflexion and Refraction of Light. Ill 



= Dcos — Ij-x + my — Yt > 





V : 



_Da 

 X 



m 



2m 

 cos — 



A 



Also 













du Q _ 

 dy 



_dv _ 

 dx 





X 



•(2) 



K) J J 



2tt r Zn\ 



Dsin x{ x *+ m y-vtV. (3) 



while v vanishes compared with u when a is zero ; and in 

 this case we have merely a surface wave of normal displace- 

 ment given by u travelling over the boundary. 



Let us suppose the same to be the case at the common sur- 

 face of an isotropic medium and a crystal ; we know that 

 the normal wave has no effect in the interior, we shall find 

 shortly that all the surface conditions are satisfied by the 

 hypothesis of a normal displacement over the surface of the 

 same form as at the boundary of two isotropic media. 



Let S be the amplitude of the optical disturbance in the 

 incident wave so that the disturbance is 



S cos — (Ix + my — V*) , 



A 



ot, /3, 7 the angles it makes with the axes, I, m, n the direction 

 cosines of the wave-normal. Let S 1; a 1? /3 l7 <y u l h m 1} n x refer 

 to the reflected wave ; S', a!, &o. to one refracted wave ; S", 

 a" to the other. Let %' and %" be the angles between the 

 rays and the wave-normals. 



The conditions at the surface are that u, v, w, N l7 T 2 , T 3 

 are all continuous. The last three taken in order give, since 

 we suppose the rigidity the same in all media, and since 

 u, v, w are not functions of z, 



dv du dv , dw 

 dy* dy dx* dx 



continuous. The first of these three is already satisfied by the 

 continuity of v ; and we have thus five equations to find the 

 amplitudes of the two refracted waves, the amplitude and the 

 azimuth of the plane of polarization of the reflected wave, and 

 the amplitude of the surface effect. 



These five equations may be written, if we introduce the 



values of u , -~- &c, and divide by the periodic factor, as 



follows : — 



K2 



