308 FRESNEL ON DOUBLK llEFRACTIOTV. 



We may deduce from the construction of Huygens applied 

 to equation (D.) general formulae, which give the direction of the 

 refracted rays for all directions of the incident rays, and of the 

 surface of the crystal relatively to these axes, as Malus has done 

 for Iceland spar, where the extraordinary wave is an ellipsoid of 

 revolution. I have not calculated these formulfe, of which I had 

 no need, in order to verify my theory on topaz. In general, so 

 long as we are concerned with crystals whose double refraction 

 is feeble, and when we confine ourselves to the investigation of 

 the divergence of the two beams obtained by forming the crystal 

 into a prism, it is sufficient to determine, in the first place, ap- 

 proximately the direction of the luminous ray in the interior of 

 the crystal by the law of Descartes, with the index of refraction 

 of the ordinary or extraordinary rays ; and when we thus know 

 the approximate direction of the refracted ray, we may calculate 

 the two corresponding velocities by means of equation (D.), or 

 the two velocities of the wave measured perpendicularly to its 

 plane by means of equation (C), which represents the section 

 made in the sin face of elasticity by a diametral plane parallel to 

 the wave, and in which (??2) and («) are given as soon as we know 

 the direction of the refi-acted wave. These two velocities once 

 known, it becomes easy to deduce from them the direction and 

 the divergence of the two beams, or of the two systems of emer- 

 gent waves. 



If greater accuracy however were desired, it would be necessary 

 to determine with the velocity thus calculated a new and more 

 approximate direction of the ray or of the plane of the wave in 

 the crystal, and calculate afresh the corresponding velocity by 

 the aid of equation (D.) or of equation (C), according as we wish 

 to obtain the velocity measured on the ray or the normal to the 

 plane of the wave ; then we can deduce from this the direction 

 of each of the two emergent beams. This method is quite as 

 accurate and much less laborious than employing the formulae of 

 which we have spoken, which would be doubtless very compli- 

 cated. It may aho be applied to crystals whose double refrac- 

 tion is more powerful, by repeating the operation a sufficient 

 number of times. 



When it is sought to verify the law of the velocities by an ex- 

 periment of diffraction, it is sufficient to consider the velocity of 

 propagation of the refracted wave measured perpendicularly to 

 its plane. This is even the most simple method, since the ex- 



