6 The Double Refraction of Light in a Crystallized 



Let the plane which cuts the ellipsoid intersect its circular 

 sections in the lines OR, OS', and 

 let the principal section AOC of the 

 ellipsoid cut the circular sections in 

 OR and OS ; then OR, OS, OR', 

 OS', will be all equal to the mean 

 semiaxis OB, and hence the semiaxes 

 OA, 00' of the section A'OC' will 

 bisect the acute and obtuse angles 

 made by OR' and OS'. Let a plane A( 

 through OB and OA' intersect the 

 principal plane AGO in the line OT. 

 Then, by the nature of the ellipse, we 

 have, in the ellipse A'OC', 



1 1/1 1 



Fig. 4. 



ORf 2 OA 2 \00' 2 OA 



and in the ellipse BOT, 



OT 2 



sin 2 BOA. 



Hence, observing that OB and OR' are equal, we have 



1 1 = / 1 1 \ sin 2 BOA' 



OC' 2 OA' 2 ~ \OB 2 OT 2 ) sin 2 A OR'' 



But the ellipse OAC gives 



Therefore 

 1 1 



\OC 2 OA 



(8in AOR ~ sin ' AOT} 



sin R OT sin 



1 1 \ sin 2 BOA . . V^T 



,- T5> sin ROT an SOT, 



because OJ5 and OjR are equal. 



