GENERAL THEORY OF ABERRATION
In Chapter III the subject of Aberration was treated in a simple and
geometrical manner, but it is now time to deal with it more generally.
And to do this compactly I must be content in the greater part of this
chapter to appeal chiefly to physicists.
The following general statements concerning aberration can be made:--
1. A ray of light in clear space is straight, whatever the motion of
the medium, unless eddies exist; in other words, no irrotational
disturbance of ether can deflect a ray.
2. But if the observer is in motion, the apparent ray will not be the
true ray, and his line of vision will not truly indicate the direction
of an object.
3. In a stationary ether the ray coincides with wave-normal. In a
moving ether the ray and wave-normal enclose an aberration angle ε,
such that sin ε = v/V, the ratio of the ether speed to the light
speed.
4. In all cases the line of vision depends on motion of the observer,
and on that alone. If the observer is stationary, his line of vision
is a ray. If he moves at the same rate as the ether, his line of
vision is a wave-normal.
5. Line of vision depends not at all on the motion of the ether, so
long as it has a velocity-potential. Hence if this condition is
satisfied the theory of aberration is quite simple.
_General Statement as to Negative Results in the Subject._
It is noteworthy that almost all the observations which have been made
with negative results as to the effect of the Earth's orbital motion
on the ether are equally consistent with complete connexion and
complete independence between ether and matter. If there is complete
connexion, the ether near the earth is relatively stagnant, and
negative terrestrial results are natural. If there is complete
independence, the ether is either absolutely stationary or has a
velocity-potential, and the negative results are, as has been shown,
thereby explained. Direct experiment on the subject of etherial
viscosity proves that that is either really or approximately zero, and
substantiates the "independence" explanation.
_Definition of a Ray._
A ray signifies the path of a definite or identical portion of
radiation energy--the direction of energy-flux. In other words, it
may be considered as the path of a labelled disturbance; for it is
some special feature which enables an eye to fix direction: it is that
which determines the line of collimation of a telescope.
Now in order that a disturbance from A may reach B, it is necessary
that adjacent elements of a wave front at A shall arrive at B in the
same phase; hence the path by which a disturbance travels must satisfy
this condition from point to point. This condition will be satisfied
if the time of journey down a ray and down all infinitesimally
differing paths is the same.
The equation to a ray is therefore contained in the statement that the
time taken by light to traverse it is a minimum; or
∫{A,B} ds/V = minimum
If the medium, instead of being stationary, is drifting with the
velocity _v_, at angle θ to the ray, we must substitute for V the
modified velocity V cos ε + _v_ cos θ; and so the function that has to
be a minimum, in order to give the path of a ray in a moving medium,
is
Time of
journey = ∫{A,B} ds / V(cos ε + α cos θ)
= ∫{A,B} (V cos ε - v cos θ) / (V²(1-α²)) ds = minimum
where α is the ratio v/V.
_Path of Ray, and Time of Journey, through an Irrotationally Moving
Medium._
Writing a velocity-potential φ in the above equation to a ray, that is
putting
v cos θ = dφ/ds,
and ignoring possible variations in the minute correction factor
1-α² between the points A and B, it becomes
Time of
journey = ∫{A,B} cos ε / (1 - α²) · ds/V - (φβ - φα) / V²( 1-α²)
= minimum.
Now the second term depends only on end points, and therefore has no
effect on path. The first term contains only the second power of
aberration magnitude; and hence it has much the same value as if
everything were stationary. A ray that was straight, will remain
straight in spite of motion. Whatever shape it had, that it will
retain.
Only cos ε, and variations in α², can produce any effect on path;
and effects so produced must be very small, since the value of cos ε
is
√ (1 - α²sin²θ).
A second-order effect on direction may therefore be produced by
irrotational motion, but not a first-order effect. A similar statement
applies to the time of journey round any closed periphery.
_Michelson's Experiment._
We conclude, therefore, that general etherial drift does not affect
either the path of a ray, or the time of its journey to and fro, or
round a complete contour, to any important extent. But that taking
second-order quantities into account, the time of going to and fro in
any direction inclined at angle θ to a constant drift is, from the
above expression,
T₁ + T₂ = 2T cos ε / (1-α²) = √(1 - α²sin²θ) / (1 - α²) × 2T,
where 2 T is the ordinary time of the double journey without any
drift.
Hence some slight modification of interference effects by reason of
drift would seem to be possible; since the time of a to-and-fro
light-journey depends subordinately on the inclination of ray to
drift.
The above expression applies to Michelson's remarkable experiment[10]
of sending a split beam to and fro, half along and half across the
line of the earth's motion; and is, in fact, a theory of it. There
ought to be an effect due to the difference between θ = 0 and θ = 90°.
But none can be detected. Hence, either something else happens, or the
ether near the earth is dragged with it so as not to stream through
our instruments.
_Alternative Explanation._
But if the ether is dragged along near moving matter, it behaves like
a viscous fluid, and all idea of a velocity-potential must be
abandoned. This would complicate the theory of aberration (pp. 45 and
61), and moreover is dead against the experimental evidence described
in Chapter V.
The negative result of Mr. Michelson's is, however, explicable in
another way,--namely, by the FitzGerald-Lorentz theory that the linear
dimensions of bodies are a function of their motion through the ether.
And such an effect it is reasonable to expect; since, if cohesion
forces are electrical, they must be affected by motion, to a known and
calculable amount, depending on the square of the ratio of the speed
to the velocity of light. (See end of Chap. IV.)
The theory of Professor H.A. Lorentz, accordingly, shows that the
shape of Michelson's stone supporting block will be distorted by the
motion; its dimensions across and along the line of ether drift being
affected differently. And the amount of the change will be such as
precisely to compensate and neutralise the optical effect of motion
which might otherwise be perceived. This theory is now generally
accepted.
It is this neutralising or compensatory effect,--which acts equally on
to-and-fro motion of light, to-and-fro motion of electric currents,
and on the shape of material bodies,--that renders any positive result
in experiments on ether-drift so difficult or impossible to obtain; so
that, in spite of the speed with which we are rushing through space,
no perceptible influence is felt on either electrical or optical
phenomena, except those which are due to relative motion of source and
observer.
_Some Details in the Theory of the Doppler Effect, or Effect of Motion
on Dispersion by Prism or Grating._
When light is analysed by a prism or grating into a spectrum, the
course of each ray is deflected--refracted or diffracted--by an amount
corresponding to its frequency of vibration or wave-length.
Motion of the medium, so long as it is steady, affects neither
frequency nor wave-length, and accordingly is without influence on the
result. It produces no Doppler effect except when waxing or waning.
Motion of the source alone crowds the waves together on the advancing
side and spreads them out on the receding side. An observer therefore
whom the source is approaching receives shorter waves, and one from
whom the source is receding receives longer waves, than normal. At any
fixed point waves will arrive, therefore, with modified frequency.
So long as a source is stationary the wave-lengths emitted are quite
normal, but motion of an observer may change the frequency with which
they are _received_, in an obvious way; they are swept up faster if
the receiver is approaching, they have a stern chase if it is
receding.
All this is familiar, and was geometrically illustrated in Chapter
III, but there are some minor and rather curious details which are
worthy of brief consideration.
_Grating Theory._
For suppose a 'grating' is used to analyse the light. Its effect can
depend on nothing kinetic; it must be regulated by the merely
geometric width of the ruled spaces on it. Consequently it can only
directly apprehend wave-lengths, not frequencies.
In the case of a moving _source_, therefore, when the wave-length is
really changed, a grating will appreciate the fact, and will show a
true Doppler effect. But in the case of a moving _observer_, when all
the waves received are of normal length, though swept up with abnormal
frequency, the grating must still indicate wave-length alone, and
accordingly will show no true Doppler effect.
But inasmuch as the telescope or line of vision is inclined at the
angle of dispersion to the direction of the incident ray, ordinary
aberration must come in, as it always does when an observer moves
athwart his line of vision; and so there will be a spurious or
apparent Doppler effect due to common aberration. That is to say a
spectrum line will not be seen in its true place, but will appear to
be shifted by an amount almost exactly imitative of a real Doppler
effect--the imitation being correct up to the second order of
aberration magnitude. The slight outstanding difference between them
is calculated in my _Philosophical Transactions_ paper, 1893, page
787. It is too small to observe.
It is not an important matter, but as it is rather troublesome to work
out the diffraction observed by a grating advancing towards the source
of light, it may be as well to record the result here.
The following are the diffracted rays which require attention,--with
the inclination of each to the grating-normal specified:--
The diffracted ray if all were stationary, θ₀;
The real diffracted ray when grating is advancing, φ;
The ray as perceived, allowing for aberration, θ;
The equivalent diffracted ray if all were stationary and the wave-length
really shortened, θ₁.
As an auxiliary we use the aberration angle ε, such that sin ε = α sin
θ, where α = v/V.
Among these four angles the following relations hold; so that, given one
of them, all are known.
{ θ = φ - ε
{ sin θ₁ = (1 - α) sin θ₀
{ sin φ = (1 - α vers φ) sin θ₀
Whence θ and θ₁ are very nearly but not absolutely the same. θ₁ is
the ray observed by an instrument depending primarily on frequency,
like a prism; θ is the ray observed by an instrument depending
primarily on wave-length, like a grating.
_Prism Theory._
Now let a prism be used to analyse the light; its dispersive power is
in most theories held to depend directly upon frequency--i.e. upon a
time relation between the period of a light vibration and the period
of an atomic or electronic revolution or other harmonic excursion.
Let us say, therefore, that prismatic dispersion directly indicates
frequency. It cannot depend upon wave-length, for the wave-length
inside different substances is different, and though refractive index
corresponds to this, dispersive power does not.
In the case of a prism, therefore, no distinction can be drawn between
motion of source and motion of receiver; for in both cases the
frequency with which the waves are received will be altered,--either
because they are really shorter, though arriving at normal speed, or
because they are swept up faster, although of normal length.
_Achromatic Prism._
It must be noticed that the observation of Doppler effect by a prism
depends entirely on dispersion; i.e. on waves of different length
being affected differently. But prisms can be constructed whose
dispersion is corrected and neutralised. Such achromatic prisms, if
perfectly achromatic, will treat waves of all sizes alike; and,
accordingly, the shortening of the waves from a moving source will not
produce any effect. Achromatic prisms will therefore behave to
terrestrial and to extra-terrestrial sources, i.e. to relatively
stationary and relatively moving sources, in the same way.
This must be recollected in connexion with several of the negative
results rightly obtained by some observers; such as Arago, for
instance, who applied an achromatic prism to a star which the earth
was approaching, without observing any effect. A Doppler effect should
have been observed by a dispersive prism, but not by an achromatic
one: for the refractive index of a substance is not affected by any
motion of the earth.
It is not reasonable to expect that refractive index would be
affected, since it depends in simple geometrical fashion on retarded
velocity, i.e. on optical etherial loading or apparent extra internal
density.
An achromatic _grating_, however, is (rashly speaking) an
impossibility.
EFFECT OF TRANSPARENT MATTER.
But when a ray is travelling through transparent matter, will not
motion of that matter affect its course?
If the matter is moved relatively to source and receiver, as in
Fizeau's experiment with running water, most certainly it will; to the
full effect of the loading or extra or travelling density, (μ²-1),
compared with the total density μ².
This fraction of the velocity of the material medium must directly
influence the velocity of light, for the waves will be conveyed in the
sense of the material motion _u_, with the additional speed
u(μ²-1)/μ². (See also Appendix 3.)
But if the transparent matter through which the light is going is
stationary with respect to source and receiver--only sharing with them
the general planetary motion, i.e. being subject to the opposite
all-pervading ether drift,--then no influence due to the drift can be
experienced; for the free ether of space behaves just as it would if
the matter were not there. This can be shown more elaborately by the
following calculation.
_Optical Effect of Ether Drift through Dense Stationary Bodies._
The calculation of the lag in phase caused by Fresnel's etherial
motion may proceed thus:--A dense slab of thickness _z_, which would
naturally be traversed with the velocity V/μ, is traversed with the
velocity (V/μ) cos ε + (v/μ²) cos θ; where _v_ is the relative
velocity of the ether in its neighbourhood; whence the time of journey
through it is
(μz) / V(cos ε + (α/μ) cos θ), instead of μz/V,
So the equivalent air thickness, instead of being (μ - 1)z, is
μz / (cos ε + (α/μ) cos θ) - z =
( (μ cos ε - α cos θ) / (1 - α/μ)² - 1 )z,
or, to the first order of minutiæ,
(μ - 1)z - αz cos θ;
θ being the angle between ray and ether drift inside the medium.
So the extra equivalent air layer _due to the motion_ is approximately
±α z cos θ, a quantity independent of μ.
Hence, no plan for detecting this first-order effect of motion is in
any way assisted by the use of dense stationary substances; their
extra ether, being stationary, does not affect the lag caused by
motion, except indeed in the second order of small quantities, as
shown above.
Direct experiments made by Hoek,[11] and by Mascart, on the effect of
introducing tubes of water into the path of half-beams of light, are
in entire accord with this negative conclusion.
Thus, then, we find that no general motion of the entire medium can be
detected by changes in direction, or in frequency, or in phase; for on
none of them has it any appreciable (i.e. first-order) effect, even
when assisted by dense matter.
* * * * *
Another mode of stating the matter is to say that the behaviour of
ether inside matter is such as to enable a potential-function,
∫ μ²v cosθ ds,
to exist throughout all transparent space, so far as motion of ether
alone is concerned. (See Appendix 3.)
The existence of this potential function readily accounts for the
absence of all effect on direction due to the general drift of the
medium, whether in the presence of dense matter (such as water-filled
telescopes) or otherwise. Whatever may be the path of a ray by reason
of reflexion or refraction in a stationary ether, it is precisely the
same in a moving one if this condition is satisfied, although the
wave-normals and wave-fronts are definitely shifted.
However matter affects or loads the ether inside it, it cannot on this
theory be said either to hold it still, or to carry it with it. The
general ether stream must remain unaffected, not only near, but inside
matter, if rays are to retain precisely the same course as if it were
relatively stationary.
But it must be understood that the etherial motion here contemplated
is the _general drift of the entire medium_; or its correlative, the
uniform motion of all the matter concerned. There is nothing to be
said against aberration effects being producible or modifiable by
motion of _parts_ of the medium, or by the artificial motion of
transparent bodies and other partitioned-off regions. _Artificial_
motion of matter may readily alter both the time of journey and the
path of a ray, for it has no potential conditions to satisfy; it may
easily describe a closed contour, and may take part in conveying
light.
But I must repeat that this conveyance of light by moving matter is an
effect due to the material load only; it represents no disturbance of
the ether of space. Fresnel's law, in fact, definitely means that
moving transparent matter does _not_ appreciably disturb the ether of
space. Direct experiment, as recorded in Chapter V, shows that close
to rapidly-moving opaque matter there is no disturbance either.
I regard the non-disturbance of the ether of space by moving matter as
established.
FOOTNOTES:
[10] _Philosophical Magazine_, Dec., 1887.
[11] _Archives Néerlandaises_ (1869), Vol. IV, p. 443, or _Nature_,
Vol XXVI, p. 500. Also Chapter IV above.
SUMMARY.
The estimates of this book, and of _Modern Views of Electricity_, are
that the ether of space is a continuous, incompressible, stationary,
fundamental substance or perfect fluid, with what is equivalent to an
inertia-coefficient of 10¹² grammes per c.c.; that _matter_ is
composed of modified and electrified specks, or minute structures of
ether, which are amenable to mechanical as well as to electrical force
and add to the optical or electric density of the medium; and that
elastic-rigidity and all potential energy are due to excessively
fine-grained etherial circulation, with an intrinsic kinetic energy of
the order 10³³ ergs per cubic centimetre.
APPENDIX 1
ON GRAVITY AND ETHERIAL TENSION
In the arithmetical examples of Chapter IX we reckon merely the force
between two bodies; but the Newtonian tension mentioned in Chapter
VIII does not signify that force, but rather a certain condition or
state of the medium, to variations in which, from place to place, the
force is due. This Newtonian tension is a much greater quantity than
the force to which it gives rise; and, moreover, it exists at every
point of space, instead of being integrated all through an attracted
body.
It rises to a maximum value near the surface of any spherical mass;
and if the radius be R and the gravitational intensity is _g_, the
tension at the surface is T₀ = gR. At any distance _r_, further
away, the tension is T = gR²/r.
This follows at once thus:--
Stating the law of gravitation as F = γmm´/r², the meaning here
adopted for etherial tension at the surface of the earth is
T = ∫{R,∞} γE/r² dr = γE/R;
so that the ordinary intensity of gravity is
g = -dT/dR = γE/R² = 4/3πργR.
Accordingly, near the surface of a planet the tension is T₀ = gR, or
for different planets is proportional to ρR².
The velocity of free fall from infinity to such a planet is √(2T₀);
the velocity of free fall from circumference to centre, assuming
uniform distribution of density, is √(T₀); and from infinity to
centre it is √(3T₀).
Expanding all this into words:--
The etherial tension near the earth's surface, required to explain
gravity by its rate of variation, is of the order 6 × 10¹¹ c.g.s.
units. The tension near the sun is 2500 times as great (p. 103). With
different spheres in general, it is proportional to the density and to
the superficial area. Hence, near a bullet one inch in diameter, it is
of the order 10⁻⁶; and near an atom or an electron about 10⁻²¹ c.g.s.
If ever the tension rose to equal the constitutional elasticity or
intrinsic kinetic energy of the ether,--which we have seen is 10³³
dynes per square centimetre (or ergs per c.c.) or 10²² tons weight
per square millimetre,--it seems likely that something would give way.
But no known mass of matter is able to cause anything like such a
tension.
A smaller aggregate of matter would be able to generate the velocity
of light in bodies falling towards it from a great distance; and it
may be doubted whether any mass so great as to be able to do even that
can exist in one lump.
In order to set up a tension equal to what is here suspected of being
a critical, or presumably disruptive, stress in the ether [10³³
c.g.s.], a globe of the density of the earth would have to have a
radius of eight light years. In order to generate a velocity of free
fall under gravity equal to the velocity of light, a globe of the
earth's density would have to be equal in radius to the distance of
the earth from the sun, or say 26,000 times the earth's radius. If the
density were less, the superficial area would have to be increased in
proportion, so as to keep ρ R² constant.
The whole visible universe within a parallax of 1/1000 second of arc,
estimated by Lord Kelvin as the equivalent of 10⁹ suns, would be
quite incompetent to raise etherial tension to the critical point
10³³ c.g.s. unless it were concentrated to an absurd degree; but it
could generate the velocity of light with a density comparable to that
of water, if _mass_ were constant.
If the average density of the above visible universe (which may be
taken as 1.6 × 10⁻²³ grammes per c.c.) continued without limit, a
disruptive tension of the ether would be reached when the radius was
comparable to 10¹³ light years; and the velocity of light would be
generated by it when the radius was 10⁷ light years. But
heterogeneity would enable these values to be reached _more_ easily.
_Gravitation_ is thus supposed to be the result of a mechanical
tension inherently, and perhaps instantaneously, set up throughout
space whenever the etherial structure called an electric charge comes
into existence; the tension being directly proportional to the square
of the charge and inversely as its linear dimensions. _Cohesion_ is
quite different, and is due to a residual electrical attraction
between groups of neutral molecules across molecular distances: a
variant or modification of chemical affinity.
APPENDIX 2
CALCULATIONS IN CONNEXION WITH ETHER DENSITY
Just as the rigidity of the ether is of a purely electric character,
and is not felt mechanically--since mechanically it is perfectly
fluid,--so its density is likewise of an electromagnetic character,
and again is not felt mechanically, because it cannot be moved by
mechanical means. It is by far the most stationary body in existence;
though it is endowed with high intrinsic energy of local movement,
analogous to turbulence, conferring on it gyrostatic properties.
Optically, its rigidity and density are both felt, since optical
disturbances are essentially electromotive. Matter loads the ether
optically, in accordance with the recognised fraction (μ²-1) / μ²;
and this loading, being part and parcel of the _matter_, of course
travels with it. It is the only part amenable to mechanical force.
The mechanical density of matter is a very small portion of the
etherial density; whereas the optical or electrical density of
matter--being really that of ether affected by the intrinsic or
constitutional electricity of matter--is not so small. The relative
optical virtual density of the ether inside matter is measured by
μ²; but it may be really a defect of elasticity, at least in
non-magnetic materials.
Electrical and optical effects depend upon _e_. Mechanical or inertia
effects depend upon e². Electric charges can load the ether
optically, quite appreciably; but as regards mechanical loading, the
densest matter known is trivial and gossamer-like compared with the
unmodified ether in the same space.
_Massiveness of the Ether deduced from Electrical Principles._
Each electron, moving like a sphere through a fluid, has a certain
mass associated with it; dependent on its size, and, at very high
speeds, on its velocity also.
If we treat the electron merely as a sphere moving through a perfect
liquid, its behaviour is exactly as if its mass were increased by half
that of the fluid displaced and the surrounding fluid were
annihilated.
Ether being incompressible, the density of fluid inside and outside an
electron must be the same. So, dealing with it in this simplest
fashion, the resultant inertia is half as great again as that of the
volume of fluid corresponding to the electron: that is to say the
effective mass is 2πρα³, where ρ is the uniform density. If an
electron is of some other shape than a sphere, then the numerical part
is modified, but remains of the same order of magnitude, so long as
there are no sharp edges.
* * * * *
If, however, we consider the moving electron as generating circular
lines of magnetic induction, by reason of some rotational property of
the ether, and if we attribute all the magnetic inertia to the
magnetic whirl thus caused round its path,--provisionally treating
this whirl as an actual circulation of fluid excited by the
locomotion,--then we shall proceed thus:--
Let a spherical electron _e_ of radius _a_ be flying at moderate speed
_u_, so that the magnetic field at any point, _rθ_, outside, is
H = eu sinθ / r²,
and the energy per unit volume everywhere is μH²/8π.
But a magnetic field has been thought of by many mathematicians as a
circulation of fluid along the lines of magnetic induction--which are
always closed curves--at some unknown velocity _w_.
So consider the energy per unit volume anywhere: it can be represented
by the equivalent expressions
½ρw² = μH²/8π = μ/8π · e²u²sin²θ / r²;
wherefore
w/u = √(μ/4πρ) · e sinθ / r².
The velocity of the hypothetical circulation must be a maximum at the
equator of the sphere, where r=a and θ=90; so, calling this _w₀_,
w₀/u = √(μ/4πρ) e / a²,
and
w/w₀ = a² sinθ / r²;
wherefore the major part of the circulation is limited to a region not
far removed from the surface of the electron.
The energy of this motion is
½ρ ∫{0,π} ∫{a,∞} w² · 2π r sin θ · rdθ · dr,
whence, substituting the above value of _w_, the energy comes out
equal to 4/3 πρa³ w₀².
Comparing this with a mass moving with speed _u_,
m = 8/3 πρa³(w₀/u)².
This agrees with the simple hydrodynamic estimate of effective inertia
if w₀ = ½√3·u; that is to say, if the whirl in contact
with the equator of the sphere is of the same order of magnitude as
the velocity of the sphere.
Now for the real relation between _w₀_ and _u_ we must make a
hypothesis. If the two are considered equal, the effectively disturbed
mass comes out as twice that of the bulk of the electron. If _w₀_ is
smaller than _u_, then the mass of the effectively disturbed fluid is
less even than the bulk of an electron; and in that case the estimate
of the fluid-density ρ must be _exaggerated_ in order to supply the
required energy. It is difficult to suppose the equatorial circulation
_w₀_ _greater_ than _u_, since it is generated by it; and it is most
reasonable to treat them both as of the same order of magnitude. So,
taking them as equal,
e = a² √(4πρ/μ)
and m = twice the spherical mass.
Hence all the estimates of the effective inertia of an electron are of
the same order of magnitude, being all comparable with that of a mass
of ether equal to the electron in bulk. But the linear dimension of an
electron is 10⁻¹³ centimetre diameter, and its mass is of the order
10⁻²⁷ gram. Consequently the density of its material must be of the
order 10¹² grams per cubic centimetre.
This, truly, is enormous, but any reduction in the estimate of the
circulation-speed, below that of an electron, would only go to
increase it. And, since electrons move sometimes at a speed not far
below that of light, we cannot be accused of under-estimating the
probable velocity of magnetic spin by treating it as of the same order
of magnitude, at the bounding surface of the electron, as its own
speed: a relation suggested, though not enforced, by gyrostatic
analogies.
_Some Consequences of this Great Density._
The amplitude of a wave of light, in a place where it is most intense,
namely near the sun where its energy amounts to 2 ergs per c.c., comes
out only about 10⁻¹⁷ of the wave-length. The maximum tangential
stress called out by such strain is of the order 10¹¹ atmospheres.
The hypothetical luminous circulation-velocity, conferring momentum on
a wave-front, in accordance with Poynting's investigation, comes out
10⁻²² cm. per sec. These calculations are given in the concluding
chapter of the new edition of _Modern Views of Electricity_.
The supposed magnetic etherial drift, along the axis of a solenoid or
other magnetic field, if it exist, is comparable to ·003 centim. per
sec., or 4 inches an hour, for a field of intensity 12,000 c.g.s.
But it is not to be supposed that this hypothetical velocity is slow
everywhere. Close to an electron the speed of magnetic drift is
comparable to the locomotion-velocity of the electron itself, and may
therefore rise to something near the speed of light; say 1/30th of
that speed: but in spite of that, at a distance of only 1 millimetre
away, it is reduced to practical stagnation, being less than a
millimicron per century.
In any solenoid, the ampere-turns per linear inch furnish a measure of
the speed of the supposed magnetic circulation along the axis--no
matter what the material of the core may be--in millimicrons per sec.
[1 micron = 10⁻⁶ metre; 1 millimicron is 10⁻⁹ metre =
10⁻⁷ centimetre, or a millionth of a millimetre.]
To get up an etherial speed of 1 centimetre per second--such as might
be detected experimentally by refined optical appliances, through its
effect in accelerating or retarding the speed of light sent along the
lines of magnetic force,--would need a solenoid of great length, round
every centimetre of which 1000 amperes circulated 3000 times. That is
to say, a long field of four million c.g.s. units of intensity.
In other words, any streaming along magnetic lines of force, such as
could account for the energy of a magnetic field, must be comparable,
in centimetres per second, to one four-millionth of the number of
c.g.s. units of intensity in the magnetic field.
APPENDIX 3
FRESNEL'S LAW A SPECIAL CASE OF A UNIVERSAL POTENTIAL FUNCTION
The modern view of Fresnel's Law may be worded thus:--
Inside a region occupied by matter, in addition to the universal ether
of space, are certain modified or electrified specks, which build up
the material atoms. These charged particles, when they move, have
specific inertia, due to the magnetic field surrounding each of them.
And by reason of this property, and as a consequence of their
discontinuity, they virtually increase the optical density of the
ether of space, acting in analogy with weights distributed along a
flexible cord. Thus they reduce the velocity of light in the ratio of
the refractive index μ:1, and therefore may be taken as increasing the
virtual density of the ether in the ratio 1:μ².
That is to say, their loading makes the ether behave to optical waves
as if--being a homogeneous medium without these discontinuous
loads--it had a density μ² times that which it has in space outside
matter. Calling the density outside 1, the extra density inside must
be μ²-1, so as to make up the total to μ².
The μ²-1 portion is that which we call "matter," and this portion
is readily susceptible to locomotion, being subject to--that is,
accelerated by--mechanical force. The free portion of normal density 1
is absolutely stationary as regards locomotion, whether it be inside
or outside a region occupied by ordinary matter, for it is not
amenable to either mechanical or electric forces. They are transmitted
by it, but never terminate upon it; except, indeed, at the peculiar
structure called a wave-front, which simulates some of the properties
of matter.
(If free or unmodified ether can ever be moved at all, it must be by
means of a magnetic field; along the lines of which it has, in several
theories, been supposed to circulate. Even this, however, is not real
locomotion.)
Fizeau tested that straightforward consequence of this theory which is
known as Fresnel's Law, and ascertained by experiment that a beam of
light was accelerated or retarded by a stream of water, according as
it travelled with or against the stream. And he found the magnitude of
the effect precisely in accordance with the ratio of the locomotive
portion of the ether to the whole,--the fraction (μ²-1)/μ² of
the speed of the water being added to or subtracted from the velocity
of light, when a beam was sent down or up the stream.
But even if another mode of expression be adopted, the result to be
anticipated from this experiment would be the same.
For instead of saying that a modified portion of the ether is moving
with the full velocity of the body while the rest is stationary, it is
permissible for some purposes to treat the whole internal ether as
moving with a fraction of the velocity of the body.
On this method of statement the ether outside a moving body is still
absolutely stationary, but, as the body advances, ether may be
thought of as continually condensing in front, and, as it were,
evaporating behind; while, inside, it is streaming through the body in
its condensed condition at a pace such that what is equivalent to the
normal quantity of ether in space may remain absolutely stationary. To
this end its speed backwards relative to the body must be u/μ²
and accordingly its speed forward in space must be u(1-1/μ²).
For consider a slab of matter moving flatways with velocity _u_; let
its internal etherial density be μ², and let the external ether of
density 1 be stationary. Let the forward speed of the internal ether
through space be _xu_, so that a beam of light therein would be
hurried forward with this velocity. Then consider two imaginary
parallel planes moving with the slab, one in advance of it and the
other inside it, and express the fact that the amount of ether between
those two planes must continue constant. The amount streaming
relatively backwards through the first plane as it moves will be
measured by _u_ times the external density, while the amount similarly
streaming backwards through the second plane will be (u-xu) times
the internal density. But this latter amount must equal the former
amount. In other words,
u×1 must equal (u-xu) × μ².
Consequently _x_ comes out x = (μ²-1)/μ²; which is Fresnel's
incontrovertible law for the convective effect of moving transparent
matter on light inside it.
The whole subject, however, may be treated more generally, and for
every direction of the ray, on the lines of Chapter X, thus:--
Inside a transparent body light travels at a speed V/μ; and the ether,
which outside drifts at velocity _v_, making an angle θ with the ray,
inside may be drifting with velocity _v´_ and angle θ´.
Hence the equation to a ray inside such matter is
T´ = ∫ ds / ((V/μ) cos ε´ + v´ cos θ´) = min.,
where sin ε´/sin θ´ = v´/(V/μ) = α´.
This may be written
T´ = ∫ cos ε´ ds / (V/μ (1-α´²)) - ∫ v´ cos θ´ ds / (V²/μ² (1-α´²));
the second term alone involves the first power of the motion, and
assuming that μ²v´ cos θ´ = dφ´/ds, and treating α´ as a
quantity too small for its possible variations to need attention, the
expression becomes
T´ = μT cos ε´ / (1 - α´²) - (φ´B - φ´A) / (V²(1 - α´²)),
T being the time of travel through the same space when empty. Now, if
the time of journey and course of ray, however they be affected by the
dense body, are not to be _more_ affected by reason of etherial drift
through it than if it were so much empty space, it is necessary that
the difference of potential between two points A and B should be the
same whether the space between is filled with dense matter or not (or,
say, whether the ray-path is taken through or outside a portion of
dense medium). In other words (calling φ the outside and φ´ the inside
potential function), in order to secure that T´ shall not differ from
μT by anything depending on the first power of motion, it is necessary
that φ´B-φ´A shall equal φB-φA: i.e. that the potential inside and
outside matter shall be the same up to a constant, or that
μ²v´ cos θ´ = v cos θ; which for the case of drift along a ray is
precisely Fresnel's hypothesis.
Another way of putting the matter is to say that to the first power of
drift velocity
T´ = μ T - ∫ (μ² v´ cos θ´ - v cos θ) ds/V²,
and that the second or disturbing term must vanish.
Hence Fresnel's hypothesis as to the behaviour of ether inside matter
is equivalent to the assumption that a potential function,
∫ μ² v cos θ ds , exists throughout all transparent space, so far
as motion of ether alone is concerned.
Given that condition, no first-order interference effect due to drift
can be obtained from stationary matter by sending rays round any kind
of closed contour; nor can the path of a ray be altered by etherial
drift through any stationary matter. Hence filling a telescope tube
with water cannot modify the observed amount of stellar aberration.
The equation to a ray in transparent matter moving with velocity _u_
in a direction φ, and subject to an independent ether drift of speed
_v_ in direction θ, is
∫ ds / (V/μ cos ε + v/μ² cos θ + u[1 - (1/μ²)] cos φ) = const.
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Transcriber's Notes
Parentheses have been added to mathematical equations for
clarity and correctness. Letters representing mathematical
variables are not marked as italic when this would impair
clarity in expressions and equations.
The lower and upper limits of definite integrals are indicated
by curly brackets, e.g. ∫{0,π}...dθ represents a definite
integral with values of θ ranging from 0 to π.
The caret character ^ is used in the expression (17/16)^{3/2}
to signify raising the fraction 17/16 to the power 3/2. Other
exponents are represented using superscripted numerals, e.g. x².
Italic text is denoted by _underscores_ and bold text by
=equal signs=.
Obvious punctuation and spelling errors and inconsistent
hyphenation have been repaired.
In ambiguous cases, the text has been left as it appears in
the original book.