FRESNEL ON DOUBLE REFRACTION. 



313 



the second for the axis of [z], the 

 mean diameter coinciding with the 

 axis of (y), projected in A, the centre 

 of the eUipsoid. If we give the 

 name of optic axes of the medium 

 to the directions along which the 

 luminous rays which traverse it can 

 have only one velocity, those which 

 possess this property are, according 

 to the construction which deter- 

 mines the velocity of the luminous rays, the two diameters of the 

 ellipsoid perpendicular to the circular sections. Next, let the 

 equation to the elUpsoid be 



If in this we put ?/ = 0, we shall have fx"^ + A ^r^ = 1 for the 

 equation to the ellipse C M B N C M' B' N' situated in the plane 

 of the figure, which we shall suppose to coincide with that oi xz. 

 The two diametral planes M M' and N N', which cut the ellipsoid 

 in a circle, pass through the mean axis projected in A, and must 

 be inclined to the axis of x at an angle (i), such that the semi- 

 diameters A M and A N may be equal to the mean semi-axis of 

 the eUipsoid, or that the squares of the former may be equal to 



the square of the latter, which is — . Denote AM or A N by 



(r), we shall have 



z =: r sin i, and x = r cos i. 

 Substituting these values in the equation to the ellipse fx^ 

 + hz^ = 1, we have 



fr"^ . cos^ i + h r^ sin^ i = 1 ; 



or, since r^ = — , 



whence we obtain 



fcos^ i + h . sin^ i = ff; 



^f-9 



_ g — h 



tan' 



Hence the equation to the plane AM is r = jr A/ ' -,'■> and 



that to the plane AN of the other circular section z=—x\ /*~^. 



V (j-k 



Y 2 



