FRESNEL ON DOUBLE REFRACTION. 311 



which has started from E will be that which will arrive there 

 the first ; and by reason of the general property of maxima and 

 minima, all the differences will be equal and symmetrical at a 

 small distance round the shortest path E D ; that is to say, the 

 elementaiy waves which have started from points (m) and {n) 

 equally distant from E will be found behindhand by the same 

 quantity at D relatively to the wave which started from E, and 

 will therefore arrive at D in the same time. It is also in the 

 neighbourhood of a minimum or maximum of a function that its 

 variations are the least sensible ; D will therefore be the point 

 for which there will be the smallest possible differences between 

 the paths described at the same instant by the elementary waves 

 which have started from the opening m n ; and consequently it 

 is there that the most perfect accordance between their vibrations 

 will exist, if, as we have supposed, the greatest differences do 

 not exceed a semi-undulation. It is at D therefore that the 

 maximum of light will be found, and consequently E D will be, 

 for this reason as well as for all the others, the direction of the 

 luminous ray in the crystal. Now if the screen be removed, it 

 will still be true that the refracted rays which start from the 

 various points of the incident wave, considered then as indefinite, 

 are parallel to E D, that is to say, to the radius vector directed 

 towards that point of the surface of an interior wave for which 

 the tangent plane is parallel to the refracted wave. 



The meaning to be attached to the word " luminous ray" being 

 thus settled, we see that the ellipsoid constructed on the same 

 rectangular axes as the surface of elasticity, gives rigorously, 

 by the two semi-axes of its diametral section, the velocities of 

 the refracted rays perpendicular to this section, as the analogous 

 construction made in the surface of elasticity gives the velocities 

 of propagation of waves parallel to the diametral section, these 

 velocities being reckoned perpendicularly to the plane of the 

 waves. Thus understood, the first construction is a mathematical 

 consequence of the second, and represents the pha2nomena in as 

 rigorous a manner, whatever may be the energy of the double 

 refraction or the inequality of the three axes a, b, c. 



In translating into the language of the emission system the 

 law of Huygens for the double refraction of Iceland spar, M. da 

 Laplace has found, by an elegant application of the principle of 

 least action, that the difference between the squares of the velo- 



VOL. V. PART XVIII. Y 



