378 VOURTH RKPORT — 1834. 



Snellius ; and the formulae of refraction to which lie arrived 

 in the latter were found to be identical with those furnished by 

 the construction of Huygens. 



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

 the I'eciprocal 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 it- 

 self into the principle of least time, — and that whatever be the 

 form of the wave-surface ; and as tlie 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 sui'faces be described whose radii-vec- 

 tores are inversely as the velocities of the incident and refracted 

 rays in their directions ; and if the incident and refracted rays 

 be produced to meet these surfaces, and tangent planes be 

 drawn at the points of meeting, the line of intersection of these 

 planes will be on the separating surface of the two media*. 

 Hence the position of the refracted ray is determined when that 

 of the incident ray is known ; and the construction thus sup- 

 plied for its determination is obviously the generalization of the 

 construction of Huygens already alluded to, if only the radii- 

 vectores be taken in the direct ratio of the velocities, instead of 

 the inverse. 



It is obvious, then, that the problem of double refraction, con- 

 sidered as a physical question, resolves itself into the determina- 

 tion of the law of velocities. Newton showed that the constant 

 ratio of the velocities in ordinary media, and therefore the law 

 of the sines, could be explained on the supposition that the lu- 

 minous molecules are solicited by attracting forces emanating 

 from the molecules of the refracting body, and sensible only at 

 very small distances. The phenomenon of extraordinary re- 

 fraction, in like manner, was ascribed by Laplace to the operation 

 * Mem. Inst. 1815. 



