Transparent Inactive Crystal Plates. 



171 



of Fresnel's ellipsoid (F). Each point S of the ray surface (2) 

 defines a ray direction ; the normal OQ to the tangent plane 

 SQ through S of the ray surface is then the radius vector of 

 the normal tu the wave producing the ray, S. (Fig. 3.) The 

 extension of this wave normal vector to its reciprocal length 

 ON determines a point 1ST on the index surface. The two points 

 N and S are said to be corresponding points, and the plane 

 NOS is normal to the polarization direction. 



Similarly, a point p of the index ellipsoid (I) is the corre- 

 sponding point P of Fresnel's ellipsoid (F), (fig. 4), if its radius 

 vector coincides in direction with the normal op' to the tan- 

 gent plane Pp' through F and is equal in length to the recip- 

 rocal of the normal Op'=q. The radius vector, OP repre- 

 sents a ray of velocity s (fig. 4) while the radius vector Op = - is 



the reciprocal of the corresponding wave normal velocity q. The 

 normal to the plane Pop is then the polar- 

 ization direction. By obtaining the coor- 

 dinates of such corresponding points on the 

 two ellipsoids (F) and (I), Potier discovered 

 a simple relation between the expressions 

 Z, m, n, p, of equations (18a) which has 



Fig. 4. 



of 



proved of great value in the solution 

 problems of reflection and refraction. 



For the sate of simplicity, let the equa- 

 tion of Fresnel's ellipsoid and the index 

 ellipsoid be referred to the principal axes; 

 Fresnel's ellipsoid is then represented by 



2F=- + 



y 



z 

 c 



:l 



21: 



the index ellipsoid by 



(22) 



(23) 



If the coordinates of a point P on Fresnel's ellipsoid be x° ', 

 2/°!, s°j, then the equation of the tangent planes through P is : 



<~M?D 



+ {y— V 



•€) 



+ (*-3°J 



\9 x/& 



= 



The equations of the normal to this plane are 

 x y z 



)z° 



(i y \ 



9x Jx c 



(f\ 



(24) 



9y/y° 1 \9zjz 



By definition, the point p is common both to the normal and 

 the index surface and its coordinates, aj„ y,, s,, are readily 

 found by means of (22), (23), and (24), to be : 



