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.