Refracted Bays of Light in Crystalline Media. 271 



of the ray in the medium be 90°, and the angle in the crystal 

 cf) • then by the ordinary sine law the refractive index of the 

 medium is' 1*3484. On the other hand, if the angle in the 

 medium be 90°, and the angle in the crystal be a, as given 

 above, then the refractive index of the medium is 1*5539. 

 Or, in other words, if the refractive index of the medium is 

 less than 1*3484, there can only be one ray which will enter 

 the crystal, however it is cut, without divergence, and that is 

 along the optic axis. If 1*3484 < l/d< 1*5539, there are two 

 directions of non-divergence, and if l/d> 1*5539, three ; one 

 always being in the direction of the optic axis. 



Having established the necessary conditions for the bound- 

 ing medium, it is possible to determine a section of Iceland 

 spar which will give three coincident pairs of ordinary and 

 extraordinary rays. The equation of the envelope (J>) is put 

 for calculation in the form 



( s i n 56>_ 77?0 2 s i n 2 a ) | A sin «0 + B s in 4 <9 + C sin 2 + D\ = 0, 



where 



A = — 1 /cos l a, 



B= fW^a m , 1 j 



(, cos a cos 2 « J 



C = - ( 2m 2 (m 2 -f- m e *) ?2j? -f (m 2 -f m*) 2 } ■ 



D = 77? 2 sin 2 a (m 2 — m e 2 ) 2 . 



From the refractive indices of spar given above, wi = 

 1*65846/1*63034 and m e = 1*48654/1*63034. a is taken equal 

 to 77°. Substituting these values in the equation, 



(sin 2 0-w o 2 sin 2 a) {390-524 sin c 0-457-418sin 4 



+ 75*941 sin 2 £-0*0308} = 0. 



The roots by approximation are :— 6>'=80° 13', 0" = 26° 33', 

 f// =x— 82° 23'. The last value gives the direction of a ray 

 which will be refracted undivided along the optic axis. The 

 fourth pair of roots is due to the intersection of the surface 

 with the dotted branch of the locus, and is here, as afterwards, 

 omitted. 



If the surface is tangent to the locus, there are only two 

 solutions. Here <*=69° 32 / 30" ; 0' = 0" = 55° 47' 45" j and 

 6>'"=-72°22 / 50". 



