**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 μ_{ab} the suffix *ab* signifying that the light passes from
medium *a* to medium *b*; similarly μ_{ba} 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＝μ_{ab}. 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 α＝μ, 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
12*i*, and the angles of incidence and emergence φ, ψ are each equal
and connected with φ′ by Snell’s law, *i.e.* sin φ＝μ sin φ′. Also the
deviation D is 2 (φ−φ′). Hence μ＝sin φ/sin φ′＝sin12 (D+*i*)/sin12*i*.

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.

More readily manipulated and of superior accuracy are refractometers
depending on total reflection. The Abbe refractometer
(fig. 3) essentially consists of a double Abbe prism AB to contain
the substance to be experimented with; and a telescope F to observe
the border line of the total reflection. The prisms, which are
right-angled and made of the same flint glass, are mounted in a
hinged frame such that the lower prism, which is used for purposes
of illumination, can be locked so that the hypotenuse faces are
distant by about 0·15 mm., or rotated away from the upper prism.
The double prism is used in examining liquids, a few drops being
placed between the prisms; the single prism is used when solids
or plastic bodies are employed. The mount is capable of rotation
about a horizontal axis by an alidade *J*. The telescope is provided
with a reticule, which can be brought into exact coincidence
with the observed border line, and is rigidly fastened to a sector
S graduated directly in refractive indices. The reading is effected
by a lens L. Beneath the prisms is a mirror for reflecting light

Fig. 3.
into the apparatus. To use the apparatus, the liquid having been
inserted between the prisms, or the solid attached by its own
adhesiveness or by a drop of monobromnaphthalene to the upper
prism, the prism case is rotated until the field of view consists of a
light and dark portion, and the border line is now brought into
coincidence with the reticule of the telescope. In using a lamp
or daylight this border is coloured, and hence a compensator,
consisting of two equal Amici prisms, is placed between the
objective and the prisms. These Amici prisms can be rotated,
in opposite directions, until they produce a dispersion opposite
in sign to that originally seen, and hence the border line now
appears perfectly sharp and colourless. When at zero the alidade
corresponds to a refractive index of 1·3, and any other reading
gives the corresponding index correct to about 2 units in the 4th
decimal place. Since temperature markedly affects the refractive
index, this apparatus is provided with a device for heating the prisms.
Figs. 4 and 5 show the course of the rays when a solid and liquid

Fig. 4.Fig. 5.

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 S*c*, 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 iaws 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 lumiere* (Paris, 1904); and P. Drude’s *Lehrbuch der Optik*,
2^{te} 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. Briinnow,
*Sphärischen Astronomie*; S. Newcomb, *Spherical Astronomy*; R.
Radau, “Recherches sur la théorie des refractions 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.)