FRESNEL ON DOUBLE REFRACTION. 29l 



axes. They are reduced to one only when two of the semi-axes 

 a, b, c of the surface of elasticity are equal to each other ; when 

 a =^ b, for example, A = 0, the two circular sections coincide 

 with the plane oi xy; and the two optic axes which are per- 

 pendicular to them coincide with the axis of z, or the axis (c) 

 of the surface of elasticity, which becomes then a surface of 

 revolution. 



This is the case of those crystals which are designated by the 

 name of uni-axal crystals, such as calcareous spar. When the 

 three axes of elasticity are equal to each other, the equation of 

 the surface of elasticity becomes that of a sphere ; the forces no 

 longer vary with the direction of the molecular displacements, 

 and the vibrating medium no longer possesses the property of 

 double refraction. This is what appears to be the case in all 

 bodies crystallizing in cubes. 



As yet we have calculated only the velocity of propagation of 

 luminous waves measured perpendicularly to their tangent plane, 

 without seeking to determine the form of the waves in the inte- 

 rior of the crystal and the inclination of the rays to their surface. 

 Whilst it is sought only to calculate the effects of the double 

 refraction for incident waves which are sensibly plane, that is to 

 say, which emanate from a luminous point sufficiently far off, it 

 is sufficient to determine the relative directions of the plane of 

 the wave within and without the crystal, since we thus find the 

 angle which the emergent wave makes with the incident wave, 

 and consequently the mutual inclination of the two lines along 

 which the visual ray or the axis of a telescope must be suc- 

 cessively directed, in order to obtain the line of sight {voir le 

 point de mire), first directly, and then across the prism of cry- 

 stal ; T say the prism, for if the plate of crystal had its faces 

 parallel, the emergent wave would be parallel to the incident 

 wave, in the case we are considering, where the luminous point 

 is supposed at an infinite distance, whatever in other respects 

 might be the energy of the double refraction and the law of the 

 velocities of propagation in the interior of the crystal. 



There cannot therefore be any sensible angular separation of 

 the ordinary and extraordinary images in this case, except inas- 

 far as the crystallized plate is prismatic ; and to calculate the 

 angles of deviation of the ordinary and extraordinary beams, 

 which by their difference give the angle of divergence of the two 

 images, it is sufficient to determine the velocity of propagation 



