OPTICAL PROPERTIES IN CRYSTALS 



177 



will occur when /3n is a maximum and the other when /3n is a minimum. 

 Using the transformation equations (31) and the direction cosines of (27), 

 we find that fin is given by the equations 



Ju = fill cos 6 cos" (f cos" l/' 



sin 2(p sin 2^ cos d , .2 . " , 

 • + sm (f sm" \p 



+ i3i2[sin 2(p cos 2\l/ — sin" ^ sin 2(p cos i/* + cos 6 sin 2i/' cos 2^] 



+ iSi3[ — sin 26 cos ^ cos ;/' + sin ^ sin ^ sin 2\p\ (73) 



to r 2.-2 2 , , cos ^ sin 2ip sin 2i/' , 2 • 2 , ~| 

 + 1822 cos d sm ^ cos i/' + 1- cos (p s\n rj/ \ 



+ |823[ — sin 20 sin (y? cos" 4/ — sin cos tp sin 2i/'] + fi^^^ sm 9 cos^ ^ 



sb' 



Differentiating with respect to \p and setting — ~ = 0, we find an ex- 



pression for tan 2\f/ in the form 

 tan 2\J/ = 



— fill sin 2(p cos 6 + 2fii2 cos 6 cos 2^ 

 + 2(Si3 sin <p sin 9 -}- fioo cos ^ sin 2(p — 2/323 sin 6 cos (^ 



/3ii[cos" cos (p — sin ^] + |8i2[(l + cos" 9) sin 2<p] 

 — fiu sin" 6 Ck s(p -\- /322(cos" ^ sin ^ — cos" (p) 

 — 1S23 sin 29 sin ^ + fiss sin^ ^ 



(74) 



Inserting this value back in equation (73) we find that the two extreme values 

 of fill are given by the equation 



2fi'ii = 2^22 + (fin - /322)(cos2 9 cos2 cp + sin2 <p) + 0333 - fi-i^) sin' 9 



— ;Si2 sin^ 9 sin 2^ — fin sin 20 cos^ — 1823 sin 29 sin ^ 



± 



1/ 



(/3ii - /322)'(cos2 9 cos2 ^ + sin2 cpY + 2(^ii - /?22)(/333 - ^22) sin2 9X 



(cos2 9 cos2 ^ - sin2 <p) + (/333 - /322)' sin^ 9 - 2(fiii - fi2o)X 



[(Si2(sin 2<p sin^ 0(cos"^ 9 cos^ ^ + sin- (p) -{- fin sin 20 cos ^X 



(cos^ cos <p -\- sinV) — fi2z sin 20 sin ^(1 + cos^ <p sin^ 0)] 



+ 2(/333 — fe) sin^ 9\fii2 sin 2^(1 + cos- 0) — ;3i3 sin 20 cos (p 



4 (75) 



— ;323 sin 20 sin (p] + (2,Si2)2[sin sin^ ^ cos- 93 -f cos- 0] 



- 4i3i2i8i3sin2 sin^[cos2 cosV + sin"^ <p] - 4(^i2fe) 

 [sin 20 cos ^(sin- (p cos- 0+ cos- (p)] + (2/3i3)- sin- 0X 



(cos^ cos^ ^ + sin^ (p) — 4:fiizfi2z sin 2(p sin 



-f (2/323)^ sin^ 0(cos- sin^ (p + cos^ <p) 



