54 Geometrical Propositions, <fbY. 



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

 semidiameters, we have the equations 



1 1 _ 1 1 1 /cos 2 g cos 2 cos 2 y 



- ~- -- 



1 _ 2 COS 2 O + i 2 COS -I- C 2 COS 2 7 _ 



" ~ 



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



in which p denotes indifferently either semidiameter, Tor Or, of the hiaxal sur- 



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



P P P 



cos a, cos j3, cos 7, and # 2 + y 2 -f z 2 instead of p 2 , we obtain, for the equation of the 

 biaxal surface, 



c 2 z 2 ) - 2 (i 2 + c 2 ) a; 2 - J 2 (a 2 



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 fin 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 

 7, 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. 



