324 



FRESNEL ON DOUBLE REFRACTION. 



in E and O, the points of contact of this plane with the surface 

 of the wave ; hence the radii vectores A O and A E are the direc- 

 tions of the ordinary and ex- 

 traordinary rays which corre- 

 spond to the plane wave T S, 

 parallel to the circular section 

 ofthe surface of elasticity; and 

 they traverse the plate t st' s' 

 in the same interval of time, 

 although by following differ- 

 ent routes. The radius vec- 

 tor AL, drawn to the point 

 of intersection of the ellipse 

 and circle, and for which the 

 two values obtained from the 

 equation to the wave become 

 equal, is the direction along 

 which the luminous rays can 

 have only one velocity, and 

 consequently that of the nor- 

 mal to the circular section of the ellipsoid, which we have called 

 the optic axis. We find for the tangents of the angles which 

 these three radii vectores make with the axis of x, 



tanQAT = ^^^,--p,tanLAT=-^^^-p, 



tan EAT = y^^-,-^. 



We see that these expressions differ only by the factors -g, — , 



which in most crystals are very nearly equal to unity. 



All the ordinary and extraordinary rays parallel to L A tra- 

 verse the crystal in the same interval of time and with the same 

 velocity*, because they also follow the same path ; but they 

 necessarily diverge outside the ci'ystal, because the two tangent 

 planes drawn tarough the point L to the two sheets of the wave 

 surface make w-ith each other a sensible angle. On the con- 

 trary, the rays A E and A O, which take also the same time in 



* Whatever be the directions of the faces of entrance and emergence, since 

 these rays follow the same route L A ; whilst the rays E A and O A do not take 

 exactly the same time to traverse the crystalline plate, except when its faces 

 ts and <'«' are parallel to one ofthe circular sections ofthe surface of elasticity. 



