Rays, inVrj/sialslxiith one andt'woAxesofHouhleRefrastioii. 1 37 



axes, because it is onl}^ by this elegant theory of Fresnel that we 

 can find the directions in which the prisms must be cut. Fresnel 

 founded his theory on two hypotheses, viz. 1. That in doubly 

 refracting crystals the elasticity of the vibrating medium is 

 diflPerent in different directions; and 2. That the vibrations of 

 the light polarized are at the same time perpendicular to the 

 direction of its propagation and to the plane of polarization. 



He supposes that in every crystallized substance there are 

 three directions perpendicular to each other, called axes of 

 elasticity, according to which the elasticity may in general be 

 different. If the elasticity is the same in all these three direc- 

 tions, the crystal belongs to the regular system, and has no 

 double refraction. If it is equal in two directions, the crystal 

 refracts doubly, and has one optical axis ; and if the elasticity 

 is unequal in all the three directions, the crystal has two optical 

 axes. From this difference of elasticity there results for light 

 a different velocity, which ought necessarily, in general, to 

 become unequal for the two rays into which the light becomes 

 divided itself, and whose planes of polarization are perpen- 

 dicular to each other. There are in crystals with two optical 

 axes only two directions ; that of the axes themselves, in which 

 the two rays are propagated with the same velocity. Con- 

 sequently, in order to appreciate the velocity of the two rajrs 

 in any direction, we must determine their planes of polariza- 

 tion, which is done by the following considerations. The 

 plane in which the two optical axes are situated contains also 

 two of the axes of ci-ystallization, one of which bisects the acute, 

 and the other the obtuse angle of the optical axes. If we con- 

 ceive, then, two planes passing through the direction in which 

 we wish to have the velocity of the two rays, and respectively 

 through each of the optical axes, the plane which bisects the 

 angle formed by these two planes will be the plane of polari- 

 zation of one of the rays, that of the other being perpendicular 

 to this plane, and passing through the given direction. 



It follows from this, that if the light comes in a direction 

 perpendicular to one of the axes of crystallization, one of the 

 rays ought to have its plane of polarization perpendicular to 

 this axis. The velocity with which these vibrations are pro- 

 pagated, depending only on the elasticity in the direction of 

 this axis, it is evident that it remains the same whatever be 

 the direction of the ray in the plane perpendicular to the axis. 

 Tlie other ray, on the contrary, whose plane of polarization 

 passes through the axes, and consequently changes with its 

 direction, will have different velocities in different directions, 

 because its vibrations being always made in the plane of the 

 other two axes of crystallization, may become successively pa- 



Third Series. Vol.. 1. No. 2. Auiiust 1832. T 



