Refracted Rays of Light in Crystalline Media. 2G9 



for a nniaxal crystal, 



x 2 = h 2 , 



„_ a 2 !> 2 

 Z " d l + b r 



These points, since a>b> c, are essentially real, and there 

 are in each case four intersections with the axes ; two, at 

 least, must be singular points, and if the curve of the locus 

 does not cross an axis all four have this property. They and 

 the asvmptotes have been located approximately on the 

 diagrams. 



An examination of the figures shows that a line through 

 the origin of coordinates may cut the locus in four pairs of 

 points. Since this line forms the trace of the surface of the 

 crystal with a coordinate plane, there are, at most, three 

 directions in which the ordinary and extraordinary rays 

 coincide for this single section. There are, of course, an 

 infinite number of these sections. It should be noted that the 

 two intersections with the horizontal branch on the right 

 must be taken with the intersection with the vertical branch 

 on the left, and vice versa. If the surface of the crystal is cut 

 tangent to the horizontal branch of the locus, two directions 

 of coincidence are possible; for greater angles, only one. 

 With a uniaxal crystal one of these solutions is due to a ray 

 along the optic axis. As biaxal crystals, generally, have two 

 of their in lices of refraction nearly equal in value, one branch 

 of the locus approximates closely to a straight line ; so one 

 pair of coincident rays is found near the axis of the ellipse 

 which is more nearly equal to the radius of the circle. 



Hitherto we have discussei only the conditions for the 

 coincidence of the two rays inside the crystal. To complete 

 the investigation the direction of the corresponding incident 

 ray in the bounding medium must be determined. Huyghens' 

 principle gives at once the solution. A circle is drawn, 

 shown dotted in fig. 1 (p. 2o3), whose radius is proportional 

 to the velocity of light in the medium, and a tangent is drawn 

 to it from the point S'. The ray is then perpendicular to the 

 tangent. Let d be the radius of this circle, the angle of 

 incidence of a ray in the bounding medium, and a the angle 

 between the normal to the surface and the axis X. Then 

 (6 + a) is the angle made by the ray and X, and it' x, z are the 

 coordinates of a point on the envelope, 



x cos ex + z sin a = 0. 

 x cos (a + 6) + z sin (a f 6) — d. 



