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AETHEE AND MATTER
aont)on: C J. CLAY AND SONS,
CAMBEIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
ffilnstjoto: r,o, AVBLLINGTON STREET.
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ILcipiis: F. A. BROCKHAUS.
i^CM lork: THE MACMILLAN COMPANY.
Bombnu: E SEYMOUR HALE.
^ AETHER AND MATTER
A DEVELOPMENT OF THE DYNAMICAL RELATIONS OF THE AETHER TO MATERIAL SYSTEMS
ON THE BASIS OF THE
ATOMIC CONSTITUTION OF MATTEE
INCLUDING A DISCUSSION OF THE INFLUENCE OF THE EARTH'S MOTION ON OPTICAL PHENOMENA
BEING AN ADAMS PRIZE ESSAY IN THE UNIVERSITY OF CAMBRIDGE
BY
JOSEPH LARMOR, M.A., F.E.S.
FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE
CAMBRIDGE AT THE UNIVERSITY PRESS
1900
[All Rights reserved]
(oil
L2)
Cambritige :
PKINTEB BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
PREFACE
Aedijicium autem hujiis universi structura sua, inteUechii humano contem- planti, instar lahyrinthi est. — F. Bacon
The following Essay was originally undertaken mainly as a contribution towards the systematic theoretical development of the standpoint which considers electricit}', as well as matter, to be constituted on an atomic basis.
This is, as regards its general idea, no recent hypothesis. Within ten years of the publication of the fundamental dis- coveries of Volta, in the year 1800, which first revealed the existence of permanent electric currents. Sir Humphry Davy had been led to maintain the proposition that "chemical and electrical attractions were produced by the same causes, acting in the one case on particles and in the other on masses*," as the outcome of the researches on electro-chemistry which are recorded in his earlier Bakerian Lectures : and a little later the foundation for a complete system of chemistry was sought by Berzelius in a distinction between electro-positive and electro- negative atoms. Although it would be of course wrong to read back our precise modern knowledge into the general views Avhich a survey of the facts impressed on Davy's mind, — just as it would be erroneous to consider his more widely known views on the nature of heat as an anticipation of modern exact thermal theory — yet his striking pronouncements show how
* See Appendix D, infra, p. 317.
vi PREFACE
rapidly the times became ripe for the quantitative electro- chemical investigations of his successor Faraday. Since Faraday's work on electrolysis the notion of the atomic constitution of electrification, in its electro-chemical aspect, has never been entirely absent: it has been insisted on by Maxwell and more particularly by von Helmholtz : it was adapted, in the most natural manner, to the ideas of the Weberian electrodynamics, which treated of moving electric particles: but it is only recently that any efforts have been made towards the development of the Maxwellian aether-theory on that basis.
The form under which the atomic electric theory is intro- duced in this Essay, originally presented itself — as it happened — in a quite different connexion, in the course of an inquiry into the competence of the aether devised by Mac CuUagh to serve for electrical purposes as well as optical ones. It was found, reasoning entirely from abstract principles, that the only possible way of representing electrification, in an elastic aether, was as a system of discrete or isolated electric charges, consti- tuting singular points involving intrinsic strain in the structure of the medium. If the propagation of disturbances in the aether, with their ascertained optical properties, is to be explained dynamically at all, that is by the interaction of inertia and motion, the elastic reaction of this medium to displacement is almost restricted, as Mac Cullagh showed, to be effectively (even if not fundamentally*) a purely rotational one ; an essential confirmation of this rotational character of its elasticity here presents itself in the fact that in a medium
* It is not superfluous to repeat here that the object of a gyrostatie model of the rotational aether is not to represent its actual structure, but to help us to realize that the scheme of mathematical relations which defines its activity is a legitimate conception. Matter may be and likely is a structure in the aether, but certainly aether is not a structure made of matter. This introduction of a suprasensual aethereal medium, which is not the same as matter, may of course be described as leaving reality behind us : and so in fact may every result of thought be described which is more than a record or comparison of sensations.
PREFACE VU
thus constituted the structure of an atomic electric charge can be directly specified, so far at any rate as is required for a knowledge of its interaction with other electric charges at sensible distance, whereas the absence of any conception of what constitutes electrification had previously been commonly regarded as the fundamental obstacle to the electrical develop- ment of aether-theory. The order of ideas thus indicated, when carried through up to its logical extent, involves the explanation of the atomic character of matter itself: matter must be constituted of isolated portions each of which is of necessity a permanent nucleus or singularity in and belonging to the aether, of some such type as is represented for example by a minute vortex ring in perfect fluid or a centre of permanent strain in a rotationally elastic medium. It is thus natural to infer that the ultimate atom of electricity is one aspect of the entity which constitutes the ultimate atom of matter, a con- clusion which (foreshadowed as above in set terms by Davy) is almost demonstrated by Faraday's electro-chemical law ex- pressing an exact numerical connexion between them. The question must of course remain open as to whether other forms of activity besides this electrical one can be recognized in the constitution of the atom of matter: as yet nothing seems to have been found in the ascertained types of general physical and chemical phenomena which demands a further amplification, so that any advance in that direction would at present be premature if not gratuitous. For it is to be borne in mind that the proper^ aim of an atomic theory is not to attempt the impossible task of reducing once for all the whole complex of physical activity to rule, but is rather to improve and connect accepted methods of explanation of the various main regular types of interaction that have been brought to light in this field of knowledge. It is incumbent on us to recognize an a,ejthereal substratum to matter, in so far as this proves con- ducive to simplicity and logical consistency in our scheme of physical relations, and helpful towards the discovery of hitherto
Vlll PREFACE
unnoticed ones ; but it would be a breach of scientific method to complicate its properties by any hypothesis, as distinct from logical development, beyond what is required for this purpose. It may therefore be held that, in so far as theories of the ultiniate cojmexion of different physical agencies are allowed to be legitimate at all, they should deyelope along the lines of a purely electric aether until critics of such a simple scheme are able to point to a definite group of phenomena that require the assumption of a new set of properties and that moreover can be reduced to logical order thereby : a charge of incompleteness without indication of a better way, is not effective criticism in questions of this kind, because, owing to the imperfection of our perceptions and the limited range of our intellectual operations, finality can never be attained.
It appears to be not without value, as regards clearness and definiteness of view, that the conception of an elastic aethereal medium that had been originally evolved from consideration of purely optical phenomena, is capable of direct natural develop- ment so as to pass into line with the much wider and more recent electrodynamic theory which was constructed by Maxwell on the basis of purely electrical phenomena, — in fact largely as the dynamical representation and development of Faraday's idea of a varying electrotonic state in space, determined by the changing lines of force.
In the following discussions some care has been taken to trace the connexion with the historical course of physical ideas. While there has been a steady gain in precision, the trend of fundamental physical speculation has always been much the same, and thus does not in its main features pass out of date : but owing to the rapid accumulation of new phenomena and the consequent necessity of formal treatises digesting the state of actual knowledge, there is less and less opportunity to become critically acquainted with the points of view of the original discoverers in physical science, and the general lines of thought in which the definite recorded advances are often only
PREFACE IX
incidents. The accumulation of experimental data, pointing more or less exactly to physical relations of which no sufficiently precise theoretical account has yet been forthcoming, is doubt- less largely responsible for the prevalent doctrine that theoretical development is of value only as an auxiliary to experiment, and cannot be trusted to effect anything constructive from its own resources except of the kind that can be tested by immediate experimental means. The mind will however not readily give up the attempt to apprehend the exact formal character of the latent connexions between different physical agencies : and the history of discovery may be held perhaps to supply the strongest reason for estimating effort towards clearness of thought as of not less importance in its own sphere than exploration of phenomena. Thus for example, the present view of the atomic character of electricity, which is at length coming within the scope of direct experiment, has been in evidence with gradually increasing precision ever since theoretical formulations were attempted on this subject: in fact if the considerations ex- plained below (§ 46) are valid, it is the only view that could logically be entertained without involving either a reconstruc- tion of the accepted basis of physical representation or else an admission of its partial or merely illustrative character.
Some of the following chapters may be regarded as a re-statement in improved form of investigations already de- veloped in a_series of memoirs, Phil. Trans. A 1894-6-7, They cover only a part of the survey that was there attempted, being concerned mainly with the systematic construction of general electric theory on the basis of intrinsic atomic charges. Judging from some criticisms which that method has attracted, it would appear that misconception has existed owing to difficulty in attaining the point of view, such as may possibly arise from the circumstance that the fundamental concept of the electron did not there present itself at the beginning of the discussion, but was introduced subsequently in an appendix. In estimating the value of an undertaking of this kind, it should of course be
X PREFACE
judged as a connected argument. Single departments of electric theory can be, and usually are, more concisely formulated when the nature of their connexion with other departments is not a question of immediate interest. Here the foundation is intended to be definite and universal, thus precluding the adoption of an independent standpoint on each branch of the subject, to be developed as far as it will go, with the help if need be of empirical modification irrespective of the requirements of other branches : and the main question to be determined is not whether the presentation is entirely free from flaw, but whether and in what respects it is an advance sufficient to merit further attention with a view to its improvement. The time has fully arrived when, if theoretical physics is not to remain content with being merely a systematic record of phenomena, some definite idea of the connexion between aether and matter is essential to progress ; the discussions which follow are largely concerned with the development of the consequences of one such conception. It seems reasonable to hold that the utility of such a discussion will not be entirely removed should this conception turn out to be practically incomplete, or even incoherent : for it is concerned with a body of ideas relating to the ultimate activity of molecules which is on a different plane from the indefinite notion of mutual forces to which dynamical molecular science has mostly been restricted.
It is recognized of course that every attempt at improve- ment in scientific exposition must have a limited range, and that the chief critical interest will soon be transferred from what can be explained by any new formulation to what it has not shown itself competent to include. Yet it has turned out, to take a famous instance, to be no insuperable objection to Dalton's chemical development of the atomic theory, that he could form no idea as to how his atoms held each other in combination : although in fact Wollaston, and to some extent Davy, hesitated about accepting the atoms while holding to the laws of combination in definite and multiple proportions that
PREFACE XI
were suggested by them, because the theory did not explain how an atom is constituted. An instructive illustration of the practical divergence of view that can arise in this manner is afforded by the difference of opinion (cf Appendix D infra) between Newton and Huygens regarding the scope of the law of gravitation.
The general conception of a kinetic molecular structure for matter invites a reconstruction of the theoretical basis of ordin- ary mechanics itself The customary development of abstract dynamics rests on the foundation of forces acting between particles so as to satisfy Newton's three laws of motion. The only meaning that can here be assigned to such a particle is an unchanging element of mass whose volume is large enough to contain a vast assemblage of molecules. Even in the hands of Kirchhoff, who is considered to have in his treatment notably broadened the subject, the dynamics of finite bodies is derived through the principle of Least Action from the conception that they are constituted of particles acting on each other in this way ; thus in fact adhering to Lagrange's original procedure developed in the year 1760. It seems not too much to say, in the light of molecular science, that such a constitution for a material body is purely imaginary, and even meaningless when applied to such subjects as stress and strain in elastic matter. ^ The real foundation of general abstract mechanics is either the j principle of Action in its general form, assumed as a descriptive analytical formulation of the course of phenomena, or else the principle of d'Alembert which is analytically equivalent to it.^' The latter is doubtless the simpler basis when we are dealing only with elements of mass in the ordinary sense ; for it is merely the statement that the irregular or uncoordinated part of the internal motions and strains in a stable or permanently existing element of mass has nothing to do with the changes of configuration of that element considered as a whole ; and the preparation for its application consists in the process of picking out and specifying the coordinated parts so far as is needed for
XU PREFACE
the problem in hand. In electrodynamic problems however the individual electrons in the element of volume come into con- sideration, at any rate as regards their main division into positive and negative : and then the Action principle, as being the more universal, must almost of necessity be employed. If further we assume that the material molecule is wholly of aethereal constitution, this general dynamical principle of Action must itself be involved (cf Appendix B) as a direct consequence of whatever scheme of properties is assigned to the free aether, irrespective of special hypothesis : so that, reasoning back from its truth, we may gain some general knowledge as to the nature of the interactions exerted by the ultimate material atoms across the aether. Its significance would then consist, not in any directly ultimate character such as it was originally sup- posed to possess, but in its being a derived analytical formulation sufficiently comprehensive to cover by itself the domain of mechanical science, as well as (in a minor degree) in its aptitude for analytical transformation to suit whatever aspect of the phenomena is under discussion, after the manner illus- trated for example by the analysis of Chapter vi infra. In the course of such an abstract development of the dynamics of mechanical systems from the Action principle, the idea of force would be introduced through the coefficients in the completed variation of the Action, to which it is necessary for purposes of physical discussions and comparisons to give a name, that name which is in fact suggested from our sensation of muscular effort. It is possible that these considerations are insisted on in various connexions to an extent that will be tedious* : but they are at
* At one time it was customary to appeal to absolute dynamical principles relating to forces, as the fixed unchanging datum of physical science. The interest in fundamental discussions regarding the mode of formulation of dynamics has however revived in recent years, mainly through the writings of James Thomson, and of Hertz and Boltzmann, and of the school of physicists which advocates, restriction to a purely descriptive science of energetics. The conception of the subject that is propounded here is different from the points of view of these writers; while it aims at defining the domain (including all that of steady or very slowly varying states) within which the simpler principles of
PREFACE xm
any rate not more prominent than the cognate fundamental discussions on convergency and functionality have become in pure mathematical analysis. A field in which these two types of approximation towards ideal exact procedure have something in common is sketched in the analysis of Appendix A: it is now recognized that, in a strictly rigorous presentation of the theory of gravitational and other agencies, it is necessary to inquire (at considerable length) how far a potential is a function that can legitimately be differentiated at a point inside the attracting mass : it is here explained, among other things, that in a physical problem the potential about which these mathe- matical discussions arise is not the potential of the actual molecular distribution of matter but that of an ideal smoothed- out or continuous distribution, in fact a mechanical* repre- sentation which is, only in certain definite respects, equivalent to it. Refinements of this kind are no doubt foreign to an empirical formulation of mathematical physics, where the aim is merely a concise expression of the facts of observation ; but it seems not unlikely that in the final theory of the transition from molecular to mechanical dynamics they will be of funda- mental import.
The main general principle in this domain, which is con- sidered in some aspects more at length in the memoirs above referred to" is that the mechanical and the molecular properties of a material system, which is not undergoing constitutive change, are independent of each other and not mutually in- volved :' so that the mechanical interactions of a system can be developed independently of a knowledge of its molecular constitution. ^^ This principle may even be sufficiently deep- seated to have a bearing on the solution of the philosophical question of ultimate mechanical determinism.
energetics can supply an adequate clue to the course of physical and chemical change.
* Throughout this Essay the term mechanical is used in antithesis to mole- cular: mechanics is the dynamics of matter in bulk, in contrast with molecular dynamics.
L. C
xiv PREFACE
No apology is offered for what may possibly be considered as the non-mathematical character of a considerable portion of the book. The physical hypothesis has been kept intentionally in the foreground, and algebraic results have been where possible translated into their descriptive equivalents. This synthetic course, while more flexible, doubtless makes the ex- position more difficult to follow, and apparently less exact, than would be a mathematical development from a system of initial hypotheses, in which attention would be demanded for the analytical processes alone: but on the other hand it exposes the whole situation, and conduces to the direct detection and examination of discrepancies that might possibly otherwise remain latent : it is thus, notwithstanding the somewhat im- perfect focussing of the subject, more suitable for a theoretical procedure which is in the constructive stage.
The Essay in its present form was completed at the end of the year 1898, except as regards the Appendices D, E, and F, and the articles and footnotes distinguished by a double asterisk or other mark. In the revision of the sheets before publication the writer has however had the good fortune to obtain the collaboration of his friend Prof. A. E. H. Love, of Oxford, whose acute and vigilant criticism has led to many improve- ments in the exposition as well as the correction of various mistakes, thereby adding very substantially to the value of the work. For similar most valuable services relating to the latter half of the book, and for the greater part of the list oi corrigenda — as regards some of which special apology must be made — the writer is indebted to his friend Prof W. M^F. Orr, of the Royal College of Science, Dublin. Notwithstanding much increased confidence in the general validity of the argument, arising from the expert assistance and criticism thus most kindly afforded, many serious defects doubtless remain. Various questions not ripe for final definite treatment, and matters on which opinions can differ, have been passed under consideration: for instance, the discussion on the molecular basis of mechanics
PREFACE XV
will have largely served its purpose if it draws attention to the point of view. In questions of this kind the most fruitful source of progress is perhaps the process of mutual approximation of different standpoints ; any single attempt at effective adaptation of fundamental conceptions must involve detail that is only provisional, and leave points of difficulty unsolved or imperfectly analyzed, while in many cases it will originate more problems than it settles. This latter feature of general theoretical physics cannot in fact be better illustrated than by the original found- ation of all such theories as given by Maxwell, which in its broad outlines was the culmination of the greatest advance in modern times : yet it presented itself with so many gaps in its formal development, and raised so many new questions for discussion, that finality with regard to the mode of formulation of the subject is possibly yet far off.
Acknowledgment is due to the officials of the University Press for the exact and obliging manner in which they have executed the printing and corrections.
St John's College, Cambridge 2Iar. 6, 1900
CONTENTS
PAGE
Chapter I Introduction ...... 1
SECTION I
Chapter II Historical survey ..... 6
Astronomical aberration of light : Bradley's guiding ideas. Oi^tical refraction uninfluenced by the Earth's motion : Fresnel's explanation. Aberration as measured by a water-telescope. Theoretical views of Cauchy. Views of Sir George Stokes as to constitution of the aether : suggested partial analogy of material substances like pitch ; irrotational viscous motion unstable, also involves dissipation of energy: irrotational character of finite motions of the aether exiDlained by its high rigidity or rotational elasticity. Maxwell's formulation of Fresnel's theoiy. The general absence of any optical influence of the Earth's motion suggests that the aether may move along witli the Earth : the difiiculties of this mode of explanation. Maxwell's discussion of electrodynamic theory relative to moving bodies. Introduction of atomic electric charges into the aether theory. Aether stagnant. Alleged difiiculties of the electric theory. Maxwell's original scheme effective for systems at rest : nature of its adaptation to moving systems. A constitutive aether affords the only explanation of an atomic constitution of matter : the various atomic theories : electric aspect of an atom. MacCuUagh's optical aether is the electric aether. Molecules may be systems of electrons : the duality of positive and negative dynamically necessary. Maxwell's elimination of ' electricity ' from the theory is available for ordinary electrodynamics, but cannot be extended to problems involving radiation.
Chapter III General kinematic theory of optical rays
IN MOVING media ....... 30
Specification of a ray: ray-velocity: wave-velocity: wave-front: principle of Least Time: application to a moving medium. Theories of aberration : if the aether moves sensibly its motion must be irrot-
xviii CONTENTS
ational : influence of its elasticity in this direction : evidence of water- telescope. Influence of convection of the material medium on velocity of propagation ; Fresnel's law necessary whether there is aethereal flow or not. Eay-velocity in moving medium. Influence of convection on positions of optical foci: on period of the light: on phenomena of difl'raction and interference, application to diffraction-grating. Inclusion of the second order of small quantities : theory of Michelson's inter- ference experiment; law of reflexion by rotating mirror. General analysis of interference relative to moving media: the retardation may be calculated on the undisturbed path: applied to Michelson's arrangement.
Chapter IV The problem of optical convection : indi- cations TOWARDS A DYNAMICAL THEORY ... 54
Examples of convection of wave-trains by simple media: sound waves in air, waves on stretched cord. Maxwell's equations for the compound medium, aether and matter: various possible constitutive hypotheses considered. Fresnel's law obtained; the electric theory of moving media which it requires ; same expressed without the aid of potentials. Sketch of results to be derived from a precise molecular theory : Michelson's interference experiment gives a clue as regards the constitution of a molecule. Magnetic effect of convection of an electric charge with the Earth : Eontgen's null result explained by countervailing charges : the electric effect of the convection of a magnet must be similarly null.
SECTION II
Chapter Y On method in general physical theory . 68
On the scientific utility of hypothesis : illustrated from the history of the corpuscular theory of light, and of the Weberian theory of electrodynamics. Helmholtz's criticism of the latter not destructive ; it is now included in modified form in the aether theory: the same applies to MacCullagh's optical theories, which were at one time rejected. On vector terminology. Hypothesis of aethereal constitution of matter; necessary in order to avoid an irreconcilable duality of matter and aether; historical. ' Phenomena are expressible in terms of matter alone, unless velocities or alternations comparable with those of radiation are concerned. This electric theory of aether and matter is precisely formulated; thus even if itself incomplete, it will throw- light on the possibilities of correlation in that remote region : ordinary molecular theories suppose the molecules so far apart that their inter- actions can be expressed by ' forces. '
CONTENTS XIX
PAGE
Chapter VI Dynamical theory of electrical actions . 82
Least Action, fundamental in general dynamics. Dynamical equations for free aether, derived from its energy function : the two fundamental vectors, circuitally related : velocity of elastic propagation. Introduction of electrons, as point-charges. Modification of relation of aethereal strain to displacement in a medium containing electrons : displacement of an electron equivalent to a local aether strain : two independent variables, electric flux and aether strain : combined they form Maxwell's circuital total current. Generalization of Stokes' theorem of circuit and barrier integration, for a medium containing singularities. In ordinary electrodynamics, aether strain inconsiderable compared with electric flux : expression of the kinetic energy of the aether in terms of electric flux, viz. in terms of true flux of electrons and ' apparent ' flux the equivalent of change of aether-strain, involves the introduction of the vector potential of the electric flux : the potential energy involves the aether strain alone. Application of Action principle to the energy as thus specified : introduction of the additional condition ^ that each electron is a pole of the aether strain: variation j^erformed as regards a moving electron :' interpretation of result as giving the electric ^' force acting on the electrons and the aethereal force straining the aether: modification required in magnetized media to obtain the ' mechanical ' part of the electric force which excludes the purely local part : the mechanical force on an electric current^ Electric currents of conduction, due to motions of ions, constituted half by positive and half by negative ions. Currents producing material electric polarization. Current arising from convection of charged bodies. Current arising from convection of a polarized dielectric, expressible primarily as a 5((«sj-magnetization. Any magnetization identical mechanically with a distribution of current. Total mechanical force on the material medium, as constituted by the force on the true current, that on the magnetism, that on the electric polarization and the true electric charge: the part of it arising from the convection of the medium. The vector potential of magnetism must be defined as that of the equivalent distribution of electric flow: this equivalence extends to magnetic induction, not to magnetic force.
Chapter VII Review of the electrodynamic equations
OF A material medium . . . . . .109
Exact dynamical relations. Exact relations inherent in the con- 7 stitution of the medium.^^ Consequences: the static electric force due to the actual distribution adds to the kinetically induced force. Approximate constitutive relations of the material medium : dielectric ~and~rnagnetic coefficients : analysis of aeolotropic conduction, any rotational character must ^15e due to extraneous vector influence. Elimination of mathematical potentials : the circuital relations. Max- well's |3urely abstract fifh^mp; determinate for media at rest and
XX CONTENTS
identical with the present scheme : illustration from theory of double refraction, compared with Helmholtz's wider theory. '^'^ The aether -;^ sufficiently defined by its dynamical equations. The transition from molecular to mechanical theory: the latter involves only the principal values of the integrals expressing the potentials.
Chapter VIII Optical and other developments relating
TO ENERGY AND STRESS . . . . . .127
Mechanical electrodynamic forces expressed in terms of a stress- system: limited validity of Maxwell's expressions, their physical meaning : his dielectric stress invalid except in free aether. Application to repulsion of conducting masses by ntagnetic alternators: copper filings lie along lines of force. Contrast between relations of obstacles to electric waves and to sound waves. Mechanical pressure of radiation ; on a black body is given by Maxwell's law. Absorption of radiation : character of surface of a perfectly black body; of a perfect reflector. Kelatiou of the complete radiation to the temperature: Boltzmaun's proof of Stefan's law. Dynamical and material symme^try_^ general deductions regarding perversion, reversibility, chirality : influence of convection through the aether on structure. Adiabatic comiDression of radiation : its mechanical value: legitimacy of an ideal screen impervious to radiation but pervious to aether.
SECTION III
Chapter IX Influence op steady motion on an electro- static MATERIAL SYSTEM . . . . . .149
Existence of an electric potential in every steady, s_tate. Equations for case of uniform translation : charaKterTsfTfTegTiat.inn of f.he pntent^ial, solved : electric distribution and force unaffected, but magnetic field altered: correlation with a stationary system. Null results of con- vection of magnets. Convection of a dielectric system. Uniform rotation of an electrostatic system : electric potential not constant in conductor : solution for a rotating dielectric; for a spherical con- ductor. ~
Chapter X General problem op moving matter treated
in relation to the individual molecules . . 161
Analytical specification of an electron as a moving pole in the electric field necessitating a singularity in the magnetic field. ""^The problem, formulated soas to take cognizance of the electrons indi- vidually, is determinate in terms of the aether alone if matter is
CONTENTS XXI
constituted of electrons: arguments in favour of its being mainly so constituted.' The equations relative to the convected material system; transformed back to the standard form : resulting correlation with same system stationary, but with a time-origin varying from point to point; includes the previous special investigations: in static distributions the charges and electric forces are the same in both systems to the first order, but not the aethereal displacements and magnetic forces.
Chapter XI Moving material system : approximation
CARRIED TO THE SECOND ORDER ..... 173
Electrodynamic equations, taking account of individual electrons : referred to moving system : restored to standard form by change of scale in space and time : resulting correlation. Second-order shrinkage in an aethereally constituted system arising from convection. Optical propagation in moving matter : Earth's motion optically inoperative. ? Null effect on structure of a molecule. Null effect on conductivity of the medium. Discussion_of null effect_to second order in Michelson's interference experiments. Are the linear equations of the aether-exact? Inference as to structure from the definiteness of atomic masses : gravitation not involved with the present subject.
SECTION IV Chapter XII On optical rotations magnetic and
STRUCTURAL . . . . . . . .194
Magneto-optic energy term: deduction of relation between polar- ization and electric force : the magnetic influence purely rotational. Equations of propagation : coefficient of rotation. A hypothesis as to relation of rotation to density : not experimentally verified. Physical explanation of the rotation. Theory of magneto-optic reflexion and the Kerr effect. Structural rotation : the equations : its kinetic origin. Chiral relations of ions. Kotation produced by artificial twisted structure. See Appendix F.
Chapter XIII Influence op the earth's motion on
ROTATIONAL OPTICAL PHENOMENA . . . . .211
Equations for moving rotational medium : influence of convection on velocity of propagatioTT TThe convective effect simply superposed. Verification of Mascart's null result as regards striictural rotation : consistent with the electron theory. Null result as regards magnetic rotation.
XXll CONTENTS
SECTION V
Chapter XIV On the mechanism of molecular radiation 221
The aethereal disturbance propagated from a moving electron. The field of a vibrating doublet ; of an electron describing an orbit. The type of radiation emitted from a moving electron : its intensity : a uniformly moving electron does not radiate. Isolated pulse caused by sudden disturbance: its energy conserved as it moves onward^ mode of establishment of a steady magnetic field as the trail of such a^ulse. Conditions for absence of radiation from a molecule : concentration of its energy
Chapter XV On the nature of ordinary radiation, and
ITS synthesis into regular wave-trains . . . 235
Eontgen radiation, not of vibratory origin : its intensity per mole- cule : absorption proportional to density. The Fourier analysis, how far objective. Analysis of radiation by a series of simple damped receivers : response to a discontinuous putse, to a damped wave-train : case of Hertzian radiation. Character of radiation from gases : periods, effective phase, interference. Penodicity created or intensified by the analyzer; case of a grating; case of a prism. Eontgen radiation would analyze into high periods.
Appendix A On the principles of the theory of mag- netic and electric polarity : and on the mechanical significance op divergent integrals .... 252
Definition of polarity: its relation to the Maxwellian electric dis- placement: polarization of dielectric matter. Distribution of polarity replaced by a distribution of density. Formal laws of induced polar- ization. Transition from the aggregate of molecular polar elements to the polarized mechanical medium. The potential of mechanical theory defined : the actual magnetic vector potential must be modified to a " mechanical form. The integrals occurring in mechanical theory are defined by their principal values.^ The formula for electric force always consists of an electrokinetic part added to the electrostatic force.
Appendix B On the scope op mechanical explanation :
AND ON THE IDEA OF FORCE ..... 268
General mechanics based on principle of virtual work and principle of d'Alembert. Generalized law of mechanical reaction. Scope and physical reality of statics: idea of force fundamental and prior to mechanical motions. The method of molecular dynamics. Mechanics a growing science as regards its principles. General kinetics formulated
CONTENTS xxm
with regard to relative motions alone, in special cases : in other cases the stagnant aether is the ultimate datum of reference. Kirchhoff' s exposition of the Lagrangian formulation on the basis of the Action principle applied to systems of ' particles.' The Lagrangian scheme implies that the system is conservative as regards its energy, and involves that it is conservative (e.g. reversible, dynamically permanent) in other respects. Hertz's objection that rolling motions are not included. Physical concepts are abstract ideas cultivated under the guidance of observation. If matter is aethereally constituted, all dynamics ultimately rests on that of the aether : mechanical Action principle in this way deduced. The approximate inclusion of actual conservative material systems under the ideal Action principle involves observation and experiment. Thermodynamics a branch of statics : its aim is the formulation of the available energy, on which alone mechanical effect depends. Physical explanation of law of uniformity of temperature: temperature not a dynamical concept. Mechanical analogies. Mechanics of permanent systems is independent of mole- cular dynamics : illustrated by theory of osmotic pressure. Energy not an ultimate concept. Vital activity not mechanical as regards its stimulus.
Appendix C On electrolysis : and the molecular
CHARACTER OP ELECTRIC CONDUCTION .... 289
Laws of Faraday and Kohlrausch : they require that electrolysis is accompanied by convection of the electrolyte. Independent diffusion of the ions : their^gquations of transfer : |i-elation of their diffusion con- stants to their electric mobilities. Diffusivity connected with electric data : electromotive forces arising from concentration. Ions not free in metallic conduction. Ultimate steady gradient of concentration established by electrolysis. Special case of no current. Hall effect in electrolytes. Influence of motion through the aether. Equations for mixed electrolytes: electromotive forces: simple cases. Convective material flow due to bodily charge, compared with electric osmosis. Thermoelectric influence : the moving ions carry their heat along with them : the thermoelectric gradient along an unequally heated conductor hot a true voltaic effect.
Appendix D On the historical development of atomic
AND radiant theory . . . . . , .310
Fermat on Least Time or Action in optics. The aether-theory of Huygens : on transmission by waves: on the nature of the elasticity of solids, contrasted with that of gases : ideas as to elasFicity of the
aether : kinetic theory of gases, and of matter in general : matter freely permeable to aether: his hmited acceptance of the law of gravitation. Gravitation uninfluenced by structure. The aether-theory of Newton:
xxiv CONTENTS
anj,ether essential to hisjviews: compelled by absence of explanation of shadows to introduce the extraneous aid of luminiferous corpuscles: would welcome any constitutive aether that would not disturb the motions of the planets : atoms necessary to physics. Young's plea that the electric aether may "also be the medium of optical propagatioii. Davy's view that electric attraction is of the essence of the atom, and is the cause of chemical affinity. Gauss on the necessity of a medium for the_transmission of electric force between atoms. Kelvin's view that atoms are structures. Graham's view that an atom is a vortex in the aether. Fresnel's views on the optical influence of motion of material bodies: the Earth's motion does not disturb the aether : his law of optical convection deduced from his hypothesis that the aether is denser in material media : consequence that ordinary optical pheno- mena are uninfluenced by the Earth's motion.
Appendix E On kinematic and mechanical modes of
REPRESENTATION OF THE ACTIVITY OF THE AETHER . 323
Mechanical models and illustrations. Eotational elasticity : Kelvin's gyrostatic illustratr6u7~its limitations. Model of an electron in a rotational aether : its creation by a supernatural process : analogous constructions in elastic matter. Mechanical analysis of attractions between electrons ; involves dissection of the aether by strain-tubes connecting complementary electrons : illustration from the generalized form of Stokes' analytical theorem. The ultimate foundation in physical theory is the Action principle. The electron effectively a point-charge: the details of its structure unknown. All physical representations are at bottom comparative or illustrative : the scheme of a rotational aether merely consolidates the various hypotheses into a single one. Static attraction transmitted, not j)ropagated. A con- stitutive aether contrasted with an accidental one.
The electron theory essential to the formulation of ordinary electro- dynamics as well as for radiation. Neumann-Helmholtz electrodynamic potential theory invalid ; experimental tests by FitzGerald and Lodge. The expression for the electrokinetic potential of two electrons com- pared with Weber's formula : they lead to the same results in the electrodynamics of ordinary currents.
Appendix F Magnetic influences on radiation as a
CLUE to molecular CONSTITUTION .... 341
The Zeeman effect: determined for a molecule in which the mobile electrons are all negative : same results apply when there are mobile positive ions relatively very massive. Nature of magnetic polarization of a molecule, it does not involve orientation : the exceptional phenomena of the magnetic metals arise from cohesive aggregation^of molecules. The polarizations of the Zeeman Hues indicate the characters of the
I
CONTENTS XXV
PAGE
vibrations of the molecular system: the methods of dynamics of par- ticles adequate to the problem: the system referred to rotational coordinates; the condition for circular principal vibrations. Any purely constitutive potential energy for the molecule leads to the Zeeman phenomena, with the requisite generality. Inference as to effective isotropy of the molecule. The Faraday effect and Becquerel's law of dispersion deduced from the Zeeman effect _when the freely mobile electrons are all negative ; or when the dispersion is controlled by one absorption band, or by several bands for which the Zeeman constant is the same. Optical rotations necessarily of dispersional type; and therefore not simply related to material structure. Direct kinematic analysis of optical rotations for crystalline media, by refer- ence to rotating frame: law of rotation in different directions: the permanent types of vibration when rotation is superposed on double refraction : problem of refraction into a cbiral medium not deter- minate.
Index 357
XXVll
|
11 |
line 30, |
|
12 |
„ 8, |
|
32 |
„ 25, |
|
41 |
,, 17, |
|
43 |
n 8, |
|
44 |
at foot, |
|
58 |
at foot, |
CORRIGENDA
Page 11 line 30, for nj2Tr read 27ru
delete no. Thereare viscous tractions which generate heat in the fluid; though they do not affect its motion, they exhaust the energy of the moving solids
after cases insert of limited variation. Cf. p. 276 17, for v'^jV-cos-d read v'^IV-sin^d for V read v'
for OA' read O'A; delete 80 in tollov/ing lines: amend first equation on next page, and change sign of right hand in second and third equations for Kfj-jc'^ read C'7A>, and for [KixYjc read cKK/j,)^ 90 in equation (i) delete the fluxion dots : cf. footnote, p. 330 104 line 6 from foot, read 7 (v - f/) - j3 {w - h) 114 lines 9 and 10, P, Q, E should be cyclically interchanged ,, line 12, multiply by ^ 121 ,, 11, for K read K^
129 lines 20 — 29, the statement should be reversed: copper filings in a magnetic field alternating as regards intensity but not direction will set themselves along the field
134 line 4 from foot, for c^/n- read «-/c-
135 , , 4, for i-n-fj. read iTr/j. {nlp^}- ; Hue 7, for 2/7v> read 2 ; litie 10, for only the purely aethereal part read the whole of it; and line 14, after not insert all. In an undamped wave- train the energy is all propagated, up to this approximation: the difference between group velocity and wave velocity involves the theory of dispersion
136 for last two lines read only when ^ or o- is infinite or else both of them vanish
146 last line, delete -
153 lines 8 and 12 from foot, for e read e^
154 line 2, fore-^Q read e~^Q : line 2 from foot, for e-i read e~^ : and in last line multij^ly by e-
155 ,, 9 from foot, for Itt read -iTrc- 158 „ 12 from foot, for A'+ 2 read K - 2 167 „ 15, for the second /; read c
174 ,, 2, and 2 from foot, for v read v
175 lines 1 and 3, for t read t'
XXVlll CORRIGENDA
Page 179 line 3, for 3 cos- 6' read cos- 6' 183 footnote, for B read C 190 line 7, for <pjk read (pk'^\ and line 8, for and read though not
198 ,, 2 from foot, for c- read c^
199 lines 10 and 11, for a.^ read a^c- ,, line 15, for +t('3 read - tWj
200 on lines 3, 11 and 18, the multipliers before djclt should be nr)n' and m}in' respectively, where [V, ?«', ii') is the direction vector of the magnetic field ,
204 the sign of (Oj, a„, a^ has been changed in § 132
206 line 9, for C read C, equal to C ( -- — ^ ) ; and line 12, for C read
QtvC'C, and for )- read )~^. Cf. also Appendix F, § 5 308 footnote, for 1884 read 1894 350 for -S^ir, -S^arread +2^1^, -t-S^jr
I
AETHEE AND MATTEE.
ADDENDA AND CORRIGENDA.
p. 2 add to footnote %
A treatment of electrons as singularities in the aether has also been developed independently by E. Wiechert 'Die Theorie der Elektrody- namik und die Eontgen'sche Eutdeckung, ' Schriften d. phys.-okonom. Ges. z. Konigsberg, 1896, and more especially ' Grundlagen der Elektro- dynamik ' Leipzig (Teubner), 1899.
p. 60 for § 37 line 9 to end read
dR dq _ ha dy~~dz~ ~dt'
where — = — + u -— ; and (P, Q, R) is the electric force acting on the con- dt dt dx
vected charges, being equal to (P', Q'-vc, R' + vh); in which (P', Q', R'j
represents 47rc- (/, g, h) and is the force straining the quiescent aether.
Also (/', g', h') = - — .^ (P, Q, R), as the convection cannot alter the value •±7rC
of K to the first order. These equations are equivalent to the preceding. p. 86 line 21 add to footnote
More strictly, if the electron were moving with velocity v such that (u/c)- cannot be neglected, e must be detined as the value of ^(lf+iug + itIi)dS taken over any surface surrounding it alone.
p. 91 line 29 add footnote
The current then contains magnetic terms : see § 73. p. 93 add to footnote
The result of not making 4> null, as in the text, would be merely that the electric potential "ir (§ 58) would be diminished by the amount - d^ldt ; and the (very minute) distributions of electric charge arising from electro- kinetic causes would thus remain undetermined by the theory. p. 94 add to footnote
L is constant only when the squares of quantities like .ijc can be neglected. Thus Maxwell's law of pressure of radiation cannot be extended to theoretical discussions involving radiators or reflectors moving with velocity comparable with that of light. p. 102 for lines 1 — 7 from foot read
During this transfer the moment of the doublet gradually changes, thus the circuit is not a parallelogram : moreover we must make a correction by superposing on the complete circuit a creation of the doublet in its final position at the end of the transfer and an annulment in its initial position at the beginning of the transfer, in order to obtain the net eiiect of its transfer and change, free from completion of the ends of the circuit. Combining these creations and annulments for adjacent elementary circuits all traversed in the same element of time, we see that they con- stitute the polarization current already specified in § 62.
[There will be surface terms outstanding at sudden transitions, which are best evaluated independently by considering the limit of a gradual transition.
This analysis of the electric motions in the individual molecules is necessary for the correct determination of the mechanical forces acting on the system through its electrons (§ 65). But in treating of the electrodynamic pro- pagation we must ultinrately reijlace all magnetism by equivalent aggregate linear electric flux (p. 263) obtaining 8jdt (/', g', h'), where d/dt stands for djdt+jJd/dx + qdjdy + rdldz, as the total current of polarization, which thus represents the time-rate of separation ^Jtr unit volume of the positive and negative electrons in the molecules.] p. 105 line 4 from foot for dE'jdt read c~-dE'jdt. p. 112 after § 70 add a footnote
The potentials, combined in a different manner, may in fact be expressed as directly propagated from the sources of the aethereal disturbance, which are the stationary and moving electrons : see footnote to p. 227, added infra.
ADDENDA AND CORRIGENDA.
p. 116 line 7 add footnote
The true current is under all circumstances defined by the flux of electrons across fixed interfaces. At the same time it constitutes a volume dis- tribution (»j, i\, u\) 8t, as supra under p. 102.
p. l-i6 line 11 for diminished read increased and for deducted read added.
p. 147 line 4 for the second A read - A.
p. 154 in lines 8, 9 read
; but their level surfaces will linearly correspond to those of
p. 154 line 14 dele radial but.
p. 155 line 14 for force read potential.
p. 157 line 16 for force read distribution.
p. 176 line 10 for charges read volume-densities.
p. 201 add at end of § 129 See p. 354.
p. 222 lines 1, 5, 7 from foot, for c'- read C'~-.
p. 227 line 5 from foot for ?. read 4.
p. 227 add to footnote
Following Levi-Civita (Nnovo Cimento, 1897) the general result may be com- pactly and instructively expressed as follows. Let (Mj, r^, u\) denote the true electric current including the equivalent of magnetic whirls, if any, and let p be the density of true electrification. Let ii-^ {t) denote the value of ii-i at time t, so that «j {t - rjc) denotes its value at time t - rjc. Define I-\, Gj, Hj and 'I'^ by the relations
Then the state of the medium is expressible in terms of these new potentials by the same equations as give it in terms of the usual vector and electric potentials. These modified potentials here api^ear as transmitted without loss in symmetrical spherical sheets from the sources of the disturbance, which are the true electric fluxes and electrifications. Consider in fact an electron e suddenly displaced from A to B. When at rest at A it possessed an intrinsic field of electric force, derived from a potential c'-elPA : the shift of it sends out an electromagnetic pulse in a spherical sheet : after the passage of this pulse over P the intrinsic electric field at P remains that derived from the new potential c-ejPB. Then a further displacement of e may follow, producing a pulse which will again alter the intrinsic electric field at P as it passes over that point. By superposing the kinetic electric reaction of the pulse on the intrinsic field of the electron, and summing for all the electrons, the general ex- pression for the electric (or rather aethereal) force at P is immediately obtained; while the magnetic force is the curl of (Pj, Gj, H^). At foot of p. 226 the terms involving r~^ constitute the statical field of a doublet el; in the problem there considered they should be replaced by that of the single electron, namely by an electric force er~'^ along r.
p. 297 § 6 for first paragraph read
It has appeared (§ 1) that the result of the unequal drifts of the positive and negative ions, under the action of the electric force, is necessarily an electric current conveyed to an equal extent by positive and negative ions, together with a drift of the electrolyte in mass. In steady electric flow round a circuit made up of various electrolytes there would be a uniform current of this type in each of them, together with a drifting accumulation of the electrolyte continually being neutralized by steady backward diffusion. If a portion of the circuit is metallic, such an accu- mulation would also be produced in it unless (i) the positive and negative ions or electrons have equal mobilities, or (ii) their mobilities are both so great as to render the accumulation insensible.
p. 332 add footnote to line 5
The nucleus being of unknown constitution, its mobility has to be assumed ; that of its strain form is already secured.
■Ion. 19U1.
EELATIONS OF AETHEE TO MATTEE
CHAPTER I
INTRODUCTION
The scheme of this essay may be summarized as follows.
1. In the first section a historical account is given of the progress of experimental knowledge of the influence of the motion of matter through the aether on phenomena directly connected with that medium. For a long period this group of phenomena was purely optical ; the principal member of the group was the astronomical aberration of light, and the main interest of the others was centred in their use as tests of theories constructed to account for or explain aberration. The recent development of the connexion between the theories of electricity and optics has added to this domain the class of electro-optic and magneto-optic phenomena, and also various phenomena purely electric. The progress of theory is here passed under review alongside of the progress of experimental knowledge, and some attempt is made to compare the strengths and relative bearings of the positions that have been from time to time taken up by successive investigators. In this review brevity and interconnexion have been chiefly aimed at, for the subject has been historically a rather tangled one owing to the number of writers who have treated it and the variety and isolation of their standpoints: it will be seen that the estimates here given of the relative merits of the various partial theories difi'er somewhat in character from those generally current. For
L. 1
2 RELATIONS OF AETHER TO MATTER !
references to the earlier historical data Ketteler's treatise* has: proved most useful: the principal writings among these, and, all the later ones, have also been examined directly. 1
To this review is appended a general account of wave and ray propagation in moving media, which though originally written independently, necessarily follows very much the same course as the one given in Lorentz's memoir of ISSTf.
2. The second section developes the general theory of the relations between matter and aether, which is to form the basis of the treatment of moving material media. In it the electric and optical activity of the matter are assigned to the presence of electric charges associated with the material atoms : the complete scheme of electrodynamic and optical equations is then derived as a whole, on this basis, from the single founda- tion of the Principle of Least Action. The modifications which arise in the scheme owing to motion of the matter are on this hypothesis directly and definitely ascertainable, on the , supposition that the motion of the matter does not affect the quiescent aether except through the motion of the atomic electric charges carried along with it. It appears that the law of the phenomenon of astronomical aberration is fully verified, as are also all the other first order effects — mostly of a null character — of the motion of material systems, which experiment has established. This portion of the subject has been already profoundly treated by Lorentzj, by an analysis very different from the present one, but with ideas and results that are in the main in agreement with those here arrived at. In the treat- ment here given, the essential distinction between molecular theory and mechanical theory, and the principles involved in effecting the transition from the former to the latter, are carefully traced.
* ' Astronomische Undulatious-Theorie, oder die Lehre von der Aberration des Liehtes,' von Dr E. Ketteler. Bonn, 1873.
t H. A. Lorentz, ' de I'influence du mouvement de la Terre sur les jiheno- m^nes lumineuses.' ^ Archives Neerhmdaises xxi, pp. 103—176.
t ' La Th^orie Electroniagnetique de Maxwell, et son application aux corps mouvants,' Archives Neerlandaises xxv, 1892: ' Versuch eiuer Theorie der electrischeu uud optiscben Erscheinungen in bewegten Korpern,' Leiden, 1895.
INTRODUCTION 3
3, The third section enters on more speculative ground. It developes the exact consequences, as regards the influence of convection through the aether, which flow from the hypothesis that the atom of matter is constituted of an orbital system of equal primary electric point-charges (or electrons) and of nothing else : or, what comes in certain respects to the same thing in a mode of statement that may possibly be preferred, it assumes that the mass of each sub-atom is proportional to the absolute number of electrons, positive and negative, that it carries, and that the effective interatomic forces are entirely or mainly electric. From this basis a complete formal correla- tion is established between the molecular configurations of a material system at rest and the same system in uniform translatory motion, which holds good as far as the square of the ratio of the velocity of the system to the velocity of radia- tion. This correspondence carries with it as a consequence the null result, up to the second order, of the very refined experi- ments of Michelson and Morley on the influence of the Earth's motion on optical interference fringes. The correlation pre- supposes that the material atoms are independent systems that maintain their relative positions : thus in the simplest case, with which alone we are actually concerned, the material bodies are supposed to be solid, and the influence of the distantly wandering ions, if there are such, that convey electric currents, is left out of consideration as relatively negligible on account of the smallness of their number. In an appendix the mechanism of electrolytic conduction is scrutinized, primarily with a view to drawing conclusions by analogy as to the extent and character of the migrations of the ions in solid conductors : this discussion has however grown altogether out of proportion to its connexion with the present subject, and forms to some extent a connected theoretical account of electrolysis and the voltaic phenomena associated with it, such as the concentration of the electrolyte investigated by Hittorf, the electromotive forces of concentration investigated by von Helmholtz and Nernst, the electric osmosis of Quincke, and the nature of contact differences of potential.
4. It is generally held, chiefly on the ground of Lorentz's
1—2
RELATIONS OF AETHER TO MATTER
analysis, that the absence of any dependence between tl optical rotatory power of quartz and the direction in which tb light travels with regard to the Earth's motion, is in discrepanci with theoretical schemes like the present one which conside! the Earth to move through the aether without carrying thsl medium along with it. In the fourth section this question^! treated, with results however that prove to be in accord wit! the facts. As there seems to exist a feeling, — put in evidencj by Lorentz's conclusion above mentioned, which asserts a coni vective influence on rotatory power although he had showi- that there was none such on ordinary optical phenomena— tha| these rotatory phenomena are intrinsically of a class by themi selves, a view which may derive strength from their relativeb; slight or residual character, a discussion of the general natuni of the structural and the magnetic rotatory optical properties if given. An attempt to connect rotatory power with density fails, for reasons that are tentatively suggested. It is pointed out that the absence of any convective influence on the rotation' affords some independent evidence, in addition to Michelson'g result above stated, in favour of the effective validity of the view as to molecular constitution that is considered in the third section.
5. The fifth section treats of the subject of the radiation of material systems, the difficulties of which are not peculiar to any special theory of the connexion between aether and matter. The present theory, which attributes radiation to the oscillatory^ motions of electrons in the molecule, must give some acco.ujit. of why it is that molecules radiate only when they are violently disturbed ; and in particular, which concerns more closely our special subject, why it is that motion of bodies through the aether does not affect the amount or quality of their radiation, except after the merely kinematic manner of the Doppler effect. To carry out this purpose, an expression is found for the traiu^ of radiation and of general aethereal disturbance emanating from a single electron moving in any manner.
As connected with the molecular theory, and in fact demanded by it, a discussion is also given of the principles on which optical resolving apparatus is able to decompose the
INTRODUCTION 5
wholly tumultuous train of disturbance which constitutes ordinary white light into an orderly series of trains of simple harmonic waves. It is held (with Lord Rayleigh) that the train of impulses or vibrations which constitutes the Rontgen radiation would be similarly resolved into simple wave-traius of very high frequencies if we had fine enough apparatus to bring to bear upon it ; though the molecular structure of ordinary matter is probably too coarse to be sensibly effective for this purpose. Reasons are given for the view, opposed to what is now sometimes perhaps too generally stated, that counting the number of the succession of interference bands, that can be produced with the light from the whole of a sharp bright line in the spectrum of a gas, enables us to form an estimate of the degree of regularity of the vibrations of the individual mole- cules which emit the radiation : a result which is of importance for both optical and molecular theory.
SECTION I
CHAPTER II
HISTORICAL SURVEY
6. The phenomenon of the astronomical aberration of Ji^ was discovered by Bradley as the outcome of an effort, con- ducted with unusual care, to detect traces of annual parallax| in certain stars which passed near his zenith, and so wer( amenable to accurate measurement. His observations exhibitec displacements in the position of each star, which had thej expected period of a year ; but instead of being towards the i Sun they were towards a perpendicular direction, that of the i Earth's motion in its orbit, while they followed the same law of the sine of the inclination as parallax would do. After many attempts to coordinate his observations, the clue to the aber- rational method of representing them was suggested to Bradley, it is said, by casual observation of a flag floating at the mast- head of a ship ; when the ship changed its course, the flag flew in a different direction. The analogy is rather more direct when it is the drift of the clouds that is the object of remark ; the apparent direction from which they come is different from the real direction when the observer is himself in motion relative to the air that carries them. So, the observer of the star being in motion along with the Earth, the apparent direction in which the light from the star appears to him to come may be expected to be different from its real direction ; thus leading to the usual elementary representation of the
CHAP. Il] HISTORICAL SURVEY 7
aberration as a phenomenon of relative motion. This explana- tion is absolutely valid if the light is something which travels in definite rays with finite speed, itself undisturbed by the motion of the Earth : on a corpuscular theory it thus requires that the luminous corpuscles are not sensibly affected by the Earth's attraction : on the undulatory theory it involves either that the luminiferous aether is not disturbed at all by the Earth's motion through it, or else that some special adjustment of its motion holds good which gives the same result. The explanation of the phenomenon of the aberration of light thus immediately opens up the whole question of the disturbance of the aether by the motion through it of material bodies like the Earth, and also of the manner in which the reflexion and refraction of light in our observing instruments is affected by their motion along with the Earth. It is merely one particular result, — more prominent because a positive result — in the field of the mutual influence of aether and moving matter.
7. It occurred to Arago, reasoning on the lines of the cor- puscular explanation, that inasmuch as the velocity of light is different in glass from what it is in vacuum, the aberration_of its path arising from the Earth's motion would also be different in the glass, and therefore the optical deviation caused by a glass prism would vary according as the light traversed it in the direction of the Earth's motion or in the opposite direction. With the achromatic prism which he employed for testing this conclusion, he calculated that this difference might be as much as a minute of arc. The outcome of the experiments showed no difference at all. Arago worked with star light for which the Doppler effect due to relative motion would make a real difference, excessively minute however and beyond his obser- vational means : with light from a terrestrial source which (as Fresnel remarked) would do equally well for his test, the differ- ence would be absolutely null. The significance of this result, as against the then current explanation of aberration, on corpuscular ideas, was fully realized by Arago : and he com- municated the facts to Fresnel with a view to eliciting whether there was anything more satisfactory to be adduced on the basis of the wave theory, which he was then engaged in developing
8 THE ABERRATION OF LIGHT [SECT. I
with the support of Arago's powerful advocacy. The pheno- menon to be accounted for w^as that the motion of thai Earth does not affect the laws of reflexion and refraction of' lio-ht. In Fresnel's reply*, which is one of the fundamental documents on the present subject, he pointed out that a simple^; answer was possible, namely to assume that the surrounding \ aether is carried along completely by the Earth so that all ' relative phenomena would be the same as if the Earth were at rest : but he went on to say that this view could not be enter- tained on account of the facts of astronomical aberration, of which he could form no intelligible conception except on the hypothesis that the aether remained absolutely stagnant as the Earth moved through it. On this latter hypothesis the velocity of light outside a transparent body must have the normal value : and it was an easy problem to find whether it was possible for any law of modification of the velocity of light inside the body, arising from its motion, to make the laws of refraction and reflexion relative to the moving body the same as for matter at rest, as Arago's experiment required. It appeared that there is such a law, the conditions being all satisfied if the absolute velocity of light inside a transparent medium of index /x is increased by the fraction 1 — yu,~- of the velocity of the medium resolved in its direction. This supposition, adopted on tlie** above grounds by Fresnel, keeps the paths of the rays relative to the moving bodies unaltered, and at the same time satisfies the facts of aberration. The attempt made by Fresnel to provide for it a dynamical foundation suffers from the same kind of obscurity as did his later dynamical theory of crystalline re- fraction: and though the sul)sequent views of Boussinesq and Sellmeier, on the part played by the matter in the mechanism of refraction and dispersion, allow a valid meaning to be read into Fresnel's explanations, yet they perhaps form no very essential part of his achievement in this field. Afterward Sir George Stokes showed in detail that Fresnel's hypothesis not only left the relative paths of rays unaltered, but the phenomena of interference as well, some of which had been urged against it by Babinet.
* See Appendix D.
CHAP. Il] HISTORICAL SURVEY 9
The result obtained by Arago suggested a wide field of experimental inquiry as to whether other optical phenomena as well as refraction were independent of the direction of the Earth's motion through space. In most cases the experimental test is very precise and delicate ; for the apparatus exhibiting the optical effect has only to be installed in the most sensitive manner possible, and note taken as to whether the gradual change of absolute direction of._the light passing through it, arising from the Earth's movement of rotation, causes any diurnal inequality in the results. The negative results of theory have gradually been extended, by special investigations, to other optical phenomena, such as dispersion and crystalline interference, as these were successively found by experiment to be uninfluenced by the Earth's motion in space. It will be seen that the modern or electric view of the aether supplies a succinct dynamical foundation for the whole matter.
8. Long before Arago's time it had occurred to Boscovich, reasoning from Bradley's original point of view, that inasmuch as the velocity of light in water is different from what it is in air, the aberration produced by the Earth's motion in the apparent path of a ray travelling through water should be different from the normal astronomical amount : he suggested the use of a telescope with its tube filled with water to find out by star observations whether this is the case, in the expectation that the line of collimation would be different, in order that the relative rays in the water should focus on the cross-wires, from what it would be if the interior of the tube contained only air. In recent times Sir George Airy has actually had such an instrument temporarily installed at the Greenwich Observatory : he has found that observations with it, continued over a con- siderable time, gave the ordinary value of the constant of aberration, the different aberration of the ray in water being thus compensated by a modification of the ordinary law of refraction on the passage of the light into that moving medium. This experiment had already been discussed by Fresnel in his letter to Arago, with the remark that there is no occasion to complicate the result by aberration, as a terrestrial object might equally well be focussed on the cross- wires of the instrument
!
10 THE ABERRATION OF LIGHT [SECT. I
placed transverse to the direction of the Earth's motion ; and that the observations might even be carried out by a microscope, reversible by hand, sighted on a bright point attached to its frame. It was pointed out by Fresnel that Wilson had shown that on the corpuscular theory no effect was to be expected : and he added a demonstration that the same was the case on his own view of the undulatory theory, thus predicting on a consensus of all points of view a negative result.
9. The theory was next taken up, from the undulatory standpoint, by Cauchy, who preferred the first of Fresnel's alternatives, that the Earth in its orbital motion pushes the aether in front of it so that the portions near the surface travel along with the Earth, as he was unwilling to admit that the heavenly bodies could move through the aether without dis- turbing it at all. He pointed out that astronomical aberration was then to be explained, not probably by any effect of changed aethereal elasticity or inertia, but merely by a kinematic slewing round of the advancing wave-fronts (or rather absence thereof) *j owing to the translatory motion of the medium in which the waves are propagated. The disturbance of the aether itself owing to the motion of the Earth he was prepared to regard as the source of the electric and cognate phenomena associated with that body. This mere preference of Cauchy's did nothing towards removing Fresnel's difficulty as to how such a motion of the aether is to be imagined as exactly adjusted so as to involve the correct amount of aberration in accordance with Bradley's law : but the view subsequently became a real theory in the hands of Sir George Stokes. That physicist had just had cause, in his hydrodynamic researches, to analyze the differential change of form, arising from the state of motion in a fluid medium around any given point, into pure strain made up of three superposed elongations, combined with pure^ rota- tion: and it became clear that if the latter component is absent in the aether, so that the motion of the aether is differ- entially irrotational, the advancing wave-fronts will not be slewed round at all, and therefore the waves will travel througlT"' space in straight lines as if the aether were at rest. When these rectilinear waves get into the resfion of aether imme-
CHAP. Il] HISTORICAL SURVEY ]1
diately around the observer, which is carried on with him, they will be affected relative to him with the full aberrational change of direction arising from his motion, just as a moving corpuscle would be. Now the irrotational quality of aethereal motion thus pointed to, is, by Lagrange's fundamental hydrodynamical theorem, the characteristic of the motion of frictionless fluid which has been originally at rest : thus the material for a physical theory lies at hand. The aether is, as regards slow motions in bulk, simply assumed to have the properties of frictionless continuous fluid substance, while for the excessively rapid small vibrations of light it has solid elastic quality. The question remained how far these two sets of qualities can coexist in the same medium: an affirmative answer was de- fended, or rather illustrated, by an objective appeal to the actual properties of a substance such as pitch, which flows like Avater if sufficient time is allowed, while at the same time it can be moulded into an efficient tuning-fork for small vibrations as frequent as those of sound. There appears to be good ground for demurring against the mutual consistency of the properties imputed to a simple, permanent, and flawless medium like the aether being settled by an appeal to the approximate behaviour of a highly complex and viscous body like pitch : the principle that is involved can however be expressed in a purely abstract manner. If any term in the analytical dynamical equations of the aether is made up of two parts, so as to be of type such as cm + hdhijdt- where u represents displacement, then when h is very small compared with a the first part au will practically represent the term for slow motions, while on the other hand for simple vibratory motion of excessively high frequency ?i, dhi/dt- being then equal to (i>4./2?r)- u, the second term is the all-important one. The objection to this kind of explanation, which substitutes a very close approximation for the exact term, is that we have actually to provide in the aether for a transparency which is adequate to convey the light of the most distant stars, which points rather to exact abstract mathematical relations than to complex and approxi- mate physical laws of elasticity.
10. At the same time Sir George Stokes expressed his
12 THE ABERRATION OF LIGHT [sECT. I
belief that it would not do to actually take the aether to be an ordinary fluid, on the ground that this ideal motion of irrota- tional quality would then be unstable. In a subsequent note {Phil. Mag. 1848) he advanced as proof of this instability the fact that the mathematical solution for the steady motion of a sphere through a viscous fluid, w^hich he had just obtained, is the same however slight may be the degree of viscidity of the fluid. Now an irrotational motion calls out "iil^ viscous reaction throughout the mass, and therefore satisfies the conditions of viscous as well as of perfect flow : but there is one circum- stance which destroys its claim to be a solution in the former case, namely the presence of slip at the surfaces of the solids. If the surfaces of the solids were ideally frictionless this would not matter : but if when the irrotational flow has there been fully established, the actual frictional character of the surface were restored, laminar rotational motion would spread out from each surface in the same manner as heat would spread out by diffusive conduction from a hot body, until a new state of steady motion would supervene. The solution of Stokes shows (as is also clear from general principles) that however small the viscosity, this new steady state is wholly different from the ideal irrotational steady state belonging to mathematical absence of viscosity and friction : and it might appear to follow that this state is, not precisely an unstable one, but rather one which could not exist at all in the fluid. The term unstable is however appropriate because, if the solids are impulsively started into their steady state of motion, the initial state of motion of the fluid will {assuming that there is no such thing as impulsive friction*) be the irrotational one, which will gradually be transformed by diffusion of vortex motion from the surfaces at a rate which is the slower the less the viscosity of the fluid. This conclusion follows as a special case of Lord Kelvin's general dynamical principle that when a material system is impulsively set into motion by imposing given velocities at the requisite numbeT^oT
* The direct proof from the hydrodynamical equations is not however limited in this way, if the law of impulsive viscosity may be assumed to be linear.
CHAP. Il] HISTORICAL SURVEY 13
points or surfaces, (namely in this case given component of velocity normal to the boundaries) the state of motion instan- taneously assumed by it is that one for which the kinetic energy is least, which is easily shown to be the irrotational one in the case of a liquid.
As Sir George Stokes was not disposed to admit that the aether could pass freely through the interstices of material bodies in the manner required by Fresnel's views, and as any other theory of its motion which could be consistent with the fact of astronomical aberration required irrotational flow, an explanation of the limitation to that flow had, he considered, to be found. He pointed out that the existence of tangential stress depending not alone (like viscosity) on relative velocities, but also (like elastic stresses) on relative displacements, would make the flow irrotational ; for any deviation from irrotational quality would now be propagated away not by diffusion but by waves of transverse displacement, and the coefficient of Jthe elastic part of the force, and consequently the velocity of this propagation, may be assumed so great that the slightest beginning of rotational motion is immediately shed off and dispersed. This chain of argument, that motion of bodies disturbs the aether, that aberration requires the disturbance to be differentially irrotational, that this can only be explained by the dispersion of incipient rotational disturbance by trans- verse waves, and further that radiation itself involves transverse undulation, he regards as mutually consistent and self-support- ing, and therefore as forming distinct evidence in favour of this view of the constitution of the aether*. The coexistence of fluidity on a large scale with perfect elasticity on a small scale he illustrates by the ordinary phenomena of pitch or glue, passing on to a limit through jelHes of gradually diminishing consistency until perfect fluidity is reached : the chief difficulty here is (as already mentioned) that absolute mathematical
* It would thus appear that the slip at the surface of the moving solids, which is offered as a decisive objecti^ to Stokes' view by Lorentz, is not really- fatal to such a view of aberration, teken by itself, except in so far as it leads to continual radiation from the surface of the moving body and therefore to resistance to its motion.
14 THE ABERRATION OF LIGHT [SECT. I
transparency of the aether is replaced by approximate trans- parency, such as would involve ultimate decay of all structures existing in it.
11. The problem of the relative motion of the Earth and the aether was treated by Clerk Maxwell in 1867, in a letter to Sir W. Huggins which has been incorporated in the funda- mental memoir of the latter on the spectroscopic determination of the velocity of movement of stars in the line of sight*. It is there pointed out that there are two independent subjects for examination. The Doppler alteration of the period of the light from a star is quite definite, and independent of the special details of the form of undulatory theory that may be adopted. But there is a second question as to whether the index of refraction depends on the orientation of the ray with reference to the direction of the Earth's motion, in which the observer and all his apparatus participate: this involves the physical nature of the undulations : here, as Fresnel had already remarked, the sources of light may just as well be terrestrial as astronomical. According to Arago's original experimental result, which had been closely tested by a more delicate arrangement by Maxwell himself working with homogeneous light, some years before this time, there is no influence on the index of refraction arising from the Earth's motion. As refraction depends solely on retardation in time owing to the smaller velocity of propagation in the refracting medium, the relative retardation must therefore be unaltered by the Earth's motion. If V be the velocity of a ray in air, and v the velocity of the aether in air relative to the observer, and if F' be the velocity of the same ray in a dense medium and v' the velocity of the aether in that medium relative to the observerf , then across a thickness a of this medium the light is retarded with respect to air by a time
V' + v' V+v'
* Phil. Trans. 1868, p. 532.
t This is Maxwell's phrase, no doubt interpreting Fresnel : on a wider and more modern view r' is the amount by which V is altered owing to the motion of the aether relative to the medium.
CHAP. Il] HISTORICAL SURVEY 15
which is equal to
that is to
v\v' v[v' W"^F^vT^"7^;^'
As this is, by the experimental evidence, to be independent of V and v' to the first order, we must have v'/v= V'-jV-; or, expressed in words, the effects of the Earth's motion on the velocities of the ray relative to the observer in the two media are proportional to the squares of the ray- velocities for the ray under consideration. In the moving refracting mediuni ^e absolute velocity of the ray is therefore increased hy v — v\ that is by -v (1 — V'^jV-), where v is the velocity of the medium in the direction of the ray. When the medium is isotropic, VjV is equal to the index of refraction /jl, thus the alteration of the velocity is v{\ — ix~-), as Fresnel originally found.
According to the ideas underlying Fresnel's general optical theory, refraction depends on change of density of the aether. Thus the density of the aether in the refracting substance would be proportional to ijl- : and if the aether is imagined as flowing across the refracting substance in its relative motion, its velocit}^ in that substance must by the equation of conti- nuity be /x~^ of its velocity in air outside. Thus on the hypothesis that the change of velocity is solely due to a convection by the moving aether, we are led from Fresnel's general notions to the same law as Arago's experiments demands*.
In the same place Maxwell remarks on the great instru- mental difficulty, and also the absence of confirmation, of the
o
experiments of Fizeau and Angstrom indicating displacement of the plane of polarization by passage through a pile of glass plates and by diffraction respectively, depending on the orientation of the apparatus with regard to the direction of the Earth's motion. /
* This remark is given by Maxwell. I do not find it in Fresnel's letter to Arago, but it occurs in part in the paper by Sir G. Stokes, Phil. Mag. 1846.
16 THE ABERRATION OF LIGHT [SECT. I
12. It results from very various experimental investigations some of which are mentioned above that, with a very doubtful but unique exception in the case of Fizeau's experiments on piles of glass plates, the most varied optical phenomena, whether of ray paths or of refraction, dispersion, interference, diffraction, rotation of plane of polarization, have no relation .. to the direction of the Earth's motion through space, though! for many of them the test has been made with great precision, j The most obvious conclusion from this consensus of evidence taken by itself would be the view that the Earth's motion carries the aether completely along with it, and that all the relative optical and other phenomena are therefore just thfi, same as they would be with both the Earth and the aether at rest. Such a view is also very temptingly suggested by the absolutely negative result, up to the second order, of the Earth's motion on the Michelson interference experiment. If then we could assume that the Earth's motion produces flow, differentially irrotational according to Sir George Stokes' criterion, in the surrounding aether, but such that in all regions near the Earth's surface, up to the greatest distance at which we can explore, the aether is practically carried along bodily with the Earth, the requirements both of astronomical aberration and of the mass of negative optical results would be fully satisfied. But here we are met by various difficulties. If we assume that the aether around the Earth near its surface is carried on by the Earth as that body traverses its orbit, and also assume that at a great distance the aether is at rest, these states of motion cannot be connected without discontin- uity by any possible irrotational motion of the intervening aether. The irrotational motion set up by the motion of the Earth and the surrounding shell of aether, supposed attached to it, is the same as would be set up by a moving solid in ideal frictionless liquid : the continuity of normal flow can be pre- served, but there must be tangential discontinuity (slip) either at the boundary of the solid or somewhere else : this is the case whether incompressibility of the aether is assumed or not, the two sets of conditions continuity of normal flow and continuity of tangential flow being more than can be simultaneously
CHAP. Il] HISTORICAL SURVEY 17
satisfied. This way of surmounting the discrepancies is there- fore, on the very threshold of our present wider survey, ilkisory. Were it not so, it would only be necessary to proceed a step further in order to encounter fresh difficulties. If the aether were carried on bodily by the Earth, we must assume that the aether very near a mass moving along the Earth's surface is at any rate partially carried along by that mass. This point has been tested directly with great precision by Lodge f, who tried to detect whether the aether between two whirling steel discs partook to any extent in their motion ; and the result has been decisively negative. The only possibility of escape from this result, that the aether is not carried on by the Earth's motion, would be in an assumption that the large mass of the Earth controls wholly the motion of the aether in its neighbourhood somehow as it does gravitation, so that the smaller mass of the rotating discs is inoperative in comparison. In any case the former difficulty remains decisive : we might indeed be tempted to replace the absolutely irrotational motion of the surrounding aether, involving surfaces of slip, by very slightly rotational motion such as would evade all tangential slip : but the law of astronomical aberratioi^ would thereby be upset, since the smaller the rotation thus imposed the greater the distance to which it must extend, while the resulting aberration is proportional to these quantities jointly*. A hypothesis that would allow the aether to be moved in any degree by material bodies passing across it thus has small cjiance of correspondence with the body of ascertained optical facts.
We are therefore thrown back on Fresnel's view that the aether is not itself set in motion by the movement of material systems across it, or, in terms of the simile of Young, that it passes through the interstices of material bodies like the wind through a grove of trees.
t ' Aberration Problems ' Phil. Trails. 1893 a.
* It has been suggested by Des Coudres, as a way out of the difficulty, that the aether is possibly subject to gravity : but that would merely produce a balancing hydrostatic pressure without altering the irrotational character of the motion.
18 THE ABERRATION OF LIGHT [SECT. I
13. The next important theoretical contribution to our subject is implicitly contained in §§ 600, 601 of Maxwell's "Treatise on Electricity and Magnetism" (1872). It is there verified, by direct transfoi-mation, that the type of the equations of electromotive disturbance is the same whether they are! referred to axes of coordinates at rest in the aether or to axes which are in motion after the manner of a solid body. Thei principle is here involved, as FitzGerald-f- was the first to point out, and as was no doubt in Maxwell's own mind considering his recent occupation with the subject, that in treating of the electromotive disturbance which constitutes light we are per- mitted to make use of axes of coordinates which move along with the Earth without having to alter in any way the fofrn of the analytical equations. This statement covers as a special case Bradley's law of astronomical aberration. It also directly includes in its entirety the principle of Arago and Fresnel that the laws of geometrical optics are not affected by the Earth's motion: it ought therefore to involve as a consequence FresneTs expression for the change of velocity of radiation produced by motion of the material medium which it traverses. The latter question was examined directly from Maxwell's analytical equations by J. J. Thomson* with a result different from Fresnel's, namely, that the acceleration of velocity is always half that of the moving material medium, being the same for all kinds of matter. This discrejDancy is one of several which indicate that for extremely rapid disturbances like optical waves, the analytical scheme of Maxwell does not sufficiently take into account the influence of the material medium on the propagation. A contradiction of some kind is also suggested by the circumstance that Maxwell's theorem does too much by making the optical properties independent of uniform velocity of rotation of the material medium, as well as of uniform velocity of translation ; we shall see (§ 23) that the possibility of exact independence in both respects is negatived by the general nature of rays. The necessary amendment of the scheme of Maxwell has been independently arrived at lyy more than one writer, but somewhat earliest in point of time
t Tra7is. Royal Dublin Society, 1882.
* Proc. Cmnh. Phil. Soc. v., 1885, p. 250.
:hap. ii] historical survey 19
Dy H. A. Lorentz; and it involves the general electrodynamic ;onsiderations, including the discrete distribution of electricity imong the molecules of matter, on which the present essay is )ased. Shortly after Lorentz the subject was taken up from I similar point of view by von Helmholtz, primarily in relation io the theory of optical dispersion : but his equations, derived rom a difficult abstract procedure in connexion with attempted generalizations of the principle of Least Action, were soon bund to be at fault in the matter of moving media just as nuch as was the original scheme of Maxwell. Possibly there- )y incited, von Helmholtz considers directly the question of notion of the aether in his last published memoir: he finds, IS Hertz had done some years before, that the mechanical brce as given by the equations of Maxwell cannot by itself ceep the aether in equilibrium if we suppose this force to act )n it as well as on matter: and on the assumption that the lether is fluid as regards movements arising from extended listurbance, and of very small density, he obtains differential ;quations for the determination of the steady state of aethereal notion that must on that hypothesis exist in an electrodynamic ield. The existence of any finite motion of this sort, unless t is very minute, has been negatived by the elaborate experi- nents of Lodge, and also by more recent observations on the lame plan by Henry and Henderson which were inspired from Toii Helmholtz's theory.
14. Quite recently a general summary of the state of the [uestion of the mutual relations of aether and moving matter las been published by W. Wien*, as a guide to a discussion of ;he subject at the annual meeting of the German Scientific Association. He there works out some special cases of von ielmholtz's theory just mentioned, arriving at the result that f the density of the aether is absolutely null there can exist a iteady translational motion of electric charge through the lether which will not involve any disturbance of that medium, vhile if the density is very small the disturbance thus involved vill be very slight : but in motions not steady, for example the miform separation of the components of a stationary electric
* Wied. Annalen Ixv., July 1898.
20 THE ABERRATION OF LIGHT [SECT. E
doublet, infinite velocities of disturbance of the aether will enter at the very beginning of the motion, so that the steady state cannot be originated.
On the other hand, the abstract theory to be here giveni may be translated into a concrete scheme which identifies! electrodynamic energy with the translatory kinetic energy of| the aether considered as possessing inertia : to make the aetheri remain practically quiescent under all conditions it is thenj necessary and sufficient to take its inertia to be sufficiently great: in fact if this were not secured, the electrodynamic equations instead of being linear would involve the very great complication of non-linear terms with which we are familiar in theoretical hydrodynamics.
In summing up at the end of the above-mentioned essay,' Wien formulates three outstanding objections to the hypothesis ' of a quiescent aether ; ~~~
(i) the observed absence of any magnetic efiect of the motion of electrically charged bodies carried along by the Earth,
(ii) the absence of any influence of the Earth's motion on the optical rotatory property of quartz,
(iii) Fizeau's experiment, in which he found some evid- ence for changes, arising from the Earth's motion, in the displacement of the plane of polarization of light produced by passage through a pile of glass plates.
As regards these objections, the first appears to be a point in favour of the theory instead of against it (§ 40 infra) : the second is based on a theoretical investigation of Lorentz, which appears to be at fault (Ch. XIII.) so that the result is again in favour : while the conclusion in the third case was regarded as doubtful by Fizeau himself on account of the extreme difficulty experienced in excluding disturbing causes (a doubt which has been shared by most authorities who have since examined the matter, including Maxwell and Rayleigh), and the experiment has not been repeated.
I
BAP. Il] HISTORICAL SURVEY 21
Electrodynamic view of the Aether.
15. The astronomical aberration of light is one of the small roup of phenomena in which the reactions between matter ad aether depend sensibly on the state of motion of the latter. Disturbances originated in the aether are equalized nd smoothed out with such great speed, that the aether-field round a body, which is moving with any attainable velocity, ; practically at each instant in the same equilibrium condition 3 if the body were at rest : it is therefore only in the case f very rapidly alternating phenomena such as radiation that lere is any practical occasion to pass beyond a mere theory of mvection of aethereal effect along with the molecules of the latter. It is owing to this circumstance that the electro- ynamic theories of Ampere and Weber represented so well the hole range of phenomena then open to experiment, even to le extent of giving in Kirchhotf's hands the correct velocity ;hat of radiation) for the transmission of electric waves of very igh frequency guided along a wire : and that, as regards the eeper questions of propagation of electric effect in time, leory has been, chiefly in Maxwell's hands, uniformly so ir in advance of the means of verification.
The logical validity of the older electrodynamics was con- ned to systems of uniform currents streaming round closed athsj and all investigations purporting to deduce from experi- lental data expressions for the electromotive forces induced 1 open circuits, or for mechanical forces acting on separate ortions of circuits carrying currents, were necessarily illusory •om the fact that such portions were practically unknown 3 separate independent entities. The new departure instit- ted by Maxwell came, when expressed mathematically, to a :atement that dynamically all electric discharges are effect- ^ely of the nature and possess the properties of systems of losed currents, being completed when necessary by so-called isplacement-currents in free space and in dielectric media ; in Lct that the consideration of the electrodynamics of unclosed ircuits never arises. That theory, as left by its author, works lit by adapting the established Amperean theory of closed
22 ELECTRODYNAMICS OF MOVING MATTER [SECT. I
currents to the new ideas; and there still remains the same ambiguity in respect to mechanical forces on portions of flexiMe or extensible current-circuits*. There is however no ultimate ambiguity as regards electromotive phenomena in bodies at rest, the equations of the theory sufficing to eliminate the arbitrary element that initially must be introduced, and thus to give a definite determination of the electric force at each point of space. New difficulties, of practical importance only in the theory of radiation, occur when the material medium which carries the current is in motion : and the theory for that case was left in the form of a first approximation, which assumed that the aethereal disturbance was simply convected by the moving matter, that being amply sufficient for ordinary electrodynamic applications. The dynamical methods were however sketched by Maxwell which would have to be employed to work out a more definite scheme of the relation of aether to the matter at rest in it or moving through it : and quantities of dynamical origin or suggestion, such as the vector potential of electric j currents, which have sometimes been considered so great a complication by subsequent writers as to justify their summary abolition, turn out in fact to be of the essence of a more thorough analysis.
16. The dynamical scheme which thus in Maxwell's hands furnished a formulation of the electrodjTiamics of material systems at rest in the aether, completely effective except as regard the material mechanical forces acting on the matter carrying the currents, was one of continuous differential analysis: the matter was taken as .simply modifying, where it existed, the effective constants in the formula for the spacial distribu- tion of electric energy : when the aether did not move with any finite speed or the matter move across it, there Avas no pressing occasion to separate the energy into a part belonging to and propagated by the aether and a part attached to the molecules of matter. The theory, at the stage at which it was^ left by Maxwell, being a theory of complete electric circuits,
* Cf. Phil. Tranx. 1895 a, pp. 697—701, for a demonstration that the ponderomotive forces cannot be directly deduced from a single energy-function without the aid of molecular analysis.
CHAP. Il] HISTORICAL SURVEY 23
the total current was a continuous streaming flow; there proved t<i be no necessity, in the case of systems at rest, for keeping distinct the current of conduction, the current arising from changing electric polarization in a dielectric substance, and the displacement current belonging to free aether apart from matter altogether: the only hypothesis he required was that there is an aethereal current of such amount as to complete into a single circuital stream all the t}^es of true electric flux which are associated with matter. These distinctions however become essential as soon as the theory is to take cognizance of the motion of the matter, especially in the domain of radiation where a mere equilibrium theory, contemplating the convection unaltered of its electric field along with the matter, is not a valid approximation. Then convection, relative to the aether, of electric charge and of dielectric jjolarization, contributes to the total current, as well as the change of aethereal elastic displacement and of material polarization. The problem thus presents itself in the form of two media, the aether and the matter, each with its own motion, but both occupying the same space ; and some idea has to be formed of the interconnexions b}' which they influence each other. If Ave treat them both by the methods of continuous analysis, the only way open is to assume the most general linear relations between the two sets uf variables representing the properties and states of the two media, and subsequently try to reduce the generality by aid of experimental indications. This is a well-tried course of pro- cedure in abstract physics, and has been very effective under simpler and more easily grasped conditions : but even if suc- cessful it could hardly help us to mentally realize the connexion between aether and matter, while on the other hand the jihilosophical objections to filling the same space with several different media have been widely felt and emphasized.
17. Possibly the only sound procedure is the one which recommends itself on purely philosophical grounds. From remote ages the great question with which, since Newton's time, we have been familiar under the somewhat misleading antithesis of contact vet'sus distance actions, has engaged specul- ation,—4iow_i_tj.s that portions of matter can interact on each
24 ELECTRODYNAMICS OF MOVING MATTER [SECT. I
other which seem to have no means of connexion between them. Can a body act where it is not ? If we answer directly in the negative, the spacial limitations of substance are to a large extent removed, and the complication is increased. The simplest solution is involved in a view that has come down from the early period of Greek physical speculation, and forms one of the most striking items in the stock of first principles of knowledge which had been struck out by the genius of that age. In that mode of thought the ultimate reality is trans- ferred from sensible matter to a uniform medium which is a plenum filling all space : all events occur and are propagated in this plenum, the ultimate elements of matter consisting of permanently existing vortices or other singularities of motion and strain located in the primordial medium, which are capable of motion through it with continuity of existence so that they can never arise or disappear. This view of physical phenomena, which was no doubt suggested by rough observation of the comparative permanence and the mutual actions of actual whirls in water and air, was quite probably, even at that time, not the mere idle philosophizing which has sometimes been- supposed. It at any rate involves the fundamental consequence' that the structure of matter is discrete or atomic — that con- trary to a priori impression matter is not divisible without limit : and it perhaps enables us to form some idea of the line of development of those views on the constitution of matter which, as Democritus and Lucretius described them, were con- siderably ahead of anything advanced in modern times until the age of Descartes and Newton. The same doctrine was prob- ably the ideal towards which Descartes was striving when he identified space and matter, and elaborated his picture of the Solar System as a compound vortex. In Newton's cautious hands, the relation of material atoms to aether is not dealt with : his establishment of an exact law of gravitation indeed originated the school of action at a distance, which held bluntly that matter can be considered as acting where it is not, and whose influence lasted throughout the sevententh century through Boscovich and the French astronomers and mathema- ticians, until the time of Faraday. This doctrine of the finality
CHAP. Il] HISTORICAL SURVEY 25
of action at a distance was however strongly repudiated by Newton himself, and hardly ever became influential in the English school of abstract physics represented by investigators of the type of Cavendish and Young. More recently, the following out into modern developments of the mere idea of continuous transmission of physical actions gained for Faraday a rich harvest of fundamental experimental discoveries : while the general results obtained by von Helmholtz in the abstract theory of fluid motion have enabled Lord Kelvin to reconstruct on a precise scientific basis the notions of Leucippus and Descartes on the relation of matter to aether**.
Meanwhile, irrespective of such general cosmical views, the development of electrical theory itself has been steadily tending to an atomic standpoint. It has been noted by Maxwell, and was afterwards very fully enforced by von Helmholtz, that the interpretation of Faraday's quantitative laws of electrolysis could only be that electricity is distributed in an atomic manner, that each atom of matter has its definite electric equivalent, the same for all kinds of atoms : and even the expressive phrase " an atom of electricity " was imported into the theory by Maxwell. The only difficulty in this mode of formulation related to the mechanism of transference of these atomic charges or electrons from one molecule of matter to another. The order of ideas to be presently followed out will however require us to hold that the atomic charge is of the essence f 7 of each of the ultimate subatoms, or as we may call them protions of which an aggregation, in stable orbital motion round each other, go to make up the ordinary molecule of matter : so that the transference of electric charge will involve transference or interchange of these constituent protions them- selves between the molecules, that is it will always involve chemical change, as Faraday held on experimental grounds must be the case.
** Of. Appendix D. It may well be that too favourable a view is taken in the text of the earlier physical atomic theories, which up to the period of Lord Kelvin's vortex atoms could only have been hypothetical speculations.
tt Cf. Sir Humphry Davy, in Appendix D.
26 ELECTRODYNAMICS OF MOVING MATTER [SECT. I
18. The fluid vortex atom of Lord Kelvin faithfully repres- ents in various ways the permanence and mobility of these subatoms of matter : but it entirely fails to include an electric charge as part of their constitution. According to any aether- theory static electric attraction must be conveyed by elastic action across the aether, and an electric field must be a field of strain : hence each subatom with its permanent electric charge must be surrounded by a field of permanent or intrinsic aethereal strain, which implies elastic quality in the aether instead of complete fluidity : the protion must therefore be in whole or in part a nucleus of intrinsic strain in the aether, a place at which the continuity of the medium has been broken and cemented together again (to use a crude but effective image) without accurately fitting the parts, so that there is a residual strain all round the place.
The assumption of elasticity of some kind in the aether is of course absolutely essential to its optical functions : and the elucidation from the optical phenomena, as a purely abstract problem in analytical dynamics, of the mathematical type of this elasticity, was acomplished in 1839 by MacCullagh* in an investigation which may fairly claim to rank amongst the classical achievements of mathematical physics. The type of elasticity which he arrived at was one wholly rotational, so that the aether would be perfectly fluid for all motions of irrotational type, but would resist elastically, by a reacting torque, any differential rotations of the elements of volume, somewhat after the manner that a spinning fly-wheel resists any angular deflexion of its axis. Here then we have the specification of an ideal medium that would behave as a fluid to solid bodies moving through it, because its irrotational motion would be precisely the same as that of a fluid in the corresponding circumstances : it would not resist the motion of such solids any more than the aether resists the motion of the heavenly bodies or of material masses generally : moreover vortex rings could permanently exist in it and persist according to the well- known laws of abstract hydrodynamics. But these tempting
* 'An Essay towards a Dynamical Theory of Crystalline Reflexion and Refraction,' Trans, li. I. A., vol. xxi : Collected Works, p. l-lo.
i
CHAP. Il] HISTORICAL SURVEY 27
indications must be put aside in favour of a track lying in a rather different direction, the ultimate element of material constitution being taken to be an electric charge or nucleus of permanent aethereal strain instead of a vortex ring.
A view of the constitution of matter, which proves to be sufficient over an extensive range of physical theory and must not be made any more complex until it proves insufficient in some definite feature, asserts that the molecule is composed simply of a system, probably large in number, of positive and negative protions in a state of steady orbital motion round each other. Nothing has yet been done directly to examine 1 how wide a field of possibility of different types of molecules and molecular combinations is thus opened up : but it is easy to recognize that the range is more extensive than would be offered by a Boscovichian system of attracting points, or of attracting polar molecules as in A. M. Mayer's illustrative experiments with magnetic elements, or by fluid vortex rings. Thus for example a system of electrons ranged along a circle, and moving round it with the speed appropriate for steadiness, constitutes a vortex ring in the surrounding aether: it will therefore enjoy to some extent the well-known wide limits of stability of such a ring*: and the stability will probably be maintained even when there are only a few electrons circulating at equal intervals round the ring. Again, a positive and a negative electron can describe circular orbits round each other, stable except as regards radiation, thus forming a simple type of molecule devoid of magnetic moment : or again, we might have a ring formed of electrons alternately positive and negative. And moreover we may imagine complex structures composed of these primary systems as units, for example successive con- centric rings of positive or negative electrons sustaining each other in position.
The duality arising from the assumption of two kinds of electrons, only differing chirally so that one is the reflexion of
* It is here implied that the electrons are constrained by the attraction of an electron of opposite sign at the centre of the ring : as otherwise their mutual repulsions and the centrifugal forces would produce their dispersion. On the question of loss of energy by radiation from such a system, cf. Ch. xiv. infra.
28 ELECTRODYNAMICS OF MOVING MATTER [SECT. I
the other in a plane mirror, will present nothing strange to those physicists who regard with equanimity even the hypo- thesis of the possible existence of both positive and negative matter.
On this view of the constitution of atoms the transit of a material body through the aether does not involye_any: disturbance in bulk or pushing aside of that medium,, unless the body carries an electric charge or is electrically or mag- netically polarized.
19. In Maxwell's final presentation of electric theory, in his "Treatise," he deals with displacement but not with any- thing called electricity**: so that a diagram of molecular polarization is foreign to it. When electric current (recognized electrodynamically) flows from ^ to jB along a wire, the circuity is completed by displacement from B to A through the di- electric : and the notion of charges at A and B is (but only to this limited extent) irrelevant. At the same time there is little doubt that this scheme was the outcome of consideration of the theory of Kelvin and Mossotti, who were the first (in 1845) to extend Poisson's theory of magnetic polarization to dielectrics, of which the electric activity had then just been rediscovered by Faraday : and it seems possible that this notion of electrically polar molecules was dropped by Maxwell because his model of the electrodynamic field did not suggest to him any means of representing the structure of a per- manently existing electric pole.
This agnostic attitude as to the nature of electric dis- placement and electric charge does not however limit the application of his theory on the electromotive side, so far as regards bodies at rest ; for on any view the most that can be made of conduction in bodies at rest amounts to the direct application of Ohm's law, while the electrodynamics of stationary circuital currents had been already made out by Ampere,
** This statement does not however apply to the memoir 'A Dynamical Theory of the Electro-magnetic Field,' Phil. Trans. 1864, in which the theory of discrete electric charges is distinctly indicated; cf. §§ 78, 79. For the demonstration that electrons can have a permanent existence in the rotational aether, cf. Appendix E at the end of this volume.
1
CHAP. Il] HISTORICAL SURVEY 29
Faraday, and Neumann. But to the case of bodies in motion such a scheme can give no clue, except the first approximation based on the assumption of an equilibrium state of the sur- rounding field at each instant of the motion. And it can give no account of mechanical or ponderomotive forces, nor therefore of electrostatic phenomena in general, except by the empirical formation in simple cases of a fragment of a mechanical-energy function by taking advantage of the indications of independent observation and experiment.
CHAPTEE III
GENERAL KINEMATIC THEORY OF OPTICAL RAYS IN MOVING MEDIA
20.
Specification of a Ray The relation between the direction of the ray and that
in which the radiant waves are travelling is the fundamental conception of optical science. When the material medium transmitting the radiation is at rest in the aether, the ray, or path of the radiant energy, is the same relative to the matter as relative to the aether; and its direction is determined by the wave-surface construction of Huygens, in a manner of which the precise rationale is due chiefly to Fresnel. If the point 0 becomes a centre of radiant activity owing to a train_ of regular waves advancing on it, the radiation sent on from it travels out into the surrounding space, so that the locus at which a given phase of it has arrived at any instant is a surface S surrounding 0, called the wave-surface. Suppose
now that a train of waves is £ass_- ing across the point 0 and let the plane F be tangential to a wave- front : draw a parallel plane tan- gential to the wave-surface on the onward side, the point of contact being Q : then the radiant energy of the portion of this wave-train which passes across 0 is propagated in the direction OQ. For, if we draw the wave-surface with Q as centre which passes through 0, the plane F will be tangential to it at 0: hence
CHAP. Ill] EXISTENCE OF OPTICAL RAYS 31
the actual radiation sent on from all points of the element of the wave-front F situated at 0, which is tangential to that Avave-surface, will reach Q in the same phase, as follows from the definition of the wave-surface and the reversibility of the radiation: hence the effects due to all parts of this element of wave-front situated at 0 will reinforce each other at Q, while those of any other element of the same order of magnitude will obliterate each other owing to differences of phase: thus it is only the portion of the wave-front around 0 that sends radiation to Q, and the other parts of it may be shut off by screens without altering the effect at Q. It is here tacitly assumed that the medium is homogeneous, so that wave-surfaces of all magnitudes round 0 are similar, and the ray OQ is tlierefore a straight line. When there is heterogeneity we must take a wave-surface of very small dimensions, correspond- ing to a very short time of transit, so that OQ is an element of arc of the )-ay; the next element of arc starting from Q ■will now be in an infinitesimally different direction ; and thus the ray will be a curved line. The path of a ray between two })oints P and P' is of course actually explored by placing a source at P and gradually limiting the beam by screens so as nut to affect the illumination at P' : so long as the screens do not cross the curve PP' constructed as above this will not be affected.
It remains to express these kinematical ideas in analytical form. With a view to this object, we must distinguish, when the medium is of crystalline quality, between the wave-velocity and the 7'ay-velocity corresponding to any given direction. Thus in the diagram the plane wave-front F is propagated in the direction normal to itself with the wave-velocity ap- propriate to that direction: but the radiant energy of that \vave travels in the direction OQ with the ray-velocity, which IS greater than the former in the ratio of OQ to OT, where OT is perpendicular to the tangent plane at Q. When the form of the wave-surface round 0 is known, the wave-velocities and ray-velocities corresponding to all directions are thereby determined.
Now the wave-surface 8 marks the outer boundary of the
32 PRINCIPLE OF LEAST TIME [SECT. I
region which a radiant disturbance, initiated at 0, can affect in a given time. Each ray of the disturbance by following its natural path, with the ray-velocity proper to its direction at each instant, can travel to that bounding surface in the time ; but if it is constrained to follow some other path, it cannot get so far in the time. Thus any point on the ray-path OQ is the farthest point on that path that a ray starting from 0 and guided by any constraint, could possibly reach in the time; and the disturbance actually reaches that point by travelling along the ray itself That is, the path of a ray from P to P' is that path along which the energy of the disturbance, travelling at each instant with the ray- velocity appropriate to its direction, can pass from P to P' in the least time. This is the generalization, afforded by the theory of undulations, of Fermat's empirical principle*, which asserted that a ray of light travels from one point to another along such path as would make its time of transit least.
This principle remains precisely a principle of least time « for paths from P up to all points P' such that the successive wave-fronts between P and P' belonging to a radiant disturb- ance maintained at P do not develope any singularity along the course of the ray. But when P' lies beyond a place of infinite curvature (cuspidal edge) on the wave-front the principle becomes merely one of stationary time: in certain cases it may be even a principle of maximum time. A suf- ,1/ ficient illustration is aftbrded by the simple case ^ of rays y diverging from P, which after any series of refractions finally emerge into an isotropic medium as straight rays at right angles to a wave-front 8. Let the ray from P to P' cross this
wave-front at Q : then by definition the time for the ray from P to Q is the same as the time for the ra}^ from P to any consecutive point Q' on this wave-front : in comparing the times for the ray PQP' and a consecutive ray PQ'P' we have thus only to
• Cf. Appendix D.
I
CHAP. Ill] RAYS IN MOVING MEDIA 33
compare the times for the straight segments QP' and Q'P', that is we have only to compare the lengths of these segments. Now clearly from the point P' on a normal QP' to a surface S of double curvatures, P'Q is the line of least length that can be draAvn to meet the surface in the neighbourhood of Q, so long as the centres of both the principal curvatures at Q are beyond P'; it is the line of greatest length when P' is beyond both these centres ; and it is only of stationa7^y length, neither maximum nor minimum, when P' is between these centres.
21, Let us consider the form that these principles will assume when the matter across which the radiation is travelling- is itself in motion. The radiation is now not reversible, and the demonstration of the law of ray-direction must be expressed difterently from the above. This however is easily done.
The time from 0 to Q is the same as from 0 to Q', hence is the same as from 0' to Q, where 00' is equal and parallel to QQ', each of them being infinitesimal compared with OQ. Hence the disturbances from all points 0', near 0 on the plane wave-front, reach Q at the same time and therefore in common phase, and therefore accumulate, while at a
point in any direction other than OQ they would annul each other. Thus the path of a ray is still determined by the principle of stationary time: but the path from P to P' is not the same as the path from P' to P because the velocity of propagation relative to absolute space is altered on reversing the direction of the ray.
In circumstances of moving matter there are moreover two kinds of rays to be distinguished, one of them being the paths of the radiant energy with respect to the particles of the moving matter, the other the absolute paths of the radiant energy in the stagnant aether, or as we may say in space. As radiation is revealed to us wholly by its action on matter, including therein the parts of the eye itself, it is the former
L. 3
34 ABERRATION OF A RAY [SECT. I
type of rays that is of objective importance. In determining the course of the ray among the elements of moving matter, when it is thus referred to these elements, the principle of stationary time is to be employed using therein a ray velocity which is at each point the velocity of the ray relative to the matter there situated. For that principle ensures that, as the element of matter at Q moves, all the radiant energy arriving at it nearly in the direction of the ray reaches it at each instant in the same phase and thus accumulates : on account of the sameness of time, the path of the relative ray from P to Q is not affected by altering the motions of these terminal points alone.
22. Consider then radiation travelling in a medium of varjang density so that the velocity at the point (x, i/, z) is V: and let us examine the type of the varying velocity {p, q, r) that may be imparted to the medium, supposed isotropic, without disturbing the forms of the paths in space along which the radiant energy travels. We assume, for the present, that the velocity of the ray in space is affected by a fraction k of the velocity of convection of the medium in its direction, the value of k depending on the index of refraction at the place, and being unity for the free aether. When the medium is stationary, the paths of the rays are to be 'determined by the equation of variations S/(c?s/F) = 0; for the vanishing of this variation ensures that consecutive rays starting in the same phase from one limiting point of the integral shall reach the other limiting point in identical phases, and therefore reinforce each other. When the medium is in motion, the equation of the ray-path in space becomes
* =0,
V -\- k {Ip + mq + nr)
where (/, m, n) is the direction-vector of the element of arc ds, and {p, q, r) is the velocity of the material medium. Thus to the first order of approximation
Bit
Jy~^JT'^ (^i^ + "^'l + »'■) = ^'
CHAP. Ill] IRROTATIONAL THEORY OF SIR G. STOKES 35
that is
SJ V-'ds - BJk V-"' (pdx + qdy + rdz) = 0,
where V is proportional to ix~'^, the reciprocal of the refractive index.
The ray-paths in space cannot remain unaltered unless the second of these terms depends only on the limits of the integral, that is unless kfi" (pdx + qdy + rdz) is an exact differ- ential. While if this condition be satisfied, the change in the time of passage of the ray between the two terminal points, arising from the motion of the medium, depends only on the limits of the integral and so is the same for all rays: thus all phenomena of interference between pairs of rays will be unaltered.
We can directly apply this result to a theory of aberration which supposes that the Earth in its orbital motion pushes the free aether in front of it and so sets up a velocity (p, q, r) in it. Our hypothesis will by the principle of relative motion be exact for free aether, k then being unity; thus, taking fi fur air to be practically unity, we see that the paths of rays in space will be the same as if the aether were at rest, will tlicrefore be straight, provided pdx + qdy + rdz is an exact differential, that is provided the aethereal motion set up around the Earth is of the differentially irrotational type, in agreement with Sir George Stokes' result.
Thus if the aether around the Earth were set into irrota- tional motion by the Earth's progress through it, the rays of linht from the stars would still travel through space in straight lines: their velocity in space would however be the standard velocity of radiation combined with the velocity of Convection of the aether, and so would be affected near the Earth to the order of the ratio of the velocity of the Earth's motion to the velocity of radiation. If now an observer esti- mated the direction of these rays by looking along two sights situated in free space, or what is practically the same, in open air, his motion and that of the sights would, on the ordinary ! principles of relative motion, involve an aberrational change in their direction when adjusted to catch the ray, which would 1h' the same as the existing astronomical aberration, except
3—2
36 AETHER FLUID FOR LARGE MOTIONS [SECT. I
that its coefficient, being the ratio of the velocity of the Earth to the effective velocity of the ray in space, would, on account of its convection by the moving aether, vary for different parts of the Earth and different times of day by about one part in 10^, an amount which would not be detected by astronomical observation.
The present object is to make out the best case possible for this type of theory of aberrational effect, which assumes the aether to be set in motion : so we must try to assign a cause for the irrotational quality of motion thus demanded. In free space we have merely to postulate that the aether possesses the properties of the ideal perfect fluid : then by Lagrange's funda- mental theorem in fluid motion no convective motion that can be propagated into that medium can be other than irrotational The question then arises how far this explanation will extend to the case in which the aether is entrained by the matter that is moving through it. Attention has already been drawn (§ 10) to Sir George Stokes' considerations which would make the luminiferous property itself prevent the initiation of any rotational motion in the aether. It is in fact not difficult to prove that the energy of strain of a rigid incompressible medium of the type of ordinary matter may be expressed as a volume integral involving only the differential rotation, together with surface integrals extended over boundaries : and it follows that any local beginnings of rotational motion in an aether of elastic- solid type would be immediately carried off and distributed by transverse waves, so that if the rigidity is great enough no trace of rotational motion of the medium in bulk can ever accumulate. In opaque media, however, such waves would not be effectively propagated. The coexistence in the same medium of liquidity for large-scale motions and rigidity for light-waves would on this view be the thing to be explained.
We have been proceeding on the supposition that the Earth's atmosphere moves through the aether without disturb- ing the motion of the latter, or rather that any disturbance thereby produced does not destroy the differentially irrotational character of its motion. Suppose the observer fixes the direc- tion of the star by an observing telescope instead of by simple
C HAP. Ill] DIRECT INFLUENCE OF MATTER 37
sights, the astronomical law of aberration will still hold provided the motion of the aether in the region inside the telescope retains the same irrotational character as in free space, and not otherwise. But when the tube of the telescope by which the direction of the ray is determined is tilled with water instead of air, then if the continuously irrotational character of the aethereal motion were maintained in the water as well as in air, on the lines of Sir George Stokes' dynamical explanation, the Course of the ray referred to space, when inside the tube, would not be altered by the motion, and therefore the coefficient of aberration relative to the observer would be reduced in the ratio of the velocity of radiation in water to that in air.
23. This conclusion is contrary to fact. The preservation of irrotational continuity of motion, thus dynamically suggested, must therefore be abandoned: and we are compelled to treat of two interacting media, aether and matter, instead of a simple modified aether. From this new standpoint, in addition to the convection of radiation along Avith the moving aether, there will have to be a first-order influence on its velocity of propagation in the aether, arising from the relative motion of the matter through it and proportional to its relative velocity. This effect Could vanish only if the aether moved along with the matter : whereas if it did so its motion could not be continuous, and also irrotational outside the matter, as in any case it is required to be.
We therefore proceed, for the case of terrestrial rays passing in part through dense media, to develope this wider hypothesis of interacting media. The effective velocity of the rays is now made up of the standard velocity V which would obtain for conditions of rest, diminished by the velocity (P, Q, R) of the space attached to and moving along with the material observing .system, increased by the absolute velocity (p, q, r) of the aether itself, and increased by k times the velocity {p, q\ r') with which the matter transmitting the radiation is moving through the aether, all measured at the point under consideration and referred to axes fixed relative to the undisturbed distant aether : here k is a constant depending on the nature of the matter, which for an isotropic medium must be scalar.
38 NECESSITY OF FRESNEL's LAW [SECT. I
Thus the velocity, relative to the observing system, of a ray travelling in the direction (I, m, n) is
V- (IP + mQ + nR) 4- {Ip + mq + nr) + k {Ip' + mq + nr).
Let us first consider the usual case in which all the matter is at rest relative to the observing material system, so that
{p'+p, q' + q, r' + r) = {P,Q,M);
the velocity of the ray is now
V-(l-k)(lp' + viq' + no'').
Just as before, the condition that the ray-paths relative to the observing system are unaltered to the first order by the common motion of all the matter is that
fM- (1 — ^) (p'dx + q dy + r'dz)
shall be the exact differential of a continuous function of position. As {p , q, r') has no necessary connexion with the value of fx, this requires that yLi-(l —k) shall be a constant A, so that k—\— Afi~'- : as moreover k must tend to a null value for very rare material media for which //, is practically , unity, we must haVe A equal to unity so that k = l—/jL~^. The condition further requires only that p'dx + q'dy + r'dz shall be an exact differential ; so that there is still room for motion of the matter relative to the aether, provided it is differentially irrotational. For example, if we suppose that the aether is stationary, and that the velocity of optical transmission in it is at each point specifically altered after Fresnel's manner, by the fraction 1 — fxr- of the velocity of the matter there moving through it, then the ray-paths are unaltered as to form % and as to relative phases when the material system to which they are referred and the space attached to it are set into any state of continuous irrotational motion, — the rectilinear motion of the Earth along its orbit furnishing a case in point. This argument shows that the law of Fresnel is on any view required in order to account for Arago's principle.
Let us then proceed, on the basis of Fresnel's value of k thus demonstrated, to the general case in which the material system that transmits the light is in motion with velocity
I
CHAP. Ill] AETHEREAL FLOW UNNECESSARY 39
(Pi, qi, i\) relative to the observing system. The velocity of the ray relative to the observing system is now
V + (ipi + mqi + ni\) — yT" {Ip + mq' + nr).
Thus the condition that the relative ray-paths are unaltered is that
IJ? (pidx + q^di/ + i\dz) — {p'dx + qdy + r'dz)
should be an exact differential : that is, in addition to the con- dition already obtained that the absolute motion of the a,ether itself should be differentially irrotational, we must also have
fi-pi dx -f- /i-^-i dy + /jr)\ dz
the exact differential of a continuous function. In a region of constant index this condition requires that the motion of the matter transmitting' the light must relative to the space of the observing system be continuous and irrotational ; but at the transition between different substances the tangential com- ponents of this motion must be discontinuous so that on the two sides of the interface they are inversely as the squares of the indices of refraction on those sides. These relations are extremely unlikely to l)e satisfied in actual circumstances.
It appears then that there is no optical method of detecting a differentially irrotational flow of the aether superposed on the necessarily existing Fresnel influence of relative motion of the matter on the velocity of propagation in the aether, unless that flow be of cyclic character in the region considered. The Earth's motion might thus, so far as we have yet gone, cause or control such a flow of the surrounding aether, provided however we are willing to admit that bodies of ordinary size at the surface of the Earth are powerless to sensibly deflect it. Such a flow is not required by any optical facts, though Fresnel's effect is so demanded : it is therefore gratuitous to introduce it, especially as it does not in any way simplify our conception of the disturbance imparted to the aether by bodies moving through it : and it will in fact appear that its presence to any sensible extent would introduce excessive complication into a theoretical ptheme otherwise simple.
40 RELATED OPTICAL PHENOMENA [SECT. I
24. The argument above given proves not merely that the principle of Fresnel forms a sufficient basis for the usually received facts of astronomical aberration, but in effect shows that it is necessitated by them. 1'hus its experimental verifi- cation, by Fizeau and by Michelson, was of value rather as a confirmation of the general validity of this line of physical reasoning, than as a special proof of Fresnel's principle itself: if Bradley's law of aberration is granted, in connexion with the observed absence of influence of the Earth's motion on terrestrial ray-paths, that principle follows deductively.
Irrespective then of any experimental evidence relating to the effect of the Earth's motion on interference, diffraction, double refraction, polarization by reflexion, rotatory polarization, or other physical phenomena depending on the dynamical nature of the radiation, considerations of a merely geometrical or kinematical character have restricted the influence of motion of the material medium on the propagation of radiation to the definite relation of Fresnel. 1 1 is necessary to find a dynamical basis for that relation, and to show that this basis is in keeping with the relations to the Earth's motion of the other types of phenomena here enumerated, remembering that it is demon- strated only up to the first order of the ratio of the velocity of the material system to that of radiation. Of these other phe- nomena it may here be noted that interference experiments involve only the relative phases of the rays, and so are included in the above discussion : while the remaining effects above enumerated involve more intimately the dynamics of the waves.
25. In this consideration of ray-paths relative to the moving matter, it has been necessary to include only the first order of small quantities, so that it was unnecessary to dis- tinguish for isotropic media between ray- velocity and wave- velocity. In some discussions which follow, in which the second-order terms are included, it is of course ray-velocity with which we are concerned : and the difference between the two velocities must therefore be determined. When the ob- serving system is moving across aether with uniform translatory velocity v, the velocity of propagation relative to the matter of
(HAP, III] RAY-VELOCITY RELATIVE TO THE EARTH 41
<i wave-front travelling in free aether in a direction inclined to that of V at an angle 6' is V—vcoad': hence the relative wave-snrface, being the envelope of simultaneous wave-fronts, is exactly a sphere, say of radius unity, referred to an origin 0 situated at a distance vjV from its centre G measured in the direction of v. Corresponding to a point P on this sphere, the relative ray-velocity is V multiplied by the vector OP ; while the relative wave- velocity is as usual V multiplied by the vector perpendicular from 0 to the tangent plane at P. Now, 6 being the angle between the directions of the ray OP and V, we have
OP' + GO' -f- 20P . CO cos 6^ = 1 ;
hence OP = {I -GO' sin^ Of - GO cos 6 ; thus the magnitude of the ray-velocity is {V — v' sin^ 6f — v cos d, or up to the second order V— v cos 6 — ^v'j V .-gq^^Bt Also, the disturbance relative To the moving matter, that is the ray, is propagated in a direction inclined to the wave-normal at an angle equal, to the first order, to v/F. sin 0 measured away from the direction of V. These results are on the hypothesis that the propagation relative to the aether itself is isotropic, so that V is independent of direction : otherwise there will also be terms involving inter- action between the velocity of convection and the aeolotropic quality.
26. As the axial rotation of the Earth does not come under the restriction above made to an irrotational motion, it follows that our results will not strictly apply as regards the diurnal aberration. In this case there would be an accumulated change of direction in the relative ray-path, in dense matter, for example down a water-telescope, of the order of magnitude of the ratio of the greatest transverse change of the velocity of the matter along the ray to the velocity of radiation : but under no practical circumstances could this be of any importance.
The discussion above given applies to the paths of single ravs : it exhibits the conditions under Avhich the time of
ty-'
42 INFLUENX'E OF ABERRATION ON FOCI [SECT. I
passage of the ray from a fixed point A to another fixed point B on it is unaffected to the first order by motion, say of uniform translation, of the matter through which it travels. The con- dition that A and B should be conjugate foci on the ray is that, for a slight variation of the path of the ray, the time of passage shall be unaffected to the second order of this varia- tion: thus we should be prepared to find that uniform trans- latory motion of the matter would have a first-order influence on the position along the ray of the focus conjugate to a given one. But there is no means of recognizing an effect of this kind.
It is easy to extend the results here obtained to the case in which deviation is produced by a diffraction grating instead of by reflexion or refraction. The direction of the diffracted ray is determined by the principle that the difference between times of passage from J. to i? by way of successive physical elements of the grating shall be a period of the light or a multiple of that period : to the first order this difference of times or phases will not be affected by a uniform motion of translation. Thus the principle of Arago and Fresnel extends also to the use of grating spectroscopes. The discussion of the question for more complicated cases of diffraction will be best conducted by aid of the dynamical theory.
Influence of Convection on Radiant Periods: Doppler Effect
27. The disturbance excited in the aether by the uniform motion through it of the radiating body and the Earth cannot affect the change of relative period due to approach of the observer and radiant source : for such disturbance will be steady, and therefore the number of vibrations which leave the source in a given time must be the same as the number experienced by the observer, except in so far as his distance from the source has changed in the interval. If we take the aether to be quiescent, the emitted wave-length \ is shortened to (1 - vjv) X, where v is the velocity of the source resolved in the direction towards the observer and C is the velocity of radiation in vacuum ; while the observer, moving towaj-ds
1
f
CHAP. Ill] DOPPLER EFFECT 43
the source with velocity whose component is v, picks up these waves with a period r , which is related to the true period, t or (1 — ^c) \/c, by the equation
r' (c + v) = rC.
, 1 - v/'g X Hence t = :; ,-, -
1 + V /c C
so that the combined motions affect the period in a ratio, which to the first order of small quantities, to which our knowledge is in most respects confined, is equal to \ — {v + v')/c, thus de- pending only on the relative motion.
The only first-order difference that could ai-ise, according to whether or not the moving body jjushed the aether in front of it, would be a difference of wave-length. No such effect would be produced by the source moving the aether in this way, if the receiver is far enough off to be out of the range of the disturbance : but if the velocity v' of approach of the receiver involves a velocity kv' of the aether around it, the wave-length of the radiation relative to the receiver will be altered in the ratio 1 — kv'/c. At first sight it appears as if this change of wave-length might be revealed on analysis of the light, for example by the use of a grating which is the type of analysis in which the theoretical conditions are simplest : if that were the case the analysis of light from a terrestrial source, such as a sodium or thallium flame, would show a change of wave-length depending wholly on the relative motion of the matter and the aether around it. But it will be shown generally that uniform relative motion of matter and aether can produce no first-order change whatever either in refractive or diffractive effects : so that no such influence can arise. It seems however worth while to give in brief a special analysis for a grating, in order to illustrate more fully the origin of this absence of effect.
28. The problem under consideration is one in which the observer and all his apparatus are in uniform motion of trans- lation along with the Earth : we have therefore to consider lay-paths and vibration-periods relative to the Earth's motion
44
THEORY OF MOVING DIFFRACTION-GRATING [SECT. I
through space Avith velocity v'. Suppose that the aether takes part in this motion to the extent hv' : then the relative velocity of radiation in air which is travelling in a direction making an angle 6 with the direction OX of the Earth's motion is V — {v — kv') cos 0, say V—v cos 6 where v = (l — k) v'.
We have to determine the law of diffraction from a grating under these circumstances. Let t denote the angle of incidence of the ray OA, say from a sodium or thallium source at 0, and i the angle of diffraction of the diffracted ray AO', which travels at an angle TT — 0' with the direction OX of the Earth's motion. If we compare the ray OAO' with the ray OBO' which comes from the next reflecting space of the grating, the difference of their times of transit must be a whole number n of complete periods of the radiation, say nr: this will be true for any point 0' on the diffracted ray, up to the first order of the ratio of AB to OA, while for the focus of a curved grating it will be true up to the second order. Thus we have
OA
+
AO'
V +v cos 6 V—v cos 6'
OB
+
BO'
V+vcos{d + Bd)^ V-vcosie' + Bd') where 6 + 6' = t + t ; or, transposing,
■K + nr
I
OA
OB
F+vcos^ V+vcos(e + 86)
O'A
T/ +
O'B
7^ + nr,
F- V cos ^' F - y cos (6' + 86') so that to the first order of approximation, for which f
86 = AB cos i/OA, 86' = - AB cos c'/OAi, q'A
OB=OA+ABsmc.86, f,
CHAP. Ill] TEST OF ARAGO'S PRINCIPLE 45
\\ e have
- V.ABsmi-OA.vsind.Se-ABsini.VcosdSe
V- + 2Vv cos e
_ - V. AB sin i'- OB . v sin 6'. 80'- AB sin l'. Fcos d'SO' - V'--2VvcosJ' ■^'*'"'
and therefore
sin t ( 1 — pP C0SC7 + -5^ cot L sin
f V , V \ V
— sm I [\ +^ffr cos 6' — YF cot i' sin 6' = -7^ ?iT, V F V J AB
which gives, to the first order in v/V,
. , V nX
sin I — sm I = -~r^ m = -r^ . AB AB
Thus the law of diffraction does not involve the relative velocity v', which is the required result : it is only changes of period that can affect the law.
For a curved grating the relative motion would affect the focussing, which depends on adjustment of the reduced paths up to the second order : this effect will be of the first order in V c, but of course wholly beyond the range of detection.
29. It is worthy of remark that, whether the radiation is analysed by a grating or by prisms, the consistency of the results of astronomical measurements of velocities in the line of sight involves incidentally a delicate test of the hypothesis of Arago and Fresnel that uniform motion of the Earth through the aether does not affect the laws of geometrical optics. An uncertainty in the measures of velocity of about one mile per second, which seems at present to be the superior limit in a process that is rapidly improving, would permit discrepancies amounting to only about one second of arc in the optical laws; to recognise this amount in ordinary terrestrial measurement, very exact appliances would be required.
There is one point, however, essential to the theory of measurements of celestial velocities by this means, that has to be settled. How do we make sure that the motion of the Source through the aether does not affect the intrinsic periods
?
46
INFLUENCE OF EARTH S MOTION ON
[sect. I
of its radiant vibrations ? The answer is that the effect, such as it is, must be the same when the translatory motion of the molecules which act as source is reversed, because there is no other velocity or directed quantity connected with the moving molecules such as could enter into combination with their translatory motion : thus the effect on the free periods must depend on the square of the ratio of the translatory velocity to the velocity of radiation, and therefore be far below the order of actual measurements.
Detailed Theory of the Michelson-Morley Interference Experiment
30. The theory of the Michelson and Morley interference experiment* which is fundamental in this subject, will form an illustration of the principles explained above. A ray of light
from a source S, proceeding in the direction SG of the Earth's motion, is divided by a glass lamina at G inclined to it at an angle ^tt; the reflected part traces the path GBHI in space, being returned by a mirror at B which is parallel to the direction of the Earth's motion, and reaching the lamina again when the point G of it has moved on to iT ; the transmitted part traces the path SGAHI in space, being returned by a mirror at A at right angles to the direction of the Earth's
* Phil. Mag. Dec. 1887.
■: iU
CHAP. Ill] OPTICAL INTERFERENCE 47
motion. It is still to be proved that the paths in space as thus specified are correct ; it has to be shown that the ray SG will be reflected along GB, and that the ray which is returned along AH will be reflected along HI. That being assumed for the present, these two rays, adjusted so as to be both travelling exactly along HI, will be the rays that produce interference fringes in an observing telescope directed along IH, when the mirrors A and B are almost exactly equidistant from G ; and the circumstances of the interference will be determined by computing the difference of times from G where the rays are separated to H where they are united again. By ray-paths in space we mean for the present ray-paths relative to the aether, which may or may not (so far as we are here concerned) itself be in uniform translatory motion : also GH is the distance the point G of the dividing lamina has moved, relative to the aether, while the reflected portion of the light has traversed the path GBH. If V is the velocity of radiation through aether and v the velocity of the material system relative to the aether, we have the angle GBH equal to IvjV, say 2^; and the angles of incidence and reflexion at B are equal. If the distance of the mirror B from G is L, we have GB = BH =l,{l +^6-), and the velocity along each of these lines is V: hence the time over the path GBH is 2y F. (1 + ^6-). To find the time over the path GAH, it is easiest to work with velocity of the radiation relative to the material system, otherwise the position of A at the instant of reflexion would have to be found : now if Ij is the distance of the mirror A from G, the relative velocity Ifrom G to A is V-v and from A back to iT is V+v, hence the time is
ir^ + w-i-' ^'hat is, ^jl'a + e').
V — V V + v V
rhe coefficients multiplying l^ and L in the two cases differ by '^- V, where 6 is equal to v/V: thus if the adjustment to equality 'f time is made for the system in any position, that adjustment vill be disturbed when the whole system is turned through a •i£;ht angle. The effect of slow steady rotation of the system \niild thus be a procession of interference bands across the Id of the observing telescope, which would reverse four times
ic
48 REFLEXION BY A MOVING MIRROR [SECT. I
(§ 34) in a complete revolution, the number of bands that have crossed between two reversals corresponding to a time-difference of 6-1/ V or Iv-jV^, where I represents l^ or L. But according to the experiments, which have recently been repeated with a refinement that leaves no room for doubt, this effect depending on the square of v/Fis entirely absent.
31. It yet remains however to complete the above demon- stration by showing that the ray AH \& reflected along HI : if that were not so we should have to seek the ray AH' that would be reflected in the direction of HI, and the difference of times up to the focus of the telescope would be affected to the second order in v/F if the inclination of AH! to AH were of the first order. We have thus to determine the law of reflexion, at an advancing mirror, of a ray-system referred to the aether: consider two parallel rays of which one meets the mirror in H; the other would meet it in K if the mirror had not moved forward in the meantime, but really meets it in K' where KK' : KL = v:V, v being the velocity of advance of the mirror towards the light: therefore the reflexion is the same as if the mirror were HK' fixed in aether, thus being turned through an angle KHK' or e, where, t being the true angle of incidence,
tan {t — e) _V —V tan I ~ V ' so that, e being small,
v e = ^ -f^ sin 2t.
In the present case t is ^tt; and rotation of the mirror through 6 would rotate the reflected ray through 2d ; therefore the ray AH is reflected along HI. In the same way the ray SG is reflected along OB. Moreover it follows from the principle of continuity that practically the same value for the retardation would be obtained by taking any adjacent pair of interfering rays instead of the pair in the diagram. The bands usually observed will naturally correspond to reflexion at the first face of the lamina in each case
I
CHAP. Ill] RAY-VELOCITY IN A MOVING MEDIUM 49
32, Wave-Velocitij and Ray-Velocity iyi an isotropic Moving Medium. — In a material medium of" index of refraction /x, in uniform translation along the direction of the axis of angular measurement with velocity v, the relative velocity of a train of light-waves travel- ling in a direction making an angle 6' Avith the direction v is /ii~^V — kv cos 6', where on Fresnel's hypothesis k is equal to /jT-. As this velocity must be equal to the perpendicular fr(jm the origin on the tangent to the wave-surface constructed relative to the moving medium, it follows that this surface is exactly a sphere of radius fjr^ V with its centre C at a distance kv behind the origin 0. The ray- velocity relative to the moving system, in any direction OP, is represented by the radius vector OP of the wave-surface : thus if this ray OP makes an angle 0 with the direction of v, we have
OP' + 2kvOP cos d + k-v' = fi-'V-^ giving
0P=- kv cos 0 + (fx-'-V - t-v"- sin2 0f
= kv cos 0 — h -rr
fl ^ V
sm-
correct up to the second order.
The path of a ray relative to the moving material system would be determined by making the variation of its time of ti-ansit between any initial and any final point on the path vanish, using this value of the ray-velocity.
But it is important to remark that the correctness of this si'cond-order term in the relative ray-velocity depends on the I -sumption that the relative velocity of wave-propagation is as al)ove stated, and thus involves no term depending on (v/V)-. Such a term would be independent of reversal of the direction of V, and therefore could only arise from a constitutive change in the material medium itself, produced by its translation through the aether.
The expression above obtained is correct to the second order i' >v relative ray-velocity in free aether, as in § 25 ; though, as
L. 4
50 INTERFERENCE IN TERMS OF RAY-PATHS [SECT. I
has just been seen, there is no reason why it should be correct to that order for a moving material medium, so that any de- velopments derived from it are for ponderable media mainly illustrative. An alteration of the second-order terms in the expression would not however assist towards the explanation of the null result of the Michelson-Morley interference experiment, for the paths of the divided ray are there wholly in air, which is for the present argument practically the same as free aether: thus we are still confined, for the explanation of that result, to the equally reasonable hypothesis of a second-order change in the linear dimensions of the solid material system of the ex- periment, arising from its motion through the aether (§ 112).
33. General Analysis of Interference in Moving Media. — The problem of optical interference in moving material media may be treated in a quite general manner. It has already been seen that, on Fresnel's hypothesis, the relative paths of the rays in a uniformly moving material medium of varying density are the same as if the system were at rest, up to the first order of small quantities inclusive. Further it has been shown that the relative velocity of the ray which travels at an angle 6 with the direction of the uniform translational velocity v of the system is jjT^V {\ — he co^ 6 — ^k'-e- ain- 6), where e is ecpial to tivjV, while on Fresnel's hypothesis k is equal to jm^-. Thus if hs denote an element of any ray-path of continuous curvature, relative to the system in motion, and hs the corresponding element of the ray-path if the system were at rest, the time of t passage of the ray for the moving system is
I ^y- (1 - ke cos d - p-e- sin- 6)-',
*
which is equal up to the second order inclusive to J
I ^y (1 - /.-f cos e - llc'e' sin-' er' + I ^ (r/s' - ds).
Of this expression the second term is the difference of the times of transit of a ray over the path s' and over the natural path s, when the medium is at rest. Now Format's principle of least time shows that if these paths differed in jmsition up
CHAP. Ill] RELATIVE TO THE MOVING SYSTEM 51
to the first order, the times of transit would differ only by the second order: but actually the paths differ in position only up to the second order, hence this term is negligible up to that order. We can therefore calculate the time of passage of a ray relative to the moving material system, correctly up to the second order, by assigning to it the path s that would actually belong to it when the material medium is at rest.
This proposition holds good however abrupt the transition of density may be at certain surface-loci : hence it really |ij includes as limiting cases those in which the continuity of curvature of the ray is disturbed by a finite number of re- flexions or refractions. It is however easy to see independently that these cases do not introduce any disturbance into the I result. For consider a refraction as in the diagram, P'p' being the actual ray relative I to the moving medium, and pP that which it is proposed to substitute for it, namely the corresponding ray in the medium at rest. We have seen that the length PP' ilis of the second order of small quantities. The special effect of the refraction on this substitution is to add the element of arc Pp to the integral for the time of passage, and to take away the element of arc P'p' : these are both of the second order, thus to that order the change produced is to add to jthe integral /Xj . Fp, and to subtract ix.,.P'p', where /Ai and fx., fjare the refractive indices above and below the interface : but these terms are equal, each representing the lime of passage (from the wave -front P'p to the wave-front i'*^;' : hence there is no change in tiine here introduced, up to the second order.
34. Now when the medium is at rest the paths of the [interfering rays in the Michelson-Morley arrangement are as in :he diagram, each of the rays being reflected straight back to the nirror which originally divided them. Thus if the velocity v )f the material system makes an angle 6 with the dii-eetion of the ncident light, the times of transit of the two rays ivlative to he moving system are, up to the second order, nklV{l - ke cos 6 - ^/.-V sin-^ 6)
+ tilJV(\ + ke c<^s 6 - We" sin'-' 0)
52
APPLICATION TO
[sect. I
and
fil/V (1 - k€ sin e - We^ cos- 0)
+ fil,IV(l + ke sin e - i^'-e- cos- ^) ;
these are equal to
and
V
(1 + Z;V - lA-V sin^- 0)
-^- (1 + ^'^e- — ^A--e- cos- 6) respectively.
/
n
I
The difference of times of transit is thus
2/x — yt^ + /^ f" ^"^' ^^^ "^'
since in the second-order terms we may write I for l^ or L. Thus, as the apparatus revolves, the fringes pass backwards and furwards across the field of view of the observing telescope with a simple harmonic oscillation, moving in the same direction during a quadrant of the revolution. The total change of phase betAveen the two positions, in Avhich the light is incident along and at right angles to the Earth's motion, is
V ' ,mV"-'
The effect of inserting a tube of water in one of the arms, so that part of the path is in air and part is in water, may be
CHAP. Ill] MICHELSON's EXPERIMENT 53
easily estimated in this way ; the result will still of course be of the second order. Thus if the path l^ consists of a part l^' in air and a part li" in water of index fi, while the path L is all in air, the difference of times of transit would come out as
1 e^
constant — ^ (4 + ^i' + /xkHi") ^ cos 20,
where approximately
But it is to be borne in mind, as above explained, that this result neglects the second-order effect on the velocity of radiation in the material medium due to constitutive change in it arising from its motion through the aether, and also the effect on the linear dimensions of the material system arising from the same cause. When these effects are included, the result will probably, on any view, be quite different : accord- ing to the general molecular theory to be explained later, it will always be null.
CHAPTEE IV
THE PROBLEM OF OPTICAL CONVECTION : INDICATIONS TOWARDS A DYNAMICAL THEORY
35, Consider in the first place the propagation of waves — or indeed the course of any kind of disturbance — in a single self-contained medium, with a view to determining the effect on it of a velocity of uniform translation imparted to the medium. The principle of relative motion supplies the solution. Impart to the whole system a velocity equal and opposite to that of the medium ; and, because this uniform velocity intro- duces no new kinetic reactions, the phenomena of the relative motion will pursue the same course as before, but they will now be relative to the medium at rest instead of in motion. In all such cases, therefore, the disturbances in the medium are simply carried on along with the medium itself, with its full velocity of translation, and in other respects pursue their course unaltered.
For example, the velocity of translation of the air through which sound is propagated is added (in algebraic sense) at each instant to the velocity of the sound itself. In the same way, if, adopting the view discussed by Sir G. Stokes, we considered the surrounding free aether as disturbed by the Earth's motion, its velocity at each point would have to be added on to the intrinsic velocity of radiation through it. This is true whether the motion of the medium is uniform or not : when however it is not uniform a simple wave will no longer travel as a simple wave, and to that extent the meaning of the term velocity of the wave is indefinite.
CHAP. IV] EXAMPLES OF SIMPLE SYSTEMS 55
In expressing the velocity of a particle of the moving system relative to coordinate axes travelling with the system
didt + vdjdx
must replace djdt, where v is the velocity of the system and :s taken to be parallel to the axis of x. We may illustrate ay the simple case of the propagation of waves along a stretched )ord which is carried through two fixed eyelets, and runs through them with a uniform velocity of translation v. Con- lidering transverse waves, if 77 denote the transverse displace- nent at a distance x along the tight cord, the transverse velocity is {djdt + vdjdx) tj, and the transverse acceleration of his element of the cord is {d/dt + vdjdx)- tj. The tension T in he cord is uniform because v is uniform ; in fact any slight [ifference of tension, however initiated, is smoothed out by Dngitudinal waves which are assumed to travel very much ister than the transverse waves under consideration. Thus
he restoring force is as usual T-j- -p^_8x per length Sx: and
tie equation of propagation is
\di dxj dx dx
fd d\ _ .d-T]
\dt dxj dx'^ '
here C'-, equal to Tjp, is the normal velocity of propagation, ssuming a solution in the form of the simple wave-train
27r 77 = ?;o exp i — {x - Vt),
lis gives the relation {V - v)- = c-, so that V=c+v exactly, was to be anticipated.
For purely longitudinal waves, of displacement ^, the equa- )n of propagation is
fd d y ^ d ( r. d^\ P[dt'-''dx)^=dx[^
dxj '
lere E is the longitudinal elasticity, and (Ejp)- is the un- iturbed velocity of propagation. As in the previous case,
56 COMPOUND SYSTEM, AETHER AND MATTER [SECT. I |
this velocity is increased by the velocity of translation of the cord measured positive towards the direction of propagation of the waves,
36. Let us proceed now to the propagation of electric waves across a dielectric medium, which is moving with uniform velocity v parallel to the axis of x. If we follow Maxwell's scheme of equations, and his notation, we have
„ dF d-^ ^ dG d^ ^ , dH f/^ ,.,
dt dx ^ dt dy ' dt dz ^ '
where (a b c)=i^^ -— — -^ ^^_^V \dy dz ' dz dx ' dx dy) '
, . , . _ fd'y d/3 da dy d/3 da
\dy dz ' dz dx ' dx dy
and (a, b, c) = /x (a, ^, 7) ;
yielding V^ {F, G,H) = - 4<7rfi (u, v, w)* (ii).
These equations are satisfied by the propagation of a train of transverse waves along the axis of x, in which P and a and u and F are null, while "^ is a function of y, z: thus for such a wave-train they give
d_ _i\p_(l^ ,dt dx) dy
Q = -[^. + v^^G-':^ I
\dt dx) dz '
. dF dG dH .
As -r- + -7- + -^ IS null ni accordance with (li), any theory that
makes
dP dO dR
j — y ^
dx dy dz null will also make V-^ null, that is will make ^ merely the static potential of the electric charges in the field. Then we shall have
* Cf. § 55, aud end of Appendix A.
CHAP. IV] VARIOUS POSSIBLE HYPOTHESES 57
and the remainder of the analysis will depend on the relation that is adopted between the dielectric current (u, v, lu) in the moving medium and the electric force (P, Q, R).
(1) If we were to assume the ordinary relation for a medium at rest, namely
the above condition of nullity of
dP dQ dR
1 H
dx dy dz
\vould be satisfied, and the equations of propagation would be
On introducing the type of a simple wave-train
{Q, R) = {Q„ R,) exp i Y (*■ - ^^^' they would give V {V— v)= j^-
so that V= — ^ + 2 ^ + i ^ — ^-^ ^^'^ approximately ;
thus the velocity of the wave-train would be increased, to a first approximation, by half the velocity of translation of the medium.
(2) If we assumed that the whole of the system, aether and matter, that is polarized or otherwise affected by the electric force, moves together, with the uniform velocity v, and that the change of its actual polarization constitutes the di- electric current, we should have
(».».«')= 4^. (I +" a) (-p. ft -R);
and the equation of propagation would be
58 VARIOUS POSSIBLE HYPOTHESES [SECT. I
The waves would then partake of the whole of the velocity of translation of the medium.
This rather than the previous result in (1) is what we should expect on the dynamical principle of relative motion, Avhen the whole system transmitting the waves is involved in the translatory motion. We therefore conclude that on such an aspect of Maxwell's theory — one namely which considers everything to partake in the motion — the present relation between the dielectric current and the electric force in the moving medium would be the right one.
But neither of these results is in agreement with the facts, which in very rare media such as gases make the influence on the velocity of the waves extremely small. So we conclude that in a rare medium the main part of the electric flux, which is then the part connected with the aether itself, is not con- vected at all with the moving material system. We are there- fore led to another hypothesis,
(3) which divides the total dielectric current into an
Id
aethereal part — — ^ -j- (P, Q, R), Avhich is not convected pre-
sumably because the aether does not participate at all in the motion of the matter, and another part depending on material polarization which is convected to the full extent and therefore
is of the form -r^ ( "t. + ^^ -7- 1 (P> Q, ^)- Thus for the total 4!7rC- \dt dxj ^ ' ^' ^
current, referred to axes at rest, we would have leading to equations of propagation The equation for the velocity is now
(K ">- so that F= — 1- ^ (1 — K~^) V, approximately.
%'(''
;hap. iv] fresnel's law obtained 59
Che change of velocity of the waves is now just half of that ^iven by the formula of Fresnel, which has been fully verified )v experiment. Thus we are impelled a stage further, and led 0 inquire why the displacement current in the stagnant aether hould have to do at all with the electric force (P, Q, B), which nvolves in its constitution the velocity v of the matter carrying he electric charges on which alone electric force operates. If ve assume that this aethereal electric current is excited by the aine cause as produces it when there is no matter present, or \ hen the matter is at rest, namely by what we may call the ethereal force (P', Q', R), comiected with the electric force by he relation
(P', Q', R) = {P,Q+ vc, R - vb),
ve shall have to combine an aethereal displacement current
.nd a material polarization current
u order to obtain the total dielectric current, which will thus be
SdW putting "^ null for purely transverse waves, which will )e found to cause no discrepancy, we have
{Q',R) = -j-^{G,H);
Lud, keeping now to G, H as more convenient independent aiiables, we derive
vith the similar equation for H. These equations give for the elocity of propagation
60 FINAL SCHEME FORMULATED [SECT. I |
or KV- -2(K- 1) vV=-- (K-l) v\
H'
that is F = (1 - K-') ^+[-j^ - —jr. - ^■
= T^i + (1 - A'-^u - (^^y (1 - K-') ^, approximately,
agreeing to the first order of small quantities with Fresnel's formula.
The principles to which the above cursory preliminary sketch has pointed, form the basis of the definite dynamical theory of the electrical and optical relations of moving material media, which will be worked out in detail in the following pages.
37. If we determined to avoid the introduction of the auxiliary vector potential {F, G, H), this argument would have to be expressed as follows. Let (/, g, h) denote the aethereal part and (/', g', h') the material part of the total electric dis- placement of Maxwell ; the circuital electrodynamic relations, for the moving material medium will, when referred to axes at rest in the quiescent aether, be of types
dy dz \dt dt
dR_dXl__d[a dy dz dt '
where (P', Q', R) represents 47r(7- (/, g, h), and where d'jdt when it operates on (/, g, h) is the same as djdt, but when it operates on (/', g', h') is the same as hjdt or djdt + vdjdx.
Further {/+/', g + g', h + h') = K (/, g, h), just as when the material medium is at rest : for its motion cannot alter the value of K to the first order of small quantities.
This scheme of equations forms a sufficient basis for the theory, without any direct assumption as to the relation between (/', g', h') and (P', Q', R') : the relation previously given
(/. 9\ h') = ^^ (P', Q' - vc, R + vh)
is in fact implicitly involved in this scheme. Thus the dis- tinction that is necessary in moving media between the electric
JIIAP. IV] WITHOUT POTENTIAL FUNCTIONS 61
orce and the aethereal force is involved in the circuital relations IS here expressed.
The first of this system of equations gives, in the general )roblem of dielectric propagation, equations of type
dy dt dz dt \df' dty '
A hence by substitution from the second we obtain for a homo- geneous medium the three equations of type
:hat is
with the similar equations for (a, /3, 7). These lead to Fresnel's L'Xjjression for the convection-effect, as before.
38. It will be shown later (Ch. x), more generally, in connexion with molecular theory, that if any system of electrons exist at rest in the aether with ideal rigid connexions between tht'Ui, and its state is compared with that of the same rigidly connected system of electrons in motion with uniform trans- latory velocity v through the aether, then, when the square of v/a is neglected, (i) the forces which act on the individual electrons are the same in the two cases, (ii) a correspondence can be established between aethereal disturbances propagated across the system from one group of electrons to another in the two cases, so that though electric and magnetic displace- ments do not correspond yet relative wave-fronts do, and a place where there is no disturbance in the one system corre- s]i(jnds to a place where there is no disturbance in the other. Xow all, or almost all, exact electrical and optical measure- ments are made by null methods : that is, a moveable piece of apparatus is introduced into the system and so becomes p;ut of it, and observation is made of its position when a certain kind of disturbance is just obliterated. All such experimental determinations will therefore be the same, up to the first power "* This simply means that the rate of change of the integral of magnetic force ; round a small fixed circuit is equal to 47r times the rate of change of the current I through its aperture.
62 SKETCH OF KESULTS [SECT. I
of v/c, in the fixed and the moving system : there will be no possibility, except it may be as regards the second order of v/o, of deciding whether the system is at rest or in uniform motion through the aether, by means of phenomena which occur w^holly within the system itself.
As an illustration, consider the rotation of the plane of polarization of light in passing through quartz. On formulating a direct analytical theory of the effect, and transforming the equations to axes moving with the matter, assuming that the value of the rotatory coefficient of the matter is not altered by the motion, we should obtain, according to Lorentz's analysis, a first-order effect arising fnmi the motion of the Earth through space, which is greater than would escape detection. Yet consider the system formed of polarizer, quartz plate, analyzer : if it is so arranged as to prevent the incident luminous dis- turbance from getting through when the Earth is at rest, it should, by the above general result, remain thus arranged, correctly up to the first power of v/u, when the motion of the Earth intervenes : and thus change of direction of the Earth's motion should not have any first-order effect on the adjust- ment. This is in keeping with Mascart's experimental result, of which the validity up to the first order of v/c admits of little doubt. According however to Lorentz's analysis there ought to be a first-order effect when the optical rotatory coefficient is supposed unaltered. If that were so we should, in the light of the general principle, have to compensate this effect by assuming an alteration in the rotatory coefficient arising from the motion. It will be seen (§ 92) that a priori there is no formal objection to the existence of a new constituent of the rotatory powx'r, arising from this cause, which would be of the first order: but this neAV term would be related to a directed (piantity, namely the velocity of the motion, and therefore it would be of the type of magnetic rotation, and thus could not, except accidentally in a particular case, com- pen.sate an effect that is structural, as Lorentz's term is. It will appear (Ch. xili.) that the discrepancy is cleared up by the existence of error in Lorentz's analysis : and that Mascart's result indicates that there is in fact no first-order modification
CHAP. IV] OF THE MOLECULAR THEORY 63
at all in rotatory optical quality arising from convection of the material medium.
A very large number of optical phenomena have been examined by various experimenters with a view to detecting an influence on them of the Earth's velocity of translation. The only such influence that has been announced is that found by Fizeau on the displacement of the plane of polarization of light, produced by transmission through a pile of glass plates : according to Fizeau's own view the experiment was uncertain owing to the numerous disturbing causes that had to be guarded against ; and this doubt as to the feasibility of the observation has been fully shared by Maxwell and most other authorities who have considered the matter.
The only cases in which a first-order effect of the motion of the medium is to be anticipated theoretically are those in which the optical or other disturbance that is examined comes from outside the uniformly moving system which includes the observer. The known instances, which are fully covered and explained by the first-order theory just mentioned, are the Doppler effect of change of optical period arising from the relative motion of the source and the observer, and the astro- nomical aberration of light.
. 39. An interference experiment on the difference of the times of propagation round two cyclic paths, originally sug- gested by Maxwell, has been carried out by Michelson and Morley (§ 30) with the negative result anticipated on all theories as far as the first order is concerned. It occurred to them that by aid of very high refinement in the experimental arrancrements the terms of the second order, which can effect a discrimination, might be successfully examined : the result of the experiments, which have recently been repeated with still further refinement and delicacy, has been to make it reasonably certain that the terms of the second order also vanish, that in fact the time of propagation is independent of the Earth's motion not merely to a first approximation but to a higher order. The theory as hitherto developed in terms of the physical constants of the material media has nothing to say
64 AN EXPERIMENTAL CLUE [SECT. I
to this result, because it is not in a position to assign the second-order changes that the Earth's motion produces in the physical constants of material media and in the instrumental arrangements. But the purely negative result is in itself an important clue towards an extension of theory to the second order, in which we must necessarily deal with the molecular structure of the medium.
Hitherto in treating in this molecular manner of the change of distribution of a free electric charge owing to the Earth's motion, the electrons of the charge have been supposed to be rigidly fixed in the positions they would occup}- when the conductors are at rest, and the additional forces to which the motion would subject them are calculated. It is found that these forces vanish up to the first order : so that to that order no change in the distribution will result. But in proceeding to higher orders we must deal with the problem as one of pure aether in which each electron is a singular point. It is found, on transformation to axes of coordinates (,/•', y , z') moving with the electrons, that corresponding to each resting configuration, expressed by functions of x, y, z and t there is a moving one, expressed by the same functions of e^x , y, z and t , where t =t — vxjc"' and e = 1 + y-jc'-, which has the same electrons in corresponding positions, and also the w^ave-fronts of radiation traversing it in corresponding positions. The inference is made that the change from x to e^x is a real shrinkage of the material system, and that after this has happened the courses of all the phenomena above mentioned are identical in the two systems up to the second order. This inference rests on the hypothesis that an electron is nothing more than a point- snigulanty or pole in the electrodynamic and optical aether, and that the atoms of matter are constituted of aggregations of such poles. Should it turn out that the atoms have also inertia and mutual forces of other kinds than this view involves, which however must arise from another entirely different set of properties of the aether to which no clue has yet appeared, the argument would lose its validity even were this extraneous inertia proportional to the intrinsic electric charge. It is inferred that Michelson's negative result supports the widely
CHAP. IV] MAGNETIC IXFLUENCE OF MOVING CHARGES 65
held conclusion that the main part of the actions, chemical and other, between molecules and between the constituent parts of molecules, is of electrodynamic type : that if gravit- ation and possibly some actions of cohesion stand outside that type, their importance, considered as regards molecular relations, is slight compared with that of the electric actions.
Can the convection of electrically charged bodies along with the Earth affect a magnetometer ?
40. Under no circumstances can the motion through space, with uniform velocity of translation, in which a system of charged conductors participates along with the Earth, produce any magnetic force in a region shielded by a conducting screen from outside electrostatic influence. For consider the influence of a single point-charge q of the system, which is moving with uniform velocity v parallel to the axis of oc : the magnetic force due to it at a point whose distance r makes an angle 6 with v is, to the first order, qvr~' sin 6 tending around the direction of motion of the charge. Thus, taking the point as origin, it is made up of components qvr~'^z parallel to the axis of y and — qvr~^y parallel to the axis of 2 ; while for any system of such charges the effect is obtained by summation. Now at a ^joint inside a conductor in a steady state, situated in a magnetic field (tto, bo, Co), the total electric force, which is thus equal to
(c- %qr~^x, c^l.qr'^y — vCq, C'Xqr~^z + vbu),
must vanish. Hence the magnetic force due to translation of the charged bodies with uniform velocity vanishes to the first order of v/c, compared with that of the field, throughout any space shielded off from the charges by a conducting body ; the reason being that a countervailing charge is induced on the surface of this conducting screen.
This accounts for the negative result of Rontgen's ex- periments, in which he tested whether the convection of a charged body along with the Earth affected the orientation of a compass needle in its neighbourhood. The charged body here induces a countervailing electric charge on the electric screen })rotecting the compass needle, or on the surface of the needle
T. 5
QQ COUNTERVAILING CHARGE [SECT. I
itself, such that whatever be the direction of the Earth's velocity through the aether, the actions of these two charges on the magnetic elements w^hich make up the body of the needle exactly neutralize each other. At first sight it might appear that when the countervailing charge is on the surface of the needle itself, it could exert no resultant influence, because the action of the needle on this charge would be equal and opposite to that of the charge on the needle. But it will appear (§ 41) that the electric force arising from the convection of the mag- netic needle is derived from a potential — vF, and therefore is also countervailed by an electric distribution in the needle and on its surface (cf. § 67) which prevents it from affecting the superficial charge, and is itself not affected because there is no electric force. The effect of the motion of a permanent magnet on its own constitution must be of the second order of small quantities and so will not enter here, because that effect is not altered by a reversal of the velocity.
A more delicate question, though not a practical one, arises if we imagine the permanent magnet to be made of dielectric material, and not screened off electrostatically from the moving charges. In this case, as before, the magnetic force at any point due to the convection of the electric charge with uniform velocity v parallel to the axis of cc is o~^{0, — vR, vQ)*; where (P, Q, R) is the electric force due to the charges, which is not now compensated by the shielding of an induced superficial charge. Thus there would appear to be in this case a real magnetic force throughout the magnet arising from the convection of the charges ; so that, if there could be such a dielectric permanent magnet (and if the direct electrostatic action could be experi- mentally allowed for), a convection effect of the kind here con- sidered might be expected.
41. In the same w^ay a converse influence of the uniform translatory motion of the magnets, through the aether along with the Earth, on the electric force might at first sight be
* More generally a system of electrons or charged bodies whose electric field is (P, Q, E) will, when moving with steady uniform velocity {p, q, ;•), produce a magnetic field c--' {qll - rQ, rP-pR, pQ - qP).
CHAP. IV] ELECTRIC EFFECT OF MOVING MAGNETS 67
anticipated. Generally the kinetic part of the electric force (i.e. of the force which acts on the electrons of material bodies) is by § 59 (7^ — ^z — F, az — 'yx— G, ^x — ay— H) ; thus when as before {x, y, z) = (v, 0, 0) and the system is in a steady state of translation, so that F + vdF/dx is null, there is a direct change in the electric force of amount (vdF/dx, vdF/dy, vdFjdz) arising from the motion, assuming as is natural that the value of (F, G, H) at any point is not sensibly affected thereby. This additional term in the electric force is derived from a potential and so will not disturb electric currents. It will not even tend to alter the electric distributions on conductors in the neighbour- hood : for it can be represented as arising from an ideal electric distribution within the magnets and on their surfaces (§ 67), so that if these magnets are conducting bodies, what would happen would be that actual electric distributions would be induced throughout their volumes and over their surfaces which would neutralize this part of the electric force for their interiors and ^ the same time shield them off from the surrounding space : thus here again no effect would arise*.
* The existence of an effect, of this kind also, is suggested by Wien, Wied. Ann., July 1898.
5—2
SECTION II CHAPTER Y
ON METHOD IN GENERAL PHYSICAL THEORY
On the Scientific Use of Hypotheses
42. The cultivation of d priori physical theories of purely abstract type is not merely an affair of philosophical speculation. Their practical necessitj^ for scientific progress, as also the amount of uncertainty that is inherent in them, may con- veniently be illustrated by a review of some chapters of the scientific history of our present subject.
The master idea of Roemer that the delay in the observed eclipses of Jupiter's satellites, when the planet is in the part of its orbit furthest removed from the Earth, is due to the interval
of time required by light to transmit the event across the inter- vening space to the terrestrial observer, was at the time when it was enunciated an effort of pure scientific imagination, for which the evidence lay solely in the intellectual simplicity^ the explanation which it afforded**. This evidence was many years afterwards very materially strengthened by Bradley's cardinal discovery of the astronomical aberration of light : for
** The idea that light ma}- travel with finite velocity seems to have origin- ated, 80 far as regards modern physics, with Galileo, who had an intention of submitting the subject to experiment. According to Descartes' ideas, light was a sort of impulsive pressure which spread out instantaneously throughout space, in favour of which view he claimed that if the velocity were finite, eclipses would be seen at an interval after their real times of occurrence ; this is precisely the principle that guided Roemer to his estimate of the velocity of light.
CHAP. V] CORPUSCULAR OPTICAL THEORY 69
it was recognized that if light consists of corpuscles moving towards the observer, with a definite speed for each medium, then the apparent direction from which they come must be affected by motion of the observer exactly as Bradley's law requires. The corroboration thus obtained for the hypothesis of the finite velocity of light was powerful and legitimate, and the ideas involved in that hypothesis had much to do with the evolution of Bradley's great discovery, notwithstanding that the physical scheme involved in his use of it, that namely of the corpuscular theory of light, was not merely imperfect but positively erroneous. Had there been independent means at that time of arriving at a tolerable estimate of the Sun's distance, this train of physical deduction would have had some- thing very substantial to confirm the net of pure hypotheses on which it was supported : for it would then have been possible to verify the identity of two values of the velocity of light <lerived from entirely independent sources. In the cognate case of the electrodynamic theory of radiation, a numerical <•! uroboration of this kind was, for a considerable series of years, the only experimental evidence that was forthcoming for a scheme which originated with Maxwell as a train of purely hypothetical deduction, and was on that ground refused acceptance by weighty authorities. Our present object, how- »ver, is to notice that as matters stood at the beginning of the present century, there was in existence a compact and reasoned theory of the finite propagation of light, constructed wholly on the corpuscular view : that this theory, though actually on wrong lines and not merely incomplete, was yet a useful hypothesis in its day, in that it gave a constitution to radiation that in certain ways was so analogous to its actual constitution that it served as a basis for great practical advances in astro- nomical and optical science. We have thus an illustration of the fact that a hypothetical scheme may serve as a useful instrument for the progress of Natural Philosophy, notwith- standing that more minute scrutiny may subsequently prove it to be not merely imperfect but quite on a wrong track. There are in fact two ways in which such a hypothesis may work : it may lead readily to deductions which are really
70 ANALOGICAL ASPECT OF HYPOTHESES [SECT. II i
logically involved in the facts that suggested the hypothesis,} and which will therefore be verified by observation and lead on j to deeper knowledge : but on the other hand it may lead to results which intrinsically depend on the hypothetical inter- . pretation as well as the facts themselves, and by these it will ' be amended or rejected. The corpuscular theory represented existing knowledge as regards the propagation of light with sufficient completeness in its day, to be able to indicate the direction of attack for the development of new knowledge and new relations : in so far it had all the utility of a valid scientific hypothesis : but as the science became enlarged the features in which it was unavailing rose into the more prominent place, so much that it became degi'aded to the position of an analogy reaching only over a portion of the field of phenomena some time before the crucial experimental determinations of the velocity of h'ght in material media decisively robbed it of all higher claim, by proving that in one department of the phenomena its analogy was in error.
It is not superfluous to consider sometimes what there is to prevent many of the scientific hypotheses of physics, chemistry, and other branches of Natural Philosophy, which are at present effective and successful, from being similarly of a merely provisional and analogical character. The uni- formities which it is customary to call laws of nature are often just as much laws of mind : they form an expression of the implications between mind and matter, by means of which" material phenomena are mentally grasped. The mere effort of the mind after a wider formulation of these implications ivill not be wholly fortuitous and useless for progress even when It leads temporarily towards error, for that effort is itself an orderly development taking place in the cosmos of interacting mnul and matter, of which successive stages must have widei^ and deeper ramifications than appear on the surface. The formal analogies between the mathematical theories of different branches of physics perhaps originate as much in the nature of the necessary processes of thought as in the nature of the external things : for ' the mind sees in all things that which it bi-ings with it the faculty of seeing.' ' "^
CHAP. V] HISTORY OF ELECTRODYNAMIC THEORY 71
43. A glance at the order of historical development of electrical theory will serve for further illustration. Here the point of view under which an exact theory was developed, throughout the greater part of the present century, was that of the various portions of a permanent entity called electricity exerting mutual forces at a distance across empty space, after the analogy of the law of gravitation. This scheme was ab- solutely complete for all the usual electrostatic applications. In the domain of electrodynamics and magnetism it explained and coordinated, in the hands of Ampere, Neumann, and Weber, a vast range of otherwise extremely complicated phenomena : von Helmholtz and Lord Kelvin showed how it might have anticipated Faraday's cardinal discovery of the electromag- netic induction of electric currents : Kirchhoff found by calculation that according to it waves of very high period would be propagated along a metallic wire with a velocity which according to Weber's fundamental electric determinations comes out to be about the same as the velocity of light. The scheme also, in Weber's hands, gave a definite and rational account of the mechanical attraction between portions of matter carrying electric currents, and between portions of magnetized matter. As elaborated by Weber it was in fact a complete formulation of the whole domain of the experimental electric science of the time : the circumstance that it was insufficient for the case of bodies moving with velocities at all approxim- ating to that of light, or for vibrations with frequency so high as to approach that of light, was unknown because the production of such experimental conditions had not then been attempted, while the continuity between electrodynamic and optical phenomena had only been vaguely guessed at*. So far as existing knowledge went, the only kind of objection to which the Weberian electrodynamics was exposed was a critical attack on its foundations. This was carried out with strong insistence by von Helmholtz : but his arguments perhaps only brought into clear relief the circumstance that when velocities
* Cf. an interesting early appreciation by Maxwell of Weber's theory, in his memoir On Faraday's Lines of Force, Camb. Phil. T/vn/s. 1855 ; Colh'cted Pai)erg, 1. p. 208.
72 SCOPE OF THE WEBERIAN METHOD [SECT. II
or vibration-frequencies comparable with those of radiation were contemplated, the Weberian scheme was incomplete, and could not in such extreme cases stand, in the light of general dynamical criticism, without fundamental modification.
There was no experimental knowledge in existence in electro- dynamics, previous to Hertz's quite recent classical researches, that could not fairly be collated under the Weberian doctrine : and the preference expressed by Gauss for the notion of an action propagated in time from one moving electric particle to another, instead of a law of instantaneous attraction across space, must be based rather upon his "subjective conviction" as regards the probable nature and fitness of things, and the striving after a view that would lend itself to orderly develop- ment into regions beyond the limit of actual experience, than upon any inadequacy of the Weberian type of formula to include and explain all that was then actually known of electro- dynamic actions. " In a very interesting letter from Gauss to W. Weber (March 1845) he refers to the electrodynamic specu- lations with which he had been occupied long before, and which he would have published if he could then have established that which he considered the real keystone of electrodynamics, namely the deduction of the force acting between electric particles in motion from the consideration of an action between them, not instantaneous, but propagated in time, in a similar manner to that of light. He had not succeeded in making this deduction when he gave up his electrodynamic researches, and he had a subjective conviction that it would be necessary in the first place to form a consistent representation of the manner in which the propagation takes place " (Maxwell, 'Treatise,' § 861)*.
* Cf. Appendix D. Two other attempts at theories of propagation, of a dilTerent kind, are noticed by Maxwell, ' Treatise,' § 862.
That of lliemann depends on an assumed propagation of an electric potential V according to the formula
d-'^ (I-Vldt- = V-V+iwp, p being electric density. This is really the equation of propagation of pressure in compressible fluid, in which there is a distribution of sources of strengths amounting to p (1 - a'- dVldt) per unit volume, or simply of strength p when the fhiid is nearly incompressible. Though this theory of ideal fluid motion and
CHAP. V] MEETING-POINT WITH OPTICAL THEORY 73
44. The consistent representation thus aimed at, of the mode in which electrodynamic action is propagated across free space, — the absence of which formed a barrier to Gauss' progress, — is simply in set terms a dynamical, or, if the term is preferred, an analytical theory of the activity of the lumin- iferous medium. The solution of that problem was developing along different lines of its own, at the same time as these speculations on laws of electrodynamic action across space were being initiated : a complete analytical scheme of the vibratory activity of the aether, constructed on the basis of a masterly discussion of the optical facts, was actually obtained by MacCullagh in 1839, though he fully admitted that it was not such a solution as had anything in common with analogies of the dynamical propagation of waves across material sub- stances. But the times were not then ripe for a new departure transcending in this way all known material analogy, perhaps owing to the circumstance that the more familiar possibilities could hardly have been considered to be exhausted : and it seems to have been only in the vivid and unconventional in- tellect of Macquorn Rankine* that the potentialities of MacCullagh's doctrine obtained clear recognition and develop- ment. The solution thus given by MacCullagh, of the problem of aethereal constitution, was spelled out through examination of the optical interaction between free aether and aether modi- fied by the presence of matter, isotropic or crystalline : precisely the same solution was independently arrived at by Clerk Maxwell twenty years later through an examination, of quite analogous nature, of the accumulated knowledge of electrical interactions across the aether as modified by the presence of different kinds of matter. Most students would probably be struck by the similarity between the analytical methods and
vibration, with mobile sources and sinks, would lead to interesting hydro- dynamic analysis, it cannot afford a sufficiently wide basis on which to construct the much more complex electric theory.
The other attempt, made by Betti (Naovo Vimento 1868), assumes that an electric current is made up of polarized elements like elementary twists in a solid elastic medium. The extent to which such an analogy carries iu electro dynamics has been specified, Pidl. Trans. 1897 A, p. 212.
* Miscellaneous Scientijic Papers, pp. 63, 160.
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7(3 STREAMS AND GRADIENTS [SECT. II
stream can be divided up into tubes of flow, each of which has the properties of an independent pipe or channel devoid of leakage. In certain cases — in fact in all the ordinary cases in which the originating disturbance is local and does not involve the creation of new sources — these channels are ringshaped and the flow in them is thus a circulation round the channels : the flow may then be called a cyclic stream.
A vector of the type — {d/dw, d/dy, d/'dz) x it is proposed to call a gradient vector (the simple lamellar of Maxwell), as it is the gradient or slope of the scalar quantity ^. If this scalar is multiple-valued, its lines of slope will be ring-shaped curves returning into themselves ; and the vector may then be called a cyclic gradient. The total gradient from one point to another is estimated as a line integral along a path connecting them : its value round a complete circuit back to the point of starting is called, after Lord Kelvin, the circulation in that circuit and is null, or else equal to a cyclic constant of the scalar function ;^ of which the vector is the gradient. A circuit in which there is circulation encloses of necessity, is linked with, a core of some kind around which the circulation is established ; and this core must itself be either of infinite length or ringshaped.
The term circuital, as introduced by Lord Kelvin, is synonymous with stream, thus including cases in which cir- culation (jf the stream is not contemplated ; it is therefore entirely distinct from cyclic.
Aethereal Constitution of Matter
46. The difficulty of imagining a definite uniform limit of divisibility of matter will always be a philosophical obstacle to an atomic theory, so long as atoms are regarded as discrete particles moving in empty space. But as soon as we take the next step in ph3'sical development, that of ceasing to regard space as mere empty geometrical continuity, the atomic con- stitution of matter (each ultimate atom consisting of parts which are incapable of separate existence, as Lucretius held) is laisc'd to a natural and necessary consequence of the new standpoint. We may even reverse the argument, and derive
CHAP. V] A PLENUM NECESSARY TO AN ATOMIC THEORY 77
from the ascertained atomic constitution of matter a philo- sophical necessity for the assumption of a plenum, in which the ultimate atoms exist as the nuclei which determine its strains and motions**.
This idea of a plenum with uniform properties throughout all extension, but permeated by intrinsic singular points, each of which determines and, so to speak, locks up permanently a surrounding steady state of strain or other disturbance, forms the ultimate basis of all developments relating to the con- stitution of aether and matter such as are here attempted.
To make a beginning in the direct or synthetical manner, it is necessary to assign a working scheme of properties to the plenum. One way of starting off is to rely on optical theory. The plenum must be the medium of transmission of radiation, >^ with its known finite velocity. It must therefore be specified, in dynamical terms, as possessing, when disturbed, energy of strain and energy of inertia ; for it is only by the interaction of these that propagation in time can be conceived under a dynamical scheme, which takes account of nothing except substance and motion. The precise formal nature of these endowments of the plenum was first unravelled by MacCullagh in his masterly analysis of the optical phenomena of crystals. But he realized very clearly that nothing of the nature of such a type of strain as he was led to postulate, can be thought of as associated with ordinary matter ; so he retained his specifi- cation of the dynamical constitution of the plenum as a purely analytical scheme, that is, as a consistent scheme of properties of this M^^ra-material medium which he could not illustrate from the behaviour of elastic matter. Shortly afterwards Rankine, never timid in his speculations, expounded MacCullagh's analytical scheme soundly and clearly, in full contrast with the elastic properties of matter, as representing a uniform medium or plenum endowed with ordinar}- inertia but with elasticity of purely rotational type. This conception has recently been revived by Lord Kelvin, who illuminated the whole matter by
** It is perhaps not superfluous to point out the argument here involved against any tendency we might have to assign to the aether itself an atomic structure.
78 THE PLENUM AN ULTIMATE BASIS OF PHYSICS [sECT. II
showing how by aid of gyrostatic systems the abstract con- ception of a rotationally elastic medium could be illustrated and closely copied in a material model*.
It is curious that, although the idea of an intimate connexion between the propagation of electric and of optical effects- has"' always been present to sjjeculative physics, yet no attempt was made to ascertain whether MacCullagh's plenum could in addition to its vibratory functions take up such a state of permanent strain as would represent the electrostatic actions between charged conductors, or such state of motion as would represent the electrodynamic action between currents. The first hint on this side of the matter was FitzGerald's passing remark in 1880 f that MacCullagh's optical equations are identical with those of the electrodynamic theory of optics developed by Maxwell.
47. The basis of the present scientific procedure thus rests on the view, derivable as a consequence of general philosophical ideas, that the master-key to a complete unravelling of the general dynamical and physical relations of matter lies in the fact that it is constituted as a discrete molecular aggregate existing in the aether. At the same time all that is known (or perhaps need be known) of the aether itself may be formulated as a scheme of differential equations defining the properties of a continuum in space, which it would be gratuitous to further explain by any complication of structure ; though we can with great advantage employ our stock of ordinary dynamical concepts in describing the succession of different states thereby defined.
(-)n account of the very high velocity Q of ti-ansmission and equilibration of elastic disturbances in the aether, it follows that (on the assumption of a stagnant aether) the motion of material systems across it produces no sensible deviation from the mere succession of equilibrium states of that medium which correspond to the separate configurations of the matter as they arise, so long as the velocity of the matter is not comparable to that of radiiitioii. It is for this reason that simple convection
* Cf. Appendix E.
t ' On the Electromagnetic Theory of Light,' Phil. Trans. 1880.
CHAP. V] WIDE RANGE OF AN EQUILIBRIUM THEORY 79
(if the electric fields belonging to material bodies furnishes so good a first approximation to the laws of the electrodynamics of bodies in motion : although in some cases, such as unipolar induction due to the spinning of a magnet round an axis of symmetry, care must be taken to realize in forming a physical picture of the phenomenon that the effective moving elements are the separate independent electrons, not material bodies as a whole*.
In the case of a homogeneous body moving with uniform velocity v, there will occur changes in the velocity of radiation across it of the order of the first power of v/c, because this velocity is, like v, a directed phenomenon. But the changes of the scalar properties and dimensions of the body itself are of the order of the square of v/c: for example, it is found as a matter of observation, that the relative free periods corresponding to the spectral lines of gases are not altered to the first order by translatory motion of the vibrating molecules along with the Earth and the Solar System.
Mutual aid of electrical and general molecular theory
48. As thus formulated in terras of the aethereal constitu- tion of the individual atoms, the problem of the aethereal relations of material media is one of molecular dynamics ; and it shares in all the difficult and refined considerations of averaging which belong to that branch of physics. But it may be held that its discussion contributes moi-e to the principles of general molecular dynamics than it receives from them. The laws of electrical phenomena have been primarily ascertained in their larger features by a process of mixed induction and <leduction, which proceeded, for more than three-quarters of a century, on wholly different lines from those laid down here. These laws, thus independently and in part empirically ascer- tained, must be derivable from the molecular standpoint ; and their demonstration in that manner confirms and vividly illus- trates the principles by which a transition is made from the dynamics of systems of discrete molecules to the dynamics of their aggregates treated as continuous matter. Practically the
* Cf. Phil. Trans. 1895 A, pp. 727—31.,
80 PRECISION OF THE MOLECULAR PROBLEM [SECT. II
only field (outside the theory of gases) on which these under- lying principles connecting molecular theory with general mechanics have hitherto had scope, is the theory of capillarityj^ and this constitutes an application that has been held not to be free from difficulty. Now in the problem of a cloud of mutually influencing electrons we are on a clearer basis of physical reality than in a discussion of particles acting on each j other with hypothetical forces at a distance obeying undeter- I mined laws. The constitution of an electron is quite definite ; I and its reaction on its neighbours is quite definite, for the : reason that its energy is located in a definite manner in the j surrounding aether. Precision reigns everywhere in the ^a to ; and the transition from the separate electrons to the aggi-egates forming the material medium, treated as continuous as it is presented to the perceptions of sense, must therefore be a definite logical process capable of explicit and precise formu- lation.
The theory of the dynamical interaction between the aether and the matter Avhich subsists in it is on a different plane from a mere formal adaptation of the equations which represent the constitution and activity of the free aether to the case where its properties are modified by the presence of matter. Such an empirical adaptation has worked well for the case in which the matter is at rest*: but for the case in which it is moving with velocity yielding appreciable influence on the phenomena, that is with velocity not wholly insensible com- pared with the speed of radiation, the adaptation in this way has involved the merest guesswork.
The ultimate inadequacy of a method of treating material media, based on merely empirical or speculative additions to the ascertained equations of free aether, had indeed been clearly recognized by von Helmholtz for the last decade of his
* "...And if we attempt to extend our theory [of radiation] to the case of dense media, we become involved not only in all the ordinary dilKculties of molecular theories, but in the deeper mystery of the relation of the molecules to the electromagnetic medium. To evade these difficulties we shall assume that in certain media the specific capacity of electrostatic induction is different in difl'ereut directions,..." Maxwell, ' Treatise,' ii § 794.
!l
CHAP. V] HELMHOLTZ'S ACTION-THEORY 81
life. It would appear that one main object of the close scrutiny of the analytical foundations of dynamics, particularly of the single principle of Action which may be made to cover their whole extent, with which he occupied himself