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NINETEENTH - CENTURY AETHER THEORIES
by Kenneth F. Schaffner
Associate Professor of Philosophy and of History and Philosophy of Science, New Collegiate Division, The University of Chicago
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PREFACE
THIS book is an account of a group of theories which occupied the attention of some of the best physicists of the nineteenth century, but which is largely only of historical and philosophical interest today. During the previous century the various aether theories held an importance for physicists which general relativity and SU 3 symmetry, for example, hold today. Fresnel, Cauchy, Green, MacCullagh, Stokes, Kirchhoff, and Lord Kelvin, to mention only a few physicists, developed elastic solid theories of the aether. Maxwell, Fitzgerald, Heaviside, Sommerfeld, and Larmor engaged in serious research on the mechanical characterization of the electromagnetic aether. The famous Michelson and Morley experiment, when first performed, was thought to support Stokes' theory of aberration rather than Fresnel's view. Subsequently the experiment was reinterpreted to support the Fresnel-Lorentz theory of the aether. It took over two decades before the interferometer experiment came, after Einstein's work, to signify a confirmation of his theory of relativity—a theory which employed no aether at all.
The analyses in this book constitute the first stage of an investigation into some of the important ideas and experiments which led to Einstein's special theory of relativity. Certainly, though, the analyses of the aether theories and the various experiments associated with them can be considered on their own terms, rather than as leading toward any particular goal. For reasons which I shall spell out in some detail in the first chapter, I believe that this book should be of interest to scientists, historians of science, and philosophers of science.
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PREFACE

Selections from among what I believe to be the most important primary sources in aether theory constitute the second part of this book, and of course they are necessarily selective. Sufficient references should be given in the commentary of Part 1, however, to guide the reader to most other primary sources. There are omissions of some aether theories, such as the vortex sponge aether and C. Bjerknes' and A. Korn's pulsating sphere theory, and the interested reader is advised to consult E. T. Whittaker's (1960) important monograph on the history of aether theories for a more comprehensive account. Though Whittaker's work is the most important secondary source on the history of aether theory, and though this author is in great debt to Whittaker's book, I cannot advise that it be read alone. There are some severe defects in Whittaker's monograph, particularly in connection with his unwarranted idolization of MacCullagh's aether, his failure to consider many developments in aether theories during the years 18601880 in both England and on the Continent, and—in what is perhaps his most famous blunder—his unfair treatment of Einstein's special theory of relativity. For these and other reasons I have found that Whittaker had to be supplemented with other secondary works to obtain an accurate view of nineteenth-century aether theories. The most useful of these supplementary works are in the Reports to the British Association for the Advancement of Science by H. Lloyd (1834), G. G. Stokes (1862), and R. T. Glazebrook (1885). A book by H. A. Lorentz (1901) which re-presents a series of lectures which he delivered in 1901-2 on various aether theories and aether models is also very useful. Essays by Rosenfeld (1956) and Bromberg (1968) have also proved to be stimulating and helpful in obtaining a more accurate overview of developments in the electromagnetic theories. Papers by G. Holton (1960, 1964) and by T. Hirosige (1962, 1965, 1966) concerned with Lorentz and with relativity theory are highly recommended. I have also had the privilege of seeing some unpublished papers by S. Goldberg and by R. McCormmach, which it is hoped, will soon be more widely

PREFACE

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available.f Some of the philosophical aspects connected with the aether and relativity are developed in the important works of M. B. Hesse (1965) and A. Grünbaum (1963).
I should like to express my thanks to the directors and librarians of the institutions which assisted me in my research, among them the History and Philosophy of Physics Center at the American Institute of Physics, New York; the Institution of Electrical Engineers, London; the Algemeen Rijksarchief in the Hague; and the Deutsches Museum in Munich. I am also indebted to Messrs. Taylor and Francis for permission to reprint the Michelson-Morley paper from the Philosophical Magazine, to Cambridge University Press to reprint the Larmor selections, and to Michael G. Heaviside of the Oliver Heaviside Educational Foundation for permission to repruit the Heaviside selection. Finally I should like to express special appreciation to Melba Phillips and to Dudley Shapere for their critical reading of the commentary, to William Lycan for translation help, to Carl Dolnick for mathematical assistance, to Jan Jones for secretarial help, and to my wife for assistance with the manuscript. I am also indebted to the National Science Foundation for support of the research for this book.
K. F. S.

t R. McCormmach, "H. A. Lorentz and the Electromagnetic View of Nature" (mimeo); "Einstein, Lorentz, and the Electron Theory" Historical Studies in the Physical Sciences, II. (in press); and S. Goldberg "In Defense of Ether: The British Response to Einstein's Special Theory of Relativity, 1905-1911 " (mimeo) ; "The Lorentz Theory of Electrons and Einstein's Theory of Relativity" Am. J. Phys., Oct. 1970.

CHAPTER I
INTRODUCTION: THE FUNCTIONS OF THE AETHER
THROUGH the nineteenth century, various ideas of the aether dominated much of optical and electromagnetic theory. Though aether theories had been proposed in previous centuries, it was primarily through the development and acceptance of a powerful wave theory of light that more and more attention became focused on the nature of the optical medium. At first aether theories were attempts to explain mechanically various optical laws and optical phenomena. Later, with the development in the latter part of the nineteenth century of Maxwell's electromagnetic theory of light, a number of scientists tried to formulate mechanical aether theories adequate to explain Maxwell's theory, and derivatively, physical optics.
The aether approach offered a means of applying the elegant Lagrangian and, later, the Hamiltonian forms of mechanics to optics and electromagnetism. It seemed that a "unified field theory", to use more current jargon, might be the result of careful research into the nature of the aether. Maxwell's theory, at least in the British Isles, inspired some scientists to point their research in this direction. Among them I might mention Oliver Heaviside, George Francis Fitzgerald, and Sir Joseph Larmor. Larmor, the author of the influential Aether and Matter (1900), surveyed the field of aether research in 1907 and wrote in a statement that is typical even though it came in the twilight of the aether approach :
It [the aether] must be a medium which can be effective for transmitting all the types of physical action known to us; it would be worse than no
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solution to have one medium to transmit gravitation, another to transmit electric effects, another to transmit light and so on. Thus the attempt to find out a constitution for the aether will involve a synthesis of intimate correlation of the various types of physical agencies, which appear so different to us mainly because we perceive them through different senses.

It should also be noted, with reservations to be explored in Chapter V of this Commentary, that such a "unified field theory" would have been a mechanical theory, thus accomplishing a complete unification of physics that has often been an inspiration and goal to physicists from Oersted and Faraday to Einstein and Wheeler.
The aether also played a most significant role in the evolution and revolution of ideas of space and time. As the medium of optical, and then electromagnetic activity, the aether was assumed by many to constitute the absolute frame of reference in which the equations of the optical aether and Maxwell's equations would have their simplest form. As the Earth was clearly in motion about the Sun, effects of this motion were conceivable, depending on the aether theory held, which would be experimentally accessible. But serious difficulties appeared in connection with a consistent aether theoretic explanation of stellar aberration, the partial dragging of light waves by moving transparent media, and the null result of the Michelson interferometer experiment. Eventually, to accommodate the last result, Fitzgerald and Lorentz proposed a contraction hypothesis in which the length of an object depended on its velocity through the aether. Soon after this, Lorentz and Larmor developed more radical hypotheses by which certain compensating effects, including an alteration of time, cooperated in eliminating most aether wind effects. Finally, Einstein, knowing of the null effects of various aether drift experiments and of the dependence of electromagnetic induction of relative velocities only, and cognizant of some of Lorentz's ideas, took a most revolutionary step. He articulated a "principle of relativity" and built a theory on it which showed that the Galilean and Newtonian ideas of space and time were in error,

INTRODUCTION I THE FUNCTIONS OF THE AETHER

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and that a simple and consistent electromagnetic theory would require the elimination of the notion of an aether rest frame and require a new understanding of simultaneity and the way in which the spatial and temporal aspects of processes were connected. Though this book does not present anything like a careful analysis of Einstein's special theory of relativity, it does develop the aether concept to the point where it can be shown, with the help of some of Einstein's theory, why the aether was eliminated from physics.
It is hoped that this book, in addition to informing scientists about some of the discarded foundations of electromagnetism and special relativity, might have some effect in stimulating a renewed interest in the history of the aether by historians of science. It is also hoped that it might provide a number of philosophers of science with an insight into the sophisticated complexities of mechanical explanations and mechanical "models", the relation between theory and experiment, theory change, ad hoc modifications of theories, and scientific revolutions. The last topic has been the focus of considerable interest following the publication of T. S. Kuhn's The Structure of Scientific Revolutions (1962) and some essays by P. K. Feyerabend (e.g. 1962, 1965a, 1965b). One of the implicit theses developed in the present book is a critique of those views, such as Kuhn's, which argue that logic and experimental evidence are of little weight in the process of theory replacement. It has become more and more evident to me in doing the research for this book that Kuhn's claims—and also to some extent Feyerabend's—regarding the lack of rationality and experimental control in the development of science constitute serious oversimplifications of the history of science. Point by point refutations of Kuhn and Feyerabend are not the function of this essay, but it is hoped that a careful and historically accurate account of the rise and fall of various nineteenth-century aether theories will constitute the ground on which one can be built.1*

t I do not want either to imply that Kuhn's and Feyerabend's views of the nature of theory replacement are identical, nor that I agree with none of

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their claims. For an example of their differences consider that Kuhn's view of scientific development argues that one paradigm at a time characterizes a science after the pre-paradigm, pre-scientific state, and that this paradigm is instrumental in a dialectical process of creating its sole nemesis and consequently its own downfall. On the other hand, Feyerabend's view of proper scientific development has it that there are sets of partially overlapping but mutually inconsistent theories, each element of which competes for the allegiance of a scientist, their relative merits apparently depending on their relative ability to survive falsification. The last point is, however, somewhat questionable within Feyerabend's philosophy of science, because of the tight connections and influences between theory and observation, but he does claim in various places (1965b) that his approach is built on Popper's work, and unless it totally transcends its Popperean foundations, falsifiability and falsification must play a central role in theory evaluation. See Sir Karl Popper's important (1959) monograph. Since this manuscript was completed, I have formulated an account of the logic of scientific development, and have applied it to the case of theory competition between Lorentz' absolute theory of the electrodynamics of moving bodies (discussed below in Chapter VI) and Einstein's relativistic theory. See my (1970) essay, "Outlines of a Logic of Comparative Theory Evaluation with Special Attention to Pre- and Post- Relativistic Electrodynamics," in Minn. Stud, in Phil, of Sei., 5, ed. R. Stuewer, University of Minnesota Press, Minneapolis.

CHAPTER II

THE HISTORICAL BACKGROUND OF THE NINETEENTH-CENTURY AETHER T H R O U G H Y O U N G AND FRESNEL

THE aether played an important role in natural philosophy and physics from ancient times to the beginning of the twentieth century. Conceived in its most general terms as a "thin subtile matter, or medium, much finer and rarer than air", it was believed by most to fill the celestial regions, and by some to also pervade the air and even solid bodies. Some natural philosophers considered it a fifth element, or "quintessence", not reducible to combinations of the four elementary substances of earth, air, water, and fire. The question as to whether the aether was "ponderable", or subject to the forces of gravity, was also occasionally debated, the most recent proponent of its ponderability being Max Planck, the founder of quantum mechanics.1"
In its long history, the aether has had a number of different tasks assigned to it, such ascriptions occasionally leading scientists to perhaps multiply aethers beyond necessity.î For Descartes, the aether was a form of matter that was transparent and which filled those regions where the matter of the Earth and Sun were not. Such an aether was required by Descartes since "extension" was the essential property of matter or the Res Extensa, and the Universe, in order to exist physically, was required to be a plenum.

t Planck's theory of the ponderable compressible aether is briefly discussed in Chapter III of this book.
t See Whittaker (1960), I, pp. 99-100, for a discussion of the problem of one versus many aethers.

S-N.C.A.T. 2

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NINETEENTH-CENTURY AETHER THEORIES

Not much later Isaac Newton utilized the aether for various other purposes. For Newton an aether was required to explain the transmission of heat through a vacuum, and Newton also attempted, unsuccessfully, to develop an explanation of his inverse square law of gravitation using the aether. In addition, Newton employed an aether in his optical theory, not as a medium through which light could be propagated like sound is propagated through air, but rather as a medium which could interact with light corpuscles to produce refraction phenomena and Newton's rings.
In direct opposition to Newton's approach, the central task of the nineteenth-century aether, with which this book is primarily concerned, is to provide a medium for the propagation of light waves. But wave theories of light and theories of the optical aether had been developed before and during Newton's period by Robert Hooke in his Micrographia (1665) and by Christiaan Huygens in his well-known Traite de la Lumière... (1690). It will be important for reasons to be made clearer later to discuss one of these theories briefly, before we consider the work of Young and Fresnel. Since Hooke's work was rather vague and unquantitative, and because Hooke, unlike Huygens, had little influence on nineteenth-century aether theories, I shall begin with Huygens' contributions and with Newton's criticisms of them.

1. Huygens
Huygens worked within the Cartesian tradition of physics, though he did differ with Descartes on certain points, notably on the finite velocity of light propagation. Huygens proposed that light must be a mechanical motion conveyed from a luminous body to the eye. For Huygens (1690), "in the true philosophy... one believes all natural phenomena to be mechanical effects We must admit this or else give up all hope of ever understanding anything in physics." As we shall see later, a belief in the funda-

HISTORICAL BACKGROUND

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mental character of mechanics is a common assumption held by almost all aether theorists.
Since light is motion and since two light beams could cross one another at any angle without disturbing one another, Huygens did not believe light could be the motion of material particles translated from the luminous object to the eye. Rather, Huygens (1690) argued: "Light is propagated in some other manner, an understanding of which we may obtain from our knowledge of the manner in which sound travels through air".
Huygens, accordingly, proposed a medium he called the aether, which he conceived of as a dense collection of very small, very rigid elastic spheres, through which light was propagated. These spheres filled all space and even penetrated into "solid" material bodies through their hidden porous structure. Furthermore, Huygens proposed that the aether and the matter interacted so as to affect the velocity of the light in the bodies, since in an intermingled state the total elasticity of the medium could be considered diminished thus retarding the velocity of propagation of a wave through it. Explanations of refraction and even double refraction were founded on these ieas, and worked out by Huygens using his own recently discovered principle of secondary wave propagation.
For Huygens, light waves were very much like sound waves, for even though they were propagated with a considerably higher velocity than sound, they were strictly longitudinal in form. Isaac Newton, whose optical investigations both preceded and postdated Huygens' work, could not accept a wave theory of light as he was unable to see how well-defined rays and sharp shadows could be explained by such a theory. Newton favored a corpuscular theory of light, such as Huygens had rejected, which did not require an aether for quite the same reasons that Huygens' theory did, though Newton, as noted above, did use an aether for other purposes in his optics. Later, in 1717, Newton felt confirmed in the wisdom of his rejection of the wave theory when he discovered
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NINETEENTH-CENTURY AETHER THEORIES

that a ray of light which had been obtained by double refraction differed from ordinary light in that the former possessed a directional orientational property the latter did not have. Newton talks of the ray obtained from double refraction as having "Sides", such as may be possessed by a rectangle but not by a circle. (Later Malus termed this property "polarization".) Newton was quite convinced that a wave theory of light could not explain such a property, though a corpuscular theory might, as corpuscles themselves could have sides.
Such objections, together with Newton's growing authority in physics, brought hard days on the proponents of the wave theory of light during the eighteenth century, and accordingly on the development and acceptance of aether theories in which the aether functioned as the light medium. The rejection of the aether during the eighteenth century was apparently also aided by the influence of the "philosophical" or methodological preface which Roger Cotes introduced into Newton's second edition of the Principia in 1713. In this preface Cotes polemized against Descartes and the Cartesians who :

When they take a liberty of imagining at pleasure unknown figures and magnitudes, and uncertain situations and motions of the parts, and moreover, of supposing occult fluids, freely pervading the pores of bodies, endued with an all-performing subtility, and agitated with occult motions, . . . run out into dreams and chimeras, and neglect the true constitution of things, which certainly is not to be derived from fallacious conjectures, when we can scarce reach it by the most certain observations.

Cotes distinguished natural philosophers into three camps: (1) the Newtonians, who founded their science on experiments and observations, and who subscribed to action at a distance and a "void" in their gravitational theory, (2) the Aristotelians and Scholastics whom Cotes summarily dismissed, and (3) the Cartesians, who fill the void with vortices and subtile matter to the detriment of true scientific philosophy. It is easy to see what might have hap-

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pened to an aether theory approach as such a positivistic philosophy of science became widespread during the eighteenth century.1"

2. Young
The authority which Newton's theories eventually came to exercise over physics by the beginning of the nineteenth century is well displayed in comments in the writings of Thomas Young, a physician who began to work on sound and light in the closing years of the eighteenth century. In 1800 Young published his first thoughts on a wave theory of light as a small part of a paper discussing some experiments on sound. In the section of this paper titled "Of the Analogy between Light and Sound", Young proposed a wave theory of light rather similar to Huygens' theory, except that the difference in velocities of light in media was ascribed to differences in the aether's density rather than to differences in rigidity.
About two years later Young (1802) worked out his thoughts in somewhat more detail. Apparently in an attempt to win his ideas a fair hearing from the Newtonians, Young quoted Newtonian "scripture" in support of the hypotheses of his wave theory. The spirit of the times is perhaps aptly caught in Young's (1802) statement that :
Those who are attached, as they may be with the greatest justice, to every doctrine which is stamped with the Newtonian approbation, will probably be disposed to bestow on these considerations so much the more of their attention as they appear to coincide more nearly with Newton's own opinions.
Young collected passages from Newton's scattered writings on optics and the aether in support of the four hypotheses of his own aether and wave theory. These hypotheses, which indicate quite
t The influence of Cotes' preface is discussed by Whittaker (1960), I, pp. 30-31.

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clearly the connection between a theory of optics and an aether at the beginning of the nineteenth century were :
1. A luminiferous aether pervades the universe, rare and elastic in a high degree.
2. Undulations are excited in this aether whenever a body becomes luminous.
3. The sensation of different colors depends on the different frequencies of vibrations excited by light in the retina.
4. All material bodies have an attraction for the aethereal medium, by means of which it is accumulated in their substance and for a small distance around them, in a state of greater density but not of greater elasticity.
Young deduced several propositions from these hypotheses concerning the common velocity of light waves in a medium, and spherical form of the wave, the refractive capacity of a medium as a function of its aether density, and a principle of interference of waves. Young continued his work in the next few years and in a volume published in 1807 (Young 1807) he discussed his famous two-slit interference experiment which is so often cited as proving the existence of waves of light. Young also attempted provisional explanations of inflection or diffraction, and did some excellent work on the colors of thin plates and Newton's rings. Later, in 1809, he defended Huygens' theory of double refraction, along with several modifications of his own, against Laplace's corpuscular theory of double refraction.
Though Young's experiments were brilliantly conceived and executed, the arguments which he gave in support of his theory did not go much beyond what Huygens had accomplished. Young apparently was somewhat deficient in training in mechanics and higher mathematics, and was not able to bring the sophisticated theoretical developments of recent science to work in his favor.1"

t See Crew's (1900) brief biography of Young in which he suggests this.

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The Newtonian school was still exceedingly strong in the beginning of the nineteenth century, and to say that Young's theory did not attract many followers would perhaps, on the evidence of his biographer, be understating the situation.1" Even after Young's fundamental papers on the wave theory and its experimental foundations had appeared, Herschel and Laplace continued to develop optics in the corpuscular manner, and to ignore Young's contributions.
There were some reasons for this other than simple scientific inertia. In the years 1808-10 E.-L. Malus performed a number of experiments on the intensity of light that was reflected from a transparent body's surface. Malus analyzed the light using a double refracting crystal of Iceland spar and noted that light reflected from the surface of transparent media possessed the same property of having "Sides" which Newton had noticed in connection with doubly refracted rays. Malus gave the name "polarization" to this property, and attributed it, on the basis of a particle theory of light, to light corpuscles having their sides all turned in the same direction, much as a magnet turns a series of needles all to the same side. Subsequent to Malus' publication, the French physicist Biot developed a more complex corpuscular theory of polarization.
In England in the years 1814-19, David Brewster conducted several experiments, some of them similar to Malus', and obtained results which were of considerable significance to the future of optics. Among these results was a formula connecting the angle of complete polarization of the reflected ray with the refractive index of the media—a relation which Malus had sought but could not determine. Brewster also discovered the existence of biaxal crystals in which there were two axes along which double refraction did not occur, rather than one axis as in Iceland spar. The immediate effect of Brewster's discovery was to call into serious question Huygens' analysis of double refraction and the wave theory of

t Whittaker refers to Peacock's Life of Young in recounting the incident in which Young's pamphlet replying to a scathing attack on his wave theory in the Edinburgh Review only sold one copy.

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light in general since Huygens' construction no longer sufficed to account for the refraction in the more complex biaxal crystals. Brewster also empirically obtained what have come to be called Fresnel's sine and tangent laws, about which I shall have more to say below.

3. Fresnel
It was clear that polarization was a problem which was exceedingly difficult to explain on the basis of the wave theory, as long as light was conceived on the analogy with sound. Sound waves, as longitudinal waves, could not account for the "sidedness" displayed in polarization phenomena. Young, purportedly reflecting on Brewster's experiments and on the results of an experiment carried out in France by Arago and Fresnel,1" was the first to suggest a possible explanation of polarization on the basis of the wave theory. In a letter to Arago dated 17 January, 1917, Young proposed that if light waves were conceived of as transverse waves, they could admit of polarization. Not long after, in another letter to Arago, Young compared light waves to the motions of a cord which has one of its ends agitated in a plane.
Arago showed this letter to Fresnel who at once seized upon the hypothesis of transverse waves as one with which he could explain polarization. Subsequently, Fresnel made this hypothesis the basis of his most influential dynamical theories of double refraction and reflection and refraction.
In the years between 1814 and 1818, Fresnel had already made very important contributions to the wave theory of light. His great memoir on diffraction (Fresnel, 1826) was developed gradually during these years and is worked out on the basis of the older
t This is the experiment in which two pencils of light polarized in planes at right angles to one another cannot be made to interfere under any condition of path-length difference. The results were not published until 1819 though the experiment had been done several years earlier.

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longitudinal wave theory of light. But since this inquiry was essentially "kinematical" and not concerned with the true motions of the aethereal medium, the shift to an aether which would support transverse vibrations did not vitiate his diffraction theory.1' Fresnel also developed in the year 1818 another theory which was not strictly dependent on the structure of the medium and the type of wave it would support. This is Fresnel's famous explanation of aberration phenomena and includes the derivation of his partial dragging coefficient. I shall have more to say about this inquiry in the next chapter.
I now turn to consider, somewhat sketchily, Fresnel's two important dynamical theories of light. My intention here is not to present an adequate account of Fresnel's dynamical theories, but rather to emphasize the important but unsatisfactory aspects of these so as to prepare the reader for the more adequate aether theories of the later chapters. Fresnel never actually worked out an acceptable mechanical theory of light, though this seems to have been his intention, and his accounts are at best quasi-mechanical or quasi-dynamical attempts at the analysis of wave motions of the aether. Nevertheless these attempts were of the greatest significance because of their plausibility and simplicity, their agreement with experiment, and their ability to predict new experimentally confirmable phenomena. The fact that Fresnel could so effectively systematize optics from the point of view of the transverse wave theory soon resulted in the almost complete acceptance of the wave theory and the consequent rejection of the corpuscular approach.
Fresnel'sfirstattempt at a dynamical theory of the aether medium focused on the problem of double refraction which had been raised anew by Brewster's biaxal crystals. The result of Fresnel's (1821) inquiry was a theory of double refraction which was characterized in 1834 in a report to the British Association for the Advancement of Science in the following eulogistic terms :
t Fresnel's diffraction theory was not given an appropriate dynamical basis until G. G. Stokes' essay (1849) "On the Dynamical Theory of Diffraction".

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The theory [of double refraction] of Fresnel to which I now proceed,— and which not only embraced all the known phenomena, but has even outstripped observation, and predicted consequences which were afterwards fully verified—will I am persuaded, be regarded as thefinestgeneralization in physical science which has been made since the discovery of universal gravitation.t

FresnePs theory is an "elastic solid theory" in that it assumes that the aethereal medium is so constituted as to permit transverse waves to be propagated through it. It is not a continuum but, like Huygens' aether, consists of a huge number of very small aether molecules with forces acting between them. There is nothing inconsistent with using this type of molecular hypothesis as the basis of a general mechanical theory of the aether, and it was later employed by Cauchy and by Green, whom I shall consider in Chapter IV. Fresnel did not pursue a true mechanical approach, however, but rather introduced hypotheses additional both to the mechanical laws of motion and to the force functions expected in an elastic medium.t
One of Fresnel's additional hypotheses was innocuous and concerned the relation between the vibrations of the medium and the direction of the plane of polarization; it assumed that the vibrations of polarized light were at right angles to the plane of polarization. Fresnel's three other assumptions, however, were somewhat artificial and even inconsistent with a true elastic solid theory. For example, his second hypothesis, that the elastic forces produced by the propagation of a transverse plane wave were equal to the product of the elastic force produced by the displacement of a single molecule of the aether multiplied by some constant which is independent of the direction of the wave, is not true in the case of a mechanical elastic solid. In elastic solids, the elastic forces

t The report was authored by Humphrey Lloyd (1834). The various phenomena that were accounted for and predicted de novo are given in Lloyd's essay.
% The assumptions of Fresnel's theory were analyzed by E. Verdet (1869), on whose work the following account of Fresnel's double refraction theory is based.

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are functions of the particle's displacement relative to its neighbors, and the implied restriction of the elastic force to act only along the line of displacement is also false. In still another hypothesis, Fresnel stipulated that only the component of the elastic force parallel to the wave front was to be considered effective in the propagation of a light wave. This amounts to an ad hoc elimination of the longitudinal wave that should also be propagated in a disturbed elastic solid. (The problem of the longitudinal wave will be considered in detail in Chapter IV.) Finally, Fresnel's fourth hypothesis asserted that the velocity of a plane wave is proportional to the square root of the effective component of the elastic force developed by the wave But this hypothesis had for its foundation only the analogy that such a relation holds in the case of transverse vibrations of a stretched string.*
Fresnel's theory of double refraction is clearly not a reduction of optics to mechanics, and Fresnel himself expressed concern about the security of foundation of his hypotheses. Nevertheless, he thought them warranted by the consequences which could be drawn from them, and draw them he did. He obtained acceptable explanations of double refraction, solving the problem of biaxal crystals and showing that Huygens' construction for uniaxal crystals was a special case of his own more general wave surface. It was subsequently determined by Hamilton that the Fresnel theory would imply the unanticipated phenomena of conical refraction, which was then sought for and found. At the time all available experimental tests of Fresnel's theory showed striking agreement between theory and experiment. Later, however, more precise optical experiments showed Fresnel's wave surface to be only a very close approximation to the actual surface.t

t These criticisms are in part based on Verdet's (1869) work and Preston's (1895) account, pp. 318-23, but are paralleled by similar ones in Whittaker (1960), I, p. 119.
Î See Whittaker (1960), I, pp. 121-2, for more detail on this topic.

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Two years after he had developed the theory of double refraction, Fresnel (1832) proposed another quasi-dynamical theory: this one to take account of reflection, refraction, and the polarizing properties of the surfaces of transparent media. In this theory Fresnel proceeded similarly to the way he had two years earlier. He made use of whatever dynamical principles he could use, e.g. the conservation of vis viva; he violated other mechanical principles, such as the continuity of the normal component of the aether displacement across an interface—this resulted in a covert inconsistency with his other boundary conditions. Fresnel also added the unwarranted but plausible assumption of Young that refraction depended on the differences in density and not in rigidity. He obtained valuable results though, among them derivations of the important laws connecting the relative amplitudes of the reflected and incident waves. If the wave was polarized in the plane of reflection he obtained :
Ai __ sin(/—r) Ar ~ sin(f-hr)
in which / is the angle of incidence of the normal of the plane wave and r the angle of refraction of the normal of the refracted waven. If the wave were considered to be polarized perpendicular to the plane of reflection, then the amplitude ratios became:
Ai _ tan(/—r) Ar ~~ tan (/+/·) *
In spite of the dynamical insecurity of the foundations of these theories of double refraction and reflection, it would be unfair to criticize Fresnel for failing to avail himself of the proper mechanical bases. For at the time when Fresnel was constructing these theories, the dynamics of an elastic solid were only beginning to be developed and were not worked out in any satisfactory form. In fact, it was due in part to FfesneFs groping attempts at such theories that the French mathematical physicists Navier, Poisson, and Cauchy were stimulated to perfect an adequate mechanical theory of

HISTORICAL BACKGROUND

19

the motions of an elastic solid. This is a topic to which I shall return in Chapter IV.
This historical introduction, then, brings us somewhat into the nineteenth century, and relates how the wave theory of light together with its elastic solid aether developed in the early quarter of the century of our concern. But with the exception of a brief aside, I have ignored one basic problem which confronted the wave theory of light from 1727 on, and which is now considered to have been adequately solved only by Einstein's special theory of relativity in 1905. This is the problem of aberration, first discovered in starlight by Bradley, and later experimentally examined in both stellar and terrestrial cases by Arago, Airy, and Michelson and Morley. In the next Chapter I shall consider this problem and its connection with the aether of the wave theory of light. I shall return to the dynamical problems of the optical aether in the following chapter where I shall consider in some detail the elastic solid theories of Green, MacCullagh, and Lord Kelvin.

CHAPTER III
ABERRATION FROM BRADLEY TO MICHELSON AND MORLEY
IN THE previous chapter I outlined in fairly broad terms the development of the elastic solid aether in the early nineteenth century. I now move to more detailed investigations of the nineteenthcentury aether theories. Following the plan of most nineteenthcentury aether monographs, I shall begin a close scrutiny of the aether by examining the connections that were thought to exist between aether and ponderable matter. Later I shall discuss the hypothesized nature of the aether in free space. I shall not consider all of the aether-matter connections in this chapter, but will primarily be concerned with the aberration observations and experiments performed during the eighteenth and nineteenth centuries.1* Problems of dispersion, in which the aether interacts with matter, will be referred to near the conclusion of the next chapter. There are, of course, certain problems which arose in connection with light and matter, such as the photoelectric effect, which never obtained an explanation on the basis of an aether theory.
Aberration problems are concerned with the effect of the motion of ponderable matter on aether phenomena, such as the velocity of light waves in the aether as viewed from moving ponderable
t The alteration in the observed frequency of the light waves produced by relative motion, and at one time it was thought absolute motion, will not be discussed here, as it does not seem to have been very important in the development and replacement of aether theories. This phenomenon, known as the Doppler effect, is different in the aether and relativity theories, however, and would have to be discussed in any thorough comparison of these two approaches.
20

ABERRATION FROM BRADLEY

21

matter, or the effect of the motion of ponderable matter on light moving within it. Throughout most of the nineteenth century, all known aberration phenomena were optical. Toward the close of the century, however, with the development of Maxwell's and Lorentz' theories, some electrical and magnetic aberration experiments were performed. These were occasioned by Hertz' and Lorentz' theories and were satisfactorily accounted for by Lorentz' theory.

1. Bradley
In the years 1725-6 Samuel Molyneux, with some assistance by James Bradley, the Professor of Astronomy at Oxford, attempted to carry out a careful experiment designed to detect traces of the annual parallax of the "fixt Stars". Beginning on 3 December, 1725 and intermittently over a period of twelve months, measurements were made of the position of the star y Draconis, and an apparent causal influence of the Earth's motion on the direction of the star was observed. It was not, however, the sort of result which the two astronomers had been expecting. Bradley (1728) noted that : "this sensible alteration the more surprised us, in that it was the contrary way from what it would have been, had it proceeded from the annual parallax of the Star". The apparent displacements of the star, instead of being directed towards the Sun as expected, were in a direction perpendicular to the earth's orbit.
In 1727-8, with an instrument which was less constrained in movement than Molyneux's, Bradley alone repeated the observations, this time being able to observe several stars. He again found the same perplexing phenomenon. Larmor (1900) reports that Bradley was led to the true explanation of this phenomenon by the "casual observation of a flag floating at the masthead of a ship ; when the ship changed its course, the flag flew in a different direction". Bradley does not recount this particular story in his paper, how-

22

NINETEENTH-CENTURY AETHER THEORIES

ever, which he published in the Philosophical Transactions (1728),
but rather states simply :
At last I conjectured, that all the Phaenomena hitherto mentioned, proceeded from the progressive Motion of Light and the Earth's annual Motion in its Orbit. For I perceived that if Light was propagated in Time, the apparent Place of a fixt Object would not be the same when the Eye is at Rest, as when it is moving in any other Direction, than that of the Line passing through the Eye and Object; and that, when the Eye is moving in different Directions, the apparent Place of the Object would be different.

Bradley formulated a mathematical expression relating the apparent displacements to the velocity of the earth and the velocity of light :
And in all Cases, the Sine of the Difference between the real and visible Place of the Object, will be to the Sine of the visible Inclination of the Object to the Line in which the Eye is moving, as the Velocity of the Eye to the Velocity of Light.

How this follows can be seen from Fig. III.l. The star is at A with the earth moving from B towards C. A telescope, were the Earth stationary, would be sighted parallel to line CA, but, since the Earth is in motion, the telescope must be sighted along a line parallel to BA. From the law of sines it follows that sin oc/CB = sin ß/AC. Since DB : AC : : v : c, where v is the orbital velocity of the Earth and c the velocity of light, we have sin a : sin ß : : v : c, which is Bradley's relation. If the angle a is small, and it is since v is much smaller than c, then one can write to a close approximation tan a = v/c. This is usually the way in which the law of aberration is expressed in contemporary texts. The ratio v/c is often referred to as the aberration constant.
Bradley explained his findings in terms of the corpuscular theory of light, on whose basis the addition of velocities is physically very plausible. One only need assume that the corpuscles are not affected by the Earth's gravitational attraction. On the basis of the wave theory, the true path of the light is more difficult to explain, for the explanation apparently has to involve the assumption that the

ABERRATION FROM BRADLEY

23

Earth's motion through the aether medium does not affect the motion of the medium. Thomas Young (1804) actually made this suggestion in connection with an explanation of aberration when

Direction of earth's motion
FIG. III. 1. Stellar aberration.
he wrote : "Upon considering the phenomenon of the aberration of the stars I am disposed to believe that the luminiferous aether pervades the substance of all material bodies with little or no resistance, as freely perhaps as the wind passes through a grove of trees."
2. Fresnel In the beginning of the nineteenth century the French physicist
François Arago reasoned, on the basis of the corpuscular theory of light, that the aberration of light in an optically dense medium, such as in a glass prism, would be different if the incident starlight
S-N.C.A.T. 3

24

NINETEENTH-CENTURY AETHER THEORIES

were passed through the prism in the same direction as the Earth's motion than if it were passed in an opposite direction. Arago's experiments to test this hypothesis were performed in 1808-9 but gave a null result. Though their accuracy has since been questioned, it appeared at the time that a most peculiar phenomenon was occurring. Though it was clear that the motion of the Earth affected the direction of apparent propagation of incoming light from a star, when the same light was sent through a refracting medium, the medium exerted no additional aberrational effect on it. This seemed to imply that aberration was and yet was not operative. Some years later, after Fresnel had made his initial contributions to the wave theory of light, Arago wrote to him telling him of these experiments and of his inability to conceive of a reasonable explanation on the basis of corpuscular theory. He asked Fresnel whether an explanation in terms of the wave theory might be possible.
Fresnel replied to Arago's query in a letter which was subsequently published in the Annales de Chemie... (1818). In this letter Fresnel was able to formulate a simple and elegant explanation of Arago's results on the basis of the wave theory of light; an explanation which not only accounted for aberration effects then known, but which was subsequently confirmed in a number of different ways throughout the nineteenth century.
Fresnel began his letter to Arago by considering possible alternative explanations. The corpuscular interpretation seemed improbable to Fresnel for the reason that it would require, as Arago had suggested, that a radiating body would have to impart infinitely many different velocities to light corpuscles, and that the corpuscles would affect the eye with only one of those velocities. Complete aether drag was also ruled out, for though it would explain the null effects of the Earth's motion on refraction phenomena, it apparently could not explain Bradley's aberration phenomena. Fresnel accepted Young's idea that aberration phenomena, to be explained on the basis of the wave theory, would have to allow the aether to

ABERRATION FROM BRADLEY

25

pass freely through the earth, or at least not disturb the aether's motion in the atmosphere where aberration observations had been carried out.
Fresnel proposed to account for both Arago's result and aberration phenomena by supposing a partial aether drag in which transparent bodies with refractive indices greater than a vacuum (in which the index n = 1) were conceived to have a greater aether density within them, and that only the aether density which constituted an excess over and above the aether density in the vacuum would be completely carried along by the moving body.
Fresnel's argument to support this point and to derive his partial dragging coefficient is somewhat analogical and not very convincing. He supposes, like Young before him, and as he himself does in his later paper on reflection and refraction, that the index of refraction («), the velocity of light (c), and the densities of the aether (Δ) in empty space and within the body are related by:

_£_ = ZÜ* = Ub - ^Ab

a i)

cb ~ n ~~ 1 ~ <yM

the b subscript distinguishing the c, n, and Δ within the transparent

body. In a moving body in which n > 1, only the excess of the

aether is considered dragged. Fresnel gives the following argument

in support of his reasoning that if this is so, then only a partial

augmenting of the velocity of light in the moving medium will

occur :

By analogy it would seem that when only a part of the medium is displaced, the velocity of propagation of waves can only be increased by the velocity of the centre of gravity of the system.
The principle is evident in the case where the moving part represents exactly half of the medium ; for, relating the movement of the system to its centre of gravity, which is considered for a moment as fixed, its two halves are travelling away from one another at an equal velocity in opposite directions; it follows that the waves must be slowed down in one direction as much as they are accelerated in the other, and that in relation to the centre of gravity they thus travel only at their normal velocity of propagation; or, which amounts to the same thing, they share its movement. If
3*

26

NINETEENTH-CENTURY AETHER THEORIES

the moving portion were one quarter, one eighth or one sixteenth, etc., of the medium, it could be just as easily shown that the velocity to be added to the velocity of wave propagation is one quarter, one eighth, one sixteenth, and so on, of that of the part in motion—that is to say, the exact velocity of the centre of gravity; and it is clear that a theorem which holds good in all these individual instances must be generally valid.
This being established, and the prismatic medium being in equilibrium of forces (tension) with the surrounding ether (I am supposing for the sake of simplicity that the experiment is conducted in a vacuum), any delay the light undergoes when passing through the prism when it is stationary may be considered as a result solely of its greater density... .

By Fresnel's supposition, only the excess aether density above the vacuum's density is dragged by a moving transparent body, i.e. (Δ^—Δ) which by (3.1) above equals Δ6(ΐ — 1/^). In accordance with the reasoning developed in the quote above, the increase in the velocity of the light within the moving media will be v(l — 1/nf}. The factor (1 — 1 /n2) is Fresnel's famous "partial dragging coefficient", variously called Fresnel's convection coefficient or the coefficient of entrainment. Regardless of what one may think of the argument by which it was deduced, the coefficient is of the greatest importance in aberration theory. It was noted by Lord Rayleigh as late as 1892 in connection with various aberration problems that: "It is not a little remarkable that this formula [i.e. the convection coefficient] and no other is consistent with the facts of terrestrial refraction, if we once admit that the aether in the atmosphere is at absolute rest."
In his 1818 letter Fresnel showed that his partial dragging hypothesis would adequately explain Arago's result, and, moreover, that it would also predict that filling an aberration detecting telescope with water would have no effect on the observed aberration. Such an experiment had been proposed in the previous century by Boscovich, but it was not carried out until 1871 by Airy, who did obtain Fresnel's predicted result.
Rather than presenting a reconstruction of the way in which Fresnel shows that his formula accounts for Arago's experiment— and his reasoning is not very explicit on this point, as can be seen

ABERRATION FROM BRADLEY

27

from the Fresnel selection, pp. 128-31, I propose to give instead an example which is closer to Fresnel's water-filled telescope or microscope case, but which can easily be seen to extend to the analysis of a moving prism. This example is a modification of one ofH. A. Lorentz'(1901).
We consider light from a star impinging on a moving system as is shown in Fig. III.2. The tube ABCD is empty (a vacuum) and

FIG. III.2. Stellar aberration in a vacuum and in a moving transparent medium. (After Lorentz, 1901.) The shaded area is filled with glass.

the tube CDEF is filled with glass with an index of refraction

n > 1. The light from the star follows the apparent path parallel

to AC and BD and strikes the glass at CD at angle of incidence /.

Let us suppose now, for the moment, that there is no influence

of the Earth's motion on refraction phenomena. If this were so,

then simply taking the angle i as given and applying Snell's law we

get:

n sin r = sin i,

(3.2)

the index of refraction of the vacuum being equal to 1. Let v be the velocity of the Earth through the aether, c the velocity of light in the vacuum, and cb the velocity of light in the glass. By*Bradley's

28

NINETEENTH-CENTURY AETHER THEORIES

aberration experiment just discussed, we can set tan / = v/c, or

for small angles :

tan i = sin / = v/c.

(3.3)

From (3.2) and (3.3) we then have tan r = v/nc, since r is even

smaller than /, and accordingly segment length EG = Ivjnc, if

CG = I.

Whatever the length of EG is, it represents a measure of the effect

of aberration within this system. We have now calculated its

length assuming that the motion of the Earth exercises no effect

on refraction phenomena, though we have included the effect of

the Earth's motion on aberration in a vacuum.

But this calculation above is inconsistent with the assumption of

an absolutely stationary aether through which the glass is moving,

unless there are effects which compensate for the aether wind

which is, by our hypothesis that is supported by the aberration

effect in the vacuum, blowing through the glass with a velocity

equal to — v. Let us see what the effect of the aether wind within

the glass would be if the aether passed through the glass as freely

"as the wind passes through a grove of trees". We now consider

the true direction of the light (as we did in calculating the angle

of aberration for the vacuum) as perpendicular to the interface

CD. Since the light moves, on this assumption, with velocity cb in the vertical direction and with velocity — v in the horizontal

direction we obtain the two new points, E'G\ where it intersects

with the base line of Fig. III.2. From similar triangles it follows

that:

E'G.CG = v:cb = v: c/n

(3.4)

whence E'G — Inv/c. The difference between EG and E'G represents the difference between the outcome of Arago's experiment and the predicted outcome of a wave theory without Fresnel's partial dragging coefficient. If Fresnel's hypothesis is correct, then by including the effect of the partial drag on the case we have just considered, the difference between E'G and EG should disappear.

ABERRATION FROM BRADLEY

29

By Fresnel's hypothesis, in the time l/cb, the moving glass should have dragged the light over a distance oflv/cb(l — 1/n2). This should just be equal to the discrepancy between E'G and EG or equal to E'E in the diagram. Now

E>E = E'G-EG=!^-^

(3.5)

c nc

and since cb = c/n, if we divide the numerator and denominator of both fractions by n and factor we obtain

«-Si-?)·

™

which is the desired result, indicating that the Fresnel partial dragging coefficient accounts for what is actually observed.
The Fresnel convection coefficient was subsequently confirmed for light projected through water moving relative to the surface of the Earth by Fizeau (1851) and a similar experiment was repeated by Michelson and Morley (1886) with considerably increased precision, with the convection coefficient again being confirmed.

3. Stokes
We have seen that Fresnel noted in his letter to Arago that he could not see how to possibly account for stellar aberration on the basis of the wave theory if the Earth were assumed to completely drag the aether along with it, so that the velocity of the aether would be equal to the absolute velocity of the earth. In 1845 the British physicist G. G. Stokes published a short paper in the Philosophical Magazine which showed how this could be done. The Stokes theory of aberration was of some influence during the nineteenth century, at least until about 1886-7, for reasons that have to do in part with the confirmation of Fresnel's partial dragging coefficient for moving water by Michelson and Morley (1886), and in part with a criticism of Stokes' theory by Lorentz in 1886,

30

NINETEENTH-CENTURY AETHER THEORIES

Until 1886, however, it seemed that Stokes' and Fresnel's theories

were each adequate to account for aberration phenomena.

Stokes assumed that the Earth completely dragged the aether

along with it in its orbit, but that it did so only near its surface.

The velocity of the aether is, however, claimed to be identical at

every point on the Earth's surface and apparently equal to the

absolute velocity of the Earth in the universe. Out in space, how-

ever, "at no great distance" from the Earth, the aether was supposed

to be in a state of absolute rest. With these ideas in mind, Stokes

considered how aberration phenomena might be explained.

He began by noting that the direction of the wave front of the

starlight impinging on the Earth should be dependent on both

the velocity of the light through the aether and on the velocity of

the aether streaming near the Earth. Stokes analyzed the possible

effect of the aether's supposed motion on the equation of the wave

front.*

Let w, v, w be the velocity of the aether stream in the neighbor-

hood under consideration, i.e. somewhere above the Earth. Assum-

ing that the axis z of an xyz coordinate system is in the direction

of the propagation of a plane wave, the equation of the wave front

is:

z = C+Vt+C

(3.7)

where C is some arbitrary constant, V the velocity of light, and / the time, and ζ a small quantity and a function of x, y, and t. ζ will turn out to be a measure of the rotation or aberration of the wave front as caused by u, v, w.
Stokes confined himself to first-order quantities, dropping terms involving squares of the ratio of velocity of the Earth to the velocity of light. The direction cosines of a normal to the wave front are :

cos a = —-T- 5 cos β = — — , cos y = 1. (3.8)

t Stokes' analysis is presented in more elegant terms in Lorentz (1901).

ABERRATION FROM BRADLEY

31

At a distance V dt along the normal, the coordinates will be altered from what they would be if there were no aether stream velocity. The coordinates, taking the moving aether into account, will be :

x' = x+iu — V-j-) dt,

y=y+L-v*^dt9

(3.9)

z' = z+(w+v)dt.

Substituting F(x, y, t) for C, and employing (3.7), expanding the resulting expression neglecting dt2 and the square of the aberration constant, and solving for z, Stokes obtained :

z = C+ VtH + (w+ V) dt.

(3.10)

Using (3.7) again, this time computing the wave front's equation

at time t+dt, Stokes got:

("S) z = C+VtH: (v+^\dt.

(3.11)

Comparison term by term of (3.10) and (3.11) yielded:

Y~ = w or ζ = ί wdt.

(3.12)

But since ζ is small, w dt may be approximately represented by

w dz/V9 the equation for the wavefront (3.7) becoming: /

z = C+Vt+~ iwdz.

(3.13)

Comparison of (3.13) with the equations for the direction cosines of the normal to the wavefront gives :

π 1 Γ dw . a π 1 Γ dw .
*-2 = vjn;dz> ß-2 = v)Tydz·

„
(3·

14)

32

NINETEENTH-CENTURY AETHER THEORIES

The terms under the integral sign are measures of the rotations of the wave front about the y and A: axes. Integrated, the expressions represent the components of the total rotation of the wave front— i.e. the aberration—due to the aether stream velocity. The limits of the integration must range from the Earth's surface to a point out in space where the effect of the Earth's motion on the aether is imperceptible. The equations of (3.14) will account for the aberration that is actually observed if w, v9 and w are such that udx+ + v dy+ w dz is an exact differential. Physically this amounts to the assumption that the aether is irrotational : that it has no vortices in its stream. If this is so, then :

and substitution of this in (3.14) yields:
(3.15)
Stokes applied equations (3.15) to a star. Point 1 is sufficiently distant so that u\ = 0 and v± = 0. The plane xz was chosen so that it passed through the direction of the Earth's motion. Then v2 equalled 0, and ß^—ßi also equalled zero. Consequently:
0C2— a i = w—2 which is the aberration constant, and Bradley's well-known result.
4. Michelson and Morley In 1881 when A. A. Michelson first performed his interferometer
experiment Fresnel's explanation of aberration was generally

ABERRATION FROM BRADLEY

33

accepted.* If, as in Fresnel's theory, the aether was indeed stationary with the Earth moving through it, the time it would take for a light wave to pass between two points on the surface of the Earth would be different if it were moving in the direction of the Earth's motion, or opposite to this motion. Because of cancellation effects involved in passing the light to and fro over the same path, the effect of the Earth's motion is extremely small, of the second order of v/c9 or about one part in 108. Nevertheless, Michelson discovered a means of measuring this quantity.
He constructed an apparatus known as an interferometer, which permitted two rays of light which traveled over paths at right

FIG. ΙΠ.3. A Michelson interferometer.
angles to one another to recombine and interfere. The original interferometer is diagrammed in Fig. III.3. The light from a lamp or a sodiumflamepositioned at S is divided by the partially silvered
t For a more complete exposition of Michelson's work see R. Shankland's (1964) excellent article on the interferometer experiment. Lloyd Swenson's (1962) dissertation is also worth reading in this connection.

34

NINETEENTH-CENTURY AETHER THEORIES

mirror located at A and moves to B and C, from which it is reflected and then recombined at A. If the paths AB and AC are equal, the two rays interfere along AE. The interference shows up in the eyepiece positioned at E as thin dark fringes or bands in the white or yellow light. Ordinarily monochromatic light is used for alignment and then white light can be substituted and colored fringes sought for. In the latter case the fringes disappear very easily and can be used as a careful check on the equality of the paths.
If we assume with Fresnel that the Earth moves through the aether without dragging it along, the amount that the fringes of the interferometer should shift when the interferometer is turned through an angle of 90° can be computed as follows : We let c be

FIG. III.4. The "vertical" path of light in the interferometer as considered from the point of view of Fresnel's aether theory.
the velocity of light, v the velocity of the earth through the aether, D the distance AB or AC, T the time light requires to pass from A to C, and 7Ί the time required for the light to return from C to A\. The distance AA\ is shown in Fig. III.4 and is due to the

ABERRATION FROM BRADLEY

35

movement of the interferometer during the time required for the

passage of the light from the partly silvered mirror to the reflecting

mirrors and back again. In both cases, however, the horizontal

distance traversed is Z>, with the velocity of the light in the first

case being c—v, as it "bucks" the aether wind, and c+v on its

return.

Accordingly we have :

D (c-v)'

D x" (c+v)'

The total time required for transit is then :

D

D

1 (c-vy(c+v)

2Dc c*-v*'

The total distance traversed by the light, then, is :

Sr 2Dc2
(c2

2

»

('4)

dropping terms of the fourth and higher order of v/c in the expansion.
The length of the "vertical" path can be computed in several ways. In his 1881 analysis, Michelson overlooked the fact that the vertical path was actually a triangular path. In 1887 Michelson and Morley used for the value of the "vertical" path distance the expression :
2D ][{l+5) or exPandins> 2D(l+-^)

if we neglect powers higher than the second order of v/c. This first expression differs from the expression usually employed today for the distance, which is
2D

36

NINETEENTH-CENTURY AETHER THEORIES

The difference vanishes, however, if we restrict ourselves to second order quantities, and is most likely due to an approximation involved in the calculation of the "vertical" path length.1"
The difference in path then, in the 1887 analysis, is given by the quantity D(v2/c2) which is obtained by subtracting the two path length calculations.
If the 1881 "vertical" path length distance is used, however, the path difference is equal to 2Dv2/c2. The error in Michelson's calculations was pointed out to him soon after he published the first results, and a detailed examination of the experiment was published somewhat later by H. A. Lorentz (1886) as part of a long paper on aberration.
Michelson's 1881 experiment was performed in Potsdam in April of that year, and Michelson's computation of the expected aether drift took into account the direction of the Earth's motion at that time of the year. His reasoning told him that the fringe displacement due to the aether wind should be about 0 1 of a fringe, maximum, as the interferometer was rotated. The calculated fringe shift is plotted in Fig. III.5, as a dotted line. The solid line is the observed shift. Michelson concluded his 1881 paper with the following comment, which is all the more interesting because of the reference to Stokes' theory and the quoting of Stokes' views on aberration theories :

The interpretation of these results is that there is no displacement of the interference bands. The result of the hypothesis of the stationary aether is thus shown to be incorrect, and the necessary conclusion follows that the hypothesis is erroneous.

t The difference between the contemporary expression for path length and MicheTson and Morley's expression is easily seen if we write the contemporary expression as

to which it is exactly equal. Michelson and Morley apparently disregarded this small factor in their derivation of the vertical path length.

ABERRATION FROM BRADLEY

37

005 0
-005

FIG. III.5. Graphical representation of A. A. Michelson's anticipated results (dotted line) and his experimental results (solid line). (After Michelson, 1881.) The ordinate represents the amount of fringe shift, and the abscissa the compass direction of one axis of the interferometer.
This conclusion directly contradicts the explanation of the phenomenon of aberration which has been hitherto generally accepted, and which presupposes that the Earth moves through the aether, the latter remaining at rest.
It may not be out of place to add an extract from an article published in the Philosophical Magazine by Stokes in 1846.
"All these results would follow immediately from the theory of aberration which I proposed in the July number of this magazine [this is the theory discussed under section 3 above—K. F. S.] : nor have I been able to obtain any result admitting of being compared with experiment, which would be different according to which theory we adopted. This affords a curious instance of two totally different theories running parallel to each other in the explanation of phenomena. I do not suppose that many would be disposed to maintain Fresnel's theory, when it is shown that it may be dispensed with, inasmuch as we would not be disposed to believe, without good evidence, that the ether moves quite freely through the solid mass of the Earth. Still it would have been satisfactory, if it had been possible to have put the two theories to the test of some decisive experiment."
In his article on aberration, Lorentz (1886) not only criticized Michelson's calculations by pointing out the missing factor of 2, but he also argued that if the correct values were used, that the experimental error involved would be enough to call into doubt any rejection of Fresnel's theory. Lorentz had a very specific reason for wishing to defend a version of Fresnel's theory against experimental refutation, for he had in the same article shown that Stokes'

38

NINETEENTH-CENTURY AETHER THEORIES

theory assumed boundary conditions which were inconsistent with its theoretical assumptions. Specifically, Stokes required his aether to be irrotational, that is, it had to have a velocity potential. But Stokes also assumed that the velocity of the aether everywhere on the surface of the Earth was the same, and Lorentz was able to show that this condition is inconsistent with the assumption of a velocity potential. About a dozen years later, Max Planck (see Lorentz 1899b) briefly resuscitated Stokes' aether by showing that Stokes' two assumptions could be made consistent if the aether were as compressible as a gas that follows Boyle's law, and if, accordingly, its density was great near the surface of the Earth, and smaller as the distance from the Earth increased. Such an increase in aether density could itself be accounted for if the aether were attracted by the Earth's gravity. Planck's hypothesis had no other observable consequences, however, and as the Fresnel aether had in a sense been very successfully absorbed into Lorentz' electron theory in 1892, Planck's notions did not attract very much attention. Lorentz (1899b), however, commented on them and criticized them.
In his 1886 essay Lorentz had unequivocally sided with the Fresnel aether, though he analyzed it in somewhat more sophisticated terms than Fresnel had done, ascribing a velocity potential to it that would yield the partial dragging coefficient within ponderable bodies, but assuming that the aether was stationary in empty space. Such an aether implies nearly the same positive interferometer results that Michelson had anticipated (except for the factor of 2), and Lorentz did not deny that the interferometer experiment would not have a positive result were it performed again with more precision.
At the urging of Lord Rayleigh, Michelson repeated his experiment again in 1887 with the assistance of his colleague at Case Institute, E. W. Morley. This time the calculations were corrected for the influence of the Earth's motion on the "vertical" ray. The precision of the experiment was also increased by lengthening the

ABERRATION FROM BRADLEY

39

path by using four mirrors at the extreme reflecting points rather than one. A diagram of the improved interferometer appears in the appended Michelson-Morley selection and need not be reproduced here. The observations were made in July of 1887 at different hours. This time only the Earth's orbital velocity figured in the calculations, and the predicted fringe displacement was, with the increased path, computed to be about 0-4 of a fringe. The curves of one-eighth of the predicted displacement and the displacement observed are given in Fig. 6 of the Michelson-Morley selection. The observed displacement was somewhere between one-twentieth and one-fortieth of the predicted value. Michelson and Morley's assessment of their result was as follows :

It appears from all that precedes reasonably certain that if there be any relative motion between the earth and the luminiferous aether, it must be small; quite small enough entirely to refute Fresnel's explanation of aberration. Stokes has given a theory of aberration which assumes the aether at the earth's surface to be at rest with regard to the aether, and only requires in addition that the relative velocity have a potential; but Lorentz shows these conditions are incompatible. Lorentz then proposes a modification which combines some ideas of Stokes and Fresnel, and assumes the existence of a potential, together with Fresnel's coefficient. If now it were legitimate to conclude from the present work that the aether is at rest with regard to the earth's surface, according to Lorentz then there could not be a velocity potential, and his own theory also fails.

In Chapter VII shall discuss Lorentz' response to this refutation, and also touch on Einstein's "explanation" of the null result of the interferometer experiment.

S^N.OAX 4

CHAPTER IV
THE ELASTIC SOLID AETHER
As I pointed out in Chapter I, nineteenth-century aether theories were largely attempts to formulate explanations of optical, and later, electromagnetic phenomena, in mechanical terms. In these theories some law or principle of mechanics was asserted, from which, subject to the proper boundary and symmetry conditions, laws which possessed a formal analogy with optical and electromagnetic laws were derived. This tack was not universally taken, and Lorentz' post 1892 work does constitute a significant exception. Nevertheless, in order to understand nineteenth-century aether theory in many of its aspects, a rudimentary knowledge of the theoretical mechanics of the period is essential. The most important mechanical approaches during this time are the variational formulations of Lagrangian analytical mechanics and, later in the century, the analyses presented in terms of Hamilton's principle or the "principle of least action".
1. Introduction to Nineteenth-century Mechanics
In 1788 in his Mechanique Analytique, Lagrange presented an account of mechanics which eliminated the dependence of the subject on the Newtonian geometrical reasoning. The name "analytical mechanics" has been appended to Lagrange's approach because his account was algebraic or "analytical" rather than geometrical or "synthetic". Newton's mechanics was certainly of the latter character, and many of Newton's modifiers, such as D'Alembert, the Bernoullis, and, to some extent, even Euler, util-
40

THE ELASTIC SOLID AETHER

41

ized geometrical reasoning. For more detailed discussion on this point, the reader should refer to Ernst Mach's The Science of Mechanics, pp. 560-2.
Lagrangian mechanics is logically equivalent to Newtonian mechanics, though it does represent an advance with respect to mathematical elegance. Furthermore, its approach is extremely general and quite powerful for analyzing many mechanical problems which would be cumbersome and difficult to solve in the Newtonian formulation. Similar advantages accrue to the still later analysis of mechanics by Hamilton, who in 1834 developed the science on the basis of a principle later known variously as "Hamilton's principle" or the "principle of least action", though these two notions are considered somewhat different today.
It is not very difficult to state and derive the relationships between the Lagrangian approach, the Hamilton principle, and today's least action principle. (Though the mathematically illiterate may skip the following section, they do so unadvisedly, as Lagrangian and Hamiltonian methods are extensively used by Green, MacCullagh, Fitzgerald, and Larmor in their aether theories.)
Lagrangian dynamics develops from a principle of statics known as d'Alembert's principle of virtual velocities, or better, virtual (or arbitrary) "displacements". If we have some interconnected system with various internal forces acting on the parts, if the system is to be in equilibrium, the sum of all the forces resolved into components in the x, j , and z axis directions, each multiplied by an infinitely small displacement δχ, δγ, δζ in those directions, must add up to zero. The interconnections within the system will establish constraints or relations between the infinitesimal displacements. This principle can be expressed in the formula :

Z{FX ôx+Fy Ôy+Fz δζ) = 0.

(4.1)

This principle is extended to systems in motion in the following way. Consider the system to be composed of mass points mv m2, . . . mn, and refer the system to a Cartesian coordinate system
4·

42

NINETEENTH-CENTURY AETHER THEORIES

of mutually perpendicular axes x, y, z, as in the static case. Let resolved forces act on each mass point, their values being represented by Xi, Υχ, Zi, for m±9 X2, Y2, Z%, for ra2, etc. Assume such forces produce virtual displacements δχι, ôyi, δζι, and δχ2, ôy2, όζ2, etc. Thus far nothing new has been added to the above static case except a change of symbolism. In fact equation (4.1) in this new symbolism would be :

£ (Xi ÖXi+ Yt byt+Zi bzt) = 0.

(4.2)

1=1

Now consider the forces X, Y, Z acting on each mass point as impressedforces which produce motions within the interconnected system such that particle mi moves with acceleration components
dPxx cPyi (Ρζ± IB*9 dt2'9 dt* #

If each of these acceleration terms is multiplied by mi, the product is equivalent to the net force acting on m\. These net forces are termed effective forces. They are not necessarily equivalent to the impressedforces cited above. The extension of d'Alembert's statics principle to dynamics then is made as follows: the difference between the impressed force components and the effective force components must be such that the sum of them (the differences) adds up to zero, as they produce no motion. In equation form this statement amounts to :

(4.3)

This can also be stated in more compact vector notation as :

UFi-miS)-er-°

(4·4>

where r is a displacement vector and F is a force vector. Finally, the principle can be stated in a form in which it is easily applicable

THE ELASTIC SOLID AETHER

43

to continuum mechanics, such as we shall find in the various nineteenth-century aether theories, by replacing the particles' masses by a density function multiplied by a differential volume, transposing the force function to the right hand side, and assuming that it acts on a volume element. We then obtain in an integral formulation :
JJJ < ^ t o + ^ * ^ * ) * * * = JJJ"**·

(4.5)

Equations (4.3), (4.4), and (4.5) are essentially equivalent to one

another. Because of its compactness of form, I shall use (4.4) to

show how Hamilton's principle can be obtained from the Lagrange formulation.1"

I shall restate (4.4) using dot notation for differentiation with

respect to time as :

EÇFi-mrd'tot = 0.

(4.4)

Now from the properties of δ it follows that :

where v] = r^r^ v being interpreted as velocity. Consequently:

-j- Empt · bfi = δΣ — m$\+Empi · or/

(4.6)

at

2

in which the summation runs over i from 1 to n. Integrating both

sides of (4.6) from t = t0 to / = tv we obtain, setting T = E\m{v\\

ImiVràrT = P (dT+Zmfrdri) dt.

(4.7)

It can now be assumed that the systems to be analyzed are ones whose initial and final positions—i.e. at t0 and h—are the same.
t This derivation essentially follows Lindsay and Margenau (1957), pp. 131-2.

44

NINETEENTH-CENTURY AETHER THEORIES

It follows from this restriction that ri at t0 and tx is equal to zero. But then the left side of (4.7) is zero and we obtain :

| (dT+Zmift - dri) dt = 0.

(4.8)

Suppose now, and this will be important for Green's and Mac-

Cullagh's aether investigations later, that there exists a function

V of the rectangular coordinates of the parts of the system such

that:

ZFrrt = Zmfrbri = -ÄF.

(4.9)

Then (4.8) becomes:

à Γ 1 ( Γ - Κ ) Λ = 0.

(4.10)

This is "Hamilton's principle", but it is often referred to by nineteenth-century physicists as the "Principle of Least Action". In words it can be stated as "Assuming a conservative system, the system changes in such a way as to minimize (over short intervals of time) the action integral". In contemporary works, the "principle of least action" is usually understood as the assertion that:

δ [h2Tdt = 0

(4.11)

which is closer to Maupertuis' principle of least action. Equation (4.11) is less general than (4.10) since (4.11) is restricted to cases in which, during changes in the system, the total energy U = Γ-f V9 is constant and the same over every varied path. In (4.10), however, the variations of the paths are perfectly general except at the end points.
Equation (4.11) can be stated still another way by making use of the total energy equation mentioned immediately above, and

by introducing an element of arc length ds = (2T)2 dt. We then obtain :

δ [h{U-Vf ds = 0

(4.12)

THE ELASTIC SOLID AETHER

45

which is still another way of stating the "Principle of Least Action".
The rationale for the digression into these variational formulations of mechanics is to prepare the reader for the nineteenthcentury inquiries into the aether which were conducted by Green and MacCullagh, as regards the optical aether, and by Maxwell, Fitzgerald, and Larmor, as regards the electromagnetic aether. For the latter two, at least, MacCullagh's optical aether was Maxwell's electromagnetic aether. I shall discuss the contributions of Green and MacCullagh in this chapter, and consider the others' work in the following chapter.

2. The Development of the Elastic Solid Theory of the Aether
As was discussed earlier in Chapter II, the Young and Fresnel theories of the aether were not dynamical theories in any real sense. At the time they were developed, the proper equations and solutions describing wave motion in an elastic solid were not available. It was left to the French mathematicians, Navier, Cauchy, and Poisson to develop a mathematical theory of vibrations in a mechanical elastic solid, and to Cauchy to first apply these analyses to the wave motion of light.1" Cauchy's contributions are of considerable importance in the elastic solid theory of light and his work was highly valued by his successors in this field. Cauchy first presented his molecular theory of the aether in 1830, and later in 1836 and 1839 presented two more somewhat different theories of the optical aether. Because of the brief, and therefore necessarily eclectic character of this book, however, I can do no more than cite Cauchy's contributions, and must refer the interested reader to other sources.
For various reasons, I have decided to include the full text of a paper by the English physicist and mathematician George Green.
t See Whittaker (1960), I, pp. 128-33, for a good discussion of the French mathematicians.

46

NINETEENTH-CENTURY AETHER THEORIES

Green's theory of the aether was first presented in 1837, and came to have considerable influence on the development of the elastic solid theory through the enthusiastic missionary work of William Thomson, later Lord Kelvin. The virtue of Green's theory, aside from its historical influence, lies in its simplicity, its generality, its adaptability to change, e.g. by Lord Rayleigh (1871) and Lord Kelvin (1888), and in its physical naturalness. It constitutes a particularly elegant example of a style of argument and a type of theory which was of considerable influence a little more than 100 years ago, but which is almost completely forgotten today. Some of Green's other work has enjoyed a better fate, and Green's contributions to function theory, potential theory, and electrical theory are fairly well known. "Green's functions" are also extensively used in contemporary differential and integral equation theory.

3. Green's Aether Theory
Green's approach to the elastic solid theory of the aether is through Lagrangian mechanics applied to matter in bulk. Though Green, like Cauchy, supposes a molecular structure for the aether, his analysis is sufficiently independent of this structure so as also to be able to characterize a continuous aether.
Green begins by pointing out that Cauchy's theory (apparently he is referring only to Cauchy's "first theory") involves an assumption of forces acting between aether particles in which the direction of the action of forces is always along a line joining any two particles. This assumption or principle of central forces, common among the Newtonian-influenced French physicists of the nineteenth century, seemed "rather restrictive" to Green. The assumption which Green wished to substitute in its place was a version of d'Alembert's principle as developed in Lagrange's mechanics, about which I have spoken above. Green wrote :
The principle selected as the basis of the reasoning contained in the following pages is this: In whatever way the elements of any material

THE ELASTIC SOLID AETHER

47

system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions the total sum for any assigned portion of the mass will always be the exact differential of some function.

Green then turned his attention to a difficulty which was to appear again and again in elastic solid theories. In elastic media, with some peculiar and quite questionable exceptions to be touched on in the later sections of this chapter on MacCullagh's and Kelvin's aethers, a disturbance produces two spherical waves: one is a longitudinal compressional wave, the other a transverse wave. One of the major problems that was faced by all elastic solid theorists was to eliminate the longitudinal wave, or at least to eliminate its observable consequences, for experiments, such as the Arago-Fresnel experiment cited in the previous chapter, indicated that light was a purely transverse wave. We saw how Fresnel eliminated it in his quasi-mechanical theory—simply by hypothesis. Cauchy, in his first theory, did not seek to eliminate it, as he thought it actually existed and might be degraded as heat, and that experiment would disclose its effects. This, however, was not the case.
Green's manner of eliminating the longitudinal wave is to conceive of his elastic solid aether as so rigid—in the sense of being resistant to compression—that the velocity of the longitudinal wave becomes practically equal to infinity. This resistance to compression, though, is a relative resistance as we shall see below. In the introductory section of his paper he anticipates the results of his inquiry into the aether, and tells us that the solution of the equation of motion of his medium will contain two arbitrary coefficients, A and 2?, whose values depend on the unknown internal constitution of the aether. The velocity of the longitudinal wave is proportional to Λ/Α and the velocity of the transverse wave to <\/B. The effect of the longitudinal wave must be eliminated since in Green's theory, even if it itself be incapable of affecting the eye, it will give rise to a new transverse wave at a reflecting-refracting

48

NINETEENTH-CENTURY AETHER THEORIES

surface unless the ratio A/B equals zero or infinity, which would be visible. Green argues that if A/B is less than -f-, the medium is unstable. Consequently the velocity of the longitudinal wave must be very great compared with the velocity of ordinary light, i.e. approximately infinite, (if A/B were less than |-, an increase in pressure would produce an increase in volume, and the medium might possibly explode.) The constant A is, roughly, a measure of the aether's resistance to compression or change of volume and the constant B a measure of the aether's resistance to distortion with no change in volume, e.g. to twist. Consequently the ratio of A to B then, though practically infinite, is really a relative ratio of the resistance against compression to the resistance against distortion. Accordingly the aether need not be any more resistant to compression than the rarest known gas, if the resistance to distortion is exceedingly small. Even though the aether must support transverse waves moving at a velocity approximately equal to 3X108 meters/second, such a small compressibility is not ruled out on the foregoing suppositions if we can assume that the aether density is exceedingly small. Accordingly, Green's assumptions regarding the nature of the aether were not necessarily inconsistent with what was known about the Universe when he wrote, and the planets could move through such an aether.

For the constitution of the aether Green supposed, as many had before him, that it consisted of a large number of very small aether particles interacting with one another via very short-range molecular forces. Let x, y, and z be the equilibrium coordinates of any particle, and let x+w, y+v9 and z+w be the coordinates of the same particle in a state of motion. In accordance with Green's version of d'Alembert's principle we may write:

{ d2u

d2v

d2w 1

Dm^ du+Dm^ δν+Dm— owl = Σ dcfrDV (4.13) where Dm(cPu/dt2% for example, is equivalent to the internal force exerted in the x direction, bu is the x component of the virtual

THE ELASTIC SOLID AETHER

49

displacement of the particle, and δφ is the variation of the function sought for. The summed product ΣδφΏΥ represents the work given out by the differential volume DV in passing from an equilibrium state to the new non-equilibrium state. Dm is not the mass of a single aether particle, but is, rather, the mass of a very small differential volume which nevertheless contains'a large number of aether molecules.
The equation (4.13) can be put into integral form so as to closely resemble in form equation (4.5) which was discussed above in the introduction to nineteenth-century mechanics. Green in fact does let (4.13) pass into the integral form by introducing an aether density term, ρ, and rewriting (4.13) as:

(4.14)

Green argues that φ is a function entirely dependent on the internal actions of the particles of the medium on each other, and accordingly is a function of the compression and distortion of the medium, δφ must be an exact differential for Green because if the converse held true perpetual motion would be possible. Green wrote before conservation of energy (other than conservation of vis viva) was accepted, and he seems to base his argument of the form of δφ on the conservation of work.
To obtain the form of φ Green considers the effects of an arbitrary distortion administered to the differential element of volume. He lets dx, dy, and dz represent the sides of a rectangular differential element, and dx\ dy\ and dz' the element in a state of distortion (and compression). Green introduces small quantities su s2, and £3 to represent elongations and a, /?, and y to represent angular distortions or principal shearing strains. These quantities are related

50

NINETEENTH-CENTURY AETHER THEORIES

to the sides of the differential elements by :

dx' = dx(l+si) dy' = dy(l+s2) dz' = dz(l+s3) (4.15)

/dy*

/dx'

/dx'

a = cos<

ß = cos<^

y — cos<^ (4.16)

\fe'

\fe'

\//

/dy'
The notation cos^ indicates the cosine of the angle formed by

W

the line elements dy' and dz'. Green later shows that if we neglect

higher-order quantities, these small quantities can be defined in

terms of the motion of the aether particle as :

_ du dx
_dw dv dy dz

_ dv dy
o __dw dv dx dz

_dw 3 dz
_ du dv dy dx

(4.17)

The important function φ then is considered to be a function of these six quantities or:

φ = function (su s2, 5-3, α, β9 γ).

(4.18)

The determinate form of φ is obtained by a complicated series of steps :
(1) First φ is expanded into a series:

Φ = Φ0+Φ1+Φ2+Φ3+ . ·.

(4.19)

each φι being an ith degree function of s±, s2, s$, a, /?, γ... . (2) Certain plausible boundary conditions are then imposed.

Since φο = a constant, δφο = 0. By hypothesis, at equilibrium

u = 0, v = 0, and w = 0—i.e. the medium is unstrained at equi-

librium position—so it follows that

άχάγάζδφι = 0. (If the

medium were under a pressure at equilibrium, φχ would be a func-

THE ELASTIC SOLID AETHER

51

tion with six arbitrary constants.) φζ and φ^ etc., are considered to be exceedingly small with respect to φ2, so that the general function φ reduces to φ2, which, since it is a homogeneous function of six independent variables of the second order, contains no more than twenty-one arbitrary constants in its most general form.
(3) Green then assumes that the medium is unlike a crystalline body, that is, that it is symmetrical with respect to three rectangular planes. If this is so, the twenty-one arbitrary constants reduce to nine, and we have :

M £ ) V " ( I H £ ) S + L * , + T O ° 420
ifo dw do dw du dv dy dz dx dz dx dy

where G, H, /, L, M, N9 P, <2, and R are the nine arbitrary constants.
(4) If, furthermore, φ2 is restricted to a medium with symmetry around one axis, Green shows that :

G = H = 2N+R, L =M9

(4.21)

And, finally, if φ2 is symmetrical with respect to all three mutually perpendicular Cartesian axes, that is, if the medium is isotropic, we get :
G =H = I = 2N+R9

L =M = N,

(4.22)

P = Q =R.

The equation (4.20) can accordingly be simplified by utilizing these relations among the constants. Introducing two more constants, apparently for aesthetic purposes, A = 2G and B — 2L, Green

52

NINETEENTH-CENTURY AETHER THEORIES

obtains his determinate form of φ2'· Jdu dv dw\2 J/du dv\2

(du dw\2 (dv dw\2
+ (^+^)+(7z+^)
Idv dw du dw du dv\) \dy dz dx dz dx dy)\'

,A ^
(4·23>

This is the general form of what we would now call the potential

energy function of the aether in a non-crystalline medium.

The general equation of motion, (4.14) above, is then written

for an aether disturbance moving from one substance into another

across a surface, which will be the reflecting-refracting surface,

and which is taken to be an infinite horizontal plane. If the aether

density in the upper medium is ρ and the density of the lower

medium ρν and φ2 and φ^ the respective work (or potential

energy) functions, by (4.14) we may write two equations and sum

them to obtain :
m

J J J \d2u fi

d2v fi d2w fi 1

=

άχάγάζδφ2-\-

dxdydubφψ

in which the subscripts 1 distinguish quantities in the lower media. The integration for the triple integrals extends over the whole volume of the respective aethers.
Green then uses (4.23) to substitute the determinate form of φ2 into (4.24) for both media, adding subscripts where necessary to distinguish quantities in the upper medium from their counterparts in the lower. Carrying out an integration by parts yields two com-

THE ELASTIC SOLID AETHER

53

plicated expressions: one a summed volume integral and another a surface integral. Both of these must equate separately to zero: the triple integral yielding the equations of motion of the medium, the surface integral giving the boundary conditions which hold at the interface between the two media (in this case where x = 0). The boundary conditions are obtained by the substitution of an additional requirement of continuity of the aether displacement into them. The equations of continuity are, obviously, u = wi, v = vu w = Wi, from which one also directly obtains :

ou = oui, δν — δυ±, and öw = δ\νχ.

Green obtains his equation of motion for the aether disturbance in the form of:

d2u __ d l du dv dw\

dt2

dx \dx dy dzj

nid2u d2u d /dv dw\)

/Λ ^^

+B{w>+d*-*;[dï+^)\·

(4·25)

There are, of course, three equations for each medium, and (4.25) is only one of one set. The equations of motion can be put in a more elegant form by rewriting (4.25), for example, as :

d2u ,A ^ d /du dv dw\ n 0
ρ ^ = ^-ΰ)^(^+φ7+^)+5ν2"·

,A ^./x
(4·25)

It is perhaps easier to see the wave equation present in this form of the equation of motion.
The boundary conditions determined by the theory follow readily from the surface integrals and the conditions of continuity of aether displacement, and when x = 0, as at the interface chosen,

54

NINETEENTH-CENTURY AETHER THEORIES

become :
/ du dv dw\ \dx dy dz )

I dv dw\ \dy dz)

= A /dux dvx dwA 2B /dvx dwA

\dx dy dz J

\ dy dz )

*&+£) = Bi&+lr)

\dy dxj

\ dy dx J

Having obtained his equations of motion and his boundary

conditions, Green turned his attention to attempting to derive

Fresnel's sine and tangent laws for plane polarized waves, and to

account for other optical phenomena, such as phase reversal on

reflection. He considered the two cases of the plane polarized wave

incident on his infinite horizontal plane : one in which the polari-

zation is in the plane of incidence, the second in which the polari-

zation is perpendicular to the plane of incidence. I shall only con-

sider the first case in detail, though I shall comment on Green's

second case.

The z axis is now chosen parallel to the intersection of the line

formed by the intersection of the plane of the incident light and

the interface. Polarization in this first case amounts to setting

u = 0, v = 0, and u± = 0, v± = 0, as the vibrations now occur

only in the z direction. The equation of motion, based on (4.25),

thus reduces to :

d2w n (d2w d2w) Q-W=B\^ [ dx2 ++ -dy^2\j

,. __x (4.27)

and introducing a new constant γ2 = Β/ρ Green obtains :

d2w

0 /d2w d2w\

7t—r2{w+w)

(4·28)

THE ELASTIC SOLID AETHER

55

together with a similar equation for the lower medium, the only difference being in the addition of subscripts to w, γ, and B.
The boundary conditions of continuity and of (4.26) require in this case that :
w = Wi

B— = B — 1

dx

dx

(4#29)

In order to proceed further Green makes use of an additional hypothesis concerning the B quantities. He based his argument on the fact that the quantity A, which represents the compressibility, and on which the velocity of longitudinal waves depends, is independent of the nature of a gas as in sound wave theory. On the basis of this fact Green supposed that the 2?'s in his media are also equal. It is clear that this is somewhat hypothetical, though it does agree with Young and Fresnel's views about the nature of the aether in different bodies in which refraction and wave velocity only depend on variations in aether density and not in rigidity.
Elementary differential equation theory will lead to a solution of (4.27), which is a wave equation. Green represents the solution by the equation :

w = f(ax+by+ct) = F(-ax+by+ct)

(4.30)

in the upper medium, with/representing the amplitude belonging

to the incident wave and F that of the reflected plane wave, c is

understood to be a negative quantity. In the lower medium Green

writes :

wi =fi(a!X+by+ct).

(4.31)

Substitution of these solutions into the equations yield solutions if c2 = y2(a2+b2) and c2 = y ^ + è 2 ) . Application of (4.29) then

gives :

f(by+ct)+F(jby+ct)

=fi(by+ct),

af'(by+ct)-aF'(by+ct)

= axf\by + ct)

l' '

S-N.C.A.T. 5

56

NINETEENTH-CENTURY AETHER THEORIES

from which, taking the differential coefficient of the first equation and writing the characteristic only gives :
f'=F=fi
which in conjunction with the second equation of (4.32) yields :
F_ _ 1 - a i / a __ α-αχ _ cot Θ-cot gx _ sin(fli-fl) T 7 " " 1+Λι/α~ a+cn "~ cotö+cotOi ~ ύηζθχΤθ) ( ' '
which is Fresnel's sine law for light polarized in the plane of incidence, Θ and θι being, respectively, the angles of incidence and refraction.
Green then goes on from here, showing that if the generality of the wave function can be restricted to a function similar to that which describes the motion of a cycloid pendulum, certain interesting results connected with phase shifts in reflection can be demonstrated. Green's explanation of these restrictions and the derivation of the phase shifts are clear, and the reader is referred to the Green paper included in the readings, pp. 176-7.
The above argument leading to Fresnel's sine law should be sufficient to show how Green's theory is applied. It should also be taken as roughly equivalent to the mode of analysis which many dynamical aether theorists pursued during the nineteenth century, for it shows fairly clearly how the equations of motion and the boundary conditions are obtained and how additional hypotheses are incorporated in order to obtain optical results. The arguments are sophisticated, highly mathematical, and quite definite, and are not very different in spirit or level of physical competence from those presented in theories which have survived in physics until today.
I now turn to consider, rather briefly, Green's second case of polarization in which the plane wave is polarized in a direction perpendicular to the one just considered. Though this is the more interesting case—for it does not quite give Fresnel's tangent law and it also explicitly involves the problem of the longitudinal wave—it is considerably more complicated mathematically, and

THE ELASTIC SOLID AETHER

57

cannot be dealt with in detail without going beyond the scope and space limitations of this Commentary. The case is, of course, developed in the appended selections in Green's own words, and it has been critically considered by Lord Rayleigh (1871), to whose paper the reader may repair for an alternative account.
It is important, though, to comment on some of the physical assumptions made by Green in his second case, so that Green's position in the history of aether theories can be adequately considered.
When the light is polarized at right angles to the plane of incidence, w = Wi = 0, and the boundary conditions become:

u — \i\ v = vi,
*&$)-»$-Η&τ)-»^(434)
'(3+s)-*(-S+£)·

In accordance with what was said earlier, the B's may be cancelled in the last equation.
In this case, solution of the equations of motion yields two waves, a transverse wave with a velocity equal to \/(Β/ρ), and a longitudinal wave with a velocity of \/Λ/ρ. The two waves may be produced from a purely transverse wave by reflection of that wave at the interface x = 0.
The supposition the A/B equals a very great quantity is invoked, for reasons mentioned in the earlier discussion, to all but eliminate the effect of the longitudinal wave. This supposition, together with restrictions on the form of the wave function, identical to that referred to in connection with phase shifts in the first case, yields an expression for the relative intensities of the reflected to the incident wave of:

[(μ'+ΟΟι'-οι/^+^-ΐ)*^/^ [(μ2 + l)2 (μ* + aijäf + (μ2 - 1 )4 Ρ/αψ

(435)

5*

58

NINETEENTH-CENTURY AETHER THEORIES

where μ is the index of refraction and a and b are the amplitude coefficients of the x and y components of the wave function. Green shows that (4.35), as a first approximation, gives Fresnel's tangent formula, tan (0 — 0i)/tan (0 + 0i). But Green's expression diverges sufficiently from the experimentally supported Fresnel expression for large 0's to render Green's theory inadequate in explaining this result.
Green's theory may be modified in various ways so as to eliminate this difficulty, however, the most successful of which was Lord Kelvin's modification. This was based on a reconsideration of Green's argument concerning the necessity to suppose the ratio of A to B infinitely great, and Kelvin was able to show that Green's approach, were AjB set equal to 0, could lead exactly to both of Fresnel's laws of reflection. I shall have more to say about Kelvin's views of Green's aether in a later section.
Green's aether theory was also suspect in that it would not, as presented above, account for double refraction, and Green developed a somewhat different second aether theory to explain this phenomenon. In this second theory he permitted the B terms to be functions of the strain direction in the doubly refracting medium. This was directly contrary to the assumption of the uniformity of B made in the paper we have considered, and Green never was able to effect an accommodation of his two aether theories.
Green's first theory was considered very successful, however, if a slight modification were made, in explaining the results of some subsequent experiments on reflection involving phase shifts and elliptical polarization which were done by Jamin (1850). Later Lord Rayleigh (1871), in assessing the merits of the modified Green's theory as against its various competitors, found that Green came off by far the best.

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4. MacCullagh's Aether Theory
Green's aether theory had two major competitors in the elastic solid class during the nineteenth century. One was Cauchy's (1839) "third theory" which he developed in apparent ignorance of Green's earlier results. Cauchy commented on Green's theory in 1849, however, and disagreed with its approach, particularly on the usefulness and appropriateness of using the d'AlembertLagrange principle in investigating the optical aether. Cauchy's theory was subsequently developed by Haughton, St. Venant, and Sarrau.1"
Green's other major competitor was a theory which had been developed in the years 1834-7 by James MacCullagh of Trinity College, Dublin.! MacCullagh succeeded in putting his theory, which had been more a collection of hypotheses about the aether than a unified dynamical theory, on a relatively secure dynamical basis in 1839.
MacCullagh's theory has been highly praised by E. T. Whittaker in his History of the Theories of the Aether and Electricity. Whittaker wrote :
MacCullagh . . . succeeded [in 1839] in placing his own theory, which all along had been free from reproach as far as agreement with optical experiments was concerned, on a sound dynamical basis; thereby effecting that reconciliation of the theories of light and dynamics which had been the dream of every physicist since the days of Descartes.
The central feature of MacCullagh's investigation . . . is the introduction of a new type of elastic solid. He had in fact concluded from Green's results that it was impossible to explain optical phenomena satisfactorily by comparing the aether to an elastic solid of the ordinary type, which resists compression and distortion; and he saw the only hope of the situation was to devise a medium which should be as strictly conformable to dynamical laws as Green's elastic solid, and yet should have its properties specially designed to fulfil the requirements of the theory of light.
t See Glazebrook (1885), pp. 170-5, for a discussion and references. % A very similar theory was independently developed about this time by F. Neumann (1837). MacCullagh's theory is sometimes referred to as the MacCullagh-Neumann theory.

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I will discuss MacCullagh's new type of elastic solid shortly. It might be well to note here though the reason which Whittaker offers for MacCullagh's theory being largely ignored in the nineteenth century. Whittaker wrote :

MacCullagh's work was regarded with doubt by his own and the succeeding generation of mathematical physicists, and can scarcely be said to have been properly appreciated until Fitzgerald drew attention to it forty years afterwards. But there can be no doubt that MacCullagh really solved the problem of devising a medium whose vibrations, calculated in accordance with the correct laws of dynamics, should have the same properties as the vibrations of light.
The hesitation which was felt in accepting the rotationally elastic aether [i.e. MacCullagh's medium] arose mainly from the want of any readily conceived example of a body endowed with such a property [i.e. purely rotational elasticity]. This difficulty was removed in 1889 by Sir William Thompson (Lord Kelvin) who devised mechanical models possessed of rotational elasticity.

I have quoted this much of Whittaker on MacCullagh because of Whittaker's influence on the views which many contemporaries have of nineteenth-century aether theories. I shall show below that Whittaker's characterization of MacCullagh's aether as being in accord with the laws of dynamics is incorrect, and the reasons why MacCullagh's aether was not acceptable to nineteenth-century physicists are not those which Whittaker cites. In order to understand the actual historical relations between the various aether theories in the nineteenth century, as well as to become somewhat clearer on just how successful mechanical explanations can be in explicating optical theories, I will consider MacCullagh's aether in some detail.
MacCullagh relates in his 1839 paper that certain laws of the reflection and refraction of light at the surface of crystals, about which he had previously written, and which were, he claimed, "remarkable for their simplicity and elegance, as well as for their agreement with exact experiments", were, none the less, wanting a coherent mechanical explanatory basis. In the 1839 paper, of which I have included some extracts, MacCullagh gave a theory

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which he believed to be adequate for explaining reflection, refraction, and double refraction. The theory is based on two assumptions. MacCullagh hypothesized :

Firsty that the density of the luminiferous aether is a constant quantity; in which it is implied that this density is unchanged either by the motions which produce light or by the presence of material particles, so that it is the same within all bodies as in free space, and remains the same during the most intense vibrations. Second, that the vibrations in a plane wave are rectilinear, and that, while the plane of the wave moves parallel to itself, the vibrations continue parallel to a fixed right line, the direction of this right line and the direction of a normal to the wave being functions of each other. This supposition holds in all known crystals, except quartz, in which the vibrations are elliptical.

The first assumption will, in effect, both rule out any compressional wave, as well as require that refraction be made dependent, as it was for Huygens, but not for Young, Fresnel, or Green, on the difference in the rigidities of two media. The second assumption will be utilized in the derivation of the potential energy function of the aether.
MacCuUagh's dynamical approach is roughly the same as Green's, and though Whittaker suggests that MacCullagh was aware of Green's work, I have not found any evidence that this is so. Stokes (1862), in commenting on MacCuUagh's mode of analysis, also believes that MacCullagh was unaware of Green's aether theory at this time.
Like Green, MacCullagh applies the d'Alembert-Lagrange principle to his aether and seeks to determine the appropriate energy function which would satisfy the various restrictions he has imposed on his medium. MacCullagh uses the symbol V in place of Green's φ, and, of course, its determinate form will also differ because of their different views of the aether. As MacCullagh notes, the form of V is dependent "on the assumptions stated respecting the ethereal vibrations...".
The general form of V is, of course, given as usual by the general variational equation (4.14), which in MacCuUagh's symbolism

62 becomes :

NINETEENTH-CENTURY AETHER THEORIES

(4.36)
with x, y, and z being the coordinates of an aether particle before it is disturbed, and x + | , y+η, and ζ+ζ its coordinates at time t. MacCullagh also sets the aether density, which is in his theory everywhere the same, equal to unity so that dxdydz may represent either an element of volume or of mass.
The determinate form of V is obtained by considering a system of plane waves moving through the aether, parallel to which we construct a plane x'y'. The waves are apparently polarized parallel to the y' axis, so that the disturbance of an aether particle is confined to the y' direction, and ξ' and ζ' are both equal to zero. Then an elementary differential parallelpiped is constructed, with sides dx'dy'dz' respectively parallel to the axes of x'y'z'. I have drawn the parallelpiped in Fig. IV. 1, and attempted to represent the effect of the plane wave moving up the z axis on several "slices" of aether particles. As a result of the passage of the wave, the bottom of the differential volume will be shifted parallel to the x'y' plane with respect to the top of the volume. Consequently, a line connecting the top corner with the bottom corner, formerly directly beneath it, will no longer be parallel to the z' axis, but will be inclined to it at an angle k where tan k = dr\'\dz'.
MacCullagh then argues that the function V for which he is seeking: "can only depend upon the directions of the axes of x'y'z' with respect tofixedlines in the crystal, and upon the angle k which measures the change of form produced in the parallelpiped by vibration."
This is the most general supposition which can be made concerning it. MacCullagh then uses his second assumption, quoted on p. 61 above, which implies that any one direction, say x\ determines the other two directions y' and z' because of the imposed

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63

requirement of constant orthogonality. Thus V can be written as a function of k and x' alone.
In the mathematical section II of his paper (which is not included in these selections) MacCullagh shows that analytical geometrical

FIG. IV. 1. The shearing displacement of a segment of a differential volume of MacCullagh's aether, caused by a plane polarized wave moving in
the positive Z axis direction.

considerations associated with arbitrary rotations of coordinate systems allow him to demonstrate that the angle k and the direction of x' with respect to the primary xyz coordinate system are known if some special quantities XYZ are known. These quantities XYZ are curl functions of the displacement |, η9 ζ of the aether particle considered above on p. 62, and are defined in Cartesian component terms as :

X=^- dz —dy

dx dz

dy dx (4.37)

V therefore, according to MacCullagh, may be considered a function of XYZ alone.

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MacCullagh's argument is not completely cogent, however, and in 1862 in an influential report to the British Association for the Advancement of Science, G. G. Stokes criticized MacCullagh's derivation of V, noting:
[MacCullagh's] reasoning, which is somewhat obscure, seems to me to involve a fallacy. If the form of V were known, the rectilinearity of vibration and the constancy in the direction of vibration for a system of plane waves travelling in any given direction would follow as a result of the solution of the problem. But in using equation . . . [(4.36)] we are not at liberty to substitute for Fan expression which represents that function only on the condition that the motion be what it actually is, for we have occasion to take the variation ôV of V, and this variation must be the most general that is geometrically possible though it be dynamically impossible. That the form of V arrived at by MacCullagh, is inadmissible, is I conceive, proved by its incompatibility with the form deduced by Green from the very same supposition of the perfect transversality of the transversal vibrations; for Green's reasoning is perfectly straightforward and irreproachable. Besides MacCullagh's form leads to consequences absolutely at variance with dynamical principles.

I shall comment on the dynamical deficiencies of MacCullagh's aether shortly; for now it will suffice to sketch the remaining steps of MacCullagh's argument.
This can be done quickly as it is very similar to Green's more general case. Supposing k very small, XYZ will be very small, and V can be expanded in a series. First-order terms ought to vanish assuming an initially unstrained medium, and third- and higherorder terms are neglected in comparison with the second order. Since we now have a second-degree function of three quantities— recall that Green had six—V2 will be a homogeneous function containing in its most general form terms involving the squares and products of XY and Z with six arbitrary coefficients. The coefficients associated with the product terms can be made equal to zero by choosing the proper orientation of the XYZ axes, since the quantities XYZ transform in the same manner as do axes, whence MacCullagh obtains :

V= -±(a2X2+b2Y2+c2Z2)

(4.38)

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65

in which — ~tf2, ~y£2, and — \c2 are the arbitrary coefficients. The negative sign is introduced so that the velocity of propagation can never become imaginary.
Having arrived at the form of V MacCullagh says: " . . .we may now take it for the starting point of our theory, and dismiss the assumptions by which we were conducted to it." He then develops, in a manner not unlike Green's, the equations of motion, the boundary conditions, and based on these, derivations of Fresnel's sine and tangent laws, which he obtains exactly.
Interesting though MacCuUagh's theory may be, it is not a dynamical theory in the same sense that Green's is. It was shown by Stokes (1862) and also Lorentz (1901) that the MacCullagh aether violates the dynamical principle of the equality of action and reaction in regard to moments. In Stokes' (1862) words:

The condition of moments is violated. It is not that the resultant of the forces acting on an element of the medium does not produce its proper momentum in changing the motion of translation of the element . . . but that a couple is supposed to act on each element to which there is no corresponding reacting couple.

It might also be noted here that MacCuUagh's theory was also refuted by experiment. Lorenz (1861) and later Lord Rayleigh (1871) showed that an aether which assumed constant density, as did MacCuUagh's and Neumann's, implied the existence of two polarizing angles at π/8 and 3π/8 radians, whenever the difference in the indices of refraction between two media is small. Experiments disclose only one such angle, however, and imply that MacCuUagh's theory is incorrect.
Stokes' objection against the theory was generally accepted, and eliminated MacCuUagh's theory from serious consideration as a dynamical theory. I shall have occasion to quote Larmor on this point in the next chapter.
MacCullagh, contrary to Whittaker's implications, never thought he had provided a satisfactory dynamical theory of the aether. At the conclusion of his 1839 paper MacCullagh wrote:

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NINETEENTH-CENTURY AETHER THEORIES

In this theory, everything depends on the form of the function V; and we have seen that, when that form is properly assigned, the laws by which crystals act upon light are included in the general equation of dynamics. This fact is fully proved by the preceding investigations. But the reasoning which has been used to account for the form of the function is indirect, and cannot be regarded as sufficient, in a mechanical point of view. It is, however, the only kind of reasoning which we are able to employ, as the constitution of the luminiferous medium is entirely unknown.

In a sense, the MacCullagh aether can be defended if it can be supplemented with another aether which would provide the restoring couple missing from the MacCullagh aether. This supplementation in fact is the case in models which Kelvin constructed for Green's aether in 1889 and 1890 and which Whittaker erroneously implies were realizations of MacCullagh's aether. This is a subtle point though, and I shall return to it below when I consider Kelvin's models in detail.
The MacCullagh aether is more important as an electromagnetic aether than it is as a dynamical aether. As we shall see in the next chapter, both Fitzgerald and Larmor explicitly used generalized models of the MacCullagh aether in terms of which to interpret Maxwell's electromagnetic theory.
The idea of the aether as an elastic solid was seriously pursued and developed throughout the nineteenth century until about 1890. The contributions of the many physicists who formulated various aether theories are too many and various to do little more than mention.
Stokes' more positive contributions to aether theory ought to be cited. In addition to working out the theory of aberration which was discussed in the previous chapter, Stokes also formulated a dynamical theory of diffraction (1849) which was based on Green's theory of the aether. Stokes also developed a theory of the "fluid" aether in which he introduced an important distinction between the "rigidity" and the "plasticity" of a substance. With this distinction, Stokes was able to explain how the planets could move easily through the aether, the aether behaving in this case as a

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67

fluid, while at the same time light might be rapidly propagated through it, as if it were a rigid body. Glazebrook (1885) commented on this theory of Stokes, noting that: "These same views also tend to confirm the belief that for fluids, and among them the aether, the ratio of A to B (the elastic constants of the medium in Green's notation) will be extremely great". The aether, according to Stokes, acts very much like a synthetic plastic, which is often sold for amusement purposes, and sometimes known as "Silly Putty", or "Monster Putty". This substance shatters like glass when struck sharply, but flows like a liquid when subjected to a constant force over a long period of time.
I have previously mentioned some of Lord Rayleigh's contributions to aether theory; he made a number of others. On the Continent, G. Kirchhoff (1876) pursued an interesting aether theory and also developed a particularly elegant analysis of diffraction which, however, we cannot discuss for lack of space.
Several chroniclers of nineteenth-century aether theories1" find it useful to make a distinction between theories like Green's, MacCullagh's, and Cauchy's, which are elastic solid theories, and a different type of aether theory which developed primarily during the 1870s and 1880s. This new type of aether theory took careful cognizance of the interaction between aether and matter in ways which the earlier theories had overlooked. Communication of momentum between the aether and the matter it interpenetrated had not been dealt with by these earlier theories in any thorough manner.
Boussinesq (1867) was perhaps the first aether theorist to consider this problem seriously and attempt to explain reflection, refraction, polarization, dispersion, etc., on the basis of a uniform aether which varied in rigidity and density only when it got entangled

t These are primarily Glazebrook (1885) and Basset (1892). Whittaker is not among this group, and his History is rather deficient in this regard, though he does discuss Boussinesq's (1867) aether theory. See below for more comments on this point.

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NINETEENTH-CENTURY AETHER THEORIES

with matter. (Actually the properties of the aether itself do not change, but the net effect is as they did.)
Other theories of this type were vigorously developed by Sellmeier (1872), Helmholtz (1875), Lommel (1878), Voigt (1883), and Ketteler (1885).* The most important example which belongs to this type, even though it was electromagnetic rather than optical in nature, is the Lorentz aether which will be discussed in Chapter VI.

5. Kelvin's Aether and his Models
In England one of the proponents of this new type of aether theory was Lord Kelvin. In 1884 Kelvin gave a series of highlevel lectures on aether theories at the Johns Hopkins University in Baltimore. The lectures were published shortly thereafter as Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. These lectures constituted an inquiry into the dynamical shortcomings of various aethers, and focused on Green's theory. Kelvin told his "coefficients", as he punningly termed the twentyone professors who attended his lectures, that Green's theory at least had to be supplemented to include the interaction with atoms in ponderable matter. In his own characteristic style, Kelvin presented various mechanical models consisting of shells and springs whose action might mimic in a "rude" manner the hypothetical interaction of aether and matter.
Kelvin was by no means satisfied with the Green aether, even as supplemented with such notions, but he trusted it far more than he did the electromagnetic theory of light which was at the time receiving more and more attention, even though Hertz' important experiments had as yet not been performed. Though there were some nineteenth-century aether theories which were strongly influenced by Maxwell's electromagnetic theory and which will be dis-
t See Glazebrook (1885) for a discussion of these theories and for references.

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69

cussed in the next two chapters, many late nineteenth-century aether theories were developed entirely independently of any relation to electromagnetism, or to Maxwell's theory, specifically. Kelvin's antipathy to Maxwell's theory is well known. In the Baltimore lectures he wrote :

If I knew what the electromagnetic theory of light is, I might be able to think of it in relation to the fundamental principles of the wave theory of light. But it seems to me that it is a rather backward step from an absolutely definite mechanical notion that is put before us by Fresnel and his followers to take up the so-called Electro-magnetic theory of light in the way it has been taken up by several writers of late. . . . I merely say this in passing, as perhaps some apology is necessary for my insisting upon the plain matter-of-fact dynamics and the true elastic solid as giving what seems to me the only tenable foundation for the wave theory of light in the present state of our knowledge.

The 1884 version of the Baltimore Lectures was somewhat inconclusive on the positive side, as it raised more problems that it solved. As Glazebrook (1885) noted, however, it did develop an interest in England in the interaction type of aether theories.
Four years later Kelvin (1888) came upon a most important theoretical discovery regarding aether theory. It is worthwhile to let him tell it in his own words :
Since thefirstpublication of Cauchy's work on the subject in 1830, and of Green's in 1837, many attempts have been made by many workers to find a dynamical foundation for Fresnel's laws of reflexion and refraction of light, but all hitherto ineffectually. On resuming my own efforts since the recent meeting of the British Association in Bath, I first ascertained that an inviscid fluid permeating among pores of an incompressible, but otherwise sponge-like, solid, does not diminish, but on the contrary augments, the deviation from Fresnel's law of reflexion for vibrations in the plane of incidence. Having thus, after a great variety of previous efforts which had been commenced in connexion with preparations for my Baltimore Lectures of this time four years ago, seemingly exhausted possibilities in respect to incompressible elastic solid, without losing faith either in light or in dynamics, and knowing that the condensational-rarefactional wave disqualifies any elastic solid of positive compressibility, I saw that nothing was left but a solid of such negative compressibility as should make the velocity of the condensational-rarefactional wave zero. So I tried

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it and immediately found that it, with other suppositions unaltered from Green's, exactly fulfils FresneFs "tangent-law" for vibrations in the plane of incidence, and his "sine-law" for vibrations perpendicular to the plane of incidence. I then noticed that homogeneous air-less foam held from collapse by adhesion to a containing vessel, which may be infinitely distant all round, exactly fulfils the condition of zero velocity for the condensational-rarefactional wave; while it has a definite rigidity and elasticity of form, and a definite velocity of distortional wave, which can easily be calculated with a fair approximation to absolute accuracy.
Green, in his original paper "On the Reflexion and Refraction of Light", had pointed out that the condensational-rarefactional wave might be got quit of in two ways, (1) by its velocity being infinitely small, (2) by its velocity being infinitely great. But he curtly dismissed the former and adopted the latter, in the following statement:—"And it is not difficult to prove that the equilibrium of our medium would be unstable unless A/B > 4/3. We are therefore compelled to adopt the latter value of A/B" (oo) "and thus to admit that in the luminiferous ether, the velocity of transmission of waves propagated by normal vibrations, is very great compared with that of ordinary light." Thus originated the "jelly" theory of ether, which has held the field for fifty years against all dynamical assailants, and yet has failed to make good its own foundation.
But let us scrutinize Green's remark about instability. Every possible infinitesimal motion of the medium is, in the elementary dynamics of the subject, proved to be resolvable into coexistent condensational-rarefactional wave-motions. Surely, then, if there is a real finite propagational velocity for each of the two kinds of wave-motion, the equilibrium must be stable! And so I find Green's own formula proves it to be provided we either suppose the medium to extend all through boundless space, or give it a fixed containing vessel as its boundary.
Kelvin's analysis thus sets Green's coefficient A equal to zero,
and proves that the medium is not unstable to the point of explo-
sion, as was thought by Green, if Kelvin's conditions of a con-
taining vessel or infinite extent of the aether hold. This type of
aether will not support a longitudinal wave, for as regards its
energy distribution :
If A = 0, as we are going to suppose for our optical problem, no work is required to give the medium any infinitely small irrotational displacement; and thus we see the explanation of the zero velocity of the condensation and rarefactional wave . . . (Kelvin, 1888).
Such an aether is sometimes referred to as a quasi-labile aether,
inasmuch as it is "labile" with respect to compression, much as

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71

a cylinder rolling on a horizontal plane is in labile equilibrium. The aether is, however, resistant to rotational forces.1"
In 1889 and 1890 Kelvin was able to develop gyrostatic mechanical models of such a quasi-labile aether. In Kelvin's (1890) words, such a model was a "mechanical realization of the medium to which I was led one and one half years ago from Green's original theory by purely optical reasons".
I have included Kelvin's article which discussed this "mechanical realization" in order to give the flavor of Kelvin's "model" type of thinking, and in order to contrast it with his more theoretical investigations which are not very different in spirit from those of the typical nineteenth-century aether theorist.
Unfortunately, Kelvin is often taken, when he is discussing a "mechanical realization" of his theories, as being typical of nineteenth-century British thinking on the aether, and this has led to a number of confusions, especially in contemporary philosophy of science, about the relation between model and theory and the reality status of theories.
Kelvin's model is important to the extent that it made a mechanical explanation of the optical aether more plausible by showing that there was nothing inconsistent in those mechanical theories which were characterized by a gyrostatic rigidity. Green's theory as modified by Kelvin has this peculiar property, as does MacCullagh's aether theory about which I spoke earlier. We shall see that the Kelvin model also, however, assumes a second interpenetrating aether against which the rotational torques of the optical aether are exerted, thus outflanking Stokes' objection against the MacCullagh aether. Larmor (1894) showed that the Green-Kelvin aether and the MacCullagh aether were intertranslatable, thus proving formally what the model might lead one to suspect.

t Soon after Kelvin's theory appeared, R. T. Glazebrook (1888) applied it to the phenomenon of double refraction and obtained satisfactory results, thus bringing together, for the first time, Green's two aethers. (See above, p. 58.)
S-N.C.A.T. 6

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Kelvin's gyrostatic aether "realization" is extremely complicated, and unfortunately Kelvin's brusque manner of presenting it does not do much to clarify the matter. I intend only to sketch the principle features of his structure, and must refer the reader who desires a thorough analysis of the model from the point of view of theoretical gyroscopic mechanics to Lorentz' lecture (1901) on Kelvin's model.
Kelvin's model consists of a three-dimensional array of connected tetrahedrons, essentially built up on the basis of the plane system of equilateral triangles shown in Fig. IV.2. If the nonshaded triangles in the figure are taken as the bases of the tetrahedrons, we can consider PQRS and T as the top vertices of these

FIG. IV.2. A schematic representation of Kelvin's gyrostatic aether. (After Lorentz, 1901.)
tetrahedrons, themselves constituting the base points of a second level of tetrahedrons. If we carry out the building up and out of these levels systematically, every corner point in the system will be the common vertex of four tetrahedrons.
To imagine the mode of connection of these tetrahedrons which will give him the labile property of non-resistance to compression,

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Kelvin proposes that at each vertex there is a "ball and twelve socket mechanism" from which issue "six fine straight rods and six straight tubes, all of the same length, the internal diameter of the tubes exactly equal to the external diameter of the rods". The bars issuing from one of the balls fit into and slide without friction in the tubes of the other balls, and vice versa, the interconnected tubes and rods thus now constituting the edges of the tetrahedrons. This interconnected system is the framework into each tetrahedron of which Kelvin then introduces "a rigid frame G [consisting] of three rods [which can expand or contract] fixed together at right angles to one another through one point 0". This G frame is so positioned that three of its bars are put into permanent but sliding frictionless contact with the three pairs of rigid sides of any tetrahedron of the framework. Lorentz (1901) notes that "in a regular tetrahedron these bars coincide with the lines joining the midpoints of the opposite edges, but also in any tetrahedron whatever, a set of mutually orthogonal intersecting lines joining pairs of opposite edges can always be assigned". Kelvin then proposes to proliferate G frames throughout his framework so that the G frames constitute a "second homogeneous assemblage".
It can be shown that if the system of framework and G frames is subjected to an infinitely small homogeneous irrotational distortion, that the G frames do not undergo any rotation, though they do translate. Kelvin argues, however, that should the distortion have a rotational component, such as might be produced by an arbitrary displacement of the system, that : "any infinitely small homogeneous displacement whatever of the primary assemblage [i.e. the framework] produces a rotation of each frame equal to and round the same axis as its own rotational component." This is a most important property since if a resistance to rotation alone can be conferred on the G frames, the property of being both labile for compression and at the same time resistant to rotation will exist in this system. This is exactly what Kelvin sets out to do.
He introduces resistance to rotation into the G frames by mount-
6»

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NINETEENTH-CENTURY AETHER THEORIES

ing two gyrostats on each bar of the G frame. One such gyrostat of the solid type is depicted in Fig. IV.3. Kelvin was not particular about which type of gyrostat to use, as this was only a model, and he also describes a liquid gyrostat in section 12 of the appended selection.

FIG. IV.3. A solid gyrostat. Line AB is in the axis of the G frame. The solid rotatingflywheelat O is free to turn on axis RS, while the inner ring may rotate about axis PQ J_ AB. The outermost ring is fixed in the G
frame. (After Lorentz, 1901.)
What the gyrostats do, when six of the solid or twelve of the liquid variety are introduced into each G frame, is to provide the necessary resistance to rotation but not to translation of the G frame. Consequently the system as a whole exhibits the types of properties ascribed to both the Green and MacCullagh aethers. Because of the existence of framework, however, the G frames can react on another object, and thus outflank Stokes' criticism against the MacCullagh aether.
It does not appear that Kelvin thought of this model as anything more than an analogy. He did maintain a faith in a modified Green aether until at least 1904, though from 1899 to about 1902 he thought he might have to relinquish this aether since it led to

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some unfortunate predictions concerning the amount of energy dissipated in the case of a vibrating sphere in the aether. In 1902, however, Kelvin discovered another modification of the Green aether which permitted him to revise these predictions. The story is told in the 1904 edition of the Baltimore Lectures and cannot be discussed here.
Kelvin's approach to aether theory raises, as I suggested above, some important philosophical questions. One which has not been adequately considered is the question of effecting a reduction of one theory or one science to another.* Most nineteenth-century aether theorists were seeking a reduction of the phenomena and laws—and occasionally even theory—of optics to some form of mechanical theory. Such a mechanical theory need not be much more "concrete" than the Green theory or the Kelvin-Green theory. Effecting a reduction to mechanics or—what is the same thing—finding a general explanation in terms of a mechanical theory, must be distinguished from imagining some concrete model of wheels, pulleys, gyroscopes, or whatever. Providing the latter provides an analogy and not a reduction. The nineteenth-century aether theorists understood the distinction quite clearly, and when they began to consider a mechanical explanation of Maxwell's theory, became quite self-conscious about the criteria of "mechanical explanations". We turn to consider such theories next.

t See Nagel (1961) and Schaffner (1967) for analyses of reduction.

CHAPTER V
THE ELECTROMAGNETIC AETHER
IN THE previous chapter I mentioned the lack of contact between some late nineteenth-century optical aether theories and Maxwell's electromagnetic theory. There are some reasons for this. Certainly prior to Hertz' (1888) important experimental production and detection of Maxwell's electromagnetic waves, Maxwell's theory was only one of many competing optical theories, and occupied a similar non-paramount position with respect to electromagnetic theories. We have discussed the various optical theories in the last chapter. With respect to competing electromagnetic theories, Maxwell's theory was flanked on the right by Kelvin's various attempts to characterize the electric and magnetic aethers, on the left by various continental action at a distance schools, the most prominent of which was Weber's, and from above (to extend the metaphor) by Helmholtz' influential synthetic theory. This book cannot by its very nature consider very carefully these complex competing doctrines of electromagnetic action, and must refer the reader to other literature.1" The purpose of this chapter is to sketch some of the notions of the aether that were held by Faraday, Maxwell, and some of Maxwell's followers, with emphasis on the latter, and to show how theories of the mechanical optical aether were eventually brought into close relations with Maxwell's theory.
t See Whittaker (1960), Chaps. 7-10, Thomson (1885), and Rosenfeld (1957).
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THE ELECTROMAGNETIC AETHER

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1. Faraday
Michael Faraday was persuaded by his own doctrine of "lines of force" that the mode of connection between ponderable bodies that were coupled by electrical, magnetic, gravitational, or optical interaction, was via a peculiar contact action of the bodies themselves. Bodies, for Faraday, were aggregates of Boscovichian atoms which extended indefinitely outward into space. Lines of force were apparently related to this extension, according to the statements of Faraday which I shall cite below.
Faraday was quite clear about his antipathetic view of the Fresnel and post-Fresnel optical aethers. In his seminal essay, "Thoughts on Ray Vibrations", published in 1846, Faraday wrote :
The point intended to be set forth . . . was, whether it was not possible that the vibrations which in a certain theory are assumed to account for radiation and radiant phenomena may not occur in the lines of force which connect particles, and consequently masses of matter together; a notion which as far as it is admitted, will dispense with the aether, which, in another view, is supposed to be the medium in which the vibrations take place.
Faraday, accordingly, was against the aether as well as being opposed to the "action at a distance" approach. He is, however, committed to a "medium", if this word can be used for the bodies themselves, through which electric and magnetic action travels, and travels with a velocity comparable with the velocity of light. Faraday thus suggested the identity of light waves and electromagnetic waves. Faraday has also been looked on as being the originator of "field theory", in so far as Maxwell's electromagnetic field theory follows closely both on Faraday's experimental work and the mathematization of some of Faraday's speculative ideas concerning the lines of force and the electrotonic state. This latter notion was a theoretical idea in terms of which Faraday thought he might account for electromagnetic induction.*
t See L. P. Williams' (1965) excellent study on Faraday, and also Tricker (1966) for additional material on the electrotonic state.

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Faraday contended that he was led to this possibility of an electromagnetic view of light from the Boscovichian idea of matter. This notion, in Faraday's mind, considered atoms not as "so many little bodies surrounded by forces, . . . these little particles [having] a definite form and a certain limited size"—rather, in the Boscovichian atom "that which represents size may be considered as extending to any distance to which the lines of force of the particle extend: the particle indeed is supposed to exist only by these forces, and where they are it is".

2. Maxwell
Faraday's view of the interaction of charged and magnetized bodies via lines of force and the possible identification of light with electromagnetic vibrations strongly influenced James Clerk Maxwell. In a paper written in late 1855 Maxwell (1856) first presented some of his thoughts on Faraday's lines of force, offering certain mathematical expressions drawn from fluid mechanics partly based on some of Lord Kelvin's work, in terms of which to interpret the "lines of force". For example, in his section on Faraday's electrotonic state, Maxwell defined certain complicated functions which might characterize this state. The quantity which represents the electrotonic intensity turns out to be identical with the contemporary vector potential a, which is related to the magnetic induction vector by : curl a = B. Maxwell had little faith in this tentative mathematization of Faraday's theory, however, and noted in his concluding remarks that :
In these . . . laws I have endeavored to express the idea which I believe to be the mathematical foundation of the modes of thought indicated in the Experimental Researches. I do not think that it contains even the shadow of a true physical theory; in fact its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything.
Maxwell's mode of analysis thus far had little contact with the aether theories which were discussed in the previous chapters.

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79

Over the next ten years, however, Maxwell developed theories of a mechanical-electromagnetic aether and a purified electromagnetic aether, both of which he identified at different times with the optical aether.
The evolution of Maxwell's thought on the aether can only be sketched in broad outlines. The interested reader should refer to Whittaker (1960) and to a recent paper by Joan Bromberg (1968) for more detail.
Maxwell published his first electromagnetic theory of light in 1861-2 in a series of papers that appeared in the Philosophical Magazine under the title "On Physical Lines of Force". This first theory is, as Bromberg (1968) points out, not really an "electromagnetic theory of light". Rather it is "better characterized as an electro-mechanical theory of light, for in it the equations of light are derived, not from electromagnetic laws alone, but partly from electromagnetic laws and partly from laws of mechanics".
Maxwell's 1861-2 mechanical model of the electromagnetic aether is well known, if not very well understood, and is adequately treated in a volume in this series by Tricker (1966). Suffice it to say here that the electromagnetic theory of light does not appear until the model of whirling magnetic vortices and electrically charged idle wheels, which was adequate for characterizing the relations between currents and magnetism, is altered to incorporate a representation of a static electrical field. This was done by now conceiving of the whirling vortices as static cells which are twisted from their equilibrium position by the tangential displacement of the charged idle wheels. This displacement is occasioned by an electromotive force applied to the system, and is Maxwell's wellknown "electric displacement", a change of which (because of the motion of "charges") constitutes the "displacement current". Elimination of the applied field permits the medium to return to its original undisplaced untwisted state.
Maxwell offered a kind of defense for the ascription of elasticity to these cells which foreshadows his later identification of the

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electromagnetic and optical media. Early in part III of his paper he wrote :
The substance in the cells possesses elasticity of figure . . . simi" lar to that observed in solid bodies. The undulatory theory of light requires us to admit this kind of elasticity in the luminiferous medium in order to account for transverse vibrations. We need not then be surprised if the magnetoelectric medium possesses the same property.

After incorporating this into his theory, Maxwell calculated "the rate of propagation of transverse vibrations through the elastic medium of which the cells are composed, on the supposition that its elasticity is due entirely to forces acting between pairs of particles". He found that it "agrees so exactly with the velocity of light calculated from . . . optical experiments ...that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and
magnetic phenomena". (Emphasis is Maxwell's.) Maxwell thus identified the two aethers.
Maxwell was apparently uneasy with the mechanico-electrical system on which his theory depended, and in late 1864 represented his theory on a rather different basis, emancipating it from the earlier model that aided its formulation. He did not, however, consider that he had in any way eliminated the aether, which was now conceived of as being accurately characterized by his electromagnetic field equations.
It is worth quoting Maxwell rather extensively on this point. At the beginning of his paper published in early 1865 Maxwell wrote :
[Rather than seeking an action at a distance theory] I have preferred to seek an explanation of electrostatic and [electromagnetic phenomena] by supposing them to be produced by actions which go on in the surrounding medium as well as in the excited bodies, and endeavouring to explain the action between distant bodies without assuming the existence of forces capable of acting directly at sensible distance.
(3) The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric or magnetic bodies, and it may be called a Dynamical Theory,

THE ELECTROMAGNETIC AETHER

81

because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced.
(4) The electromagnetic field is that part of space which contains and surrounds bodies in electric or magnetic conditions.
It may be filled with any kind of matter, or we may endeavour to render it empty of all gross matter, as in the case of Geissler's tubes and other so-called vacua.
There is always, however, enough of matter left to receive and transmit the undulations of light and heat, and it is because the transmission of these radiations is not greatly altered when transparent bodies of measurable density are substituted for the socalled vacuum, that we are obliged to admit that the undulations are those of an aethereal substance, and not of the gross matter, the presence of which merely modifies in some way the motion of the aether.
We have therefore some reason to believe, from the phenomena of light and heat, that there is an aethereal medium filling space and permeating bodies, capable of being set in motion and of transmitting that motion from one part to another, and of communicating that motion to gross matter so as to heat it and affect it in various ways.
(5) Now the energy communicated to the body in heating it must have formerly existed in the moving medium, for the undulations had left the source of heat some time before they reached the body, and during that time the energy must have been half in the form of motion of the medium and half in the form of elastic resilience. From these considerations Professor W. Thomson has argued, that the medium must have a density capable of comparison with that of gross matter, and has even assigned an inferior limit to that density.
(6) We may therefore receive, as a datum derived from a branch of science independent of that with which we have to deal, the existence of a pervading medium, of small but real density, capable of being set in motion, and of transmitting motion from one part to another with great, but not infinite, velocity.
Hence the parts of this medium must be so connected that the motion of one part depends in some way on the motion of the rest; and at the same time these connexions must be capable of a certain kind of elastic yielding, since the communication of motion is not instantaneous, but occupies time.
The medium is therefore capable of receiving and storing up two kinds of energy, namely, the "actual" energy depending on the motions of its parts, and "potential" energy, consisting of the work which the medium will do in recovering from displacement in virtue of its elasticity.
The propagation of undulations consists in the continual transformation of one of these forms of energy into the other alternately, and at any instant the amount of energy in the whole medium is equally divided, so that half is energy of motion, and half is elastic resilience.

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(7) A medium having such a constitution may be capable of other kinds of motion and displacement than those which produce the phenomena of light and heat, and some of these may be of such a kind that they may be evidenced to our senses by the phenomena they produce.

The theory which Maxwell presented in a systematic form in 1865 contained twenty equations in twenty unknowns. Even this form, however, which made use of the vector and scalar potentials in its basic equations, could have been somewhat "simplified" had Maxwell used vector equations rather than writing out the equations component-wise. Nevertheless, the theory was exceedingly complicated, and Maxwell's mode of presentation did not add to the clarity. There is sufficient testimony on this matter by prominent scientists, such as Ehrenfest (1916) and Sommerfeld (1933), who have commented on the difficulties of reading Maxwell in the original, and it seems that the simplification of his theory by Heaviside and Hertz in 1889 and 1890 had almost as much impact on the acceptance of the Maxwell theory as did Hertz' experiments.
The theory which Maxwell gave in 1865 is non-mechanical, though it is clearly not anti-mechanical. Maxwell's difference in orientation in his two papers (i.e. 1862 and 1865) is brought out very clearly by comments which he makes near the end of Part III of his later paper. Maxwell (1865) wrote:
(73) I have on a former occasion attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory.
(74) In speaking of the Energy of the field, however, I wish to be understood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in the electrified bodies, conducting circuits, and magnets, in the form of an un-

THE ELECTROMAGNETIC AETHER

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known quality called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and is in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probably hypothesis, as the motion and the strain of one and the same medium.

Maxwell's theory was clearly a kind of aether theory, but what kind of an aether it was—i.e. was it a mechanically explicable elastic solid aether?—was less clear. Maxwell did not concern himself with this question to any great extent, but rather attempted to formulate explanations of electrical, magnetic, and optical phenomena on the basis of the theory itself. In 1873 he re-presented his views in the monumental Treatise on Electricity and Magnetism. It is not necessary to go into this work in any detail except to point out the expressions for the energy of the electromagnetic field which he gives there. Though these are essentially the same as in the 1865 paper, the references of Maxwell's followers with whom we shall be concerned are usually to the Treatise.
In the sections 630-8 Maxwell investigated the distribution of energies in the field, and concluded :

The energy of thefieldtherefore consists of two parts only, the electrostatic or potential energy:

W = ■§ j | J (Pf+Qg + Rh)dxdydz

(5.1)

[where P, Q, and R are the components of the electromotive force intensity

and /, g, h those of the electric displacement], and the electromagnetic

or kinetic energy :

T = ~L· J J 7 (aoi+bP+cri dxdy dz

(5·2)

[in which a, b> and c represent the components of the magnetic induction and α, β9 and γ those of the magnetic force.]
Later in the Treatise Maxwell reasserted his commitment to the aether, citing again the connection between the optical aether and the electromagnetic aether :

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NINETEENTH-CENTURY AETHER THEORIES

In the theory of electricity and magnetism adopted in this treatise, two forms of energy are recognized, the electrostatic and the electrokinetic . . . , and these are supposed to have their seat, not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to act. Hence our theory agrees with the undulatory theory in assuming the existence of a medium which is capable of becoming a receptacle of two forms of energy.

3. Fitzgerald's Electromagnetic Interpretation of MacCullagh's Aether
G. F. Fitzgerald represents a fairly typical Maxwell follower of the late nineteenth century. On the one hand, he was convinced of the worth of Maxwell's theory as he found it presented in the 1873 Treatise, though there were, of course, rough patches to smooth over and undeveloped areas in which to apply the theory. On the other hand, Fitzgerald also exhibits a belief fairly widespread during this period that the discovery of the mechanical aether theoretical basis of Maxwell's theory would constitute a significant step in the advancement of the theory.* Working in the "Dublin tradition" of optics and aether theory Fitzgerald took over MacCullagh's aether theory and the application of that theory to reflection and refraction, and showed how it could be used within Maxwell's theory. Maxwell had not been able to work out an explanation of reflection and refraction on the basis of his electromagnetic theory, as he was not able to satisfy himself as to the boundary conditions which should hold in his theory. Such an explanation was first presented by H. A. Lorentz in his doctoral dissertation in 1875, and I shall comment on this in the next chapter. Fitzgerald, not knowing of Lorentz' work, developed his own electromagnetic account of reflection and refraction in 1878. Maxwell refereed Fitzgerald's paper embodying this theory for the Philosophical Transactions of the Royal Society and commented
t See Glazebrook's (1885) comments on deficiencies in Maxwell's theory, and the letter by Heaviside to Hertz quoted below, on p. 90.

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85

favorably on it, noting that though Lorentz had anticipated Fitzgerald on a number of points, the paper made several new contributions to the electromagnetic theory, particularly as regards reflection and refraction in magnetized mediae
Fitzgerald's method of attacking the problem, to characterize it in very broad terms, is (1) to use Maxwell's expressions for the kinetic and potential energies of the medium, then (2) to map the basic quantities of the electromagnetic theory into MacCullagh's aether and show that there is a parallelism in the energy equations of Maxwell's and MacCullagh's media, and finally, (3) to obtain both the equations of vibratory motion and the boundary conditions by using these energy expressions in Hamilton's Principle of Least Action.
Fitzgerald uses quaternion notation as well as Cartesian component notation in his essay and though I will not pretend to explain quaternions with any degree of adequacy, I will make some comments on the effective relation of them to the vector notation.
Following Maxwell, Fitzgerald defines the potential or electrostatic energy as :

W = - \ ( ( ( S(E-D)dxdydz

= * ÎSÎ (Pf+Qg+Rh)dxdydz. (5.3)

The first expression is in quaternion notation, S(E*D) representing for our purposes the quaternion analogue of the negative of the vector dot product.! E and D represent, of course, the electromotive force and electric displacement respectively. E and D are
t See the report by Maxwell on Fitzgerald's (1880) essay which is part of the Joseph Larmor Collection, Anderson Room, Cambridge University Library.
% See Bork (1966b) for a concise discussion of the quaternion notation and some disputes that arose over its use.

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understood to be related by :

E = φΌ

(5.4)

where φ is a dielectric function of the medium, not necessarily isotropic. The kinetic energy is as in Maxwell's writings given by:

T = - i - f f ( S(B.H)dxdydz

^L· f f \(a*+bß+cr)dxdydz

(5·5)

where B is the magnetic induction, H the magnetic force, and ae usual, B = μΗ where μ is the coefficient of magnetic inductivs capacity.
Fitzgerald also makes explicit use of one of Maxwell's "curl equations", to use later parlance, which Fitzgerald writes as:

Anb = VvH

(5.6)

which is the same, for our purposes, as :

4π-^- = ν Χ # .

(5.7)

At this point in his discussion, Fitzgerald introduces MacCullagh's aether. He does this by defining what amounts to an aether displacement vector R, which is MacCullagh's |, η, ζ, such that :

R= Γ Hdt,

(5.8)

which is equivalent, differentiating both sides with respect to

time, to :

R = H.

(5.9)

Substituting this definition in (5.6) or (5.7) above gives:

AnD = VvR

(5.10)

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87

if we integrate both sides of the result in the substitution in order to eliminate the differentiation with respect to time. In component notation (5.10) becomes:

4nf=%—ÈL 4 ™ - ^ L · ^ 47th-Él^

*nJ dy dz> ^g~dz

dx> *m~dx

f5in
dy-V·11'

Either (5.10) or (5.11) tells us that Maxwell's dielectric displace-

ment is the rotation of this "elastic solid" aether, and from (5.9)

we can see that the magnetic force is the velocity of an aether

stream in this aether.

Substitution of the R term into the Maxwell expressions for T

and W reveals a striking analogy with the MacCullagh aether with

respect to the form of the potential energy of the medium. In

MacCullagh's aether we found that:

V = -|(a2X2+Z>2y2+c2Z2).

(4.38)

If we compare (5.11) and MacCullagh's definitions of X, Y, and Z (see (4.37)) we see that the rotations are taken in an opposite sense, thus accounting for the difference in sign. The expressions then will differ by constant factors which can be adjusted so that the expressions (4.38) and (5.1) are equivalent.
Fitzgerald's actual expression for W in his version of MacCullagh's aether is obtained by substitution of (5.4) and (5.10) in (5.3) which yields:

W = —^ f f f S(VvR^VvR)dxdydz.

(5.12)

But Fitzgerald had to generalize MacCullagh's medium before he could build an analogue of the magnetic force into it. This was done, as suggested above, by the introduction of the streaming motion into the aether. The energy of this stream is, of course, kinetic, being energy of motion. Substitution of (5.9) into (5.2) yields :

Γ = -^- [[[ μ&άχάγάζ

(5.13)

S-N.C.A.T.7

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which is an acceptable expression for the mechanical kinetic energy of the aether if μ can be taken as a measure of the aether's inertial resistance.1" (Fitzgerald's actual expression for (5.13) has a minus sign because of quaternion conventions.)
Having obtained expressions for the potential and kinetic energy of his aethereal medium, Fitzgerald applied the Hamilton formulation of the Principle of Least Action :

δ f (T-W)dt = 0

(5.14)

which was given in essentially the same form as (4.10) in the previous chapter. Substitution of (5.12) and (5.13) into (5.14) gives a complicated expression which can then be developed by standard mechanical methods, such as were followed by Green and MacCullagh in their earlier aether theories. Integration by parts gives rise, as usual, to two sets of integrals: (a) triple or volume integrals, which Fitzgerald refers to as "general integrals", and (b) double or surface integrals which Fitzgerald calls "superficial integrals".
The reader should already be able to see the similarity to the Green approach considered in some detail in Chapter IV. As with Green, and also with MacCullagh, the "general" integrals will yield the equation of motion of any disturbance, and can easily be made to give the equation of a plane wave by imposing the appropriate restrictions. The "superficial" integrals give the boundary conditions for reflection and refraction which hold at an interface between any two media, subject of course to the limitations of the assumptions made at the very beginning of Fitzgerald's paper which limited his inquiry to nonconductors with an isotropic μ.
By following MacCullagh's (1839) analysis fairly closely, Fitz-
t Whittaker (1960) presents Fitzgerald's theory in a somewhat different manner than what I have done here, my treatment being closer to Fitzgerald's original approach.

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gerald was able to obtain the law of reflection, Snell's law of refraction, and the Fresnel sine and tangent laws exactly.
I have only included the first part of Fitzgerald's paper in which he introduces his generalization of the MacCullagh aether, and not that part in which he obtains the laws of reflection, refraction, and Fresnel's laws. No new principles are introduced here, and as the mathematics is carried through mostly in the quaternion form, with translations made to the Cartesian form from time to time, it would be diflicult for the average reader to follow.
The reason for the introduction of the Fitzgerald analysis is to show one example in which the optical aether and the electromagnetic aether were brought together. The Fitzgerald analysis was later explored in a more detailed manner by Joseph Larmor, to whom I shall turn in a momenta
It would, however, be somewhat misleading to argue that analyses like Fitzgerald's were completely satisfactory. We have already seen that the MacCullagh aether is deficient in an important dynamical way, so that even if an adequate reduction of the Maxwell theory to the generalized MacCullagh aether could be carried out, it would not constitute a reduction of the electromagnetic theory to mechanics. Larmor puzzled over this deficiency in the MacCullagh aether and eventually came to terms with it in a most interesting way as we shall see below. But to a growing number of late nineteenth-century physicists it was not clear that these rotational aethers were much better than interesting analogies.
Oliver Heaviside, who made a number of contributions to the development of Maxwell's theory, was one such person. Heaviside, though he lived until 1925, never gave up his belief in an aether and after Einstein's special theory of relativity had been accepted by most physicists, continued to criticize it as being too "abstract" and as wanting an aether. Earlier, in a letter to Hertz dated 13 September, 1889,î Heaviside displayed the widespread feeling

t Sommerfeld (1892) also considered an aether similar to Fitzgerald's. J This letter is located in the Deutsches Museum, Munich.
7*

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NINETEENTH-CENTURY AETHER THEORIES

which I mentioned above that Maxwell's theory required a clearer aether theoretical basis. Heaviside wrote:
I believe it quite possible to frame a mechanical theory of a compressible aether which should lead to Maxwell's equations. But no doubt Maxwell's theory of displacement and induction in ether must remain (in spite of your and similar experiments to come) a Paper-Theory—as long as we do not know what functions of the ether D and B are! .

Later in the same letter Heaviside foreshadowed an aether yet to be developed by Joseph Larmor, writing:

It often occurs to me that we may be all wrong in thinking of the ether as a kind of matter (elastic solid for instance) accounting for its properties by those of the matter in bulk with which we are acquainted; and that the true way could we only see how to do it, is to explain matter in terms of the ether, going from the simpler to the more complex.

Two years later Heaviside wrote up a short paper on an aether which was essentially the same as Fitzgerald's generalization of MacCullagh's aether. Heaviside, however, thinks of it more in line with a generalization of Kelvin's quasi-labile, sometimes called quasi-rigid, aether which I discussed in the last chapter.1" The mode of presentation of this aether by Heaviside is also quite different from the more abstract work of Fitzgerald. Heaviside's approach, rather than proceeding through energy expressions and the Principle of Least Action, utilizes Newton's law of motion for translation and for rotation : F — ma, and torque = /a, as expressed in their most elementary formulations.
Though the Heaviside rotational aether is not very important from the point of view of the history of aether theory, it does afford a different approach to the electromagnetic aether, and also points out how it may be applied to practical electromagnetic problems, such as telegraphy. It also shows in what ways the parallelism between such an aether and Maxwell's electromagnetic

t In 1890 Kelvin had applied his aether to magnetism. See his collected papers, vol. iii, p. 465.

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theory begins to break down.1" It is for these reasons that I have included it in the selections.
We may now move to consider the most highly developed opticalmechanical-electromagnetic aether theory which came out of the nineteenth century. This was Joseph Larmor's aether, which Whittaker suggested, I believe erroneously, was able "to withstand . . . criticisms based on the principle of relativity, which shattered practically all rival concepts of the aether".

4. Joseph Larmor's Aether and the Electron
Larmor is perhaps not so well known today as the previously cited writers on the electromagnetic aether. Nevertheless, it was Larmor who not only brought the mechanical-electromagnetic aether to its most developed state, but who also was the first person (1897) to incorporate the Lorentz-Fitzgerald contraction within a general electromagnetic explanation of aberration phenomena. I will discuss the first of Larmor's contributions in this chapter, as it forms a natural conclusion to the work of Maxwell and Fitzgerald. For reasons which will become clearer later, Larmor's account of aberration best belongs in the next chapter after a preliminary discussion of Lorentz' work.
Larmor was trained at Queens College, Belfast, and at St. John's College, Cambridge, where he was first wrangler in 1880, followed by J. J. Thompson. After three years as professor at Queen's College, Galway, he returned to Cambridge, first as a lecturer. He assumed G. G. Stokes' Lucasian Professorship in 1903 which he held until he retired in 1932. He spent the remaining ten years of his life in Holywood near Belfast. In Larmor's obituary, A. E. Eddington (1942) discusses Larmor's strong feelings for the Irish
t This form of the rotational aether also had doubt east on it by the experiment of O. Lodge (1897) which showed that a strong magnetic field would not influence the velocity of light, as one would expect to be the case if, as in this type of aether, a magnetic field is represented by an aether stream.

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NINETEENTH-CENTURY AETHER THEORIES

nation and countryside, and suggests that "it is no accident that Aether and Matter [Larmor's major work] is so largely a development of his countrymen MacCullagh, Hamilton, [and] Fitzgerald".
For whatever the reason, Larmor was a most vigorous proponent of MacCullagh's aether as understood through Fitzgerald's electromagnetic interpretation and Hamilton's Principle of Least Action. Larmor's inquiry into this aether began in 1893 and developed through the next five years. He republished a revised version of his essays in book form as Aether and Matter in 1900 for which he was awarded the Adam's Prize.
Larmor from the inception (1893) of his inquiry had felt that "our notions of what constitute electric and magnetic phenomena are of the vaguest as compared with our ideas of what constitutes radiation" and he believed that "many obstacles may be removed by beginning at the other end, by explaining electric actions on the basis of a mechanical theory of radiation, instead of radiation on the basis of electric actions".
Larmor's specific approach was, as he himself acknowledged, identical with that of Fitzgerald as regards the free aether. He used Fitzgerald's expressions for the potential and kinetic energy of the media which could variously be interpreted in either Maxwell's aether or MacCullagh's generalized aether. Larmor employed the same transformations as Fitzgerald to obtain a relation between the basic quantities of the "two" aethers, identifying electric displacement with aether rotation and magnetic force with aether velocity. The Principle of Least Action was also used to obtain the equations of motion of a disturbance propagated through the aether.
What is interesting and new about Larmor's analyses, other than the introduction of the electron which I shall discuss later, is the blend of critical self awareness about the problem of mechanical explanation together with an almost unshakable faith in the applicability of Hamilton's Principle to the aether. Larmor's position

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on the relation between mechanics and MacCullagh's aether apparently underwent some evolution between 1893 and 1900—an evolution which is worth discussing not only in connection with the evolution of the concept of the aether, but also because of certain parallels which it exhibits with the development of an electromagnetic view of nature, or electromagnetic "Weltbild", on the Continent.
I cited Stokes' criticism of MacCullagh's aether in the previous chapter. Larmor was well aware of these difficulties and in 1893 in his first paper on MacCullagh's aether Larmor wrote:

As regards the rotational elasticity of this hydrodynamical aether on which we have made all radiative and electrical phenomena depend, it was objected in 1862 by Sir George Stokes to MacCullagh's aether, that a medium of that kind would leave unbalanced the tangential surface traction on an element of volume, and therefore could not be in internal equilibrium; and this objection has usually been recognized, and has led to MacCullagh's theory of light being put aside, at any rate in this country.

Larmor's (1893) reply to Stokes' criticism was rather unsatisfactory. It depended on conceiving of gravitation as a non-aethereal process which would provide the missing restoring force. Larmor himself called this use of gravitation a "saving hypothesis" and a "useful deus ex Machina", and he did not employ it again in his later analysis in Aether and Matter?
In Aether and Matter, rather, instead of attempting to reduce electromagnetism to mechanics via the mechanical aether, Larmor suggested, as Heaviside had before him, that the mechanics of matter might be reduced to the actions of the electromagnetic aether, the latter conceived of as a kind of ultra-primitive matter or, if I may use the term, an C/r-aether. Larmor did not have a very clear idea of this C/r-aether, though in several places he

t See the letter from Larmor to Heaviside, 12 October, 1893, in the Heaviside Collection at the Institute for Electrical Engineers, London. A. Sommerfeld (1950) suggested that MacCullagh's aether involved "a 'quasielastic' body
. responsive to rotations relative to absolute space" !

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suggests that the actions of ordinary matter might be explained on its basis. For example, he indicates that the inertia of a particle of ordinary matter might well be understood in terms of the aether as characterized by Maxwell's equations or the Fitzgerald generalization of MacCullagh's theory. Elsewhere in the book he says: "An aether of the present type can hardly on any scheme be other than a medium, or mental construction if that term is preferred, prior to matter and therefore not expressible in terms of matter." In a long obituary which Larmor (1908) wrote on Lord Kelvin, Larmor discussed the Kelvin aether model which was presented in the previous chapter. His comments illuminate both Kelvin's model as well as his own views about the t/r-aether:
It has come to pass that by making a model, with ordinary matter, of an elastic medium that has not the properties of ordinary matter, Lord Kelvin has vindicated to many minds if not entirely to his own, the power and cogency of mathematical analysis which can reach away without effort from the actual to the theoretically possible, and for example, make a mental picture of an aether which is not matter for the simple reason that it is something antecedent to matter.
It is essential, then, to distinguish in the selections I have extracted from Aether and Matter between a reduction to "dynamics" in Larmor's sense of the term, which is nothing more than the characterization of this C/r-aether with the aid of the Principle of Least Action, and a different kind of aether approach, such as we encountered in Green's theory. This latter approach, according to Larmor, "virtually identifies aether with a species of [ordinary] matter". Such an approach has led to difficulties which seem insoluble, but such difficulties can at least be "deferred" in the case of his own aether, Larmor maintains:
if we are willing to admit without explanation the scheme of equations derived [in the first of the appended Larmor selections] from the form of energy functions for the aether, supposed stagnant, which is then postulated, in combination with the Principle of Least Action, and as a corollary, with an atomic structure of matter, involving electrons in its specification.
The aether had, during the seven years in which Larmor developed his theory, become very "aethereal" indeed, and Larmor,

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relinquishing the search for an ordinary dynamical foundation for his aether, seems to have moved in the same direction as those physicists who were developing an electromagnetic foundation of nature. Such physicists, among them W. Wien and M. Abraham, argued that since the "mass" of an electron could be understood in terms of the self-action of the electron's field on its own charge, rather than attempting to reduce electromagnetics and optics to mechanics, perhaps mechanics ought to be reduced to an electromagnetic theory.1"
We have seen that Larmor largely followed in MacCullagh's and Fitzgerald's footsteps. But Larmor carried the aether theory considerably beyond what he had found. The most original contribution of Larmor's aether theory in its initial stages was, according to Fitzgerald himself, the introduction of the vortex atom of Lord Kelvin into such an aether. Fitzgerald wrote to Heaviside on 8 February, 1895 candidly commenting on Larmor's recent work and noted : "Larmor... has made a decided advance in pointing out that M'Cullagh's medium may have a common irrotational flow without any stresses so that vortex rings might exist in it I anyway had not appreciated this before."!
Ironically enough Larmor gave up the vortex atom idea almost immediately and in its place substituted the mobile electron. The electron, or natural unit of electricity, which had received its name several years before from G. Johnstone Stoney, was conceived by Larmor to be a singularity in his aether, or in his own words, "analogous t o . . . a simple pole in t h e . . . theory of a function of a complex variable". The vortex atom had been an important contribution of Lord Kelvin who had based his thoughts on some work by Helmholtz. Kelvin had conceived of the possibility of

t See M. Jammer's (1961) book for abrief discussion of this trend in physics. R. McCormmach also has an excellent unpublished paper on this topic (see Preface, p. ix).
X The Fitzgerald letter is at the Institute for Electrical Engineers, London, in the Heaviside Collection.

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material atoms being permanent vortex rings in a structureless homogeneous, and frictionless medium.1" In his first full paper (1894) on the MacCullagh aether—Larmor had published an abstract of his theory in 1893—Larmor believed he could explain Ampere's theory of permanent magnetism by conceiving of the vortex atoms in his aether as electric currents. However, in an Appendix to this paper, dated 13 August, 1894, he admitted that vortex rings would not account for the magnetism and instead was driven to assume the existence of permanent charges, or electrons, and to conceive of magnetic molecules as "a single positive or right handed electron and a single negative or left handed one revolving round each other". Larmor realized that such a system should radiate, but as experiments revealed no such radiation, he proposed that no radiation should be released except when the "steady motion" of such a system was disturbed. This was somewhat ad hoc at the time, though something like it was introduced later by Bohr in his quantum theory of the hydrogen atom.
Larmor's electrons are freely mobile aether singularities which move through the aether "much in the way that a knot slips along a rope". Larmor's electron theory that resulted from the introduction of the electron hypothesis, along with the required appropriate modifications of the Fitzgerald equations of the aether, is very much like the electron theory of H. A. Lorentz. If it can be said—and reservations were expressed about this—that Fitzgerald found a generalization of the optical aether which was adequate to take into account Maxwell's electromagnetic aether, it can also be said that Larmor discovered a means of absorbing within a "dynamical" aether theory, the electron theory of Lorentz. Larmor claims that his own electron theory was independent of Lorentz', about which more will be said in the next chapter, but it is clear that in certain ways Larmor knew of and built on Lorentz' contributions. The question of independence is rather unclear though, and Lar-

t See Whittaker (1960), I, pp. 293 ff., for a discussion of the vortex atom-