12 The Double Refraction of Light in a Crystallized 



" Describe an ellipsoid whose axes are in the directions of the 

 axes of elasticity, the squares of their lengths being directly as 

 the elasticities in their respective directions ; cut the ellipsoid by 

 a plane through its centre, as QOq, and in a perpendicular to 

 that plane take two portions OT and V equal to the semiaxes 

 OQ and Oq of the section. The double surface which is the locus 

 of the points T and Fis the surface of the wave." 



As to the planes of polarization of the rays belonging to the 

 two parts of the wave, Fresnel has shown how to find them by 

 means of a surface which he calls the surface of elasticity. But 

 it is desirable to be able to find them by means of the same 

 ellipsoid which serves to find the surface of the wave. Now TS, 

 parallel to OR, is the direction of the vibrations of the ray OT, 

 and the tangent plane at Q is perpendicular to OR, and there- 

 fore parallel to the plane of polarization of the ray OT. In like 

 manner the plane of polarization of the ray V is parallel to 

 the tangent plane at q. Hence the planes of polarization of two 

 different rays, having a common direction, are parallel to the 

 planes which touch the ellipsoid at the extremities of the semi- 

 axes of the diametral section perpendicular to their common 

 direction. 



When two of the axes are equal, the ellipsoid becomes a 

 spheroid, and the crystal is said to be uniaxal, the double refrac- 

 tion being regulated by the third axis which 

 is perpendicular to their plane. Let A. OB 

 be a section of the spheroid through the 

 third axis OA, which is its axis of revolu- 

 tion ; take OK = OB, and OA' = OA, and 

 let the ellipse A OK and the circle BOB' 

 revolve about OB" as an axis ; they will 

 describe a surface compounded of a sphe- Fig. 6. 



roid and sphere, which will in this case be the surface of the 

 double wave. " For if OH be the direction of a ray, and if a 

 plane perpendicular to OM cut the ellipse AOB in OQ, and the 

 equator of the spheroid in Oq, the lines OQ and Oq, of which 

 the latter is equal to OB, will be the semiaxes of the section 



