Polarization in Biaxial Crystals. 365 



bright point on a darker ground will be seen. These pheno- 

 mena are seen equally well in the case of a non-rotatory 

 biaxial crystal. The parts FGr and BC respectively belong 

 to waves diverging from and converging to foci. These foci 

 probably cannot be observed^ on account of their lying on or 

 near to the focal lines of the rest of the wave-surface. 



5. In order to investigate the phenomena exhibited by 

 plates in convergent light we must find the directions of 

 vibration. Dividing (4) by TVaA and (5) by TVj3X and adding 

 we have 



H II (^ +^V.X(UVaX + UVj3X) + S^X.(UVaX-UVj3X) 



-«S 7 X.(UVaX + UV/3X\ . . (7) 



This is of the form £-\-cor], where f and r) are perpendicular 

 vectors which lie in the wave-front and bisect the angle 

 between the intersection of the wave-front with planes con- 

 taining the normal to the wave-front and either optic axis; 

 i. e., % and r] are what the directions of vibration would be if 

 the crystal had no rotatory power. 



On substituting in (1) and rejecting the imaginary terms, 

 we find that er is of the form 



£ cospt -\-ri$mpt ; 



and hence the simple vibrations are elliptical, f and rj being 

 parallel and proportional to the axes of the ellipse. Since 

 (6) gives two values for X 2 , there are two such elliptic vibrations. 

 Let Ta = a, T/3 = /y, TX = l, and let the angles that X makes 

 with the optic axes be 6 1 , 2 , and let the angle between the 

 planes of Xa, and X/3 be ^. Then on substituting for X 2 from 

 (6) the ratio of the axes is 



— qM ran ft sin 0, + */ [aW sin 2 B x sin 2 9 + S 2 r X) 



for one ellipse, while the ratio, h, for the other ellipse is ob- 

 tained by changing the sign of the radical. Hence k x k 2 = — 1, 

 so that the ellipses are similar, but with the corresponding 

 axes perpendicular, and are traced out in opposite senses. 

 We have 



1 _ s/(a 2 b 2 l 2 sin 2 6>x sin 2 6 2 + S V ) 

 * 1+ *i ~ S r \ 



Let D equal the relative retardation of the two waves. 

 Then D varies as the difference of the reciprocals of the velo- 

 cities, or, with sufficient accuracy, as the difference of the 



