Transparent Inactive Crystal Plates. 167 



ways, the angle of refraction or reflection of any light waves 

 in the crystal can be calculated. 



From 15 (a), the following equation is readily derived : 



tfff= a «__ 



« ia cos r— a 3S sin r 



which may be written 



to ip - (*'—*») fr' r - g « ■ 



wherein k = . . as in (16). From (17) the azimuth of the 

 sin i 



plane of polarization can be determined, provided r be known. 



Equation (16) is biquadratic and indicates that in a crystal 



there are four possible waves, of equal significance, — two 



reflected and two refracted waves, — which must be taken into 



account in the general boundary conditions for the crystal. 



The general equations (11), (12) and (14) for the magnetic 



light vector on passage through the boundary between two 



inactive, transparent crystal plates may therefore be written :' 



4 4 



X A k cos r k cos \f/ k = £ A' k cos r\ cos \p' k (from (t<), = (w) 2 ) 



4 



£A k sin «/, k = £ A' k si" "A'k (from (w) i = («) s ) (18) 



% A k sin r k cos *// k = % A' k sin r' k cos \j/' k (from (w) x = (w) t ) 



i i 



* A k sin r k . 



2 s (s' n "Ak (« n cos r k — « 1S sin r k ) + a lo cos xp k ) = 



1 # k 



x\ ^ Sill ?' k / • t / i f i • /\ 



2 n (sm ^ k (a n cos r k - o „ sin r k ) + a 12 cos f k ) 



(H?J=(f),) 



In the last equation of this set both sides of the equation 

 have been multiplied by the equality k = — r -^ 



2k <? k 



In the first three equations, the factors of the amplitudes, 

 Aj . . A 4 , are the direction cosines, I, m, n, of the line of 

 vibration it with the axes x', y', z' ; if the factors of the 

 amplitudes in the fourth equation be indicated by p, the 

 equations can be written in the abbreviated form, 2 



1 G. Kirehhoff, Ges. Abhandhmgen. 367-370, 1882. 



2 A. Potier, Journ. Phys. (2), x, 350, 1891. P. Kaeuimerer, N. J., Beil. 

 Bd. xx, 174, 1905. 



