322 FRESNEL ON' DOUBLE REFRACTIOX. 



diametral section made in the ellipsoid perpendicularly to the 

 direction of the luminous ray, do not now give the direction of its 

 vibrations ; so that the planes which are perpendicular to them 

 are no longer the true planes of polarization of the refracted 

 waves. The rule of M. Biot therefore does not rigorously agree 

 with our theory. But it must be recollected, — 1st, that in the 

 crystals employed by him, the normals to the circular sections 

 of the surface of elasticity differ so little from the direction of the 

 true optic axes, that they might be confounded without producing 

 any sensible error in the direction of the planes of polarization ; 

 2nd, that in the same crystals the rays directed along the 

 optic axes are nearly normal to the corresponding waves ; 3rd, 

 that this skilful experimenter could only determine directly the 

 plane of polarization of the incident or emergent beams, and not 

 that of the refracted rays. The small differences which are here 

 indicated to us by the theory, would doubtless be very difficult 

 to observe, even in those bi-axal crystals whose double refraction 

 is most powerful; for we cannot detevijii«»»»very accurately by 

 the known methods the direction of the plane of polarization of 

 a luminous ray, and there is here an additional difficulty, that 

 of fixing the direction of the plane of polarization in the interior 

 of the crystal from observations made on the emergent rays. 

 Hence, far from seeing an objection against our theory in the 

 rule given by M. Biot, it ought rather to be considered as being 

 a confirmation of it, since the small discordance which exists 

 between them must necessarily have escaped his observations. 



Most crystals present but little difference between the planes of 

 the circular sections of the surface of elasticity and of the 

 ellipsoid constructed on the same axes. 



The two circular sections of the surface of elasticity are equally 

 inclined to the plane of xy, which passes through the mean 

 axis, and the tangent of this inclination is, as we have seen, 



\ / ,^ _ j ; the tangent of the angle which the two circular 

 sections of the ellipsoid make with the same plane is equal to 



«2 _ /^2 



-5. We see by these formulas, that when the double 

 c^ 



refraction is not very powerful, that is, when (c) differs little from. 



c . 

 (a), — being nearly equal to unity, the planes of the circular 



c_ f a^- 

 a \/ b^- 



1 



