OPTICAL PROPERTIES IN CRYSTALS 169 



For a given value on the right-hand side there are two values of \p, 90° apart, 

 that will satisfy the equation and hence we have two directions of vibration 

 at right angles to each other. Inserting (34) in {?>?)) the values of /3u and 

 /3ii for these two directions are 



2|5ii = /3i(sin- ip sin'' d-\- cos- 6) + /32(cos- <p sin- d + cos- 6) + /33 sin- 6 



, , /(/3i - 182)2 (cos2 d cos2 ^ + sin2 ^y + 2(/3i - ^2)(^3 - ^2) 

 =*" y • sin2 ^ (cos2 cos2 <p - sin^ <p) + (^3 - ^^Y sin^ d. 



Since /3i corresponds to a^, etc., this equation agrees with the two velocities 

 given in equation (28) and shows that the directions of vibration correspond 

 with the maximum and minimum values of /3u . 



It can also be shown that the two directions of electric displacement co- 

 incide with the two values of i// given by equation (34). Transforming the 

 electrical displacements to the X\ Y', Z' set of axes we have 



d[ = pD,i- p D2 -i-pD, = hD, + m,D, -f mD^ 



dxi 6X2 0X3 



D2 = pD, + pD, + pD3 = l,D, + m,D, + n,D^ (35) 

 ox\ 0X2 ox? 



Dz = pDr + P D2 -VpD^- UD, + mzD2 + nM. 

 dxi 0x2 dxz 



Hence, inserting the values of A, A, D^ from equation (18), we find 



D[ = IM?2 - /3n)(^3 - /3n) + mmzifix ~ ^'nWz - ^n) 



+ nM^i - /3n)(^2 - iSn) 



D', = (.M^2 - /3n)(/33 - ^n) + nvmzi&i - I3n)((3s - /3n) 



+ fhtpM - ^n)(02 - /3ii) 



Ds = (1(132 - ^n)(^3 - /3n) + ml(0i - /3n)(/33 - /3n) 



+ «3(ft - /3n)^2 - ^11). 



From equation (20) with /3i2 = 1813 = 1823 = 0, it is evident that the Dz com- 

 ponent vanishes and hence the two values of electric displacement lie in a 

 plane perpendicular to Z'. By inserting the values of jSn and the value of 

 \p found from equation (34) we find that A = and hence the electric dis- 

 placement lies along the directions of the greatest value of (3n • Similarly, 

 from the second value of (3n , A vanishes and hence the second wave is per- 

 pendicular to the first and in the direction of the smallest value of jSn . 



(36) 



