MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 63 



whence 



KT = K~ + me gives us 



KT + (wi2 cos 6' + Wi3 sin ^' — niu sin 0' cos ^0^2 



Ki + (wii cos ^' + mi3 sin^ 0' + mu sin 0' cos 0')^ 



Ki^ + (w3i cos 5' + W33 sin^ 0')?2 



— (w4i cos 0' + W44 sin 6' cos (^')^ 











K-' 



A small transformation removes the 4th term without altering the others to 

 the first power of small quantities. 



To obtain the birefringence along the width xs, we set 6 = 6' in equation 

 (18.65): 



By = {fX3 — Ml) sin^ 6' -]- — < W12 cos^ 6' — nin cos^ d' + (mi3 — ^33) sin'' 6' 



■ 2 „, 1 + COS^ d' .2^/2 „,1 / 



— mu sm 6 — W31 sm d cos d > €2 



which, for e' = 18° is b'z = .00087 + .20 62 



For the birefringence along the length .T2 we set 



^ = 90° + d' in (6e) giving: 



3 

 B = {nz — Ml) cos" G' -\- -^ {^12 cos^ ^' — mn cos* 0' + (wi3 — mu — mss) 



sin^ d' cos^ 0' - mi4 sin d' cos 0'(1 + sin^ 6')}^ 

 which, for 0' = 18° is ^2 = .00824 + .049 e}. 



SECTION 20 

 Transverse Isotropy 



A material that has identical properties in all directions normal to a given 

 line is called transversely isotropic. Any line parallel to this line may be 

 considered as an axis of transverse isotropy. 



Dielectric Properties, Optical Properties, Thermal Expansion 



With respect to these, a transversely isotropic material behaves as does a 

 uniaxial crystal, only two constants being needed to describe each. For 

 example, the displacement current in terms of the electric field and the dielec- 



