Transparent Inactive Crystal Plates. 165 



2tt ( a:' sin r + z' cos r\ 



u = — A cos v cos r cos -^ I 2 — 1 



T \ 1 i 



2?r / cc' sin r + g' cos r\ , . 

 y = A sin if) cos -7pr( < ) ( 14 ) 



2ir / a;' sin r+z' cos r\ 



,0 = A cos *}) sin r cos -=- ( £ - I 



T \ q / 



For z' = it is evident from the boundary conditions, 

 (u) l =(w) a , («), = (w) 2 of equations (11) and (14), that for all pos- 

 sible reflected or refracted waves at the limiting surface, T, the 

 period of vibration (color) remains constant (T, = T a ) ; also 



sin t ^in v 



L = - — - , which is the sine law of wave normals ; while 



fi = signifies that all wave normals lie in the plane of 

 incidence. 



By means of equations (14), the general differential equations 

 (10) can be solved and the fundamental formulas obtained for 

 the refraction, reflection, and polarization of light waves in 

 crystals. Thus from equations (4) and (14) 



, 9w 9v . 



C = 7T~: — ~z~, = — K sin if> cos r 

 9y 9z' 



9u 9 to „ 



ri= 9z'~9^=~ Kcos,p 



9v 9u . 



^=9x'~9j' =Ksmipsmr - 



Wherein 



7r / x' sin r + z' cos r\ 



r(«-- ~ q ) 



„ » 2 7T . 2tt / x' sin r + z' cos 

 K = A . == — . sin 



T.q T 



From these expressions, we find : 



9 

 7fc, K I + a„ v + a 23 £) = 



C 



— j cos r ( — « a] sin y cos r — c? 22 cos y + « aa sin if> sm r) 



3 



^7 (ff„ £ + « a3 1? + « 3 s = 



c . , 



—j- sin r (— « J3 sin ^ cos r — « 23 cos i/< + a 3S sin i/> sin r) 



9 

 g z r(a u £ + a iiV + a 13 £) = 



C , . 



— - cos r (— a,, sin v cos r — a la cos y + « 13 sin y sin r) 



1 



