FRESXEL ON DOUBLE KEFRACTION. 327 



zation perpendicular to these vibrations. Only, the velocities 

 \vhich we should then consider would no longer be those of the 

 luminous rays, but those of the waves measured on the normal 

 to their surface ; and the two planes forming the acute and ob- 

 tuse dihedral angles which the planes of polarization divide each 

 into two equal parts, instead of passing through the luminous 

 ray and the optic axes properly so called, would be drawn along 

 the normal to the wave and the normals to the two circular sec- 

 tions of the surface of elasticity. The tangent of the inclination 



of these sections to the semi-major axis (a) is equal to a / -j^ ^ 



an expression less than unity when a^ — 6^ is Zb'^ — c^, and 

 greater when {a^ — b^) is y {b^ — c^), or, which comes nearly to 

 the same thing, when {a — b) y (b — c). In this second case, 

 the angle of the two circular sections, or of their normals which 

 contains the minor axis (c), is therefore obtuse, whilst it is acute 

 in the first case. 



Hence the Avaves whose planes of polarization are comprised 

 in the acute angle between the two planes drawn along the nor- 

 mal to the wave, and the normals to the planes of the circular 

 sections are those whose velocities of propagation vary between 

 the narrowest limits, whilst the velocities of the waves whose 

 planes of polarization pass within the obtuse dihedral angle un- 

 dergo more extensive variations. It is therefore natural to call 

 the rays corresponding to the former ordinary rays, and those 

 of the other waves extraordinary rays, as M. Biot and Sir David 

 Brewster have done. 



Particular case where there would no longer be any reasons for 

 giving the name of ordinary ray to one of the two beams rather 

 than to the other. 



A case is conceivable in which, the two beams undergoing 

 variations of velocity equally extensive, there would no longer be 

 any reason for giving the name of ordinary beam to one rather 

 than the other ; this would be the case if the two optic axes 

 were perpendicular to each other, because then we should have 



-^ \/fr3^ = Ij or c2 {a^ - b^) = a^ (^^ - c^) ; which sup- 

 poses that (a — b) is very nearly equal to {b — c), since we may 



VOL. V. PART XVIII. Z 



