Transparent Inactive Crystal Plates. 181 



In uniaxial crystals the usual formula for tg s e is 



e 2 — o 2 

 tg s e = — r-j— cos <j> e sin </> e (a) 



where <£ e is the angle Q e z of fig. 6. From the spherical tri- 

 angle Q e z z\ fig. 6, 



cos r e sin d e — cot to cos <J e .,. 



cot <j> e — — — - ; (0) 



^ sin r e v ' 



COS (J 



also cot d> e = r^ — - (cot cos r. +■ cos co sin r e ) . (b 1 ) 



sin <o 



Furthermore, in the principal section through Q e and s 



a e 2 = or cos 2 <jf) e + <? 2 sin 2 <£ e 



from which expression, we find (c) 



e 2 — or . . , , o 2 sinV„ sin (r e — r„) sin (r e + r ) 



, sin 2 <f> e = 1 — — = 1 :— 5 — = — . 



q* r q e sin r e sin - r e 



By means of the three equations (a), (b), (c), equation (44) 

 becomes 



sin (r t — ?■„) sin (r. + r ) (cot to -cos r e ^ *.) 



ifa f = tq o COS (l — r.) -\ : t. r— ; . , , 



y y e v e/ sin (Hn) sin r, (44a) 



or from equations (a), (b r ), (c) 



tg e e = tg <5 e cos (i— r e ) + 



sin (r e — ?•„) sin (r e -H' n )(cot cos r, + cos a> sin r e ) 

 sin (£ + »%) sin w (446) 



Having thus found e and e e , the azimuths 8' and B' e are 

 readily obtained by the use of equations (37). 



<S'„ = c e ±90° <5' e = e o ±90° (45) 



Isotropic Plates. 



In the case of isotropic plates, the index ellipsoid reduces 

 to a sphere and the constants of the equation become 



Equations (16), (35a), (36a), then reduce (if the surrounding 

 medium be air q =l) 



sin i 



(a) sin r = q sin % = 



(b) tg ei = , 



(c) tg 8\ = 



n 

 (b) tg e, = tg d i cos (i — r) 



— cot <S„ 



cos (i—r) (46) 



