Behaviour oj Pleochroilic Crystals alone/ Optic Asses. 91 



The relative position of the two pairs A l5 A 2 and Bj, B 2 of 

 axes is in general as impossible to assign as that of the tensor- 

 triplets rt l9 a 2 , a d and di, b 2 , b % . It is only when the crystal 

 possesses special symmetry that it becomes possible to say 

 something more or less definite. Fig. 1 gives some idea 



Fig. 1. 



regarding the general position of these axes ; all the charac- 

 teristic lines are drawn through the centre of a sphere, and their 

 intersections with the spherical surface indicated. 



3. The fundamental formula) in the theory of plane wave 

 propagation are obtained most simply by introducing a 

 system of coordinate axes .r, y, z, of which z coincides with 

 the direction of propagation. Jf, then, we denote the com- 

 ponents of the two tensor-triplets along these axes by a u , a 22j 

 «33, ^28? a 3b **i2 an d &n, h 22 , b dd , & 23 , hu 6)2*? and if we write 

 shortly 



<*hk + ihk = CM, (4) 



where i= »J — 1, then 



(^11 — ^) (C-22-V 2 ) = C 2 rp 



r 



+ 



— C22 9 _ 1 



(5) 

 (6) 



where v stands for the so-called complex velocity, gjf for the 

 ratio of the complex amplitudes for Neumann's vibration- 

 vector along the y and x axes respectively. Expressed in 



* If oh, j8a, yh are the direction-cosines of the triplet a 1} a 2 , a 3 rela- 

 tively to the axes x, y, z, then 



a u = a l a ] 2 +as./+a :i a.r, .... 



