Refraction of Light at the Surface of a Crystal. 113 



Let ON be the wave-normal, OP the direction of vibration 

 in the incident wave, and let this wave cut the plane z = 

 in OQ. 



Then clearly 



cos a= —sin 4> sin 6 



cosj(3= cos $ sin 0^- .... (10) 



cos 7= cos 



cos ol x = — sin $ sin 1 



cos jSj = — cos (f> sin X J» .... (11) 



cos 7i= cos 



in<9V . . 



os6>J 



ostfj 



= sin 4> ~\ 



= sin <£ > 

 = sin $' J 



£ = cos (j), m =sin (j> 



?i=— cos </>, m 1 = sin <£ }- . . (12) 



Z' = cos </>', 



Again, for the refracted wave, S' is equivalent to S' cos %' 

 in the wave-front, and S' sin rf along the wave-normal. 

 S ; cos 'rf is equivalent to S' cos %' cos 0' along Oz, and 

 S' cos %' sin 0' along the intersection of the wave and the 

 plane xy, and this last is equivalent to S' cos %' sin 0' cos (/>' 

 along Oy, and — S' cos %' sin & sin $' along Ox. Again, the 

 component S ; sin y' along the wave-normal gives S' sin %'cos (/>' 

 along Ox, and S' sin %' sin (j)' along Oy. 



Hence 



cos a ' = —cos %' sin 0' sin 0' + sin yj cos <£ ^ 



cos /3' = cos ^' sin 0' cos <£' + sin yj sin <// > • • • (13) 



cos t / = cos %' cos 0' ' 



On substituting these values in equations (6), (8), (9), and 

 (5) respectively we obtain 



S cos + Si cos 1 = S' cos %' cos & + terms in S" &c. . (14) 



(S cos — Si cos 0J cot = S' cos v' cos & cot 6' + terms in S" 



(15) 



(S sin + Si sin X ) cosec <f> = 



S' cos x' sin #' cosec <£' + terms in S". . (16) 

 (S sin 9 — Si sin #j) cos <fi = 



S' cos %' (sin 0' cos (// + sin <f>' tan %') + terms in S". . (17) 



The corresponding equations on the electromagnetic theory 

 are given in the same form in a paper by myself in the ' Pro- 

 ceedings of the Cambridge Philosophical Society ' (vol. iv. 

 p. 165, equations 24-27).- If we suppose the magnetic per- 

 meability the same in the two media, and write 2 for the 



