Chapter 10 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

∫AB 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 = ∫AB ds / V(cos ε + α cos θ)

= ∫AB (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 = δφ / δs,

and ignoring possible variations in the minute correction factor 1?α2 between the points A and B, it becomes

Time of journey = ∫AB cos ε / (1 ? α2) · ds / V ? (φB ? φA) / 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 2T 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)

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 minuti?,

(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 cos θds,

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.

* * *

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

?ρw2 = μH2 / 8π = μ / 8π · e2u2 sin2θ / 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 / w? = 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

?ρ ∫0π ∫a∞ w2 · 2π r sin θ · rdθ · dr,

whence, substituting the above value of w, the energy comes out equal to 4/3πρa3w02.

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 = ? √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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FOOTNOTES:

[1] See Lodge and Howard, Philosophical Magazine for July, 1889. See also Phil. Mag., August, 1888, page 229.

[2] Cf. sections 157A, 143, 187, and chap. xvi., of my Modern Views of Electricity.

[3] Radian is the name given by Prof. James Thomson to a unit angle of circular measure, an angle whose arc equals its radius, or about 57°.

[4] The word "stationary" is ambiguous. I propose to use "stagnant," as meaning stationary with respect to the earth, i.e. as opposed to stationary in space.

[5] Lord Rayleigh, Nature, March 25, 1892.

[6] It does not seem to have been noticed that in Query 22, quoted in the Introduction to the present book, Newton seems to throw out a curious hint in this same direction,-though he immediately abandons it again. He does not appear to have carefully edited his queries; probably they were published posthumously.

[7] On doing the arithmetic, however, I find the necessary concentration absurdly great, showing that such a mass is quite insufficient. (See Appendix 1.)

[8] See Lodge, Philosophical Magazine, April, 1907. Also Appendix 2 below.

[9] Address to Section A of British Association at Montreal, 1884.

[10] Philosophical Magazine, Dec., 1887.

[11] Archives Néerlandaises (1869), Vol. IV, p. 443, or Nature, Vol XXVI, p. 500. Also Chapter IV above.

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