Transparent Inactive Crystal Plates. 179 



[o>(x" + z"')-l][a u x' t + a 33 s ,a + 2a 3l z'x'-i~\=0. (38) 



In this formula as' and z' can, by virtue of the sine law, be 

 readily expressed in polar coordinates ; if i be the angle of 

 incidence and r the angle of refraction and n the refractive 

 index of the isotropic medium, then as' = n sin i, and 



z' = - 2 — - . With these values (38) reduces to 

 Ml r 



[o 2 >C sin 2 { (1 + tg*r)-tg>r][(a„n * sin 2 i-l)tg'r + 



2a 31 n * sin a i tgr + a 33 n* sin 2 i]=0. (38a) 



From the first half of this equation (38a) 



sin r =n o sin i. 



To evaluate the coefficients of tg r in the second half of the 

 equation (38a), let the plane x' y' in fig. 6 be the boundary 



surface of the crystal plate ; x' z', the plane of incidence, the 

 positive direction of z' being on the crystal side of the bound- 

 ary surface ; as, y, z, the principal ellipsoidal axes of the crys- 

 tal ; 6 the polar angle zz', and a>, the azimuth of the principal 

 plane z z' ; let also the angles of inclination of the wave normals, 

 Qo> Qei with the z' axis be Q z'=r , Q e z'=r e \ with the z axis, 

 be Q z = cf> , Q e s=$ e - hi this case the direction cosines of as', 

 y', z' with x, y, z, are respectively 



p 1 = — COS COS CD 



p t = sin to 



jt? 3 = sin 6 cos w 



q 1 = — cos sin w 



q t = — cos a) 



q 3 = sin sin a> 



r. = sin 6 



r. = 



r. = cos 



also, in equations (9) a' = e 2 , J 2 = e 2 , c 2 



= o* 



