54 Geometrical Propositions, &c. 



an ellipsoid is constant, as well as the parallelepiped described on three conjugate 

 semidiameters, we have the equations 



1 1 1 1 1 1 



1 _a i cos 2 a + 2 cos j8 + c 2 cos 2 7 _ 

 r' 2 r" = * i 2 c 2 



Whence it appears that /, /', are the values of p in the equation 



in which p denotes indifferently either semidiameter, OT or OF, of the biaxal sur- 



face. Therefore putting for M and N their values, and writing -, -, -, instead of 



P P P 



cos a, cos j8, cos 7, and # 2 + y z + z 3 instead of p 2 , we obtain, for the equation of the 

 biaxal surface, 



This is the equation of the surface of refraction for a biaxal crystal in which 

 a, b, c, are (54) the three principal indices of refraction, taking OS the radius of 

 the sphere to be unity. The left-hand member of the equation is therefore the ex- 

 pression supplied by FRESNEL for the function 7 in Art. 51. 



"When the faces of the crystal are parallel to any of the principal planes of the 

 ellipsoid to the plane of xy for example the nature of the ring -trace may be 

 found very easily. For if the difference of the two values of z, deduced from the 

 preceding equation of the surface of refraction, be put equal to a constant quantity 

 J, the result, when cleared of radicals, will be an equation of the fourth degree in 

 x and y, which will be the equation of the corresponding ring-trace. This is a case 

 that occurs frequently in practice ; the crystal being often cut with its faces per- 

 pendicular to the axis of x or of z, because these lines bisect the angles made by the 

 optic axes. 



