T7 



sation of the ray OT. In like manner the plane of polarisation of 

 the ray OV is parallel to the tangent plane at q. Hence the planes 

 of polarisation of two different rays, having a common direction, 

 are parallel to the planes which touch the ellipsoid at the extremities 

 of the semiaxes of the diametral section perpendicular to their com- 

 mon direction. 



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

 and the crystal is said to be uniaxal, the double refraction being re- 

 gulated by the third axis which is perpendicular to their plane. Let 

 AOB be a section of the spheroid through the third 

 axis OA, which is its axis of revolution ; take OB' 

 = OB, and OA' = OA, and let the ellipse A'OB' 

 and the circle BOB' revolve about OB' as an axis ; 

 they will describe a surface compounded of a sphe- 

 roid and sphere, which will in this case be the sur- 

 face of the double wave. For if OM be the direc- 

 tion 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 QOq. Taking therefore OT and OV equal to OQ and 

 OB, the locus of V will evidently be the circle BOB' ; and the locus 

 of T will be the ellipse A'OB', since the angle B'OT is equal to the 

 angle BOQ. 



In the general case, OT and OV, which are equal to OQ and Oq, 

 represent the velocities of the two different sorts of rays having a com- 

 mon direction in the crystal ; and two right lines, which lie in the 

 plane of the greatest and least axes of the ellipsoid, and are perpen- 

 dicular to its two circular sections, are called the optic axes. It 

 appears, therefore, by the 6th lemma, that the difference of the 

 squares of the reciprocals of the velocities of the two rays having a 



