Transparent Inactive Crystal Plates. 169 



z'. In these equations i, E, e, q of the incident wave are 

 known ; also by calculation (equations (16) and (17)), r„ r„ and 

 S„ S., of the refracted waves ; and i, q of the reflected wave ; 

 unknowns are R, p of the reflected wave and D„ D 2 of the 

 refracted waves W„ W 2 . 



In abbreviated form, corresponding to (18a), these equations 

 may be written :' 



D/, + D 2 Z 2 = (E cos £— R cos p) cos i 



D,???,, + D 2 m 2 = E sin e + R cos p 



D^?, + D 2 re„ = (E cos € + Rcos p) sin i 



D^! + D 2 p 2 = (E sin e— R sin p) sin i cos i (19a) 



At the second boundary surface where the two refracted 

 waves emerge from the crystal plate into the isotropic medium, 

 two sets of boundary equations obtain, one for each refracted 

 wave, W, and W 2 . At this surface, there are for each incident 

 wave, W, and W,, two reflected waves and one refracted wave 

 as indicated in fig. 2. 



For the refracted wave W, the boundry conditions reduce to 



D, cos Sj cos i\ + R', cos p\ cos r\ + R", cos p" t cos r\ = 



D', cos S\ cos i. 



D l sin 8, +R\ sin p\ + R\ sin p\ --= 



D', sin B\. 



D, cos 8\ sin j'jH-R'j cos p\ sin r\ +K\ cos p\ sin r\ = 



D\ cos 8\ sin i. (20) 



sin v I I 



D, — ~\ sin 8, ('/„cos r-a is sin »■,) + «„ cos 8, + 



R', sin »•' r . , . , , . , . 



sin p I (<r n cos ?■ ,— a i3 sm r J +a ia cos p J + 



2 > L 



R" sin r" 

 — L-,5 ? [sin p", (a n cos !•*,-«„ sin ?•",) + «,, cos r # 1 ).= 



D' a sin 8'j cos i sin £. 



wherein, for the incident wave V, the faster reflected wave 

 W ia , the slower reflected wave W ]b , and the refracted wave W',, 

 respectively, D l5 R', R", D', are the amplitudes ; S,, p\, p", B\ 

 the polarization azimuths ; r r r\, r'\, i the inclination of the 

 wave normals with z' ; and q i: q\, q'\, q , the wave normal 

 velocities. 



Similarly, for the slower refracted wave W„ the boundary 

 equations are 



D 2 cos 8 2 cos r 2 + R' 2 cos p' 2 cos r' 2 + R" 2 cos p" 2 cos r'\ = 



D' 2 cos 8' 2 cos i 

 D 2 sin8 2 +R' 1 Binp' i +R" 2 sin p\ 



T>\ sin 8' 2 



1 P. Kaemmerer, Nenes Jabrb., Beil. Bd. xx, 176, 1905. 



