Volcano Usbima, Ilzn. 



63 



lows: r>ï'>f^<"-- For the determination of the limiting values 

 of i3, the following method was adopted. 



The method is based upon the principle that the value of 3 

 lies always between the two values of the refractive indices to bo 

 observed on a crystal grain of any orientation. 



To prove this principle, let N^ and N-i be respectively the greater 

 and the smaller normal velocities in any direction N (Fig. 33) in a 

 crystal, that makes the angles, ^p and ^' (where c5>^' without any 

 loss of generality), with the two poles of the optic binormals, A 

 and B ; and let a and c be respectively the greatest and the 

 smallest principal velocities. Then iVi and Ni -may be expressed by 



1 1 



N{ = ——(a- + C-) + -^— (a" — q) cos (^ — cp'), 



iV,- = —-(a' + G-) + ---" {a- - C-) cos (f + ^'). 1 

 'A "A I 



■(1) 



Since the values of the normal 

 velocities are the reciprocals of those 

 of the refractive indices, we have the 

 relations 



N,-- 



N, = - 



a = - 



Fig. 33. 



1 



n, 



a 

 1 



Wehere iii and ri: are the refractive indices for the faster and the 

 slower waves respoctively. From those relations we may trans- 

 form (1) into 



