UNIVERSITY OF CALIFORNIA ANDREW SMITH HALLIDIC: 1868 ^Hk3 1901 ELECTEIC WAVES HonHon: C. J. CLAY AND SONS, CAMBEIDGE UNIVEKSITY PKESS WAEEHOUSE, AVE MAKIA LANE. laggofo: 50, WELLINGTON STREET. F. A. BROCKHAUS. $eta lork: THE MACMILLAN COMPANY. Botnbag anU Calcutta : MACMILLAN AND CO., LTD. [All Rights reserved.] ELECTRIC WAVES BEING AN ADAMS PRIZE ESSAY IN THE UNIVERSITY OF CAMBRIDGE BY H. M. MACDONALD, M.A., F.KS. FELLOW OF CLARE COLLEGE, CAMBRIDGE. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1902 HALLIDIE (Camfortoge: PRINTED BY J. AND C. P. CLAT, AT THE UNIVERSITY PRESS. PREFACE. nnHE following essay was undertaken with the object of discussing the possibility of obtaining directly from Faraday's laws a consistent scheme for the representation of electrical phenomena, and of applying the results to obtain the quantitative relations which exist in certain cases of the propagation of electrical effects. Maxwell's memoir on " A dynamical theory of the Electro- magnetic Field," communicated to the Royal Society in October, 1864, marks a new departure in electrical theory. In it the analytical representation of Faraday's laws is systematically developed and applied, as also the analytical formulation of the electromagnetic theory of light which had already been proposed by Faraday in 1846 ; but these contributions to electrical theory, though of great importance, are subservient to the main object of the paper, which is to shew that the laws of electrical phenomena obey the same general principle as the laws of mechanics. It has not been sufficiently noticed that Maxwell presented his theory under two forms, in one of w hich the electrokinetic energy is expressed in terms of currents, and in the other in terms of magnetic force and induction. The first form is the one used throughout the paper, the second form being given without application; and in his Treatise the first form is used in the discussion of the general theory, the second form being given later and only applied to discuss the 1 03596 VI PKEFACE pressure of radiation and magneto-optic relations. Subsequent writers, FitzGerald, Heaviside, Hertz and others, have taken the second form of the energy function as the starting-point of their investigations. The fact, that in certain cases the direct application of Faraday's laws gives without ambiguity results different from those which appear to follow from the latter form of Maxwell's theory, led the writer to suspect that there must be some flaw in the process of reasoning by which this form of Maxwell's theory is deduced from Faraday's laws. This suggested the procedure adopted in this essay, to begin by applying Faraday's laws, without the intervention of any dynamical theory, to the different cases which can arise, and to examine whether the results obtained are consistent with observation. A satisfactory scheme having been developed in this way, a short account of Maxwell's procedure is given with the view of discovering the source of the discrepancy, and the result of examination is to shew that the first form in which Maxwell presented his theory is a logical consequence of Faraday's laws while the second form is not. That this has not been noticed earlier is to be explained by the fact that Maxwell did not use the second form of his theory to obtain results whereas subsequent writers have used it in preference to the first form, and that, so far as the applications made by Maxwell are concerned, the same results follow from either form of the theory. That the second form of the theory is easier to work with is obvious, as in this form the Lagrangian function is of the same type as the Lagrangian function of a material medium, thus allowing the argument from analogy to be used, and this explains the preference shewn for this form of the theory. Having found that the second form of the theory is not logically involved in the first, it becomes necessary to ascertain PREFACE Vll what assumptions its use has led to, and it appears that many of the received ideas as to the nature of the aether, as for example the doctrine of a fixed aether, have arisen in this way. These assumptions not being directly concerned in the first form of Maxwell's theory, and this form being the one which logically follows from Faraday's laws, the ideas that have arisen from the assumptions cannot be regarded as being required by the facts. The next step is to examine whether the form of Maxwell's theory thus adopted is consistent with the laws of dynamics, and the logical development of this form of the theory is then resumed at the point where it was left by Maxwell. This latter is perhaps unnecessary in view of the fact that earlier in the essay it is shewn that Faraday's laws are sufficient in themselves for the development of a scheme of representation ; but it was thought desirable to add it for the sake of com- pleteness. The object of the second part of the essay is the application of the general theory to some of the problems that present themselves in connection with the propagation of electrical effects. It appears that there is an essential difference between a simply-connected and a multiply-connected space in respect of the propagation of electrical effects, there being no permanent free oscillations such as would not be dissipated by radiation be- longing to an indefinitely extended simply-connected space, while there are such permanent free oscillations associated with each of the conducting circuits that render an indefinitely extended space a multiply-connected region. . The explicit recognition of this fact makes it possible to simplify the mathematical theory of electric waves. The complete determination of the circum- stances of propagation of waves of the latter class can be reduced to the solution of linear differential equations involving one independent variable : the dependent variable belonging to any Vlll PREFACE conducting path returns to the same value on going once round the path when closed, and the waves circulate without decay. In an open path the dependent variable vanishes at both ends and the energy is dissipated in radiation. The principal appli- cation of the theory is to the effects investigated by Hertz in his experiments on electric waves. The development of the analysis gives results which are in close agreement with the observations of Sarasin and de la Rive, who repeated Hertz's experiments under more favourable conditions ; it also gives a satisfactory explanation of the discrepancies in some of Hertz's own obser- vations. The investigation thus given for the free periods of resonators is only approximate ; the accurate equation for the determination of the periods of a resonator can be easily deduced from the general theory of these waves, but this equation is such that the calculation of the periods from it would involve great labour, and as the error involved in using the approximate theory, in which the distance between suc- cessive nodes is approximately half a wave-length, must be extremely small, the gain in theoretical accuracy did not appear to be sufficient compensation for the additional labour. The essay in its present form was completed at the end of 1900, with the exception of Chapter VIII, the paragraph relating to it in Chapter I, and the Appendices. The para- graphs have also been renumbered throughout and the cross references altered accordingly. Chapter VIII has been added with the view of developing the manner in which the energy of permanent vibrations associated with closed circuits is distri- buted, and the distinction between them and the waves due to open oscillators. To effect this it was found desirable for the sake of uniform treatment to give a short account of radiation, starting from the first form of Maxwell's theory. Appendices A and B are intended to elucidate the point of view adopted in the first part of the essay. Appendix C supplies the analysis PREFACE IX belonging to Chapter V, which was omitted on the ground that it might unduly interfere with the course of the argument. Appendix D contains an application of a method developed in the essay to a dynamical problem in diffraction which includes all the cases hitherto solved; this appendix also contains the analytical investigation of a result stated without proof in Chapter VI. In the theoretical discussion references are given to other writings only when these bear directly on the argument, this being thought sufficient in an essay. On the other hand, where special problems are dealt with, references have, as far as is known, been given to all the previous investigations. To Mr J. Larmor and to Prof. A. E. H. Love, whose assist- ance in revising the sheets for the press he was fortunate enough to obtain, the writer is greatly indebted. The many valuable criticisms and suggestions received from them have done much to remove obscurities and improve the book in other ways. Acknowledgement is due to the officials of the University Press for the careful and obliging manner in which they have done their share of the work. CLARE COLLEGE, CAMBRIDGE. August 4, 1902. CONTENTS. PAGE CHAPTER I. INTRODUCTION 1 CHAPTER II. THE EQUATIONS OF ELECTRODYNAMICS . Faraday's laws. The equations obtained from them. Modi- fication of the equations. Propagation of electrical oscillations. Integrals of the equations suitable for this case. Perfect conductors. Forms of the solutions for perfect conductors. Solution expressed in terms of the electric and magnetic forces given over a closed surface. Propagation of arbitrary disturb- ances. CHAPTER III. CONVECTION CURRENTS IN MOTION AND MATERIAL MEDIA .... 20 Electric force acting on a convection current. Material media. Propagation of waves through transparent material media. Influence of motion of translation of medium on the 7 velocity of propagation : Fresnel's law. Astronomical aberration. Modification of the equations so as to apply to any material medium. CHAPTER IV. MAXWELL'S DYNAMICAL THEORY OF ELECTRICITY 28 Maxwell's statement of what he proposes to do. His pro- cedure. Electric displacement and displacement current. Electric force. The Lagrangian function. Maxwell's transformed ex- pression for the electrokinetic energy. His transformation ille- gitimate. Relation of Maxwell's theory of light to MacCullagh's. Xll CONTENTS PAGE CHAPTEK V. DYNAMICAL THEORY . . . .35 Object of a dynamical theory. All the degrees of freedom of a dynamical system are not necessarily in evidence. Influence of the unobserved degrees of freedom. The different cases which can arise. Potential energy the energy of concealed motions. The modified equations. Application to the aether. Material media. Discussion of the theory. Comparison of various theories. CHAPTER VI. PROPAGATION OF ELECTRICAL EFFECTS IN SIMPLY-CONNECTED SPACES . .48 Impossibility of permanent free electrical oscillations in a simply-connected space external to a conductor or to a system of conductors. Illustration : the space between two concentric spherical conducting surfaces. Condensers : free periods. Effect of open ends. Effect of joining the faces of a condenser by a wire. Effect of removing or setting up constraints. Non- permanent free oscillations. CHAPTER VII. PROPAGATION OF ELECTRICAL EFFECTS IN MULTIPLY-CONNECTED SPACES . . 62 Possibility of permanent free oscillations in multiply-con- nected spaces, whether these spaces are infinitely extended or not. Free periods of the waves associated with a circuit. The case of any number of indefinitely extended parallel straight conductors. Thin wires in the form of closed circuits. Ex- pressions for the components of the electric force due to the waves associated with the circuits. The effect of finite cross- section of a wire. Induced vibrations. CHAPTER VIII. RADIATION 69 Rate of change of the electric energy inside a closed surface. Application to the simple Hertzian oscillator. Rate of decay of the oscillations sent out by an oscillator. Density of the electric energy at any point due to an oscillator. Relation of temperature to radiation. Conditions of permanency of a group of ions. The force which one permanent group exerts on another. Permanence of waves associated with circuits. CONTENTS xiii PAGE CHAPTER IX. OPEN CIRCUITS 85 The conditions to be satisfied at an end and along the circuit. The solution of the case of the terminated straight circuit suf- ficient. Solution of this treated as the limit of a right circular cone. Formation of the equations. Integration of the equation in series. Determination of the arbitrary constants. The form of the solution when the angle of the cone is very small. Its form at a distance from the circuit when the sources are not near to the circuit. Its form near the circuit : summation of the series in this case. Case where the sources are near the circuit : summation of the series. Modification of the results when there is a small conducting sphere at the end of the circuit. Alter- native method for determining the waves at any point of the circuit. Application to the case of any disturbance symmetrical with respect to the circuit. CHAPTER X. STATIONARY WAVES IN OPEN CIRCUITS . 106 Expression for the stationary waves in any open circuit. Case where there are no sources or other free ends near to the free end. Case of a resonator. Determination of the positions of the nodes. Calculation of the wave-lengths belonging to the fundamental tone and the overtones of a resonator. Comparison of theory with the experimental results of Sarasin and de la Rive, Effect on the wave-length of the spheres at the free ends of a resonator. Effect of plates at the free ends. The rate of decay of the oscillations in a resonator. Positions of the nodes on a straight wire along which there are waves. Comparison of the results with experiment. Form of the wave-fronts in the neighbourhood of a straight wire. Comparison of theory with the observations of Birkeland and Sarasin. The experiments of Hertz : the mirror experiment ; the interference experiment. The experimental evidence for the existence of overtones of a resonator. APPENDIX A. THE RELATION OF THEORETICAL TO EXPERIMENTAL PHYSICS . . .134 APPENDIX B. CONTINUOUS MEDIA .... 143 APPENDIX C. THE ELECTRODYNAMICS OF MOVING MEDIA 160 APPENDIX D. DIFFRACTION . 186 ^OTE LiSl UNIVERSITY ,FOS CHAPTER I. INTRODUCTION. 1. THE object of the following essay is the discussion of the circumstances of propagation of electrical effects. The first part consisting of the second, third, fourth and fifth chapters is devoted to the discussion of the general theory. The equations of electrodynamics for a single medium in which there are conductors at rest relatively to each other are deduced from Faraday's laws, the current at a point being, in accordance with Maxwell's views, supposed to be made up of two parts, the aethereal displacement current and the convection current at that point. The integrals of these equations, when waves of definite period are being propagated in the medium, are obtained in terms of the distribution of the convection currents throughout the space ; the electric and magnetic forces at each point of the medium, which belong to waves of the period chosen, are then expressed in terms of the distribution of convection currents having the same period. The expressions in the general case, where there are waves of different periods, are sums of these. These integrals are then modified to suit the case of perfect conductors, and it appears that, in this case, the expressions for the electric and magnetic forces at each point of the medium take the form of integrals over the surface of the conductors, the unknown quantity being the magnetic force at the surface of a conductor. It is a conse- quence of these results that, when the components of the magnetic and electric forces tangential to a surface at each point of it are known, this surface being closed and enclosing M. E. W. 1 2 INTRODUCTION [l all the sources of the waves, the electric and magnetic forces at any point can be immediately deduced. The propagation of any disturbance through the medium is shortly discussed, and it follows from these investigations that the way, in which Maxwell's aethereal displacement is introduced, involves the assumption that it, as well as the electric force conceived of as belonging to it, are independent of the motion of the aether. 2~ In the third chapter convection currents in motion and material media are considered, the difference between material media and the "aether being taken to be the presence of con- vection currents. Faraday's laws are applied on this assumption to obtain the equations of propagation of electric waves in transparent media, and the results so arrived at are in agree- ment with Fresnel's formula for the effect of moving material media. The application of these laws to any material medium leads to equations determining the circumstances and to expres- sions for the mechanical forces acting on material media which are the same as those used by Larmor, Phil. Trans. A. 1897. 3. Maxwell's dynamical theory of electricity is discussed in the fourth chapter, where the manner in which the various functions arise is considered and in particular it is shewn that on his theory aethereal displacement is a vector which defines the electrical degrees of freedom of the aether at each point of space, this vector being conceived of as associated with the points of space not with definite elements of the aether. It is then observed that the Lagrangian function of the motions of the aether which is obtained on this theory is a modified Lagrangian function, the coordinates specifying degrees of freedom of the aether other than electrical having been eliminated by the process by which the function has been built up ; so that although this function, when the convection currents are completely known, is sufficient to determine the electrical relations of the aether to material media, it does not supply sufficient knowledge from which to develop the dynamics of the aether. Maxwell's second expression for the electrokinetic energy, which expresses it in terms of the magnetic force and the magnetic induction, is shewn to have I] INTRODUCTION 3 been obtained from the previous one by a transformation which is in general invalid. It then appears that it is the use of this second expression which has led to the assumption that Maxwell's electromagnetic theory is the same as MacCullagh's optical theory and thence to the identification of the magnetic induction as the velocity of the aether. This again has given rise to difficulties connected with the effect of a magnetic field on the propagation of light and to difficulties connected with permanent magnets which have compelled the further as- sumption that the density of the aether is indefinitely great. This second expression for the electrokinetic energy being illegitimate, it follows that Maxwell's theory is essentially different from MacCullagh's and that the conclusions referred to above, which are based on the identification of the two theories, are not a necessary consequence of Maxwell's theory. 4. The conclusion having been arrived at that Maxwell's theory, though sufficient for the treatment of electrical changes did not furnish a complete representation of the motions of the aether, it became necessary to inquire what form a dynamical theory could take when only part of all the possible motions can be taken into account and whether Maxwell's theory is of a form which could arise in this way. This inquiry forms the subject of the fifth chapter ; the fundamental assumption made is that if all the degrees of freedom everywhere existing could be put in evidence, the Lagrangian function of all the corresponding motions would be a homogeneous quadratic function of the velocities belonging to all the degrees of freedom, the coefficients in this expression being functions of the coordinates which specify the degrees of freedom, and, further, the motions are assumed to obey the principle of Least Action. The existence of a class of motions, in which the knowledge of a single function, viz. a modified Lagrangian function of the form T V, is sufficient to deter- mine completely the motions corresponding to the degrees of freedom, the coordinates belonging to which have been retained, is demonstrated ; the motions belonging to those which have been eliminated cannot be determined from a 12 4 INTRODUCTION [l knowledge of this function. The expression F, which occurs in this modified Lagrangian function, is equal to the energy of the motions which correspond to the degrees of freedom whose coordinates have been eliminated, so that the potential energy is the energy of the concealed motions. The forms which can arise under other circumstances are also discussed. The Lagrangian function, which is derived from observation, is, in every case, a modified function in which only the observed degrees of freedom ' appear explicitly. In attempting to apply a dynamical theory to the aether it has to be remembered that the energy of the motions of a continuous medium, such as it must be postulated to be, cannot be represented by a sum of the form fffp(u* + v* + w^dxdydz; to do so would be to endow the aether with atomic structure. It then follows that the application of the previous results to the aether, all its possible degrees of freedom being taken into account, can give rise to a modified Lagrangian function the same as that arrived at on the Faraday-Maxwell theory. The manner, in which the effect of material media can be taken into account on this theory, is then discussed and the theory is compared with the other theories which have been proposed. 5. It having been shewn that the Faraday-Maxwell theory gives a consistent account of electrical phenomena and is itself consistent with dynamical theory, and the effect of the motion of an observer and his apparatus having been ascertained, this effect being, in the case of the propagation of electric waves of the kind which occur in experiments such as those of Hertz, negligible, the propagation of electrical effects of this kind is discussed in the second part on this assumption. The propaga- tion of electrical effects in a simply connected space is dealt with in the sixth chapter, where it is shewn that permanent free electrical oscillations in an indefinitely extended simply connected space are impossible. The case of the space between two concentric spherical surfaces is treated in detail in illus- tration of the general theory. The oscillations belonging to condensers are discussed and the result of an investigation to determine the effect of an open end is given. The effect of I] INTRODUCTION 5 joining the opposite faces of a condenser by a thin wire is investigated and it is shewn that, if originally the faces are not closed surfaces, the periods are practically unaltered and no new period is introduced, but that, if originally the faces are closed surfaces, the already existing periods are approximately unaltered and a new period, which is very long in comparison with the others, is introduced. The effect of removing or setting up constraints is discussed, more particularly in the case where the space throughout which the constraint is re- moved or set up is not finite. Non-permanent free oscillations are discussed, the case of a change in the dielectric medium being treated in detail. The principal result obtained is that reflexion is the essential condition for the existence of free oscillations in a simply connected space. 6. In the seventh chapter the propagation of electrical effects in multiply connected spaces is considered, and it is shewn that permanent free oscillations can exist in a multiply connected space whether the space is infinitely extended or not, the possible free periods for the infinitely extended doubly connected space in which there is a single circuit being s/nV, where s is the length of the circuit, V the velocity of radiation and n any integer. It is then shewn that the problem, in the case of waves travelling along any number of parallel straight cylindrical conductors, is reducible to one of conformal repre- sentation. The case of circuits of any form is discussed and expressions for the components of the electric force at any point due to the waves belonging to a single circuit are obtained. The effect of finite cross-section is then considered and it is proved that, taking the cross-section of the wire to be circular, of radius small compared with the radius of curvature of the wire and with the wave-length of the oscillations, the results obtained for the circuit are still applicable. The electric waves induced in a thin closed wire are discussed ; and it follows that for any number of thin closed wires the expressions for the components of the electric force at any point are sums of the expressions found for a single closed wire, whence solvable cases for the effect of conductors can be deduced. The waves N 6 INTRODUCTION [l belonging to a circuit are unaltered by any others which exist along with it, the effect of conductors being to super- pose on the waves belonging to a circuit waves belonging to other circuits which form the image in the conductors of the first circuit. 7. The radiation of electric energy is treated of in the eighth chapter. In this connexion it was found necessary to investigate the expression for the rate of transfer of energy across a closed surface using Maxwell's first expression for the electrokinetic energy. The result is an addition to Poynting's expression of a part which, integrated throughout a complete period, vanishes so that results, which take account of average radiation only, are unaffected. The intensity of the radiation from, and the density of the distribution of energy due to, a simple oscillator are investigated, and the results are applied to obtain relations between the temperature and the intensity of radiation. The condition of permanence of a group of ions is obtained and the law of the force between permanent groups is deduced, the force between a pair, neither of which possesses free electricity, being at most of the order of the inverse fourth power of their distance apart. It is then shewn that no energy is radiated away from a circuit, as was to be anticipated from the results of the previous chapter. 8. The existence of permanent free oscillations associated with circuits having been established it follows that, if a small gap be. made in a circuit, oscillations of this kind, though they will not then be permanent, will be set up in it ; the treatment of these open circuits forms the subject of the ninth chapter. It is first proved that, if the case of a straight wire with a free end can be solved, the case of any form of thin wire whose curvature is continuous can be solved, as the waves can only get into the circuit at the open end. The waves set up by the simplest kind of source, which can produce waves of the kind sought in a straight wire with a free end, are then found. The effect of a small sphere placed at the free end is discussed and shewn to be negligible if the radius of the sphere is small compared with the length of the waves. In the tenth chapter I] INTRODUCTION 7 these results are applied to the case of resonators and it appears that the fundamental wave-length of a resonator depends on its length only, the result for a resonator in the form of a circle being that the fundamental wave-length is 7*95 D, where D is the diameter of the resonator, which agrees with the results of the experiments of Sarasin and de la Rive, who obtained the value 8 D for a circular resonator. The rate of decay of the oscillations belonging to a resonator is investigated and shewn to be very small, which agrees with the results of Bjerknes' experiments. The case of a wire with an open end, from which the waves are radiating freely, is then discussed and it is found that, for stationary waves, the distance of the first node from the free end is "192 X, where X is the wave-length of the waves which are being observed, this result again agreeing with the result of the experiments of Sarasin and de la Rive. The forms of the wave-fronts in the neighbourhood of a straight wire, from the end of which the waves are being freely radiated, are then investigated and it appears that, in travelling along the wire from the free end, the wave-fronts change from being para- boloids of revolution with the free end as focus and the wire as axis to being planes perpendicular to the wire ; this agrees exactly with the observations of Birkeland and Sarasin. Fur- ther, on the side away from the wire the wave-fronts tend to set themselves at right angles to the radius vector from the free end, which was found to be the case by the same observers. The effect of the upper harmonics of a resonator is considered, and it would seem that they provide an explanation of some results observed by Sarasin and de la Rive and that they are to some extent the cause of the result obtained by Hertz in his interference experiment for comparing the velocity of electric waves in air with their velocity along a wire. The result of these investigations relative to electric waves in wires and resonators is to shew that there is complete quantitative agreement between experiment and theory as well as qualitative agreement. CHAPTER II. THE EQUATIONS OF ELECTEODYNAMICS. 9. THE equations which determine the magnetic and electric forces at each point of space, conceived of as filled by a medium in which there are conductors at rest relatively to each other, can be deduced from Faraday's laws that (1) a closed current is equivalent as regards the magnetic field produced by it to a magnetic shell of the same strength bounded by it, (2) the electromotive force in a closed circuit is given by v dw - dt> where W is the number of lines of magnetic induction which pass through it. Further, on this scheme all currents are regarded as closed. Denoting the components of magnetic force at any point #, y, z by a, J3, 7, the components of magnetic induction by a, b, c, the components of electric force by X, Y, Z and the components of total current strength by u, v, w, the axes of reference being fixed relatively to the conductors, Faraday's laws are expressed by the equations I a. dx + @ dy + 7 dz = 4?r II (lu + mv + nw) dS... (1), JX dx + Ydy + Zdz = - jf (la + mb + nc) dS. . .(2), Il] THE EQUATIONS OF ELECTRODYNAMICS 9 where the surface integrals are taken over any surface bounded by an edge, I, m, n are the direction cosines of the normal to the surface at any point and the line integrals are taken round the edge in the positive direction. The relations (1) and (2) are equivalent to the systems of equations 8 7 8/3 -- L dy dz * - 8/3 da 4>7TW = f ox dy da = dY _dZ dt dz dy ,- _ (2'). dt dx cz _ _ dt ~~ dy dx ' In addition to these equations the relations between magnetic force and magnetic induction and between electric force and current strength must be known in order to effect a solution in any case. For the applications in view the magnetic force may be taken to be everywhere identical with the magnetic induc- tion, so that a = a, b = /3, = 7. The total current at any point is, according to Maxwell's view, made up of two parts, the displacement current and the convection current. The displacement current has components /, g, h and is connected with the electric force at the point by the relations a /= Z, 4nrV*g = Y, 4>7rV 2 h = Z, where V is the velocity of propagation of electric effects through the medium ; the convection current at a point consists of the conduction current at the point, such a current being conceived of as associated with a closed circuit forming a circuital discontinuity in the medium, and an element of current 10 THE EQUATIONS OF ELECTRODYNAMICS [ll pp, pq, pr, where p, q, r are the velocities of the free charge of volume density p at the point, such a current element being a point discontinuity in the medium and the closed current, of which it must be regarded as forming part, being completed by displacement currents through the medium. The components of total current are then given by 1 dX 1 dY I dZ ~* where u, v, w are the components of the convection current at the point #, y, z. In any case the convection current may be regarded as everywhere known or as given by a knowledge of the relation between the electric force and the conduction current, and of the distribution of free charges and their velocities. 10. Before proceeding to discuss the integration of the equations it is convenient to obtain them in a modified form. For this purpose let a vector, whose components are F, G, H and which is connected with the magnetic induction by the relations _dH _dG " dy dz' dF dH b = 5 5~ 02 OX dG dF c = be introduced. C = o o~~> ox oy da d*H d*G Inen 7 , = --x, ^r^, > dt dydt dzdt dY dZ that is -= -- 5 = K. 5-5: > dz oy dyot ozot 8 f v dG\ d ( d\ or 5- Y+ ^- = _- [Z+ -rr , dz \ dtj dy\ dt J II] THE EQUATIONS OF ELECTRODYNAMICS with two similar relations; therefore ^T7 ^ dt dx 11 dt dt (3). Now equations (I') are equivalent to where whence from (3) 8JP dG dH =- + -f -^- .(4). 11. Proceeding now to the integration of the equations in the case where an unlimited train of electrical oscillations is being propagated through the medium, those parts only of the electric and magnetic forces which depend on the time need be obtained, and the remaining parts if any will accordingly be omitted. The distribution of the convection currents will be assumed to be known and it will be proved that the electric 12 THE EQUATIONS OF ELECTRODYNAMICS [ll and magnetic forces at any point can be expressed in terms of these currents. The components F, G, H of the vector intro- duced above are so far restricted by two independent relations, so that they can be subjected to any other condition not incon- sistent with these. Assume as a further condition T- " F 2 dt ' where now only those parts of the various quantities involved which depend on the time are considered ; then equations (4) become .(4')- Let r denote the distance between the points x, y, z and f, 97, f, then the integrals of equations (4'), which correspond to the propagation of an unlimited train of electrical oscillations of period 27r//cF, are given by where u = u l e l " vt ) v = v 1 e t * vt , w = w 1 e ueFif , u, v, w being the components of the convection current at the point f, 77, f and the integrals being taken throughout all space. It* follows from these expressions that II] whence THE EQUATIONS OF ELECTRODYNAMICS dt and therefore Hence by (3) dz >-(- & J! dxdz with two similar relations ; therefore dx 2 dxdy 7=- 9 2 8 2 iVf[[/ a Z = --- ^i^- K Jjj\ dx ^^ j ^-^^ dz dydz a 2 a 2 -f w 13 The expressions for the components of the magnetic force can be obtained in a similar way, and are a = SS!( wi ~*4) e ~^ d ^ d ^ etc - v These expressions only include the parts of the electric and magnetic forces which depend on the electrical oscillations of period 2?r//cF. When there are oscillations of more than one period, the components of convection current u, v, w will be expressible in the form and the expressions for the electric and magnetic forces are 14 THE EQUATIONS OF ELECTRODYNAMICS [ll obtained from the above by summation. The F, G, H of the above investigation is different from Maxwell's vector potential, the difference being a vector whose components are of the form , P*, 5* . The integrals in the form (5) admit of easv ox oy oz interpretation. From the expressions given in (5) it im- mediately appears that the components of the electric force at a point are the sums of the components of the electric forces due to a distribution of Hertzian elements throughout the space where there are convection currents, the direction and strength of an element being that of the convection current at the point. The solution in this form is thus the analogue of the solution V= In-d^d^d^ in the theory of potential. 12. It is known that even when the vibrations are slow the currents in a conductor tend to concentrate in the neighbour- hood of the surface, and, as the vibrations become faster, the currents come to be practically confined to a thin layer at the surface ; the limit of this state of affairs is a perfect conductor. In such a conductor the magnetic and electric forces will be zero at each point inside it and at the surface the electric force will be normal to it and the magnetic force tangential to it. The currents are on the surface and are measured by the dis- continuity in the magnetic force in crossing the surface; the relation between current strength and magnetic force at the surface is found as follows. Let the axis of z be the normal to the surface at a point on it drawn outwards, the axes of x and y being in the tangent plane to the surface at the point. Then imagining the current to be the limit of a current distributed throughout a small thickness, the relations between current and magnetic force at any point throughout this thickness are given by 87 8/3 where u is indefinitely great, the thickness being supposed indefinitely small. Integrating throughout the thickness, the Il] THE EQUATIONS OF ELECTRODYNAMICS 15 first of the above relations becomes r i /-) /? The integral 1^ dz vanishes in the limit, for the quantity under the sign of integration remains finite while the range of integration ultimately vanishes; similarly the integral ~ dz vanishes and therefore 4f7T judz = ft, where ft is the value of ft on the surface, its value at an internal point being zero. In the same way it may be shewn that where a is the value of a on the surface. Therefore the current on the surface is measured by the magnetic force at the surface divided by 4?r, their directions being perpendicular and such that the directions of the magnetic force, the current and the outward drawn normal to the surface form a right-handed system of axes. In what follows perfect conductors will be chiefly considered, as in the case of waves whose vibrations are fairly fast the effect of imperfect conduction is a slight dissipation of the energy of vibration into the conductors, which may in general be left out of account. 13. The expressions obtained, 11, for the electric force will, so far as they depend on conductors, be simplified when the conductors are taken to be perfect. In this case the currents, as was seen above, are zero everywhere throughout the volume of a conductor and exist only on the surface, so that the volume integrals in (5) due to the presence of perfect con- ductors become surface integrals. Let ^,m t , n^ be the direction cosines of the current at a point f, rj, on the surface of a conductor and M the magnetic force at the point, then the values of X, Y, Z, the components of the electric force 16 THE EQUATIONS OF ELECTRODYNAMICS [ll at any point x, y, z of the medium due to the conductors, become 4>7r/cJJ V d^ 2 dxdy i 5 o ^io ^ ' ^ ^^i ) tto ,- ( ), 8/ 8y8^ J r iV rr / P) 2 ^ 2 J^ 2 \ p-tw II l/f /7i 2\ ^7O 4>7TKj J \ l dxdz l dydz 1 dz 2 1 j r where the integrals are now taken over the surfaces of the conductors. The components of the magnetic force due to the conductors are given by dZ; (7). //-( Thus, when there are perfect conductors only in the medium, the problem of finding the circumstances of propagation of waves of given period through it is reduced to finding the magnetic force at each point of the surface of the conductors. 14. In the preceding investigation of the integrals of the equations everything is expressed in terms of the convection currents which may be looked upon as discontinuities in the time rate of change of the electric force ; in exactly the same way integrals of the equations can be found expressing every- thing in terms of discontinuities in the time rate of change of the magnetic induction. The expressions which correspond to (6) and (7) now make the magnetic force tangential to the surface vanish, while the electric force tangential to it is discontinuous. From this it appears that, if at each point of a surface, which encloses all the sources of the waves, the electric and magnetic forces tangential to it be known, the electric and magnetic forces at every point outside it can be II] THE EQUATIONS OF ELECTRODYNAMICS 17 expressed in terms of them, the components of the electric force being given by , da? dxdy dxdz d \ p- iicr - -d, etc, 1 f f /- 47rJJ oy where E and M are the electric and magnetic forces tangential to the surface, l lt m lt n-^ the direction cosines of the tangent to the surface perpendicular to M, and //, m/, n^ the direction cosines of the tangent perpendicular to E. 15. The electric and magnetic forces at any point when an arbitrary disturbance is being propagated through the medium can be obtained in a similar way. As in 11 only those parts of the electric and magnetic forces, which belong to the pro- pagation of the disturbance, will be taken into account. Assume, as in that case, JL 8 ^ " F 2 a* ' then the equations to be satisfied by F, 6r, H are (4/), whence where u^v^Wi are the values of u, v, w at the point f, 77, f at a time r/ V before the time t under consideration. Now and, writing M. E. W. 18 THE EQUATIONS OF ELECTRODYNAMICS [ll the parts of the components of the electric force which depend on the propagation of the disturbance are given by V* These expressions can be modified for the case of perfect conductors as in 13. The part of the electric force at any point which depends on the disturbance is thus expressed in terms of the convection currents which existed at the various points at a time t r/V, where r is the distance of the point under consideration from any other point. The effect of a disturbance at a point A travels out from it in spherical waves, arriving at a point P in the time AP/V, when P takes up the disturbance, after which time it comes to rest*. 16. In the above a single medium filling all space in which there are present what have been termed convection currents is contemplated, and on this hypothesis it has been shewn that, assuming Faraday's laws, electrical effects, whether the pro- pagation of oscillations of definite periods or of an arbitrary disturbance is considered, are propagated in the medium with a definite velocity, the directions of propagation being straight lines and the displacements strictly transverse. Further it has been shewn that the circumstances of the propagation of electrical effects can be completely expressed in terms of the distribution, supposed known for all time, of the convection currents, these convection currents corresponding mathe- matically to the singularities of the necessary functions. The effect at a point P is the sum of the effects, due to all points Q, which depend on the state of affairs at Q at a previous time * For the complete expressions for the components of electric force when the disturbances are supposed to be due to moving charges, see Appendix C. II] THE EQUATIONS OF ELECTRODYNAMICS 19 t PQ/V; the assumption* is thus implicitly involved that Maxwell's aethereal displacement current is independent of the motion of the aether if there is such a motion. By the application of Faraday's law, the electric force which would act on an element of a circuit at any point of the aether can be obtained, and this is on Maxwell's theory taken to be the electric force at that point in the aether, the element of circuit being conceived of as fixed. * See 22. 22 CHAPTER III. CONVECTION CURRENTS IN MOTION AND MATERIAL MEDIA. 17. To obtain expressions for the electric force on an element of a convection current moving in a known manner, let an element of such a current at a point x, y, z, the axes of reference being supposed to be fixed in space*, be considered and let I, ra, n be the direction cosines of the element. Applying Faraday's law and remembering that it is supposed to hold for closed circuits, the line integral of the electric force in the direction of the element acting on it is where F, G, H define Maxwell's vector-potential or electro- kinetic momentum, cr is a coordinate defining the order of the element in the circuit, and where, since the convection current is not supposed to be at rest, /, m, n may vary with the time. The electric force in the direction of the element is therefore given by the expression ,fdF dF dF dF\ ,'dG dG dG dG\ M ^7 +P o~ + <7 ^~ + r -^- w* I -51 +P o~ + j- +n-^ r )-F/-G^ L -H^- -3f dy dz J os ds os os which is equivalent to -~ Hence the components of the electric force, which acts on an element of convection current at the point #, y y z moving with a velocity whose components are p, q, r, are given by* * It will be observed that it is the time rate of variation of the vector F, G, H in a direction, which itself changes with the motion, that is taken, not the time rate of variation in a fixed direction. It is the first of these that is appropriate to electrodynamics, the experimental laws (Faraday's) of which are expressed in terms of the behaviour of closed circuits. Some writers have erroneously used the second; their results however are in general correct, as for the case usually treated of, that of uniform motion, the expressions are the same. 22 CONVECTION CURRENTS IN MOTION [ill ._ ___ where ^ = Fp + 9 + /7r + <#>. 18. On any theory all electrical effects must be explicable on the hypothesis of one uniform medium in which there are convection currents ; thus the difference between the aether and any material medium must be the presence of convection currents, that is, wherever there is matter there are these convection currents distributed in the aether, which are to be regarded as discontinuities in it. A complete knowledge of the distribution and strengths of these convection currents for all time would make it possible to determine completely all the electrical circumstances, but such knowledge it is impossible to obtain, and as in any case what can be observed is the effect of an aggregate of these convection currents, it is necessary to make hypotheses which shall replace a knowledge of the in- dividual by something which shall represent the effect of the aggregate. It has been remarked by Maxwell* that " if we attempt to extend our theory to the case of dense media, we become involved not only in all the ordinary difficulties of molecular theories, but in the deeper mystery of the relation of the molecules to the electromagnetic medium." This remark is clearly equally applicable to the case of any material medium ; and the assumption made in his next paragraph, it being remembered that throughout this chapter Maxwell is thinking of the medium as at rest except in so far as there are motions due to electrical disturbances, is equivalent to assuming that for transparent media, the aggregate effect of the convection currents can be represented by displacement currents in the * Treatise, Vol. n. 794. Ill] AND MATERIAL MEDIA 23 aether. These displacement currents, which replace the con- vection currents and which, to distinguish them, may be termed material displacement currents, differ from aethereal displace- ment currents inasmuch as they must be conceived of as moving with the velocity of the aggregate of the convection currents which they replace, and as being acted on by the electric force which would act on a convection current moving with that velocity. This, combined with the assumption that the relation between the electric force acting on material dis- placement current and material displacement is of the same kind as that existing between aethereal electric force and aethereal displacement, suffices to determine the circumstances of the propagation of electrical effects through any transparent material medium. 19. Let /, g, h denote the components of the aethereal displacement at the point x, y, z referred to axes fixed in space, f\ ><7i>^i the components of the material displacement which replaces the convection currents distributed throughout the medium, and F, G, H the components of the electrokinetic momentum. The components of the aethereal displacement current then are CM. * dJ. ' dJ- * Ot Ov Ov and the components of the electric force in the aether are ~~dt~dx' ~~di~dy' ~dt~'dz ) giving dF d(j) \ ~Hi ~diic __a<7 _a0 dt dy dH d y, z, the axes of reference being supposed fixed in space, the components of the corresponding '""/* O O L current at the point are jfc. . . Maxwell's electric dis- ot ot ot placement in the aether is thus a vector defining the coordinates which specify the electrical degrees of freedom of the aether at a point of space and is quite distinct from the electric displace- ment associated with material media; the difference between aethereal electric displacement and material electric displace- ment has already been emphasised*. 23. In obtaining the electric force at a point in the aether the same considerations apply as above ; therefore if F, G, H are the components of the electrokinetic momentum at the point x, y, z, the components of the electric force at the point are _aP_cty _? ox oy oz which is zero unless there is a discontinuity in the aether at the point. In obtaining the potential energy expressed as a function of /, g, h it is sufficient to find it for the electrostatic case ; in this case since charge and potential are linearly related the portion of the potential energy contributed by the aether at the point x, y, z is \ (Xf+ Yg + Zh) dx dy dz, where X, Y, Z are the components of the electric force at the point and are proportional to /, g, h. Hence the part of the Lagrangian function belonging to the aether at the point x, y, z is |~ df dg jjdh, _ . _ ~j Ou ut vt I 1 and in this form it is properly expressed in terms of the electrical coordinates and their time rates of variation. It ought to be observed however that this Lagrangian function is not expressed in terms of all the coordinates which specify degrees of freedom of the aether ; it is a modified* Lagrangian function expressed in terms of coordinates less in number, the original coordinates having been in part eliminated by the process by which the function has been built up. A knowledge of this Lagrangian function does not then supply sufficient data * See 31. 32 MAXWELL'S DYNAMICAL THEORY OF ELECTRICITY [iv from which to develop the dynamics of the aether ; when how- ever in addition the convection currents are completely known, the data are sufficient to determine the relations of the aether to material media so far as these relations are electrical. 24. The expression for the Lagrangian function given above is the one which is natural to Maxwell's dynamical theory as presented in Chapters v. to IX. of the second volume of his treatise, but in 635 he gives the second expression for the electrokinetic energy, the part contributed by the aether at the point #, y y z being 8w, In 636 he assigns a reason for choosing this second expression as the proper one when he says : " The electrokinetic energy of the system may therefore be expressed either as an integral to be taken where there are electric currents, or as an integral to be taken over every part of the field in which magnetic force exists. The first integral, however, is the natural expression of the theory which supposes the currents to act upon each other directly at a distance, while the second is appropriate to the theory which endeavours to explain the action between the currents by means of some intermediate action in the space between them." Now in either case it is a modified Lagran- gian function which is obtained and the kinetic energy portion of such a function is made up of portions contributed from every place where there are time rates of variation of the coordinates in terms of which the function is expressed, not of portions contributed from every place where there is motion ; for example, in the theory of the motion of bodies through a liquid the modified Lagrangian function which is there used does not express the kinetic energy as an integral taken throughout the liquid but in terms of the coordinates which iv] MAXWELL'S DYNAMICAL THEORY OF ELECTRICITY 33 specify the degrees of freedom of the bodies. Thus the ex- pression of the kinetic energy as a sum to which every place where there is motion contributes is unnecessary ; as has been already stated, the electrokinetic energy and the kinetic energy of the aether are different things. Further the transformation by which the second expression for the electrokinetic energy is obtained from the first is in general invalid. Maxwell's pro- cedure 635 is to write for the currents in the first expression their equivalents in terms of the magnetic force and then integrate by parts ; in this way the electrokinetic energy is expressed as a volume integral taken throughout all space and a surface integral taken over the surface of the infinitely distant boundary, this latter being omitted on the ground that at a great distance r from the system the components of the magnetic force are of the order of magnitude r~ 3 ; but this is only true for the case when all the currents are steady, which is the case Maxwell appears to be thinking of, and then the transformation is legitimate. When the case is that of the propagation of waves, the components of the magnetic force at a great distance r contain terms of the type e tlcr /r, as do the components of the electrokinetic momentum, and the surface integral is no longer negligible, its value being really indeter- minate. The argument which has been sometimes used to justify a transformation of this kind in the case of waves that if all the sources of the disturbance are at a finite distance from the origin, the surface integral over an infinitely distant boundary cannot influence the state of affairs at a finite distance neglects the fact that the mathematical treatment of trains of waves postulates an infinite time during which the dis- turbances have been going on, and that therefore, however remote the boundary may be taken, the disturbances have already produced their effect there*. It follows then that * It is also clear that, if instead of a train of waves a number of disturbances supposed to have been set up at definite times be considered, the transformation is still invalid as the functions necessary to represent this state of affairs are discontinuous and the integration by parts required for the transformation cannot be effected. M. E. W. 3 34 MAXWELL'S DYNAMICAL THEORY OF ELECTRICITY [iv Maxwell's second expression for the electrokinetic energy is inadmissible. 25. The use of Maxwell's second expression for the electro- kinetic energy has led to the assumption that Maxwell's electro- magnetic theory and MacCullagh's optical theory are the same, the Lagrangian functions of both theories being then identical. This has further led to the identification (tentatively) of the magnetic induction as the velocity of the aether; from what has been said above it follows that both these assumptions are illegitimate, and that conclusions based on them must be rejected. The identification of the magnetic induction as the velocity of the aether has led to the result that the velocity of propagation of light ought to be altered, though possibly to an insensible extent, by a magnetic field* ; the use of the proper form of the Lagrangian function (Maxwell's first form), however, leads to equations to determine the pro- pagation of electrical effects, which are unaltered by the introduction of a magnetic field, so that the velocity of propagation of light is unaltered, agreeing with Lodge's experimental result f. The difficulties concerning permanent magnets due to the identification of Maxwell's theory with MacCullagh's likewise disappear. The conclusion then is that MacCullagh's theory is essentially different from Maxwell's and that Maxwell's theory being in agreement with the phenomena is the one which ought to be retained. * Larmor, Phil. Trans. (A), 1894. t If a 4 , /3i , 7! are the components of the magnetic force and f\ , G x , H l the components of the vector potential of the imposed magnetic field, f\, G lt H lt a i> i "Vi are independent of the time. The total components of the electro- kinetic momentum are F + F lt G + G lt H + H l and of the magnetic force a + an i y +7i > whence if /, g, h are the components of the electric displacement --. --. = dy dz dz dx dx dy ' since cij , /3 a , 7j are derivable from a potential function and _. ,=-- dt dy dt dz Therefore the equations to determine a, /3, y or /, g, h are the same as when there is no imposed magnetic field. CHAPTER V. DYNAMICAL THEORY. 26. THE tendency of physical investigations has in general been towards the construction of a dynamical theory which shall give a consistent account of phenomena, the path pursued being to arrive at such a result by inductive methods. The fundamental idea underlying attempts at a theory of this kind is that direct knowledge is confined to a knowledge of motions, the other ideas of dynamics, such as force, being inferences which are useful aids in classifying and explaining phenomena in terms of those phenomena which are more intimately known and over which there is more immediate control. Instead of trying to construct a dynamical theory inductively, another mode of proceeding is possible, to assume that all phenomena are to be explained on the basis of a dynamical theory and to proceed from this deductively. The starting-point then is that the Lagrangian function is a homogeneous quadratic function of the time rates of variation of the coordinates which specify all the degrees of freedom, the coefficients of the expression being functions of these co- ordinates, and that the time integral of this function taken between any two definite times is stationary for the actual motion. This Lagrangian function is necessarily constant for all time and the principles of the Conservation of Energy and of Least Action are included in this statement. What obser- vation reveals in any case is a certain number of degrees of 32 36 DYNAMICAL THEORY [V freedom of motion and corresponding motions taking place; knowledge is thus confined to a part only of all the degrees of freedom and the question then arises, What form does the dynamical theory take to fit in with this limited knowledge ? For convenience the discussion will be divided into several cases. 27. The degrees of freedom are divided into two sets, one set being specified by coordinates y which are known, the other set being specified by coordinates x which are unknown. The first case to be discussed is that where the Lagrangian function contains no terms of the type Cxy and the coefficients are functions of the y coordinates only. The Lagrangian function L is given by L = T X + T vt where T x = tA u x? + ^ and the coefficients A u , A Ut B u , S Kt etc. are functions of the y coordinates only. By the well-known process of ignoration of coordinates the x coordinates can be eliminated and a modified Lagrangian function L f obtained, which is sufficient to determine the motions so far as the y coordinates are concerned. The coordinates x have to be eliminated from the equations dfiL\_9L_ Si (Sy) dy by means of the relations az dx~-*' where the quantities f are constant. Writing L = L' + Sf 4 V] DYNAMICAL THEORY 37 it may be shewn that the first set of equations is replaced by d/az'x az'_ ~ ..................... ( )( - Now equations (2) are equivalent to A u x 1 + A 12 x 2 +... = % 1 .................. (2'), etc.; hence 2r = A u x^ + ZA^x^ + . . . , that is 2fc=2Z T a ., and therefore L' = T y - T x , in which T x is supposed to be expressed in terms of the quantities f by means of equations (2') and hence is a function of the y coordinates. Thus the motions so far as they depend on the y coordinates are completely determined by the equations d ftL\ _ dL' _ dt(dy) dy = where // = T- F, T is a homogeneous quadratic function of the velocities y, being that part of the total energy which is due to the motions corresponding to the degrees of freedom specified by the y coordinates, and F is a function of the y coordinates which is equal to that part of the total energy which is due to the motions corresponding to the degrees of freedom whose co- ordinates have been eliminated. From this follows the possi- bility of the existence of a class of motions whose complete history can be determined from a knowledge of one function, this function being a modified Lagrangian function. An example of this class of motions is furnished by the mechanics of a conservative system of rigid bodies, the kinetic energy of 38 DYNAMICAL THEORY [V the system being the function denoted above by T and the potential energy the function denoted by V. The Lagrangian function of the motion is T - V and from the above T + V is constant, being equal to the Lagrangian function which is expressed in terms of all the degrees of freedom. On this view then potential energy is the energy of what may be termed the concealed motions, that is the energy of those motions which correspond to degrees of freedom which are not directly observed. Another example is that of a rigid body or a number of rigid bodies moving through a liquid, the space occupied by the liquid being simply connected and the motion of the liquid irrotational ; in this case attention is confined to the degrees of freedom of the moving bodies. In all such cases certain motions corresponding to degrees of freedom which can be specified by coordinates y are observed, and to determine these motions completely it is only necessary to obtain a know- ledge of the modified Lagrangian function which is the difference of two functions, these being what are usually termed the kinetic and potential energy functions of the motion. This knowledge though sufficient to determine the motions depending on the observed degrees of freedom, does not suffice to determine the motions depending on the concealed degrees of freedom, the coordinates corresponding to which have been eliminated. The information which is obtained concerning them, when the motion depending on the observed degrees of freedom is of the character here discussed, is that the coordinates specifying these concealed degrees of freedom enter the original Lagrangian function as velocities only, and there are no terms in it which contain a product of velocities one of which belongs to the observed degrees of freedom and the other to the concealed ones. 28. The second case is that in which the Lagrangian function contains no terms of the type Cxy but both kinds of coordinates occur. The Lagrangian function is in this case given by V] DYNAMICAL THEORY 39 where T x = A n x-^ -\-^ and the coefficients A u , A lz , B n , B 12 , etc. are functions of both kinds of coordinates x and y. The equations of motion are d l\ SL and writing L = L f + 2f#, where f = (3), the equations (1) and (2) are replaced by d /a/A a// az/ , az/ where as in 27 L' = T V - T x , T x being expressed as a function of the x and y coordinates and the momenta f by means of equations (2). In the motions which belong to this class T y is the energy due to the time rates of variation of the coordinates y which are the observed ones, and T x ( V) is what is termed the potential energy, but, instead of being expressed as a function of the x and y co- ordinates and the momenta (, it will appear as a function of the x and y coordinates, the momenta f being replaced by their equivalents as functions of the x coordinates. Thus in this case the equations of motion &\J2^Q* dt\dy) 'by where L' = T-V, * The experimental data in this case do not necessarily give the equations of motion in this form though it is always a possible one. 40 DYNAMICAL THEORY [V and is a function of the x and y coordinates and the velocities y, do not in general suffice to completely determine the motion of the system so far as it depends on the y coordinates. The time rates of variation of the x coordinates expressed as functions of the time must in addition be known ; an important case is that where the motion in respect of those of the x coordinates which occur in the coefficients is steady, this steady motion being completely known. A system of linear circuits, in which there are electric currents, moving in a given manner is an example of this class of motions. 29. The third case is that in which the Lagrangian function involves terms of the type Cxy but the coefficients are functions of the y coordinates only. In this case the modified Lagrangian function L', which results when the velocities x are eliminated by means of the relations dL_t a*"* is no longer of the form T V, where T is a homogeneous quadratic function of the velocities y and V is a function of the coordinates y\ there are present in addition terms which are linear in the velocities y. As in 27 a knowledge of the modified Lagrangian function is sufficient to completely deter- mine the motion so far as it depends on the y coordinates. The motion of a system of rigid bodies to which there are attached a number of gyrostats furnishes an example of this class of motions. A further example is that of a number of solids moving through a liquid, the space occupied by the liquid being multiply connected. 30. The remaining case is that in which the Lagrangian function contains terms of the type Cxy and the coefficients are functions of both kinds of coordinates. In this case the modified Lagrangian function is of the same kind as in 29, but it involves the x coordinates, so that as in 28 a know- ledge of this function, when, as in cases where it results from V] DYNAMICAL THEORY 41 observation, it appears as a function of the x and y coordinates and the y velocities, is not in itself sufficient to completely determine the motion so far as it depends on the y coordinates. The time rates of variation of the x coordinates as functions of the time must in addition be known or relations which are equivalent. 31. When the number of degrees of freedom is finite the application of the Lagrangian method presents no difficulties. When, however, there are an infinite number of degrees of freedom, some means of identifying the coordinate which belongs to a particular degree of freedom becomes necessary. For example, if the coordinates can be arranged in a definite order forming a numberable aggregate or if each is associated with a definite point of a straight line, or more strictly with the element of length of the straight line at the point, this length being measured from a fixed point on it, the coordinate specifying each degree of freedom can be identified and the Lagrangian method can be applied. In the first case the Lagrangian function has the form of an infinite series and in the second of a simple integral. In the same way a coordinate or a finite number of coordinates specifying degrees of freedom can be associated with each point of a given space ; the Lagrangian function is then a triple integral taken throughout the given space, the coordinates x, y, z of any point are to be treated as numbers serving to identify the dynamical coordinates and, in the formation of the dynamical equation by varying the Lagrangian function, are to be taken as independent of the time. The degrees of freedom of an element of a continuous medium can be specified by means of the coordinates of the point at which the element is taken and coordinates determining the change in position, size and shape of the element as it changes its position*. When the Lagrangian function of the motion is completely known as a function of the velocities belonging to all these coordinates, the coordinates will be determined by the resulting equations as functions of the time * Appendix B. 42 DYNAMICAL THEORY [V and of the position of each element at a given time. Now this method of specifying the degrees of freedom of a continuous medium, though convenient when the Lagrangian function is completely known, may be unsuitable when only part of the motions can be observed. When part of the motions which are not directly observed is the motion of the medium in bulk, it is more convenient to conceive of the degrees of freedom as specified by the coordinates of each element and coordinates determining the change in size and shape of the element which at any time occupies a given position, these latter coordinates 6 being then associated with the points of space, not with the elements of the medium. If from the coordinates 6 certain of them denoted by can be chosen so that the Lagrangian function has no terms occurring in it which are products of the velocities cj> and of any of the other velocities, then by 27, 28 the modified Lagrangian function which results from the elimination of all these latter velocities is of the form T V t where T is a homogeneous quadratic function of the velocities they become (7rsin0)- (0r)\] 4t7TV rsin da. 80 .(3), and da dt db ~| (Fr) ) .(4). The equations are now in a form which is convenient for the treatment of the space between two concentric spherical sur- faces. In the cases to be here treated the concentric spherical surfaces bounding the space will be supposed to be perfectly Vl] IN SIMPLY-CONNECTED SPACES 51 conducting surfaces and the medium occupying the space will be supposed to be non- magnetic. The magnetic force and the magnetic induction are then identical and the current strength is given in terms of the electric force by the relations tl/ ^~ - T~r n r\ . ___ U - A TTrt *-N . where V is the velocity of propagation of electrical effects through the medium. The conditions to be satisfied at a bounding surface are F=0, Z=0. It is not difficult to see that any case can be built up from the following two simple cases. First X = 0, F = throughout the space and on the boundaries, secondly a = 0, /3 = throughout the space and on the boundaries ; these correspond respectively to the typical cases in which the wave fronts are first tangential to the bounding surfaces, secondly, orthogonal to them. Taking the case where only the Z component of the electric force does not vanish, the equations (3) and (4) give da 90' 9 . . % r sm o dt r 2 sm a/s i- a/ r ; .. ^7 = -- : - a ^ (% r Sln dt r sin dr ^ I- Hence that is, writing ^ for cos 6, 42 52 PROPAGATION OF ELECTRICAL EFFECTS [VI 27T -- Assuming Zr sin 6 = %e T and putting ^ = *, % satisfies the equation Therefore where P n (/z) is the zonal harmonic of order n, and ?i is an integer, because ^ has to be finite for all values of //. from 1 to 1 inclusive. The function R n is given by the equation that is R n = r { J. / n+i (*r) + BJ. n ^ (*r)} . If r = r and r = r^ define the bounding surfaces the boundary conditions give A J n+ i (KT O ) + BJ-n.% (tcr ) = 0, A J n+ i (KT,) + BJ_ n _i (KT,) = 0, whence J n+ (KT O ) J- n -$ (/cr,) = J n+ % (tcr^ J- n -i (/cr ), an equation to determine K and thence the possible free periods. When r and r^ are both finite, this equation has an infinite number of real roots and no others; an interesting particular case is that when r 0, the equation then is J n+ i On) = o, and gives the possible free periods of electrical oscillations in a spherical space. When r x becomes indefinitely great J" n +j(/cr a ) tends to the value Li - cos ( KT-^ --- TT) and y 7T ' KT-^ \ i J 7_n_j (/crj) to the value . / cos (tc^ -4- ^ TTJ, so that the equation to determine the free periods becomes r t H7r \ T ( W + 1 \ /n+i (**) COS 1 4BT, + |-J = J_n-i (r ) COS I K^ -- 7T 1 , VI] IN SIMPLY-CONNECTED SPACES 53 an equation which is satisfied by all values of tc when r t becomes infinite. The interpretation of this is that belonging to the space exterior to a sphere there are no free periods of electrical oscillations of this type, but that waves of any period can be propagated in it. Taking now the case a = 0, /3 = 0, where the wave fronts are orthogonal to the bounding surfaces, the equations become dt rl8r v 8 and it is clear that 7 satisfies the same equation as Z does above, hence where R n = r* [AJ n+ (XT) + BJ_ n _ (or)], the boundary conditions now being . | I when r = r and r = r l . Therefore corresponding to the harmonic of order n the equation to determine K and thence the periods is ^ {rJJn+i 00} fr {rfJ^n- an equation possessing, when r and n are both finite, an 54 PROPAGATION OF ELECTRICAL EFFECTS [VI infinite number of real roots and no others. When r vanishes the equation is giving the free periods belonging to a spherical space for this type of wave. As before, when r^ becomes indefinitely great the equation is ultimately satisfied by all values of K. Thus in the space exterior to a sphere there can be no permanent free electrical oscillations. 37. Corresponding to any closed surface there are, as was seen in 35, definite free periods belonging to the possible free electrical oscillations which can permanently exist in the space inside. Condensers are a particular case of this, and when the two faces of the condenser are very close together the free periods can in a great number of cases be simply determined. The results for various forms of thin condensers have been given by Larmor*. Taking equations (1) and (2) 36 and choosing the surfaces f so that f = f > ?= ?i are the two bounding surfaces of the condenser, a choice which can always be made when the electrostatic problem is solved, the boundary conditions are X = 0, 7=0 at both surfaces. Two cases arise, the one being that for which X and Y both vanish throughout the condenser and for which the wave fronts cut the surfaces of the condenser orthogonally, the other being the conjugate case for which a and fi vanish throughout and for which the wave fronts are tangential to the bounding surfaces. In the second case it is clear that the wave lengths will be very short and therefore the periods will be very high, so that the first is the important case; for it equations (1) and (2) become a 0\ a Z d Z * Proc. Lvnd. Math. Soc. Vol. xxvi. 1894. VI] IN SIMPLY-CONNECTED SPACES 55 whence a jMi a (Z\\ d_ (M 3 a_ /s\n i IX a? Us/) + ^ I ^ a*, U/J J ' When the surfaces determined by => and f=fi are closed surfaces, this equation suffices to determine completely the circumstances, the condition that Z should be everywhere in the medium finite determining the possible free periods. An example is furnished by the case of a spherical condenser ; the result in this case is at once deduced from the frequency equation given in 36 which becomes, when ^ - r Q is very small, /t 2 r 2 = ft(/i + l), n being an integer. When the bounding surfaces are not closed surfaces, the contours bounding the surfaces may be supposed to be joined by a perfectly conducting surface determined by some relation between f and 77, the tangential electric force over this surface would have to vanish and this condition would give an equation to determine the free periods. In actual condensers the medium between the two faces is in communication with the medium outside, the ends not being closed; if waves proper to the space closed as above be supposed to be set up and then left to themselves, their energy would be rapidly radiated into space, as the conditions imposed suppose them to be reflected at the ends. If the faces at the ends be supposed to be so close together that their distance apart at any point is small compared with the wave length of a possible oscillation, an approximation to the free periods of these possible oscillations can be obtained on the assumption that the ends of the condenser are loops, the condition then being that the magnetic force tangential to the edge of a face vanishes. When the condenser is formed by two nearly closed surfaces the free periods so found will differ but slightly from those found on the assumption that the surfaces are closed. For example, in the case of the condenser formed by two concentric 56 PROPAGATION OF ELECTRICAL EFFECTS [VI spheres pierced by small apertures, one in each sphere opposite each other, the frequency equation is where n is not now an integer but is determined by the condition that at the edge of the aperture. If a be the angle subtended by the radius of the aperture, supposed circular and very small, at the centre of the sphere, the roots of this equation are given by* where k has all positive integral values including zero and these values differ but little from the integral values of n. It may be inferred by reasoning similar to that of 35 that in the case of a partially enclosed space, the boundary being supposed to be perfectly conducting, free oscillations with a certain degree of persistence are possible, the persistence being considerable if the linear dimensions of the aperture are small compared with the wave length, or if the aperture is a slit whose width is everywhere small compared with the wave length. An approximation can be obtained in the case of a condenser whose faces are parallel plates at a distance d apart small compared with the wave length X. The correction for the open end is approximately a which is given byf and the magnetic induction in the space between the plates is approximately given by I" . 27T, ird ZTTX . 1 = A sm (x a) cos nt cos sin nt , \ A/ A< * Macdonald, Proc. Lond. Math. Soc. Vol. xxxi. 1899. t Appendix D. VI] IN SIMPLY-CONNECTED SPACES 57 where 2?r/n is the period ; whence the radiation from the open ends can be calculated. 38. If the two bounding surfaces of the condenser are not closed surfaces, the free periods will be practically unaltered by joining the opposite faces by a very thin wire, whether the ends be supposed closed by a perfectly conducting surface or not. For example, take the case of the condenser formed by two circular plates of equal area, the one being exactly opposite to the other. If r^ be the radius of either plate, r the radius of a thin wire joining them at their centres, the frequency equation, when the ends are supposed closed by a perfectly conducting surface, is. J n (/cr,) = J 1 KT \ Y n On), -* n \ K ' o) where n is any integer, and when the ends are not closed is T ' (.. v \ _ *** ( Kr o) Y ' (ifv \ ^n \*Tl) Y (*. \ n \ ltr ^/' 1 n \K r o) In either case, r being very small, no new free period is introduced and the alteration in those already existing is very small. When the bounding surfaces of the condenser are closed surfaces a new period is introduced by joining the faces, this period being very long. In the case of the condenser formed by two concentric spherical surfaces, let the wire, supposed to be very thin, be taken as the axis of the harmonics, then for oscillations in which the nodal lines are circles of latitude, the frequency equation is n(n + I) = K*r*, where n satisfies the equation P n ( cos a) = 0, a being the angle subtended at the centre of the sphere by the radius of the wire. The roots of this equation are given by* I Macdonald, loc. cit. 58 PROPAGATION OF ELECTRICAL EFFECTS [VI where k has all integral values including zero; and corresponding 1 to n there is a possible oscillation of very long period. 39. When a constraint is suddenly set up or removed at any point of the medium, the corresponding disturbance is propagated outwards from this point in all directions with the velocity of radiation, each point taking it up as it arrives there and reverting to a steady state after it has passed over it. The complete solution in any case where the original disturbance is confined to a finite space is to be found by the application of the results of 15 ; examples have been worked out in detail by Heaviside*. The distinctive feature of such cases is that the disturbance produced at any point lasts for a definite time, which can be very simply determined. When, however, the space in which the disturbance originates is not finite there is 110 such definite time. An example of this kind is furnished by the suppression of a field of electric force external to a conducting surface, the space outside the conductor being infinitely extended. In this case a continuous series of disturbances arrive at the surface of the conductor, are reflected there and then propagated outwards into space. A knowledge of the reflected disturbances is sufficient to determine the distribution on the surface at any time subsequent to the suppression of the field of force. An example of this kind is that where a uniform field of electric force external to a sphere is suppressed ; the expression for the magnetic force due to the reflected disturbances is that given by J. J. Thomson f * Electrical Papers, Vol. n. pp. 375467. f Proc. Lond. Math. Soc. Vol. xv. p. 210, 1884; Recent Researches in Electricity and Magnetism, p. 370. VI] IN SIMPLY-CONNECTED SPACES 59 where = ^- ( Vt r), . r a TT tan 6 = - tan - ; r + a 3' and this expression exists up to a distance from the centre of the sphere a + Vt, which defines the greatest distance to which the reflected disturbances have attained at a time t subsequent to the suppression of the field of force. The electrical distri- bution on the surface of the sphere at any instant immediately follows. The example of an ellipsoidal conductor under the same circumstances has, more particularly for the case when the ellipsoid is of revolution about its greatest axis, been investigated by Abraham*. 40. It appears from 37 that, when there are condensers whose bounding surfaces are not closed or partially enclosed spaces, the bounding surfaces being in both cases supposed to be perfectly conducting, there are free oscillations belonging to these spaces, which after having been set up have their energy radiated out from the ends of the condenser or through the aperture of the partially enclosed space, the rate at which the oscillations die away depending on the nature of the aperture. The possibility of successive reflexions of the waves in the simply-connected space is the essential condition for the continued existence of free oscillations whether permanent or not ; it therefore follows that there will be possible free oscillations, though not permanent, when there is a space occupied by a dielectric medium which is different from the medium surrounding it. As an illustration let there be two dielectric media, one of them occupying the space inside the surface of a sphere of radius r , the other the space outside this surface, and let the ratio of the squares of the velocities of radiation in the two * Annalen der Physik und Chemie, Bd. 66, 1898. 60 PROPAGATION OF ELECTRICAL EFFECTS [VI media be K. Then as in 36 the solution proper to the space inside the sphere, omitting the time factor, is given by ap yr sin 6 = Ar* J n+ L (/cr) (1 u?) -^- , dp that case being considered where the wave fronts are orthogonal to the surface of the sphere and where 47T 2 V being the velocity of radiation in the medium inside the sphere. The solution for the space outside the sphere is given by yr sin = A'l* (/__* (\r) - J^ (\r)} (1-yu, 2 ) ^ , where X 2 = = KK\ Both media being assumed to be non-magnetic, the boundary conditions are A Jn +t (*r ) = A' { /_ n _ } (Xr ) - *-<+*> / n+ j (Xr )}, A j- [r$ J w+i (*r )} = ^' ^ [r * {/-n-i (Xr ) - er<*+* /, +i (Xr )}], O/o ^' whence the equation to determine K is j J__ 4 (\r.) - - J^t (Xr.)} Jr W A,+ ()} = |- [r,i (/__! (Xr.) - e - w+ i' " J rt (Xr.)}] J +t (r,X cr that is after reduction -i (Xr ) - e- (w+ i' " Jn+t (Xr )} r _ n+i (Xr )4-e- ( ' J n _ (Xr )}. It will be sufficient as an illustration to consider the case when n = 1 ; in this case the equation becomes (1 \ /sin tcr \ i ) + X. I cos tcr = 0, XT O / V. KT O / that is tan tcr Q = , K A< IK h - A /V 1S/Y* A.r KT VI] IN SIMPLY-CONNECTED SPACES 61 or tan /cr = KL + The roots of this equation have their imaginary parts negative, these being very great when K differs but little from unity; thus where the media have their velocities of radiation nearly equal free oscillations die away almost instan- taneously, when the velocities of radiation differ considerably the oscillations will persist for some time. 41. The result of the preceding is that, when a space bounded by conductors is simply connected, there are free oscillations belonging to finite spaces occupied by dielectric media, for any given finite space there being definite free periods which in general will be infinite in number. When there are spaces, occupied by dielectric media, which are partially enclosed by conducting surfaces free oscillations are possible, these oscillations dying away with a rapidity depending on the dimensions of the aperture by which the medium in the partially enclosed space communicates with that outside. When there are closed spaces in the medium occupied by different dielectric media, free oscillations are possible which decay with a rapidity depending on the difference between the velocities of radiation in the medium occupying a closed space and in the surrounding medium. These results can be Utilised to form an idea of what is taking place when electrical disturbances are generated as in the case of Hertz' oscillator. The disturbances generated are propagated outwards from it with the velocity of radiation of the surrounding medium, and if this medium were everywhere the same and there were no partially enclosed spaces bounded by conducting surfaces, the disturbances would cease instantaneously to be propagated outwards when they ceased to be generated. This agrees with Bjerknes'* experi- ments where the disturbances are found to die away almost at once. * Annalen der Physik und Chemie, Ed. 44, 1891. CHAPTER VII. PROPAGATION OF ELECTRICAL EFFECTS IN MULTIPLY-CONNECTED SPACES. 42. IF there is a single perfect conductor whose surface is such that the space outside it is doubly connected, then resuming the argument of 35 and considering the case where the wave fronts cut the surface of the conductor orthogonally, it follows that starting from any position of a wave front cutting the surface of the conductor in a closed curve A, and travelling with this wave front, the curve A will in no position become evanescent, and free permanent electrical oscillations will be possible, their periods being determined by the condition that, when the wave front has returned to the position from which it started, the wave is in the same phase. This result as well as that of 35 .can be deduced from the equations (6) of 13. The result can be extended at once to the case where there are any number of conductors and the order of connexion of the space is any integer greater than two. The simplest case is that of a very thin wire in the form of a closed circuit ; the free periods in this case are given by s/nV, where s is the length of the wire, V is the velocity of radiation in the medium outside the wire and n is any integer. 43. A solution can easily be obtained for the case of waves propagated in the direction of any number of parallel straight conductors. Let the axis of z be chosen in the direction of the conductors and let the transformation VII] PROPAGATION OF ELECTRICAL EFFECTS, ETC. 63 be determined so that the curves ?? = const, include the bound- aries of the cross sections of the conductors by any plane parallel to z = ; the equations expressed in terms of the coordinates f, rj, z are, 36, the medium being supposed non-magnetic, ^- 877 _,_3 W " d_ iZ_\ _ d_ fy(hj dz\h,J\' 1^-1 (Z 9fU In this case the wave fronts are plane and are given by z = const. Hence the equations become, remembering that h s =1, h T = h 2 = J, J 9 1(1} fo\JJ' . -_ "8?U/ 877 U 64 PROPAGATION OF ELECTRICAL EFFECTS [VII From the last of these it follows that X and Y can be derived from a potential ty, that is X- T d ^ V- T d ^ A. J ^zr , JL = J -7T- , 3f drj and the other equations become __ j _ \ F 2 3^8* " 8* 7; ' 8 /A 8 / _ 8 a = = __ = 8^ " which are all satisfied if __ F 2 _ 8^ The boundary conditions are satisfied if ^ is constant over the boundary. The type of solution is then given by thus whenever the transformation can be found, the corre- sponding problem can be solved. For any number of thin wires the solution is where r is the distance from one of the wires and the summation is extended to all the wires. This result can be used to obtain the effect of a cylindrical conductor on the waves in the wire. For example, the effect of a circular cylindrical conductor, when VII] IN MULTIPLY-CONNECTED SPACES 65 there is one wire outside it parallel to it, is obtained by sup- posing the cylinder to be removed and another thin wire placed at the image of the former in the cylinder. The solution for the case where the medium is not the same throughout can also be obtained in two cases, viz. when the surfaces separating two different media are planes perpendicular to the wires, and when the surfaces separating them are cylindrical, their gene- rators being parallel to the wires*. 44. It was shewn in 1 3 that, if the magnetic force at each point of the surface of the perfect conductors were known, the electric and magnetic forces at each point of the medium could be expressed in the form of surface integrals taken over the surfaces of the conductors. When a conductor is a very thin wire, these surface integrals become, in the limit, line integrals taken along the wire and equations (6) 13 take the form X= 1 dxdy dxdz 9 2 3 2 r/ 22 p)2 32 \ a <.*r Z= (I, , V + m x ^- + r* ? + A^J L - cfo, J\ dzdx clzdy dz* ) r where, to a factor, L is the line integral of the magnetic force taken round the section of the thin wire. Observing that / 1} m lt n t are now the direction cosines of the tangent to the wire at a point (a? , 2/o> z o) on it and that _ _ = r ' e the expressions for X, Y, Z become piKr~\ K *m,L e - -Ids, dy\ ds r } r J /Tr) ( r) 0~ iltr } f>~ tltr ~\ =\^\-L~. e \+^n l L e Ids, J \_dz{ ds r ) r J * Cf. Lord Rayleigh, Phil. Mag. August, 1897. M. E. w. 66 PROPAGATION OF ELECTRICAL EFFECTS [VII which, since the wire forms a closed circuit, are equivalent to d [e~ iKr dL , ( e~ iKr , ^ -- =- ds + tc 2 /! L -- ds, dxj r ds J r v d [e-^'bL , J T err Y = ^- ~ ds + K* Im^L -- ds, dy J r ds J r d fer^dLj 2 f T Z = ^ -- ^-ds + /c* n^L dz J r ds J ds. It remains now to determine L to satisfy the boundary con- ditions at the surface of the wire. Assuming that the radius of the wire r is small in comparison with the wave length, at a point on the wire -.^S r2 = ^, hence the density per unit volume at the point a, y, z is 4" /c 4 5* 2 3[ 4/c 2 |r 2 "^ rJ + 7^ which is equivalent to At a distance r from the origin, great compared with the i A *i! - J * ^ 2 " 4 sin 2 ., , 27r 3 ^ 2 sin 2 l9 wave length, the density is -- , that is -- , and for any number of the same oscillators, orientated indif- ferently, placed at the origin, the density at a distance r will # 2 Z 2 be B - , where B is some number. VIIl] RADIATION 79 52. If there are in a space a number of ions describing orbits*, the wave lengths belonging to them being all the same, and the space be supposed to be contracted in such a way, that all lengths are diminished in the same ratio, the orbits in the contracted space will have their linear dimensions diminished in this ratio, and the wave lengths will also be diminished in the same ratio. Hence, denoting the density of the energy at any point in the original space by (7/X 4 , the density in the new space at the corresponding point will be (7/X' 4 , where X' is the wave length appropriate to the new space. Now unit volume in the original space becomes a volume X /3 /X 3 in the new space, therefore the amount of work, per unit volume of the original space, which has to be done, to pass from the original state to the new state, is C/X'X 3 - (7/X 4 , and therefore, if T and T are the temperatures corresponding to the original and new states respectively, T'-T ( C _C_\/G_ T ~ U'X 3 XV/ X 4 ' whence \T is constant, and the density of the energy is proportional to T 4 , which results have been previously ob- tained f. Further, if energy is being radiated away from the distribution of ions, the rate of this radiation is ( 49) pro- portional to X~ 4 and therefore to T 4 . 53. The condition that a group of ions should be perma- nent, that is that no energy is radiated away from the group, can be obtained as follows. The components of the magnetic force due to the revolving ions are given byj ~ d*dt r fadt r' .J 2 v^.JLs*! 7 dxdt r dydt * r ' * An ion describing an orbit can be replaced by a number of oscillators (Appendix C). t Wien, Berlin. Sitzungsberichte, 1893; Larmor, Aether and Matter, p. 137, 1900. J Appendix C. 80 RADIATION [VIII where f, rj, are the displacements of an ion at the time t -y, these displacements being supposed small, and the sum- mation being extended to all the ions of the group. Now the condition, that there should be no radiation of energy away from the group, is that ' the integral -nft)+G(na-ly) + H (Iff - ma)} dS + ^ // [X (my - 7i/3) + F (noi -ly) + Z (1/3 - ma)} dS, taken over the surface of a sphere at a great distance from the group, should vanish ; the condition will be satisfied if the terms in a, ft, y, d, ft, y, involving the inverse power of the distance, vanish, and this requires = 0, where f , 77, are the maximum displacements of an ion in the directions of the axes of reference. 54. The force, which one permanent group exerts on another, can be obtained from the preceding results. Taking first the case where there is no free electricity, the condition %e = will be satisfied for each group in addition to the conditions given above. The conditions 2ef = ^erj = %e = being satisfied, the lowest power of 1/r, which can occur in the components of the magnetic force due to a permanent group at a distance r from it, is 1/r 3 , and, the condition 2e = being also satisfied for the group, the lowest power of 1/r which can occur in the components of the electric force due to it is 1/r 3 . The components of the force, exerted by this group on any other group, are 2 Ye + 2ea - where X, Y, Z, a, ft, y are the components of the electric and magnetic forces due to the first group, and the summation extends to all the ions of the second group. If the second group is permanent and there is no free electricity belonging to it, the lowest power of 1/r, which can occur in the above VIll] RADIATION 81 expressions for the components of the force, is 1/r 4 and the order of magnitude of this part of the force is e*l*/r 4 , where I is the diameter of an orbit, so that only the groups, which are in the near neighbourhood of any given group, exert a sensible force on it. When there is free electricity belonging to one of the two groups, the lowest power of 1/r, which can occur in the expressions for the components of the force, is 1/r 3 , and the order of magnitude of this part is eH/r*. When there is free electricity belonging to both groups, the lowest power of 1/r, which can occur in the expressions for the force between them, is 1/r 2 , the order of magnitude of this part is e*/r* and its direction is that of the line joining them. It follows from the preceding that, in any material medium in which there is no free electricity present, the forces, which the groups of ions, which constitute the material medium, exert on each other are only sensible at very small distances. When there is free electricity present in the material medium, the part of the forces between the groups of ions, which is sensible at a finite distance, consists of forces between pairs of groups which possess free electricity, the force between any pair varying inversely as the square of their distance apart, its direction being that of the line joining them, and being re- pulsive or attractive according as the free electricity in the two groups consists of an excess of ions of the same or opposite signs. In different material media the configurations of the groups will be different, and, between any two different material media in contact, there will be a transition layer in which there are groups belonging to both kinds of matter. The absolute values of the forces between groups of different kinds will be different from that between groups of the same kind, and to this is to be ascribed the phenomena of capillarity and other phenomena associated with the contact of different media. So long as the groups are permanent, whether there is free electricity present or not, there is no radiation of energy away from the system ; this only takes place when the groups are being broken up to form new groups, and the intensity of this radiation for a definite wave length varies inversely as the M. E. w. 6 82 RADIATION [VIII fourth power of the wave length, as has been established above. 55. The results of the previous chapter, as to the perma- nence of waves associated with closed circuits, can also be established from considerations relative to the radiation of energy from the circuits. Taking the case of waves travelling along an infinitely extended straight wire, the expressions for the components of the electric force at any point are ( 43) Z=0, where the axis of z is along the wire, ZTT/K is the wave length of the waves under consideration, A is a constant and r* 2 = x* + y a . The values of the components of the electrokinetic momentum are ~.- K V 'i K r T and the components of the magnetic force are = 0. The rate of radiation of energy, across the surface of a circular cylinder having the wire as axis, is c -n{3)+G (not - ly) + H(l/3- ma)] dS OTT at + ^ // [X (my -n{3)+Y (na -ly) + Z (10 - ma)] dS, rp /it where 1 = -, m = -, n = 0. r r Now 7 = 0, JGT=0, Z=0, and also ^ _ n ft = 0, no. - ly = 0, my - nfi = 0, net - Ij = ; therefore the above expression for the time rate of radiation of energy across the surface vanishes and, as was seen before, the waves are permanent. VIII] RADIATION 83 The density of the distribution of the energy of the waves at any point is per unit volume, which is equivalent to that is to A^/STT F 2 r 2 , so that it varies inversely as the square of the distance of the point from the wire. It follows immediately that, for any number of parallel wires along which waves are travelling, the rate of radiation across any parallel cylindrical surface vanishes. 56. Taking now the case of waves travelling along any circuit, let the tubular surface generated by a sphere of small radius p, whose centre moves along the circuit, be considered. The electric force at a point on this surface, due to the waves in the circuit, consists of a part, which varies as l/p and whose direction is along the normal to the surface, together with a part, which involves p as a factor since it must vanish with p* ; the same is true of the electrokinetic momentum. The magnetic force consists of a part, varying as l//a, in the plane through the point perpendicular to the circuit and tangential to the tubular surface, the remaining part not being infinite when p vanishes. Through two adjacent points P and Q of the circuit let planes perpendicular to PQ be drawn determining a strip on the tubular surface. Choosing PQ as axis of Z, the components of the electric force, at a point on the strip, are given by X the components of the electrokinetic momentum by , and the components of the magnetic force by * This follows from the expressions for the electric force, 44. 62 84 RADIATION [VIII where X lt Y lJ Z lt F ly G l} H ly a lt /3 lt y r do not become infinite when p vanishes. Then F (my -np) + G(na-ly) + H (1/3 - ma) = 7l (yF, - xG,) + H l Oft - y,), and X (my - n/3) + Y(not - ly) + Z(l$- ma) hence the rate of radiation of energy across the strip is p\, where % does not become infinite when p vanishes, and there- fore, for the whole surface, the rate of radiation of energy across it is /r^/r, where ty does not become infinite when p vanishes. Now ( 47) the rate of radiation of energy across all surfaces, which enclose all the sources of the waves, is the same, and therefore in the above ty vanishes, as otherwise the rate of radiation of energy, across the surface, would depend on its size ; it therefore follows that the rate of radiation of energy across the tubular surface vanishes, and the waves are per- manent as before. CHAPTER IX. OPEN CIRCUITS. 57. IT appears from the investigation of Chapter vn. that, to excite the waves of simple character discussed in 44, some means must be found of enabling closed lines of magnetic force to thread the circuit. This can only be effected by cutting the circuit, so that these closed lines of magnetic force can pass over the open ends. It thus becomes necessary to investigate the effect of an open end on the propagation of waves along a circuit, and only those waves, for which the function denoted by L in 44 has a finite value, need be discussed, as the others, in the case of a very thin wire, have been shewn to be comparatively unimportant and, in the limit, they have no existence. The waves, in the case of an open end, must be maintained by some external means, and the function L) at any point of the wire, is to be determined by the condition ^2 T that -=-y -f K?L is proportional to the electric force, tangential OS to the wire, due to the external disturbances, and L vanishes at the open end, this being necessary on account of the manner in which the expression for the tangential electric force was derived in 44. When the function L has been determined, the radiation of the energy of the waves from the open end, from which only radiation of energy takes place by the previous investigations, can be obtained, so that, conversely, if the nature of the radiation from the open end were known, the function L could be obtained and the necessary sources ; further, for any circuit having a given open end from which radiation is taking place in a given manner, the function L will be the same at the 86 OPEN CIRCUITS [IX same distance along the circuit from the open end, provided that the curvature of the circuit is everywhere continuous. The same result will hold for a very thin wire, provided that the radius of its cross-section is everywhere small compared with the wave length and with the radius of curvature of the wire. It is therefore sufficient, when the circumstances of the radiation from the open end are known, to investigate the case where the circuit is a semi-infinite straight line. To solve the problem of radiation from a semi-infinite straight line, it is convenient to solve the problem for the case of a right circular cone, and deduce that of the semi-infinite straight line as the limit, when the vertical angle of the cone becomes indefinitely small. 58. The space to be considered is that outside the right circular cone which is determined by = # where 6 is the angle made by a vector drawn from the vertex of the cone with a fixed direction, and the space occupied by the cone is that for which 6 lies between and TT; then, choosing polar coordinates of which 6 is one, and remembering that, in view of the preceding, the case required is that in which the lines of magnetic force are circles having their centres on the axis of the cone, the equations in the notation of 36 are 4>7TW = 0, fa the medium being assumed to be non-magnetic. Hence, using the relations the magnetic force y satisfies the equation 111 i i^- n ^l J .Il/J_l/_^^l_J_?? r dr (sin03r IX] OPEN CIRCUITS 87 \vhich, writing cos 6 = //- and yr sin = -\Jr, is equivalent to Therefore, for oscillations of definite wave length 2?r//c, the equation satisfied by A/T is at all points at which the relations (M, v, w) = (/ #, A) are satisfied, that is at all points at which there are no sources. For the case under consideration the sources are distributed symmetrically with respect to the axis, and the discontinuities of the electric force are therefore symmetrically distributed with respect to the axis, but, instead of supposing the sources represented by discontinuities of the electric force, it is more convenient in this case to suppose them represented by dis- continuities of the magnetic force as in 14. The equation to be satisfied by the magnetic force at all points of a circle, whose centre is on the axis of the cone, and along which sources of this type of equal strength are uniformly distributed, is where w is a constant. The surface of the cone being taken to be perfectly conducting, the condition to be satisfied at its surface is that the tangential components of the electric force vanish, and this requires that when = Q . 59. The solution of equation (2) is given by where P n is the zonal harmonic of /A of order n, and where the 88 OPEN CIRCUITS [IX summation has to be taken so as to include all the functions of /j, whose inclusion is consistent with the condition that 9/A vanishes, when 6 . Now and therefore the functions P n , which can occur, are those which vanish when /-t = /z , that is, those for which n is a zero of P n (fj, ), where /u, = cos # . Hence where the summation extends to all the values of n which are zeros of P n (jj, ) ; these values are all real. The functions R n are then to be determined from the relation -')" + l -f?L a- n 8p " *> " ( M 2a,r sin Q = 0, Oi which is equivalent to Now and therefore, writing that is IX] OPEN CIRCUITS 89 Hence, writing for y and y' their values and reducing, <' - n) (n' + n + 1) f ' (1 - & ^"^ - / n+i (r)}, as at an infinite distance Rn cannot involve a term in which & KT occurs, there being no reflexion there. Now, jvhen these expressions are substituted in the expression for -v/r, two series are obtained for i/r, one of which is applicable when r is less than r lt and the other when r is greater than r lt these series converging for all values of r and which occur, except when r==r 1 and 6=6^ simultaneously. Therefore, when r=r lt the two series must be identical ; whence An ,/+ 1 (*T,) = B n [J_ n ^ (*n) - and the solution of (5) may be written R n = a n r* J n+i (*r) { r x 60. It now remains to determine C n . The direct deter- mination of (7 W , though possible, involves somewhat complicated analysis, which can be evaded by the observation that C n is independent of the wave length, that is of K. To establish this, it is sufficient to prove it for a particular case, which can be chosen to be that in which there is no cone and only the circle of sources radiating freely into space. In this case the values IX] OPEN CIRCUITS 91 of n which occur are all the positive integers. It can be easily shewn that equation (2) is equivalent to the equation where is the third polar coordinate, viz. the longitude. The solution of equation (2') in the particular case, where there is no cone, is given by 1 fffe- tlc = 2^ JJJ "IT where the integration is taken so as to include all places where &> has a value different from zero, and jR 2 = r 2 H- n 2 2rTj {cos cos 0! + sin 6 sin 0j cos (< Now, writing cos 7 = cos 6 cos 0! -f sin 6 sin ^ cos (< - c^), the integrand is expressed as a series involving spherical harmonics by means of the relations* when r r a ; whence . #+ (AicrO J" n+i (/cr) P n (cos 7), when r < r 1} and for a circle of sources, writing q = cory* sin l d^ 1 dr lt 7r /27T . / n+J (*r)l P n (cos 7) sin ^ * Heine, Handbuch der Kugelfunctionen, i. p. 346 ; Macdonald, Proc. Lond. Math. Soc. Vol. xxxn. 1900. 92 OPEN CIRCUITS [IX Therefore, since P. (cos 7 ) sin fcdfc = n( ^ Y) sin 6 sin 0, ^|j sin *, it follows that iri oo ttirt n ', i O p O p ^r = 0r>rrM S e 2 -^-rr\ sin2 sin ^ ir" ^ when r < rj .............................. (7), and when r > n (7'). Remembering that and comparing with the solutions obtained by substituting the expressions (6) and (6 X ) for R n in it follows that (7 n is independent of /c, that is of the wave length. 61. When K is zero, the equation (5) becomes n(n ^ nn (i ~ IJv) ~ Zn + l v dn = 2r which, putting R n = r*L n , and writing becomes 3 2 Z n 1 aZ n (n, + V - ft) Sin -- r 9r IX] OPEN CIRCUITS 93 that is, making the substitution r ae~, It is convenient to solve this equation for the range of f from zero to f and then to proceed to the limit; the conditions, corresponding to those to be satisfied in the limiting case, are L n = Q, when f = and when f=f , hence it may be assumed that -, ft where m is an integer. Substituting in the equation for L n) it becomes 5| and therefore For a single circle of sources, writing <7 = a>a*e~~*** sin O^O^ the circle being defined by * = * f-6, this becomes and therefore . m-Trf . sm sin fi- 94 OPEN CIRCUITS [IX Effecting the summation, L n is given by . dP n sinh (n + when > f i and by oPn 8 - sinh (n + when f < fi- Making f infinite, the expression for L becomes when f > fi, and f, when f < fi, that is L n = ~(2 n +\\*r\(::} -('-^} [-smfli^, when r r^ In the case required, the range of r is from to oo , and this case is obtained from the above by making a infinite; hence when r< r 1? and IX] OPEN CIRCUITS 95 when r > ?v Now, when K = 0, equation (6) becomes R n = C.t* n (n + ^) " 2~ n ~* n ( n when r < r a , that is, since r r sn C when r < TJ, and similarly from equation (6') sin (n + |) TT "" n (n + V>* whenr>r x j therefore sin(n + 4)7T a' 62. The solution, for a circle of sources emitting waves of wave length 2-7T//C, the circle being defined by r = r 1 , 6 = 0-^ and being situated in the space external to the cone defined by 6 = # , is therefore given by when r < rj .............................. (8), and by IT = - q S (-V / n+i (/crO {/^-j (*r) \TI/ 7T when r>n .............................. (8 7 ), where the time-factor is included in q y N is written for ~dPn*P* ^ ' and the summation is extended to all the positive values of n which make P n (/i ) vanish. 96 OPEN CIRCUITS [IX In the particular case where = -=- , which is the case of a Zi circle of sources in the space bounded by an infinite conducting plane, the values of n which occur are all the positive odd ?)P ?)P integers, the value of -^-? -^ - is 1 and, remembering equations (7) and (7') above, the expression for ifr is seen to be the sum of the ty due to the circle of sources and the i|r due to the image of this circle of sources with respect to the plane, a result which is otherwise immediately obvious. 63. The case of an indefinitely thin semi-infinite straight wire can be obtained from the above by taking TT e as the value of # , where e is very small. The values of n which occur are then given by n = k + 7? 2 where 2n log - = 1, and k may be any positive integer or zero. Further, since 2 Pn (~ AO = COS n-JT P n ( - - Sin W7T Q it follows that dPn(p) 2 . dQn(p) = cos mr ^ sm mr ~*"^- / ; OfJL 7T 0/JL that is, writing di Again* . _ cos ^ + sn d/u-o IT i P n Oi) log - + H' (0) - n (n) n (-7?.- l)II(r) n(r) where A r = 2 , m * Macdonald, Proc. Lond. Math. Soc. Vol. xxxi. 1899. IX] OPEN CIRCUITS 97 hence, omitting terms of the first order of small quantities, and, observing that (1 p 9 > n (n + s) cos g?r 87 n (n - 5) n M. E. W. 98 OPEN CIRCUITS [IX that is 8/i 2 7 V 2 neglecting small quantities of the second order, hence, to this order, For the case of an indefinitely thin wire, the sources of the waves not being very close to the wire, that is 8 l not being nearly equal to , substituting the above values in the equations (8) and (8'), ^r is given by = q when rn (9 X ), where yu, is not nearly equal to /v It thus appears, comparing this with equations (7) and (7'), that the effect of the indefinitely thin wire at points, whose distance from the free end of the wire is not very great compared with r lt and for which p is not nearly equal to /* , is negligible in comparison with the effect of the source. At a point, whose distance from the free end is great compared with ^ and for which /z is not nearly equal to fiQ, the expression in (9 7 ) has to have the term n e~ tltr f(/ji) added to it, which represents the effect of the free end. 64. In the immediate neighbourhood of the wire //. differs but slightly from /i and then (1- .a^ w M) IX] OPEN CIRCUITS that is and the first term of the series in (8) and (8') is no longer negligible in comparison with the others. The value of the first term in (8) now is 7T that is and (8) becomes . sin ft sin fere' 1 "* -,- J , 1 + I when r < r lt with a similar expression for ty when r > r lt giving the value of i/r along the wire. This series is capable of summation; writing E 2 = r 2 + n 2 + 2rr! cos ft, the expansion theorem for becomes*, remembering that Pa, (- cos ft) = (-)* P* (cos ft), when r - - e- l * r * sin tcr, K K. that is C '== - e" 1 ^ cos KT ; and therefore sin KT 4- 1 cos #r -- e~ l * x . The value of ^ along the wire, when ri sm KT 4- 1 cos /cr) /cr x ( 1 4- fjii sin ft whence 2 ? n o* U- t < r^ , and therefore all along the wire (r+n) _ tcTi sn j This result can be simply interpreted ; the term sn represents the -fy at the point of the wire at a distance r from its end due to the source, and the term /cr x sin c/! represents the effect of the free end of the wire. It ought to be observed that the electric force perpendicular to the wire, being given by F 2 1 d^ IK. ' r sin 6 dr ' is large compared with ^r and, since x log x tends to zero with x, large compared with the electric force in the incident waves. 65. Taking now the case where the sources are close to the wire, that is where 6 l is nearly equal to TT, the first term of the expressions for ^ in equations (8) and (8') is of the same order as the others and is given by : TP G l Sin KT ( L -~~ LL ), KT^ sm #! when r < r^ The remaining part of the series is S ! 0* when r < r l} which as in 64 is equivalent to %-*-& [e~ lKri (IJL sin /cr + 1 cos /cr) - ie~ iKR ], where R* = r 2 + n 2 + SrTj cos ^ ; 102 OPEN CIRCUITS [IX and therefore, in this case, ty, for all values of //,, is given by ^ = MI sin 0! ' 6 when r r 1 , that is *:?*! sin 6/i The first term of this expression represents the effect of the free end and the second the effect of the source, the two parts being now of the same order at all points. 66. The preceding analysis can be easily modified so as to apply to the case where there is a sphere at the end of the wire, the wire passing through the centre of the sphere. The radius of the sphere being r , the condition to be satisfied by ty at the surface of the sphere is when r r . Hence the equation, which corresponds to (8) , is now a , + (*r)-K n +i(ucr) when rr!. When r is small compared with the wave length is of the order (/cr ) 2n+1 when n has any of its values which is not TI O , and of the order (*r ) 2n+1 when n has the value n , so IX] OPEN CIRCUITS 103 that, when r is sufficiently small, the terms due to the presence of the sphere are negligible in comparison with the others, and therefore the effect of a small sphere at the end of the wire is negligible. The terms, which are due to the presence of the sphere, represent reflexion of the waves by the sphere, and, in practice, the conditions to be satisfied, in order that this should be negligible, will be that the radius of the sphere should be small compared with the wave length and not too great compared with the radius of the cross section of the wire. 67. The results of 64, 65 can be obtained by a much simpler process, although the more direct analysis given above is preferable in the general case and necessary in the case of the investigation of 66. It has been shewn that the waves associated with a closed circuit are permanent, and that there- fore, in the case of an open circuit, radiation takes place from the free end only. The value, at any point on the straight wire, of ty is therefore made up of two parts, one due to the source and the other to the free end, and these two together must be such that ^ vanishes at the free end. Taking the wire as axis of z, the value of -\/r at any point, if there were no wire, is given, 60, by "27T a -l K R where p is the distance of the point from the wire, the circle of sources is chosen to be in the plane z 0, and - cos - where p l is the radius of the circle of sources. This is equi- valent to qp r2v e -R + = ^r Jo -1T C where When there is a wire, the value of ty along it is found by taking p in the above expression to be a small quantity. Writing 104 OPEN CIRCUITS [IX it follows that e~ l * R _ e rzr = ^7[7 and I o "* that is I =- cos and therefore the value of <) sin 6 l ^ If, through the free end of the wire, a plane be drawn perpendicular to the wire, and this plane be supposed to be perfectly conducting, the value of i/r along the wire is now obtained by adding, to the above expression, the corresponding X] STATIONARY WAVES IN OPEN CIRCUITS 107 one due to the image of the circle of sources in the plane, and is therefore given by A ^r = . -r- (cos tcr cos (/cr cos ^)}. This result also follows from the expressions (8) and (8'), 62, it being remembered that, when //, is approximately zero, P n (/z) vanishes for the values of n which are given by n = n + 2k + 1, where k is an integer, and, the wire being very thin, n is very small. If, on the system of waves diverging from the circle of sources, there be superposed a system of waves, of the same period and amplitude, converging to the circle, there results a system of stationary waves, and in this case the value of ^ along the wire is given by A sin /cr sin (/cr cos Oj)}. sin From this it follows that, in the case of stationary waves along the straight wire, the part of ty, which corresponds to radiation from the free end in the direction making an angle TT - 6 l with the wire, is given by sin tcr sin (/cr cos ^)}, sin the system of sources, necessary to maintain the waves along the wire stationary, being taken into account. It appears from 57 that, at the same distance from a free' end, measured along the circuit, L, in the case of stationary waves in the circuit, is the same for any circuit, whose curvature is everywhere continuous, as for a straight circuit, if the circumstances of the radiation from the free end are the same in the two cases. Hence, in the case of any open circuit, the part of L, at any point on the circuit, which corresponds to radiation from the free end in a direction making an angle r rr 6 l with the tangent 108 STATIONARY WAVES IN OPEN CIRCUITS [X to the circuit at the free end, this tangent being drawn in the direction of the circuit, is given by D L = - 5- {sin KS sin (KS cos ^ 1 )}, sin u l where s is the distance of the point from the free end measured along the circuit, and B includes the time factor. Considering the surface, which is formed by supposing a small sphere to move with its centre on the circuit and to come to rest when the centre has reached the free end, there is no radiation across the tubular part of the surface, all the radiation takes place across the surface of that hemisphere, which closes the tube and does not intersect the circuit. Therefore the radiation from the free end of the circuit takes place in the directions which make, with the tangent to the circuit at the free end, this tangent being drawn in the direction of the circuit, angles which lie between Tr/2 and TT. The value of L at any point of the circuit is therefore given by 7T f* L = I (sin KS sin (KS cos ^)J /(^) d6 l , J o where B=f(0 l )sme i . When there are no sources or any other free end of a circuit near to the free end, the radiation will be the same in all directions, and therefore the distribution of magnetic discontinuities will be uniform over the surface of the infinitely distant hemisphere. Now B is proportional to q, and 60, q is proportional to co sin 1 dB^ ; hence, in this case, since o> is constant, /(0j) is constant, and therefore L is given by 7T L = C I {sin KS sin (KS cos 6^} dd^. Jo When there is another free end of a circuit near to the free end, the waves from the two free ends will, if of the same period, interfere, and the resultant radiation from either free end will depend on the difference of phase of the radiation from the two free ends and on the direction of the circuit or circuits at the two free ends. X] STATIONARY WAVES IN OPEN CIRCUITS 109 69. In the case of a resonator the directions of the circuit at its two free ends are opposite, and, for the waves observed by means of the sparks which pass between the two free ends, the radiations are in opposite phases ; hence the circumstances of the radiation are the same as those of the radiation from a Hertzian oscillator. To obtain the value of /(#i) for this case, it is necessary to find the distribution of magnetic discontinuity which is equivalent to the oscillator, this distribution being over the surface of a sphere, whose centre is at the oscillator. The value of -v/r due to a circle of sources at r = r lf 6 = d l} is, 60, given "by where r>r l} and for a distribution of magnetic discontinuity on the surface of the sphere r = r-^ given by q = g (0^ d^, by . 2 sm For a Hertzian oscillator -^ is given by Sm. ty^Ae^r^Ki (i/cr) sin 2 6 ; hence in the above ?(4)-*crniL4i fcri r ig4 giving ty=C' T- J| (/crj) ^T| (tier) sin 2 6, !*! and therefore the equivalent distribution of magnetic dis- continuity over the surface of the sphere is given by g = -(7sin 2 M0i. Hence, for the waves in a resonator, which are observed by means of the sparks, B-Osinftd^, where G only involves the time, and therefore IT rl Z = G \ (sin 5 sin (KS cos ^)} sin ftdft , J o 110 STATIONARY WAVES IN OPEN CIRCUITS [x , , . T n ( 1 ~ cos that is L = C ]sm/cs -- ( KS 70. The electric force perpendicular to the circuit, at any point on it, is proportional to -=- , that is, to rt ( sin KS 1 cos KS teC < cos KS -- - H -- - ( KS K 2 S 2 The points at which there are nodes are the points at which the electric force vanishes, that is, the points at which sin KS 1 cos KS _ COSKS -- - + - -=0 ............... (1). KS K*S 2 Writing KS x, the equation whose roots are required is sin# 1 -cos a? , cos# --- H --- - =0 ............... (1); X Otj and this equation is equivalent to the pair COS X = The roots of (2) and (3) occur alternately, the least root of (2) being less than the least root of (3). The nth root of (2) in ascending order of magnitude can be shewn to be given by # n = 2(K,-l)7r + 7r/2-f ............... (4), where = sn f n ({) denoting the value of V(^ -!)/{* + V(* - 1)}, when for x is written 2 (n 1) TT + Tr/2 f . The nth root of (3) in ascending order of magnitude is given by # n '=(4tt-l)7r/2-r .................. (5), where F - sin- !/'(<))) + I i {sin-/,' (F)) 1 + .-., / B '(F) denoting the value of - V(^-l)/{ 2 -V(^-l)}, where X] STATIONARY WAVES IN OPEN CIRCUITS 111 for x is written (4n 1 ) Tr/2 f '. The roots of equations (2) and (3) can, with the exception of the first root in each case, be quickly calculated from the formulae (4) and (5), the first two terms of the series for f and f giving a sufficient approximation. To obtain the values of a^ and a?/, it is more convenient to use the equation (I'), the formulae (4) and (5) being used in these cases to indicate the neighbourhood of the roots required. The formulae (4) and (5) shew that the least root of equation (2) is in the neighbourhood of 71-7T/180, and that the least root of equation (3) is in the neighbourhood of 2537T/180. Using the tables of the circular functions, and writing sin a? 1 cos x v cos # + - - = X , X X* it appears that, when # = 71-&7r/180. X = '0001593..., and when # = 71&7r/180, X = - -0000305... ; hence the least root of equation (V) lies between 10677T/2700 and 1423?r/3600. Again, when tf=253f7r/180, X =-'0001473..., and when # =253fj7r/180, X '0001127...; hence the next root of equation (!') lies between 12687T/900 and 152177T/10800. The third root of equation (!') is approximately 1663?r/675, and so on. 71. For any resonator, the fundamental wave length is that for which there is only one node on the resonator, that node being at the point which is equidistant from the two ends, the distances being measured along the circuit. Hence if I is the length of the resonator, the fundamental wave length X satisfies the relation 14237T 27T l_ 10677T 3600 > X ' 2 > 2700 ' ,, , - 1423 I 1067 thatlS ' 3600^ 2700' or 2-5328 > ^ > 2-52928. Therefore X = 2*53J, the difference between this expression for A and the accurate one being less than '04 per cent. The wave length, corresponding to the first overtone, is that for 112 STATIONARY WAVES IN OPEN CIRCUITS [x which there are three nodes on the resonator; denoting this wave length by \ lf it follows from the above that 15217 j^ 1268 10800 > \ > ~900~ ' that is, -709779 > ^ > 709732 ; and therefore X 1 = '7097. Similarly the wave length X^ corresponding to the second overtone, is given by \^ = *4059, the values of the wave lengths corresponding to the successive overtones tending to the values given by \ n = 2l/(2n + 1), as n increases. The relations between the successive wave lengths are X a = '280\ , X 2 = -160X = '572^, etc. For a circular resonator, neglecting the gap, I = TrD, where 7) is the diameter of the resonator, and the fundamental wave length in this case is, by the above, given byX =7'95Z), this expression differing from the accurate one by less than '025 per cent. 72. It has been shewn experimentally by Sarasin and de la Rive* that, when electric waves are being propagated along a wire with a free end, the waves detected by a resonator are those which have the same period as the oscillations which belong to the resonator. The resonators used by them were circular, their diameters being '75, '50, and '35, in metres respectively, and the observed distances between the first and second nodes along the wire from the free end were 2'95', 1'95 and 1'43. It will be shewn later, 76, that the distance between the first and second nodes is very approximately half a wave length, so that the observed distances are approximately half the fundamental wave lengths of the corresponding resonators. The values of the half wave lengths of the resonators, as calculated from the formula X = 7'95A are 2'98, 1'98 and 1'38 respectively, which agree well with the observed values. In their later experiments-f the wave length in air was observed by detecting the nodes and loops in the stationary * Comptes Rendus, ex. 1890. t Comptes Rendus, CXIL 1891. x] STATIONARY WAVES IN OPEN CIRCUITS 113 waves due to reflexion at a large plane reflector. In discussing these experiments the distance of the first node from the reflector will be taken to be half the wave length, it being easier to observe accurately a node than a loop, and the position of the first node being less affected than that of the second by the finite extent of the reflector. Diameter of reso- nator circle 1 metre stout wire 1 cm. in diameter 75m. stout wire 50m. stout wire 35m. stout wire 35m. fine wire 2 mm. diameter 25m. stout wire 25m. fine wire 20 m. stout wire 20m. fine wire 10m. stout wire 41 20 Dist. of first node 4-14 3-01 2-22* 1-49 1*51 94 1-17 80 93 observed value 4 2-07 1-50 1-11 74 75 47 58 40 46 calculated value 4 1-98 1-49 99 69 69 49 49 39 39 19 * No first node observed, only the first loop. In the above table the calculated values of the quarter wave lengths have been obtained from the formula X = 7*95Z), and it will be seen that the agreement between calculated and observed values is in general very close. These experiments were repeated by Sarasin and de la Rivef under more favour- able conditions. The results obtained for circular resonators of diameters *75 m. and '50 m. were that the distance between a loop and a node in the two cases were 1'50 m. and TOO m. respectively. These observed values give the formula X = 8D for the wave length, the difference between which and the formula which results from the theory being about '6 per cent. 73. In the preceding investigation of the wave lengths of resonators, it has been assumed that the effect of the small spheres, which are placed at the ends to make the sparks more definite, is negligible, and also that the damping of the oscil- lations in the resonator is negligible. The expression for ty in the case of a straight wire, with a sphere of radius r at the free end, under the influence of a circle of magnetic discontinuities, f Comptes Rendws, cxv. 1892. M. E. W. 8 114 STATIONARY WAVES IN OPEN CIRCUITS has been given, 66, and it appears from it that the difference, between the term involving n in this case and the term involving n in the case when there is no sphere, is of the order (ro/X) 271 * 1 , when n has any of its values other than n . The most effective of these terms is that for which rc = n + l, and in this case the ratio of the part, due to a sphere of radius "01 m., to the whole term lies between 10~ 5 and 10~ 6 for oscil- lations whose wave length is 2 m. The spheres at the ends of resonators have diameters lying between 1 cm. and 2 cm., so that, for a circular resonator of '25 m. diameter, the effect of the small spheres, as far as any term in the series for i/r, and therefore for L, other than the first, is concerned, is negligible, and the same result will hold for larger resonators. The value of o/r for the straight wire, with a small sphere of radius r a at the free end, under the influence of a circle of magnetic discontinuity at r = r l} = 6 lt is therefore, 66, given by 9 ( 1 r \ d_ 8^0 a/* -g 2 2 = n +l 7T .n4(l-J^p.y. As in 64, the value of ty along the wire is given by , _ 2g?y , _ t(ei A+W* oin rr ' a 7T X] STATIONARY WAVES IN OPEN CIRCUITS 115 that is, by /c?*i sin where N sin When tcr is small compared with n 0) v tends to zero, and when n is small compared with KT Q , v tends to the value te tlcr o cos KT O . The value of v in general is e te', where the absolute value of v is always less than unity. When #r is so small compared with n , that (*T /ft ) 2 can be neglected, v = e, and the value of L for a resonator under these conditions is given by n L = A I (sin KS + e (1 fr) cos KS - sin J o that is, by , A ( . e 1 cos KS) L = J. J sin KS + ^ cos 5 -- [ . I 2 9 j The nodes are therefore determined by the roots of the equation e . sin/es 1 -cos5 . . cos^-gsm/w -- + / ^ 2 = ......... (6). Denoting by # the least root of equation (I 7 ), which is what (6) becomes, when e = 0, the least root of equation (6) is approximately # 7e. Hence, if e is less than 10~ 3 , the fundamental wave length of the resonator is increased by less than *1 per cent. In general, when v = e ie, the value of L for a resonator is given by cos KS t\ L = A I [{1 - e (1 - /^)) sin KS + e (1 J o that is, by 116 STATIONARY WAVES IN OPEN CIRCUITS [X The nodes are therefore determined by the roots of the equation H) e . sin /cs 1 cos tcs A /tyN cos KS - sm KS -+- = 0...(7). 2 KS /cV The least root of this equation is less than the least root of equation (1), and therefore the fundamental wave length of the resonator is greater, that is, the effect of the small spheres at the ends of the resonator is always to increase the fundamental wave length. In their second series of experiments* Sarasin and de la Rive used resonators, which were made of two different kinds of wire, one being stout wire of diameter 1 cm., the other fine wire of diameter 2 mm. The radius of the small spheres being about 1 cm., the reflexion of the waves due to the small spheres, in the case of the stout wire, will be small, but, in the case of the fine wire, appreciable. It might therefore be expected, that the wave lengths of a resonator made from the fine wire, would be greater than that of a resonator of the same size, made from the stout wire, more especially in the case of the resonators of less diameter, as the effect also depends on the ratio r /X. This difference is well marked in the results of the experiments, the increase, in the case of the resonator of '20 m. diameter, being rather more than 15 per cent. In practice, this cause of difference could be got rid of by diminishing the amount of reflexion due to the small bodies at the ends of the wire, and this could be effected by using, instead of spheres, pear-shaped bodies, the wire being fitted on to the narrower ends of these bodies. 74. In some experiments the ends of the resonator have been fitted with small plates, instead of with spheres, and it is of some importance to find out what their effect would be. Although this problem cannot be solved directly, the effect can be very approximately determined by means of the preceding analysis. Since the result depends on the amount of the reflexion of the waves by the plate, it is clear that the effect of the plate is the same as that of a small sphere, when the * Comptes Rendus, cxn. 1891. X] STATIONARY WAVES IN OPEN CIRCUITS 117 wire is made indefinitely thin, and this is the case where, in the above analysis, v = le^cos/cr^, KT O being very small. The function L, for the resonator, is therefore given by (,, 1 ~ cos/cs] L = A-<(1 \ cos 2 KT Q ) sin KS + $ sin /cr cos /cr cos /cs -- k I KS ) and the nodes are determined by the roots of the equation (1 ^ cos 2 KTo) cos KS ^ sin /er cos rcr sin KS sin KS 1 cos KS If the wave length be supposed given, the least root of equation (8) is very small, and therefore the distance of the first node from the end is very small. In the case of a resonator, oscillating in the fundamental mode, the distance of the first node from the end is half the length of the resonator, and the fundamental wave length is therefore very great. Now, in a resonator, the oscillations, which are observed, are those, whose wave length is least different from that of the oscillator, as the intensity of the oscillations in the resonator depends, approximately, on the inverse of the square root of the difference of the squares of the reciprocals of the wave lengths of the oscillator and the resonator. Hence, in this case, the oscillations, corresponding to the fundamental wave length, would not be observed. To determine the wave length corre- sponding to the first overtone, it will be sufficient to discuss, instead of equation (8), the equation sin KS 1 cos KS _ /0/ . ICOSKS-- + ^ 2 =0 ........... (8'). The least root of equation (8') is given by KS = 0, which has already been considered. Writing sin x 1 cos x cos*- + =Z, it appears that, when x = 234 T V Tr/180, X = - '000014, and when #=234 1 J TF 7r/180, X= -000097, hence the next root of equation (8') lies between 234^77/180 and 234^/180. Taking the second of these two values, which will give a sufficiently 118 STATIONARY WAVES IN OPEN CIRCUITS [x accurate result, the wave length \ lt corresponding to the first overtone, is given by I _2341 \i~1800' where I is the length of the resonator, that is, \ = *7689. The wave length, corresponding to the first overtone, in the case where the effect of the small spheres at the end is negligible, was determined above, and is X x = "7097/, so that the wave length, corresponding to the first overtone, when plates, instead of spheres, are used at the ends of the resonator, is between 7 and 8 per cent, greater. The oscillators, which have been used in different experiments, appear to emit waves of wave lengths lying between 5 m. and 8 m., so that, when a circular resonator, of diameter 1 m. or less is used, the ends being fitted with plates, the oscillations observed will be those whose wave length is \ l} where \ 2*41 D. 75. Limits for the rate of decay of the oscillations in a resonator can be found as follows. The resonator being, in respect of the radiation from it, equivalent to a Hertzian oscillator, the rate of radiation of energy from it, during any period, is, 49, IG-Tr^yF/.SX 4 , where g is the effective gap, E is the maximum charge at either end of the resonator during the period under consideration, X is the wave length of the oscillations, and V is the velocity of radiation in the surrounding medium. Hence, if W denotes the total energy in the resonator at any time, dW dt 3X 4 Now, at any instant of time, W=E 2 /l, where I is a length, which lies between the greatest distance between any two points of the resonator and the distance between the effective free ends of the resonator ; therefore dW_ dt s and W X] STATIONARY WAVES IN OPEN CIRCUITS 119 where k IQ^g-lV/SX 4 . Hence the time, which elapses before the amplitude of the oscillations falls to l/e of its initial value, is 3X 4 /87r 4 ^F, that is 3XT/8wyZ, where T is the time of a complete oscillation. In the case of a circular resonator, of diameter D, D>l>g, and the time t, which elapses before the amplitude of the oscillations falls to l/e of its initial value, satisfies the relation 3X'!T/87ry > t When the ends of the resonator are spheres, the above relation is approximately l'92T(D/g) s >t>l'92T(D/g)*, and, in the case of the resonator, of diameter "70 m., used by Hertz, the distance between the centres of the small spheres being 2 cm., the number of complete oscillations, executed before the amplitude falls to l/e of its initial value, lies between 2352 and 82320. When the ends of the resonator are plates, the time t satisfies the relation '0533 T (D/g) 3 > t> '0533 T(D/g)*, which, in the case of a resonator of the same size as that used by Hertz, shews that, when the ends are plates, the number of complete Oscillations, executed before the amplitude falls to l/e of its initial value, lies between 65 and 2253. Bjerknes* found, by experiment, that the number of complete oscillations, executed before the amplitude falls to l/e of its initial value, was between 500 and 600, in the case of a resonator, whose fundamental wave length was about 8 metres, but, in comparing the results of his experiments with those of theory, it has to be remembered that, in his arrangement, sparks were prevented from passing between the ends of the resonator. Now, sparks can be prevented from passing in two ways, by increasing the effective gap or by arranging that there shall be nodes near the ends of the resonator, in which case the potential difference in the gap will be much less. It at once follows from the above, that an increase in the length of the effective gap greatly increases the rate of decay. The second method of preventing sparks is equivalent to preventing the resonator from executing vibrations of fundamental wave length, which can be effected by using plates at the ends, instead of spheres, and in this case> * Annalen der Physik und Chemie, Bd. 44, 1891. 120 STATIONARY WAVES IN OPEN CIRCUITS [x as was seen above, the rate of decay is about 36 times faster. It can, therefore, be concluded, that the rate of decay of the oscillations in the resonator, under the conditions of Bjerknes' experiments, is considerably greater than it is, when the resonator is arranged as in the experiments of Hertz or in those of Sarasin and de la Rive, and, taking into account the influence of the arrangement of the resonator on the rate of decay, the rate observed by Bjerknes is of the order indicated by theory. It follows, also, from the above, that the rate of decay of the oscillations in the resonator, which correspond to the first and higher overtones, is very much greater than that of the oscillations of fundamental wave length. The influence of the rate of decay on the wave length of the fundamental oscillations of the resonator will be negligible, as was assumed 71, this rate being very small. 76. Proceeding now to the case of waves along a straight wire with a'free end, there being no sources or other free ends near to the free end, it was seen, 68, that the value of L, at a^ point on the wire at a distance s, measured along the wire, from the free end is given by IT [2 L = A \ (sin KS sin (KS cos 0)} dO. J o The points at which the electric force perpendicular to the wire vanishes, that is the points at which there are nodes, are determined by the roots of the equation a ^ cos KS - I cos (KS cos 0) cos 0d0 = 0. ^ JO Writing w x- cos x \ cos (x cos 0) cos Odd = X, * Jo it appears that, when x 0, X = Tr/2 1. As a? increases from zero, cos x diminishes and vanishes when x = Tr/2, the integral cos (x cos 0) cos 0d0 also diminishes and remains positive o X] STATIONARY WAVES IN OPEN CIRCUITS 121 until its first zero, which can easily be shewn to be less than \/3, is attained ; hence, the least value of #, for which X vanishes, lies between and ?r/2. Writing the integral in the expression for X in the form of a series, X is given by 7T /, X* !& X 6 ~2 V 3 ' 3 2 .5 3 2 .5 2 .7 ' and, using this expression, it can be shewn that, when x = 23-7T/60, X = -0015. .., and, when x = 69J7T/180, X = -'0022. . .; therefore the least zero of X lies between 237T/60 and 208?r/540. For waves of wave length X, the first node is therefore at a distance d from the end of the wire, where 104X 23X T40 : > 120' whence d = '192X, this result differing from the accurate one by less than one-two-thousandth part of a wave length. It can also be shewn that the next zero is greater than TT, and that, when # = 2487T/180, X = - "0032..., and when x = 248 J?r/180, X = '00027...; hence, if d l is the distance of the second node, di satisfies the relation 1489X 124 >*>T57^> and therefore d^ '689X, this result differing from the accurate one by less than 10~ 3 X/4. The distance between the first and second nodes is given by "497X, which differs from half a wave length by "003X, justifying the assumption made 72. From the formula X = 7*95.Z), for the fundamental wave length of a circular resonator of diameter D, 71, it follows that the semi- circumference of the circle is *197X, so that the semi-circum- ference of the resonator is very nearly equal to the distance of the first node from the free end of a wire along which waves are travelling, the node being that which belongs to oscillations of the same period as those of fundamental wave length in the resonator. This was observed to be the case by Sarasin and de la Rive* in their experiments. The result of taking into * Comptes Rendus, ex. 1890. 122 STATIONARY WAVES IN OPEN CIRCUITS [X account the effect of the small spheres at the ends of the resonator would be to make the coincidence closer. To determine the positions of the other nodes along the wire, V ra the semi-convergent form of the integral I cos (x cos 0) cos Odd, J o can be used, and the expression whose zeros are required, is, in this form, 3.5.1 3.5.7.9.1.3.5 \ 3 3.5.7.1.3 3 3 2 .5 The zeros of this expression ultimately tend to (2& + 1) Tr/2, where k is an integer ; the distance between the second and third nodes is greater than half a wave length, and so on. 77. The form of the wave front in the neighbourhood of the free end of a wire can be obtained as follows. Taking T in equation (8), 62, to be very small compared with the wave length, the function ijr, in the neighbourhood of the free end, is given by and from the same equation, whence the components of the electric force R and are given by L _ o p F 2 dt ~ r* d ~ r n < Wt X] STATIONARY WAVES IN OPEN CIRCUITS 123 Therefore in the immediate neighbourhood of the free end, - R 6 that is, R or, if be the angle which the radius vector makes with the wire, 7? Hence the equation to the wave fronts near to the free end is approximately /a r* cos = const., A that is, they are portions of paraboloids of revolution, which have the free end as focus and their vertices on the wire. 78. To obtain the form of the wave front in the neighbour- hood of the wire, at a great distance from the free end, let the wire be taken as the axis of z, the axes of x and y being perpendicular to it, and the origin at the free end. The expressions for the components of the electric force at the point x, y, z are then, 44, 57, d re-"dL J- ^. X ""*!> oyJo r d*i j J ^ + Tc 2 l J dz J o r dz l where r 2 = (z - z^ + a* + 7/ 2 , and, 68, 2 r^ sin KZ I cos 2 r L = C [sin K.ZI -- I IT J 124 STATIONARY WAVES IN OPEN CIRCUITS [X Writing e* dL and for its equivalent * f - {(*-*,)+*+*> d- V2 it follows that IT . 3 [cos /c^ -- I cos (icz l cos 0) cos a* TT J o Now foo _^ (2 _ Zi)3 TOO _^ I e 2 ^i ar e the wave lengths of the fundamental tone and of the first overtone, 92 132 STATIONARY WAVES IN OPEN CIRCUITS [X "X 1 = '28X . whence 7X 1 /4 = 49X , and therefore the first node from the mirror for the fundamental tone is very near to a loop for the first overtone. If the fundamental wave length of the resonator is nearly the same, or less than, the wave length of the oscillations sent out by the oscillator, the amplitude of the oscillations belonging to any of the overtones of the resonator will be small compared with that of the oscillations belonging to the fundamental tone, so that, in such cases, the observed positions of the nodes and the loops will be very approximately the same as if the overtones of the resonator were not excited ; the only appreciable effect of the overtones will be that there will be no position in which sparks are entirely absent. If, however, the fundamental wave length of the resonator is greater than the wave length of the oscillations sent out by the oscillator, the amplitude of the oscillations belonging to the first overtone of the resonator may not be small compared with the amplitude of the oscillations belonging to the fundamental tone, as the ratio of their amplitudes is very approximately where X is the wave length of the oscillations of the oscillator, and, in such cases, the observed positions of the nodes and the loops would not be the same as if the overtones were not excited. This is probably the explanation of the fact that Sarasin and de la Rive in their third series of experiments did not obtain good results in the case of the resonator of diameter 1 m., as its fundamental tone has a wave length considerably greater than that of the oscillations of the oscillator, and the amplitude of the oscillations belonging to the first overtone of this resonator is approximately, in the case of the oscillator used, one-fifth of the amplitude of the oscillations belonging to the fundamental tone, so that, in this case, there would be a displacement of the observed positions of the nodes and the loops from the positions of the nodes and the loops belonging to the fundamental tone, which, though small, would be appreciable. Another curious effect, which they observed, X] STATIONARY WAVES IN OPEN CIRCUITS 133 was that the intensity of the minimum sparks in the resonator was greater than the intensity of the sparks, when the mirror was away*. This can be explained as follows: when there is no mirror, the almost dead beat oscillations of the oscillator produce an effect on the resonator similar to that which would be produced by a single pulse. When the mirror is present, the waves from the oscillator, after striking on the resonator, are reflected back from the mirror, as are also the waves emitted from the resonator, and these latter have a very small rate of decay. If the resonator is at a distance from the mirror which would be a loop for one of its overtones, the oscillations belonging to this overtone are continually reinforced, and after a time their amplitude will become great compared with that of the amplitude of the oscillations excited by the oscillator when there is no mirror. As the rate of the radiation of energy from a resonator varies inversely as the fourth power of the wave length, the amplitude of the oscillations belonging to an overtone will be much less in comparison w r ith the amplitude of the oscillations belonging to the fundamental tone than it is when there is no mirror, so that the results stated above, as to the effect of the overtones on the positions of the loops and the nodes, will not be affected. The effect of the overtones will be that observed by Sarasin and de la Rive; there will be no position in which the sparks totally disappear in front of a reflector, and the intensity of the sparks in the resonator, when it is in a position in which the intensity of the sparks is a minimum, will be greater than the intensity of the sparks produced in the resonator, when there is no reflector. * The effect of even slight reflexion, such as that from a wall, in increasing the intensity of the sparks in a resonator had already been observed by Hertz. APPENDIX A. THE RELATION OF THEORETICAL TO EXPERIMENTAL PHYSICS. ALL observations consist in the comparison of motions, the thing observed being always a change of position of something as, for example, of a needle, a spot of light or the hand of a watch. The function of theoretical physics is to give a con- sistent representation of these changes, and this representation has to be made in terms of certain general conceptions such as space. Now space is conceived of as possessing the property of extension only, and it is postulated of it that it is possible to assign definite positions (points) in space, to draw uniquely a line from any one point to any other, which shall have the same direction throughout, and to draw from any point uniquely a line which shall throughout have a definite assigned direction. Thus space itself is not conceived of as moving, but as being such that motion can take place in it. The possibility of the motion of a point in space involves that of its not being in motion, so that absolute position in space is a necessity of thought. If AB and CD be two parallel straight lines and points P and Q be supposed to move along AB and CD respectively so that the straight line PQ moves from the position AC to the position BD, the points P and Q are said to describe AB and CD in the same time, and further, if the straight line PQ in all intermediate positions which it occupies passes through the intersection of AC and BD, the rates of description of AB and APR A.] THEORY AND EXPERIMENT 135 CD by P and Q are said to be always in the constant ratio AB to CD. In this way rates of description of paths, that is velocities, can be compared ; hence arises the idea of a constant velocity, and time is conceived of as measured mathematically by a point which describes a definite path, as for example a circle, with a constant velocity. Mathematical time is not necessarily the same as the time of actual experience, no amount of experience can ever prove that they are the same or that they are different. What can be asserted is that there are actual timekeepers which measure time in such a way that within the range of actual experience its properties are indistinguishable from those which are assigned to the time of mathematical thought. Similarly, it cannot be asserted of any velocity of actual experience that it is an absolute velocity ; what can be is that within the range of the particular experience under considera- tion its behaviour is indistinguishable from that of a mathe- matical absolute velocity or the difference of two such velocities. There can be no actual experience of the absolute space, velocity or time of mathematical thought, but it does not thence follow as some writers have held that we can have no knowledge of them. These conceptions have been evolved by the mind to suit its mode of action which impels it to represent as far as it can the phenomena of actual experience in terms of things which can be thought of as forming parts of the whole con- ceptual system and at the same time as existing independently. Anything capable of actual motion was originally termed matter, and the motions observed were discussed on the assump- tion that they could be represented by a moving point or points. If the laws of motion according to which such a point moved were known, the complete history of the motion could be traced, the position of the point at any instant being then expressible in terms of its position and velocity at a definite time. In attempting to formulate the laws of motion of moving points, whose motions should represent actual motions, it was natural to make use of the ideas already arrived at concerning matter. Matter was thought of as indestructible and measur- 136 THE RELATION OF THEORETICAL [APR A. able, the quantity of matter in a body being independent of its position and unalterable with the time. Further, motion could be communicated to a portion of matter, the simplest case being that where motion is communicated from one moving body to another, the communication of motion being supposed to take place instantaneously. At first the quantity of matter in a body was determined by its weight; it was only when observation had shewn that the weight of a body was not independent of its position that it became clear that quantity of matter was not identical with weight. Matter was recognized as being of different kinds, and bodies, the material of which was the same, could be compared in respect of quantity by the volumes they occupied ; comparison between bodies composed of different kinds of matter would be meaningless except in respect of some property which all kinds of matter possessed in common. Their common property being that they can move, the comparison between them must be made in respect of the communication of motion from one body to another, and in the theoretical comparison each body is supposed to be capable of being represented by a moving point. Observation shews that the circumstances of the motion of two bodies after the communication of motion between them depend on the directions of their motions before, so that the simplest case is that in which the motions of the two bodies are such that they can be represented by two points moving in the same straight line, the object of discovery being some quantity which is unaltered. It is then found that there is a linear function of the velocities which remains constant, that when the two bodies are of the same material the coefficients of their velocities in this expression are in the same ratio as their volumes, and also that, if communication of motion takes place between a body A l and a body B of different material and between a body A z of the same material as A l and the body B, if the coefficient of the velocity belonging to B be taken to be the same in the two cases, the ratio of the coefficients of the velocities belonging to A l and A z is the same as when communication of motion takes place between A l and A 2 . Further the knowledge so far obtained of these coefficients APR A.] TO EXPERIMENTAL PHYSICS 137 makes it natural to expect that, when communication of motion takes place between two bodies which are not moving in the same straight line, the same linear function of the component velocities in any direction remains constant, and this is found to be the case. It follows, that all material bodies can be compared in respect of the communication of motion from one to another ; the coefficient belonging to any one body being kept constant, the others remain constant, having always the same ratio to the standard one. The linear function of the velocities in any direction which remains constant is termed the momentum* of the bodies in that direction, and the part of it which involves only one of the velocities, the momentum of the body possessing that velocity, these momenta being compounded like velocities. The ratio of the coefficients belonging to any two bodies is termed the mass-ratio of these two bodies, and, it being agreed that the coefficient belonging to a particular body is unity, this body is said to be of unit mass and the coefficient belonging to any other body the measure of its mass. When communication of motion takes place between two bodies the effect produced in any direction is measured by the change of momentum of the body in that direction. Observation shews that in many cases change takes place in the motion of a body without its being immediately assignable to communication with another moving body; the change which takes place is still measured in the same way, and the cause producing this change is termed force, the measure of the force being that of the change which it produces. When the changes are such that they must be considered as taking place gradually and not instantaneously, the corresponding laws are obtained by considering the gradual change to be the limit of a great number of changes taking place instantaneously at successive small intervals of time. Change of momentum is then replaced by time rate of change of momentum and force by continuous force. The preceding constitutes the Newtonian scheme for the representation of the phenomena of bodies in motion and in it the idea of momentum is fundamental. The idea of mass is * By some writers it has been termed the " quantity of motion." 138 THE RELATION OF THEORETICAL [APR A. arrived at subsequently, and it is a result of experience that the mass-ratio of two bodies of the same material is that of their volumes, just as it is a result of further experience that the masses of bodies can at the same place be compared by their weights. Instead of taking momentum as the fundamental idea another measure of the quantity of motion might have been chosen. That adopted by Huygens was the quantity now known as energy, and with it as the fundamental idea a complete scheme for the representation of the phenomena can be developed. So far as mechanics is concerned either scheme is equally convenient ; it is when dynamical methods come to be applied to the representation of other physical phenomena that the advantages of the second scheme become apparent. For some time physical phenomena such as those of heat and electricity were ascribed to the existence of special sub- stances which were different from matter, though measurable. Gradually, however, attention was directed to the fact that motion could be converted into heat and heat into motion. It was then discovered that quantity of heat could be measured in terms of motion, the energy of the corresponding motion being a measure of the quantity of heat. The principle of the conservation of energy, which was previously known to hold for mechanical phenomena, was thus extended to thermal phenomena and then to all physical phenomena. The natural inference was made that all physical phenomena are modes of motion, though in many cases these motions could not be directly observed. The fact that some of the motions can not be directly observed makes it convenient that the measure of quantity of motion should be independent of its direction, the simplest such measure being the energy of the motion. Further, it has to be recognized that all these motions are not necessarily motions of matter, the term matter being now restricted to that which possesses molecular structure. Motions of other kinds which may not be capable of being represented by moving points must be contemplated, and the idea of mass must not be associated with these motions as this idea involves the possibility of representing them by moving points. In order APP. A.] TO EXPERIMENTAL PHYSICS 139 then to apply dynamical methods to all physical phenomena the laws of dynamics must be expressed in terms of that measure of the quantity of motion which is termed energy and in a form independent of the idea of mass, these laws not being inconsistent with the laws of motion of material bodies. Now these conditions are fulfilled by that form of the Lagrangian method which is usually known as the principle of Least Action and which expresses the fact that the system moves so as to expend as much energy as it can. The object of discovery in any case is the Lagrangian function and this function has to be constructed from the results of experience. The presence of motions other than those which can be represented by moving points being contemplated, there may be degrees of freedom which cannot be specified by the coordinates of moving points, and the coefficients of the squares of the velocities corresponding to these coordinates in the Lagrangian function must be deter- mined from the results of experience just as has been done in the case of the coefficients of the squares of the velocities of the moving points which represent the motions of material bodies. On this view all forces are to be regarded as forces of motion, motion being the change underlying all the changes which constitute physical phenomena. This being so, it seems natural to inquire whether the Lagrangian functions which are constructed from the results of experience can be derived from the Lagrangian function which would involve all the degrees of freedom belonging to the motions which constitute physical phenomena and not those only which are directly observed. This is what has been done in Chapter v. above, where the forms of the modified Lagrangian function which arise from the elimination of a number of the coordinates specifying degrees of freedom have been discussed and where, in particular, it has been shewn that the modified Lagrangian function, which occurs in a great number of cases in the form of the difference of two functions one a homogeneous quadratic function of the time rates of variation of the coordinates which specify the observed degrees of freedom and the other a function of these coordinates arises from the original Lagrangian function when the latter involves the coordinates specifying the unobserved or 140 THE RELATION OF THEORETICAL [APP. A. concealed degrees of freedom in a particular way, and that the function occurring in the modified Lagrangian function which does not involve time rates of variation of the coordinates specifying observed degrees of freedom and usually termed the potential energy is the energy of the concealed motions. Now the addition to the Lagrangian function of a constant or of a linear function of the time rates of variation of the coordinates in evidence in it, the coefficients in this linear function being constants, does not alter the equations of motion ; thus the presence of degrees of freedom which are involved in this manner in the Lagrangian function does not affect the motions corresponding to the other degrees of freedom. Further, if there are degrees of freedom whose coordinates are involved in the Lagrangian function in such a way that it differs from the Lagrangian function, which would exist if they were absent, by the addition of parts which are (1) approximately a constant and (2) a linear function of the velocities belonging to the other degrees of freedom, the coefficients in this function being ap- proximately constant for the range throughout which the motions belonging to these latter degrees of freedom are being considered, these motions will not be appreciably altered. Thus, in the case of the motion of material bodies, the equations of motion will be approximately the same whether the motions be measured relatively to a body A, or to a body B which has a small motion relatively to A, provided that the time for which the motions are being considered is sufficiently short. The laws of motion of material bodies were first arrived at from observation of moving bodies in the immediate vicinity of some place on the earth's surface, the motions being measured relatively to this place. When motions of bodies near the earth's surface, these motions lasting a considerable time as in the case of bodies falling from a great height, were made the subject of observation, it was found to be convenient, in order to obtain a simple representation of the motions, to measure the motions relatively to axes supposed to be drawn through the centre of the earth, one of them being the axis about which the earth rotates relatively to the stars and the other two fixed in direction relatively to the stars. APP. A.] TO EXPERIMENTAL PHYSICS 141 Similarly in the case of the motions of the planets it was found convenient to measure the motions relatively to axes supposed drawn through the sun and fixed in direction relatively to the stars. If the first class of motions be measured relatively to the axes supposed drawn through the centre of the earth in the second case, the resulting motion relatively to the axes first drawn will be approximately the same, and similarly, if the motions of both these classes be measured relatively to the axes drawn through the sun in the third case, the resulting motions relatively to the sets of axes first chosen in the respective cases will be approximately the same, the differences in the several cases being experimentally inappreciable. In every case of the motion of material bodies the laws of motion must be the same, and if the natural assumption, that of Newton, be made, that the force between any two material bodies depends only on their relative position, there will be a set of bodies, possibly imaginary, there being no body not included in the set which possesses only a uniform motion of translation relatively to the set, such that if motions are measured relatively to them, the laws of motion will be accurately true, whilst they will only be approximately true if the motions be measured relatively to any other body not included in the set. In the case of motions generally there will be a body, possibly imaginary, such that if all motions be supposed to be measured relatively to it the laws of dynamics will be accurately true, while they are only approximately true when the motions are measured relatively to any other body. In the theory of electrodynamics Faraday's laws were first arrived at from observation of circuits conveying electric currents, the positions and motions of these circuits being measured relatively to the place on the earth's surface at which the observations were being made, and, when Maxwell inves- tigates the state of affairs in the intervening medium by exploring it by means of the secondary circuit, this circuit is in each position it occupies supposed to be fixed relatively to the place of observation. When the propagation of electrical effects across distances comparable with those of Astronomy comes to be discussed, the motions are measured relatively to the axes of 142 THEORY AND EXPERIMENT [APR A. reference of Astronomical investigations, and the secondary circuit of exploration must now in each position it occupies be supposed to be fixed relatively to these axes. The laws of electrodynamics arrived at from the previous experiences are assumed to be still true, an assumption justified by observation, and if the motions in these previous experiences be measured relatively to the Astronomical axes the differences between the results so obtained and those previously obtained are so small as to be incapable of detection by these observations. There will then in the case of electrodynamics be some body such that, if all motions be measured relatively to it, the laws of electrodynamics will be accurately true, and this case differs from that of mechanics, inasmuch as the laws of electro- dynamics would not be accurately true if the motions were measured relatively to a body which had a uniform motion of translation relatively to the body of reference. What is to be understood by the term " axes fixed in space " is a set of axes which are such that if all motions were measured relatively to them the laws of electrodynamics would be accurately true. In the comparison of theory and experiment the set of axes to be used instead of those of theory is that set which is most convenient and the use of which will lead to a sufficiently accurate representation of the phenomena under discussion. APPENDIX B. CONTINUOUS MEDIA. A CONTINUOUS medium may be defined as being a medium, which is such that in any region of space occupied by it no point can be found which is not also in the medium. The idea of continuity present in this definition is identical with that of the geometrical continuity of space. Rigid bodies and elastic solids, liquids and compressible fluids have, in order to subject them to mathematical treatment, been identified with continuous media, and it is important to inquire what the process known as " treating them as continuous " is equivalent to, so as to find out how far the mathematical treatment, which is applicable to them, can be applied to a continuous medium such as the aether must be postulated to be. When the motions of some specified physical body are mathematically investigated, this body must be replaced in imagination by some object which can be completely defined. It is usual to assert that all material media possess atomic or molecular structure, and this being assumed to be so, it is first necessary to try to state in what way a molecule or atom can be repre- sented so as to be capable of mathematical treatment. A molecule or an atom may be regarded as having a certain quantity of energy associated with it, this energy being ex- pressed in terms of a number of coordinates which are such that all possible variations of the energy can be taken account of by varying these coordinates. A complete representation of this kind has still to be obtained, but it would appear that in a large number of cases of motions of material bodies the 144 CONTINUOUS MEDIA [APP. B. following representation is sufficiently accurate. The position of the molecule or atom in space at any time may be supposed to be identified with that of some point whose coordinates are x, y, z, the axes of reference being the fixed axes of theoretical dynamics. The energy associated with the molecule or atom may be supposed to consist of three parts, a part depending on the velocity of the point which determines the position of the molecule or atom and which is \m (u? + v' 2 + w 2 ), where u, v, w are the component velocities of the point and m is a mass coefficient, a part which is a function of x, y, z and of the coordinates of the points which determine the positions of any other molecules or atoms which may be present, and a part which is a function of other coordinates not in evidence ; this latter part must be supposed to be invariable as must also the coefficients of the second part, it being conceivable that these coefficients may be functions of the other coordinates which are not in evidence. A material medium may be regarded as an aggregate of molecules or atoms, and the mathematical theory of such material media can then be developed from the following assumptions : The matter which occupies any finite volume of space is regarded as being constituted by an aggregate of molecules. The motion of each molecule can be represented by that of a moving point. The effect of all other motions, whether they are motions of the molecules which cannot be represented by moving points or motions which do not belong to molecules, can be represented by a potential energy function or by a system of forces associated with each molecule, this function or system of forces depending only on the positions of the molecules ; and this includes the case in which the effect of these other motions, instead of being immediately represented by a system of forces associated with each molecule, is represented by restrictions on the possible motions of the molecules, APR B.] CONTINUOUS MEDIA 145 The Lagrangian method is applicable to the motions of the system. The kinetic energy of a molecule is then represented by \m (u 2 + v* + w 2 ), where u, v, w are the component velocities of the point whose motion represents the motion of the molecule, and m is the mass belonging to the molecule. The kinetic n=N energy of the aggregate is represented by 2 $m n (u n *+ v^+ w n 2 ), n = l where the suffix n identifies a particular molecule and the summation extends to all the molecules, there being N of them. Similarly, the potential energy of the aggregate is represented n=N by 2 m n V n , or the work done by the systems of forces, which n=l represent the effect of the other motions, is n $ N m n (X n %xn + Y n Sy n + Z n Sz n ) n = l for small arbitrary displacements of the molecules. The application of the Lagrangian method will then lead to 3N equations involving in addition to the SN coordinates, which specify the positions of the molecules, as many un- determined multipliers as there are relations specifying restrictions on the possible motions of the molecules, and these undetermined multipliers specify the systems of forces which are equivalent to the restrictions on the possible motions of the molecules. When these 3-/V equations, the equations restricting the motions of the molecules being taken into account, have been solved, the positions and motions of the molecules at any time are expressed in terms of their positions and motions at a particular time, and the complete history of the motions can be traced. In general the number of dependent variables occurring in any one of the equations will be great, and the solution of the equations will be practically impossible, but, if the equations form a set of groups such that the number of dependent variables occurring in the equations of a group is equal to the number of equations in the group, and this number is not too large, the solution of the equations becomes more possible, the simplest case being that in which the groups are exactly M. E. w. 10 146 CONTINUOUS MEDIA [APP. B. alike. The most important case is that in which the discrete analysis sketched above can be replaced by a continuous analysis. If a surface be drawn enclosing a simply-connected space in which there is a number of molecules, the ratio of the mass of the molecules inside this surface to the volume enclosed by it may be termed the average density of the aggregate of molecules inside the surface. Now let such a surface surround- ing the point be contracted so that the volume enclosed by it is diminished, the average density inside the surface will tend towards a limit p so long as the least linear dimension of the volume enclosed is not less than a certain length depending on the distance between two adjacent molecules, but will after- wards depart altogether from that limit*. This limit p may be termed the density at the point 0, and will be a function of the position of 0, such that fjfp dxdydz taken throughout any volume differs from the sum of the masses of the molecules in that volume by a mass which is less than some standard small mass. By an exactly similar process of reasoning it may be n=N shewn that the expression 2 \m n (u n * 4- v n z + w n *) for the kinetic =i ene rgy may be replaced by the expression ///ip (u 2 + tf + w 2 ) dxdydz, where the integral is taken throughout the volume occupied by the aggregate of molecules, this integral differing from the sum by a quantity which is less than some standard small kinetic energy. These remarks also apply to the potential energy function or the systems of forces associated with the molecules, which represent the effect of all the other motions which may exist. In the discrete analysis the suffixes n serve to identify the molecules. In order then that the equations of motion obtained by the application of the Lagrangian method in the continuous analysis may legitimately replace the equations of motion in the discrete analysis, the assumption must be made that throughout its motion any point x, y, z is always identified with the same bit of matter, and, when the system under consideration is being regarded as an aggregate of molecules, * The limit is not actually attained. APR B.] CONTINUOUS MEDIA 147 this requires the following condition to be satisfied. If a surface be drawn passing through a number of points which specify the positions of molecules, and this surface move with the molecules, no molecule which is initially on one side of this surface ever comes to be on the other side of it. This may be expressed by saying that the order of arrangement of the molecules does not change. When the above conditions are satisfied, the Lagrangian method can be applied, and in this way a group of equations is derived, this group being typical of all the equations of motion of the molecules. The media for which the Lagrangian function is of the kind specified above, and to which the operations of mathematical analysis are applicable, will then consist of aggregates of molecules, for which the order of the arrangement of the molecules does not change, and which are such that the distance between any two adjacent molecules is very small. The " rigid body " of mathematical theory satisfies all the above conditions, for the distance between any two molecules of the aggregate which constitutes the body is assumed to be invariable, and therefore the order of arrangement of the molecules does not change. When the aggregate of molecules is such that the distance between any two neighbouring molecules can vary, but only to a small extent, and the distance between any two adjacent molecules is very small, the aggregate constitutes what is termed an "elastic solid." When the system of forces or the potential energy function associated with each molecule is known, it being assumed that the order of arrangement of the molecules does not alter, the Lagrangian method can be applied, and the resulting equations will determine the history of the changes which take place. The system of forces or the potential energy function associated with each molecule will consist of two parts, of these one is determined by the changes in the distances between the molecules, the other must be supposed to be given as in the case of the rigid body. The first object of discovery is then the system of forces or the potential energy function associated with each molecule, which arises from the change in the distances between the molecules. Two methods of effecting 102 148 CONTINUOUS MEDIA [APR B. this have been proposed. The first of these, due to Navier and developed by Poisson, Cauchy and St Venant, consists in assuming that the effect of the molecules on each other can be represented by forces between them, the force between any two depending only on their distance apart, and the direction of this force being that of the line joining them. The processes by which the stresses are then expressed in terms of the strains involve the assumptions that the order of arrangement of the molecules does not change and that the constitution of a molecule remains the same, that is, that during the displace- ments the molecules are not broken up and then formed into new molecules. These stresses possess a work function, and, when the medium is isotropic, this function involves only one constant. The other method of obtaining the stresses is due to- Green and consists in assuming that the stresses possess a work function, this function being expressible in terms of the strains. The application of the Lagrangian method then leads to equations which are sufficient for the determination of all the circumstances of the changes which take place ; but in order that the work function should be of the form assumed and that the Lagrangian method should be applicable, it is necessary to assume that the order of arrangement of the molecules does not change and that the molecules do not break up during the displacements. When the medium is isotropic, Green's work function involves two constants, and the distinction between his theory and that of Navier is that on Green's theory the intermolecular forces cannot be represented by forces between pairs of molecules, the magnitude of the force between a pair depending only on the distance between them. It would appear from the investigations of Chapter vni. above that, if inter- molecular forces be assumed to be of electrical origin, the force which represents the effect of a molecule on any other molecule is not necessarily a function of the distance between the two, but the forces, from the way in which they are derived, would possess a work function, so that a work function of the form assumed by Green is possible. The question, whether Navier's theory or Green's furnishes the best representation of the phenomena, has been much discussed, and attempts have APP. B.] CONTINUOUS MEDIA 149 been made to decide the question by an appeal to experiment, but the evidence from statical experiments cannot be accepted for a reason which will appear later. One of the strongest arguments in favour of Green's form of the work function is that it is the most general one, which permits of the applica- tion of the methods of mathematical analysis, and is such that the parts of the medium are in a state of relative rest when there is no external system of forces. In the above the idea, that a molecule is itself composite, has been introduced, and it is important to inquire what the re- strictions are under which the aggregates of atoms, each molecule being now thought of as consisting of an aggregate of atoms, can, in the application of the methods of mathematical analysis, be replaced by the aggregate of molecules. If each one of the atoms in the aggregate of atoms which constitutes any molecule is describing an orbit, which is periodic relative to some point x, y, z, this point denning the position of the molecule relative to the other molecules, the position of any atom at any time is specified by the coordinates x + f , y + 77, z + f. The equations of motion of the whole system will then be contained in the variational equation 22m (+f)&e+... + ... + 22ra where m is the mass of an atom, and the summations are taken for all the atoms in each molecule and then for all the molecules. If from these equations all the coordinates x, y, z be eliminated, there will result equations from which the differences g l 2 , r li~ r n^) (a ~~ ?2> of the coordinates f, 77, f can be determined. The integrals of these equations, the orbits of the atoms, relative to the points specified by the coordinates #, y, z, being all periodic, will be of the form cos I 2f---a,|; whence it may be assumed that v> =2 cos 150 CONTINUOUS MEDIA [APR B. Now the coordinates x, y, z can always be chosen so that they do not involve parts which are periodic in any of the periods T 8 , and when the above expressions for f, 77, f are supposed to be substituted in the variational equation, it will take the form where M is the mass belonging to a molecule. In this equation X', Y', Z' will represent the systems of forces associated with each molecule, provided that neither the part of them, which arises from intermolecular action, nor the part, which does not arise from such action, involves a term which is periodic in any of the periods T s . The aggregates of atoms can therefore, in the application to them of the methods of mathematical analysis, be replaced by the aggregate of molecules when the following conditions are satisfied. Each atom in a molecule describes an orbit which, relative to some point defining the position of the molecule to which it belongs, is periodic*, and the system of forces associated with each molecule, which does not arise from action between the molecules, does not contain a part which is periodic in any of the periods belonging to the orbits of the atoms. Further, when these conditions are satisfied, the part of the inter- molecular forces, which is effective in respect of the positions of the molecules, is the part which is independent of the periods belonging to the orbits of the atoms. When forces, which represent the effect of moving systems other than the aggregates of atoms, act on the atoms, the course of events can be represented as follows ; if no part of these external forces is periodic in any of the periods of the orbits of the atoms, they are equilibrated by the motional forces of the molecules and the part of the intermolecular forces which is not periodic in any of the periods of the orbits of the atoms. Their effect, in the first instance, will be to change the relative distances of the molecules ; this change produces a change in the part of the intermolecular forces which is periodic in the periods of the orbits of the atoms, and to balance this the * Such an orbit is not necessarily closed. APP. B.] CONTINUOUS MEDIA 151 dimensions of the orbits of the atoms are changed. The extent to which the dimensions of the orbits of the atoms can be changed is limited by the condition that the displaced orbits must be stable, and the changes in the orbits of all the atoms belonging to a molecule will be related in such a way that there is no radiation of energy from it. When there are external forces which are periodic in the periods of the orbits of the atoms, an analysis which only takes account of an aggregate of atoms will not be applicable; a finer analysis, which shall take account of the atoms whose orbits have among their periods the periods of the external forces, must be used. Again, the external forces may be such that an analysis which takes account of each molecule is unnecessary, and a coarser analysis which only takes account of compound mole- cules, each compound molecule being made up of a number of molecules, is sufficient. In every case the range, throughout which the continuous analysis will be applicable, is determined by the conditions that the units, whose motions are taken into account, whether these units are compound molecules, molecules, or atoms, do not change their order of arrangement, and that, if they are composite, they do not break up. The work done by the external forces is all accounted for by the changes in the positions and in the motions of the units, and therefore a continuous analysis is only strictly applicable to an aggregate of molecules or atoms, when the changes which take place are such that no energy is gained or lost in the form of heat or any other form which is not taken account of by the changes in the positions and motions of the molecules or atoms. The possibility of applying continuous analysis to an elastic solid therefore requires that the changes which take place in it shall take place adiabatically. When the elastic solid is executing small vibratory motions, this is probably true, and in this case continuous analysis can be applied to it. When an elastic solid passes from one state of strain to another by the 152 CONTINUOUS MEDIA [APP. B. application of external forces, the changes taking place in such a way that a continuous analysis is applicable, the intrinsic energy of the molecules will be altered, and, if there are no other influences present, the temperature of the body will be different in the two states. In any actual case the body is surrounded by some other medium, and the initial state of the body is one of equilibrium relative to its surroundings ; the state arrived at after the changes have taken place adia- batically will not be one of equilibrium relative to its surround- ings, to attain a state of equilibrium a transference of energy will take place between the body and the surrounding medium. This transference of energy, supposed to take place in such a way that a continuous analysis is still applicable, ought to be taken account of in the equations which give the history of the changes, and the result would be the introduction into these equations of a system of forces which would represent the effect of the surrounding medium. For example, when the Young's modulus of any material is determined by stretching a wire made of this material by a weight attached to it, the work done by the weight is not all used up in stretching the wire, some of it is used in altering the condition of the surrounding medium, and therefore the Young's modulus of the material so determined will differ from the true one. When an aggregate of molecules is such that the molecules can move freely and the distance between any two adjacent molecules is very small, the aggregate forms what is termed a fluid. As in the case of an elastic solid a continuous analysis will only be applicable when the molecules move in such a way that their order of arrangement does not alter and no energy is gained or lost in the form of heat. Since the order of arrangement of the molecules does not alter, the same mole- cules will always be at the boundary, no molecule which is not originally at the boundary will ever come there, and no molecule which is originally at the boundary will ever cease to be there. Further, if at a definite time t a certain number of molecules occupy an element of volume at the point # , i/ , , this point defining, for the purposes of continuous analysis, the position of these molecules, the same molecules will at a time t occupy an APP. B.] CONTINUOUS MEDIA 153 element of volume at the point x, y, z, where this point defines the position of the same molecules at the time t, and therefore, the mass belonging to the molecules being unalterable, the relation a fey,*) 9 (#o, 2/o, *o) will be satisfied at each point of the space occupied by the fluid. This relation expresses the correspondence which exists between the spaces occupied by the fluid at two different times, in consequence of the restriction on the possible motions of the molecules. If the forces which represent the effect of other systems on the aggregate of molecules possess a work function, the Lagraugian function of the motions will have the form /// tip (* + f + f)-pV} dxdydz. For a liquid p p Q , p being a constant for a homogeneous liquid and a known function of x , y , z , for a heterogeneous liquid. The relation, which expresses the restriction on the possible motions of the molecules of the liquid, is 8 Q, y, z) =1 8(^0,2/0,^0) and the application of the Lagrangian method gives subject to the above relation, that is f) - P.F+K-' dx^dy^dt = where K is an undetermined function of x, y, z. This is equivalent to . dx^dy^dz^dt = 0, that is, jjj pxSxdxdydzj + f ' f(lic&KdSdt+ ... + ... 154 CONTINUOUS MEDIA [APP. B. The quadruple and the triple integrals must separately vanish ; that the quadruple integrals may vanish, the equations a* must be satisfied throughout the space occupied by the liquid at any time, and the equations which result from the vanishing of the triple integrals give the dynamical conditions satisfied at the boundary at any time, the form of these conditions depending on the nature of the boundary. The function K which occurs in these equations represents the internal system of forces which arises from the restrictions on the motions of the molecules and is known as the pressure. There are two kinds of possible motions of a liquid ; the motions which can be generated by external conservative forces, and the motions which cannot be so generated. In the first of these the motion is what is termed irrotational and in the second rotational or vortex motion. The possibility of applying continuous analysis to the motions of a liquid involves the result that vortex motion is always associated with the same molecules. Further, every motion of a liquid due to external causes can be represented as the effect of a vortex sheet, whose surface is the boundary of the liquid to which the continuous analysis is applicable. The representation can also be effected in terms of other distributions of vortex motion external to the liquid ; for example, when a portion of the liquid is in a state of motion which is not necessarily capable of being treated by a continuous analysis, the effect of this portion on the remainder of the liquid, supposed to be moving in such a way that a continuous analysis is applicable to it, can be represented by a distribution of vorticity throughout the first portion, but it cannot be thence inferred that this vortex motion constitutes a geometrical representation of what is going on in this first portion of the liquid. When an actual liquid is in contact with an actual solid, in passing from the solid to the liquid a transition is made APP. B.] CONTINUOUS MEDIA 155 from a place where the molecules are, relatively to each other, approximately immobile, to a place where the molecules approximate to a condition of perfect mobility. The nature of the forces which hold the molecules together will therefore, throughout a certain space in the neighbourhood of the surface of the solid, change their character; in the solid they will be of a nature similar to that of the internal forces in the theoretical elastic solid, and in the liquid, at some distance from the surface, they will be capable of being represented by a pressure; in the portion of the liquid in the immediate neighbourhood of the solid they cannot be so represented. The continuous analysis which is applicable to the motions of the theoretical liquid cannot be applied to this portion, but the effect of the solid and of this portion of the liquid on the portion of the liquid to which the continuous analysis can be applied, can be represented as being due to a vortex sheet whose surface is the boundary of that portion of the liquid to which the analysis is not applicable. If this surface can be taken so near to the solid that the distance of any point in it from the surface of the solid is so small as to be negligible, the condition to be satisfied at the surface of the solid will be that the velocities of the solid and of the liquid normal to this surface are continuous. When a solid is moving through a liquid such as water, which maybe taken to be perfectly mobile, the portion of the liquid in the neighbourhood of the solid, throughout which the internal forces cannot be represented by a pressure, will be bounded by a surface very near to the solid, provided the velocity of the solid relative to the liquid at some distance from it is not too great ; therefore a sufficiently good representation of the motion will be obtained by taking the above condition to be that which is satisfied at the boundary of the solid, and this representation will be consistent with the fact that no slipping takes place at the surface of the solid, the transition from the solid to the liquid being supposed to take place in a thin layer whose thickness is negligible. When one liquid is in contact with another, there will be a transition layer throughout which the internal forces cannot be represented by a pressure ; as in the previous case, provided the velocity of the one liquid relative to the other is not too 156 CONTINUOUS MEDIA [APP. B. great, this transition layer can be taken to be very thin, and its effect can be represented by supposing a vortex sheet to coincide with the surface of separation of the liquids. The motions of jets and of heterogeneous liquids belong to this class. In a liquid the condition, that the same molecules always occupy an element of volume of the same size, determines the nature of the internal constraint, that it can be represented by a pressure. In a fluid this condition does not hold ; the relation 8(a?,y, z) only expresses the fact that the distribution of the molecules is always such that the distance between any two adjacent molecules is very small and that the total number of molecules is unalterable ; the nature of the internal forces cannot be determined from this relation. It has to be assumed that the internal forces in a fluid are of the same kind as in a liquid, and that they can be represented by a pressure, the existence of internal forces in the nature of shearing stresses being incompatible with the idea of perfect mobility. This assumption having been made, it is further necessary, in order that the transformation of the function JJJTi P (& + 2 + -z 2 ) ~ P V + *] dxdydz from the space occupied by the fluid at time t to the space occupied by it at time t can be performed, that K should be a function of p only. When this further condition is satisfied, the variation can be performed, and the equations of motion are the same as in the case of a liquid, so that the statements made above concerning the motions of liquids apply equally to the motions of fluids. Just as in the case of the elastic solid, the application of continuous analysis to obtain the equations of motion involves the assumption that the changes which take place in the fluid take place adiabatically. Thus, it being assumed that a continuous analysis can be applied to determine the circumstances of the propagation of waves of sound, the law connecting pressure and density must be taken to be" the adiabatic law. When a mass of fluid is in motion, the course of events may be such that the fluid, supposed initially to occupy a APR B.] CONTINUOUS MEDIA 157 space bounded by one closed surface, tends to divide itself into two separate portions, each occupying a space bounded by a closed surface. It is at once clear that a continuous analysis cannot be applied to trace the history of the changes that take place while the fluid passes from the state in which it consists of one portion, to the state in which it consists of two portions, as during the transition the distances between some pairs of previously adjacent molecules will become sensible. The mathematical functions, which express the circumstances of the motion, will become discontinuous as the instant of separation is approached, and will thus determine the limits up to which a continuous analysis can be applied, but it is probable that the analysis ceases to be applicable some time before this limit is reached, owing to change in the order of arrangement of the molecules. A somewhat similar state of affairs exists when plane waves of condensation and rarefaction, the amplitude of these waves being finite, are being propagated through an elastic fluid. The mathematical expressions which, in this case, specify the course of events can be obtained. If u denote the velocity and y the position of a molecule of the fluid at time t, the relation is where 7 is the ratio of the specific heats of the fluid and a is. the velocity of propagation of waves of small amplitude. From this it appears that, as the wave advances, the condensation in one part of the wave and the rarefaction in another part of the wave increase without limit ; the above expression will there- fore cease to represent what is going on in an actual fluid some time before the expression becomes discontinuous, either because the distance between adjacent molecules in the rarefied part of the fluid has become sensible, or because the law connecting pressure and density no longer represents the internal changes, whichever of the two should first happen. From the above discussion of material media it would appear that the process known as " treating them as continu- ous" is one of approximation, the limits of accuracy of the approximation depending on the distances between adjacent pairs of molecules ; and the comparison of the results of obser- 158 CONTINUOUS MEDIA [APP. B. vation and theory shews that, within limits, the treatment gives sufficiently accurate results. If, instead of being regarded as a method of approximation suitable for the treatment of material media, the above analysis be regarded as representing the changes which take place in a truly continuous medium the assumption is involved that the notion of mass can be trans- ferred from a material medium to a truly continuous medium ; but, as associated with a truly continuous medium, this notion involves the actual attainment of the limit p for the " density at a point " (cf. p. 146, supra) and it further involves the assumption of the existence at each point of a vector u, v, w in terms of which the kinetic energy of the continuous medium can be represented by an expression of the form ///i/> (^ 2 + tf + w 2 ) dxdydz. A continuous medium which is such that its kinetic energy can be represented in this way is not of the most general type of continuous medium conceivable. To distinguish it from other possible continuous media such a medium may be said to possess atomic structure. It has been seen that, when a material medium is in motion subject to the laws of dynamics, the condition that the medium should continue to exist as a medium to which continuous analysis can be applied requires restrictions to be placed on the possible motions. These restrictions have to be provided for by the existence of internal forces in the medium, such internal forces not being forces of motion of the medium. The question then naturally arises, what is the origin of these internal forces ? Two answers can be given to this question, that these internal forces are inherent in the medium, or that they are due to the motions of some other medium which occupies the same space. The statement, that the internal forces are inherent in the medium, is equivalent to the statement that their origin is unknown. There remains then the hypothesis that the internal forces of a material medium are due to the motions of some other medium which occupies the same space. In view of the fact that ex- perience of physical changes is confined to the comparison of motions it would seem natural to expect that the energy of the ultimate medium is all kinetic energy, but a continuous medium APP. B.] CONTINUOUS MEDIA 159 which possesses atomic structure cannot, in virtue of its own motions, produce any effect on the motions of a material medium occupying the same space, for such media, so far as their motions are concerned, are dynamically independent systems. It follows therefore that, if the internal forces of a material medium are due to the presence of another medium in the same space, this other medium is not a continuous medium possessing atomic structure, that is, its kinetic energy is not expressible in the form fff%p(u*+ v 2 + w*)dxdydz. It must therefore be a continuous medium of a different type. That a continuous medium may exist which is capable of providing the necessary internal forces in a material medium which is imbedded in it would appear to follow from geometrical con- siderations, for a continuous medium can be imagined which is such that there is no point in it which is not affected by a change taking place at any other point in it. The laws of motion of such a medium, if they were known, would be expressed geometrically, and, in order to submit the circum- stances of the motions to calculation, it may be assumed that, although these laws are not known, the motions possess a Lagrangian function, this function being a homogeneous quadratic function of the time rates of variation of all the coordinates which are necessary for the specification of the geometrical changes which can take place in any small volume. It need not be assumed that this Lagrangian function com- pletely represents the motions of the aether; it is sufficient to assume that it represents the effects of the motions of the aether so far as they affect the motions of arithmetically continuous media. This is the assumption that has been made in Chapter v. as to the Lagrangian function of the motions of the aether, and it has been shewn there that the modified Lagrangian function, which is arrived at in the Faraday-Maxwell theory of the electrical behaviour of the aether and which involves only those coordinates whose changes are associated with electrical effects in material media, is con- sistent with this assumption. This modified Lagrangian function is the analytical representation of Faraday's laws, and in it the axes of reference of the motions are those axes for which Faraday's laws are accurately true. APPENDIX C. THE ELECTRODYNAMICS OF MOVING MEDIA. IN Chapter u. the equations of electrodynamics were obtained directly from Faraday's laws, and these equations involved a certain vector F, G, H, which was there obtained in terms of the convection currents. It was remarked that this vector F, G, H differed from Maxwell's electrokinetic momentum by a vector whose components are the differential coefficients of a scalar function. Maxwell's electrokinetic mo- mentum is defined as the vector whose components F, G, H are double the coefficients of the current strength in the expression for the electrical energy. The application of the Lagrangian method will therefore give equations which will determine F, G, H uniquely and the result will put in evidence the difference between the F, G y H which are the components of the electrokinetic momentum and the F, G, H of Chapter II. The Lagrangian function of the motions of the system, which consists of the aether and moving electric charges, is + L, where F, G, H are the components of the electrokinetic momentum, X, Y, Z are the components of the electric force, f t g, h are the components of the aethereal electric displace- ment, u, v, w are the components of the convection current at the point x, y, z, and L is the part of the Lagrangian function which is independent of /, g y h, the axes of reference being those for which Faraday's laws are accurately true. In this APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 161 expression the coordinates which specify degrees of freedom are the coordinates f, g, h belonging to the aether and the coordinates x, y, z which at any instant of time define the positions of the moving charges. In order that it may be possible to apply the methods of continuous analysis, the moving charges must be supposed to be replaced by a .distri- bution of volume-density p at the point x, y, z, such that p and its differential coefficients are finite and continuous every- where, and it will appear later how, after the equations, which determine the history of the changes, have been integrated, a transition can be made from the distribution p to the isolated charges. The relations between the other quantities, which occur in the Lagrangian function, and the coordinates are (X, Y,Z) = +irV*(f,g,h) ............... (1), SH dG dF dff 3 (Sfdydz + Igdzdx + Bhdxdy) dt + + Zh dxdydzdt = 0. ot dzj The expression JJ[ I tl (FBf+ GSg + HBh) dxdydz vanishes identically on account of the conditions under which the variations are performed. The remaining triple integral and the quadruple integral must separately vanish. The con- dition, that the triple integral I 1 1 should be continuous everywhere and vanish at the infinitely distant boundary. The condition, that the quadruple integral should vanish, requires that the equations APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 163 (7), should be satisfied at every point. The equations of motion of given by the moving charges are (g + v) + H (h + w)} where the coordinates which are now varied are the coordinates which define the moving charges. In order to effect the variation, the integrals must be transformed to an integral taken throughout the same space at each instant, and the space may be chosen to be that occupied by the moving charges at the time t . Writing and effecting the transformation, this becomes v) + H(h+w)] dx.dy.dz.dt, where x , y , z at the time t Q are the coordinates of the point belonging to a moving charge which, at the time t, occupies the position x y y, z, and p is the corresponding volume-density, it being remembered that the total charge is unalterable. Now F, G, H are linear in / + u, g -f v, h + w, and therefore, performing the variation, F8x+GSy + HSz+ lj.^ + y+&^ \ (if f/v ficr (o JTI 3/T o V \ / ^ JT' Ci/^ ^TT\ "1 . C/JP . V(JT . oft \ 5, / . CAT . (?Cr . Oil \ ^ 9^ ^2/ dy ' \ dz dz dz J J ./?o dx dy dz dt, 112 164 THE ELECTRODYNAMICS OF MOVING MEDIA [APR C. where F, G, H must now be regarded as assuming the succession of values they take as the point x, y, z moves through the aether, not the succession of values they take at a fixed point. Integrating by parts with respect to the time, this becomes [FSx + GSy .dF .SO SH .d& .8.ff where the symbol is used to indicate that the time rates of ot variation of F t G, H are to be calculated on the understanding that they assume the succession of values specified above. Now in the equations (7) F, G, H are regarded as assuming a succession of values at a fixed point, and, as it is by means of these equations that F, G, H have to be determined, it is convenient to regard F t G, H as expressed in this way throughout. On this understanding SF c)F .dF .dF .dF* x ++ z > etc - and therefore m > etc>; hence [FSx + GSy + HBz] po dx,dy.dz ya) Be * Lagrange, Mcanique Analytique, 3rd edition, n. p. 263. APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 165 The triple integral vanishes on account of the conditions under which the variations are performed, and therefore + [-ST- x$ + ya } Bz \ p dxdydzdt \ot / J Again, writing and transforming to the space occupied by the charges at the time t 0) it becomes = 8 2 J 2/o, that is 9 (x, By, z) 9 (#, y, 2/o, it being remembered that f, g, h are independent of the coordinates defining the positions of the moving charges. Denoting by 5% 1} 8^ 2 , ^ 3 the parts of this expression which involve Sx, Sy, Sz respectively, f*i which, integrating by parts and transforming to the space occupied by the moving charges at the time t, becomes 166 THE ELECTRODYNAMICS OF MOVING MEDIA [APR C. Hence $X= I 1 1 [Sxdydz + Sydzdx 4- Szdxdy] pdt where the triple integral vanishes on account of the conditions which have already been found for <, and therefore Writing there results l-fffL'd*dyd*, and therefore the equations of motion for the moving charges are d/d ^...(8). Writing a* ay 1 dz ' ) (9), X', Y' t Z' are termed the components of the electric force acting on an element of the moving charge, and the equations (8) become d f dL'\ dL' -- A ddL d /dL'\ __ dL^ dt\dl)~"W (80. APR C.J THE ELECTRODYNAMICS OF MOVING MEDIA 167 Before proceeding with the discussion of these equations, it is convenient to obtain, by means of the equations (1), (2), (3), (5) and (7), expressions for F, G, H and the related quantities in terms of the distribution of the moving charges. It follows from equations (2) and (3) that . * 3 dF dG dH T dF c)G J = ^ -- r ^ -- H -7T- , dx dy dz with two similar equations, that is, writing and Again, from (1) and (7), that is, remembering that < is independent of the time, &F and therefore = dx ~ ' The solutions* of these equations are %, 9**+% H-F + *. * See 15 above. 168 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. where dx.dy.dz,, G' = dx.dy.dz,, r being the distance of the point x, y, z from the point x.,y.,z,, and HI, v., w l denoting the values of u, v, w at the point x., y lt z. at the time t r\V\ further ^ satisfies the equation l 8 ^ JI_J>v.J>K-K V* dt* "*" dx ^ dy dz " Now, writing ap; w dir = dx + d H " 'bz ' 3 3 3 ~~\ dx r dy r dz rj that is hence .ai a i , for - = ^ -- and ti. v. w are continuous. 9* r 9i r The unalterability of the total charge gives the relation 9/Pi ,9^1 , 9^ , ^i , is given by The components of the electric force are therefore given by and the components of the magnetic force by 4? (11).* Proceeding now to the discussion of special cases of moving distributions, the first case to be considered is that in which an electrical distribution is moving with a uniform velocity v in a given direction. Choosing this direction as the axis of x, the * Cf. Levi-Civita, II Nuovo Cimento, S. 4, t. vi. 1897. 170 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. coordinates of any point of the moving distribution at the time t will be a^, y lt z ly where x Q , 2/ , ZQ denoting the coordinates of the same point at the time from which t is measured. In this case p" is zero and the expressions to be evaluated are detMd*;, Iff ?**'&& where r'* = (x- Xl J + (y - y/) 2 + 0* ~ z,J, and u l , v- i) w l ,p l denote the values of u, v, w, p at the point #/> yi> z \ at the time t r'/V, this being the point of the dis- tribution which was at the point # , y 0) z initially. Then that is vr' / / __ / Now v = 0, w = 0, therefore [f[v l , f[[ w i JJJ T' ' JJj r' and the remaining two integrals have now to be transformed so that they are expressed in terms of the positions of the distribution at the time t. Writing K v/F, r' is given by r"> = (x - x l + icr') z + (y y^ + (z ^) 2 , whence where APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 171 The element of volume dx^dy^dz^ is that which was occupied at times t r'jV by the distribution which now occupies the element dx l dy l dz l , these times being different for different points of the distribution ; hence Uidvi'dyidfi = pv -^~ ' c ' ~ dx 1 dy l dz 1 . v \&i > 2/i > z \) T 3a?/ 3r' Now 5 = 1 K 5 , , 3#! hence ~ a that is therefore Further, ^T! *, ,;i , an ?> 1} . ' z z 8^1 3^! therefore ^ * ? ^ 3 Hence r' ^- = 1 Writing and substituting in equations (10) and (11), the components of the electric and magnetic forces are given by x = -^-v&, 7 v, z--r*%, dt dx dy oz 172 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. that is, remembering that 91 91 91 .-(12), where T/T satisfies the differential equation The case of a charged conductor moving through the aether with a uniform velocity can be deduced from the above. The electrical distribution must now be supposed to be the limit of a distribution throughout a thin layer over the surface of the conductor, this distribution being such that p and its first differential coefficients tend to zero at the boundary of this layer. The components of the electric and magnetic forces at any point external to the conductor will then be given by equations (12) and (13), and will vanish at an internal point, where ^ now satisfies the differential equation throughout the external space and is constant over the bounding surface of the conductor. A simple example is that of a charged conducting sphere of radius a moving with uniform velocity v through the aether. In this case ty is given by ET i/r = log- coth ^ , /ca 2 where a? sech 2 77 + (1 - /c 2 ) (f 4- 2 2 ) cosech 2 77 = /c 2 a 2 , and E is the charge on the sphere. The surface density of the distribution over the surface of the sphere is uniform. * Lorentz, Vermeil einer theorie der electrischen und optischen erscheinungen in bewegten kvrpern, 22, Leiden 1895. Larmor, Phil. Trans. A, 1897 ; Aether and Matter, ch. ix., 1900. APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 173 The next case to be considered is that in which the electrical distribution moves in such a way that any point of it describes an orbit which is periodic with reference to some fixed point and whose dimensions are very small. The coordinates of any point of the distribution at the time t will be #! + f, ift -f 77, z 1 + f, where x lt y^ ^ are the coordinates of the point about which it describes a small periodic orbit, and the expressions to be evaluated will be where ry> = (a- - ^ - f ,) 2 + (y ~ yi - %) 2 + (z - *i - ) 2 , and (-1,^1, ?!, MI, v lt w l , p t are the values of f, ?;, f, w, v, w, p at the time t-rJV. Now, if f =/(*), then fi=/(* - r a /F), that is where r 2 = (a; - x, - %f + (y - y, - ^ + (z - ^ - ?) 2 ; and writing r=/(-f),eto., / -(-,- rr + (y - y, - V) 2 + (* - , - r' r / = FT, it follows that it being assumed that (f/X) 2 , etc. are negligible where f, etc. denote the maximum amplitudes corresponding to a period for which the wave length in the aether is X. To the same order of magnitude a(fe,iyi, )_-, ^ii^.ii^i..?! ?r? : " ' 174 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. that is , i i a(fc 17,0 a* ay and therefore, to this order of magnitude, where, on the right-hand side, ^17,, ?i. fi, ^, ?i, n, ^ now denote the values of f, 77, f, j, 97, , r, p corresponding to the r' time t -y . The expressions for the components of the electric arid magnetic forces are then obtained by substituting the above expressions in the equations (10) and (11). The electrical distribution in the above investigation has been assumed to be such that the equations (10) and (11) are applicable. Now the validity of the processes involved in the investigation, by means of which these equations were arrived at, is secured, provided the distribution is such that p and its first differential coefficients are everywhere continuous. The transition from a volume distribution to a distribution of isolated electric charges can then be effected in the following way. Let P be a point at which there is an electric charge e, and let a closed surface be drawn enclosing P and no other point at which there is an electric charge, which is always possible as the charges are isolated. Then throughout the volume enclosed by this surface there may be supposed to be a distribution of volume-density p, which is such that p and its first differential coefficients are continuous, tend to zero as the bounding surface is approached, and vanish at every point on it, and which is such that jjjpdxdydz = e. The distribution APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 175 arrived at, when this has been done for each one of the charges, is such that the preceding investigations are applicable to it, and the results obtained will hold however small the volumes enclosed by these surfaces may be chosen to be, provided that throughout them p can be chosen so as to satisfy the conditions specified above. The electric and magnetic forces due to a distribution of isolated electric charges may therefore be taken to be those given by equations (10) and (11), when the integrals on the right-hand side are replaced by the sums of the limits of those integrals taken throughout the small volumes, these volumes being supposed to be ultimately evanescent, the limits being proceeded to on the supposition, that p and its first differential coefficients are continuous throughout any one of these small volumes, vanish at their boundaries, and that The electric and magnetic forces due to a number of electric charges, each of which is describing, with respect to some fixed point which is not necessarily the same for different charges, a periodic orbit of small linear dimensions, can be obtained from the preceding results by evaluating the limit of the integrals etc. Now it appears from 48 that the amount of the energy radiated from the distribution depends only on those parts of electric and magnetic forces which, at a great distance, are of the order 1/r; hence, no part which can contribute to the radiation being omitted, the previous integrals are given by ///* *** -///[ - f - y\ + *?> z \ ~*~ ? tne coordinates at the time t of a point of the distribution which is describing, with respect to the point x \*y\> z \> a sma U periodic orbit, and by ^/4-fj, 2/ 1 + 7 7, z \ + %\ the coordinates of the same point at the time t rJV, where r,' = ( - < - f ,)* + (y - y, - T?,) 2 + (* - *, - ), x, y, z being the coordinates of any other point, the integrals on the right-hand side of the equations (10) and (11) which, in this case, have to be evaluated are in which /?/, f /, ^/, ?/ denote the values of p, %, TJ, at the point ^i' + f,, y 1 + 7 1 , ^ 1 + f 1 at the time t rJV, d^drj^'d^ denotes the element of volume at that point and &/ = v. Transforming these integrals so that the element of volume dfidqfdfi at the point #/ + fi, 2/1 + ^1, ^"i + fi is replaced by the element of volume d^d^d^ at the point #i + i, 3/1 + %, Zi + fi *, the Jacobian of the transformation is by a previous investigation given by r^jR^ V(l 2 )> in which /c = v/F and ^ 2 = ( - x, - f x )V(i - **) + (y - yi * The transformation is equivalent to substituting for the fixed axes of reference a set of axes fixed relatively to the moving points 0. M. E. W. 12 178 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. and the above integrals become Now hence, f(t) denoting some function of t, f (t r -W{< -' that is, neglecting /c 3 arid higher powers of K, ' T which may be written V. Then f lt q ly denoting the values of f, 97, J at the time t RJV, the integral and, since p and its first differential coefficients vanish at the boundaries of the small volumes occupied by the distribution which replaces the charges, the second and third integrals on the right-hand side vanish, and therefore APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 179 If, as in the previous case, it be assumed that, I denoting the linear dimensions of an orbit, I* is negligible in comparison with the square of any length considered, the integrals involved become where now # = ( X - x,JI(\ - and J!, T) I , lt fj, %, ?! denote the values of f, 37, f, f, 77, f, when t-RjV is written for t. Then writing the component of the electric force in the direction of the axis of x is given by which is equivalent to ' in which ^- is supposed to operate on f x , rj lt & only but not now on #!. Similarly evaluating the other components, the com- ponents of the electric and magnetic forces are given by ...(17), 122 180 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. a = tydt +vl(4>- uzdt dxdt cz i/ y 2 v y i ^ / j 7 = -d^-^- v ^r.(<-- (18). Returning to equations (S 7 ), integrating them throughout the small volume occupied by the distribution which is supposed to replace a moving charge, and proceeding to the limit, the equations of motion of the moving charge are _ = dt (dx) dx ___ dt(dy ~ i(*j)-Sr-^. - in which the first % refers to the group which is being con- sidered, the second 2 to any other group, and r denotes the distance AB, where A is one of the fixed points about which a charge belonging to the first group is describing its orbit and B is such a point belonging to the second group. When there is no free electricity, the condition 2e = will be satisfied for each group, and the forces P n Q lt R l are of the order V 2 e 2 d z /r*, where r is the distance between two groups and d is a length depending on the distances between the orbits of the same group. When there is free electricity the forces on a group for which 2e is not zero are those of electrostatic theory. In every APP. C.] THE ELECTRODYNAMICS OF MOVING MEDIA 183 case the forces P 2 , ft, R 2 , are of the order F 2 e 2 2 d 2 /\ 2 r 4 , where I is a length depending on the dimensions of an orbit and X is the wave length in the aether corresponding to a period in the orbit; thus the forces P 2 , ft, R^ are small compared with the forces P 1? ft, R,*. It appears from the preceding that, when the groups and the distances between the orbits in the groups are arranged so that the forces P lt ft, R^ form with the non-electrical forces an equilibrating system, the orbital motions will arrange them- selves so that the forces P 2) ft, R 2 form an equilibrating system, as 2P 2 , 2ft, 2-# 2 vanish when the conditions of permanence of the groups are satisfied. Hence, in investi- gating the law of distribution of the groups, it is sufficient to consider the forces P lt ft, jRj and the non-electrical forces. If now it be assumed that the non-electrical forces j~ are such that they are not sensible at insensible distances, which is equivalent to assuming that the inter-atomic forces are wholly electrical, the forces P lt ft, R l will form an equilibrating system when there is no free electricity, and if, when there is free electricity, P/, Q/, Rf t denote the parts of P lt ft, R^ which remain after the removal of the forces belonging to the free electricity, the forces P/, ft', RI will form an equilibrating system. Therefore, when this assumption is made, a knowledge of the forces P lt ft, RI will suffice for the determination of the law of distribution of the groups. Considering now the case where the points, about which the charges are describing their orbits, are all moving with a uniform velocity v in the direction of the axis of x, the equations (20) become, since v is constant, K! + X = 0, Z, + Z = 0, the same equations as in the previous case. The same argu- ments will apply as in that case, and the law of distribution * Cf. 53, 54. t The force of gravitation satisfies the condition. 184 THE ELECTRODYNAMICS OF MOVING MEDIA [APP. C. of the groups can be determined from a knowledge of the forces P 1} Q lt R lt By a preceding investigation* the values of PJ, Qi, RI for this case are given by where } and a;, y, 2 being the coordinates of the point about which a charge in the first group is describing its orbit and x lt y lt z^ a similar point for the second group. Writing the equations (24) become P = */Q - > 8 s \~ * D' where #2 = (a/ _ ^'ja + (y _ y x )2 + (^ - ^) 2 , and putting P a = P/ Vl 2 , it being remembered that ' //1 /c 2 ), these equations become *--**> ^ F 2 l ~ e .(25). * p. 177. APR C.] THE ELECTRODYNAMICS OF MOVING MEDIA 185 Now the equations (25) give the values of P/, Q l} j?^ which belong to a distribution, in which the points about which the orbits are being described are at rest ; therefore, if this system is in equilibrium, the groups which form the system to which the equations (24) belong will also be in a state of relative -equilibrium. Hence the relation between the configuration of the groups, when moving with a uniform velocity v, and, when they are at rest is given by the equation x = x' \/(l /c 2 ), where /c = v/F; that is, the material medium, which consists of these groups, is contracted in the direction of its motion in the ratio V(l - v*/F a ) : 1*. In the investigation it has been assumed that there is no electric charge which is migrating freely among the groups, so that the result will not necessarily apply to the case of a material medium in which there are electric currents. It may be shewn by similar reasoning that the result holds for a material medium in which there are no electric currents, when waves due to external causes are being propagated through it, provided none of the periods of these waves are the same as those belonging to the orbits of the charges which form the groups. The medium would then, if electrically isotropic when at rest, not be so when in motion, the specific inductive capacity of the medium in the direction of the motion being altered by an amount depending on 2 and higher powers of K; and to obtain the effect of the motion of a material medium on the velocity of propagation of waves through it, when # 2 is not negligible, this alteration would have to be taken into account. * This result has been obtained for the case of charges moving with a uniform velocity by Lorentz and Larmor. APPENDIX D. DIFFRACTION. WHEN electric waves are incident on any body, other waves due to the presence of the body are set up, and to determine completely the superposed system of waves it would be necessary to know the distribution of electric charges which constitute the body; the motion would then be expressed in terms of coordinates specifying the degrees of freedom of these electric charges and the electrical degrees of freedom of the aether. In the treatment of any class of cases it is convenient to suppose that the effects of the motions of the electric charges can be expressed in terms of a smaller number of coordinates, or be represented by kinematical conditions. In the case where the body is transparent these effects are suitably represented by two coordinates for each point of the body, these coordinates being specified by a vector/i, g ly h lt which satisfies the relation &.&.a*._ a"T" r\ I O ~~ V x oy 02 at every point of the body ; this vector is termed the material electric displacement. In the case of bodies which are not transparent this representation is not suitable, but two limiting cases can be recognized, which will furnish a clue to the effect on the waves of any such body. The first extreme supposition is that the electric charges at the surface of the body move in such a way that waves of the same kind and of the same amount are radiated out from them as are incident on the surface of the body, so that no part of the energy of the waves APP. D.] DIFFRACTION 187 in the medium is absorbed by the body ; this is the case in which the body is a perfect conductor. The other extreme supposition is that all the energy of the waves incident on the body is absorbed by it. The problem of the diffraction of waves by a transparent body has been solved for the case of a circular cylinder* and for that of a sphere f, the velocity of radiation in the body differing by a finite amount from that in the surrounding medium. The problem has been solved for the general case, when the difference between the velocities of radiation is very small J. The problem of the diffraction of waves by a perfectly conducting body has been solved for the case of a circular cylinder, a sphere ||, and an indefinitely thin wedge in the form of a semi-infinite planell". When the body is perfectly absorbing the problem can always be solved. One of the most important cases is that in which the diffracting body consists of a plane screen with an aperture in it. This case has been treated in detail by Stokes** and by Lorenz ff, who have assumed that the secondary or diffracted waves depend only on the disturbance over the aperture ; this assumption is equivalent to assuming that the screen is a perfect absorber. If the problem could be solved for a perfectly conducting screen, the comparison of the two results would shew in what respect either representation is ineffective and possibly give information as to where improvement might be made. The problem has so far only been solved for the case of the semi-infinite plane, and its importance makes it desirable to have a solution as direct and simple as possible. The application of the methods used in the ninth chapter of this essay leads directly and easily to * Lord Rayleigh, Phil. Mag. xn. 1881. t Lorenz, Vidensk. Selsk. Skr. Copenhagen, 1890. J Lord Rayleigh, I.e. J. J. Thomson, Recent Researches in Electricity and Magnetism, p. 428. 1893. || J. J. Thomson, I.e. p. 437. IT Poincare, Acta Mathematica, Vol. xvi. 1892-3, Vol. xx. 1897. Sommerfeld, Math. Ann. Vol. XLVH. 1896. ** Camb. Phil. Trans. Vol. ix. 1849. ft Pogg. Ann. cxi. 1860; Crelle, LVIII. 1861. 188 DIFFRACTION [APR D. the solution of a more general case, viz. that in which the diffracting body is a wedge of any angle. Let the edge of the wedge be chosen as the axis of z, and let r, z, be cylindrical coordinates of a point so that the faces of the wedge are given by 6 = 0, 6 = a, and the space occupied by it is that between 6 = a. and 6 2?r. Taking first the case in which the electric force is parallel to the edge of the wedge, the differential equation satisfied by Z at all points at which there are no sources is that is, for waves of wave length 2?r//c, idz i&z a~i - o ~^ 53 ............... (1). dr 2 r dr r 2 90 2 For waves of the kind considered the sources can be represented by lines of discontinuity of electric force parallel to the axis of z and at points on such a line Z will satisfy the differential equation The condition to be satisfied at the surface of the wedge is that Z vanishes when 6 = and when 6 = a. The solution of (2) is therefore D . n-rrO R n sm --- i where n has all positive integral values and R n is a function of r to be determined from the relation - (&R n , 1 dR n ( . nV\ p ) . mrO 2 i 3~r- H --- 5 -- H * 2 -- - }R n } sin --- h 2-TT/o = ; ! ( 9? <2 r dr \ aV 2 / j a whence R n satisfies the equation It is sufficient to consider the case in which there is a single APP. D.] DIFFRACTION 189 line of sources determined by r = r lt = ft; the solution of (3) is given by when r<7\, for R n must be finite when r = 0, and by nrftj. R n = B n [J_ * (KT) - e - J (tcr)}, a a when r>r lt for R n cannot involve &* r , there being no reflexion at the infinitely distant boundary. Hence Z = 2 A n J (fcr) sin , ! a when rr 1 ' J now both these series will converge and be identical, when r = r l} except when = ft, in which case they will diverge ; therefore nn^i A n Jn* (icr^) = B n {/_^ (KT^ - e a J (/en)}, a a a and the solution can be written 00 nn z t ' s\ Z^^CnJn* ( K r) [J_nn (**,) - 6^ J (fCT,)} sin - - , 1 a a a Ct when rr l . It appears at once, as in 60, by considering the case where a = TT which corresponds to reflexion at a plane surface, that C n is independent of /e, and it is therefore sufficient to determine C n for the case in which K = 0. In this case the problem to be solved is the electrostatic one of a line charge influencing a conducting wedge, and the potential due to a line charge of strength ra is known to be ?5 ?5 7T r a + r a _ 2 (rr Y cos _ (0 + ) m, a v 2 g ~ ^ ~ ' r a + r x a - 2 (rr!)* 1 cos - (0 - 00 190 DIFFRACTION [APP. D. Therefore, when K = 0, V n l i r \ a U7r Q n7T0 l Z = 2m 2, - sin sin , i n \rj a. a. when r r lt whence a that is nn nn m sin 1 2m/r\ a mrO, ""1TW -^. and therefore 7T . a . W7T 2 sm - a . sm M Hence, when there is a line of electric discontinuity of strength m at r = r 1} # = #i, the electric force is given by ^ = ^ ^n7r (*) {t/"_ nTrC/crj) e a J nrr (icr-i)} 1 a a a sin . mrO . .sm -- sm , when r TV Now WIT* i Jnn (XT) [J_ n,, to) - 6 a /^ (/KTj)} 2 ; T sin a J c-i * Proc. Lond. Math. Soc. Vol. xxxn. p. 155, 1900. APP. D.] DIFFRACTION 191 where c is a real positive quantit}^, therefore a i for all values of r. Again where the real parts of oo e^/t, oo et'^t are negative, whence, writing s = e~ 1 *, where f= f + vn, where 2?r > 7 > TT, > 7' > - -r, and therefore ^ = - r. p^ ^ 1 J C OOiJ QOt+ +y . 717T0 . . sm - sm that is = _mt T i^+y' e * r 4*OtJ C OOlJ 004 + kr cos v OL sm a . 7T? sin a cos - where the path of integration lies wholly in the upper half of the f plane. Changing the order of integration and using the known result /O t z 2 Jf ,<*) J / i-S^, ^ C QOi * Proc. jLowd. Ifa^. Soc. Vol. xxx. p. 167, 1899. 192 DIFFRACTION [APP. D. the above becomes sin r% cos (0 #j) sin E? } \ ,7," a OL When a is an integral submultiple of TT, a = TT/U, this is equivalent to Z= m "Z K, \uc A /lr 2 + r? - 2rr, cos ( 6 - 0, - ~ =o L V I V w w l / f / o _ - m S ^ U^ A / 1^ 2 + n 2 - 2rr x cos + 6 l + - =o V ( \ n which, remembering that the effect of a line of electric dis- continuity of strength m in an unbounded space is mK (ucR), where R is the distance of any point from this line, is the result which would be arrived at by the method of images. When a is an integral multiple of TT, the above expression be- comes identical with that given by Sommerfeld* for this case. The solution in the case of plane waves can be obtained from this by making r a indefinitely great. The value to which KQ [iK \/(r 2 + TV 2 2rr a cos f)J tends is 6 * hence, corresponding to plane waves incident in the direction 6^ and in which the electric force is given by e lKrcos(e - e >\ the solution for the space bounded by the planes = 0, = a is where 2?r > 7 > TT, > 7' > TT. * I.e. p. 187. APP. D.] DIFFRACTION 193 When a = 2?r, the wedge becomes a semi-infinite plane and the above expression for Z becomes z= i_ p^,.^ 4t7r] oH + y sin ^ sin 4 t cos | cos ^ (0 0i) cos ^ cos The integrals can now be transformed as follows : writing sin | . root+y' ~0 7 f and putting cos | = t cos J (0 0i), X,., ,v Cfa where, if cos J (0 00 is positive, = C 00 t, j = C + 00 i, the path of integration cutting the real axis in the t plane to the right of the point = 1, and, if cosj(0 0j) is negative, the path of integration cutting the real axis in the t plane to the left of the origin. Hence, deforming the path so that it becomes the imaginary axis in the t plane, J 00 1 when cos ^ (0 0j) is positive, and / / "" dt 4ucrcos(0-0.) ' ,*iKrcos(0-0i) ,,2tcrcosH(0-0 1 ) 2 -l) ~^r e L t-i' r ooi -0i) I J _oo t _ 2?T J _oo t 1 ' when cos J (0 0j) is negative. Denoting the integral in the above relations by I lt and writing /u = V(2r) | cos ^ (6 0^) | , then M. E. W. 13 194 DIFFRACTION [APR D. that is, which becomes 7 1= - tftrcoBV-Oj f 00 ' fli^'fff-n ^ 7T Jo 2 1 ' i /* /i = - - e" > (-i) e - 7T Jo or Now therefore /j = - - #<-,) f 7T J o which, changing the order of integration and integrating with respect to t> is equivalent to 1 r oo I- __ * ^rcm(9-e l ) e - 2V-7T 6 J 0* and, putting u + t^ 2 = Av 2 , this becomes 87TI * **. A v TTt -T * e-^ct; that is T 1 ./. = 0 ~<-)+ T f " e-*dv, VTT J ^ when cos J (^ ^i) is negative, that is, in this case, * V 71 " /= ucrcos(*-*,)+^ f'^e-^dv. / -oo APP. D.] DIFFRACTION 195 Therefore 1 /* IK- f-. _i_ e i*r cos (e -0,)+ ^ L I e'^dv V 77 " J -oo where //, = ^(2fcr) cos ^ (0 - 0j), and the electric force Z is given by 1 iri /" ** 1 iri f /*' V* 73 " J -oo V* 71 " J -oo where /A' = ^(Z/cr) cos When the electric force is in the plane of incidence of the waves, the magnetic induction is parallel to the edge of the wedge, and denoting the magnetic induction by c, c will satisfy the same equation as Z above, the boundary conditions now 3c being .^ = 0, when = and when 6 = a. Thus otf " r C = 2, R n COS o which, as before, becomes 00 WTT 2 l /I c = S w J n7r (AT) { J n ^ (/en) - e a J nir (/en)) cos - - , o T ~T ^T a when r < r lt and c=^C n Jnn (tcrj [Jjn* (KT) - e tt J" nj r (r)} cos -- , a a a when r > rj, and again (7 n is independent of K. When K = 0, the problem is the hydrodynamical one of a line source in the space bounded by = 0, and = a. The solution is then the same as in the previous case, with the exception that the part involving + d l is affected with the positive instead of the negative sign, and therefore mi ron+y' ~ K. {. V(r 2 + r, - 2m cos ?)} ^2 J QOt+ sin - sin : a a cos - cos - (0 - 0^ cos - cos - (0 a a^ a a v 196 DIFFRACTION [APP. D. For plane waves incident on a semi-infinite plane this becomes c = ^^roM(9-9 l )+^l M e - ( fo+JLerco8