384 FOURTH REPORT — 1834. 



finds, are those of the greatest and least diameters of the section 

 of the surface of elasticity made by the plane of the M'ave ; and 

 if the original displacement be resolved into two, parallel to them, 

 each component will give rise to a plane wave whose velocity of 

 propagation is 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 sur- 

 face of elasticity ; while the other is perpendicular to it. This 

 conclusion does not coincide mathematically with the experi- 

 mental law of M. Biot : but the differences are much within 

 the limits of the errors of observation, and the results of expe- 

 riment 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 luave-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 con- 

 structed 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 construction of Huygens. 



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

 of the cosine could be derived from the hypothesis of transvei'sal vibrations, 

 (Ency. Brit. Chromatics, p. 161.) The subject of the experimental confirma- 

 tion of this important law has been recently brought before the French Acadenw 

 by M. Arago, and he has indicated the practical results which maybe derived 

 from this law in its application to photometry. — HerscheVs Esmy on Light : 

 French Translation, Suppl., p. 590. 



