# 1911 Encyclopædia Britannica/Refraction

REFRACTION (Lat. refringere, to break open or apart), in physics, the change in the direction of a wave of light, heat or sound which occurs when such a wave passes from one medium into another of different density.

I. Refraction of Light

When a ray of light traversing a homogeneous medium falls on the bounding surface of another transparent homogeneous medium, it is found that the direction of the transmitted ray in the second medium is different from that of the incident ray; in other words, the ray is refracted or bent at the point of incidence. The laws governing refraction are: (1) the refracted and incident rays are coplanar with the normal to the refracting surface at the point of incidence, and (2) the ratio of the sines of the angles between the normal and the incident and refracted rays is constant for the two media, but depends on the nature of the light employed, i.e. on its wave length. This constant is called the relative refractive index of the second medium, and may be denoted by μab the suffix ab signifying that the light passes from medium a to medium b; similarly μba denotes the relative refractive index of a with regard to b. The absolute refractive index is the index when the first medium is a vacuum. Elementary phenomena in refraction, such as the apparent bending of a stick when partially immersed in water, were observed in very remote times, but the laws, as stated above, were first grasped in the 17th century by W. Snell and published by Descartes, the full importance of the dependence of the refractive index on the nature of the light employed being first thoroughly realized by Newton in his famous prismatic decomposition of white light into coloured spectrum. Newton gave a theoretical interpretation of these laws on the basis of his corpuscular theory, as did also Huygens on the wave theory (see Light, II. Theory of). In this article we only consider refractions at plane surfaces, refraction at spherical surfaces being treated under Lens. The geometrical theory will be followed, the wave theory being treated in Light, Diffraction and Dispersion.

Refraction at a Plane Surface.—Let LM (fig. 1) be the surface dividing two homogeneous media A and B; let IO be a ray in the first medium incident on LM at O, and let OR be the refracted ray. Draw the normal POQ. Then by Snell’s law we have invariably sin IOP/sin QOR＝μab. Hence if two of these quantities be given the third can be calculated. The commonest question is: Given the incident ray and the refractive index to construct the refracted ray. A simple construction is to take along the incident ray OI, unit distance OC, and a distance OD equal to the refractive index in the same units. Draw CE perpendicular to LM, and draw an arc with centre O and radius OD, cutting CE in E. Then EO produced downwards is the refracted ray. The proof is left to the reader.

In the figure the given incident ray is assumed to be passing from a less dense to a denser medium, and it is seen by the construction or by examining the formula sin β＝sin α/μ that for all values of α there is a corresponding value of β. Consider the case when the light passes from a denser to a less dense medium. In the equation sin β＝sin α/μ. We have in this case μ<1. Now if sin α<μ, we have sin α/μ<1, and hence β is real. If sin αμ, then sin β＝1, i.e. β＝90°; in other words, the refracted ray in the second medium passes parallel to and grazes the bounding surface. The angle of incidence, which is given by sin δ＝1/μ, is termed the critical angle. For greater values it is obvious that sin α/μ>1 and there is no refraction into the second medium, the rays being totally reflected back into the first medium; this is called total internal reflection.

Images produced by Refraction at Plane Surfaces.—If a luminous point be situated in a medium separated from one of less density by a plane surface, the ray normal to this surface will be unrefracted, whilst the others will undergo refraction according to their angles of emergence. If the rays in the less dense medium be produced into the denser medium, they envelop a caustic, but by restricting ourselves to a small area about the normal ray it is seen that they intersect this ray in a point which is the geometrical image of the luminous source. The position of this point can be easily determined. If l be the distance of the source below the surface, l′ the distance of the image, and μ the refractive index, then l′＝l/μ. This theory provides a convenient method for determining the refractive index of a plate. A micrometer microscope, with vertical motion, is focused on a scratch on the surface of its stage; the plate, which has a fine scratch on its upper surface, is now introduced, and the microscope is successively focused on the scratch on the stage as viewed through the plate, and on the scratch on the plate. The difference between the first and third readings gives the thickness of the plate, corresponding to l above, and between the second and third readings the depth of the image, corresponding to l′.

Refraction by a Prism.—In optics a prism is a piece of transparent material bounded by two plane faces which meet at a definite angle, called the refracting angle of the prism, in a straight line called the edge of the prism; a section perpendicular to the edge is called a principal section. Parallel rays, refracted successively at the two faces, emerge from the prism as a system of parallel rays, but the direction is altered by an amount called the deviation. The deviation depends on the angles of incidence and emergence; but, since the course of a ray may always be reversed, there must be a stationary value, either a maximum, or minimum, when the ray traverses the prism symmetrically, Le. when the angles of incidence and emergence are equal. As a matter of fact, it is a minimum, and the position is called the angle of minimum deviation. The relation between the minimum deviation D, the angle of the prism i, and the refractive index μ is found as follows. Let in fig. 2, PQRS be the course of the ray through the prism; the internal angles φ′, ψ′ each equal 12i, and the angles of incidence and emergence φ, ψ are each equal and connected with φ′ by Snell’s law, i.e. sin φμ sin φ′. Also the deviation D is 2 (φφ′). Hence μ＝sin φ/sin φ′＝sin12 (D+i)/sin12i.

Fig. 2.

Refractometers.—Instruments for determining the refractive indices of media are termed refractometers.

The simplest are really spectrometers, consisting of a glass prism, usually hollow and fitted with accurately parallel glass sides, mounted on a table which carries a fixed collimation tube and a movable observing tube, the motion of the latter being recorded on a graduated circle. The collimation tube has a narrow adjustable slit at its outer end and a lens at the nearer end, so that the light leaves the tube as a parallel beam. The refracting angle of the prism, i in our previous notation, is determined by placing the prism with its refracting edge towards the collimator, and observing when the reflections of the slit in the two prism faces coincide with the cross-wires in the observing telescope; half the angle between these two positions gives i. To determine the position of minimum deviation, or D, the prism is removed, and the observing telescope is brought into line with the slit; in this position the graduation is read. The prism is replaced, and the telescope moved until it catches the refracted rays. The prism is now turned about a vertical axis until a position is found when the telescope has to be moved towards the collimator in order to catch the rays; this operation sets the prism at the angle of minimum deviation. The refractive index μ is calculated from the formula given above.

are being experimented with. Dr R. Wollny’s butter refractometer, also made by Zeiss, is constructed similarly to Abbe’s form, with the exception that the prism casing is rigidly attached to the telescope, and the observation made by noting the point where the border line intersects an appropriately graduated scale in the focal plane of the telescope objective, fractions being read by a micrometer screw attached to the objective. This apparatus is also provided with an arrangement for heating.

This method of reading is also employed in Zeiss’s “dipping refractometer” (fig. 6). This instrument consists of a telescope R having at its lower end a prism P with a refracting angle of 63°, above which and below the objective is a movable compensator A for purposes of annulling the dispersion about the border line. In the focal plane of the objective O there is a scale Sc, exact reading being made by a micrometer Z. If a large quantity of liquid be available it is sufficient to dip the refractometer perpendicularly into a beaker containing the liquid and to transmit light into the instrument by means of a mirror. If only a smaller quantity be available, it is enclosed in a metal beaker M, which forms an extension of the instrument, and the liquid is retained there by a plate D. The instrument is now placed in a trough B, containing water and having one side of ground glass G; light is reflected into the refractometer by means of a mirror S outside this trough. An accuracy of 3·7 units in the 5th decimal place is obtainable.

The Pulfrich refractometer is also largely used, especially for liquids. It consists essentially of a right-angled glass prism placed on a metal foundation with the faces at right angles horizontal and vertical, the hypotenuse face being on the support. The horizontal face is fitted with a small cylindrical vessel to hold the liquid. Light is led to the prism at grazing incidence by means of a collimator, and is refracted through the vertical face, the deviation being observed by a telescope rotating about a graduated circle. From this the refractive index is readily calculated if the refractive index of the prism for the light used be known: a fact supplied by the maker. The instrument is also available for determining the refractive index of isotropic solids. A little of the solid is placed in the vessel and a mixture of monobromnaphthalene and acetone (in which the solid must be insoluble) is added, and adjustment made by adding either one or other liquid until the border line appears sharp, i.e. until the liquid has the same index as the solid.

The Herbert Smith refractometer (fig. 7) is especially suitable for determining the refractive index of gems, a constant which is most valuable in distinguishing the precious stones. It consists of a hemisphere of very dense glass, having its plane surface fixed at a certain angle to the axis of the instrument. Light is admitted by a window on the under side, which is inclined at the same angle, but in the opposite sense, to the axis. The light on emerging from the hemisphere is received by a convex lens, in the focal plane of which is a scale graduated to read directly in refractive indices. The light then traverses a positive eye-piece. To use the instrument for a gem, a few drops of methylene iodide (the refractive index of which may be raised to 1·800 by dissolving sulphur in it) are placed on the plane surface of the hemisphere and a facet of the stone then brought into contact with the surface. If monochromatic light be used (i.e. the D line of the sodium flame) the field is sharply divided into a light and a dark portion, and the position of the line of demarcation on the scale immediately gives the refractive index. It is necessary for the liquid to have a higher refractive index than the crystal, and also that there is close contact between the facet and the lens. The range of the instrument is between 1·400 and 1·760, the results being correct to two units in the third decimal place if sodium light be used.  (C. E.*)

II. Double Refraction

That a stream of light on entry into certain media can give rise to two refracted pencils was discovered in the case of Iceland spar by Erasmus Bartholinus, who found that one pencil had a direction given by the ordinary law of refraction, but that the other was bent in accordance with a new law that he was unable to determine. This law was discovered about eight years later by Christian Huygens. According to Huygens fundamental principle, the law of refraction is determined by the form and orientation of the wave-surface in the crystal—the locus of points to which a disturbance emanating from a luminous point travels in unit time. In the case of a doubly refracting medium the Wave-surface must have two sheets, one of which is spherical, if one of the pencils obey in all cases the ordinary law of refraction. Now Huygens observed that a natural crystal of spar behaves in precisely the same way whichever pair of faces the light passes through, and inferred from this fact that the second sheet of the wave-surface must be a surface of revolution round a line equally inclined to the faces of the rhomb, i.e. round the axis of the crystal. He accordingly assumed it to be a spheroid, and finding that refraction in the direction of the axis was the same for both streams, he concluded that the sphere and the spheroid touched one another in the axis.

So far as his experimental means permitted, Huygens verified the law of refraction deduced from this hypothesis, but its correctness remained unrecognized until the measures of W. H. Wollaston in 1802 and of E. T. Malus in 1810. More recently its truth has been established with far more perfect optical appliances by R. T. Glazebrook, Ch. S. Hastings and others.

In the case of Iceland spar and several other crystals the extraordinarily refracted stream is refracted away from the axis, but Jean Baptiste Biot in 1814 discovered that in many cases the reverse occurs, and attributing the extraordinary refractions to forces that act as if they emanated from the axis, he called crystals of the latter kind “attractive,” those of the former “repulsive.” They are now termed “positive” and “negative” respectively; and Huygens’ law applies to both classes, the spheroid being prolate in the case of positive, and oblate in the case of negative crystals. It was at first supposed that Huygens’ law applied to all doubly refracting media. Sir David Brewster, however, in 1815, while examining the rings that are seen round the optic axis in polarized light, discovered a number of crystals that possess two optic axes. He showed, moreover, that such crystals belong to the rhombic, monoclinic and anorthic (triclinic) systems, those of the tetragonal and hexagonal systems being uniaxal, and those of the cubic system being optically isotropic.

Huygens found in the course of his researches that the streams that had traversed a rhomb of Iceland spar had acquired new properties with respect to transmission through a second crystal. This phenomenon is called polarization (q.v.), and the waves are said to be polarized—the ordinary in its principal plane and the extraordinary in a plane perpendicular to its principal plane, the principal plane of a wave being the plane containing its normal and the axis of the crystal. From the facts of polarization Augustin Jean Fresnel deduced that the vibrations in plane polarized light are rectilinear and in the plane of the Wave, and arguing from the symmetry of uniaxal crystals that vibrations perpendicular to the axis are propagated with the same speed in all directions, he pointed out that this would explain the existence of an ordinary wave, and the relation between its speed and that of the extraordinary wave. From these ideas Fresnel was forced to the conclusion, that he at once verified experimentally, that in biaxal crystals there is no spherical wave, since there is no single direction round which such crystals are symmetrical; and, recognizing the difficulty of a direct determination of the wave-surface, he attempted to represent the laws of double refraction by the aid of a simpler surface.

The essential problem is the determination of the propagational speeds of plane waves as dependent upon the directions of their normals. These being known, the deduction of the wave-surface follows at once, since it is to be regarded as the envelope at any subsequent time of all the plane waves that at a given instant may be supposed to pass through a given point, the ray corresponding to any tangent plane or the direction of transport of energy being by Huygens' principle the radius vector from the centre to the point of contact. Now Fresnel perceived that in uniaxal crystals the speeds of plane waves in any direction are by Huygens' law the reciprocals of the semiaxes of the central section, parallel to the wave-fronts, of a spheroid, whose polar and equatorial axes are the reciprocals of the equatorial and polar axes of the spheroidal sheet of Huygens' wave-surface, and that the plane of polarization of a Wave is perpendicular to the axis that determines its speed. Hence it occurred to him that similar relations with respect to an ellipsoid with three unequal axes would give the speeds and polarizations of the waves in a biaxal crystal, and the results thus deduced he found to be in accordance with all known facts. This ellipsoid is called the ellipsoid of polarization, the index ellipsoid and the indicatrix.

We may go a step further; for by considering the intersection of a wave-front with two waves, whose normals are indefinitely n-ear that of the first and lie in planes perpendicular and parallel respectively to its plane of polarization, it is easy to show that the ray corresponding to the wave is parallel to the line in which the former of the two planes intersects the tangent plane to the ellipsoid at the end of the semi-diameter that determines the wave-velocity; and it follows by similar triangles that the ray-Velocity is the reciprocal of the length of the perpendicular from the centre on this tangent plane. The laws of double refraction are thus contained in the following proposition. The propagational speed of a plane wave in any direction is given by the reciprocal of one of the semi-axes of the central section of the ellipsoid of polarization parallel to the wave; the plane of polarization of the wave is perpendicular to this axis; the corresponding ray is parallel to the line of intersection of the tangent plane at the end of the axis and the plane containing the axis and the wave-normal; the ray-velocity is the reciprocal of the length of the perpendicular from the centre on the tangent plane. By reciprocating with respect to a sphere of unit radius concentric with the ellipsoid, we obtain a similar proposition in which the ray takes the place of the wave-normal, the ray velocity that of the wave-slowness (the reciprocal of the velocity) and vice versa. The wave-surface is thus the apsidal surface of the reciprocal ellipsoid; this gives the simplest means of obtaining its equation, and it is readily seen that its section by each plane of optical symmetry consists of an ellipse and a circle, and that in the plane of greatest and least wave-velocity these curves intersect in four points. The radii-vectors to these points are called the ray-axes.

When the wave-front is parallel to either system of circular sections of the ellipsoid of polarization, the problem of finding the axes of the parallel central section becomes indeterminate, and all waves in this direction are propagated with the same speed, whatever may be their polarization. The normals to the circular sections are thus the optic axes. To determine the rays corresponding to an optic axis, we may note that the ray and the perpendiculars to it through the centre, in planes perpendicular and parallel to that of the ray and the optic axis, are three lines intersecting at right angles of which the two latter are confined to given planes, viz. the central circular section of the ellipsoid and the normal section of the cylinder touching the ellipsoid along this section: whence by a known proposition the ray describes a cone whose sections parallel to the given planes are circles. Thus a plane perpendicular to the optic axis touches the wave-surface along a circle. Similarly the normals to the circular sections of the reciprocal ellipsoid, or the axes of the tangent cylinders to the polarization-ellipsoid that have circular normal sections, are directions of single-ray velocity or ray-axes, and it may be shown as above that corresponding to a ray-axis there is a cone of wave-normals with circular sections parallel to the normal section of the corresponding tangent cylinder, and its plane of Contact with the ellipsoid. Hence the extremities of the ray-axes are conical points on the wave-surface. These peculiarities of the wave surface are the cause of the celebrated conical refractions discovered by Sir William Rowan Hamilton and H. Lloyd, which afford a decisive proof of the general correctness of Fresnel’s wave-surface, though they cannot, as Sir G. Gabriel Stokes (Math. and Phys. Papers, iv. 184) has pointed out, be employed to decide between theories that lead to this surface as a near approximation.

In general, both the direction and the magnitude of the axes of the polarization-ellipsoid depend upon the frequency of the light and upon the temperature, but in many cases the possible variations are limited by considerations of symmetry. Thus the optic axis of a uniaxal crystal is invariable, being determined by the principal axis of the system to which it belongs: most crystals are of the same sign for all colours, the refractive indices and their difference both increasing with the frequency, but a few crystals are of opposite sign for the extreme spectral colours, becoming isotropic for some intermediate wave-length. In crystals of the rhombic system the axes of the ellipsoid coincide in all cases with the crystallographic axes, but in a few cases their order of magnitude changes so that the plane of the optic axes for red light is at right angles to that for blue light, the crystal being uniaxal for an intermediate colour. In the case of the mono clinic system one axis is in the direction of the axis of the system, and this is generally, though there are notable exceptions, either the greatest, the least, or the intermediate axis of the ellipsoid for all colours and temperatures. In the latter case the optic axes are in the plane of symmetry, and a variation of their acute bisectrix occasions the phenomenon known as “inclined dispersion ”: in the two former cases the plane of the optic axes is perpendicular to the plane of symmetry, and if it vary with the colour of the light, the crystals exhibit “crossed” or “horizontal dispersion” according as it is the acute or the obtuse bisectrix that is in the fixed direction. The optical constants of a crystal may be determined either with a prism or by observations of total reflection. In the latter case the phenomenon is characterized by two angles-the critical angle and the angle between the plane of incidence and the line limiting the region of total reflection in the field of view. With any crystalline surface there are four cases in which this latter angle is 90°, and the principal refractive indices of the crystal are obtained from those calculated from the corresponding critical angles, by excluding that one of the mean values for which the plane of polarization of the limiting rays is perpendicular to the plane of incidence. A difficulty, however, may arise when the crystalline surface is very nearly the plane of the optic axes, as the plane of polarization in the second mean case is then also very nearly perpendicular to the plane of incidence; but since the two mean refractive indices will be very different, the ambiguity can be removed by making, as may easily be done, an approximate measure of the angle between the optic axes and comparing it with the values calculated by using in turn each of these indices (C. M. Viola, Zeit. für Kryst., 1902, 36, p. 245).

A substance originally isotropic can acquire the optical 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.)