[ 695 j Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 XX. ADynamical Theory o f Theory o f Electrons. By J oseph L arm or, F.RS, theElectric and Luminiferous Mediu Fellow Received May 16,—Read June 20, 1895. 1. I n a previous paper the concrete representation of electrical and optical pheno­ mena by means of a rotationally elastic fluid rather has been discussed.* In an Appendix it has been shown that whenever there is direct question of interaction between the molecules of matter and the fether, whether it be in the phenomena of magnetism or in the optical phenomena of dispersion and moving material media, the consideration of groups of electrons or permanent strain-centres in the rather, which form a part of, or possibly the whole of, the constitution of the atoms of matter, suffices to lead to a correlation of the various modes of activity ; while this scheme seems to be free from the chief difficulties which have pressed on other methods of representation. The present paper is chiefly concerned with the further development of the mole­ cular aspect of this theory. As a preliminary, it is maintained that a dynamical theory of electric currents, based on the ordinary conception of a current-element, must lead to expressions for the electrodynamic forces which are at variance with the facts. On the other hand, a theory which considers moving electrons to be the essential elements of the true currents in material media, gives a definite account of the genesis and the mutual relations of both types of forcive, the electromotive and the ponderomotive, and gives formulra for them, which correspond in the main with those originally deduced by M a x w e ll from consideration of the properties of his concrete model of the electric field, though they are not substantiated by his later abstract theory based on current-elements. That theory is held to be defective, in the first place on account of the discrepancy with experiment above mentioned, and in the second place, because it is not competent to describe the mode of genesis of a conduction current by electrical separation produced in the element of volume of the conductor under the influence of the field of force ; it is thus an incomplete formula­ tion of the phenomena. The application of the method of electrons to vibrational phenomena leads to formulra for optical dispersion, and optical propagation in metals, which are in general * ‘ Phil. Trans.,’ 1894, A, pp. 719-822. 22.10.95 696 MR. J. LARMOR OR A DYNAMICAL THEORY OF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 agreement with experimental knowledge. These subjects have already been treated from a similar point of view by von H elm holtz in 1892, but his equations, which are arrived at in a more abstract manner, are fundamentally at variance with those of the present theory. They lead, however, to the same type of dispersion formula when the medium is transparent, a type which has been generally accepted as in good agreement with exact measurements of dispersion. The application to the optical properties of moving media leads to F resnel’s wellknown formula, as had already been shown in tbe previous paper. If the theory of the constitution of matter which is suggested in that paper is allowed, it also leads to an explanation of the null result of the well-known second-order experiment of M ichelson and M o rley , of which previous theoretical discussions have quite failed to take cognizance. As the statistical processes connected with molecular theory are of a very delicate character, and subject to suspicion especially when pushed to the second order of small quantities, the formulae are derived by two independent methods, which supple­ ment and illuminate each other but are analytically of very different types. One of these methods, which lies nearer to recent procedure in electrodynamics, and has been used by L orentz in a comprehensive discussion of the subject of moving media, published at the beginning of the present year, is to adapt the fluxes and forces occurring in the fundamental circuital equations of the free aether, so as to make similar equations apply to the aether in ponderable media. Tbe other method is to investigate the forcive acting on a single electron, and then to build up the electric and optical phenomena out of the various types of movements of electrons that can occur in dielectric and conducting media. It is only in this latter way that a rational detailed account of the ponderomotive forces in media which are transmitting currents, or are electrically or magnetically polarized, can be derived ; as a special example, a formula for the ponderomotive pressure due to radiation is obtained. The discussion of the topics above mentioned, with the exception of the M ichelsonM orley experiment, is quite independent of any speculation as to the nature of electrons or the relation of aether to matter. It may be founded directly on M a x w ell’s equations of the electric field in free aether, as now experimentally confirmed. But when we go further, it seems to be a strong argument in favour of a rotational aether, or at any rate of an aether whose actual properties are represented with close fidelity by the scheme of a rotationally elastic fluid medium, that in this way we derive an actual physical structure for an electron, and that by the orbital motions of electrons in the atom we derive a representation of an atom as a fluid vortex, an idea which has always been present to physical speculation from L eu cippu s, through D escartes, down to the recent definite dynamical conceptions of vortices. To justify tentative adhesion to such a view, it is not necessary to be able to produce an explana­ tion of the fundamental properties of mass and gravitation in matter : in particular THE ELECTRIC AND LUMIN fEEROUS MEDIUM. 697 these are not in necessary connection with the electrical and optical phenomena. But it is easy to see that a rotational aether is not inadequate to including such properties among its relations : if the nuclei of the electrons are supposed small enough, the inertia of matter would be definitely represented by the electric inertia of the electrons ; and as the. electrons may then have vacuous nuclei and have each a free period of radial vibration in the fluid {ether, which is not subject to damping by radiation (but subject, however, to a certain instability unless the free spherical form of surface is fortified by some kind of constraint), the gravitation between them may he represented or illustrated by the hydrodynamical pulsatory theory of Bjerknes and Hicks. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 Examination o f Theories involving the Electrodynamic Potential Function. 2. Every scheme for reducing the phenomena of electric currents to a purely dynamical basis must start ultimately from the formula for the electrodynamic potential of a system of ordinary currents, discovered and first extensively applied by F. E. N eumann. In a preliminary tentative discussion of this formula, the only course open appears to be to take the current flowing in an element of volume, and the coordinates of the element, as the independent variables of a continuous analysis extended over the field of currents. If the function is treated merely as a potential of the forces of the field, we may employ the principle of N eumann, afterwards elaborated and extended by von Helmholtz, but still hypothetical in its dynamical aspect, that a variation of the potential, due to alteration of the positions of the conductors, but without an)? variation of the strength of the current flowing along any linear element of a tube of flow, leads to the ponderomotive forces tending to alter their positions : while a variation due to alteration of the strengths of the currents, the conductors beingfixed or moving, leads to electromotive forces tending to change the stiengths of the said currents, but not to alter the motion of the bodies. The dynamical theory of Maxwell aims at going further: the potential function with sign changed is assumed to be the kinetic energy residing in the latent or impalpable active medium which is associated with the current system. The induced electric forces are now taken to be reversed kinetic reactions corresponding to the currents considered as generalized electric velocity components, in accordance with the Lagrangian formula — (/—S dT dT\ ; and these together with the applied electromotive forces make up the total ones that drive the current in conformity with Ohm’s law. In the same way the ponderomotive forces acting on the con­ ductors are the reversed kinetic reactions corresponding to change of position of the material system, which require to be compensated by equal and opposite applied f>98 MR. J. LARMOR ON A DYNAMICAL THEORY OF forces if mechanical equilibrium of the conductors is to subsist. It was one of the discoveries of Maxwell that, for a system of circuited conduction currents, the potential function can actually be formally represented as the kinetic energy 0f a latent moving system, coupled with the palpable conductors and thereby influencing them—of which system the currents are to be treated as generalized velocity components corresponding to electric coordinates which do not appear themselves in the function--without thereby introducing any discrepancy into the general scheme, as already experimentally determined, of the equations of the electric field. Whichever of these methods of development of the potential function is essayed, it is a necessary preliminary with a view to a complete analysis to take the strength of the current flowing in each element of volume, or it may be in a linear element of a tube of flow, as dynamically a separate electric entity. The object of the following discussion is to consider how far this is consistent with a more detailed examination of the forcives thus derivable, particularly with the nature of that part of the internal stress in a conductor carrying a current which tends to alter its shape but not to produce motion of translation of the conductor as a whole. 3. According then to the type of theory which considers a current system to be built up of physical current-elements of the form (u, w) Sr, the energy associated with an element of volume Sr, as existing in the surrounding field and controlled by the element, is T = (Fw + G + H Sr. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 The ponderomotive force acting on the element will be derived from a potential energy function — T, by varying the coordinates of the material framework : it must in fact consist, per unit volume, of a force dF , dG , U^dx + VT * + W T c ’ dHd¥ UT y + V Ty + W-ddzyI’ and a couple (t>H —wG, wY — , uG — vY), t dG the former being derived from a translational, the latter from a rotational virtual displacement of the element.* We may simplify these expressions by taking the axis of z parallel to the current in the element Sr, so that u and v become null; then we have \ a lorce / w —dx , w — , w -d. ^ and adHcoduHple ( —wG, y d z 0). According to the Ampere-Maxwell formula, there should be simply a force at right angles to the current, specified by the general formula ( vc— ivb, wa — uc, ub — * In the previous paper*, § 120, the eouple was omitted. THE ELECTRIC AND LUMINIFEROUS MEDIUM. 699 which becomes for the present special axes of coordinates Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 The forcive at which we have here arrived thus differs from the Ampere-Maxwell one bv a f„orce / dF dG j and a couple ( — w¥. 0) ; these are equivalent to forces acting on the ends of each linear current element, equal at each end numerically to (soF, wG,per unit of cross sect the front end and negative at the rear end. They are thus of the nature of an internal stress in the medium, and are self-equilibrating for each circuital current ail’d so do not disturb the resultant forcive on the conductor as a whole due to the field in which it is situated. From Maxwell’s stress standpoint they would form an equili­ brating addition to the stress-specification in the conductor which is the formal equivalent of the electrodynamic forcive. According to the Ampere-Maxwell formula, the forcive on an element of a linear conductor carrying a current is at right angles to it, so that the tension along the conductor is constant so far as that forcive is concerned. The traction in the direction of the current, arising from the above additional stress, would introduce an addi­ tional tension, equal to the current multiplied by the component of the vector potential in its direction, which is not usually constant along the circuit, and so may be made the subject of experimental test with liquid conductors, as it would introduce differences of fluid pressure. There will also be an additional transverse shearing stress which should reveal itself in experiments on solid conductors with sliding contacts. In particular these additional forces should reveal themselves in the space sur­ rounding a closed magnetic circuit, where the ordinary Amperean force vanishes because the magnetic field is null; in that case (F, G, H) may be interpreted as the total impulsive electric force induced at any point by the making of the Circuit. Professor G. F. Fitzgerald 1ms devised an experiment in which the behaviour of a thread of mercury carrying a strong current and linked with a complete magnetic circuit was closely observed, when the circuit was made and broken. No movement was detected, whereas, when the magnetic circuit was incomplete, the ordinary Amperean forces were very prominent. According to the above analysis, the two types of forcive should be of the same order of magnitude in such a case : the result of the experiment is therefore against this theory. A like negative result has also attended an experiment by Professor 0. J. Lodge, in which he proposed to detect minute changes of level along the upper surface of a uniform mercury thread by an interference arrangement on the principle of N ewton’s rings : when the current was mdcccxcv.---A. 4 X 700 MR. J. LARMOR ON A DYNAMICAL THEORY OF turned on, the section of the thread became more nearly circular owing to the mutual attractions of the different filaments of the current, but there was no alteration in the direction of its length. This experimental evidence, combined with the fact (infra) that a theory of moving electrons, which are certainly independent physical entities, leads simply to the Ampere-Maxwell forcive, seems to justify the conclusion that either the above analysis is wrong, or else the ordinary treatment of electrodynamics in terms of a specification by current elements is physically untenable. As tending to the exclusion of the first alternative, and also as of independent critical interest, the following discussion by aid of the more usual analytical method employed by Ampere, N eumann, and von H elmholtz is given. 4. Assuming that current elements serve as a sufficient physical specification, it is known that the mutual energy (kinetic) of two such elements, iS.s* and must be Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 where

if however the molecule has several free periods, the square of the index must still, Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 THE ELECTRIC AND LUMINIFEROUS MEDIUM. 711 irrespective of special theory, be a rational function of and therefore when expressed in partial fractions must assume the form 1 -j- — ; while in the case of opaque media the constants typified by A will be complex, and other partial fractions not corresponding to absorption bands will also enter. In the case of opaque media, a peculiarity of this analysis, as compared with that of von H elmholtz* and others, is that the conduction current is derived from the electric displacement by the inverse operator k ( m d / d t c r ' ) ~ l instead of a direct operator involving a coefficient of conductivity in place of the coefficient of resist­ ance tr. Notwithstanding that there are as many as four adjustable constants, it is possible, to some extent, to crucially compare the resulting formula for K' with the experimental facts for metallic media. It is known that the broad type of formal theory which simply assigns to the metals a complex index of refraction K'= is con­ firmed by the general agreement between results deduced from calculations relating to reflexion experiments by Beer, V oigt, Drude and others, and those obtained directly from deviation experiments with thin metallic prisms by KuNDT.t It turns out however that the real part of K' is invariably negative for metallic media; and this is a fundamental difficulty in ordinary elastic theories, as it implies instability of the optical medium. On the present theory it implies that k is sufficiently large to allow the third term of K' to outweigh the first two terms. It is also found (by both methods) that if K /J = n1( — lk), then for the better conducting metals, silver, gold copper, n is less than unity, involving velocity of propagation greater than in a vacuum, while ksi a considerable number this implies that the fourth term in the formula for K' is in these cases small compared with the third, which is just what is to be expected from the smaller value of the resistance coefficient cr'. This point is*§ * V on H elmholtz, “ Electromagnetische Theorie der Farbenzerstreuung,” ‘ Wied. Ann.,’ xlviii., 1893. The analysis of vox H elmholtz consists, as usual with him, in a tentative process of fitting known electric laws into a minimum theorem which is an extended form of the Principle of Least Action. For this purpose he uses two sets of variables to represent what we have here called the true current and the displacement current of the free aether ; and he varies them independently of each other. There is no distinction drawn between the polarization and the conduction parts of the true current; so that the current of conduction appears in the potential energy function, thus being assumed to imply elastic strain of the medium, in opposition to the views that have been here set forth. It has been shown by R eiff (‘ Wied. Ann.,’ 1, 1893, p. 361) that the theory of VON H elmholtz does not lead to F resnel’s formula for the influence of moving media on the velocity of propagation, unless the asther is supposed to partially partake of the motion of the material medium; but that when certain terms are omitted from his potential energy function, it is not necessary to assume that the aether is moved with the matter. But it does not appear that any reason is assigned for such modification of the theory, which, as already remarked, seems to be intrinsically at variance with the view here taken. t K undt, ‘ Phil. Mag.,’ 1888 (2). It had been already shown by Voigt (‘Wied. Ann.,’ xxiv., 1885), that the ordinary optical formulae for prismatic deviation apply when the metallic prism is of very minute angle. + Gf.the numbers quoted from D rude in Professor J. J. Thomson’s “ Recent Researches . . . . ” § 355. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 712 MR. J. LARMOR ON A DYNAMICAL THEORY OF confirmatory of the present scheme ; for if the conduction current were derived from the displacement by a direct operator involving conductivity instead of an inverse one involving resistance as here, the opposite effect would be indicated. When the third and fourth terms in K' are predominant, the dispersion will usually be in the abnormal direction, as it is known to be for the great majority of metals, a few of the more conducting ones being however exceptions ; and the value of kwill usually increase with the frequency, which is also in accordance with fact. [Aug. 25.—The introduction of effective inertia into the equation of conduction, as above, thus appears to be essential to the theory. For the actual negative value of the real part of K/ in metals cannot arise from dielectric polarity of the material medium; that, as has often been remarked, would imply instability and consequent destruction of such polar structure. To make the other agency which is at work, namely conduction, effective for that purpose, its equation must involve inertia: and the influence of this inertia appears in fact also in other ways; thus the work done by the electric force on the ions in an electrolyte must be used up proximately in accelerating their velocities, while the increased average velocity reveals itself as Joulean heat. If there were a large number of dissociated ions along a wave-length, which is indeed the condition that the above continuous analysis be literally applicable, it is easy to see by consideration of molecular magnitudes that the effective inertia of an ion would have to be very much greater than its actual mass, or else the effect would be excessive. But this difficulty is only apparent. When the ions are more thinly scattered through the medium, there will be two sets of waves propagated ; the waves of free aether modified somewhat by the presence of the ions, but not extremely different from what they would be if the ions were held fixed in the medium ; and much slower waves propagated from ion to ion with the intervening aether nearly in an equilibrium condition at each instant. The former class alone would be sensibly excited by optical means : it may be formally repre­ sented by the above scheme of equations with the inertia coefficient large, and it is wide enough to include the phenomena both of transparent dielectrics and of opaque metallic media. It is found that, for the wave-lengths of luminous radiation, the real and imaginary parts of the square of the index of refraction are of about the same order of magnitude for all metals. This possibly indicates that the depth to which the light can penetrate in the metal, and therefore also the coefficient of absorption, depends essentially on the ratio of molecular magnitudes to wave-length, which would be about the same for all.] Refraction distinct from . 12. In the former paper, a physical foundation was assigned for MacCullaghs theory of dispersion. That theory being a statical one, must rest on the discreteness of the medium being comparable with the wave-length of the radiation; but conside­ THE ELECTRIC AND LUMINIFEROUS MEDIUM. 713 Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 rations similar to those given in that paper (§ 123) in connection with C a u ch y ’s theory, show that the number of molecules in the wave-length is too great to allow this cause to account for the magnitude of the actual dispersion. Thus by far the greater part of ordinary dispersion, as distinct from refraction, is to be assigned to the sympathetic vibrations of the molecules as here discussed. [Aug. 25.—For waves of long period and therefore great length, approaches the value zero; thus K' approaches the limiting value K and a statical theory represents the phenomena, the molecules being at each instant in the equilibrium position corresponding to the strained state of the surrounding aether. In any case, we may conveniently designate this constant part of K', the square of the index, by the name of the refraction, and the variable part, which depends on the period and ultimately vanishes when the period is long, by the name of the dispersion. In the very wide class of media for which the specific inductive capacity, that is the square of the index of refraction for very long waves, is nearly equal to the square of the index for ordinary light-waves, the dispersion is thus small compared with the refraction. In all such cases a statical theory of refraction, which, according to the argument of the previous paper, must be M acCu lla g h ’s theory, will certainly be correct; and there is ground for making this conclusion general. Thus, in particular, MacC u lla g h ’s theory of the double refraction in crystalline media will hold good as the first approximation. But just like ordinary refraction, this crystalline refraction is subject to dispersive variation when the wave-length of the light is altered. This dispersion of the optic axes in crystals is usually small compared with the double refraction itself, which justifies the present mode of treatment of it as a subsidiary effect to be joined to the main part of the double refraction. In order to obtain equations of propagation in which it shall be included, we have only to add to the effective coefficient of inertia of the aether in M acCu l l a g h ’s equations a subsidiary aeolotropic part. This part may be complex instead of real when the medium is not perfectly transparent, thus including the effect of conduction arising from the presence of free ions ; the general equations of absorbing doubly refracting media may in fact be formulated without difficulty on the lines of § 11. For transparent media, this generalization of M acC u lla g h ’s theory preserves the wave-surface, corresponding to any given period of the light, exactly F resnel’s. The character of the laws of crystalline reflexion also remains unaltered, but the constants that are involved in them are no longer exactly the same constants that occur in the equation of the wave-surface. The asymmetric refraction of higher order, which evidences itself by rotation of the plane of polarization, is obviously of a structural kind, and so is correctly represented by M acCu lla g h ’s terms. I t is itself a highly dispersive phenomenon, on account of the higher differential coefficients on which it depends ; thus the dispersion due to variation of its constants with the wave-length may usually be neglected in comparison.] Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 714 MR. ,T. LARMOR ON A DYNAMICAL THEORY OP Influence o f Motion o f the Medium on Light-Propagation. 13. To deduce F resnel’s law for moving media directly from these principles we have to remember that a movement of the material dielectric through the aether with velocity v parallel to the axis of xproduces an additional disp point fixed in the aether, which (§ 31) is equal to vd^B'/dx. Thus in Ampere’s cir­ cuital relation /dtWmust be replaced by -f- 13'. Again, from the mode in which the other circuital relation appears in the dynamical theory of the medium, d}3/dt must mean the total acceleration of velocity of the aether, due in part to change of time and in part to movement of the material dielectric; thus this also, when it operates on 10', must be replaced by d/dt-f- vd/ may neglect dispersional phenomena and so take 10 + 10' = K10. Thus finally c2V2-B = /), W = - < M ( ^ + 9 ^ + A ' £ ) r - * + ... = - c2e j ( I f + mg' + nh’) i - 1dS + cV j f £ + ^ + r ~ 1dr + ... == c r + . . ., • provided the nucleus of the electron sensibly maintains its spherical form. The resultant traction of the medium over the surface of the electron is obtained' by varying W, and is therefore a forcive 720 MR, J. LARMOR ON A DYNAMICAL THEORY OF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 (± \dx ’ ± AU yd’d a ) ’ where T' = c2S —r , which helps towards compensating the kinetic reaction aforesaid. This resnlt has been obtained by separating away the part of the stress which increases indefinitely as the electron under consideration is approached, and which therefore represents the field of that electron itself in the neighbouring aether. It is noteworthy that the resultants of’ the stresses between the electrons at finite distances apart are represented simply by their electrostatic attractions, whether the field of aether is in equilibrium or is disturbed in any manner whatever; a result which depends on the smallness of the nuclei, and the consequent intensity of the permanent strain near their surfaces. The forcive thus derived, partly kinetic and partly static, is named the electric force, because in the collocation of polarized molecules and free electrons forming an unelectrified body, it tends to produce electric separation by driving the positive electrons one way and the negative ones the opposite way, while it has no tendency to move the element of volume as a whole. But if there is an excess of one kind of electrons over the other in an element of volume of the body, so that the element has a charge q,it also provides a force, equal to multiplied by its intensity, acti the element of volume of the charged body. This mechanical forcive may in certain cases be represented as the unbalanced part of a stress in the manner of M axw ell s stress in the medium ; but in our present order of ideas it is the force itself that is the reality, and the stress is only a mathematical mode of representing it, which has no physical significance as it does not represent the actual stress either in the aether or in the material medium. In one case this analysis must go deeper, namely in deducing the traction on an element of surface of the vacuous core of an electron. One mode of procedure would be to imagine the aether to be continuous throughout the core, but unstrained inside it, thus avoiding internal boundaries,—a method which has been already applied (previous paper, §51) in more intricate cases. We must then imagine the electric charge as freely distributed over the surface of the core in order to maintain this state of equilibrium, and the traction will simply be the usual forcive on this e le c tr ic charge, namely, a normal pull equal to ct2/8tt per unit area, where cr is its surface-density, as above. The same result would be more directly derived by varying the energy in the aether with respect to a displacement of the surface of the core. 17. It remains to find the electrokinetic part of the forcive tending to increase the electric displacement (J ’ gh, ) in an element Sr of free aether. Th ciated with the element is therefore T2= ( / F + ^G + M i)Sr; S d dF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 THE ELECTRIC AND LUMINIFEROUS MEDIUM. 721 No difference is here introduced by motion of the aether such as represents magnetic induction : in fact we have seen that the kinetic reaction on an electron e moving along with the aether is—edF/dtsimply; and to a doublet moving in this way electric displacement in the element of aether itself may be assimilated. This forcive (P£, Q3, R2) strains the aether, and is compensated jointly by the stress which it thus calls forth and the kinetic reaction of the motion involved in change of strain. 18. We have now to formulate the intensity (u, v, of the total circuital current in an extended body, ais made up of conduction current, polarization current, aethereal displacement current, convection currents of free electrons and of polarized molecules, and the current which produces redistributions of electric charges in a conductor. If the vector HP' denote the aggregate result of the polarization in the body, made up of orientation of the electrons with the aethereal displacement bound to them, and HP the free displacement in the aether which excites it, the Poisson-Mossotti theory of polarization will give a relation HP -f- UP' = KIP, where the constant K denotes specific inductive capacity; thus of the total displacement IP + IP', the fraction 1 —K“] is bound to the polarization of the molecules and only the fraction K_1 is free elastic dispx lacement. When an element of volume of a conductor moves across a magnetic field with velocity (p,q, r),the force (P, Q, R) acting on its contained electrons, supposed at rest in it, is of the type P = cq — (V¥ dx ’ acting on the positive electrons one way, on the negative ones the reverse way. When the electrons are in motion forming a current, it might at first sight appear that the ones that drift in the same direction as the conductor moves would experi­ ence the greater force, and so carry most of the current. But no difficulty of this kind occurs ; for it is only the component of the velocity of the conductor at right angles to the direction of the current that is effective, and this acts equally on both sets of electrons. When a material body is in motion, the force tending to produce electric separation of its electrons, that is the electric force exerted in it, is given by this formula whether it carries a current or not. Strictly, the velocity which occurs in this formula is relative to the aether, so that it will involve the velocity of the aether itself which constitutes the magnetic field : but it follows from optical experiments that the latter is extremely minute in comparison with ordinary velocities of material bodies. On the other hand, the force tending to produce free rotational displacement in the aether is (P', Q', R'), where P '= dt dx 722 MR. J. LARMOR ON A DYNAMICAL THEORY 0 V The former force acts on 30', this one on 2D. Under steady circumstances the corresponding parts of the current must have for component K - 1 8P 1 dF 47rO3 dt 47t(J3 dt 9 in which h/dtrepresents djdt -f p -d/xf- [I + r d that mere convection of a steady polarization by motion of the matter through the aether itself constitutes a current. The total current is thus (u, ,v w) given by where K - U P , 1 dP' , u — o'P + W dt + W I t + ^ + “ o. 47rp — dF dx dQ' dy dli' dz - V2^ ; this is on the hypothesis that the material polarization is all induced and therefore circuital, and so adds nothing to the convergence of (P', Q', R'). If the velocity of the material medium is supposed uniform, we have Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 & + dy+ * = - in ^ ~ hence, when the material medium is at rest, the condition of circuitality of the current (u, v, w) is o = ( w + Kc-*J-)P - f ; so that, if initially there is any volume density of electrification, it at once diffuses on to the interfaces in the case of conducting media. The only part of this scheme of equations that is incomplete is the specification of (u0, vQi w0).that portion of the current which redistributes free electrifications. I an ordinary conductor this current is of the same order of magnitude as the polari­ zation current, which is itself wholly masked by the current of conduction, except in the case of optical phenomena. In a dielectric this current does not exist at all. In an electrolyte we may form a provisional scheme of it on the lines of Nernsts theory of migration of the ions, by putting (u0, v0, w0) = kp (P, Q, R), on the supposition that the medium is uniform, where k is analogous to a coefficient of ionic mobility : but we shall thereby destroy the linearity of the system of equations. The fact that free electrons act on each other simply by their electrostatic attractions, THE ELECTRIC AND LUMINIFEROUS MEDIUM. 723 Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 however the sether between them be disturbed, shows that they will tend to drive each other to the surfaces of the conductors, so that the current (w0, v0) wQ) will usually be a transient phase at the beginning of the settling of the disturbance, in agreement with the above, if it ever exist at a ll; and once the free charges are on the surfaces of the conductors they will remain there and be redistributed by ordinary conduction currents. This part of the current may therefore possibly be left out of account; that is, the electric density in a good conductor at rest may be taken to be always null. 19. It is the migration of the positive and negative ions through the unelectrified material medium in opposite directions that constitutes the conduction current: a movement of the medium itself carries as many positive as negative electrons along with it, and so adds nothing to the current : the medium in a sense moves through the current, without carrying it along,—in opposition to the assumption usually implied in the notion of a current element. This is true irrespective of the manner in which the current is distributed between flux of positive electrons one way, and flux of negative electrons the opposite way. In all cases, however, in which there is an electrolyte in the circuit, the law of Faraday shows, not that the current is equally divided between positive and negative ions, but that the numbers of these ions crossing in opposite directions any section of the steady current are equal,—although this necessitates extensive changes of concentration in the electrolytic solution after the current has become steady, when the ions have different rates of migration, as has been demonstrated by H ittorf. If we were to assume that such changes of concentration of ionic electrons are not important in the metallic portion of the circuit, the interface between two metallic media not being sensibly polarizable, it would follow that the velocities of migration of the positive and negative electrons in such conductors would be sensibly equal; and this equality would not be altered by the presence of a magnetic field. ( however § 23.) The conduction current does not involve elastic displacement; if it flows in a com­ plete circuit so that electrons are not allowed to accumulate and exert a back electric force, it will go on permanently, a limit being set to it only by the g'wcm-frictional resistance to the motions of the ions through the medium in the sense of the kinetic theory of gases, which is expressed by the law of Ohm. The relation to the principle of energy of the force (P, Q, R) which tends to produce electric separation in a conductor is expressed by the formula that, if there is a drift of positive electrons one way and negative the opposite way, so as to form a true current {u, , v w),' the time-rate at which the potential energy thereby exhausted is u'P + v'Q tv'R per uni introduced therefore agrees with its formulation in the ordinary elementary theory of true currents, in connection with Ohm’s law. 20. The system of electromotive equations may now be collected together. I he dynamical equations in the free sether are of type mixxjcxcv.— a . 5 A 724 MR. J. LARMOR ON A DYNAMICAL THEORY OF 4^ / + = dF dt ’ where F = \ £ dr + | ( b A _ C dr. dy) r The force acting on an electron e, moving with velocity eF= eye dF d'F — ezb— is of type e— — Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 and its coefficient of electric inertia is Le2, where L depends on the radius of its nucleus. The total current in an element of matter moving with velocity q, r) through the aether is of type T, , K - 1 FB, d f , “ - 0-1 + dt + it + pp + “»> where 1 = gp_ 7 lh _ dF dV and S/dt represents d/dt -j- p djd-j- q d/dy + r djdx ; while where d f . dy , dh P ~ d~x + d y + dz / 47rC2 47tC2 \ dt dx J * 21. When the material system is moving through the aether with uniform velocity p parallel to the axes of x, the equations become /p n p \ _i dJL ' ’ t — \at dff dx ’ “ dy pc, dR _ dt + pb Thus d~dtxF~~|,~ d~zy -T qd, d7z rf—R nV F —„V92Tt ; al. sox, m . al..l cases d--F + d-G--- \- — = 0. As the total current ( u,v, w) is circuital, we must have 1 ± , / ie _ dj \ s ? 47tC2 dt ' \ dt dt It will suffice at present to confine ourselves to a non-conducting medium, so that ) p | | + (1 - c2 V2G — — K — 1 4tt/x Ait ! d d5t 1 d r dx dG cM [J dy dt K -1) ; / — dor ld(A _ <7F\| \dxJ so that ( M - ’^v^G = fL ?L( _ (l9 _ Air dt \ dt d y ) ’ + a (1 - K -> )p^| + (1 - K-‘)p2 —do? ’ and a similar equation holds for H. Thus F, G, H all satisfy equations of precisely the same type up to and including terms of the second order in p/ c ; so that if the vector represent the vector potential, we have (M -c^vm = which is the same equation (with p substituted for v) that was found (§13) for the magnetic induction 1$ by the method of averaged fluxes. 22. Hitherto we have omitted the complication which arises in optical applications owing to dispersion. In the case when the material medium is at rest, it is however easy to include the effects of dispersion and opacity. The total current is made up, as before, according to the specification where, as in § 11, ° 7 « = * < * £ + «D (P.Q .R ). ? = at~h 10 -h 7 Ip d~T&' 15 + 10' = KID - K i + (uo>vo> 726 MR. J. LARMOR ON A DYNAMICAL THEORY OF The condition of circulation shows, as in § 18, that a volume-density subsides in a conducting medium in a non-vibrational manner, so that 'T and p may be left out of account. Then (P, Q, R) = — djdt (F, G, H), and we arrive at the same equations as previously in § 11. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 PonderomotiveForces. 23. When a conductor carrying a current is in a magnetic held, the positive and negative electrons* are urged equally by the magnetic force in a common direction perpendicular to the current and to the field, when they are themselves drifting with equal velocities in opposite directions. A difference in their velocities of migration, such as has actually been found to exist in electrolytes, would lead to a greater force on one kind than the other, and so produce a small transverse electric force repre­ senting the H all effect. When the flow in the magnetic field has become steady, the distribution of drifting free positive and negative electrons across the section will not be quite uniform, thus involving also change of resistance. In any case the aggregate of the transverse forcives acting on the free electrons will constitute a mechanical electrodynamic f'orcive per unit volume, (X, Y, Z) = (?/c — w'b, — u acting on the conductor, where however ( , v', w) is the true current, namely (u —f , v — g, w—h). There will in addition be a mechanical if the element of volume contain an excess of one kind of electrons, including as a limiting case, a normal traction |crN over the surface of each charged conductor; and there will be the forcives acting on the magnetic and electric polarization of the element, derived respectively from potential functions -f- j82-f- y2) and (K — l ) / 87r . (P2+ Q2 + R2), leaving out of account the part which merely produces molecular stress; while, if part of the magnetic or electric polarity is permanent, there will also be couples as in §§ 33, 35. 24. We have thus attained to a complete scheme of equations of the electric field, simply on the assumption that a material medium contains electrons, as many positive as negative in the element of volume unless it is electrified; that these electrons care in part combined into systems which are, or belong to, the molecules, some of these molecules being neutral, and some having an excess of positive or negative electrons, being therefore ions. The only other assumption is that the nuclei of the electrons occupy a negligibly small part of the whole space. As regards the manner in which the electromotive and the ponderomotive forcives in the electric field are accounted for, a similarity with W e b er ’s molecular theory may be remarked.! In that theory electric molecules act on one another directly at a distance according to a law of force * The argument in theso sections is equally applicable, whether the current in metallic conductors is supposed to he carried by material ions, or by electrons considered as immaterial. f W. W eber, “ Electrodynamische Maasbestimmungen Maxwell, “ Treatise,” §§ 846-860. THE ELECTRIC AND LUM INIFEROUS MEDIUM. 7 27 Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 which involves their relative velocity; on the present theory actions are transmitted from one moving electron to another solely by the intervention of the aether. The Weberian theory has been subjected to destructive criticism by von H elmholtz, on the ground that it implies the possibility of a perpetual motion ; the mode of genesis of the present theory obviates such a criticism. The features of the Weberian theory above mentioned were characterized by Maxwell as “ eminently successful ” (“Treatise,” § 856), although this commendation is afterwards limited by the assertion that these features are necessarily connected by the principle of energy. Tt is now, however, recognized that this principle cannot by itself furnish more than one relation between the various quantities that enter into the problem of electrodynamic induc­ tion, so that the fact that any theory, not otherwise discredited, accounts for the two types of forcive in the electric field, ought to weigh strongly in its favour. The range of a theory of moving electrons of the present type, with its underlying aether, is of* course much wider than that covered by W eber. When the mechanical forcive is exerted only on the discrete electrons contained in the element of volume, it clearly cannot involve directly in its constitution an equili­ brating internal stress, such as we found must be included if we take the current element as a connected physical en tity ; it is for this reason that W eber’s theory, though in other respects different from the present one, agrees with it in giving the Ampere-Maxwell ponderomotive force, involving the current however, not the total current as in Maxwell’s formula. Unipolar Induction. 25. The phenomenon of unipolar induction, in which a current is induced when a magnet revolves round its axis of symmetry through its own field of force, is deprived of all difficulty or ambiguity when it is considered under the present point of view. The electrons in the magnet, as they are moved across the magnetic field of the fether, are each subject to a forcive which is proportional to the component magnetic intensity in the meridian plane, and which produces electric separation by drifting the positive ions towards the axis and in the direction of the length of the magnet one way, and negative ions the opposite way. This constitutes an electromotive force along the revolving magnet. It follows for instance that a magnet symmetrical around its principal axis will, on rotation round that axis in its own field, acquire an electrification of excessively minute amount when the circuit is incomplete, but still sufficient to compensate the electric force induced by the motion. We can utilize Maxwell’s equations of electric force, modified so as to refer to a system of axes moving through the aether (“ Treatise,” § 600, or as in ‘Phil. Mag.,’ Jan., 1884, p. 12), to infer at once that for a solid magnet of any form, in motion of any type, the induced electric force is derived from a potential —(F p|- GqHr), where (jp, , r) is the ve of the element of the magnet at the point considered ; so that it can at each instant be 728 MR. J. TiARMOR OR A D Y N A M IC A L T H E O R Y OF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 compensated by the static force due to a minute induced electrification. The maximum difference of potential to be thus compensated, between the axial and circumferential parts of the rotating magnet, is of the order 10-4 volt in the case of the Earth. If the force were not thus derivable from a potential, a magnet in motion would induce currents in itself even when there is no extraneous field, and the energy of the absolute motion through the aether, of the Earth and all other magnets, would gradually be converted into heat owing to this cause ; as things are, only the energy of relative motion of magnets is subject to dissipation in this way. [Aug. 3, 1895.—26. The case of a solid conductor spinning steadily round an axis of symmetry, in a magnetic field which is also symmetrical round that axis, has an important bearing on theory on account of the simplicity of the conditions. If it were legitimate to specify the electrodynamic energy of the system in terms of current elements and their mutual configurations, then in this case the energy belonging to a current element associated with an element of volume of the conductor would remain invariable, on account of the steady configuration of the motion ; therefore there would be no electric force induced in such an element. Thus, for a conductor spinning in this manner there could then be no differences of electric potential caused by the motion. And, moreover, if a so-called unipolar circuit is completed by means of a fixed conducting wire, attached to the spinning conductor by sliding contacts at its equator and one of its poles, there could be no electric force induced in this wire either; and the electromotive force of the current which actually flows round this circuit would have to be sought for wholly in the sliding contacts, which, considering the definiteness of this electromotive force and the great variety of types of contact that are possible, seems to be an untenable alternative. On the other hand, the formulae of the present paper, which considers an electri­ fication to be made up of discrete elements each surrounded by free aether, make the induced electromotive force along any open line of material particles consist of two parts, (i) a part due to motion of this line of particles with respect to the quiescent aether, and equal to the time-rate at which it cuts across the tubes of magnetic induc­ tion, themselves supposed to be stationary in this computation, and (ii) a part equal to the line integral of — d/dt (F, G, H), to be computed by the integral formula for (F, G, H) given above, which represents the effect of change in the inducing system. When the electromotive force is taken round a complete circuit, all theories are of course in agreement. In a case of steady motion, such as is now under consideration, the second of these parts is null, and the electromotive force induced even in an open circuit is given by Faraday’s original rule, in terms of the number of tubes of magnetic induction which cut across the current. In the present order of ideas, a distinction has to be observed between (i) the electric stress in the aether, which is the tangential shear derived from its potential energy and represents the whole forcive acting on free aether, and (ii) the electric force which acts on electrons and moves them through the aether, thus polarizing THE ELECTRIC AND LUM INIFEROUS MEDIUM. 729 Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 dielectric media and producing true electric currents in conductors. In the hands of Faraday and Maxwell, the current of conduction was completed or rendered circuital by an effect across dielectrics, which was equivalent to a current, and there was no difference contemplated in this effect depending on whether the dielectric was a materia] substance or free aether, both being considered to be merely polarizable ; on the present more complete view this dielectric action has to be divided into the true polarization current in the material dielectric, which is excited by the electric force orientating its polar molecules, and the rotational strain in the aether which is regulated by the laws of elasticity of that medium. Thus, the question whether there is electric force induced in the free aether itself is, on our present view, nugatory, there being no electrons on which it could operate. The calculations given in ‘ Phil. Mag./ Jan. 1884, of the differences of potential, and the consequent electrification, induced in a sphere or other conductor of revolution, rotating in a symmetrical magnetic field, will thus, on the present view, be absolutely correct when the conductor is moving in a vacuum. When it is rotating in a gaseous dielectric, like air, there will be aerial flow produced owing to viscosity, after the manner of the action of a fan, and there will therefore be electric force induced in the moving portions of the air ; but the capacity of air for dielectric polarization is so small that the polarity thereby induced will not sensibly affect the state of electrification of the system. If, on the other hand, the conductor were rotating in a liquid dielectric, the polarity induced in the moving parts of the liquid would depend in part on its motion, which would thus very materially influence the distribution of electrification in a maimer which it would not be difficult to calculate if it were necessary to do so. The unipolar current obtained on completing the conducting circuit will, of course, in any case, unless the conductivity is almost evanescent, be practically independent of the nature of the surrounding dielectric. The analytical theory of Maxwell’s “ Treat,ise ” being based on current elements, that theory should, when correctly developed, give in the rotating conductor the null electrification of the beginning of this section, instead of, as here, an electrification based on Faraday’s rule. And this is easily verified if to Maxwell’s original equations of electric force are added, in accordance with von Helmholtz’s correction, terms derived from a potential function which is the scalar product of the vector potential and the velocity of the material medium ; in computing the radial force, which is in cases of symmetrical rotation the total force, these terms exactly cancel the Faraday part. rihe discrepancy between these two theories, is put to a test in a classical experi­ ment already made by von Helmiioltz.# He found that when a conductor was * ‘Berlin Monatsber.,’ 1875; ‘ Ges. Abhandl.,’ I., p. 783. Maxwell bad very early considered the question whether in his total kinetic energy as specified in terms of the current elements and the velocities of the conductors, there are any terms which involve products of these quantities, and he had drawn a negative conclusion from experiment; ‘Treatise,’ II., chap, vi., secs. 568—577. The 730 MR. J. L A R M O R ON A D Y N A M IC A L T H E O R Y OF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 spun in a magnetic field symmetrical around the axis of rotation, there was a difference of potential induced between the axial and circumferential parts as evidenced by resulting electrification, which agreed with F a ra d a y ’s rule within three per cent., a quantity well inside the limits of uncertainty of measurement. He drew the conclusion that an electrodynamic potential theory of the N eum ann type is thus proved inadmissible unless it recognizes polarization currents in the dielectric. The considerations stated at the beginning of this section seem to show that it could not even thereby be helped out ; and that the true inference from the experiment must be a wider one, namely the abandonment of a theory of currents ultimately continuous in favour of one which regards them as made up of discrete electrons separated from each other by free aether. 27. The electrification induced in a conductor by rotation in a symmetrical magnetic field has just been examined; we pass on naturally to the conjugate problem of the magnetic field induced by a rotating electrified conductor, where similar considerations must crop up. The difficulties involved in the interpretation of the results of R ow land’s experiment, on the hypothesis that an electric charge in a conductor is a continuous distribution of electricity, and an electric current a continuous flux, have been already considered in the previous paper,* and proved surmountable only on the extremely precarious assumption that the rotating gilded glass discs of the experiments were divided into mutually insulated segments by the scratches which were intended to prevent F oucault currents. The translation of an isolated electric charge carries on its own surrounding electric field and so alters the electric intensity at each point in the oether; and can therefore certainly induce a magnetic field with consequent reaction on the moving charge. But the steady rotation round its axis of a charged conductor of revolution in no respect affects the field of electric strain in the surrounding aether; that remains steady, and therefore no magnetic, that is kinetic, energy can be locally generated anywhere in it. And there is also the related difficulty previously enforced, that if a charged conductor connotes merely a field of self-locked electric strain in the surrounding dielectric, the elasticity breaking down when the surface of the conductor is reached, the rotation of the conductor round its axis of symmetry could exert no grip on this field of strain, which would therefore not be affected at all. These difficulties will not vanish unless the electric charge on the conductor is made up of discrete portions, separated by dielectric spaces however narrow. The circumstances will then be no longer symmetrical with respect to the axis of rotation so far as these spaces are concerned ; the magnetic field induced, at a place at finite distance from the conductor, will depend not only on the change of electric intensity at that place, which is null as before, but also on the surface condi­ tions which obtain along the boundaries of the dielectric region. electrification in von H elmholtz’s crucial experiment might, however, be formally expressed on that system as due to energy terms of this mixed type, but, of course, extremely minute. * ‘ Phil. Trans.,’ A, 1894, p. 764. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 THE ELECTRIC AND LUM INIFEROUS MEDIUM. 731 The phenomena are thus accounted for on the hypothesis that the electric charge on a conductor consists of a distribution of discrete electrons over its surface. It is true that the facts of electric vibrations on conductors show that these electrons must be extremely mobile and sensiti ve to electric force : this is because of the very strong charges involved in them, but it does not imply that when the conductor is rotated, the electrons will slip backward over its surface and remain where they were, owing to the electric inertia. They will not take up the motion of the conductor just at once ; but there is no electric force which would tend to prevent that result, and the same steady viscous agencies that produce electric resistance when a steady current of electrons is flowing in a fixed conductor, will ultimately make them move with the rotating conductor, after a time relatively long but probably absolutely very short. It does not, in fact, appear that there are any of the hitherto outstanding difficulties of pure electrodynamic theory that are not removed by the hypothesis of moving electrons, to which, from the consideration of several distinct classes of phenomena, and apart altogether from electrochemical theory, we have been com­ pelled to resort. This hypothesis, in its wider aspect involving the nature of matter itself, seems also to have a philosophical necessity ; for the location of causes of disturbance of the uniform all-pervading medium in permanent discrete singularities or nuclei of strain or motion, that belong to it and can move about through it, is the only way of avoiding the introduction into theory of either direct distance actions, or else those assumptions of independent media superposed in the same space and discharging different functions, which violate the maxims of modern physics. Without a precise conception of the causes which produce distur bances in it and form one side of the play of action and reaction, a theory of the aether can be merely descriptive ; while any assumed causes that are not of the nature of singularities arising from the constitution of the medium itself, must introduce a foreign element and so deprive the theory of its interconnection and self-contained character.] Mechanical Pressure o f Radiation. 28. An application of the expression for the ponderomotive forcive will be to the examination of Maxwell’s mechanical pressure of radiation (“ Treatise,” §§ 792, 793). Let us consider a train, of plane waves moving along the axis of x, with their magnetic induction c along z, and their current along y. The circuital relations give dcdc . while the equation connecting electric force with current is MDCCCXCV,— A. 5 B 732 Hence leading to where MR. J. LARM O R ON A D Y N A M IC A L T H E O R Y OF v o-Q fi­ le 47tg2 dt dh Tr _ 9 d*c , , dc d:£ ~ ^" d2 477(774 dt c =c{)e p*cos (n£ — (/.x), ( p -f- u /)2 = K / x c 2 — 47rcr/x,7U. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 Of the total current v, the part F = « 1 47rC2 dt is the true current, derived from motion of electrons and not including the cethereal rotation, c being the velocity of radiation in free aether. The mechanical forcive per unit volume of the material medium is thus X = v c ; hence _ fc Sirft U ttC2 dt Let us first take the case of a transparent medium, for which so that (X(fc d Q _ 4ttc2 dt = i\ V ~ c2 dc /Tk ’ where m is the index of refraction. The value of this indefinite integral, taken over an exact number of half wave lengths in a homogeneous medium, gives a, null result for the total forcive; but if an interface between two media is included in the volume of integration, there will also c 2 be terms in it which may be represented by averaged tractions ^ (1 ~) exerted by this interface on its two sides, or what is the same thing, equal pressures acting on the interface. In air this pressure practically vanishes. This is the result which replaces Maxwell’s formula c02/ 87r/x for the mechanical pressure produced by radiation falling on the surface separating two media. 29 When the train of waves falls on an absorbing medium, the circumstances are different. From the value of c above given, it follows that so that (IQ dt A e+~px cos (nt — qx - e'), when tan e = and Hence THE ELECTRIC AND LUM INIFEROUS MEDIUM. 733 ,—Hj)X ^ - w2 '•'0 /(„p32 + (?2)i {cos (2n£ — 2qx— e') -f- cos e'}, r_i_ a ,2r - ^ J cos(2nt - - o J47TC3 clt C 87TG3 2 (^*2 -f ^3) cos 6r (p2 + '/)* j * | x ,Jx = s L m s 3 u t + m e~** l • ' • Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 in which the lower limit is at the interface, and the upper one at a place where the radiation has been practically extinguished. Thus here there is an average pressure I acting on the interface towards the opaque material medium, which is half of Maxwell’s result. In terfacilConditions. 30. It is important to emphasize the dynamical distinction between electric force and electric displacement in a dielectric. The electric force is that vector which occurs in the rotational stress in the aether, and which, divided by the proper elastic coefficient C/Sn, gives the rotational strain or electric displacement in it. This is not circuital when there are moving electrons present, but by adding a fictitious electric displacement representing the drift of the electrons, a circuital total displacement is obtained. This fictitious displacement or true current represents, however, the rotation in a hydrodynamical flow of the aether which the motion of the electrons sets up, which belongs to the kinetic energy and in ordinary electrodynamical applications constitutes practically the whole of it. At an interface separating media of different dielectric qualities, the electric stress in the aether must be continuous, and as this is tangential, equal to the tangential component of the electric force but at right angles to its direction, it follows that the tangential components of the electric force must be continuous. Again, in a magnetic medium, the magnetic induction represents the smoothed-out velocity of flow due partly to motions communicated by distant disturbances, and partly to the circulatory motions of the magnetic vortices that are caused by the very rapid orbital rotations of the electrons in the molecule. The magnetic induction is therefore always circuital, so that its normal component is continuous at an interface. But as regards the tangential component, we must divide the whole induction into two parts, one of them representing the effect of the vortices in the immediate neighbourhood of the point under consideration, and the other including all the rest of the flow. The latter part is the magnetic force, in the ordinary phraseology, and it is clearly only this part whose tangential component must be continuous, when we cross an interface into a region in which the distribution of the magnetic molecular vortices is different. 5 n 2 734 MR. J. L A R M O R ON A D Y N A M I C A L T H E O R Y OF It is a check on the averaged form of the dynamical equations for a material medium that these six interfacial conditions regarding force and flux should be consistent with the incompressibility of the aether ; and in fact the relations given above show that only four of the six are independent. It is on this account, for example, that the general requirements of the problem of physical optics are satisfied. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 M olecular Current Systems replaced by Equivalent Continuous Currents. 31. To secure a perfectly homogeneous specification of the electrodynamjc field in a dynamical theory, there is no alternative but to reduce magnetism to molecular electric currents. As, however, our analytical equations for an extended medium involve (u, ,v w), the current according to a volume specification, while the m currents are minute whirls not involving continuous flow in any direction, it is necessary to determine the volume specification of currents that shall be their equi­ valent when the element of volume is so great as to contain a large number of whirls of which the average effect only is required. Such a specification is clearly possible; and when we bear in mind chat the volumes occupied by the cores of the whirling electrons form only an excessively small part of the total space, it is also clear that the type of magnetic force which represents the velocity of the incompressible {ether must be a circuital one, that is, the magnetic induction of M a x w e l l . There is no difficulty in verifying by direct analysis that the magnetic force at a point due to a system of molecular currents is the same as the magnetic induction due to the equi­ valent magnet; in fact it is shown (previous paper, § 120) that the said force is equal to the curl of a vector potential z — C d/dy,C d/dx — the same vector potential from which the magnetic induction of the equivalent magnet is derived. Now let us consider the projections of the molecular circuits parallel to the plane of x y : let these projections swell out in area until they come into contact filling up the whole plane, the currents round them being reduced in the same ratio as their areas are increased. Along each edge common to two circuits there will be a differential current flowing, and by making the enlarged circuits rectangular of the form 8x By, it becomes clear that the aggregate of these differential currents make up a volume distribution of currents (— dA jdy, d B /dx, 0) toge the boundary. Hence, adding up for the projections on all three coordinate planes, we obtain the distribution represented by —curl (A, B, C) together with a current sheet on the external bounding surface equal to (Bw —Cm, Cl — An, A m — Bl) per unit area. The validity of this substitution is verified by the analytical transforma­ tion THE ELECTRIC A N D LUM INIFEROUS MEDIUM. 735 dC\ 1 dy ) r dr. Thus if the magnetism is wholly induced, so that (A, B, C) = (a, /3, y), the equivalent volume distribution of currents is v , w), provided k is constant throughout the field; hence the induced magnetism may be ignored if in estimating the induction we multiply the current system by 1 + 47tk or More generally, if a part (A0, B0, C0) of the magnetism is permanent, we have in a region of constant p _ , - rt , . t /dC() (/Bft\ v F = “ 4^ “ + i7r (ify ~ ■ * ) ; of which the last term vanishes when there is no permanent magnetism, or when it is of lamellar type. At the interface between two different media (F, G, H) must be continuous. These relations are sufficient to determine (F, G, H) completely in terms of (u,v, w). Again, a steady electric polarization ( /', li) in a dielectric, moving across the aether with velocity ('p, rq, ), is equivalent in its kinetic effects (previous paper, § 1 to a molecular current system or magnetization — qti, pli — rf', q f — together with a current sheet over the bounding surface, provided the velocity is uniform. In any case, however, it is equivalent to the current system Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 Mechanical Forcivesacting on M agnetically a n d Polarized . 32. We have still to calculate the steady forcive on a molecular current, The torce on a revolving electron c has been shown to be (P, Q, R) where P = c , y - b z - 1 (- - . {x. y,z) being its velocity, and 'F the potential of the static part of the electric held due to such free charges as may exist in the material media. The first two terms, involving the velocity of the electron, will give rise to magnetic forcive, the remaining part will make up into the electric forcive, in a polarized medium. The averaged value of the first two terms, over the orbit of a single electron, gives i \ { c d y — hdz), where i the equivalent current is equal to c multiplied by its velocity v and divided by the length of the orbit ; this is the same as 1j (l da /d x + mdh/dx + n dc/dx) t/S over a surface bounded by the orbit, by Stokes’ theorem in conjunction with the circuital character of (a, b, c). Summing up for all 736 MR. J. LARMOR ON A DYNAM ICAL THEORY OF the orbits of electrons, this would give for the translational magnetic forcive per unit volume (X', Y', Z'), where x' = a =ddax + b d£x + c dx Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 and (A, B, C) is the magnetization of the medium. Now the actual magnetic forcive must certainly, from the most cursory observation of its intensity for iron, involve the magnetic force (a, fi, y) and not the induction c ) ; whence then this discrepancy ? The forcive on an electron involves correctly (a, c), for the velocity of the incom­ pressible aether is circuital. But it must be borne in mind that the molecule contains as many negative revolving electrons as positive, and therefore that the forcive on the molecule is a differential one, positive and negative electrons pulling on the average opposite ways; so that in treating of single molecules we cannot take the actual velocity of the aether to be an averaged function o f position such as ( , b, c). 33. To find the forcive on a molecule, we must in the first place divide this velocity into two parts, one independent of the immediate surroundings of the point considered and depending only on the general character of the field, the other representing the effect of local configuration. The former is the magnetic force as usually represented by (a, /3, )y; it is made up of the force due to the distribution of ideal magn matter introduced by P oisson into the theory, together with that due to the distri­ bution of electric flow. The former element in it is independent of local peculiarities by the ordinary theory of. the gravitation potential; the latter also has never any term involving the flow at the point considered, for similar reasons. The purely local portion of the velocity of the aether consists itself of two parts, a steady one, and one of varying type with very rapid alternation characteristic of the orbits of the electrons ; the part last mentioned averages to nothing, while the former part adds on to the magnetic force to produce the total averaged velocity of flow of the mther, that is the magnetic induction (a, b, c). Now even if instead of the actual velocity of the asther, rapidly varying from point to point, we substituted this average velocity (a, b, c), we should still obtain both a forcive which depends on the general character of the field, and one (though only the regular part of it) which depends on the interaction of the molecule with its immediate surroundings. The latter forcive, which cannot be completely expressed in terms of the quantities of the present theory, belongs to an intermolecular stress of cohesive type ; it is self-equilibrating and contributes nothing to the forcive on the material medium as a whole, though, as we shall see later, it produces deformation ot its parts. If we agree to consider it as a separate molecular forcive, we have for the magnetic forcive proper (X', Y', Z') the formula X' dx f b dx _|_ 0 di dx THE ELECTRIC AND LUM INIFEROUS MEDIUM. 737 There will also be a magnetic couple (L', M', N') per unit volume, where 1/ = By —0/3 ; but there will clearly be no additional molecular couple acting in the element of volume. 34. The steady magnetic force and couple thus obtained may be provisionally considered as derived from an energy function (kinetic) -f + Cy per unit volume, in which (A, B, C) is unvaried (cf. Maxwell, “ Treatise,” § 639); the regular part of the molecular forcive may be considered as involving an internal energy function 27t (A2 + B2 + C2) in which (A, B, 0 ) is varied. More fundamentally, if we divide (a, ft,y) into a part ft', y itself and a p a rt (a0, ft0,y0) due to the inducing field, the actual energy associate with magnetic force is i (Aa' + B/T + Cy ) + Aa0 -f- B/30 -f- Cy0, Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 in which (A, B, 0) is now to be varied ; and this, when there is no permanent magnetism, leads to the same mechanical forcive as an energy function tJ«-(a24 - ftz-f- y2), where k is the coefficient of magnetization, the part of it not thus compensated being connected with the internal work of orientation of the molecules. This energy function is not the whole kinetic energy of the aether; that would be in the present units the space-integral of (a2 -f- b24- c2)/87t together with a molecula the space-integral of [act-j- bft4 cy)/87r 4' («3 4" 4 " y3), of which the l term is exactly compensated by the above mechanical forces acting on the magnetized body, while the other term which remains over goes to produce electrodynamic effects, and in part to represent intrinsic energy of magnetization, and is in fact the electro­ dynamic energy formulated in M a x w e l l ’s scheme. It is to be remarked that the magnetic vector potential, obtained in the previous paper as the vector potential of the Amperean currents in the form F= f (Bd/z - C implies that the orbital velocities of the electrons do not approximate very closely to the velocity of radiation, a condition which in reality is sufficiently satisfied : if that were not so, this expression would require correction for an sethereal dis­ placement part of the molecular current. 35. The magnetic forcive (X', Y', Z ) acting on the actual magnetization I is different from the forcive that would act on the three components of the magnetiza­ tion A, B, C, in case there are currents flowing in the magnet; so that in this connection it is not permissible to treat the magnetization as a vector and resolve it into components. The reason here is that we have actually to do with molecular currents; and if we replace a molecular circuit by its three components we thereby alter the character of its linking with the lines of flow of the current flowing across the medium. But there is a consideration of a less special kind which shows that 738 MR. J. LARMOB ON A DYNAM ICAL TH EO R Y OF this cannot be done : if we were to resolve the magnetization, we should obtain results for the forcive which would be physically different according to the system of coordinate axes that is adopted. If, however, we sum up forcives on separate magnetic poles, instead of calculating forcives on resolved magnetic moments, we arrive again at the correct result, independent of the coordinate system : as will be illustrated immediately in connection with the forcive of electric origin. 36. The ponderomotive forcive of electric origin, acting on a dielectrically polarized medium, is made up of a bodily force (X, Y, Z) and a bodily couple (L, M, N), where X= / f + V ' d£ + h ' d£ ’ Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 in these formulae (/', g', ih!)s the total polarization of the material of the ordinary induced polarization (K — 1)/4tt. (P, Q, R) and, under circumstances of residual charge, also a permanent part. When the magnetic field is varying, so that (P, Q, II) is not derived from a potential, there will be delicate considerations con­ cerning this force, similar to those discussed above in connection with magnetization. The formula here given for X is however correct, because it is what is directly obtained by grouping the actual electrons of the neutral molecule in pairs to form polar doublets ; this is a legitimate procedure although the resolution of the averaged electric moment of an element of volume into three components proves not to be such. Thus the forcive on a doublet SD lying along the axis of a? is a force (3D. dP/dx, 0, 0) and a couple (0, — SD . R, 3D. Q ); and when the axes of co ordinates are changed so that SD becomes 3 ( , , li) these expressions change into the ones given above, because P S/*' -f- Q Sg'+ R change of axes.'* 37. The stress of molecular type, produced by magnetization, offers some points of interest. It has been remarked by yon H elmholtz! long ago that in a polarized medium there exists a material tension along the lines of polarization and a pressure at right angles to them. Each of these is, however, proportional to the square of the susceptibility! of the medium, and they are not necessarily equal; for media * [The circumstances of electrified and magnetized media are not parallel, in the sense that K corre­ sponds to jx. In a magnetized medium the circuital vector, namely the induction (a, b, c), is the smoothed out velocity of the aether; and the magnetic force 7) is that part of it which is inde­ pendent; of the immediately adjacent vortices or Amperean currents. In a polarized dielectric (/, g, h), — (P, Q, R)/4tt, is the smoothed out total elastic rotation or electric displacement in the aether; ( f , 0 tb’) is the time-integral of the polarization current of the material medium which is constituted of movements of orientation of the electrons ; while it is the sum of these vectors that is now circuital, representing the total electric current arising from aethcreal strain and movement of electrons com­ bined. —August 25.] t ‘ Berlin Monatsber.,’ Feb., 1881 ; ‘ Wissen. Abhandl.,* 1, p. 779. t‘Proc. Roy. Soc.,’ April, 1892, p. 63. Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 THE ELECTRIC AND LUM INIFEROUS MEDIUM. 739 of very high susceptibility they are thus far more intense than Maxwell’s hypothetical stress which depends on its first power. Their origin is very con­ veniently exhibited in a chain of iron nails hanging end to end from a pole of a magnet: the nails hold together longitudinally but repel each other transversely. Now consider a longitudinally magnetized bar : this straining together of opposite polarities in neighbouring molecular groups will produce an internal stress in the bar proportional to I 2, which will usually tend to shorten it, but may conceivably in some cases do the reverse. Furthermore the orientation of the molecular groups due to magnetization will directly alter the length to an extent depending on the first power of I. Hence the whole increase of length will be AI -f BI3, in which A and B may each be either positive or negative. When A and B have opposite signs there will be a value of I at which the total effect will reverse its sign by passing through a null value. It appears from experiment that for iron A is positive and B negative; for nickel, A is negative and B negative; for cobalt, A is negative and B positive.* [Aug. 25.—When A and B have opposite signs there will be an intensity of magnetization, I = — A/2B, which corresponds to maximum or minimum elongation. Near this intensity a small change in the magnetization will not affect the length of the bar ; and therefore conversely a change of length produced by tension will have no influence on the magnetization, any changes in the two being independent of each other. It follows that the magnetization at which the influence of tension vanishes is only half the V il l a r i critical magnetization at which the elongation of the bar is null. This is, of course, on the assumption that the phenomenon is a regular and reversible one, and that there is no sensible term in the elongation involving I 3.] There must be similar strain effects connected with the polarization of a dielectric in an electric field, though of course they would be far more difficult to detect. It seems probable that the greater part of the phenomena of electrostriction and magnetostriction is of this character, and that only a small part is due to strain of the material by the direct effect of electric and magnetic attractions between finite portions of the medium. General Considerations. 38. It has been one aim of the preseut analysis to examine how far it is possible to identify, in the play of kinetic and potential energy between the electrons, the main phenomena of matter. In order to comply with the requirements of negative optical experiments by L odge and others, it is necessary to assume the inertia of the tether to be at least comparable numerically with the inertia of derivative character which belongs to dense m atter; and the velocities of its movements are thus ex­ tremely slow. On the other hand, in the descriptive electric theories that are now commonly held, it is usual to consider the aether to be like ordinary matter as regards mdcccxcv.— A. * Cf. E wing, “ Magnetic Induction . . chapter 9. 5 C 740 MR, J. LA R M O R ON A D Y N A M IC A L T H E O R Y OF Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 its inertia, and also as regards capacity for electric and magnetic polarizations. Its density, for astronomical and other reasons, must then be excessively small; and therefore small forces, if unbalanced, will set it into very rapid motion. It has been recently shown by von Helmholtz* that it is not possible for an electrically polarizable medium of small inertia so to adjust itself by finite motions, that the electric and magnetic forcives on it shall be in internal equilibrium and thereby avoid producing very intense movements in it,—unless there exists finite slip at the surfaces of the moving bodies which set up the surrounding electric field; and it seems difficult to see how this slip could be allowed, or what circumstances would regulate its magnitude. On the present view, the inertia of matter is different in kind from that of the aether, and is possibly to be found in the electric inertia which is possessed by elec­ trons. The well-known considerations advanced by Lord Kelvin, which find in the magnetic rotation of light evidence of a rotatory motion round the lines of magnetic force, still obtain a place here, but in a modified form ; it is not the aether itself which is in rotation round the lines of force, but the electrons of the ponderable medium; rotation of the free aether would not here affect the elastic propagation except convectively. We should thus expect the magnetization of the material medium to rotate the plane of polarization without altering to any corresponding extent the mean velocity of radiation, just as in Lord Kelvin's theory : and the present theory is not open to objection on the score that the Faraday rotation is easy to observe, while convection of the radiation by a magnetic field has hitherto been sought for in vain. It is to be observed that a primary condition for the permanence of material phenomena according to a scheme such as has here been sketched, is that the aether should be absolutely devoid of friction. If there were the slightest amount of dissipa­ tion, the motions by whose stability the system hangs together would gradually diminish, and finally the positive and negative electrons would fall into each other and thus suffer complete extinction; the whole material universe would in fact gradually vanish, and leave no trace behind. All formulae with regard to the conservation, under certain circumstances, of linear or angula,r momentum, or of momenta of more general type as in the case of cyclic fluid motions, are bound up essentially with this absence of friction in the aethereal medium. If friction were present, the relative motions of parts of the system would always be transferring momentum as well as energy out of the system into the aether, and nothing could remain absolutely steady or permanent. 39. In a recent memoir by H. A. Lorentz,+ on electrical and optical phenomena in moving bodies, the author arrives, starting from the ordinary equations of the electric t * “ Folgevungen aus Maxwell’s Theorie iiber die Bewegungen des reinen Aethers,” ‘ Sitz. Berl. Acad.,’ July, 1893, *Wiss. Abhandl.,’ 3, p. 526. f “ Versucli einer Theorie . . . in bewegten Korperu,” Leiden, 1895. THE ELECTRIC AND LUM INIFEROUS MEDIUM. 741 Downloaded from https://royalsocietypublishing.org/ on 09 March 2024 field, at fundamental equations of the same type as are given by the present theory. The method adopted by him is, as here, to find the cause of electrical phenomena in the motion of ions. He considers them merely as volume distributions of electricity confined to limited spaces, and with this guiding idea determines the extension of Maxwell’s fundamental equations that will most conveniently cover the extension of the problem to include their movements. By transforming, as in § 14, to axes of co-ordinates which partake of the uniform motion of the material system, various results relating to the independence between electrodynamic phenomena and such motion are deduced, and Fresnel’s formula for the effect on the velocity of radiation is shown to be involved. The conception of a molecule, electrically polar by reason of its positive and negative electrons, of its magnetism due to the orbital motions of these electrons, and of its diamagnetism due to changes in the orbits, and the conception of conduction by a