314 FRESXEL OX DOUBLE REFRACTION. 



Let y = p X -\- qz be the equation to a diametral plane drawn 

 perpendicularly to a luminous ray of any direction ; we have to 

 calculate the difference between the two quotients of unity divided 

 successively by the squares of the semi-axes of its elliptical sec- 

 tion, as a function of the angles which this plane makes with the 

 two circular sections ; for these angles are equal to those which 

 the normal to this plane, or the luminous ray, makes with the 

 normals to the two circular sections, that is to say, with the two 

 optic axes of the crystal. Now if we denote by [m) the angle 

 contained between the plane y = p x ■\- qz, and the circular sec- 

 tion M M', and by [n) the angle which it makes with the other 

 circular section N N', we have 



P '^.f — ff — q ^ff — h 



and 



V/-A X \^l -Vp'^ + q' 

 j s/JZT-g + q Vg - h 



f — g) (cos n — cos rtif 

 p^ [g — h) (cos n + cos irif' 



whence we have 



q^ _ [f — g) (cos n — cos ni 



and 



I 



■{f—h){g — h){cos7i + cosm)^—{/—g){f—h){cos7i — cosm)'^ + 'i{f—g){g—l ^ 

 {/— h){g — h) {cos n-\- cos m)^ 

 Let us now calculate the two diameters of the elliptical section, t 

 which give the velocities of the ordinary and extraordinary ray I 

 perpendicularly to the plane of this section. To this end it is 

 sufficient to form the polar equation to the ellipsoid, and to seek 

 the maximum and minimum values of the radius vector in this 

 plane. Let x = uy and z = fty he the general equations to the 

 radius vector ; the square of its length will be equal to ^-^ + y^ + z^ 

 or to ?/- (1 +«^ + /3^), [y) corresponding to the point of intersec- 

 tion of the straight line with the surface of the ellipsoid. The , 

 equations to the straight line and to the surface being true at j I 

 the same time for this point, we have y'^ [fa.'^ + h^^^ + g) = I; 



whence we obtain y^ = x-^5- — r— 15 , and consequently the 



/a^ + h ^^ + g' ^ •' 



square of the radius vector is equal to -r-^ — %-„t^ — , an ex- 



