HO REPORT ON PHYSICAL OPTICS. 



lines and uniform velocities; and when the dependence of these 

 velocities on the directions is assumed, or given, the principle in 

 question furnishes a relation between the directions of the two 

 portions of the trajectory. Such was the problem whose solution 

 was given by Laplace, in his memoir on the motion of light in 

 transparent media ;* and he has arrived at two equations in which 

 that solution is completely contained. Laplace applied these re- 

 sults to two cases : one in which the difference of the squares of 

 the velocities of the incident and refracted rays is constant, and 

 the other in which that difference is equal to a constant quantity, 

 plus another varying as the square of the cosine of the inclination 

 of the refracted ray to the optic axis. In the former of these 

 cases he obtained the known law of Snellius ; and the formulae of 

 refraction at which he arrived in the latter were found to be iden- 

 tical with those furnished by the construction of Huygens. 



The velocity of the extraordinary ray, assumed by Laplace, is 

 the reciprocal of the radius-vector of the ellipsoid of Huygens, and 

 therefore the inverse of the assumed velocity in the wave-theory. 

 But Laplace himself has shown that the construction suggested by 

 that theory, and employed by Huygens for the determination of 

 the direction of the refracted ray, resolves itself into the principle 

 of least time, and that whatever be the form of the wave-surface ; 

 and as the law of least action and that of least time are identical, 

 provided the assumed velocities be reciprocal, it ceases to be strange 

 that two such very different methods should lead precisely to the 

 same result. The difference between Huygens and Laplace, as to 

 the mode of deducing the law of extraordinary refraction, is in 

 fact precisely the same as that which existed formerly between 

 Fermat and Maupertuis with regard to the ordinary law of the 

 sines. 



This identity of the results afforded by the two theories has 

 since been more distinctly pointed out by M. Ampere. By means 

 of the principle of least action he has arrived at the following 

 general conclusion, whatever be the assumed law of the veloci- 

 ties, that if from the point of incidence on any extraordinary 

 medium, as centre, two surfaces be described whose radii- vectores 

 are inversely as the velocities of the incident and refracted rays in 

 their directions, and if the incident and refracted rays be pro- 



* Mem. Inst. 1809. 



