NATURE 



58: 



THURSDAY, OCTOBER 20, 1892. 



FRESNELS THEORY OF DOUBLE 

 REFRACTION. 

 The Optical Indicatrix and the Transmission of Light 

 in Crystals. By L. Fletcher. (Oxford University 

 Press Warehouse, 1892.) 



MR. FLETCHER has given us a valuable and most 

 interesting book. He has attacked some parts 

 of the theory of double refraction in a manner which is 

 free from many of the objections that may be made to 

 the method more usually adopted, and which has the 

 advantage of closely resembling that by which Fresnel 

 himself made his great discovery. 



In dealing with this discovery we must carefully bear 

 in mind two facts. The result of Fresnel's theory— the 

 determination of the actual form of the wave-surface in a 

 biaxal crystal— is undoubtedly true ; the mechanical 

 reasoning on which, in his second memoir on double | 

 refraction, he bases that result, is as undoubtedly false. I 

 In recent times the explanation of the properties of the | 

 wave-surface has been usually based on the erroneous 

 mechanical reasoning. Mr. Fletcher, following Fresnel 

 himself, has shown us how all these properties may be ! 

 deduced from certain experimental results without the I 

 introduction of the false mechanics. In one sense this is a \ 

 backward step, but since the advance had been in the I 

 wrong direction, it was necessary to retreat, and though 

 Mr. Fletcher's book teaches us nothing new of the 

 mechanism of double refraction, it puts the kinematics 

 of the theory and the geometrical results which it entails 

 on a sure foundation, and this is an achievement for 

 which the author deserves our warmest thanks. 



The history of the discovery of the law of double re- 

 fraction, as given in Fresnel's own papers, is of the 

 deepest interest. The formof the wave- surface, in a uniaxal 

 crystal, a sphere and spheroid, had been discovered 

 by Huyghens. Up to the time at which Fresnel wrote 

 it was generally supposed that even in a biaxal crystal 

 one ray obeyed the ordinary law of refraction. In 1816 

 Fresnel and Arago proved that in an isotropic medium 

 the direction of the periodic disturbance to which light 

 is due is transverse to the ray, and from that time 

 Fresnel set himself to investigate the laws of double 

 refraction. He soon saw that the reasoning which led 

 him to expect that in a uniaxal crystal the velocity of one 

 wave must be constant could not apply to crystals with 

 two optic axes, and that in such, contrary to the usual 

 belief, there could be no ordinary ray. This was 

 announced in June 1820. The first memoir on double 

 refraction, presented to the Academy November 19, 

 i82i,butnot printed till 1868, gives a detailed account 

 of the experiments by which this result of his theory was 

 verified, and contains the earliest development of the 

 theory itself. According to Fresnel's views at this time 

 the ether displacement in a crystal was at right angles 

 to the ray — it is noteworthy that the latest develop- 

 ment of the mechanical theory leads to this same 

 result — and the velocity of propagation depended 

 on the elasticity of the medium in the direction of dis- 

 placement. Since, then, for any ray there are two pos- 

 NO. I 199, VOL. 46] 



sible velocities there must for that ray be two and only 

 two directions of displacement, the light must be po- 

 larized in one of two directions. Since, also, the velo- 

 cities are different in different directions, in Fresnel's view 

 the elasticity corresponding to a given direction of 

 vibration must depend on the direction, and a surface of 

 elasticity can be drawn each radius of which shall be 

 proportional to the elasticity in its own direction 

 When once this surface is known, the rest of the prob- 

 lem can be readily solved. Now, in the memoir we 

 are referring to (" Premier Mdmoire sur la Double 

 Refraction, CEuvres Completes d'Augustin Fresnel," 

 T. ii. No. xxxviii.) Fresnel proved, for ttie case in which 

 the double refraction was not strong, for which therefore 

 the distinction between the ray and the wave normal 

 might be neglected, that if for a uniaxal crystal the 

 surface of elasticity were a spheroid of revolution the 

 wave surface and the laws of double refraction would be 

 those discovered by Huyghens. It was an easy step 

 from this to generalize and to suppose that in a biaxal 

 crystal the surface of elasticity might be an ellipsoid, and 

 to examine the results. This gave him at once, to the 

 same degree of approximation as in the previous case, 

 the form of the wave surface known by his name and 

 the experimental laws of biaxial refraction discovered by 

 Brewster and by Biot. As yet the result was only ap- 

 proximate ; worked out more completely the theory can 

 be shown to lead to the same form of the wave surface 

 as that developed by Lord Rayleigh {Phil. Mag., June, 

 1871). Huyghens' laws and those of Biot and Brewster 

 are not accurately obeyed. 



But Fresnel's mind moved rapidly, and a week later, 

 November 26, 1821 (" Extrait d'un M^moire sur la 

 Double Refraction," CEuvres, Vol. ii., No. xxxix.), he 

 corrected his first results and announced his complete 

 theory. The modifications required were not great ; in 

 his original theory he had supposed the displacement to 

 be at right angles to the ray, in the final form it is sup- 

 posed to be at right angles to the wave normal, i.e., in 

 the wave front, while the elasticity in any given direction, 

 and therefore the corresponding velocity, is given by the 

 reciprocal of the radius vector of a certain ellipsoid in- 

 stead of by the radius vector itself. With these modifi- 

 cations the wave surface is accurately Fresnel's surface 

 and the experimental laws of double refraction are accur- 

 ately obeyed. To this ellipsoid (of which the inverse is 

 Fresnel's surface of elasticity) Mr. Fletcher has given the 

 name of the " Indicatrix." According to Fresnel's theory 

 the two possible velocities of wave propagation in any 

 direction are the reciprocals of the axes of the section 

 of the indicatrix by the wave front. This law has been 

 verified to a high degree of accuracy by direct measure- 

 ment, and is thus made by Mr. Fletcher the basis of his 

 treatment of the problem. These Memoirs of Fresnel's 

 remained unpublished till 1868. The only printed an- 

 nouncement of the results was a notice in the Moniteur, 

 December 12, 1821, with a view of claiming priority for 

 the discovery, and the Report of the Referees, Fourier, 

 Ampi^re, and Arago, who (August 19, 1822) recom- 

 mended that it should be printed m full in the " Recueil 

 des Savants Etrangers." 



Instead, however, the Second Memoir on Double 

 Refraction (CEuvres, tome ii.. No. xlvii.) was printed 



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