PRINOIPLES OF CRYSTALLOGRAPHY. 



259 



ellipse, C P C, one axis of which coincides with the known axis of elas- 

 ticity, and the other axis, o P, is perpendicular to it in the principal 

 section A B o. 



A plane M N o, (Fig. 27,) inclined to all three of the axes of elasticity, 

 cuts the ellipsoid iu an ellipse, whose axes are not parallel to any of the 

 axes of elasticity. In general, a > & > c is true in the principal section 

 A C, (Fig. 29,) whenever 



there is a radius, O &, whose ^'3-^ a ^--2-=--___ a« 



length is equal to the mid- 

 dle axis of elasticity, O B. 

 If a plane, B o &, is passed 

 through this last and this 

 radius, their section of the 

 ellipsoid is a circle; the 

 normal o a to this circle- 

 face lies in the principal 

 section of the largest and 

 smallest axes of elasticity, 

 o A C, and is called an op- 

 tica? axis. This ellipsoid, 

 which has three axes, has two optical axes, o a and o «', (Fig. 29,) which 

 are in the planes of the greatest and smallest axes of elasticity, and are 

 situated symmetrically with regard to both. 



The optical axes form with each other two supplementary angles, an 

 acute, 2 Y a, and an obtuse, 2 Y o, so that 2 Y a = 180° — 2 Y o, which 

 are equally divided by the axes A and C ; that axis which divides the 

 acute-angle axis is called the first middle line, fbisectrix,J and the one 

 which divides the obtuse-angle axis is called the second middle line, so 

 that two cases are again possible : 



First middle line ^ second middle line c : negative crystal. 

 First middle line ^ second middle line a : positive crystal. 



The first case is assumed in Fig. 29. Fig. 30 shows a sketch of the last. 



According as the nature of double refrac- /iaja. 

 tion consists in a difterence of the velocity 

 of propagation and of the direction of vi- 

 bration of the two beams of light capable 

 of being propagated in the same direction, 

 it is at once clear that the double refraction 

 must disappear along the optical axes. The 

 plane normal to a beam of light, that is, the 

 one which propagates itself in the direction 

 of an optical axis, cuts the ellipsoid in a 

 circle; the velocities of propagation of the 

 beams of light given by two radii are equal B 

 to each other; the directions of vibration are undetermined, i. €., remain 

 unchanged, as they were before their entrance into the crystalline medium. 



