Chapter XXII: Act 1871: , the jurisdiction, duties and command exercised by the lord (4)

This conclusion can also be arrived at by a mode of reasoning that is
independent of the theory of diffraction.[15] If linear differential
equations admit a solution of the form (5) with A constant, they can
also be satisfied by making A a function of the coordinates, such
that, in a wave-front, it changes very little over a distance equal to
the wave-length [lambda], and that it is constant along each line
conjugate with the wave-fronts. In cases of this kind the disturbance
may truly be said to travel along lines of the said direction, and an
observer who is unable to discern lengths of the order of [lambda],
and who uses an opening of much larger dimensions, may very well have
the impression of a cylindrical beam with a sharp boundary.

A similar result is found for curved waves. If the additional
restriction is made that their radii of curvature be very much larger
than the wave-length, Huygens's construction may confidently be
employed. The amplitudes all along a ray are determined by, and
proportional to, the amplitude at one of its points.

14. _Polarized Light._--As the theorems used in the explanation of interference and diffraction are true for all kinds of vibratory motions, these phenomena can give us no clue to the special kind of vibrations in light-waves. Further information, however, may be drawn from experiments on plane polarized light. The properties of a beam of this kind are completely known when the position of a certain plane passing through the direction of the rays, and _in_ which the beam is said to be polarized, is given. "This plane of polarization," as it is called, coincides with the plane of incidence in those cases where the light has been polarized by reflection on a glass surface under an angle of incidence whose tangent is equal to the index of refraction (Brewster's law).

The researches of Fresnel and Arago left no doubt as to the direction of the vibrations in polarized light with respect to that of the rays themselves. In isotropic bodies at least, the vibrations are exactly transverse, i.e. perpendicular to the rays, either in the plane of polarization or at right angles to it. The first part of this statement also applies to unpolarized light, as this can always be dissolved into polarized components.

Much experimental work has been done on the production of polarized rays by double refraction and on the reflection of polarized light, either by isotropic or by anisotropic transparent bodies, the object of inquiry being in the latter case to determine the position of the plane of polarization of the reflected rays and their intensity.

In this way a large amount of evidence has been gathered by which it has been possible to test different theories concerning the nature of light and that of the medium through which it is propagated. A common feature of nearly all these theories is that the aether is supposed to exist not only in spaces void of matter, but also in the interior of ponderable bodies.

15. _Fresnel's Theory._--Fresnel and his immediate successors assimilated the aether to an elastic solid, so that the velocity of propagation of transverse vibrations could be determined by the formula v = [root](K/[rho]), where K denotes the modulus of rigidity and [rho] the density. According to this equation the different properties of various isotropic transparent bodies may arise from different values of K, of [rho], or of both. It has, however, been found that if both K and [rho] are supposed to change from one substance to another, it is impossible to obtain the right reflection formulae. Assuming the constancy of K Fresnel was led to equations which agreed with the observed properties of the reflected light, if he made the further assumption (to be mentioned in what follows as "Fresnel's assumption") that the vibrations of plane polarized light are perpendicular to the plane of polarization.

Let the indices p and n relate to the two principal cases in which the
incident (and, consequently, the reflected) light is polarized in the
plane of incidence, or normally to it, and let positive directions h
and h´ be chosen for the disturbance (at the surface itself) in the
incident and for that in the reflected beam, in such a manner that, by
a common rotation, h and the incident ray prolonged may be made to
coincide with h´ and the reflected ray. Then, if [alpha]1 and [alpha]2
are the angles of incidence and refraction, Fresnel shows that, in
order to get the reflected disturbance, the incident one must be
multiplied by

[alpha]_p = -sin ([alpha]1 - [alpha]2) / sin ([alpha]1 + [alpha]2) (9)

in the first, and by

[alpha]_n = tan ([alpha]1 - [alpha]2) / tan ([alpha]1 + [alpha]2) (10)

in the second principal case.

As to double refraction, Fresnel made it depend on the unequal elasticity of the aether in different directions. He came to the conclusion that, for a given direction of the waves, there are two possible directions of vibration (§6), lying in the wave-front, at right angles to each other, and he determined the form of the wave-surface, both in uniaxal and in biaxal crystals.

Though objections may be urged against the dynamic part of Fresnel's theory, he admirably succeeded in adapting it to the facts.

16. Electromagnetic Theory.--We here leave the historical order and pass on to Maxwell's theory of light.

James Clerk Maxwell, who had set himself the task of mathematically
working out Michael Faraday's views, and who, both by doing so and by
introducing many new ideas of his own, became the founder of the
modern science of electricity,[16] recognized that, at every point of
an electromagnetic field, the state of things can be defined by two
vector quantities, the "electric force" E and the "magnetic force" H,
the former of which is the force acting on unit of electricity and the
latter that which acts on a magnetic pole of unit strength. In a
non-conductor (dielectric) the force E produces a state that may be
described as a displacement of electricity from its position of
equilibrium. This state is represented by a vector D ("dielectric
displacement") whose magnitude is measured by the quantity of
electricity reckoned per unit area which has traversed an element of
surface perpendicular to D itself. Similarly, there is a vector
quantity B (the "magnetic induction") intimately connected with the
magnetic force H. Changes of the dielectric displacement constitute an
electric current measured by the rate of change of D, and represented
in vector notation by

C = D (11)

Periodic changes of D and B may be called "electric" and "magnetic
vibrations." Properly choosing the units, the axes of coordinates (in
the first proposition also the positive direction of s and n), and
denoting components of vectors by suitable indices, we can express in
the following way the fundamental propositions of the theory.

(a) Let s be a closed line, [sigma] a surface bounded by it, n the
normal to [sigma]. Then, for all bodies,
_ _ _ _
/ 1 / / 1 d /
| H_s ds = --- | C_n d[sigma], | E_s ds = - --- --- | B_n d[sigma],
_/ c _/ _/ c dt _/

where the constant c means the ratio between the electro-magnet and
the electrostatic unit of electricity.

From these equations we can deduce:

([alpha]) For the interior of a body, the equations

[dP]H_z [dP]H_y 1
------- - ------- = --- C_x,
[dP]y [dP]z c

[dP]H_x [dP]H_z 1
------- - ------- = --- C_y,
[dP]z [dP]x c

[dP]H_y [dP]H_x 1
------- - ------- = --- C_z (12)
[dP]x [dP]y c

[dP]E_z [dP]E_y 1 [dP]B_x
------- - ------- = - --- -------,
[dP]y [dP]z c [dP]t

[dP]E_x [dP]E_z 1 [dP]B_y
------- - ------- = - --- -------,
[dP]z [dP]x c [dP]t

[dP]E_y [dP]E_x 1 [dP]B_z
------- - ------- = - --- -------; (13)
[dP]x [dP]y c [dP]t

(ß) For a surface of separation, the continuity of the tangential
components of E and H;

([gamma]) The solenoidal distribution of C and B, and in a dielectric
that of D. A solenoidal distribution of a vector is one corresponding
to that of the velocity in an incompressible fluid. It involves the
continuity, at a surface, of the normal component of the vector.

(b) The relation between the electric force and the dielectric
displacement is expressed by

D_x = [epsilon]1 E_x, D_y = [epsilon]2 E_y, D_z = [epsilon]3 E_z, (14)

the constants [epsilon]1, [epsilon]2, [epsilon]3 (dielectric
constants) depending on the properties of the body considered. In an
isotropic medium they have a common value [epsilon], which is equal to
unity for the free aether, so that for this medium D = E.

(c) There is a relation similar to (14) between the magnetic force and
the magnetic induction. For the aether, however, and for all
ponderable bodies with which this article is concerned, we may write
B = H.

It follows from these principles that, in an isotropic dielectric,
transverse electric vibrations can be propagated with a velocity

v = c/[root][epsilon]. (15)

Indeed, all conditions are satisfied if we put

D_x = 0, D_y = a cos n(t - xv^(-1) + l), D_z = 0,

H_x = 0, H_y = 0 , H_z = avc^(-1) cos n(t - xv^{-1} + l) (16)

For the free aether the velocity has the value c. Now it had been
found that the ratio c between the two units of electricity agrees
within the limits of experimental errors with the numerical value of
the velocity of light in aether. (The mean result of the most exact
determinations[17] of c is 3,001·10^10 cm./sec., the largest
deviations being about 0,008·10^10; and Cornu[18] gives 3,001·10^10 ±
0,003·10^10 as the most probable value of the velocity of light.) By
this Maxwell was led to suppose that light consists of transverse
electromagnetic disturbances. On this assumption, the equations (16)
represent a beam of plane polarized light. They show that, in such a
beam, there are at the same time electric and magnetic vibrations,
both transverse, and at right angles to each other.

It must be added that the electromagnetic field is the seat of two
kinds of energy distinguished by the names of electric and magnetic
energy, and that, according to a beautiful theorem due to J. H.
Poynting,[19] the energy may be conceived to flow in a direction
perpendicular both to the electric and to the magnetic force. The
amounts per unit of volume of the electric and the magnetic energy are
given by the expressions

½(E_x D_x + E_y D_y + E_z D_z), (17)

and

½(H_x B_x + H_y B_y + H_z B_z) = ½H², (18)

whose mean values for a full period are equal in every beam of light.

The formula (15) shows that the index of refraction of a body is given
by [root][epsilon], a result that has been verified by Ludwig
Boltzmann's measurements[20] of the dielectric constants of gases.
Thus Maxwell's theory can assign the true cause of the different
optical properties of various transparent bodies. It also leads to the
reflection formulae (9) and (10), provided the electric vibrations of
polarized light be supposed to be perpendicular to the plane of
polarization, which implies that the magnetic vibrations are parallel
to that plane.

Following the same assumption Maxwell deduced the laws of double
refraction, which he ascribes to the unequality of [epsilon]1,
[epsilon]2, [epsilon]3. His results agree with those of Fresnel and
the theory has been confirmed by Boltzmann,[21] who measured the three
coefficients in the case of crystallized sulphur, and compared them
with the principal indices of refraction. Subsequently the problem of
crystalline reflection has been completely solved and it has been
shown that, in a crystal, Poynting's flow of energy has the direction
of the rays as determined by Huygens's construction.

Two further verifications must here be mentioned. In the first place,
though we shall speak almost exclusively of the propagation of light
in transparent dielectrics, a few words may be said about the optical
properties of conductors. The simplest assumption concerning the
electric current C in a metallic body is expressed by the equation C =
[sigma]E, where [sigma] is the coefficient of conductivity. Combining
this with his other formulae (we may say with (12) and (13)), Maxwell
found that there must be an absorption of light, a result that can be
readily understood since the motion of electricity in a conductor
gives rise to a development of heat. But, though Maxwell accounted in
this way for the fundamental fact that metals are opaque bodies, there
remained a wide divergence between the values of the coefficient of
absorption as directly measured and as calculated from the electrical
conductivity; but in 1903 it was shown by E. Hagen and H. Rubens[22]
that the agreement is very satisfactory in the case of the extreme
infra-red rays.

In the second place, the electromagnetic theory requires that a
surface struck by a beam of light shall experience a certain pressure.
If the beam falls normally on a plane disk, the pressure is normal
too; its total amount is given by c^{-1}(i1 + i2 - i3), if i1, i2 and
i3 are the quantities of energy that are carried forward per unit of
time by the incident, the reflected, and the transmitted light. This
result has been quantitatively verified by E. F. Nicholls and G. F.
Hull.[23]

Maxwell's predictions have been splendidly confirmed by the
experiments of Heinrich Hertz[24] and others on electromagnetic waves;
by diminishing the length of these to the utmost, some physicists have
been able to reproduce with them all phenomena of reflection,
refraction (single and double), interference, and polarization.[25] A
table of the wave-lengths observed in the aether now has to contain,
besides the numbers given in § 11, the lengths of the waves produced
by electromagnetic apparatus and extending from the long waves used in
wireless telegraphy down to about 0.6 cm.

17. _Mechanical Models of the Electromagnetic Medium._--From the results already enumerated, a clear idea can be formed of the difficulties which were encountered in the older form of the wave-theory. Whereas, in Maxwell's theory, longitudinal vibrations are excluded _ab initio_ by the solenoidal distribution of the electric current, the elastic-solid theory had to take them into account, unless, as was often done, one made them disappear by supposing them to have a very great velocity of propagation, so that the aether was considered to be practically incompressible. Even on this assumption, however, much in Fresnel's theory remained questionable. Thus George Green,[26] who was the first to apply the theory of elasticity in an unobjectionable manner, arrived on Fresnel's assumption at a formula for the reflection coefficient A_n sensibly differing from (10).

In the theory of double refraction the difficulties are no less serious. As a general rule there are in an anisotropic elastic solid three possible directions of vibration (§ 6), at right angles to each other, for a given direction of the waves, but none of these lies in the wave-front. In order to make two of them do so and to find Fresnel's form for the wave-surface, new hypotheses are required. On Fresnel's assumption it is even necessary, as was observed by Green, to suppose that in the absence of all vibrations there is already a certain state of pressure in the medium.

If we adhere to Fresnel's assumption, it is indeed scarcely possible
to construct an elastic model of the electromagnetic medium. It may be
done, however, if the velocities of the particles in the model are
taken to represent the magnetic force H, which, of course, implies
that the vibrations of the particles are parallel to the plane of
polarization, and that the magnetic energy is represented by the
kinetic energy in the model. Considering further that, in the case of
two bodies connected with each other, there is continuity of H in the
electromagnetic system, and continuity of the velocity of the
particles in the model, it becomes clear that the representation of H
by that velocity must be on the same scale in all substances, so that,
if [xi], [eta], [zeta] are the displacements of a particle and g a
universal constant, we may write

[dP][xi] [dP][eta] [dP][zeta]
H_x = g --------, H_y = g ---------, H_z = g ----------. (19)
[dP]t [dP]t [dP]t

By this the magnetic energy per unit of volume becomes
_ _
| /[dP][xi]\² /[dP][eta]\² /[dP][zeta]\² |
½g² | ( -------- ) + ( --------- ) + ( ---------- ) |,
|_ \ [dP]t / \ [dP]t / \ [dP]t / _|

and since this must be the kinetic energy of the elastic medium, the
density of the latter must be taken equal to g², so that it must be
the same in all substances.

It may further be asked what value we have to assign to the potential
energy in the model, which must correspond to the electric energy in
the electromagnetic field. Now, on account of (11) and (19), we can
satisfy the equations (12) by putting D_x = gc ([dP][zeta]/[dP]y -
[dP][eta]/[dP]z), &c., so that the electric energy (17) per unit of
volume becomes
_
| 1 /[dP][zeta] [dP][eta]\²
½g²c² | ---------- ( ---------- - --------- ) +
|_[epsilon]1 \ [dP]y [dP]z /

1 /[dP][xi] [dP][zeta]\²
---------- ( -------- - ---------- ) +
[epsilon]2 \ [dP]z [dP]x /
_
1 /[dP][eta] [dP][xi]\² |
---------- ( --------- - -------- ) |.
[epsilon]3 \ [dP]x [dP]y / _|

This, therefore, must be the potential energy in the model.

It may be shown, indeed, that, if the aether has a uniform constant
density, and is so constituted that in any system, whether homogeneous
or not, its potential energy per unit of volume can be represented by
an expression of the form

_
| /[dP][zeta] [dP][eta]\²
½ | L ( ---------- - --------- ) +
|_ \ [dP]y [dP]z /

/[dP][xi] [dP][zeta]\²
M ( -------- - ---------- ) +
\ [dP]z [dP]x /
_
/[dP][eta] [dP][xi]\² |
N ( --------- - -------- ) |, (20)
\ [dP]x [dP]y / _|

where L, M, N are coefficients depending on the physical properties of
the substance considered, the equations of motion will exactly
correspond to the equations of the electromagnetic field.

18. _Theories of Neumann, Green, and MacCullagh._--A theory of light in which the elastic aether has a uniform density, and in which the vibrations are supposed to be parallel to the plane of polarization, was developed by Franz Ernst Neumann,[27] who gave the first deduction of the formulas for crystalline reflection. Like Fresnel, he was, however, obliged to introduce some illegitimate assumptions and simplifications. Here again Green indicated a more rigorous treatment.

By specializing the formula for the potential energy of an anisotropic
body he arrives at an expression which, if some of his coefficients
are made to vanish and if the medium is supposed to be incompressible,
differs from (20) only by the additional terms
_
| /[dP][zeta] [dP][eta] [dP][eta] [dP][zeta]\
2 | L ( ---------- --------- - --------- ---------- ) +
|_ \ [dP]y [dP]z [dP]y [dP]z /

/[dP][xi] [dP][zeta] [dP][zeta] [dP][xi]\
M ( -------- ---------- - ---------- -------- ) +
\ [dP]z [dP]x [dP]z [dP]x /
_
/[dP][eta] [dP][xi] [dP][xi] [dP][eta]\ |
N ( --------- -------- - -------- --------- ) |. (21)
\ [dP]x [dP]y [dP]x [dP]y / _|

If [xi], [eta], [zeta] vanish at infinite distance the integral of
this expression over all space is zero, when L, M, N are constants,
and the same will be true when these coefficients change from point to
point, provided we add to (21) certain terms containing the
differential coefficients of L, M, N, the physical meaning of these
terms being that, besides the ordinary elastic forces, there is some
extraneous force (called into play by the displacement) acting on all
those elements of volume where L, M, N are not constant. We may
conclude from this that all phenomena can be explained if we admit the
existence of this latter force, which, in the case of two contingent
bodies, reduces to a surface-action on their common boundary.

James MacCullagh[28] avoided this complication by simply assuming an
expression of the form (20) for the potential energy. He thus
established a theory that is perfectly consistent in itself, and may
be said to have foreshadowed the electromagnetic theory as regards the
form of the equations for transparent bodies. Lord Kelvin afterwards
interpreted MacCullagh's assumption by supposing the only action which
is called forth by a displacement to consist in certain couples acting
on the elements of volume and proportional to the components
½{([dP][zeta]/[dP]y) - ([dP][eta]/[dP]z)}, &c., of their rotation from
the natural position. He also showed[29] that this "rotational
elasticity" can be produced by certain hidden rotations going on in
the medium.

We cannot dwell here upon other models that have been proposed, and most of which are of rather limited applicability. A mechanism of a more general kind ought, of course, to be adapted to what is known of the molecular constitution of bodies, and to the highly probable assumption of the perfect permeability for the aether of all ponderable matter, an assumption by which it has been possible to escape from one of the objections raised by Newton (§ 4) (see AETHER).

The possibility of a truly satisfactory model certainly cannot be denied. But it would, in all probability, be extremely complicated. For this reason many physicists rest content, as regards the free aether, with some such general form of the electromagnetic theory as has been sketched in § 16.

19. _Optical Properties of Ponderable Bodies. Theory of Electrons._--If we want to form an adequate representation of optical phenomena in ponderable bodies, the conceptions of the molecular and atomistic theories naturally suggest themselves. Already, in the elastic theory, it had been imagined that certain material particles are set vibrating by incident waves of light. These particles had been supposed to be acted on by an elastic force by which they are drawn back towards their positions of equilibrium, so that they can perform free vibrations of their own, and by a resistance that can be represented by terms proportional to the velocity in the equations of motion, and may be physically understood if the vibrations are supposed to be converted in one way or another into a disorderly heat-motion. In this way it had been found possible to explain the phenomena of dispersion and (selective) absorption, and the connexion between them (anomalous dispersion).[30] These ideas have been also embodied into the electromagnetic theory. In its more recent development the extremely small, electrically charged particles, to which the name of "electrons" has been given, and which are supposed to exist in the interior of all bodies, are considered as forming the connecting links between aether and matter, and as determining by their arrangement and their motion all optical phenomena that are not confined to the free aether.[31]

It has thus become clear why the relations that had been established between optical and electrical properties have been found to hold only in some simple cases (§ 16). In fact it cannot be doubted that, for rapidly alternating electric fields, the formulae expressing the connexion between the motion of electricity and the electric force take a form that is less simple than the one previously admitted, and is to be determined in each case by elaborate investigation. However, the general boundary conditions given in § 16 seem to require no alteration. For this reason it has been possible, for example, to establish a satisfactory theory of metallic reflection, though the propagation of light in the interior of a metal is only imperfectly understood.

One of the fundamental propositions of the theory of electrons is that an electron becomes a centre of radiation whenever its velocity changes either in direction or in magnitude. Thus the production of Röntgen rays, regarded as consisting of very short and irregular electromagnetic impulses, is traced to the impacts of the electrons of the cathode-rays against the anti-cathode, and the lines of an emission spectrum indicate the existence in the radiating body of as many kinds of regular vibrations, the knowledge of which is the ultimate object of our investigations about the structure of the spectra. The shifting of the lines caused, according to Doppler's law, by a motion of the source of light, may easily be accounted for, as only general principles are involved in the explanation. To a certain extent we can also elucidate the changes in the emission that are observed when the radiating source is exposed to external magnetic forces ("Zeeman-effect"; see MAGNETO-OPTICS).

20. _Various Kinds of Light-motion._--(a) If the disturbance is
represented by

P_x = 0, P_y = a cos (nt - kx + f), P_z = a´ cos (nt - kx + f´),

so that the end of the vector P describes an ellipse in a plane
perpendicular to the direction of propagation, the light is said to be
elliptically, or in special cases circularly, polarized. Light of this
kind can be dissolved in many different ways into plane polarized
components.

There are cases in which plane waves must be elliptically or
circularly polarized in order to show the simple propagation of phase
that is expressed by formulae like (5). Instances of this kind occur
in bodies having the property of rotating the plane of polarization,
either on account of their constitution, or under the influence of a
magnetic field. For a given direction of the wave-front there are in
general two kinds of elliptic vibrations, each having a definite form,
orientation, and direction of motion, and a determinate velocity of
propagation. All that has been said about Huygens's construction
applies to these cases.

(b) In a perfect spectroscope a sharp line would only be observed if
an endless regular succession of simple harmonic vibrations were
admitted into the instrument. In any other case the light will occupy
a certain extent in the spectrum, and in order to determine its
distribution we have to decompose into simple harmonic functions of
the time the components of the disturbance, at a point of the slit for
instance. This may be done by means of Fourier's theorem.

An extreme case is that of the unpolarized light emitted by
incandescent solid bodies, consisting of disturbances whose variations
are highly irregular, and giving a continuous spectrum. But even with
what is commonly called homogeneous light, no perfectly sharp line
will be seen. There is no source of light in which the vibrations of
the particles remain for ever undisturbed, and a particle will never
emit an endless succession of uninterrupted vibrations, but at best a
series of vibrations whose form, phase and intensity are changed at
irregular intervals. The result must be a broadening of the spectral
line.

In cases of this kind one must distinguish between the velocity of
propagation of the phase of regular vibrations and the velocity with
which the said changes travel onward (see below, iii. _Velocity of
Light_).

(c) In a train of plane waves of definite frequency the disturbance is
represented by means of goniometric functions of the time and the
coordinates. Since the fundamental equations are linear, there are
also solutions in which one or more of the coordinates occur in an
exponential function. These solutions are of interest because the
motions corresponding to them are widely different from those of which
we have thus far spoken. If, for example, the formulae contain the
factor

e^(-rx) cos (nt - sy + l),

with the positive constant r, the disturbance is no longer periodic
with respect to x, but steadily diminishes as x increases. A state of
things of this kind, in which the vibrations rapidly die away as we
leave the surface, exists in the air adjacent to the face of a glass
prism by which a beam of light is totally reflected. It furnishes us
an explanation of Newton's experiment mentioned in § 2. (H. A. L.)

III. VELOCITY OF LIGHT

The fact that light is propagated with a definite speed was first brought out by Ole Roemer at Paris, in 1676, through observations of the eclipses of Jupiter's satellites, made in different relative positions of the Earth and Jupiter in their respective orbits. It is possible in this way to determine the time required for light to pass across the orbit of the earth. The dimensions of this orbit, or the distance of the sun, being taken as known, the actual speed of light could be computed. Since this computation requires a knowledge of the sun's distance, which has not yet been acquired with certainty, the actual speed is now determined by experiments made on the earth's surface. Were it possible by any system of signals to compare with absolute precision the times at two different stations, the speed could be determined by finding how long was required for light to pass from one station to another at the greatest visible distance. But this is impracticable, because no natural agent is under our control by which a signal could be communicated with a greater velocity than that of light. It is therefore necessary to reflect a ray back to the point of observation and to determine the time which the light requires to go and come. Two systems have been devised for this purpose. One is that of Fizeau, in which the vital appliance is a rapidly revolving toothed wheel; the other is that of Foucault, in which the corresponding appliance is a mirror revolving on an axis in, or parallel to, its own plane.

Fizeau.

The principle underlying Fizeau's method is shown in the accompanying
figs. 1 and 2. Fig. 1 shows the course of a ray of light which,
emanating from a luminous point L, strikes the plane surface of a
plate of glass M at an angle of about 45°. A fraction of the light is
reflected from the two surfaces of the glass to a distant reflector R,
the plane of which is at right angles to the course of the ray. The
latter is thus reflected back on its own course and, passing through
the glass M on its return, reaches a point E behind the glass. An
observer with his eye at E looking through the glass sees the return
ray as a distant luminous point in the reflector R, after the light
has passed over the course in both directions.

In actual practice it is necessary to interpose the object glass of a
telescope at a point O, at a distance from M nearly equal to its focal
length. The function of this appliance is to render the diverging
rays, shown by the dotted lines, nearly parallel, in order that more
light may reach R and be thrown back again. But the principle may be
conceived without respect to the telescope, all the rays being ignored
except the central one, which passes over the course we have
described.

Conceiving the apparatus arranged in such a way that the observer sees
the light reflected from the distant mirror R, a fine toothed wheel WX
is placed immediately in front of the glass M, with its plane
perpendicular to the course of the ray, in such a way that the ray
goes out and returns through an opening between two adjacent teeth.
This wheel is represented in section by WX in fig. 1, and a part of
its circumference, with the teeth as viewed by the observer, is shown
in fig. 2. We conceive that the latter sees the luminous point between
two of the teeth at K. Now, conceive that the wheel is set in
revolution. The ray is then interrupted as every tooth passes, so that
what is sent out is a succession of flashes. Conceive that the speed
of the mirror is such that while the flash is going to the distant
mirror and returning again, each tooth of the wheel takes the place of
an opening between the teeth. Then each flash sent out will, on its
return, be intercepted by the adjacent tooth, and will therefore
become invisible. If the speed be now doubled, so that the teeth pass
at intervals equal to the time required for the light to go and come,
each flash sent through an opening will return through the adjacent
opening, and will therefore be seen with full brightness. If the speed
be continuously increased the result will be successive disappearances
and reappearances of the light, according as a tooth is or is not
interposed when the ray reaches the apparatus on its return. The
computation of the time of passage and return is then very simple. The
speed of the wheel being known, the number of teeth passing in one
second can be computed. The order of the disappearance, or the number
of teeth which have passed while the light is going and coming, being
also determined in each case, the interval of time is computed by a
simple formula.

Cornu.

The most elaborate determination yet made by Fizeau's method was that
of Cornu. The station of observation was at the Paris Observatory. The
distant reflector, a telescope with a reflector at its focus, was at
Montlhéry, distant 22,910 metres from the toothed wheel. Of the wheels
most used one had 150 teeth, and was 35 millimetres in diameter; the
other had 200 teeth, with a diameter of 45 mm. The highest speed
attained was about 900 revolutions per second. At this speed, 135,000
(or 180,000) teeth would pass per second, and about 20 (or 28) would
pass while the light was going and coming. But the actual speed
attained was generally less than this. The definitive result derived
by Cornu from the entire series of experiments was 300,400 kilometres
per second. Further details of this work need not be set forth because
the method is in several ways deficient in precision. The eclipses and
subsequent reappearances of the light taking place gradually, it is
impossible to fix with entire precision upon the moment of complete
eclipse. The speed of the wheel is continually varying, and it is
impossible to determine with precision what it was at the instant of
an eclipse.

The defect would be lessened were the speed of the toothed wheel
placed under control of the observer who, by action in one direction
or the other, could continually check or accelerate it, so as to keep
the return point of light at the required phase of brightness. If the
phase of complete extinction is chosen for this purpose a definite
result cannot be reached; but by choosing the moment when the light is
of a certain definite brightness, before or after an eclipse, the
observer will know at each instant whether the speed should be
accelerated or retarded, and can act accordingly. The nearly constant
speed through as long a period as is deemed necessary would then be
found by dividing the entire number of revolutions of the wheel by the
time through which the light was kept constant. But even with these
improvements, which were not actually tried by Cornu, the estimate of
the brightness on which the whole result depends would necessarily be
uncertain. The outcome is that, although Cornu's discussion of his
experiments is a model in the care taken to determine so far as
practicable every source of error, his definitive result is shown by
other determinations to have been too great by about {1/1000} part of
its whole amount.

Young and Forbes.

An important improvement on the Fizeau method was made in 1880 by
James Young and George Forbes at Glasgow. This consisted in using two
distant reflectors which were placed nearly in the same straight line,
and at unequal distances. The ratio of the distances was nearly 12:13.
The phase observed was not that of complete extinction of either
light, but that when the two lights appeared equal in intensity. But
it does not appear that the very necessary device of placing the speed
of the toothed wheel under control of the observer was adopted. The
accordance between the different measures was far from satisfactory,
and it will suffice to mention the result which was

_Velocity in vacuo_ = 301,382 km. per second.

These experimenters also found a difference of 2% between the speed of
red and blue light, a result which can only be attributed to some
unexplained source of error.

The Foucault system is much more precise, because it rests upon the
measurement of an angle, which can be made with great precision.

Foucault.

The vital appliance is a rapidly revolving mirror. Let AB (fig. 3) be
a section of this mirror, which we shall first suppose at rest. A ray
of light LM emanating from a source at L, is reflected in the
direction MQR to a distant mirror R, from which it is perpendicularly
reflected back upon its original course. This mirror R should be
slightly concave, with the centre of curvature near M, so that the ray
shall always be reflected back to M on whatever point of R it may
fall. Conceiving the revolving mirror M as at rest, the return ray
will after three reflections, at M, R and M again, be returned along
its original course to the point L from which it emanated. An
important point is that the return ray will always follow the fixed
line ML no matter what the position of the movable mirror M, provided
there is a distant reflector to send the ray back. Now, suppose that,
while the ray is going and coming, the mirror M, being set in
revolution, has turned from the position in which the ray was
reflected to that shown by the dotted line. If [alpha] be the angle
through which the surface has turned, the course of the return ray,
after reflection, will then deviate from ML by the angle 2[alpha], and
so be thrown to a point E, such that the angle LME = 2[alpha]. If the
mirror is in rapid rotation the ray reflected from it will strike the
distant mirror as a series of flashes, each formed by the light
reflected when the mirror was in the position AB. If the speed of
rotation is uniform, the reflected rays from the successive flashes
while the mirror is in the dotted position will thus all follow the
same direction ME after their second reflection from the mirror. If
the motion is sufficiently rapid an eye observing the reflected ray
will see the flashes as an invariable point of light so long as the
speed of revolution remains constant. The time required for the light
to go and come is then equal to that required by the mirror to turn
through half the angle LME, which is therefore to be measured. In
practice it is necessary on this system, as well as on that of Fizeau,
to condense the light by means of a lens, Q, so placed that L and R
shall be at conjugate foci. The position of the lens may be either
between the luminous point L and the mirror M, or between M and R, the
latter being the only one shown in the figure. This position has the
advantage that more light can be concentrated, but it has the
disadvantage that, with a given magnifying power, the effect of
atmospheric undulation, when the concave reflector is situated at a
great distance, is increased in the ratio of the focal length of the
lens to the distance LM from the light to the mirror. To state the
fact in another form, the amplitude of the disturbances produced by
the air in linear measure are proportional to the focal distance of
the lens, while the magnification required increases in the inverse
ratio of the distance LM. Another difficulty associated with the
Foucault system in the form in which its originator used it is that if
the axis of the mirror is at right angles to the course of the ray,
the light from the source L will be flashed directly into the eye of
the observer, on every passage of the revolving mirror through the
position in which its normal bisects the two courses of the ray. This
may be avoided by inclining the axis of the mirror.

In Foucault's determination the measures were not made upon a luminous
point, but upon a reticule, the image of which could not be seen
unless the reflector was quite near the revolving mirror. Indeed the
whole apparatus was contained in his laboratory. The effective
distance was increased by using several reflectors; but the entire
course of the ray measured only 20 metres. The result reached by
Foucault for the velocity of light was 298,000 kilometres per second.

Michelson.

The first marked advance on Foucault's determination was made by
Albert A. Michelson, then a young officer on duty at the U.S. Naval
Academy, Annapolis. The improvement consisted in using the image of a
slit through which the rays of the sun passed after reflection from a
heliostat. In this way it was found possible to see the image of the
slit reflected from the distant mirror when the latter was nearly 600
metres from the station of observation. The essentials of the
arrangement are those we have used in fig. 3, L being the slit. It
will be seen that the revolving mirror is here interposed between the
lens and its focus. It was driven by an air turbine, the blast of
which was under the control of the observer, so that it could be kept
at any required speed. The speed was determined by the vibrations of
two tuning forks. One of these was an electric fork, making about 120
vibrations per second, with which the mirror was kept in unison by a
system of rays reflected from it and the fork. The speed of this fork
was determined by comparison with a freely vibrating fork from time to
time. The speed of the revolving mirror was generally about 275 turns
per second, and the deflection of the image of the slit about 112.5
mm. The mean result of nearly 100 fairly accordant determinations
was:--

Velocity of light in air 299,828 km. per sec.
Reduction to a vacuum +82
Velocity of light in a vacuum 299,910 ± 50

Newcomb.

While this work was in progress Simon Newcomb obtained the official
support necessary to make a determination on a yet larger scale. The
most important modifications made in the Foucault-Michelson system
were the following:--

1. Placing the reflector at the much greater distance of several
kilometres.

2. In order that the disturbances of the return image due to the
passage of the ray through more than 7 km. of air might be reduced to
a minimum, an ordinary telescope of the "broken back" form was used to
send the ray to the revolving mirror.

3. The speed of the mirror was, as in Michelson's experiments,
completely under control of the observer, so that by drawing one or
the other of two cords held in the hand the return image could be kept
in any required position. In making each measure the receiving
telescope hereafter described was placed in a fixed position and
during the "run" the image was kept as nearly as practicable upon a
vertical thread passing through its focus. A "run" generally lasted
about two minutes, during which time the mirror commonly made between
25,000 and 30,000 revolutions. The speed per second was found by
dividing the entire number of revolutions by the number of seconds in
the "run." The extreme deviations between the times of transmission of
the light, as derived from any two runs, never approached to the
thousandth part of its entire amount. The average deviation from the
mean was indeed less than {1/5000} part of the whole.

To avoid the injurious effect of the directly reflected flash, as well
as to render unnecessary a comparison between the directions of the
outgoing and the return ray, a second telescope, turning horizontally
on an axis coincident with that of the revolving mirror, was used to
receive the return ray after reflection. This required the use of an
elongated mirror of which the upper half of the surface reflected the
outgoing ray, and the lower other half received and reflected the ray
on its return. On this system it was not necessary to incline the
mirror in order to avoid the direct reflection of the return ray. The
greatest advantage of this system was that the revolving mirror could
be turned in either direction without break of continuity, so that
the angular measures were made between the directions of the return
ray after reflection when the mirror moved in opposite directions. In
this way the speed of the mirror was as good as doubled, and the
possible constant errors inherent in the reference to a fixed
direction for the sending telescope were eliminated. The essentials of
the apparatus are shown in fig. 4. The revolving mirror was a
rectangular prism M of steel, 3 in. high and 1½ in. on a side in cross
section, which was driven by a blast of air acting on two fan-wheels,
not shown in the fig., one at the top, the other at the bottom of the
mirror. NPO is the object-end of the fixed sending telescope the rays
passing through it being reflected to the mirror by a prism P. The
receiving telescope ABO is straight, and has its objective under O. It
was attached to a frame which could turn around the same axis as the
mirror. The angle through which it moved was measured by a divided arc
immediately below its eye-piece, which is not shown in the figure. The
position AB is that for receiving the ray during a rotation of the
mirror in the anti-clockwise direction; the position A´B´ that for a
clockwise rotation.

In these measures the observing station was at Fort Myer, on a hill
above the west bank of the Potomac river. The distant reflector was
first placed in the grounds of the Naval Observatory, at a distance of
2551 metres. But the definitive measures were made with the reflector
at the base of the Washington monument, 3721 metres distant. The
revolving mirror was of nickel-plated steel, polished on all four
vertical sides. Thus four reflections of the ray were received during
each turn of the mirror, which would be coincident were the form of
the mirror invariable. During the preliminary series of measures it
was found that two images of the return ray were sometimes formed,
which would result in two different conclusions as to the velocity of
light, according as one or the other was observed. The only
explanation of this defect which presented itself was a tortional
vibration of the revolving mirror, coinciding in period with that of
revolution, but it was first thought that the effect was only
occasional.

In the summer of 1881 the distant reflector was removed from the
Observatory to the Monument station. Six measures made in August and
September showed a systematic deviation of +67 km. per second from the
result of the Observatory series. This difference led to measures for
eliminating the defect from which it was supposed to arise. The pivots
of the mirror were reground, and a change made in the arrangement,
which would permit of the effect of the vibration being determined and
eliminated. This consisted in making the relative position of the
sending and receiving telescopes interchangeable. In this way, if the
measured deflection was too great in one position of the telescopes,
it would be too small by an equal amount in the reverse position. As a
matter of fact, when the definitive measures were made, it was found
that with the improved pivots the mean result was the same in the two
positions. But the new result differed systematically from both the
former ones. Thirteen measures were made from the Monument in the
summer of 1882, the results of which will first be stated in the form
of the time required by the ray to go and come. Expressed in
millionths of a second this was:--

Least result of the 13 measures 24.819
Greatest result 24.831
Double distance between mirrors 7.44242 km.

Applying a correction of +12 km. for a slight convexity in the face of
the revolving mirror, this gives as the mean result for the speed of
light in air, 299,778 km. per second. The mean results for the three
series were:--

Observatory, 1880-1881 V in air = 299,627
Monument, 1881 V " = 299,694
Monument, 1882 V " = 299,778

The last result being the only one from which the effect of distortion
was completely eliminated, has been adopted as definitive. For
reduction to a vacuum it requires a correction of +82 km. Thus the
final result was concluded to be

_Velocity of light in vacuo_ = 299,860 km. per second.

This result being less by 50 km. than that of Michelson, the latter
made another determination with improved apparatus and arrangements at
the Case School of Applied Science in Cleveland. The result was

_Velocity in vacuo_ = 299,853 km. per second.

So far as could be determined from the discordance of the separate
measures, the mean error of Newcomb's result would be less than ±10
km. But making allowance for the various sources of systematic error
the actual probable error was estimated at ±30 km.

It seems remarkable that since these determinations were made, a period during which great improvements have become possible in every part of the apparatus, no complete redetermination of this fundamental physical constant has been carried out.

The experimental measures thus far cited have been primarily those of the velocity of light in air, the reduction to a vacuum being derived from theory alone. The fundamental constant at the basis of the whole theory is the speed of light in a vacuum, such as the celestial spaces. The question of the relation between the velocity in vacuo, and in a transparent medium of any sort, belongs to the domain of physical optics. Referring to the preceding section for the principles at play we shall in the present part of the article confine ourselves to the experimental results. With the theory of the effect of a transparent medium is associated that of the possible differences in the speed of light of different colours.

Velocity and wave-length.

The question whether the speed of light in vacuo varies with its wave-length seems to be settled with entire certainty by observations of variable stars. These are situated at different distances, some being so far that light must be several centuries in reaching us from them. Were there any difference in the speed of light of various colours it would be shown by a change in the colour of the star as its light waxed and waned. The light of greatest speed preceding that of lesser speed would, when emanated during the rising phase, impress its own colour on that which it overtook. The slower light would predominate during the falling phase. If there were a difference of 10 minutes in the time at which light from the two ends of the visible spectrum arrived, it would be shown by this test. As not the slightest effect of the kind has ever been seen, it seems certain that the difference, if any, cannot approximate to {1/1.000.000} part of the entire speed. The case is different when light passes through a refracting medium. It is a theoretical result of the undulatory theory of light that its velocity in such a medium is inversely proportional to the refractive index of the medium. This being different for different colours, we must expect a corresponding difference in the velocity.

Foucault and Michelson have tested these results of the undulatory theory by comparing the time required for a ray of light to pass through a tube filled with a refracting medium, and through air. Foucault thus found, in a general way, that there actually was a retardation; but his observations took account only of the mean retardation of light of all the wave-lengths, which he found to correspond with the undulatory theory. Michelson went further by determining the retardation of light of various wave-lengths in carbon bisulphide. He made two series of experiments, one with light near the brightest part of the spectrum; the other with red and blue light. Putting V for the speed in a vacuum and V1 for that in the medium, his result was

Yellow light V : V1 = 1.758
Refractive index for yellow 1.64
Difference from theory +0.12

The estimated uncertainty was only 0.02, or 1/6 of the difference between observation and theory.

The comparison of red and blue light was made differentially. The colours selected were of wave-length about 0.62 for red and 0.49 for blue. Putting V_r and V_b for the speeds of red and blue light respectively in bisulphide of carbon, the mean result compares with theory as follows:--

Observed value of the ratio V_r, V_b 1.0245
Theoretical value (Verdet) 1.025

This agreement may be regarded as perfect. It shows that the divergence of the speed of yellow light in the medium from theory, as found above, holds through the entire spectrum.