REPORT ON PHYSICAL OPTICS. 377 



M. Biot made the important discovery that in many crystals 

 the extraordinary index was greater than the ordinary, and the 

 extraordinary ray therefore refracted towards the axis. Cry- 

 stals of the latter kind he called attractive, while those of the 

 former were called repulsive ; the extraordinary refraction being 

 ascribed, in the theoryof emission, to attractive or repulsive forces 

 which act as if they emanated from the axis*. These crystals 

 are now generally distinguished by the denominations positive 

 and negative. The Huygenian law applies to positive as well as 

 to negative crystals ; the spheroid being prolate in the former 

 case and ohlate in the latter. 



The construction given by Huygens for the direction of the 

 two refracted rays is, it has been stated, an immediate conse- 

 quence of the assumed form of the wave-surface. It easily 

 appears, from the principle of Huygens already adverted to, 

 that the same construction will apply in all cases, whatever be 

 the form of the wave or the law of the velocity of propagation 

 within the crystal ; — so that the law of direction is determined 

 when that of velocity is known. A similar connexion between 

 the velocity of the molecule and its path is established, in 

 the theory of emission, by the law of least action. This 

 principle, we know, holds generally in the motion of a point 

 subjected to the action of attracting or repelling forces ; and in 

 applying it to the case of a luminous molecule acted on by 

 forces emanating from the particles of the body which it meets, 

 we may leave out of consideration the insensible curvilinear por- 

 tion of the trajectory described in the passage from one medium 

 into another of different density, — provided we assume, with 

 Newton, that the forces exerted by the molecules of body on 

 those of light are sensible only at insensible distances. In this 

 simplification of the prcjblem we have to deal only with straight 

 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 solu- 

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

 in transparent media f ; and he has arrived at two equations, in 

 which that solution is completely contained. Laplace applied 

 these results 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 



• Mem. Inst. 1814. t Ibid. 1809. 



