DOUBLE REFRACTION. 117 



represented by that diameter, and the vibrations in each wave will 

 preserve constantly the same direction. 



Thus it appears that a polarized plane wave will be resolved 

 into two within the crystal ; and these will be propagated with 

 different velocities, and consequently follow different paths. The 

 amplitudes of the component vibrations are as the cosines of the 

 angles which the direction of the original vibration contains with 

 the two fixed rectangular directions ; and as the squares of these 

 amplitudes represent the intensities of the two pencils, the law 

 of Malus respecting these intensities follows as an immediate 

 consequence.* Again, the planes perpendicular to these two 

 directions are the planes of polarization of the two pencils ; and 

 it is easily inferred that one of them must bisect the dihedral 

 angle contained by the two planes passing through the normal to 

 the wave, and the normals to the circular sections of the surface of 

 elasticity, while the other is perpendicular to it. This conclusion 

 does not coincide mathematically with the experimental law of 

 M. Biot : but the differences are much within the limits of the 

 errors of observation, and the results of experiment must be 

 regarded as confirmatory of the theory. 



The velocity of propagation of a plane wave in any direction 

 being known, the form of the wave-surface diverging from any 

 point within the crystal may be found. For if we conceive an 

 indefinite number of plane waves, which, at the commencement of 

 the time, all pass through the point which is considered as the 

 centre of disturbance, the wave-surface will be that touched by all 

 these planes at any instant. This surface is of the fourth order. 

 Fresnel has deduced its equation, although in an indirect manner ; 

 and he has shown that it may be geometrically constructed by 

 means of an ellipsoid whose semiaxes are the same as those of the 

 surface of elasticity. The form of the wave-surface being known, 

 the directions of the two refracted rays are given by the construc- 

 tion of Huygens. 



From the construction now alluded to it appears that there 



* Young seems to have been the first to observe that the law of the square of the 

 cosine could be derivecTfrom the hypothesis of transversal vibrations, (Encyc. Brtt., 

 CHROMATICS, p. 161). The subject of the experimental confirmation of this importan 

 law has been recently brought before the French Academy by M. Arago, and he has 

 indicated the practical results which mny be derive 1 from this law in its application to 

 photometry. llersclicr* Essay on Lijht : French Transaeion, Suppl., p. 690. 



