Refracted Rays of Light in Crystalline Media. 2G5 



they should be. Unless otherwise stated, they will be omitted 

 from the future discussion of the problem. 



The analytical discussion has been limited to the properties 

 of the locus in the three principal planes of the crystal. Let 

 X be the axis of greatest elasticity, Y of mean, and Z of the 

 least elasticity; and let a, b, c be the velocities of light along 

 these three axes respectively : then a>b>c> If x\ y, 7} and 

 #", y", 2" are the intersections of two radii vector es with the 

 two sheets of FresneFs wave-surface, F = 0, tangents at these 

 points are expressed by the equations 



<-*>w +<*-*& + <*-? f =°> 



Then the locus, F = 0, of the intersections of these tangents is 

 the required envelope. Using the section in the plane XZ 

 we have the following equations : — 



x[ _ zT 



x" " *"' 



since the two vectors are assumed to coincide ; also the 

 equations of the circle and the ellipse, 



x ,2 + z ,2 = h 2 , 



a?x"? + c 2 z" 2 = a 2 c 2 ; 



lastly tangents to these 



xx' + zz f = b 2 , 

 a?jcx" + c 1 zz ,f = ci 2 c\ 



Changing to polar coordinates, where b and p are the lengths 

 of the vectors and a the angle with the axis X, then 



9 a 2 c 2 (a 2 c 2 ) 2 



P 2 = 



a 2 cos 2 a. + c 2 sin 2 a (a 2 x cos a -j- c 2 z sin «) 2 ' 

 x cosnc + ^sin a — b, 

 cos 2 a -f sin 2 a =1. 



