304 FRESNEL ON DOUBLE REFRACTION. 



Let z = px + qy he the equation of the cutting plane ; the 

 squares of the two axes of the section are given by the following 

 relation, 



fl2 ( J2 _ ^2) (^2 - r^) / + J2 (^2 _ ^2) (^2 _ ^2) ^2 



+ c^ (a2 _ 7-2) (6^ - r-2) = 0, 



in which (r) represents the gi'eatest and least radius vector of 

 this elliptical section. 



The equations of a straight line drawn through the centre of 

 the ellipsoid perpendicular to the cutting plane, are 



X = — pz, and y ■= — qz; 



XV 



whence jo= j9'= "~— 5 and substituting these values in 



the above equation, we have 



^2 ^2 (^2 _ y2) (^2 _ ^2) ^ ^2 ^2 (^2 _ ,^2) (^2 _ ^2') 

 + C2^2(a2_r2) (62-^2) ^Q. 



or, effecting the multiplications, 



(«2.r2+ 6'2?/2 + 6-2-2)^4 



- [a2 (^2 + C2) ^2 + ^2 (^2 ^ ^2-) ^2 + ^2 (^2 ^ ^2) _^2-] ^2 

 + a2 ^2 ^2 (^2 + 2/2 + ^2) ^ 0. 



Finally, observing that r^ = x^ + «/■ + ^^ and suppressing the 

 common factor {x"^ + tj- + ;r'), we arrive at the equation (D.), 



(^,2 + ^2 + ^2) (^2^2 ^ ljiy<i + ^2 .2) _ ^2 ^^2 ^ ^2) ^2 

 - 52 (o2 ^ ^2') j/2 _ ^2 (^2 ^ J2) _^2 ^ ^2 ^2 ^2 = Q. 



If we wish to refer the surface of the wave to polar coordinates, 

 we must put (r^) in the place of [x^ + ^^ + z^), and substitute 

 for .r^, y-, z^ their values r^ cos^ X, r^ cos^ Y, r^ cos^ Z, which 

 gives the following equation, 

 (a2 cos2 X + ^2 cos2 Y + c2 . cos^ Z) t-" 



- [«2 (62 + c2) cos2 X + 62 (a2 + c2) cos2 Y + c2 (a2 + 62) cos2 Z] r2 



+ «2J2c2= 0, 



by the aid of which we may calculate the length of the radius 

 vector of the wave, that is to say, its velocity of propagation 

 reckoned along the direction of the luminous ray itself, when we 

 know the angles which this latter makes with the axes of elasti- 

 city of the crystal. 



It is easy to assure oui'selves that the intersections of the sur- 

 face represented by the equation (D.) with the coordinate planes 



