OPTICAL PROPERTIES IN CRYSTALS 167 



and the equation for the velocities becomes 



2v~ = a'isin (p sin' 6 + cos"" 6) + ft"(cos" (p sin' 6 -\- cos' d) + c' sin' d 

 /(o2 - 62)2(cos2 cos'^ (^ + sin2 <py + 2(a2 - b-'){c^ - b'-) 

 ^ y sin2 ^(cos2 d cos2 ^ _ gin^ ^) + (c^ - b^y sin* ^ 



(28) 



A very elegant construction for the wave-velocities and the directions of 

 vibration is the Fresnel index ellipsoid. Consider the ellipsoid 



aV + by- + ch'- =1 (29) 



Then FresneP showed that, for any diametral plane perpendicular to the 

 wave normal, the two principal axes of the ellipse were the directions of the 

 two permitted vibrations, while the wave velocities were the reciprocals of 

 the principal semi-axes. 



We wish to show now that the maximum and minimum values of the im- 

 permeability constants in a plane perpendicular to the direction of the 

 wave normal determine the directions of vibration and the values of the two 

 velocities. To show this we make use of the fact that /?,•,■ is a second rank 

 tensor and transforms according to the tensor transformation formula 



0;, = p ^ ft, (,,0) 



dXk dxt 



where the partial derivatives are the direction cosines 



Expanding equation (30) the six transformation equations become 



/3ll = <^l/3u + 2AWl/3i2 + 2A»l/3l3 + Wi/?22 + 2oti»ijS23 + "1/333 



0'u = ('Mn + (dm, + w/2)/5l2 + (A«2 + w/2)/3i3 + mim^22 



+ (niirh + WlW2)i323 + "l«2^33 



+ ("1W3 + WiH3)/323 + "l"3iS33 (31) 



' See for example 'Thotoelasticity," Coker and Filon, Caml)riclge University Press, 

 pages 17 and 18. 



