properties of a crystal under the issuance of homogeneous strain, the principal axes of the wave-surface being parallel to those of the strain, and the medium being uniaxal, if the strain be symmetrical. John Kerr also found that a dielectric under electric stress behaves as an uniaxal crystal with its optic axis parallel to the electric force, glass acting as a negative and bi sulphide of carbon as a positive crystal (Phil. Mag., 1875 (4), L. 337).
Not content with determining the laws of double refraction, Fresnel also attempted to give their mechanical explanation. He supposed that the aether consists of a system of distinct material points symmetrically arranged and acting on one another by forces that depend for a given pair only on their distance. If in such a system a single molecule be displaced, the projection of the force of restitution on the direction of displacement is proportional to the inverse square of the parallel radius-vector of an ellipsoid; and of all displacements that can occur in a given plane, only those in the direction of the axes of the parallel central section of the quadric develop forces whose projection on the plane is along the displacement. In undulations, however, we are concerned with the elastic forces due to relative displacements, and, accordingly, Fresnel assumed that the forces called into play during the propagation of a system of plane waves (of rectilinear transverse vibrations) differ from those developed by the parallel displacement of a single molecule only by a constant factor, independent of the plane of the wave. Next, regarding the aether as incompressible, he assumed that the components of the elastic forces parallel to the wave-front are alone operative, and finally, on the analogy of stretched string, that the propagational speed of a plane Wave of permanent type is proportional to the square root of the effective force developed by the vibrations. With these hypotheses we immediately obtain the laws of double refraction, as given by the ellipsoid of polarization, with the result that the vibrations are perpendicular to the plane of polarization.
In its dynamical foundations Fresnel’s theory, though of considerable historical interest, is clearly defective in rigour, and a strict treatment of the aether as a crystalline elastic solid does not lead naturally to Fresnel’s laws of double refraction. On the other hand, Lord Kelvin’s rotational aether (Math. and Phys. Papers, iii. 442)—a medium that has no true rigidity but possesses a quasi-rigidity due to elastic resistance to absolute rotation-gives these laws at once, if We abolish the resistance to compression and, regarding it as gyrostatically isotropic, attribute to it aeolotropic inertia. The equations then obtained are the same as those deduced in the electro-magnetic theory from the circuital laws of A. M. Ampére and Michael Faraday, when the specific inductive capacity is supposed aeolotropic. In order to account for dispersion, it is necessary to take into account the interaction with the radiation of the intra-molecular vibrations of the crystalline substance: thus the total current on the electro-magnetic theory must be regarded as made up of the current of displacement and that due to the oscillations of the electrons within the molecules of the crystal.
Bibliography.—An interesting and instructive account of Fresnel’s work on double refraction has been given by Emile Verdet in his introduction to Fresnel’s works: Œuvres d’Augustin Fresnel, i. 75 (Paris, 1866); Œuvres de E. Verdet, i. 360 (Paris, 1872). For an account of theories of double refraction see the reports of H. Lloyd, Sir G. G. Stokes and R. T. Glazebrook in the Brit. Ass. Reports for 1834, 1862 and 1885, and Lord Kelvin’s Baltimore Lectures (1904). An exposition of the rotational theory of the aether has been given by H. Chipart, Théorie gyrostatique de la lumière (Paris, 1904); and P. Drude’s Lehrbuch der Optik, 2te Auf. (1906), the first German edition of which was translated by C. Riborg Mann and R. A. Milliken in 1902, treats the subject from the standpoint of the electro-magnetic theory. The methods of determining the optical constants of crystals will be found in Th. Liebisch’s Physikalische Krystallographie (1891); F. Pockel’s Lehrbuch der Kristalloptik (1906); and J. Walker’s Analytical Theory of Light (1904). A detailed list of papers on the geometry of the wave-surface has been published by E. Wolliing, Bibl. Math., 1902 (3), iii. 361; and a general account of the subject will be found in the ollowing treatises: L. Fletcher, The Optical Indicatrix (1892); Th. Preston, The Theory of Light, 3rd ed. by C. J. Joly (1901); A. Schuster, An Introduction to the Theory of Optics (1904); R. W. Wood, Physical Optics (1905); E. Mascart, Traité d’optique (1889); A. Winkelmann, Handbuch der Physik. (J. Wal.)
III. Astronomical Refraction
The refraction of a ray of light by the atmosphere as it passes from a heavenly body to an observer on the earth's surface, is called “astronomical.” A knowledge of its amount is a necessary datum in the exact determination of the direction of the body. In its investigation the fundamental hypothesis is that the strata. of the air are in equilibrium, which implies that the surfaces of equal density- are horizontal. But this condition is being continually disturbed by aerial currents, which produce continual slight fluctuations in the actual refraction, and commonly give to the image of a star a tremulous motion. Except for this slight motion the refraction is always in the vertical direction; that is, the actual zenith distance of the star is always greater than its apparent distance. The refracting power of the air is nearly proportional to its density. Consequently the amount of the refraction varies with the temperature and barometric pressure, being greater the higher the barometer and the lower the temperature.
At moderate zenith distances, the amount of the refraction varies nearly as the tangent of the zenith distance. Under ordinary conditions of pressure and temperature it is, near the zenith, about 1″ for each degree of zenith distance. As the tangent increases at a greater rate than the angle, the increase of the refraction soon exceeds 1″ for each degree. At 45° from the zenith .the tangent is 1 and the mean refraction is about 58″. As the horizon is approached the tangent increases more and more rapidly, becoming infinite at the horizon; but the refraction now increases at a less rate, and, when the observed ray is horizontal, or when the object appears on the horizon, the refraction is about 34′, or a little greater than the diameter of the sun or moon. It follows that when either of these objects is seen on the horizon their actual direction is entirely below it. One result is that the length of the day is increased by refraction to the extent of about five minutes in low latitudes, and still more in higher latitudes. At 60° the increase is about nine minutes.
The atmosphere, like every other transparent substance, refracts the blue rays of the spectrum more than the red; consequently, when the image of a star near the horizon is observed with a telescope, it presents somewhat the appearance of a spectrum. The edge which is really highest, but seems lowest in the telescope, is blue, and the opposite one red. When the atmosphere is steady this atmospheric spectrum is very marked and renders an exact observation of the star difficult.
Bibliography.—Refraction has been a favourite subject of research. See Dr. C. Bruhns, Die astronomische Strahlenbrechung (Leipzig, 1861), gives a résumé of the various formulae of refraction which had been developed by the leading investigators up to the date 1861. Since then developments of the theory are found in: W. Chauvenet, Spherical and Practical Astronomy, i.; F. Brünnow, Sphärischen Astronomie; S. Newcomb, Spherical Astronomy; R. Radau, “Recherches sur la théorie des réfractions astronomiques" (Annales de l’observatoire de Paris, xvi., 1882), “Essai sur les refractions astronomiques” (ibid., xix., 1889).
Among the tables of refraction which have been most used are Bessel’s, derived from the observations of Bradley in Bessel’s Fundamenta Astronmniae; and Bessel’s revised tables in his Tabulae Regiomontanae, in which, however, the constant is too large, but which in an expanded form were mostly used at the observatories until 1870. The constant use of the Poulkova tables, Tabulae refractionum, which is reduced to nearly its true value, has gradually replaced that of Bessel. Later tables are those of L. de Ball; published at Leipzig in 1906. (S. N.)
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