OPTICAL THEORIES CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER EontJOn: FETTER LANE, E.G. Btnburgij ' 100 PRINCES STREET $tfo lorfe: G. P. PUTNAM'S SONS Bombag, Calcutta an* JBabras: MACMILLAN AND CO., LTD. Toronto: J. M. DENT AND SONS, LTD. Eoftgo: THE MARUZEN-KABUSHIKI-KAISHA All rig/its reserved OPTICAL THEORIES BASED ON LECTURES DELIVERED BEFORE THE CALCUTTA UNIVERSITY BY D. N. MALLIK, B.A. (Cantab.), Sc.D. (Dub.), F.R.S.E. x\ Late Scholar, Peterhouse, Cambridge Professor, Presidency College, Calcutta Fellow of the Calcutta University Cambridge : at the University Press 1917 PREFACE TN the year 1912, the University of Calcutta appointed me Reader in Physics and invited me to deliver a course of lectures to its advanced students on Optical Theories, one of the conditions of the appointment being that the lectures should be published after their delivery. The lectures were actually delivered during the months of February and March, 1912, but pressure of other work has prevented me, till now, from seeing them through the press. It has been my object in these lectures to trace the develop- ment of Optical Theories from the earliest times to the present day. I have tried to understand and help others (so far as I can) to understand the relation between the different theories, so that one may be clear as to how much is certainly known and how much is mere speculation. In the midst of the be- wildering mass of investigations that a student of luminiferous medium is confronted with at the present day, a sketch, such as the one attempted here, describing, with such details as will make the general argument intelligible, how we have been led up to the present position and what that position really is should, as it seems to me, be of considerable use. How far I have succeeded in my attempt, it is for others to judge. To the latest developments of the optical theory including the theory of relativity, no reference has been made here. I hope to deal with them in a later volume, if the present attempt proves successful. D. N. M. November, 1916. 370761 CONTENTS CHAP. PAGE I. EARLY SPECULATIONS, CORPUSCULAR THEORY, UNDULATORY THEORY ... . . 1 II. ELASTIC SOLID THEORY 33 III. ELECTRO-MAGNETIC THEORY .... 61 IV. ELECTRON THEORY 124 V. SUMMARY. CONCLUSIONS .... 162 APPENDIX I 172 II 173 III 174 IV 178 INDEX 180 CHAPTER I INTRODUCTION. CORPUSCULAR THEORY. UNDULATORY THEORY SECTION I. ANCIENT SPECULATIONS. ERRATUM Page 12, line 3, for must have read may take after all extremely limited. 2. This, indeed, is what is to be expected illustrating, we may remark in passing, the limit and scope of scientific inquiry in general. For, although the phenomena with which, to confine ourselves to optical investigations alone, we have to deal are simple and well-known, and although we can formulate the laws governing them like any other body of scientific laws which state (as Karl Pearson has put it), " in conceptual short- hand, that routine of our perceptions, which forms for us the totality of the phenomena," to which they refer, when we come to inquire into the intimate nature of the processes, associated with these laws, we are confronted with insuperable difficulties. These are so subtle and deep-seated, that they will probably always elude our grasp. We are, therefore, reduced to preparing models, that shall approximate to the actual, as far as possible, and our task consists really in improving these models, more and more, so that they may more and more nearly approximate to the actual. Thus, of the mode of propagation of light and the nature of the medium which takes part in its propagation, we, with our limitations, can never have any direct knowledge. M. o. T. 1 CHAPTEE I INTRODUCTION. CORPUSCULAR THEORY. UNDULATORY THEORY SECTION I. ANCIENT SPECULATIONS. DESCARTES. FERMAT 1. A complete theory of optics has to furnish an adequate account, not merely of the nature of light but also of the mode and mechanism of its propagation, as well as the nature of the medium in which the propagation takes place. And we shall see, as we pass in review, in historical order, the various theories that have been proposed, that our knowledge on these points is after all extremely limited. 2. This, indeed, is what is to be expected illustrating, we may remark in passing, the limit and scope of scientific inquiry in general. For, although the phenomena with which, to confine ourselves to optical investigations alone, we have to deal are simple and well-known, and although we can formulate the laws governing them like any other body of scientific laws which state (as Karl Pearson has put it), " in conceptual short- hand, that routine of our perceptions, which forms for us the totality of the phenomena," to which they refer, when we come to inquire into the intimate nature of the processes, associated with these laws, we are confronted with insuperable difficulties. These are so subtle and deep-seated, that they will probably always elude our grasp. We are, therefore, reduced to preparing models, that shall approximate to the actual, as far as possible, and our task consists really in improving these models, more and more, so that they may more and more nearly approximate to the actual. Thus, of the mode of propagation of light and the nature of the medium which takes part in its propagation, we, with our limitations, can never have any direct knowledge. M. o. T. l 2 OPTICAL THEORIES [CH. \Ve are, therefore; driven to reason by analogy and construct a picture which shall be as near a representation of the actual as we can make it. At first, only the outlines are noted ; then, the details are gradually filled in, but the picture can never fully represent the actual, unless and until our powers of perception are so sharpened that we are able to take cognizance of the order of quantities which we now call infinitesimal. 3. Although speculations as to the nature of light are of remote antiquity, anything like a clear notion as to the nature of light belongs only to comparatively modern times. 4. Among the most ancient recorded speculations on the subject of light seem to have been those of the Hindus. In the Nyaya Vasya*, there is a discussion, from the point of view of the Nyaya Philosophy, on the theory of the mirage : " In the summer, the rays of the sun coming in contact with the heat issuing from the earth vibrate upwards and downwards and gradually reaching the distant observer's eye produce a false impression of water, by an incorrect association of sight and object." It is clear that the writer of Nyaya Vasya was interested only in the metaphysics of the subject. For, with regard to this discussion, the commentary Bartika points out that in this case, although the solar rays are certainly there, as well as the vibrations, what is at fault is the impression of water the mistaken association of the phenomenon with water, etc. AH the same, it is evident from this, that the above was the generally accepted theory of the mirage at the time the Vasya was written and must have been treated of in formal books on Physics, which seem to have been lost. * The Indian schools of Philosophy (called Darsanas) are classed under two main heads. One class, called the 'Astika' (the believing), believes in the authority of the Vedas, and the other called the 'Nastika' (non-believing) does not recognize this authority. Each of these is divided into six schools : Nyaya, Vaisesika, Sankhya, Patanjal, Purva Mimansa and Uttara Mimansa (or Vedanta) belong to the first or the Astika group, while Charvak, the Jain and four Buddhistic schools belong to the latter. Nyaya Sutra, of which the reputed author was Gautoma, the founder of the Nyaya school, deals with canons of correct reasoning. Nyaya Vasya is a commentary (by Vatsayana) on Nyaya Sutra and Bartika (by Udyotaka) is a commentary on the latter. I] ANCIENT SPECULATIONS 3 5. In the same work (Nyaya Vasya), the following explana- tion occurs of the formation of images by reflection : " The ' eye rays/ striking against mirrors, return and come in contact with the face [to which the eye belongs, i.e., of the observer]. From this contact is derived the knowledge of the face. The ' rupa,' i.e., form or colour of the mirror, helps in producing this knowledge." The commentary Bartika thus amplifies the theory : " The ' eye rays ' rebound at mirrors, water, etc. On rebound- ing, they come in contact with the face. As the extremities (fore-part) of rays come into relation with the face, the face appears to be in front. This is the law relating to knowledge acquired by means of the eyes, viz., the object which comes into relation with the extremities (fore-part) of the ' eye rays ' is made out by this knowledge to be in front, e.g., the knowledge of the face of a man standing in front [of the observer]." 6. Before the days of Nyaya Sutra, a theory prevailed that every object emits rays. The author of Nyaya Sutra points out that in that case, stones, etc. should be capable of being seen at night, and the author of Nyaya Vasya further argues that one could not imagine such a thing (viz., rays emitted by stones), whereas ' eye rays ' are imaginable ! 7. On the theory of transparency, we are told, in Nyaya Sutra, that " the ' eye rays ' are not turned back by (i.e., are allowed to pass through) glass, etc. This is how objects placed beyond glass, etc. can come into relation with 'eye rays' and are seen. Opaque bodies like walls turn off ' eye rays ' and therefore bodies cannot be seen through them." 8. As to the eye rays, it is stated in Nyaya Kandali, a treatise of the Vaisesika school, that " their form cannot be seen nor can they be touched, but they go to a distance and produce the knowledge of bodies, if nothing stands in the way." More- over, the eye rays, like solar rays, are to be regarded (according to a commentary on the Vedanta Parivasa of the Vedanta school) as " transparent bodies [Art. 10] and may therefore have rapid motion" 1-2 4 OPTICAL THEORIES [CH. 9. With regard to these extracts*, it should be noted that they only incidentally occur as illustrations of principles discussed in Hindu Philosophy. They have, therefore, no further interest from our present point of view, beyond the fact that they show that optical speculations in India dated beyond the days of the Nyaya Sutra. It is interesting, moreover, to note that similar speculations appear in the first systematic European work on light, that of Empedocles (444 B.C.). According to him, light consists of particles, projected from luminous bodies, and a vision is the effect of these bodies and a visual influence, emitted by the eye itself, although Pythagoras and his followers had previously maintained that vision was caused by particles continually projected from the surfaces of objects into the pupil of the eye. If, therefore, as is maintained by some, Nyaya Sutra was written between 500 B.C. and 200 B.C., it would follow that contemporary optical ideas in Greece and India proceeded on similar lines. 10. The fallacy of the theory of a ' visual influence ' was discussed by Aristotle (350 B.C.), who argued that if a visual influence was emitted by the eye, we should be able to see in the dark. He considered it more probable that light consisted in an impulse, propagated through a continuous medium, rather than an emanation of distinct particles. Light, according to him, is the action of a -transparent substance [Art. 8] and if there were absolutely no medium between the eye and any visible object, it would be absolutely impossible that we should see it. The meaning of the latter part of the argument seems to be that if, between the luminous object and the eye receiving the impression, there did not exist something endued with the physical property that makes it capable of transmitting the influence (whatever its nature may be) emitted by the luminous object, that influence could never reach the eye. This is, also, in effect the postulate of modern science. 11. From this time up to that of Descartes, optical dis- coveries related mainly to the two fundamental phenomena * I am indebted to Mahamahopadhyaya Gurucharan Tarkadarsana-tirtha,. Professor of Nyaya, Sanskrit College, Calcutta, for these extracts. I] DESCARTES 5 of reflection and refraction. Archimedes was evidently ac- quainted with the property of burning mirrors and seems to have made some experimental investigations on this subject, while Vitellio, a Pole, developed a mathematical theory of optics*. Roger Bacon is said to have invented the magic lantern, and is regarded by some as the first to have invented the telescope also. But the first person who is certainly known to have made a telescope was Janson, a Dutchman, whose son, by accidentally placing a concave and a convex spectacle glass at a short distance from each other, observed the increased apparent magnitude of an object seen through them. It was to Galileo, however, that we owe the first construction and use of such a telescope (the Galilean) for astronomical observations. And it was to him also that we owe its theory. Galileo states f, in his Nancius Sidereus, that happening to hear that a Belgian had invented a perspective instrument by means of which distant objects appeared nearer and larger, he discovered its construction by considering the effects of refraction. Finally, Kepler worked out the true theory of the Astronomical tele- scope (a combination of convex lenses), made some experiments on the nature of coloured bodies and experimentally verified the formation of inverted images on the retina of the eye. 12. Descartes published the law of refraction, originally discovered by Snell, and deduced the law from theory : or rather an analogy. " When rays meet ponderable bodies, they are liable to be deflected or stopped in the same way as the motion of a ball or stone impinging on another body." Let a ballj thrown from A meet at B a cloth CBE, so weak that the ball is able to break through it and pass beyond, but with its resultant velocity reduced in some definite proportion, say 1 : k. Then, if El length, measured on the refracted ray = AB, the time to describe El = k x the time to describe AB. But the component velocity parallel to the cloth is un- affected. * Lectures by Th. Young. t Ency. Brit. J Whittaker, A History of Theories of the Ether and Electricity. OPTICAL THEORIES [CH, BE projection of BI, on the cloth, = k . BC, where BC is the projection of AB. BE 7 CB 7 . . sin r = -Yfjr = k . -ff-:- =k . sin i, fl >A or the sines of the angles of incidence and refraction are in a constant ratio. 13. He also propounded a theory of light. On this theory, light consists in pressure transmitted instantaneously through a medium, infinitely elastic, while colour, according to him, is due to a rotatory motion of the particles of the medium, the particles which rotate most rapidly giving the sensation of red r etc. But Descartes supposed light to pass more quickly through a denser than a rarer medium, while Fermat, main- taining the contrary view, enunciated the principle of swiftest propagation of light. 14. Fermat's argument was metaphysical " nature works by the shortest route." The result, however, is remarkably correct. For, this law [Fermat's law of swiftest propagation of light] states that S dt = 0, where t is the time of propagation of light between two given points and 8 is the operator of the calculus of variation. Now, since, on the wave theory, /-i, the index of refraction, varies inversely as the velocity of propagation, the above is obviously the same as 8 I i*ds = 0, an equation which, as we shall see, analytically embodies a complete kinematical statement of all optical phenomena. 15. Thus, 5 f fids = I fyds + I since c?s 2 = dx 2 + dy 2 + dz 2 (doc -j- d&x + ... + ... ds FERMAT Therefore, integrating the second term by parts, we have obviously f dx] 2 \it-r-\ cfoji ox dfi d ( dx f- -j-i/ji-- Sxds + . . . -f . . . = dx ds\ or d / dx\ T~ = T (P ~T > etc -> dx ds \ dsj dx while, at a surface bounding media (1, 2), p ^- , etc., are con- tinuous, giving the ordinary laws of reflection and refraction and the path of a ray in a heterogeneous singly-refracting medium. 16. Again, let PQ, QR be the incident and refracted rays at a surface of separation between the media of refractive indices p, /// and let Q"R" be a ray parallel to PQ, and PqR a ray consecutive to PQR. Then, drawing QQ'Q" perpendicular to PQ and qq', RR" perpendicular to QR, we have since pPQ + p'QR = pPq + p'qR, vQ'q = ft'Qq', or fiQ"R?' = p'QR. This gives Huyghens' construction for singly-refracting media. 17. If the second medium is doubly-refracting, QQ', RR" are the traces of the wave fronts (in the plane of the paper) and thus, since the wave surface is the envelope of the wave front, 8 OPTICAL THEORIES [CH. the wave surface can be determined, in the usual way [Art. 54], as the envelope of Ix + my + nz = v, where I, m, n are the direction-cosines of the normal to the wave front, and v the velocity of propagation, while I, m, n and v must be connected by a relation which, however, requires to be determined on some theory independent of Fermat's prin- ciple (such as Fresnel's). Thus, Fermat's principle is seen to be capable of giving a complete kinematical account of double refraction, also with the help of a subsidiary hypothesis, regarding the law of variation of /* with direction. 18. Again, since 8 pds we may take pds dV^ and we have SF=0 at each reflection or refraction. And, since ar dv dv dv - = /*, or ^ : -j- : ^ : : a : p : % ds dx dy dz where a, /3, 7 are the direction-cosines of the ray, we conclude that V - constant is a surface orthogonal to a system of rays. Accordingly, if a given surface can, at any stage, be made to coincide with a surface V = constant, a surface can always be drawn to coincide with any other member of the family V = constant : Or Any system of rays originally orthogonal to a surface will always be orthogonal to a surface, after any number of reflections and refractions. 19. Since, then, the rays of light (which are orthogonal to a surface) may be regarded as normals to a family of surfaces, and, by Sturm's theorem, all the normals to a surface in the neighbourhood of a point converge to or diverge from two focal lines at right angles to one another, each of which passes through the centre of curvature of one of the principal normal sections and is perpendicular to the plane of that section, we conclude that all the rays of a thin pencil which can be cut at right angles by a surface pass through two lines, such that the I] FERMAT planes containing either of them and the principal ray are per- pendicular to each other. 20. Again, since the equation of a surface near the origin with the axis of z along the normal at the origin is, to the second order, the characteristic function (V= constant) for a thin pencil in a medium /JL, with the axial ray proceeding along the axis of z, is approximately, if aberration is neglected. With the help ot this equation, the problem of reflection and refraction of direct and oblique pencils (aberration being neglected) can be treated in the usual way. 21. To illustrate this, consider the following example : A pencil of rays is refracted through a prism and the axis of the pencil is constantly in a principal plane of the prism : also the angular position of the focal lines at incidence and after the first and second refraction relative to the edge of the prism is defined by a, /3, 7. If, now, the distances of the initial and final foci from the first and second surfaces are u lt u 2 and v 1} V 2 , respectively, to obtain the equations of refraction. Let us take U lt 7 2 as the distances of the foci after the first refraction from the first surface, and V l} V 2 those from the second surface. Then V l U l = V 2 U 2 = length of the axial ray in the prism. Let, finally, (/>, (/>' be the angles of incidence and refraction at the first surface, and >/r, ^r those at the second surface. The edge of the prism being taken as the axis of y and the normal to the face of incidence as the axis of z, the direction- cosines of one of the focal lines before incidence are sin a cos , cos a, sin a sin <. 10 OPTICAL THEORIES [CH, Thus, the characteristic function, before incidence, becomes V= n [x sin + z cos $ \u^ (x sin a cos + y cos a 4- z sin a sin c) 2 2^2 (# cos a cos (/> y sin a + 2 cos a sin ) 2 }. From the continuity of the function, at z = 0, we get, by equa- ting coefficients of x, a?, xy> and y 2 , IJL sin < = fjf sin <', , /sin 2 3 cos 2 a sin 2 o yu- sin a cos a ( --- ) cos ^> = // sin /3 cos j3 ( -^ - ^ J cos <^' r \1^i 'ita/ v v/i U o/ and similar equations for the second refraction. 22. In order to take account of aberration, we must obtain the characteristic function up to the order zx*. Let the equation of the characteristic surface (V=0) be 2z= + + 2z (ax* + frxy + yy*) + etc. Pi p2 Now the perpendicular from the origin on the tangent plane at a? iy2 x' t y', z' to the surface 2^ = + -- is Pi P* tfrtf* v +i+i if terms of the order ^a? 2 are retained. Therefore, in 2z = - + ^ Pi PI (1 a; 2 1 ?/ 2 \ 1 - - *- ) instead of z t * PI * PzJ so that the result may be correct up to this order. Hence the equation of the characteristic surface (F = con- stant) is of the form dy 3 (say), I] FERMAT 11 the terms in z 2 and z 3 being neglected, as being of higher orders. The projection of this on the principal plane y = is 1 x 2 1 a- 2 fift+rrf;* 8 *: The equation of the normal to this at x', z is x x z z x x _ /g 1 x' 2 pi pi z % Pi 2 This meets the normal through the origin at the point Z, 0, where Z z -, . Pi pf Therefore, the aberration in the principal plane is given by 2 Pi Pi - + ^ + Pi Pi - Saxpf, rejecting terms of higher orders, Sapi 3 ^!, where = 6^. PI We conclude, therefore, that all the results of geometrical optics can be deduced from Fermat's law. 23. Proceeding, now, to the dynamical significance of Fermat's law (which, as we have seen, Fermat deduced from metaphysical considerations), we observe that the configuration of equilibrium and motion of a dynamical system is defined by 12 OPTICAL THEORIES [CH. where T = Kinetic energy, ^Jj F = Potential energy. If this is to be consistent with Fermat's law, we for the propagation of light, T- V = C (constant) (1). Also if V = Potential energy of optical disturbance, during the displacement of the disturbance through the length of a wave, Tdt=l V'dt ........ ................ (2). From (1) and (2) we get V+C=V, showing that we must postulate a certain intrinsic energy in the system whose motion is associated with the propagation of light, in order to justify the principle of swiftest propagation. [Art. 96.] 24. Again, from the principle of energy, we have T+ V=C' (constant) .................. (3). Therefore from (1) and (3) we get But this is meaningless, since the mean potential energy and the mean kinetic energy are alone constant, as these quantities are taken to mean in the above equations. Accordingly, the only conclusion that seems to be consistent with all the equations is that the optical energy is entirely kinetic. The equation (1) then becomes T= A, the equation (2) dis- appears and the equation (3) becomes T' = A', and T' T= A' A= intrinsic energy. Again, if the potential energy of deformation of the ethereal medium involved in the propagation of light is to be regarded as essentially kinetic, we are led to conclude that all energy is kinetic. 25. This view of the intimate nature of energy intrinsic or otherwise is partially accepted, in effect, in different branches of Physics. Thus, on the kinetic theory of gases, the pressure of a I] HOOKE 13 gas has a kinetic origin, and, in fact, all cases of equilibrium in molecular physics are best explained as those of mobile or con- vective equilibrium. 26. But, if the interpretation of Fermat's law, sketched above, is admissible, we are led to a further generalization, to regard the energy of mere configuration, also, as kinetic. We must, in fact, conceive some subtle ethereal and electronic [Ch. iv] motion, being associated with every given configuration of a conservative system, existing in the field; thus the so-called potential energy of a vibrating system at any moment is, in reality, kinetic energy of the field. When we consider the later theories, we shall see that there is considerable justifica- tion for the view that the energy of light is entirely kinetic*. SECTION II. HOOKE. NEWTON 27. In 1665, Hooke published a theory of light in the Micrographics According to him, light is " a quick and short vibratory motion, propagated in every way through a homo- geneous highly elastic medium in straight lines, like rays from the centre of a sphere." Hooke believed that refraction is pro- duced by the readier transmission of light through the denser medium and he gave a geometrical construction, similar to Huyghens', for the incident and refracted wave-fronts. Ac- cording to Hooke, colour is generated by the distortion of the disturbance in the course of refraction. 28. Newton f was evidently aware of this theory, a portion of which, that relating to colour, he refuted on the result of the first of his optical researches. But it was a hypothesis merely, unsupported by facts and experiments and was moreover beset with formidable difficulties. " For, to me," said Newton, " the fundamental supposition itself seems impossible, viz., that waves or vibrations of any fluid can, like the raj^s of light, be propa- gated in straight lines without continued and very extravagant spreading and bending into the quiescent medium, where they * Fermat's Law. Philosophical Magazine, July, 1913. t Glazebrook, Address as President of the Physical Section of the British Association, 1893. 14 OPTICAL THEORIES |~ CH - are terminated by it." The corpuscular theory, on the other hand, seemed to him to be capable of offering a dynamical or a quasi-dynamical explanation of known optical phenomena, and Newton, accordingly, applied it to the explanation of reflection, refraction, diffraction, colours of thin plates and even polarisa- tion. In doing so, he was forced to subtle postulates, which it would not be without interest to examine briefly. 29. (a) Reflection. Let MN limit the region of molecular activity, AB being the reflecting surface. Let XY x , be a ray, that is, the path of a . / light corpuscle, which when it \ / reaches the limit of the region Y ^V /^' of molecular activity, viz. MN at ^ -^ B Y, must be regarded as being in a condition to be repelled. Thus, the corpuscle is acted upon by a repulsive force in a direction perpendicular to AB, so that the normal velocity of the corpuscle decreases. We must assume that it becomes zero when the corpuscle reaches AB, while the tangential velocity remains unchanged. Hence, ultimately the corpuscle passes off along Y'X' inclined at the same angle to AB as XY. This is the well- known law of reflection. (6) Refraction from a rarer to a denser medium. The action of the medium on the moving corpuscle is now to be regarded as attractive ; that is, we must assume that when it enters the region of molecular activity within the denser medium, it is attracted. The path wil! 3 obviously, be bent towards the normal. The velocity in the direction of the normal increases and the tangential velocity remains unchanged. Thus, let V a = velocity in air, V g velocity in (say) glass, i = the angle of incidence, r = the angle of refraction. l] NEWTON 15 Then, V a sin i = V g sin r : .'. = ^ = //,-== index of refraction of glass, sin r V a (c) Refraction from a denser to a rarer medium. The action of the medium must be regarded as repulsive, so that the velocity perpendicular to the bounding surface must decrease but must not be zero when the corpuscle passes beyond the range of molecular action. 30. We have thus to postulate a repulsion [of suitable magnitude] to explain reflection, an attraction [of suitable magnitude] to explain refraction into a denser medium and a properly adjusted repulsion to explain refraction into a rarer medium*. 31. These postulates are thus explained by Newton {Optics, Prop. XII) : " Every ray of light, in its passage through any refracting surface, is put into a certain transient constitution or state, which in the progress of the ray returns at equal intervals and disposes the ray at every return to be easily transmitted through the next refracting surface, and between the returns to be easily reflected by it." This is manifest (according to Newton) from the arrange- ment of colours of a thin plate in the reflected and refracted light. For, one and the same sort of rays, at equal angles of incidence on any thin transparent plate, is alternately reflected and transmitted, for many successions, according as the thick- ness of the plate increases in arithmetical progression : (1, 3, 5, for reflection; 0, 2, 4, for refraction). And he proceeds: "The returns of the disposition of any ray, to be reflected, I will call its Fits of easy reflexion, and those of its disposition to be transmitted, its Fits of easy transmission, and the space it passes, between every return and its next return, the Interval * It is easy to see that we can explain these as resulting from the motion of electrons suitably interspersed in the ethereal medium. 16 OPTICAL THEORIES [CH of its Fits." And, further (Prop. XIII): "The reason why the surfaces of all thick transparent bodies reflect part of the light incident on them and refract the rest, is that some rays at their incidence are in Fits of easy reflexion and others in Fits of easy transmission." A possible explanation of these fits is supplied in Query 17. "When a ray of light falls upon the surface of any pellucid body and is then refracted and reflected, may not waves of vibrations or tremors be thereby excited in the reflecting medium and do they not overtake the rays of light and, by overtaking them successively, do they not put them into Fits of easy reflexion and easy transmission*?" As a further explanation, he suggests (Q. 26) that rays of light may have several sides four sides to explain double refraction. 32. We have seen that the colours of thin plates were explained by they, in fact, led to the hypothesis of fits and a certain polarity of the rays. But, in order to explain all the observed facts, it was necessary to suppose that the length of a fit varies as the secant of the angle of incidence. This has been held by some to emphasize the very artificial nature of the whole theory, but this conclusion is not justified, for it should be noted that it might have conceivably been the consequence of a simpler lawf. His explanation of diffraction is contained in the following query : "Do not the rays which differ in refrangibility differ also in flexibility and are they not by their different inflexions separated from one another, so as, after separation, to make the colours in these fringes" [of the diffraction pattern he had previously described] ? And again, "Are not the rays of light in passing by the edge and sides of bodies, bent several times backwards and forwards, with a motion like that of an eel ? And do not the three fringes of coloured light arise from three such bendings ? " (Q. 3.) * Cf. the theory of the X-rays. f Cf. Kepler's laws of Planetary Motion, iu particular the third law. l] NEWTON 17 33. But, although Newton worked out these explanations on the basis of the corpuscular theory, he himself had an open mind "argued t>he corporeity of light but without any absolute positiveness." He was, moreover, fully cognizant, as we have just stated, of the rival hypothesis and its merits. y Thus, in Q. 13, he proposes an explanation of colour on the wave theory: "Do not several sorts of rays make vibrations of various bignesses, which according to their bigness excite sensations of various colours, the most refrangible, the shortest vibrations ? " Again, in Q. 17 he puts forward what may be called a y combined theory as a possible explanation of the 'Fits' [Art. 31]. But in Q. 28, he points out the crucial difficulties of the wave theory the rectilineal propagation of light. "If it (light) consisted in pressure or motion, propagated either in an instant or in time, it would bend into the shadow. For pressure or motion cannot be propagated in a fluid in right lines beyond an obstacle, which stops part of the motion but will bend and spread every way into the quiescent medium which lies beyond the obstacle." Nor was it at all clear how double refraction was to be explained if light consisted in vibrations, like those of sound. 34. It was these difficulties which led him (in Q. 29) to suggest the corpuscular theory and point out how it served to explain the formation of shadows. "Are not the rays of light very small bodies emitted from shining substances ? For , such bodies will pass through a uniform medium in right lines without bending into the shadow, which is the nature of the rays of light." 35. Previously to this, however, he had proposed a compre- hensive hypothesis, which, as a mere hypothesis, he did not think it desirable to include in his treatise, though it was of an older date. In a paper sent to Oldenburg for the Royal Society but withheld from publication at his own request, he thus states his hypothesis : M. o. T. 2 18 OPTICAL THEORIES [CH. "Were I to assume an hypothesis, it should be this, if pro- pounded more .generally, so as not to assume what light is further than that it is something or other, capable of exciting vibrations of the ether. First, it is to be assumed that there is an ethereal medium, much of the same constitution with air but far rarer, subtiller and more strongly elastic. In the second place, it is to be supposed that the ether is a vibrating medium, like air; only the vibrations, far more swift and minute; those of air made by a man's ordinary voice, succeeding at more than half a foot or a foot distance, but those of ether at less distance than the hundred thousandth part of an inch. And as in air, the vibrations are some larger than others but yet all equally swift, so I suppose the ethereal vibrations differ in bigness but not in swiftness. In the fourth place, therefore, I suppose that light is neither ether nor its vibrating motion but something of a different kind propagated from lucid bodies. They that will may suppose it an aggregate of many peripatetic qualities. Others may suppose it to consist of multitudes of unimaginable small and swift corpuscles of various sizes, springing from shining bodies at great distances, one after the other, but yet without any sensible interval of time.... I would suppose it diverse from the vibrations of the ether. Fifthly, it is to be supposed that light and ether mutually act upon one another. It is from this action that reflection and refraction come about. Ethereal vibrations are, therefore, the best means by which such a subtle agent as light can shake the gross particles of solid bodies to heat them." "And now to explain colour. I suppose that as bodies excite sounds of various tones and consequently vibrations in the air of various bignesses, so when the rays of light, by im- pinging on the stiff refracting superficies excite vibrations in the ether, these rays excite vibrations of various bignesses the biggest with the strongest colours, reds and yellows ; the least with the weakest, blues and violets ; the middle with green and a confusion of all with white." 36. This is, practically, the undulatory theory, which in the hands of Huyghens, Young and Fresnel furnished a fairly I] NEWTON 19 satisfactory explanation of all the known phenomena of optics. The explanation, however, is only kinematical. As contrasted with this, the corpuscular theory, so far as it goes, has a dynamical basis. As such, it evidently has, prima facie, an advantage over the undulatory theory, and, accordingly, deserves further consideration. 37. The first and the simplest objection to the theory, which appeared almost conclusive at one time, was based on the fact that light corpuscles appeared to be absolutely without momentum. It was, however, easy to meet this objection by attributing to light corpuscles a minuteness, sufficient to evade any means that was then possessed of detecting their existence by their effects on other bodies. We now know (1) that light does exert an impulsive effect and (2) that corpuscles moving with velocities comparable with that of light are at least as real as the chemical atom. 38. On the other hand, the phenomenon of aberration is much more simply explained on this theory than on that of un- dulation. When, however, we come to consider the theory of refraction, apart from the very artificial nature of the theory of fits on which its explanation is based, it is found that it is necessary to suppose that the velocity increases with the density of the medium. But experiment shows that the reverse is the case and we have, accordingly, to reject the theory but only in the special form in which it was applied by Newton to the explanation of refraction. For the disagreement might have been due to the fact that the mode and law of interaction between 'gross' matter and corpuscles as postulated by Newton was wrong, but it would not necessarily prove that light is not propagated by means of corpuscles. 39. As to the velocity of corpuscles, modern theory postu- lates very high velocities for the cathode rays and for particles projected from radium. There was, accordingly, nothing a priori absurd in the corpuscular view. 40. But light has two characteristics, not only a velocity of propagation but also a periodicity. Newton must have felt the 22 20 OPTICAL THEORIES [CH. inadequacy of the corpuscular theory in explaining this the latter aspect of optical phenomena and it was on account of this, obviously, that he had to introduce the element of time the time of a fit in his explanations and as even that did not seem to account for all known phenomena, he definitely adopted the ' bignesses of vibrations/ as alone explaining colour. He could not, however, reconcile himself to the undulatory theory in its entirety, because of its failure to explain rectilineal propagation, although in one respect he was more precise than the advocates , of the undulatory theory themselves; for he laid down that light was neither ether nor its vibrating motion, but energy, and it was the mode of propagation of this energy through an ethereal medium for such a medium must be postulated in any case about which it was necessary to frame a hypothesis. SECTION III. UNDULATORY THEORY 41. If it is difficult to see how the element of periodicity (so essentially associated with optical phenomena) enters in the motion of corpuscles (on the corpuscular theory), periodicity enters as a fundamental element in wave motion. There was, thus, an initial justification for the theory which regards the energy of light as due to wave motion. As soon, therefore, as the crucial difficulties propounded by Newton, viz. the phenomenon of rectilineal propagation and of double refraction, were explained on this theory, it served to furnish a complete kinematical ex- planation of all optical phenomena. The phenomenon of rectilineal propagation was ultimately explained on Huyghens' principle. The principle, simply, is that each element of a wave surface may be regarded as a source of the disturbance and, therefore, (1) the wave surface at any subsequent instant is the envelope of the wave surfaces, constructed with these elements as sources; (2) the disturbance at any point is the vector-sum of the disturbances due to these elements*. The first of these principles enabled Huyghens to give a construction and a complete (kinematical) explanation of * The weight to be attached to each element was naturally left doubtful. This was supplied by Stokes [Art. 178]. I] UNDULATORY THEORY 21 refraction and reflection. It further served (on the assumption that the wave surface inside a uniaxial crystal, like iceland spar, consists of a sphere and a spheroid) to explain the pheno- menon of double refraction, also, in such crystals. 42. The second principle would, if properly interpreted, have led to the proof of the rectilineal propagation, but it was not till quite a hundred years later that this was successfully accomplished by Young and Fresnel. For the second is in reality the principle of interference, from which it follows that since the wave-length of light is infinitely small, luminous effect at any point must be due to a small portion of a whole wave front, and, therefore, light must, as a rule, be propagated in right lines but that it should (as it does, as a matter of fact) bend round apertures, even as Newton argued it ought to, if the aperture is sufficiently narrow. This mode of regarding the phenomena also effectively meets the initial difficulty to which the undulatory theory is subject, viz. that it leads to the conclusion that there ought to be shadows of sound as well as those of light. For, inasmuch as the wave-length of light is infinitesimal, as compared with that of sound, sound shadows can be produced only by large obstacles, as we know they actually are. 43. A fairly complete mathematical explanation of the ordi- nary phenomena of optics, based on the principles of the wave theory propounded by Huyghens and Young, was given by Fresnel, but his explanations were mainly kinematical. Where he attempted a dynamical explanation, that explanation on examination is found to be either arbitrary, or at any rate to involve postulates which it is difficult to interpret. 44. Before proceeding to the consideration of the attempts that have been made at a strictly dynamical explanation, we shall examine the kinematical explanations that the undulatory theory has supplied and to which reference has just been made [Art. 41]. 45. If v is the velocity of a periodic disturbance, we have v = a sin 2?r ft #\ I -- - ] , \T A./ 22 OPTICAL THEORIES [CH. where a is the amplitude, r, the periodic time, and X, the wave- length, while x is the distance at which the velocity is v, the velocity at the origin being at the same time (t) = a sin 2?r - . Then, it easily follows that the mean kinetic energy of the disturbance ^T the mass of a vibrating element being taken to be equal to unity. Consider, now, the effect at any point of two disturbances from two origins (sources) at distances x lt x 2 . If the velocities are in the same direction [Art. 48] (whatever be this direction) the resultant velocity will be . _ ft x,\ , . . ft x z \ v 1 4- v., = ! sin 2-7T + 2 sin 2?r VT X/ \T X/ where = A sin 2?r ( -J , say, A 2 = ttl 2 + 2 2 + 2 ai a 2 cos 27r ^~^- 2 ) . \ X / Therefore, (1) A = 0, or the resultant disturbance is zero, i.e. the resultant effect is one of absence of light, if and ^-^ = (2/1 + 1)^. (2) Under the same conditions, A has its maximum value or the resultant disturbance gives the maximum of illumination, if #j # 2 = n\ where n is an integer. (3) If ^varies irregularly, during the time nr, i.e. if X the disturbance at the origin changes in phase from moment to moment, independently of each other, the mean of cos 2?r 1 - X during nr, where n is large, 1 [ HT+t x l x> 2 since x l x 2 goes through all possible values during the interval. In this case A 2 = a x 2 + 2 2 , always. I] HUYGHENS 23 The third result explains why two lights cannot interfere, if they proceed from two different sources. This, indeed, is an experimental fact from which we conclude that vibrations from two sources change their character, independently of each other, from moment to moment. (1), (2) specify the conditions under which interference can take place. Exactly the same principle applies when we proceed to explain the effects observed, when light passes through a narrow aperture or a large number of narrow apertures arranged in any manner (a grating) [Art. 180J. This is the phenomena of diffraction. Generally, therefore, when light from the same source reaches a point after traversing two different paths, we have the phenomena of interference. 46. It has sometimes been argued that the interference effects as shown by the experiments of Young and Fresnel are fatal to the emission theory. It should be noted, however, that in all these cases, there is a redistribution of energy but no loss and, therefore, if we conceive light to be propagated by perfectly elastic particles (which are of the nature of revolving doublets), these by their collision would change the distribution of energy in the way in which it is actually found to do*. 47. The principle of Huyghens, then, and that of inter- ference which is really involved in it, serve to give a complete kinematical explanation of rectilineal propagation, reflection, refraction, interference and diffraction of light. 48. In explaining these, it is unnecessary to define the direction of vibration in relation to the ray. When we come to double refraction, the question of this relation comes naturally to the fore. Let us recall the fact that when a ray of ordinary light is passed through a piece of iceland spar, the emergent light is divided into two rays ; one of them is refracted according to the ordinary law of refraction and is called the ordinary ray while the other does not obey this law (the corresponding index of * Preston, Theory of Light. 24 OPTICAL THEORIES [CH. refraction depending on the direction of the incident ray) and is therefore called the extraordinary ray. It is just conceivable (on some suitable hypothesis), that even if the vibrations of light were longitudinal, the incident ray would be thus de- composed. But when these decomposed rays are again passed through another piece of iceland spar, it is found that, as the principal section of the second crystal is rotated round that of the first [Art. 57], the diverging rays (two for each ray) vary in brightness. The vibrations cannot therefore be entirely longitudinal. It is, moreover, found that when ordinary light from a given source is decomposed by means of a uniaxial crystal like iceland spar, any two of the resulting ordinary rays (or of the extraordinary rays) if allowed to pursue slightly different paths will interfere, but that an ordinary ray will not interfere with an extraordinary ray and vice versa. From this, it follows (1) that the vibrations of light cannot be even partially longitudinal, i.e. they must be perpendicular to the ray, (2) the ordinary and the extraordinary rays must each continuously vibrate in or perpendicular to a certain invariable plane and (3) that these planes must be perpendicular to each other. 49. These rays are said to be plane polarised, so that the difference between plane polarised light and ordinary light consists in this that in the latter the vibrations continuously change their azimuths round the ray, while in the former these azimuths remain unchanged. And whenever and through whatever cause, the vibrations have this character impressed on them (whether the light suffers double refraction at the same time or not) the) 7 are said to be plane polarised. Thus, double refraction consists not in creating these transverse motions but in decomposing them into two lights, polarised in two planes at right angles to each other, and separating the components. 50. The above mode of analysis of the vibrations constituting light being admitted, the explanation of elliptic and circular* * The difference between these and ordinary light again consists in this that in the case of the latter, when decomposed in two fixed directions, the elements o, a, 6, ft continuously change, while in the case of the former, these are invariable. I] FRESNEL 25 polarisations as the result of compounding two plane polarised vibrations of suitable types, namely . /27T \ 7 . /27T \ a sin f --- a) and o sin f -- /3J , rectangular or oblique, easily follows. 51. In attempting, in the manner of Fresnel, a more detailed explanation of double refraction, we note in the first place that in the case of a uniaxial crystal, the assumption seems legiti- mate that the wave surface in such a crystal consists of a sphere and spheroid. For it is, a priori, evident that luminous vibrations would be propagated in a homogeneous medium in spherical waves, and hence it is natural to admit that in a doubly-refracting medium, having an axial symmetry, they will be propagated in a sphere and a spheroid. This leads, on Huyghens' principle, to the following construction: 52. Describe a sphere and a spheroid with a point on the refracting surface as centre and the axis of revolution of the spheroid coincident with the axis of the crystal. Through any point in the plane of incidence, belonging to the refracting surface, draw tangent planes to these surfaces. The radii-vectores to the points of contact are the refracted rays and the perpen- diculars to the tangent planes represent, on a suitable scale, the velocities of propagation. 53. Let the equations of these surfaces be a; 2 4- f + z 2 = a 2 ..................... (1), and J + 2/!+f! = 1 ..................... (2). Let also /, m, n define the wave front [Art.vL7] and ^ be the velocity of propagation of the ordinary ray, while I', m, n', v 2 are similar quantities for the extraordinary ray; then, from the definition of the wave surface, we have (I 2 + m 2 + n 2 ) a* = vf, and a 2 Z' 2 + 6 2 (m' 2 + n*) = v 2 2 , or, 2 , are the roots of the quadratic in r 2 , viz. ....(3), 26 OPTICAL THEORIES [CH. V m* n? - + - + - - = 0, i.e. , are the axes of the section of the spheroid a 2 (y 2 + z 2 )=l .................... (4) by the plane Ix -f my + nz = 0. We conclude, therefore, that the reciprocals of the velocities are equal to the lengths of the axes of the central section of a certain spheroid (4) by Ix + my 54. Generalizing the above, in a biaxial crystal, it is reasonable to suppose that the surface corresponding to (4) of Art. 53 will be an ellipsoid. Let the equation of this surface be a?x 2 + b 2 y 2 + c 2 z 2 = 1. Hence, it easily follows, that I, m, n, the direction-cosines of the wave normal, will be connected by the relation V ^ * =0 v 2 - a 2 v* - b 2 v 2 - c 2 corresponding to (3) of Art. 53, and the wave surface will be the envelope [Art. 52] of Ix -f my + nz = v, where . m v I ' I __1__ __ Q v' 2 -a 2 v 2 - 6 2 v 2 - c 2 and Further, if X, //,, v be the direction-cosines of an axis of the central section of aV + b 2 y 2 + c 2 z 2 = 1 (principal axis, r), we have & = r' 2 = a 2 \ 2 + 6X + cV 2 .............................. (1), v .................... ............. (2), = (6 2 -c 2 )-|--(c 2 -a 2 )+-(a 2 -6 2 ) ........ (3). A. LL V So far, the work is entirely geometrical arid is derived by a reasonable generalization of Huyghens' assumption that in a uniaxial crystal, the wave surface consists of a sphere and a spheroid. It remains now to see whether a dynamical inter- pretation of the equations (1), (2), (3) is possible. Now, (1) obviously suggests an equation of small motion of the type pi = - %p (a 2 X 2 + 6> 2 + cV). l] FRESNEL 27 Let two of the axes of reference be the axes of the crystal and let OP = the displacement of a particle at any time ~ f = (?^> ?/*. &)- If this is supposed to bring into play a force of restitution = (a?p% X, b 2 p^fji, c*p!;v) (say) (4), the resolved part of this force along X, JJL, v (1) can, therefore, be interpreted, if we assume that the force of restitution, called into play, is of the above type (4) of which the only part that is effective in producing vibration inside the crystal is that contemplated in (5). It follows further that a, 6, c are the principal wave-velocities. (2), obviously, means that the direction of velocity is perpen- dicular to the direction of the ray. (3) is equivalent to a 2 X b' 2 /j, c 2 v = 0, X yLt V I m n which means that the ray, the displacement and the force of restitution are in the same plane. In other words, only those displacements are effective which give rise to stress in the plane containing the ray and the displacement, i.e. those that give rise to stress whose component along the wave-front is along the direction of displacement. 55. It is hardly necessary to enter on a detailed examina- tion of hypotheses which were actually* made, in order to obtain a particular analytical result. As, however, the final result deduced from them has been verified by experiment, it would be as well to examine them briefly. The hypotheses are : (a) The force of restitution is proportional to displace- ment (the constant of proportionality being the product of density and a factor depending on direction). If, however, the ethereal medium inside a crystal behaves like an elastic solid, the force should be proportional to strain * Whittaker, A History of Theories of Ether and Electricity. 28 OPTICAL THEORIES [CH. [Art. 70]. On the other hand, if the equation of motion is that of an electron [Ch. iv], this objection does not apply. (b) Only the resolute of the force along the direction of dis- placement is effective. This is obviously dynamically unsound, unless we hypothecate this as a property of the medium. (Cf. the incompressible or labile ether of the elastic solid theory.) (c) The vibrations are perpendicular to the ray. (d) The ray, the displacement and the force of restitution are in the same plane. This does not seem to have any justification, unless we regard it as defining the property of the medium, due to structure, etc. The phenomenon of conical refraction was at one time regarded as a singular verification of Fresnel's theory, but it only indicates that the wave surface should be a surface of two sheets, having conical points, and cannot tell us anything as to the special hypotheses made. 56. Fresnel's wave surface*. Let OP be perpendicular to the tangent plane to an ellipsoid 8 (centre 0), say, at the point Q, and OP' the con- secutive perpendicular. Let 5+iS* ?*.- [Art. 54] , and let S' be recipro- cal to 8. Also let ON be perpen- dicular to the central section of S' containing OP and let ON = OP, and finally let P', N' be consecutive points to P, N, the locus of Q' being the envelope of the plane perpendicular to ON. Then OQ' is equal and perpendicular to OQ. Now, if p = perpendicular to the tangent plane (X, p, v) to 8, p2 = a 2 X 2 + &> 2 + cV = v 2 , if p is drawn in the direction of displacement. Then ON= v, and as it is drawn perpendicular to the central section of 8', the wave surface is the envelope of planes perpendicular to ON [Art. 54]. * MacCullagh, Collected Works, i. I] FRESNEL 29 Hence, this surface is the locus of points at the extremities of lines drawn perpendicular and proportional to the radius vector (OQ) of the ellipsoid S, corresponding to the perpendicular OP, in the direction of displacement, to the tangent plane to & Also S' = a?x 2 +...+ ... 1=0. Now since OP is one of the axes of the central section of S', OQ will be one of the axes of the central section of S and OQ' will be the normal to that section (say, I, ra, n). Therefore, writing OQ = OQ' = r, we have a 2 ~ r 2 and Ir = x, etc., if x, y, z are the coordinates of Q'. Whence the equation of the wave surface is In the case of a uniaxial crystal, it is easy to verify that the surface becomes a sphere and a spheroid. For, since the wave surface in the general case is if b = c, we have = 0. 57. The hypotheses [Art. 55], when applied to the particular case of a uniaxial crystal, will amount to the following. Let OX be the axis of the crystal, OZ, perpendicular to OX in the wave front, and OP, the direction of displacement; let also Z ZOP = a and the dis- placement (OP) = er. Then, the resolved part of the displacement along OZ is a cos of. Since the force of restitution is in the direction of displacement from symmetry, it will give rise to a ray along the axis. And every incident ray gives rise SO OPTICAL THEORIES [CH. to a refracted ray along the axis, no matter what its direction is. This constituent is therefore the ' ordinary ray.' That is, the vibrations in the ordinary ray are perpendicular to the axis of the crystal. Consider, now, the following experiment*. A ray of light (unpolarised) is passed through a spar in the principal section which is vertical. If, now, we cut off the extraordinary ray and let the ordinary ray of intensity pass through a second spar in the principal plane, we get two images of intensities 0', E', given by 0' = cos 2 or, E' = sin 2 a, where a is the inclination of the second principal section to the vertical. Therefore if a = 0, 0' = 0, i.e. the ordinary ray is in fixed relation to the principal plane ; it is said to be polarised in that plane. Now, since .the vibrations of the ordinary ray, on Fresnel's theory, are perpendicular to the axis, Fresnel's theory involves the supposition that the vibrations of plane polarised light are perpendicular to the plane of polarisation. It is easy to see that this mode of regarding the phenomena is consistent with experimental results. For this, let us discuss the experiment cited above. 2_/ A vibration given by A cos is incident on the second crystal, inclined at a. to OZ (say) drawn perpendicular to the ray and the axis. It can be resolved into A cos a cos -- along OZ (ordinary ray) and A sin a cos perpendicular to OZ (extraordinary ray). Then, the intensities are as cos 2 a : sin 2 a. But this is the law of Malus, according to which the ratios of the intensities of the ordinary and the extraordinary rays are as cos 2 a : sin 2 a. It will be seen that this experiment does not uniquely determine the direction of vibration of plane polarised light, in relation to the plane of polarisation. 58. Fresnel's theory of reflection and refraction. The principles on which the theory is based are : (1) The vibrations are perpendicular to the plane of polarisation. * Jamin et Bouty, Cours de Physique, Tome in, 3. I] FRESXEL 31 (2) The density is different in different media, but rigidity the same and, therefore, the velocity (= */ , . "M varies V V density/ inversely as the square root of the density. (3) The tangential displacement is continuous. (4) The phase is unchanged at reflection and refraction. (5) The total energy in the reflected and refracted wave is equal to the energy of incident wave. Assuming (I), Fresnel's theory leads to the conclusion that the intensity of natural light reflected is* 1 rsin 2 (i-r) tan 2 (i-r)1 2 [sin 2 (i + r) tan 2 (i + r)J ' the first giving the intensity of light polarised in the plane of incidence, while the second, that perpendicular to this plane. When, therefore, i + r 90, i.e. at the polarising angle, we get only the first term and light is polarised entirely in the plane of incidence. This may serve as the basis of an experimental method of determining the plane of polarisation and also provides (on Fresnel's theory) the basis of the following definition, viz.: the plane of polarisation is the plane in which light is polarised by reflection. 59. Now, we have seen that when light is polarised in its passage through iceland spar, the plane of polarisation of the ordinary ray is the principal plane (of the spar) containing the ray. The ordinary ray also follows the ordinary law of refraction, viz. the plane of incidence is the same as the plane of refraction. Therefore, the plane of incidence is the same as the plane of polarisation. The principle (1) is thus seen to be consistent with the hypothesis on which double refraction is explained. (2), (3). These are also the hypotheses of the ' Labile ' ether theory [Art. 72]. (4). This is inconsistent with experiment. There is always change of phase at reflection and refraction. * Jamin et Bouty, Tome in, 3. 32 OPTICAL THEORIES [CH. I 60. Again, Fresnel's theory leads to the conclusion that for light polarised perpendicularly to the plane of incidence, the intensity of reflected light is* - 7^ - { tan 2 (i + r) Therefore when i + r = 90, i.e. at the polarising angle, there should be no reflected light. Experiment shows however that this is not the case. Lorentz has shown that a transition layer of ether of variable density will account for this as well as the change of phase [Art. 58]. Now, from the continuity of energy, we get where S, S' the areas of corresponding elements of the wave fronts in the two media, X, X' = wave-lengths, p, p' densities of the media, and (a 1} a 2 , a') = amplitudes of incident, reflected and refracted vibrations, .-. o.-.a....^:.^! (i), p cos i n sin i V X . S cos i since -: -== = n = --, and = - - , smr V X 8 cosr where n = index of refraction, V, V = velocities of propagation in the two media. Also =s Tl == -. (^)i np sin r if the rigidity is the same in both media. Now if both the tangential and normal displacements are continuous, we must have ,cosr . sn r a - : ..................... (4). sm % . -. from (2), (3) and (4) a 3 2 - a 2 2 = a' 2 . - < ^A . . .(5), ncosi .'. p = p', from (1) and (5), which is absurd. This shows that, on Fresnel's theory, both the displacements cannot be regarded as continuous. * Jamin et Bouty, Tome in, 3. CHAPTER II ELASTIC SOLID THEORY SECTION I. THEORY OF ELASTICITY. REFLECTION. REFRACTION 61. We have seen that the method of Fresnel is not strictly dynamical*. Now in order to obtain a strictly dynamical ex- planation of optical phenomena, it is necessary to postulate certain properties of the medium which takes part in the propagation of the vibrations constituting light. The simplest hypothesis that we can make in order to explain the propagation of the vibrations of light would be to suppose that the medium is an elastic medium, and since these vibrations are transverse, it must possess the property of resisting shearing strain. Now, the only bodies that we know, possessing such properties, are elastic solids. The postulation of the transfer of the transverse vibrations of light by a medium, in virtue of its elastic properties, therefore, amounts to the supposition that such a medium behaves as an elastic solid. It is the model, we naturally conceive, in order to explain the peculiar features of the phenomena. The difficulty of conceiving such a model, however, is obvious the difficulty, namely, of associating the interstellar space with the properties of an elastic solid. But it is well-known that a body behaves differently under different circumstances of motion. In the case of steady motion of bodies through air, for instance, it behaves very nearly like a perfect fluid but perfectly compressible ; yet, when the rapid vibratory motion of sound is imposed on it, it offers resistance * Or, rather, it is dynamical, on the basis of special hypotheses. M. O. T. 3 34 OPTICAL THEORIES [CH. to compression, like an elastic body. Similarly, a medium may have all the properties of a perfect fluid for steady motion like that of the heavenly bodies, but may offer resistance to shear, when extremely rapid transverse vibrations, constituting light, are imposed on it. 62. The following remarks of Stokes [Papers, Vol. n, p. 12] clearly set forth the point of view on which the elastic solid theory is based : " Suppose a small quantity of glue dissolved in a little water so as to form a stiff jelly. This forms in fact an elastic solid. It may be constrained, and will resist constraint and return to its original form when the constraint is removed, by virtue of its elasticity ; if constrained too much, it will break. Suppose the quantity of water in which the glue is dissolved is gradually increased till it is only glue-water. At last, it will be so far fluid as to mend itself again, as soon as it is dislocated. Yet, there seems hardly sufficient reason for supposing that at a certain stage of the dilution, the tangential force, whereby it resists constraint, ceases all of a sudden. In order that the medium should not be dislocated and therefore should have to be treated as an elastic solid, it is only necessary that the amount of constraint should be very small. The medium would however be what we should call a fluid, as regards the motion of solid bodies through it. The velocity of propagation of normal vibrations in our medium will be nearly the same as that of sound in water; the velocity of propagation of transverse vibrations depending as it does on the tangential elasticity will become very small. Conceive, now, a medium having similar properties, but incomparably rarer than air, and we have a medium such as we conceive ether to be a fluid as regards the motion of the earth through it, an elastic solid as regards the small vibrations which constitute light. The sluggish transverse vibrations of our thin jelly are in the case of the ether replaced by vibrations propagated with a velocity of nearly 200,000 miles per second. We should expect a priori the velocity of propagation of normal vibrations to be in- comparably greater." THEORY OF ELASTICITY 35 63*. If u, v, w be the displacements of a point (#, y, z) of an elastic solid, then x + u, y + v, z + w will be its displaced coordinates. The displaced coordinates of x + Air, etc. will be x + A# + u + ~- A#, etc.; .'.the new lengths of A#, Ay, As will be .'. the linear extension in the direction of a? is Aa? (l Lt and the voluminal expansion 'du dv dw\ 64. Let a rectangle ABDC be held fixed by the edge AB and pulled by a force along CD, AC and c being displaced to AC', BD'. Then GG' _ T Aw _ du Ay~9y' along y , " = . ~ AC if AB is taken along ar, and and equal to Ay, while 6 the circular measure of the Z GAG' . Hence, for a rectangular parallelepiped, subjected to tan- gential stress, with the plane 20 fixed, the strain dv du = to + ^ EEe *" sa y ; e xy is called a shearing strain. 65. If U, V, W be the components of displacement of any point (x + f , y + rj, z + f ), then 7r 9w ,. 9w 9w v (7 =W+ f + ;r- ?; +^- f i 9 b 9y 9^ * Love's Elasticity. 32 3G OPTICAL THEORIES [CH. .*. the relative displacement in the direction of x du du du .. = ^~ + 5- v +5- ? 9a? * dy dz * du f. i flu dv\ i fdu dv dw Hsi-stcw*" provided we take ^ ^- = 2a> 2 , etc (1) ox oy and f, ?y, f small. Therefore, the relative displacement in the direction of x. = ||-a,^ + y r (2), where 2< = e xx ^ + e^?; 2 4- e zz ^ 2 + e y ^f -f g^ff + e^f??. But when a rigid body is in motion and its displacements, are referred to axes meeting at a fixed point but moving in space with angular velocities lt 2 , O s about themselves, then, if the displacement of any point (x, y, z) in the direction of x fixed in space is X 1} and its displacement in the direction of the moving axis of x is X, X 1 = Z-% + ^ (3). Comparing (2) and (3), we conclude that the displacement at a neighbouring point (x + f , y + 77, z + f) relative to (a?, y, 2). is a certain linear displacement together with a displacement due to rotations of the axes of reference, the angular displace- ments of the axes being given by 0) X , toy, W Z ... (1). These are called molecular rotations, for obvious reasons. 66. At any point, let the elastic stresses or forces per unit area be X X) X y , etc., where X x stress in direction of x perpendicular to x = ; X y = tangential stress in the direction of x, in the plane y =0;. X z = tangential stress in the direction of x, in the plane z=Q, etc. Il] THEORY OF ELASTICITY 37 67. Considering, now, the equilibrium of an elementary cube, not acted upon by external forces, we have by taking moment about the axis of z through the c.G. of the cube where Y x ' is the value of Y x at the opposite edge and a the length of a side ; but Therefore when a is infinitesimal, we have X y Y x , ulti mately. 68. Again resolving along the axis of x, we have or x y _ JUt? ^ r f~ r ^, "> ox oy dz and two similar equations. The corresponding equations of motion obviously are dX x , dX y , dX z pu = -= -- h -r-* + - , etc. ox oy oz 69. Let such a cube be strained (under no body-forces), the strain being unaccompanied by loss or gain of heat. Let 8 W = the increase of potential energy per unit volume of the elementary volume dxdydz under displacements Su, etc. Then $W dxdydz K dSu dX x ^ \ "1 Z *^ + ^^ + "- + -'-J ^ ' rejecting terms of higher orders. oX x , d8u fc/atA since h ... + ...= 0, etc. and -j = o ( -5- = oe xx . dx dx \dx = oe 38 OPTICAL THEORIES [CH, W is called the ' strain energy function/ and the equations of motion [Art. 68] become d fdW\ d /dW\ d fdW pu = 5- - I +' sH o ) + 5~ I * ox \oe xx l oy \fle ay / oz \oe 70. Again, since X x , etc. are functions of e XX) etc., we conclude that W is a function of e xx , etc., which can be expressed in the form to + ^l + ^'2 + -"> where ->|r is a constant, ty l is a linear function of e xx , etc., and so on. If, now, powers of e xx , etc. beyond the second are neglected and the medium is free from stress when in equi- librium, i.e. i/r = 0, T/TJ = 0, we have Let a cube [side ABCD = I 2 ] of a homogeneous isotropic solid be subjected to a pull (= 1) at one of the faces perpen- dicular to BC (=1). Let the plane ABCD become strained into A'B'C'D', through the action of this force. Let e be the angle between the initial and final positions of the diagonal plane, i.e. Z BOB'. Then the figure being a section parallel to ABCD (or ABCD itself) tan (* - e] = But and while and A'D" AB A'B' = lateral contraction (1), A'D' AD = longitudinal extension X^)> lateral contraction a- = Poisson's ratio = longitudinal stress ,-, longitudinal strain ~ longitudinal extension 1 ...(3), A'D' -AD (4). AD Now, since AD = 1, AD' = 1 + -= , from (4), while AB - A'B' = ^ , from (1), (3) and (4); JtL !- 1- 1 + E c' where E is Young's modulus. Il] THEORY OF ELASTICITY 39 .'. 2e= ~tf~> ( e being very small), and the shear on the diagonal = 2e = =r corresponding to a tangential pull per unit area over the diagonal plane equal to ^ (since the resolved part of the pull along the diagonal = , and its area is V2). Also the coefficient of rigidity is by definition = ^ . If we call this /u, we have shearing strain 77 r and X?, Let stresses X x , Y y , Z z be applied to the faces of an elementary parallelepiped. Then, since the total extension = longitudinal extension lateral contraction in the same direction, we have, by Hooke's law, e xx = ~ [X x - a (Y y + Z g )], etc. ; we have X x = XA H- 2fjL'e xx . .-. 2W = (X + 2p) A 2 + ^ (e^f + e^ + ^ 2 This can be written (X + /*) A 2 4- I/A {(e xx -e yy Y -f ... + ...) 4- M^ 2 + ,.. + ... : . . for stability X 4- At should be positive and yit positive [Art. 72]. Hence, A may be = 0, consistently with stability. 71. The expression for W is capable of an interesting transformation : thus, we have, if S is the bounding surface of the region to which W refers^ [dv dw -, , , ff dw -. , ~\ f d fdw\ -,,-, AT- 7T" dxdiidz = \ fv -5- dxdz \ v . ( T- dxaydz } ty %Z {_] dz J bet . limits ^ ty \02 / 40 OPTICAL THEORIES [CH. Again dvdw where /, m, n are the direction-cosines of the normal to dS ; [dv dw , , j [dvdw j , * [ ( dw dw\ 70 .-. J ^ -^ dxdydz =J - - dxdydz + 1* (m ^-n^- y) dS. 72. Provided, therefore, the surface integral vanishes, i.e. d ^ either v = 0, or m^ -- ^ -=0, etc .................. (1), oz dy at the bounding surface, we have [dv dw , j ' [ dv dw 7 , , I -5- ^- dxdydz = / - ^-- dxdydz, J dz dy J oy dz .'. f2W dxdydz = J[(X + 2^) A 2 -f V (a),, 2 + ay* + /)] dxdydz ...... (2). This shows* that positive work is needed to bring the solid to the strained condition, typified by u, v, w, from its unstrained equilibrium. Therefore the position of unstrained equilibrium is stable. Thus, X -j- 2fj, may be zero consistently with stability. If we assume X 4- 2/i = 0, we get the labile-ether theory of Lord Kelvin, which involves the conditions (1), viz., at the bounding surface the displacement is either zero or connected by a special relation. The first may be satisfied, if the surface extends to infinity. If A = 0, or the medium is incompressible, while the above surface conditions are satisfied, we have, from (2), J2 W dxdydz = 4 //A (o^ 2 + &>/ -f a> 2 2 ) dxdydz. The medium is then said to be rotationally elastic. 73. Consider, now, the general expression 2 W = (X + 2yu) A 2 + 4/x (a> x * + to/ + a)/} If 8 is the operator of the calculus of variation, dw dv dv * Lord Kelvin, Phil. May. Vol. xxvi, p. 417 Il] THEORY OF ELASTICITY , 41 [\~fdw d&v dw dSv\ Sdv d&w dv dSw\~] . - J LU fc ~ fc ^ ) + U w - sy ^)\ dxdydz f\ f dw dw\ l\ \ n ^ -- f -5 I JL\ % ^/ dv dv 00 (/, w, ?i being direction-cosines of the normal at dS) = surface integrals only. 74. Applying the formula Sf(T- W)dtdxdydz = 0, and remembering that fT dxdydz = %fp (u* + v 2 + w*) dxdydz (T being the kinetic energy per unit volume of the [solid] medium and p the density), we get the equations of motion (under no body forces), viz. JpS [f(u* + tf + w 2 ) dtdxdydz] - o/H- (X + 2/x) A 2 + 2,a (w^ 2 + ft> y 2 + fc> 2 2 ) dtdxdydz] = 0, since the remaining terms of TT involve surface integrals only for a homogeneous and isotrbpic body. Again, ASJYt 2 dtdxdydz = [/li^^rfy^]^ jdtudxdydzbu ...... (1), and IBJ tfdtdxdydz = j dt &(jjL$u+ ... + ...} = fdt A (ISu + m while 3 fitf dtdxdydz = j dt co x B (^ - ^} dxdydz = I dtco x I Bw - Sv J dxdydz fdt x These are evidently equations of wave propagation. 75. Let A =f(lx + my -f nz - cj) ............. ..... (7), w x = F(lx + my + nz c. 2 t), etc ............. (8), restricting ourselves, for the present, to the consideration of plane waves f. * Bouth, Rigid Dynamics, Vol. n. f The equation (7) states that A has the same value f(d) over the two planes therefore, if p 1 , p 2 are the perpendiculars from the origin on these planes, ^i-P 2 = c i('i- f 2)> or the plane (l,m,n) travels parallel to itself with velocity ^ , every point of the plane being in the same state of dilatation. The plane Ix + my + nz = constant is thus the plane wave front, corresponding to the [dilatational] wave. Il] THEORY OF ELASTICITY 43 From (7), (8) we have, substituting in (5) and (6), We have thus two kinds of waves a pressured wave propagated with velocity A/-~ - and a distortional wave with velocity I - . 76. If either A = 0, or X + 2//, = (or vanishingly small), we have only one kind of wave. Now A = 0, when the medium is incompressible, i.e. when the coefficient of volume elasticity is large. Calling this quantity k, we have X = -- , so that the vanishing of A corresponds to a large value of X. In any case, if we can assume with MacCullagh fWdxdydz=ffy(a) x * + a)y* + tofidxdydz ...... (9), we get only one kind of wave. Stokes has pointed out* how- ever that this expression for energy makes X y - F^f, so that the equilibrium condition is not satisfied or the medium is unstable. This objection does not apply if only the total strain energy has the value given by (9) and the energy is not localized in each element of volume or if the ether is ' labile ' [Art. 96]. 77. The conditions at the bounding surface are different according as X + 2yu, = 0, or A = (in which latter case, X is large). In general, the conditions are (collecting surface integral * II. A. Report, 1862. t The expression for 5W without the implication X y = Y x is .: Xy = 5 , Y x - , etc. , 8%' dv x ' du dv if we write u v ==- , v x = -?- , etc. Hence from (9) we get X y - Y x . 44 OPTICAL THEORIES [CH. terms [Arts. 73 and 74] and equating the coefficients of Su, &v, &w each to zero) that i.e. X AJ + 2/u, f + m&> z - rc&jjJ , etc., du 7 9it 9% du where = l=- + m-~- + n-^- t dv ox oy 02 should be continuous. 78. In particular, at the plane x = 0, I = 1, m = 0, n = 0. du\ =r- + V continuous. Besides these, there will be continuity of displacements at the surface. We shall take two special cases : . 79. I. Let u 0, v = 0, and let the axis of z be the line of intersection of the plane of the incident wave with the reflecting surface. Let also dashed letters refer to the second medium. Then dw , dw' ' = '' (3). Note that (1) is identically satisfied, since w is independent of*. 80. II. Let w = 0, and u, v, independent of z. Then fdu dv\ du\ X ( -f x- ) + 2/i 5- I \d# 9y/ 9 3?; 9i*\ y continuous. II] REFLECTION AND REFRACTION 45 Remembering that the plane of polarisation is defined as the plane through the ray and the normal to the reflecting surface, when light is polarised by reflection, case II is that of light polarised perpendicularly to the plane of incidence. 81. There are, thus, in case II, four conditions to deter- mine two unknown quantities, viz., the intensities of reflected and refracted rays. There must, therefore, be two reflected and two refracted waves, even if the original wave was entirely transverse. But the energy of the incident wave is found* to be equal to the energy of the reflected and refracted (light or transverse) waves. Therefore no energy is absorbed by pressural waves, if they exist. 82. Green showed f moreover that the pressural wave will absorb very little energy, if - - is either very large or very A* small. He assumed it to be very large, and on this understand- ing, and on the further assumption that the rigidity is the same in different media, showed that (1) a difference of phase is introduced by reflection and refraction; (2) the intensity of the reflected wave never vanishes. As regards (1), the value of the difference of phase found does not agree with the result of experiment, while as regards (2), the residual effect shown by his theory is much too great. 83. Proceeding now to the particular cases in which A = or X + 2/-1 = 0, we observe that the conditions in case I are the same on either theory. In case II, if A = 0, the conditions are du fdv du continuous. v Green, Math. Papers. f Ibid. 46 OPTICAL THEORIES [CH. If X + 2^1 dv ^ 4. ^\ \ continuous, dx dy) but since v must be the same and y the same, yu, must be the same ; we have then the conditions dv du 3 dy\ continuous. v 84. Case I. Consider the solution w = A l e i p( lx+m y- vt) + A* e i p(~ lx+m y~ vt} , where I = cos i, I' = cos r m = sin i, m = sin r Now, since w w at x = 0, F F' jpF=/F', i.e.- = ~, where Z, Z' are the wave-lengths in the two media and pm =p'm, i.e. sin i V L Also, applying the conditions (2) and (3) of Art. 79, , a tan i a tan r while 1 // tan i + //, tan r ' A l IJL tan r /jf tan i + fj, tan r Again if E is the mean energy corresponding to w= A cos -i- ( Vt x), L* ' 11] REFLECTION AND REFRACTION 47 therefore, if E l} E 2 , E 3 be the energies of the incident, reflected, and refracted waves, _ L L L' Finally, if we take /JL = //, we get tan i tan r JLJL i) == -^J- 1 J tan i + tan r tt _ tan i + tan r 85. Case II. Light polarised perpendicularly to the plane of incidence. (Labile-ether theory.) Let v 1 = A 1 cos i e { v & cos * + * sin * ~ vt "> , (incident ray), v 2 = - A 2 cos i e* (-*<*'+ sin i-rt) f (reflected ray), v s = A s cosre^^ xcosr+ y s[nr - vt ^ (refracted ray), -. U^ V l U 2 ^2 . U S V S sin i ~ cos i ' sin i ~ cos i ' sin r cos r ' We have also #1 + ^2=^3, at x = 0. /. (A l A 2 ) cosi = A s cosr. This leads to a discontinuity of u. We shall see later that on the Electro-magnetic theory, electric displacement perpen- dicular to the surface is continuous, while along the surface the electric force alone is continuous. Continuity of u may be secured by introducing a pressural wave, propagated with zero du dv . velocity. Also, since ^- + ^- is continuous, dy dx sn Thus (A l - A 2 ) cos i = A 3 cos r, ( A l + A a ) sin r = A 3 sin i ; A l A 2 _ sin IT A l + A 2 ~~ sin 2i ' J.j tan (i + 7') whence r = ,- -p - c , A 2 tan (i - r) 2 cos i sin r A * = Al 48 OPTICAL THEORIES [CH. These are, also, Fresnel's formulae. On this theory, the rigidity is the same in all media, but the density different in different media. 86. Summarizing*, we find that we may hypothecate I. Null velocity of longitudinal waves (X + 2/j, = 0). II. Incompressibility (A = 0). For I, (1) The tangential components of displacement are continuous. (2) The tangential stresses are continuous while the normal stress is also continuous, if there is no loss of energy at reflection or refraction i.e. in the case of the labile-ether of Kelvin. This also yields Fresnel's result. For II, (1) The three displacements are continuous, giving three conditions. (2) Energy continuous, one condition. (3) Tangential stresses continuous, two conditions (which are redundant). SECTION II. THEORIES OF DOUBLE REFRACTION 87. The strain-energy function for a heterogeneous (aeleo- tropic) solid contains six linear terms and twenty-one quadratic terms in strains e xx , etc. If there are three planes of symmetry, then the terms involving the first power of shears cannot exist, since the form of this function should remain unchanged, if the signs of x, y, z are changed. Therefore, the linear function will consist of three terms, viz., P^xx + Qe y y + R^zz ( sav )> an d the quadratic function will contain 21 12 = 9 constants. We may therefore write 2W = 2 (Pe xx + Qe yy + Re zz ) + (A,B, C, F, G, H) (e xx> e yy , + Le yz * + Me zx > + Ne xy * Larmor, Dynamical Theory of Electric and Luminiferoiis Media. n] GREEN'S THEORY 49 If the body is originally unstressed, P = Q = R = ; there- fore, the equations of motion of the medium (of density p) will be 8 ( is an arbitrary function. (X, /JL, v) will then define the direction of displacement and I 2 + m 2 + ri* = 1, while where ^1 ^ 2 + JVm 2 + Jf ?? 2 = a, (H + JV) ?7iZ = /* , etc. a\ + hn + gv Hence ^- - = ... = ..., A/ showing that the directions of vibrations are along the principal axes of a certain quadric, the constants of which, however, depend on the direction of propagation itself. 89. If we assume, with Fresnel, that two of the vibrations are in the wave front, then, l\ + m/ji + nv = ............... , ..... (1). Thus, we have (al + km + gn) \ + (hi + bm +fri) p + . . . = 0. Restoring the values of a, 6, c, etc., in terms of A, B, C, we have- \l[Al* + (H+2N)m* + (G + 2M)n*]+ ... + ... =0 ...(2). M. O. T. 4 50 OFflCAL THEORIES [CH. And since (1) and (2) should be true for all values of X, //, v, we must have (where K is an undetermined constant), and two similar equa- tions. For this, it is necessary and sufficient that On this proviso, we have 2 W = K& + L (e y * - te m e xx ) + .... /: S/2 Wdxdydz = S/(#A 2 + ALaJ + 4>Ma v * + 4^o> 2 2 ) dxdydz. This leads to the equations 7 , 9 A cMtrdtov ~ IT d(o z pu = K +2M ^ - 2N -, etc., 00! 9s d i.e. /oA = /V 2 A (eliminating o^, w y , <0 Z ), T T-7o ^ I T 9ft)a; , , 9ft) u XT and ^.XVX-^|_ + ^ + ^ (eliminating A) [Art. 74]. 90. Putting now 20)^ = X^ (Z# + my + nz Vt), we have (p F 2 L) X' = - (Ll\ f -}-... + ...)/, etc., which, on account of the condition l\' -f m^ -f nv = *, yields This, of course, gives Fresnel's wave surface. We have, however, i.e. F 2 = a 2 X' 2 + 6> /2 + c 2 i/ /2 , if a, 6, c are the principal wave velocities; but, on Fresnel's hypothesis, F 2 = a^ 2 + &V 2 + c V [Art. 54], which shows that X', JJL, v, defining the axis of rotation on the present theory, coincide with X, ft, v, defining the direction of Fresnel's dis- placement. The axis is therefore perpendicular to the plane of polarisation ; that is, the direction of vibrations on Green's theory is in the plane of polarisation. * Since rotations and displacements are necessarily at right angles to each other, this also follows from the relation dw dv ^^-d^" from (1) of Art. 89. IT] MACCULLAGH'S THEORY 51 91. In order to obviate this difficulty, Green included the terms involving P, Q, R along with second order terms for e xx , etc. in the form du 1 r/d*A fa + 2 [UJ Green, further, supposed that for waves perpendicular to any two of the principal axes and propagated by vibrations in the direction of the third axis, the velocity of propagation is the same. This forced relation enabled him to obtain Fresnel's wave surface as well as the direction of vibration given by Fresnel's theory. Moreover, he took density to be the same in every direction but rigidity different in different directions. 92. In applying the theory to the case of crystalline re- fraction it is found necessary to suppose further forced relations between the rigidity in different media, for which there does not seem to be any justification. 93. MacCullagh's theory. We have seen [Art. 76] that the strain-energy function in the case of an isotropic solid can be written (if (X -f 2//,) A = 0) = 2/A (&V + ft)/ + &>/), since the other terms yield only surface integrals, when the variation of the function is taken (even if we do not assume the ether to be labile), and that MacCullagh assumed this to be the actual form of the function. Generalizing this, for a crystal, he assumed that W = 2 (Leo,? + Ma> y * + Na> g *), while the density was taken uniform and the same in all media. The equation of motion of the disturbed medium accordingly becomes or piu x = L V 2 &)2. L -_ x + M - + N - dx [_ ooc oy 'bz \ These equations of torsional vibrations are of course the same as Green's. [Art. 89.] 42 52 OPTICAL THEORIES [CH. On this theory, however, du dv dw _ ~ h 5~~ i "o == ^> CM? oy oz so that there is no compression of the medium involved, whether the ether is assumed to be compressible or not. Otherwise, we have the same difficulty as on Green's theory. 94. A hypothesis that the density is the same in all media gives correct results, but this leads to the conclusion that in an isotropic medium, there are two polarising angles [Rayleigh, loc. cit.]. 95. The general import of all this analysis may be thus summed up. Assuming that the ethereal medium is of the nature of an elastic solid, it is easy to calculate (as was done by Green) the effect of a small disturbance (in the nature of an elastic dis- placement). It is, then, found that two waves will be set up, a pressural wave and a torsional wave, the latter of which alone can constitute that of light. Green, accordingly, supposed the pressural wave to be propagated with very great (infinite) velocity. But such a supposition proves to be unacceptable, inasmuch as it is incapable of explaining the well-known phenomenon of polarising angles. In the next place, in explaining double refraction on the basis naturally of the theory of crystalline asleotropic elasticity with three planes of symmetry, he assumed that in a crystal, ethereal rigidity is different in different directions, but the (ethereal) density the same throughout. This led to the erroneous conclusion that there are two polarising angles. His theory, moreover, led to the further conclusion that vibrations of polarised light are parallel to the plane of polarisation, in contradistinction to Fresnel's, according to which the vibrations are perpendicular to this plane. It was, therefore, natural to inquire if the velocity of propa- gation of the pressural wave might not be zero consistently with the principle of energy, and Lord Kelvin showed that this is allowable, provided we postulate certain properties [Art. 72] of the ether at the bounding surface, which he called ' labile ' II] INTRINSIC ENERGY 53 (e.g. null velocity). MacCullagh had previously argued that, inasmuch as light disturbance was transversal, the strain- energy function for the ethereal medium, so far as its optical properties are concerned, must be a function (quadratic) of its molecular rotations only, and this yields the same result as Lord Kelvin's theory. 96. It follows that all the optical theories which have been at all successful in giving a fairly satisfactory account of the mechanism of light propagation depend on the assumption that the potential energy of light disturbance in a homogeneous medium is of the form (except for a constant factor) 1 1 1 ( (1% + my -f nz Vt\ etc., Px V*l' = (\ + p)l (ll f + mm' ?7l a ' i -jrr^ "~ i" 100. If the ether is labile, we may put X + %/JL 0, I 2 I 2 + m a -4- w 2 /2 TT> - r+... + ...=0, where = a 2 , etc., F 2 - a 2 p* which gives Fresnel's wave surface. If we put - X + yt pressible), we get If we put - =0 (that is, assume the ether to be incom- Comparing this with the equation of Fresnel's wave surface, + ... + ...-0 (2), r" - a 1 we notice that since in (1) V is equal to the perpendicular on the tangent plane and (I, m, ri) is the normal to the tangent plane, (2) is the pedal of (1). Fresnel's wave surface is therefore the pedal of the wave surface obtained on the theory of an incompressible ether supposed to be elastically isotropic but of seleo tropic inertia. 101. Again, we have, if X + 2^ = 0, ~ -l\l' = -l (ll f + mm + nri), etc (3). n] RAYLEIGH'S THEORY 57 Multiplying the equations in order by /, m, n and adding, we get II' mm nri Hence (multiplying the equations in order by , -^ , -, CL" C and adding) Therefore if we write L Ll'+. .. + ... = . = pr~ a* a 2 " 4 "'"" 1 "*" ...(5), we get F 2 = a 2 Z 2 4- showing that L, M, N define the direction of the displacement, on Fresnel's theory. From (3) if II' +... + ... = cos ^, therefore from (4) - J from (5). '* \- , + ..- + ... T, - - = sin 2 ^ = (Ll f + Mm' + Nn Therefore the inclination of /', m', ?i' to , m, w is the complement of the inclination of I', m' y n' to L, M, N ; that is : If OP is the wave normal, OR the direction of the ray, PR the direction of the displacement, on Fresnel's theory, PN is the direction of the displacement on the present theory, PN being perpendicular to OR. LMN Again, since ^ - l\ I' = - I (If + mm +nn\ Imn y we have ^ = ... = K (say) ...(6). (X i 58 OPTICAL THEORIES [CH. But for Fresnel's wave surface, if (x, y, z) is the point of contact of the tangent plane (I, m, n) with the surface, - -_-. i ^~ GL -m f- V / . -.x y I'x + my 4- nz = 0, a 2 # 2 since 4- . . . 4- . . . = 0, r 2 a 2 that is, (I', m, n') is perpendicular to the ray. Let be the centre of the wave surface, OP the normal, OR the ray, RP the tangent line and PN perpendicular to OR. Then by the triangle of velocities Hence, by projection, NP.L- Vl + . -=0 , (7); r r since OP=V,OR = r ('and, therefore, (W = ) ; also, Z, M, are the direction-cosines of PR. .-. from (7), NP.L = IV - V* - L M N that is, the direction of vibration is in the plane PON as on Fresnel's theory but not in the wave front. [Glazebrook, Phil. Mag. 1888.] 102. The theory is based on the hypothesis that the ether is elastically isotropic but that its optical density is different in different directions and different in different media. In attempting to analyse this hypothesis, we observe that the Il] BOUSS1NKSQ 59 equation of motion of the ether may, obviously, be written, on the theory of contractile (or labile) ether, in the form pu = fj$*u + X, where X is the force exerted by matter molecules. If p! is the density of matter and ( (7, V, W) its displace- ment, then the equation becomes pii = fjbV 2 u p l U, for X may obviously be assumed to be of the form while U, the displacement of the matter molecule may be taken to have the form given by U =/(). the nature of / depending on the nature of the crystal *. 103. Thus, if we take U = Au, etc. in an isotropic medium, we have to replace p by p 4- pi A, and the equation may be written (p + p l ^Mi = fj^^u. In the case of aeleotropy of inertia, if we write U=A l u, V = A z v, etc., we get the expression for the kinetic energy, viz., i S (pxU~ + Py& + pzW*) dx'dydz. Again in the case of media exhibiting rotatory polarisation, a form that will give the experimental result is U = Au -f B ( - TT- = ^4 2#s7 T) etc. V9s 9y/ In order to explain dispersion we may take the relation U=Au-DV*u, etc., and finally aberration may be supposed to be due to the form of X given by -a a 9 a -Ms+* S + *^ + *K' where i^, etc. are the velocities of the matter molecules. Or we may assume the form for X (in the case of isotropy) to be , 2 - n - J - * Boussinesq, ioc. cit. 60 OPTICAL THEORIES [CH. II and remembering that U must necessarily be small compared with u, if there is no absorption of energy we get the equation (p + n) u = /jiV z u, and admitting n to be different for different directions, we get the equations appropriate to seleotropy of inertia. For dispersion, if we adopt the view that it arises from absorption of energy, due to the near synchronism of the ether- period with the period of matter molecules, the force on the matter molecules may be taken to be of the form where a, 7 are constants. Thus, for the ether we have and for matter where p l is the density of the matter molecules actually taking up the vibrations * 104. The above mode of writing down differential equations in order to obtain a particular result might at first sight appear to recall Fresnel's mode of explaining double refraction; but it cannot be denied that it has what may be described as a dynamical substratum, a fuller view of which can only be obtained on a clearer apprehension of the intimate nature of the interaction between matter and ether, which these equations serve provisionally to represent. This fuller view, as we shall see, is partly supplied by the electron theory (Ch. iv]. * Glazebrook, loc. cit. CHAPTER III ELECTRO-MAGNETIC THEORY SECTION I. STATICAL AND DYNAMICAL ELECTRICITY 105. We have seen that an attempt to construct the properties of the ether from the properties of light, on the assumption that the ether behaves as an elastic solid when it takes part in the propagation of light, has been so far only partially successful. For, on any of the theories that have at all succeeded in giving an approximately satisfactory account of optical phenomena, the question is left entirely doubtful, as to how the ethereal density and the ethereal rigidity arise and what is the intimate nature of the forces that the inter- action of matter and ether brings into play. Some light may be thrown on these points if the electro- magnetic field should turn out to be, as we have ample justification for holding that it is, the ethereal medium of optical theory. " Although," to quote Maxwell, " to fill all space with a new medium, whenever any new phenomenon is to be explained, is by no means philosophical, if the study of two different branches has independently suggested the idea of a medium and if the properties which must be attributed to the medium in order to account for electro-magnetic phenomena are of the same kind as those which we attribute to the luminiferous medium, in order to account for the phenomena of light, the evidence for the physical existence of the medium will be considerably strengthened." 106. Now, the ethereal medium, for the purposes of the propagation of light, on the theory of undulation, must be a receptacle of energy. The first question that must therefore 62 OPTICAL THEORIES [CH. be considered is, Is the electro-magnetic medium also a receptacle of energy ? Faraday answered this question in the affirmative and postulated a state of strain in the electrostatic and electro- magnetic field, imaged by lines of electric and magnetic forces. To him, all electric and magnetic phenomena reduced themselves to changes in the number and distribution of lines of force, the existence of which were, according to him, effectively demonstrated by the phenomena of static and electro- magnetic induction. Maxwell studied these properties mathematically and in doing so showed that an electro- magnetic disturbance should be propagated with the velocity of light. This we proceed to consider. 107. The well-known gravitational law of force is expressed, as we know, as that of an action at a distance. And, of course, the law of electric action can also be expressed in the same way, viz., that the repulsion between two point charges e, e' oo is -T^ , where r is the distance between the charges and Kr 2 K the specific inductive capacity of the medium in which the action takes place. 108. Now from this known law of electric action in any medium and without any assumption as to the intimate nature of electric phenomena, it is easy to deduce that if a charge e is placed in any medium of specific inductive capacity K, V=ihe potential at any point of the medium, defined in the same way as gravitational potential, dS = an element of surface S at that point, dn an element of the normal, F n the electric intensity at that point, resolved normally to the surface, and dur = solid angle subtended at that point by the surface S, then 6 being the angle between r and the normal to dS. And if the surface completely encloses a charge e, Ill] STATIC AND DYNAMIC ELECTRICITY 63 Let NO and N be two numbers such that (by a choice of units) we can put 4<7re = N () , -, - = ^TV- , then / the integration being taken over S. The equation can be stated in words, thus : The normal induction over S = number of tubes of induction intercepted by S, provided we define (1) jKF n dS over S as the normal induction over S, (2) a unit tube to be the cone generated by lines of force issuing from the charge such that its solid angle = e~ l . When all the tubes are drawn round the point-charge, the field is obviously mapped out, so that electric action, due to the charge can be exhibited by means of these lines. 109. In order to exhibit the effects due to a charged conductor, which is known from well-known experiments to be an equipotential surface, we have to draw lines of force (normal to the charged conductor) along the periphery of an element dS of the surface of the conductor. Consider the tubular surface generated by these lines and limited by consecutive equipotential surfaces on either side of the conductor : The charge enclosed by the surface is adS, if a is the surface charge. Applying the theorem jKF n dS 4?re, we have (KF n \ d8 l + (KF n \ dS 2 = ^irvdS, (KF n \ dS l referring to the inner equipotential, and (KF n \ dS 2 to the outer. But (F n \ = since the potential inside a hollow conductor (which does not enclose a charge) is constant; /. (KF n \ = 47T0-, since dS 2 = dS, ultimately. Know (KF n ) 2 dS = N, then kirtrdS = N. That is, the charge on the element dS is proportional to the number of tubes of force issuing from the element. By drawing tubes of force, therefore, we can represent the electrical condition of the conductor. 64 OPTICAL THEORIES [ChL 110. If we fix our attention on the electrified conductors only, these tubes of force are to be regarded as serving merely to represent geometrically the state of the conductors. But since N depends on K, we are naturally led to conclude that electrical action must depend on the nature of the medium at any rate to some extent. These tubes issuing from a charged conductor must end somewhere. The locus of their extremities must be another conductor (or another part of the same conductor) which there- fore presents to the tubes an opposite aspect to that which is presented by the conductor or portion of the conductor from which they issue. The second conductor is said to be oppositely charged. Thus a charge of one kind means the existence of an equal and opposite charge. This is also a fundamental fact of experiment, viz., equal and opposite charges always subsist together; the charge from the conductor from which a tube issues being called (provisionally) positive, the charge of the conductor at which they end must be called negative. 111. Consider now a condenser consisting of two infinite parallel conducting plates at potentials F, and F 2 . If the medium is air, the number of tubes of force will be F F proportional to l -j -, where d is the distance between the plates. If it is a medium of specific inductive capacity K, this F F number is proportional to K - l = 2 ; that is, if K>\, there CL will be a greater aggregation of these tubes for the same electromotive intensity and a correspondingly greater accumu- lation of charge, positive and negative, on the two plates. 112. We may, therefore, describe the electric state of a condenser, as being due to a certain modification a certain mechanical condition or constraint of the medium, the con- ductors being boundaries of the medium, whose electric conditions define the boundary condition of the medium as in the theory of elasticity. Ill] ELECTROSTATIC ENERGY 65 113, From what we have seen, every case of electrifica- tion is that of a condenser, the electrified body forming one boundary, and the oppositely electrified body, the opposite boundary of the medium in which these are situated. As this view of electric phenomena can be justified on experimental grounds, it follows that (1) the electric energy is the energy due to this peculiar condition of the medium, and (2) the tubes of force are to be viewed as being related in some manner to this mechanical this strained condition, so that they must be regarded as having a physical and not merely a geometrical significance. 114. We know that along a tube of force spanning between surface charges &, a (dS being the elementary cross-section of a tube and F n = electric intensity along it) KF n dS = . where ^ is the electric intensity along a tube of force and (rfV dn an element of a line of force. If, now, dr is an element of volume of the medium, Zdr= =-lj where p is the volume density of free electricity as in Art. 121 and the electric property of the medium will be determined by the above equation, while its dielectric constant will depend on the quantity p. It will be observed that the electron theory is formally based on this hypothesis of Maxwell. 134. Further, if we describe a surface S round a charge e, then / KF n dS = 4w / - * V -x r F = R I or l = K = resistance 01 the circuit j H I = ionisation current, where I = e (q l n l + q 2 n 2 ) F 3 , e = charge on an ion, q } } q 2 = velocities of +, ions, under unit electric- intensity, n lt n z = number of ions per unit volume, F lt F z , F 3 being the electromotive intensities, in each case. 140. One point of difference between the displacement and conduction currents on the one hand and the convection Ill] IONISATION 75 current on the other is that the E.M.F. is not necessarily proportional to the current in the case of the latter, whereas it is necessarily so in the former. It will be seen, however, on examination that there is no real distinction between them. 141. To fix our ideas : Consider an electrical system consisting of a voltaic cell, the poles of which are connected to two conducting wires ending in two parallel plates with an air-gap. When the difference of potentials is greater than a certain amount*, ions (+, -) are thrown off in sufficient numbers to produce a steady current across the air-gap, and when the difference of potentials is great enough to produce sufficiently copious ionisation, the negative ions or corpuscles moving with sufficient velocity produce luminosity of the gas and a discharge passes. 142. Thus, the path of an electric discharge, silent or luminous, is the path of a current of electricity in the same sense in which we use the expression, "a current of electricity" in a closed conducting circuit joining the poles of a battery, and, therefore, since an electric discharge is obviously due to a movement of ions, an electric current must be a " procession and not an arrangement." In the voltaic cell itself, there is an actual procession going on of ordinary ponderable atoms which must be carriers of electricity. In the same way, a Leyden jar represents a continuous circuit for the flow of electricity. 143. We may thus modify, in a general manner, Maxwell's theory of electric displacement in view of the theory of ionisation in gases. As soon as the plates in Art. 136 are connected with the poles of the battery, by the action of the E.M.F. in the latter, a transference of electricity takes place in the wire charging one of the plates, say, positively, and the other negatively, and producing a difference of potential between the plates. Across the dielectric, on the other hand, an electric * Even before this even for the slightest potential'difference, it is likely that in the intervening air, there is a current or procession of ions, due to spontaneous ionisation. V6 OPTICAL THEORIES [CH. displacement and ionisation take place, the total amount of which is equivalent to the electricity transferred along the wire. The transfer or procession therefore takes place .in a closed circuit. When the plates are discharged by disconnecting them from the battery and connecting them up by a wire, the displacement and ionisation will be in the opposite direction. In either case, there is a current due to a procession of electricity, but this current is necessarily transient. When an E.M.F. continues to act after the current has once passed, a state is reached on account of the electric elasticity of the medium which is one of dynamical stability. The electric displacement thus seems to be associated with what is not of the nature of a static but of a kinetic phenomenon. 144. The state, in fact, is one of convective equilibrium. There is generation and re-combination of positive ions and corpuscles, so that there is always a current. This is small, unless the E.M.F. of the battery is sufficiently great, when ionisation may be sufficient for a strong and steady current, with or without luminosity. 145. On the whole, therefore, every case of electric current is that of procession of ions in electric or metallic conduction as well as in dielectric polarisation or gaseous convection; while every case of electrification must be held to consist in a passage of a current, instantaneous, transient or continuous. SECTION II. MAGNETISM 146. The phenomena of magnetism can also be explained by postulating the existence of stresses in the magnetic field, as in the case of electric phenomena. This we proceed to consider. 147. Since any permanent magnet can be split up into elementary magnets, whose axes are in the direction of magnetisation, the potential energy of a magnet in a field, dV whose potential at any point is V, is SM -=- , where M is the Ill] MAGNETISM 77 magnetic moment of an elementary magnet at any point, and ds = the length of the elementary magnet. And since M = Idv, I being the intensity of magnetisation and dv the volume of the elementary magnet, f dV The potential energy ( W) = I Idv -x r I J , ~ +/A-^- + I>-~-] dv, dx ^ dy if X, IJL, v define the direction of magnetisation, ~D /"f \xJ 9# 9v dz dv VdS-j Y(^ +... + ...} dv (1), where dS is an element of surface of the magnet, 6 = the inclination of the normal (I, in, n) at dS to the direction of magnetisation, while A, B, C are components of magnetisation at any point (x, y, z). Now, if we take T ft A dB dC\ /cos<9=<7, and __+_ + _ ) = P) we get If = Jo-F^ + /pFdi; ........................... (2) from (1). Or, the* energy may be conceived to be due to a surface distri- bution = /cos 0, and a volume distribution =: ( + ... + ... J \dx / of magnetism. 148. Suppose now that the potential V is due to the magnet itself to which <7, p refer. Then, since the law of force is the same as in the case of electric phenomena, V will satisfy the conditions ........................ (3), where dS 2 is an element of the surface of the magnet just outsideo-. 78 OPTICAL THEORIES [CH. From (3) we have if we write a = = \- 4nrA = a 4- $nrA , etc. ox Again, (4) may be rewritten f 9 Z dS - 4?r [ (Al + Bm + Cn) dS J on J = j ^~ - 4.7T/ cos 0\ d8 = . . .(6). If we now put 9F B n = H n + 47T/COS 6 = - ~ + 47T/COS (9 ...... (7), i.e. if ^T w = the magnetic force in the direction of the normal, we have (over any closed surface) the condition fB n dS=0 ........................ (8), or the total magnetic induction over any closed surface is 0, if we define B n dS to be the normal magnetic induction over dS, B n being the normal component of a, b, c, i.e. = al 4- bm + en. Again, (4) gives (attending to the signs with obvious notations) dS 1 ( l H^+4nrIcos0) = tH n dS ............ (9), or dS l . l B n = dS. 2 . 2 B n ......... (10), if the medium (2) separated by dS l from surface (1) has no magnetism, permanent or induced, in the neighbourhood of dS. 149. Now (7) is equivalent to the statement that the resultant of H and / is B, which is therefore the total resultant magnetic induction. Draw a small tube inside the magnet having its generating lines in the direction of resultant magnetic induction. Let us call this a tube of magnetic induction. Then, applying (8) to this tube, we get Writing this in the form B n dS = - BndS', we conclude that the normal induction along such a tube of magnetic induction drawn inside a surface is constant and Ill] MAGNETISM 79 is always directed in the same way, remembering the meaning of |^ in (6). dn Then (10) shows that the same is true for a tube drawn outside. 150. We conclude, therefore, that (a) a tube of induction forms a close circuit both inside and outside a magnetic body ; (6) the normal induction is constant throughout such a tube; (c) the tube of induction coincides with the tube of magnetic force outside a magnet in space, devoid of magnetism. Note that the lines of magnetic force are in opposite senses inside and outside the magnet. 151. If the medium surrounding the magnet is filled with a magnetic substance (all substances are more or less so), there will be induced magnetism on it, and the equation (10) will still be true ; only the magnetisation in the second medium, J. (say), being induced, it must be in the direction of H n : that is 2 1 = 2-"?l #2? where : 2 is constant (provided the magnetising force is small), the nature of which depends on the nature of the medium ; . \ 2 B n = *H n (1 + 47rx: 2 ) = yu, . z H n (say), where //, is a constant, if Ji n is small, but which in any case depends on the nature of the medium as well as on the magnitude of H and also in some cases on its direction (as in magne-crystallic phenomena). //, is the magnetic permeability. Along a tube of induction, therefore, we have B n dS = constant and for a magnetic medium, /j,H n dS constant. At the surface of separation of two magnetic media (/JL, jjf) pH n = p Hn. If one of the media is a permanent magnet, p. H n for that medium stands for 152. The properties of a tube of induction are therefore mathematically equivalent to the properties of a tube of electric induction and we conclude that the magnetic field may be mapped out metrically in the same way as an electric field. 80 OPTICAL THEORIES [CH. Again, the energy of a magnetic system due to the mutual action of its parts is ^mV, where m is the strength of a perma- nent magnetic pole. Hence, this energy = J2)m - ds = ^M , where ds is OS OS the length of an elementary magnet and 7lf=its magnetic moment. Therefore, if I I = strength of permanent magnetisation, the energy E= J//irfr( H\ dr being an elementary volume of a permanent magnet ; where A l ,B 1 ,C l are the resolved parts of I I and a, j3, 7 those of H. But if a, 6, c are the resolved parts of B, from Art. 148 a = a + 47T A = a + 4?r (^ + KOL) = /xa + 4?r J.j ; --jr- {(a- since |(aa + 6/3 + 07) rfr = - I fa ^ + ...+... J dxdyd for the surface integral reduces to terms at each element of a surface of separation and therefore necessarily vanishes, while at each point, we have the solenoidal condition |*- + ... + ...=0 [Art. 148]. Therefore, exactly as in explaining electrostatic energy, we may postulate a distribution of magnetic tubes of stress in explaining magnetic energy. SECTION III. ELECTRO-MAGNETISM 153. The experiments of (Ersted prove that an electric current produces a magnetic field. If, then, we admit that an electric current consists in a procession of charged corpuscles, since such a procession of corpuscles must carry lines of force Ill] ELECTRO-MAGNETISM 81 with them, we conclude that the motion of lines of force gives rise to or is associated with a magnetic field in which magnetic lines of induction are distributed according to the law already found. Further, since all cases of charge and discharge of electricity and, therefore, all electrical phenomena consist, on Maxwell's view, in motion of electricity in closed circuits, we may describe an electric field as follows : 154. In every part of an electric field, there is a procession of electric charges ; in a dielectric, it is transient (being, how- ever, still associated with ethereal motion, if we accept the view that all energy is kinetic); while the procession lasts, it gives rise to magnetic induction in every part of the field. The whole field, moreover, is filled with lines of electric induc- tion, which move with the motion of electricity. The field is under stress, imaged by electric and magnetic lines of induction. 155. Now, the magnetic potential at any point due to a magnetic shell is fl, where is the strength of a shell and II the solid angle, subtended by the shell, at the point. Experiments of Ampere and Weber show that a small circuit carrying a current behaves as a magnetic shell. The potential due to it will therefore be of the same form except for a constant factor. Since a circuit of finite dimensions may be conceived to be made up of elementary circuits, the potential due to a finite circuit must also be of the same form. But as the space surrounding a wire is a cyclic space, while the space surrounding a magnetic shell is a-cyclic, the potential due to a current must necessarily be a cyclic function, while the other potential is single-valued. This is verified by experiment. 156. Again, experiment shows that the force due to an infinite current i along the axis of z, at a distance r from the . . 2i current, is . r The corresponding potential will therefore be y 2i tan" 1 - + constant, x a cyclic function. M. o. T. 6 . 82 OPTICAL THEORIES [CH. Identifying this with a magnetic shell, coinciding with the infinite plane, of which the infinite current is the boundary, we conclude that the potential due to a circuit carrying a current i at any point, where O is the solid angle, subtended by the circuit at the point, and s, an integer. 157. It follows, moreover, that the mechanical force in the direction of x, between a circuit carrying a current of dN strength i and a magnetic system, is i ^- , where NfB n dS*, (jOO B n being due to the external magnetic system and dS an element of the shell with which the circuit is identified. On account of the solenoidal condition satisfied by magnetic induction, the surface integral can be replaced by a line integral [Art. 148], since induction depends only on the bounding line. We may, therefore, write fB n dS = j(Fdx + Gdy + Hdz), where B n = la + mb + nc. Hence it follows easily that a = ( -= -- "3* i ' e ^ c '' ^ O1 bviously dG\j , 1 , [[fdH dG\ 7 , "I TT U +... + ... dS = ^ -- -5- ) dydz +... + ... teJ J J L\ty dzJ l cy If, now, W is a function, such that the attraction P in the direction of x is -r (between two circuits of invariable form), then = dx dx where Mii' = W = if(F'dx +G'dy + H'dz), F', G', H' referring to i'. * Since the shell does not belong to the magnetic system, magnetic force and magnetic induction coincide. Moreover, the magnetic induction like the electric induction is given by the number of tubes of induction [Maxwell, Elect, and Mag. Art. 489]. Ill] ELECTRO-MAGNETISM 83 If u, v, w be the densities of a current i in three rectangular directions, defined by the relation idx udxdydz, etc., then W = j(F'u + G'v + H'w) dxdydz. 158. We have seen that a circuit carrying a current creates a magnetic field in its neighbourhood. If V the magnetic potential due to the current and O = the solid angle subtended by the circuit at any point, then F=ifl + 4rOT ..................... (1), i.e. j(adx + ftdy + ydz) = 4?n = ^-jrj(lu + mv -f nw) dS. . .(2), where a, /3, 7 are the resolved magnetic forces in the directions of x, y, z, the line integral being taken round a line of magnetic force, and the surface integral over a surface, of which the line is the periphery. Then, obviously we have as in Art. 157 .(3), by Stokes's theorem [Maxwell, Elect, and Mag. Art. 24], which shows further that so that the current (u, v, w) may be regarded as that of an incompressible fluid. 159. Referring to Art. 139, we observe that the current in general will consist of three parts : (1) polarisation current (/> 9> h)> ( 2 ) ohmic current (say) (p, q, r), (3) ionisation current. Ignoring the separate existence of (3), for the present, in accordance with Maxwell's theory, we get 160. Again, since the energy of a magnetic field is this must 'be the energy of such a field, if the field is due to currents, and, as such, should [Art. 142] be called electro-kinetic energy. 62 84 OPTICAL THEORIES [CH. This is ~- oTT rA , ,., ' [Art - lo7] - , from (3) of Art. 158, F, G, H being due to the total field [Art. 157]. The vector (F, G, H) is evidently of the nature of a momentum and may be called the electro-kinetic momentum, if we can regard (u, v, w) as a velocity. But this, as we have seen, we are justified in doing. This vector is called the vector- potential of magnetic induction. 161. Now ^ = ^- 8 |, etc., ; i : 87 8/8 and 4<7rw = ^-' - , etc. ; oy cz 8 /dF^dGdH\ .-. 4wttw ^ [ 5- + 5- -4- -5- ) * v f Jf. 8a? \dx oy 02 J Let F=F 1 + F 9 , such that V 2 ^ + 4>7r/jLU = 0, etc., dF.dG.dH, and '+*' Then -V^ 2 , + + 8j; V dx oy o dv which shows that F* = ^ . etc., cue where % is an arbitrary function ; therefore, all the equations will be satisfied if we take F^- dx'dy'dz', and F, = , etc. But F 2 does not enter into the expressions for a, b, c. Ill] ELECTRO-MAGNETISM 85 Moreover, Sf\ fa provided /JL is independent of the current (which, however, is not admissible), since ( X _ afj + (y- yj + ( Z - Z J = r ^ d 1 Ci n o t - = -~ , I'u + m'v' + nw = 0, and -, -- + . . . 4- . . . = 0. ox r ox r ox On the understanding, however, that fju is independent of the current, F 1 may be taken equal to F. Accordingly, idx NOTE. Observe that +... + ... =0 and ^ + ... + ... = ox ox correspond to the assumption A = of the elastic solid theory. 162. Now, the energy of an electric circuit is due not merely to the field created by other electric systems, but also to that created by itself. Hence, the energy of the circuit (supposed to be at rest) due to its own current is of the form Ni, where Ni = i B n dS = i f (la + mb + nc) dS Hdz) - t ((dxdx +... + . .. ~2J\ r remembering that each element dx or dx belonging to the same circuit occurs twice. The total electro-kinetic energy due to currents in more than one circuit is therefore T, where and f dXidx* -f ...+... f COS ft)io 7 ^ M } ,= -= - -ds^s*, etc. J r l2 j r l2 0)12 being the angle between the elements ds^ds.i of the circuits (1, 2) and r 12 the distance between them. 86 OPTICAL THEORIES [OH. 163. Consider now the flow of energy in such a circuit ; since the energy due to a current i must be equal to the sum of (1) the energy lost in frictional generation of heat, and (2) increase of electro-dynamic energy, we have where E is the impressed E.M.F. in the circuit, a- the resistance. If there is no impressed E.M.F. in the circuit, Ydy + Zdz\ if the electromotive intensity = (X, Y, Z\ .-. -~ kte+mb + nc)dS=f(Xd* + Y4y + Zdz) (Jut J J Hdz), where ty is an arbitrary function which is introduced to make the solution general. To interpret this, we notice that i.e. ijr is the potential of free electricity of density p. We have also g- = - a, etc. (by Stokes's theorem)... (1), dy dz the medium being stationary. Again, from Art. 159 we get X 47T v, U = 1- -TV- A , (7 J\ O ^7 and from Art. 158 47ryu,w=5 r- (2). dy dz Thus, for a conductor u = , K and for a dielectric u -r X (3). 4-7T Ill] ELECTRO-MAGNETISM 87 164. From (1), (2), (3) of Art. 163, if //, is constant, we get = V 2 X ; that is, in a medium (/*, K) the electric force is propagated with the velocity - . In the electro- magnetic system //, = 1, K = - 2 for air ; where v = the ratio of the electrostatic to the electro- magnetic units. Therefore, in air, the velocity of propagation of an electro- magnetic disturbance should be equal to v, while from direct experiment we find that the velocity of light is equal to this ratio. We are, therefore, justified in concluding that light is an electro-magnetic disturbance. 165. If n is equal to the index of refraction, we have ri 2 = jj,K; and as /j, is very nearly the same in all non-magnetic media, we deduce that n^ K. In testing the accuracy of the relation, we have to remember that n refers to infinitely short electro-magnetic vibrations which constitute light and therefore K should be determined in relation to such infinitely short waves. A comparison of calculated results with experimental data, therefore, cannot be instituted in the form in which theory requires it. In spite of this, n- for the longest light waves experimented upon has been found to agree with K for slowly varying fields in several instances, although it differs in others*, while a direct deter- mination of the velocity of propagation of electro-magnetic disturbance and that of K have also yielded results which are found to agree in many cases. 166. Moreover, as n depends on wave-length, while K appears as a constant (independent of wave-length) in the Air n for yellow light 1-000294 1-000295 Hydrogen C0 2 1-000138 1 -000449 1-000132 1-000473 NO 1-000346 1 -000345 H 2 Methyl alcohol 1-333 at 1-3379 at 9 5-7 88 OPTICAL THEORIES [CH. above investigation, it is obvious that the theory will require modification, in order that there may be agreement between the results of theory and experiment. Direct experiments in verification of the theory that light is an electro-magnetic disturbance were supplied by the researches of Hertz, who showed that an oscillatory electric discharge (the nature of which had been previously discussed by Lord Kelvin) induces varying electromotive forces in the field, which can be detected by means of a suitable circuit with a spark gap ; in other words, an electro-magnetic oscillation is propagated with the velocity of light. 167. In order to illustrate this, consider the case of the discharge of a Leyden jar. [W. Thomson, Phil. Mag. 4th series, v.] Let Q charge of the jar, i = the current, C = capacity, 72 = the resistance of the circuit in connection with it, L = the coefficient of self-induction. Then, E-iR=~ (Li) [Art. 1 63], while i= ST , from definition. dt Q = 0; therefore, if g>^, 73- the discharge is oscillatory and T = periodic time = . . /I # If f be negligible, then V CL 4# Such a system may serve as a source of electro-magnetic disturbance. 168. Generally*, a wire with a spark gap will serve the same purpose, if we arrange a variable electromotive force to work in the system. In that case, it may be called a vibrator. * Hertz, Ausbreitung der elrJitrischen Kraft; J.J.Thomson, Recent Researches. Ill] ELECTRIC OSCILLATION 89 A wire with a spark gap, if placed in a field of varying electromotive force, will exhibit oscillatory discharge if its natural period of oscillation coincides with the period of oscilla- tion of the source. Such an arrangement may therefore be called a resonator. If the source has a metallic (zinc) reflector behind it, the maximum effect is manifested, for the superposition of direct and reflected waves causes stationary waves, and at definite points in the region, intervening between the source and the detector, there are, as in the case of sound, loops and nodes. The electro-magnetic vibrations produce other effects also. Falling on loose iron filings, they make them cohere, so that if these filings form part of the circuit of an electric current, they allow the passage of the current through them, each time the vibrations impinge on them. An electric circuit completed through a tube containing loose iron filings and enclosing a galvanometer may also, therefore, effectively serve as a detector. With a suitable source and detector, it has been found possible to verify that optical and electro-magnetic media are identical. 169. In order that T should be infinitesimal, we must have G and L infinitely small quantities of the first order and R an infinitesimal of the second order, so that the electro- magnetic system whose oscillatory discharge will set up vibra- tions of light must be of infinitesimal dimensions. 170. From the equations of Art. 163, we can derive generally fj,K$ = V 2 (for a dielectric), where < stands for any of the quantities X, Y,Z, a,b,c, f,g,h, u,v,w. In particular, let us consider /= F 2 V 2 /, where KpV- = 1. If we have plane waves defined by f = f n sin (Ix + my +-nz Vt), etc., A. f)f then, since ~ + ... + ... = 0, ox If 4- mg + nh = 0, i.e. the electric displacement is in the wave front. 90 OPTICAL THEORIES [CH. And also since "dy dz K \dy t dz) 4?r ZTT . 7 , ZTT /7 -i rj \ = -j= . (mh ng Q ) sin - (Ix + my + nz V t), we have a = 4?r V (ng mh), etc., or la -f m@ + ny = 0. That is, the magnetic force (a, /3, 7) is also in the wave front. We have, therefore, in the wave front, both electric dis- placement and magnetic force, perpendicular to each other. There is, thus, in the wave front, two sets of effects at right angles to each other, one set in the plane of polarisation and the other perpendicular to it. The question as to which of them may be taken to correspond to the elastic displacement of the elastic solid theory cannot, therefore, be uniquely decided. 171. Let us now consider the flow of energy in the electro- magnetic field. The electrostatic energy W in a dielectric is where dr = element of volume. .-. 8 W = i /( Z8/+ YSg + Z8h) dr + $f(fiX + ... + ...) dr =f(XSf+ ... since KX = 4?r/, etc. The heat evolved by Ohm's law is H, where and (p, q, r) is the conduction current, so that H = j(Xp + Yq + Zr) dr, since vp = X. Finally, the electro-kinetic energy is jP, where Ill] FLUX OF ENERGY 91 SO that 1 / / 47T J 4>TT . provided we can take yu.cc = d. Therefore, the rate of increase of energy in a stationary element of volume is W+H+t. This ought to be expressible as a surface integral, if the energy resides in the medium. Now (the same system of units being used for all the quantities) W+ H + T = ~ dS {(my - n/3) X + ... + ... f r) since - --- -- - = - a, etc. [Poynting, Phil. Trans. 1884] 172. We thus find that the rate of increase of energy is equal to the normal component of a certain flux. The direction of this flow is evidently perpendicular to both the electric and the magnetic force, so that the direction of propagation of energy is perpendicular to th'ese directions. This direction is, evidently, the direction of a ray of light, if optical energy is the same as the energy of electro-magnetic disturbance. 173. It has been assumed in this investigation that the energy is localized in each element of volume. Now both electrostatic and electro-magnetic energies have been shown to be capable of being estimated on this postulate and so also, the ohmic energy of electric current. We conclude, therefore, that the above theory is consistent with the other parts of the electro-magnetic theory. 92 OPTICAL THEORIES [CH. SECTION IV. DIFFRACTION 174. Returning now to the equation of Art. 170, we know that if ? = c'Vt, we may put sinh (ctV) . 2, ^ being arbitrary functions of cc, y, z. Suppose, initially, $ = F, ?=/ ........................ (2), where F, f are given functions of a?, y, z. Then the solution, of the above equation may be put in the form sinh (ctV) r, f = cosh (c*V)/+ ~ J F ............ (3), where the arbitrary functions are replaced by known functions. For, obviously, T (putting t = in the solution). In order to interpret this symbolic solution, let (a, /3, 7) be a point (P) on a sphere of radius r, centre (oc, y, z), and consider the integral where dS is an element of surface of the sphere. On transformation, this can obviously be written so that r- 9 = V 2 . dz 2 If now we put z = r cos 0, dS = 27rr* sin 6 d&, the integral becomes f "" sinh T*V e rcos v sin ^ ^ = 47r?- 2 - - . rv Ill] DIFFRACTION 93 Putting now r = ct, and remembering that dS r z dl, where is the solid angle, subtended by dS at the centre, ^ sinh cV sinhctfV c v -- Writing now a = let, etc., we have the symbolic solution replaced by etc.) dfl + ~~ [< j/(* + let, etc.)l dO ...... (4). 175. Hence, to find the effect at (x, y, z) at the time t, due to an initial disturbance /, F as defined in (2), we describe a sphere of radius ct, with as centre, note the effects at all points over the sphere and take the sum as in the above expression. In other words, if U Q is the initial velocity, and f is the initial displacement in the direction of x of any point on a sphere of radius ct, having for origin, then the displacement at at any time t, in the direction of a?, can be briefly written from (4) which is Poisson's result. [Rayleigh's Sound, Vol. II.] 176. Thus, the expression for the disturbance, [f(t)] , at a point 0, due to an element subtending dl at 0, can be written from (5) . -^r- ^ + i^oK<-''x/w ...... (6), where U t > is the resultant velocity at time t' at the sphere centre and of radius c (t t') ; and the above construction shows that, if we put then dCl = cos nr, 94 OPTICAL THEORIES [CH. and Uf , = ^f( t -*:} cj I/A >:) on \ cj cos nr [cos 7i?^ = angle between r and the normal, drawn outwards]; ~U^ = -^\f(t-^\^ -W; which is Kirchhoff's solution. 177. In particular, if '^ is the distance of the centre of a diverging spherical wave from dS, and / j i rt\ -*"* I *' " v * 1 (at ao) = cos -^ l\ \ 7 in view of the fact that the disturbance (s) from a point-source has to satisfy the differential equation then it can be shown that at P (whose distance from dS is r) the disturbance is 2X /I / / Y _J_ f \ sin 2?r ( _ - ) (cos nr cos n?^) dS. [Drude.] TTi \J. A, / 178. Remembering that the direction of displacement must be always perpendicular to the direction of propagation, the above result will be correct only on the supposition that the direction of vibration is perpendicular to both r and i\. If this is not the case, we must resolve the displacement (which is necessarily perpendicular to i\) in the direction perpendicular to r (in the plane of r and the vibration). In doing so, we observe that if u f , v, w' be the components of the velocity perpendicular to r, and u, v, w those perpendicular to r lt then u' = u ql y etc., where q is the component of u, v, w along r. Ill] DIFFRACTION 95 In Poisson's solution, then, u must be changed into u q l, a corresponding change being effected in the second term. The modified equations will then be (neglecting terms in r~ 2 ) u - q l ,~ /af 7 dp \ dS = -a>S + ^- I*-*-] , etc ....... (9), r \dr drj r where p = l% + w?7 + w , and , m, ?i the direction-cosines of ?', f, 77, f being the required displacements due to f , T/O, f at c?/S^. Taking now the particular case of a displacement along the axis of z, viz. fit --- j (the direction of propagation being along the axis of #), we have, neglecting terms depending on r~' 2 , and remembering that the resolved part of the displacement perpen- dicular to r is f(t ') sin <, where

= -= nearly. On this understanding we can work out the case of oblique incidence at a narrow slit as follows : Let as be the distance of a small element dx from the centre of the slit. The disturbance reaching any point will be [since the difference of phase due to disturbance from the central /Vl element and that at a distance x is equal to - (sin i sin 0)] A, I i - 6} 1 cos i + cos (i - 0) . (I . . . \ . ft S\ = - . . --. ~ --sin - (sin i - sin 6) ) . sin 2?r (~ - ) , 4?r sin i - sm \\^ ' ) \T \) where 2?r (y T -- ) is the incident disturbance, and I is the breadth of the slit. Ill] ELECTRO-MAGNETIC THEORY 97 On either side of the zero position, the disturbance will therefore be cos i -f cos (i 0) . / 1 I - (sin i sin 0) J , sin i sin 6 which accounts for the want of symmetry observed by Mr C. V. Raman [Phil. Mag. Jan. 1909] when a diffraction band is produced at oblique incidence. SECTION V. REFLECTION AND REFRACTION 181. The equations of Maxwell are of the type : j ' r f- K X . (2) J 4-7T TO = crX C3) IL .. (4) OX dH dG \ x /> .(5) dy dz UbOL ru _fy_d$ ..m. Sy Moreover, the conditions to be satisfied at the surface of separation of two media are the continuity of tangential magnetic forces and of normal induction magnetic as well as electric. From (1), (6), (7), 4 7r ( +f\ = -( C -}- = -^ , and let a: = be the refracting surface. Then and J- must be continuous, at ^'= 0. /A 8a? 9 Let 6 X = ! sin ( Fit ^ 4- ??ii/), incident wave, A- ^> 2 = a 2 sin ( Vt my Ix), reflected wave, A< < 3 = a 3 sin - ( V't i'x + my), refracted wave. A/ The continuity of <^> yields a x + 2 = a ;5 and , = , = =-, , and 1 8 . , T cti a 2 I a s I' the continuity of ^- yields = . , , i 2 Ms tan r . A i.e. rs . . , as in Art. 84, o^! + <^ 2 /Xj tan % and a ;? = a (/A 2 tan r + /*i tan i) cos r 183. Case II. h = = a = 6. Here let <6 = - Then the surface conditions are the con- P tinuitv of and -^ ~ . And the results are obtained by K ox changing ^ into K in the above. Ill] ELECTRO-MAGNETIC THEORY 99 184. If we put /*! = //. 2 , Case I gives Fresnel's formulae for reflection and refraction for light polarised in the plane of incidence. Comparing this with the assumption made there, we conclude that the electric displacement is perpendicular to the plane of polarisation, if Fresriel's results are to be accepted and if /A 1 = / 2 . Similarly, Case II yields formulae which are also Fresnel's for light polarised perpendicularly to the plane of incidence. 185. It follows, therefore, since the plane of incidence is the plane of polarisation, that the light vector is perpendicular to the plane of polarisation, provided the electric force is identified with that vector, and ^i=.yLt 2 - If, on the other hand, the light vector is identified with magnetic force, it will be in the plane of polarisation. 186. On applying the theory to the case in which K and /u, are both variable, it is found that in order to account for the fact that the light scattered by small particles on which polarised light impinges does vanish in one direction per- pendicular to the original ray, we must make* either &K or A//, vanish, so that both cannot be variable. This corresponds to the conclusion that both inertia and rigidity of the ether, supposed to be an elastic medium, cannot vary. Electrical evidence is in favour of the variation of K. We have not equally certain experimental evidence as to the choice between rigidity and inertia. Both labile-ether theory and Green's theory necessitate the rion- variation of rigidity. 187. For metallic reflection, a- in the second metallic medium, and therefore in the second medium K is to be changed into When this is done, it is found that we obtain Cauchy's expression, but at the same time the experimental results obtained by Jamin cannot be reconciled with theory without supposing K 1 : K 2 , that is, the real part of the square of the complex refractive index negative. * Rayleigh, Phil. Mag. 1881, p. 89. 72 100 OPTICAL THEORIES [CH. Maxwell's theory, therefore, requires modification in order to account for metallic reflection. [J. J. Thomson, Recent Re- searches, Art. 356.] 188. The theory, however, is capable of fairly satisfactorily explaining most of the phenomena of optics. But the out- standing questions remain what is the intimate nature of the medium that is the seat of electro-magnetic phenomena and what is the intimate nature of electricity and magnetism, of electric displacement and electric and magnetic stress ? Optical effects are certainly due to periodic changes of some properties of a medium, which we call the ether. The direction of pro- pagation of this change is the direction of the flow of energy in the medium. But of the intimate nature of the mechanism by which this energy is produced or redistributed and propa- gated, we have no knowledge. Various attempts have been made to push the analysis of phenomena beyond this limit. It would be interesting to consider some of these. SECTION VI. CONCORDANCE BETWEEN THE ELECTRO- MAGNETIC AND ELASTIC SOLID THEORIES 189. It will be useful at this stage to compare the un- modified results of the electro-magnetic and elastic solid theories. If T is the kinetic energy of a strained elastic medium of density f + x , etc. are molecular rotations, and 8 the operator of the calculus of variation (n being the rigidity). On the other hand, the electro-magnetic energy in a medium of permeability p is (3), Ill] ELECTRO- MAGNETIC THEORY//, ; where a, 0, 7 are the components of magnetic force, while the electrostatic energy is ...(4), (f, g, h) being electric polarisation and K the specific inductive capacity. 190. On the mode of identification suggested by Larmor, we may, therefore, take (a, 0, 7) proportional to (f , r,, ) = -- (f , 77, f ) and (f,g,h) . (w^, o> y , o> 2 ), the constants being suitably adjusted so" that 27T/ fi), ,.,.,. 1 In which implies or the velocity ( = - j of electro-magnetic disturbance is equal to 4 / = that of the rotational wave in the elastic medium. V a- This also satisfies the usual relations in elasticity and electro-magnetism, since and and further implies the condition +...+...-0, a; w ), since +.._* 4. -S> _ p. da? d 9^ 10 "2 ; OPTICAL THEORIES [CH. 191. Further, the equation of small motion in an elastic medium (n, &) is 3A w z do)\ . (6), where K is the modulus of compression, and A the cubical dilatation. If A = 0, we have, putting o> 2 = 2wA, f = , and = F 2 , (7), which is the same equation as is derivable from the theorem - J* j (la + mb + nc) dS = J(Z^ + Ydy 4- ^cfo), i.e., the line-integral of E.M.F. is equal to the rate of decrease of lines of force embraced by a circuit [Maxwell, Vol. II, end of Chapter vni, note], and necessarily involves the supposition that 192. Now applying the condition 8f(T - W}dt= 0, we get from (1) and (2) & x = V*V*a x ........................ (8), where V is the velocity of propagation. Thus, we derive at once f= F 2 V 2 y, and two similar equations, in a medium denned by 3 / + ?? + ^ = o. da dy dz 193. If, however, the medium is defined, owing to the presence of electrons [Ch. iv], by + * + -<> ..................... O), where p is the volume density of electricity of polarisation ; and moreover, if Ill] ELECTRO-MAGNETIC THEORY 103 so that the total current is taken to consist of polarisation and convection (electronic) currents, (u, v, w) being the velocity of electrons, we must take w/ = 2?r (h + pw), etc ................ (11); while, under statical conditions (12). From (9) and (12) we get K^ + ^Trp = Q .................. (13), when the electrons are at rest; while from (9) and (11) doy x ' we have, since -r -- 1- ... 4- ... = 0, which is the equation of continuity of a fluid of density p, and is also the condition postulated in the electron theory of Lorentz. Now (7) gives, if we introduce the vector-potentials F, G, H, given as usual by dH dG ^ = 8^-aP etc - ^ a^ 47r/ i" _ al A dF' dG din -F- 5 1 = -F^-, where V 2 iir = = ^- + ^-+^ . dx K dx dy dz ] where co x is the molecular rotation in free ether. This may be interpreted by saying that the force tending to produce free rotational displacement of the ether is F^, etc.; while (12) shows that the force producing K 9 the total effect is proportional to /'+ ^~> etc. (a conclusion 4-7T dx otherwise arrived at by Larmor, loc. cit). 194. From (7) and (10) we have - dxdxdy Sz dy Sz 104 OPTICAL THEORIES [CH. while, also from (7) and (10) we get, since df Sg dh_ + + ~ Thus, on the whole, the motion in the general case depends on an equation of the form 195. In order to solve* we have only to find the particular integral. Proceeding in the usual way, = ~ |{cosh c (* - t')V}F'dt' = ~j{coshc(t-t f ) V}Fdt attending to the meaning of the operation cosh [c (t - 1') VF] [Art. 174], -F(t- ) dtt ~ , since c(t-1)~r [Art. 176], where dr is an element of volume, which is Lorentz's result. * Phil. Mag. Aug. 1914. Ill] ELECTRO-MAGNETIC THEORY 105 196. Consider, now, a medium for which K (k x , k y , k z ) depends on direction. W is then of the form 2?r I (k x ~\f 2 + V 1 9" + fe' 1 h *) dxdydz = 2^ ( k *~ l -&x 2 + k y ~ l iffy* + k z - 1 KZ} dxdydz, if we take 27r/= = - -i - , etc., 3y 82- since (, r y , r z ) x (_p, ^, r). Again, since (^, 6r, //") is defined by dH dG u = , in Hvaroavnamics, dy dz i -i 3^T 36r . T^,, , . while LLCL = ^ , in Electro-magnetism, 8y c<^ the analogy may be said to be complete. We observe further that these equations include the previous mode of identification [Art. 191], viz. vr x oc / and u oc f . 200. If we admit that the above analogy is not merely formal but has a physical basis, we are led to some such interpretation of electro-magnetic phenomena [Larmor, "On a Dynamical Theory of the Electric and Luminiferous Medium"]. A permanent magnetic element will be represented by a circuital cavity or channel in the (elastic) ether, along the surface of which there is a distribution of vorticity. It will, in short, be a vortex ring with a vacuum for its core. Atoms will be ordinary coreless vortices, the velocity of the primordial fluid constituting magnetic force. When a piece of Ill] ELECTRO-MAGNETIC THEORY matter is electrified, what happens is that an elastic rotational displacement is set up in the ether surrounding it, the absolute rotation at each point representing the electric displacement. A charged body in the field causes a rotational strain in the ether all around it. If the motion of the ether represents magnetic force, the fact that p is practically the same in all sensibly non- magnetic bodies as in vacuum must be taken to indicate that the ether flows with practically its full velocity in all such media, so that there is very little obstruction interposed on its motion by the presence of matter. 201. As to magneto-optic rotation, we may readily admit (with Larmor) that "the rotation of the plane of polarisation in a uniform magnetic field depends on the interaction of the uniform velocity of the ether which constitutes that field [on the above analogy] with the vibrational velocity of light disturbance. For light waves, the motion that is elastically effective is the spin (iz x , is y , iz^ = (/, g, h), and the varying part of the velocity of an element of volume containing rotational motion of the magnetic vortices is proportional to , n, f) = ^(f,i, ) (say), (%, *!> ?) being the displacement of the medium, and (cr , $>, *y ) the impressed magnetic field. This variation is caused by alteration of the vibrational velocity of a particle owing to its change of position, as it is carried along in the magnetic field, as in hydrodynamics. There may thus exist a term in the energy, resulting from this interaction, of the form ^ /+ ^a Wo) is obtained by writing for a, in (1), a -, etc. Hence the above expression (1) becomes Thence, as in Art. 210, we get the terms in the equation of motion appropriate to the Hall effect. Thus, we observe that magnetic rotation can be explained independently of the theory that magnetic force is a rotation of the luminiferous medium. 203. Again, the velocity of light is not sensibly altered by motion along a field of electric displacement. Therefore, we cannot connect electric displacement with any considerable body-velocity of the ether. But the same difficulty attaches to the present theory, for Professor Lodge has found that the velocity of light is unaffected by even a very strong field of magnetic fdrce. This leads to the conclusion, on either view, that ethereal density must be great. 204. In discussing the theory to which the above analogy leads, Larmor remarks as follows : " The foundation of the present view is the conception of a medium which has the properties of a perfectly incompressible fluid, as regards irrotational motion, but is at the same time endowed with an elasticity which allows it to be the seat of energy of strain and to propagate undulations of a trans- verse type. This idea of a fluid medium disposes of the well-known difficulties which pressed on all theories that Ill] ELECTRO-MAGNETIC THEORY 111 impose on the ether the quality of solidity. If a perfect fluid is a mathematical abstraction, the rotational elasticity with which the medium is endowed effectually prevents any slip or breach such as would be the point of failure of a simple fluid medium without some special quality to ensure continuity of motion. "The objection has been taken to the idea of identifying velocity in hydrodynamics with magnetic force in electric phenomena, that it would be impossible on such a theory for a body to acquire a charge of electricity. A cardinal feature in the electrical development of the present theory is, on the other hand, the conception of intrinsic rotational strain constituting electric charge (/= *r xt etc.), which can be asso- ciated with an atom or an electric conductor and which cannot be discharged without rupture of the continuity of the medium.... An atom is a vortex ring with associated intrinsic strain. The elastic effect of convection, through the medium of an atom thus charged, is equivalent to that of a twist round its line of movement. Such a twist is thus a physical element of an electric current." 205. It is noteworthy that we are able to derive the equations giving the fundamental assumption of the electron theory [Ch. iv] from the hypothesis of rotational strain [Art. 193]. SECTION VIII. MOLECULAR VORTEX THEORY 206. The consideration of the action of magnetism on polarised light would seem naturally to lead to the conclusion that, in a medium under the action of magnetic forces, something in the nature of an angular velocity, whose axis is in the direc- tion of the magnetic force, forms part of the phenomena, corre- sponding to that of molecular rotation of an elastic solid or of angular velocity of vortex motion in hydrodynamics. If the latter analogy is to be complete, the displacement of the medium during the propagation of light should produce a disturbance of the vortices. 112 OPTICAL THEORIES [CH, Pursuing this analogy, let us take w x , iffy, &Z (vortex spin) = magnetic force (a, /3, 7). dw dv Then we have 2vr x = -= -- =- , dy dz dH dG corresponding to IJ,CL = ..................... (1), and therefore F=%/JLU, etc ...................... (2), where u, v, w, F, G, H have their usual meaning in hydrodynamics and electro -m agne tis m ; i.e. u, v, w are the velocities of the fluid medium, and F, G, H the components of electro-kinetic mo- mentum ; therefore the electro-kinetic momentum = ^/j, (u, v, w) = the momentum of the primordial fluid, provided we admit that it, oc p, where p is the density of the fluid. This is the molecular vortex theory of Maxwell. dy a/3 , Moreover, since -^ - = 4"jrf, /JL being regarded as a constant. 207. If the primordial fluid is a viscous mass, the .equations of motion will be v dp 1,3 fdu dv div\ 7 _ pu = pX- - f + ^k U- + 5- + 5- + ^ u > etc - -( 4 ) dx 3 dx \dx dy dz ] (where p == the pressure and k = coefficient of viscosity). But since F %JJLU, etc., we have from (2), (3), (4) _ p, dx 3 p Now, (5) is of the form D -TT/x A if we write P --- / P (that is, the electric conductivity (C) / T. ) a^r M Z ^ 8^ 4 k d fdF and ^- = ~s~ ~o~^+o~ o~l5 *" a^c 2 2/3 a# 3 /o aa? va# Ill] ELECTRO-MAGNETIC THEORY 113 where . . das 2 Now since ^- + ... + ...= 0, and ^ = F 3-', etc., d* G eta? For a conductor, since ijr is evidently the potential of free O ET electricity, V^ = 0. Let, now, ^- +... + ... = J. Then o^/ (i/ -IT = 0, .'. J" = const. = X (say). But the equation of continuity gives ^ , i <> r i v/, ot ox But this is meaningless, since p is to be the density of the medium. Therefore \ must be = 0, or that is, the primordial fluid is incompressible. Further from (3) _/= -V 2 ^- 1 - V%, J 4-TTyu, STT i.e. the electric current = ^ L V 2 (the velocity of the fluid), O7T while the electric displacement = - V 2 (the displacement of OTT the medium). [Glazebrook, Phil. Mag., June, 1881.] 208. When C is very small, the viscous fluid becomes an elastic solid and the equations of motion with the body forces X, F, Zare pu = pX + A - + wV 2 f, etc. 3? 877 8f where A = ^ + x- + ^ o# dy dz and ^1, ?i are constants. M. o. T. 8 114 OPTICAL THEORIES [CH. The concordance of these equations with the electro- magnetic equations of Maxwell has 'already been noticed, with the condition between the constants, given by tJfifa = K = specific inductive capacity. 209. Consider, next, the magneto-optic energy. Assume that the kinetic energy of the medium under impressed magnetic forces contains a term of the form 2C (aw! + f}w. 2 + yw 3 ) (1), that is, a term of the form provided the surface integral vanishes. But if (, 77, f ) be the displacement of a fluid medium, and the fluid is incompressible, etc. Hence, the expression (1) becomes * T 9 /9" dvj\ n [ t> dt&x r /\ w a*. I zi ~~ ^ J ~^~ " ' ~*" ' " i , I f "^1^ +... + ..., from (2), where Gr a; = 'M; 1 , etc. [Maxwell's Elect, and Mag. Vol. IL] This suffices to explain magnetic action of light both Verdet's results as well as Hall's phenomena. But it requires that /A should vary as density, that is, postulates uniform density in contradiction to the previous theories. 210. Applying 8f(T- W)dt = 0, since B W = B J2n (n x * + OTy 2 + *r z 2 ) dxdydz, Ill], ELECTRO-MAGNETIC THEORY 115 X * ^ ~~-- p - -' since F = = and - / = - V'fc etc. Z 2 07T 3 ) 4- 4-7T (70^ - where ^r' = ^|r + "^ ( Wi + ... + . . .)^ giving the additional E.M.F. due to the impressed magnetic field. [Glazebrook, Phil Mag. June, 1881.] SECTION IX. MOVING MEDIA. ELECTRO-MAGNETIC THEORY OF ABERRATION 211. So far, we have been attempting to analyse the intimate nature of the electro-magnetic field on the basis of Maxwell's theory. Maxwell's theory, however, is not capable of giving an adequate account of phenomena which involve the interaction of matter and ether. Take aberration for example. In order to apply this theory, we must modify the equations to suit the case of a moving medium. 212. If a circuit or medium in which there is a flow of electricity is at rest, we have X = F 3-_ [Art. 163.] 82 116 OPTICAL THEORIES [CH. When the medium is in motion, we have to take account of this motion in the equation. Thus, if D expresses total increment j(Xdx + Ydy + Zdz) = ^ l(Fdx + Gdy + Hdz) D DL dL dL Let an element of circuit AB come to the B position A'B'. Now the surface integral over A'B'BA rB rB' ,'A' - I Jds- I Jds+ Jds, J A where L AS = the line integral from A to B and dL is the part of the increment of L due to motion. But dS = ABA'S' = Udt ds sin a, where AB = ds, BE' = Udt t Z BAA = Z a. dx dy dz I (la + mb -\- nc) dS = I dt u v w a b c ds = (dx, dy, dz), and U = (u y v, w). 2 f* Jds = = 2 I* Jds, J A J B for a complete circuit. Hence we have ultimately, dx dy dz since But dt -I and X = - F - ^- + (vc - bw), etc. ox 213. We are now in a position to discuss the phenomena of aberration on Maxwell's theory. The equations of Art. 181 will be modified as follows : d fc>F so\ a . dt (dy ~ + ~ Ill] ELECTRO-MAGNETIC THEORY 117 (v x , Vy, Vg) being the velocity of the medium, while we shall still have a /&- Putting ?y (the direction of propagation being along z), we get , /9 3\ 2 pW = W V 2 W - /Oj J. ( ^ + V 2 ^J W. If we, now, take w =f(z Vt\ we get = F, nearly, X = .' r . i.e. ir T7 ^-! | r \7 [/ I j, __ when X is the refractive index, for X 2 - 1 = fr^i X 2 ~ p + frA' 216. The same result follows from Fresnel's theory. For this, let [Potier, Journal de Physique, l re serie, t. v. p. 105] p = density of free ether. p = density of bound ether, i.e. ether dragged through free ether on account of motion of bodies through the ether. V = velocity in free ether of propagation of light waves. T = the periodic time. V velocity in a medium consisting of free and bound ether at rest. T' the corresponding periodic time. The elastic force (F) of restitution P u (~fr) as on Fresnel's theory of elastic ether = (p + p) u ( ^>- J , assuming that the elastic force in both cases is the same, u being the displacement of the ether during the propagation of light. ... (p + p) V' z = p F 2 , i.e. p 4- p = /oX-. Ill] ELECTRO-MAGNETIC THEORY 119 Suppose the condensed ether to move with the velocity v, in the direction of propagation of waves, V l the velocity of propagation in free ether (when the medium of free and bound ether is in motion) of vibrations of the same amplitude and the same wave-length (I) as the preceding, and V l v = velocity relative to condensed ether. Then T l = -==- , in free ether, Vi TI = v , in the medium, K v and F = pn + ,'- (p V? + p'(V l - )*]. .-. pV* = ( p + p ') V'^pVf + p(V,- V )\ If, now, e = velocity of drift of luminous waves, e = V l V, i.e. (p + p') F' 2 = p (W + e) 2 + p (V + e - v) 2 vp ^v(\*-l) ~P + P f3S x 2 as in Art. 215. This agrees with experiment. 217. It is, a priori, evident therefore that a theory of moving electrons is likely to give a better account of the phenomena than the electro- magnetic theory in its simple form. 218. Before we proceed to consider this, however, it will be desirable to discuss certain modifications of the formulae proposed by Hertz, and see whether these will serve our purpose better. Let us proceed, in the first place, to find y- I (la + mb + nc) dS=-j- I R n dS, say, ' when the medium of which S forms part is in motion or the total rate of variation fjr-j in the number of tubes of induc- tion. It will, of course, consist of two parts : (1) [ (Id + mb + nc) dS = j ~ff dS, say. (2) Rate of change L- I depending on the velocity of the tubes. 120 OPTICAL THEORIES [CH. Suppose the tubes across dS spread over dS' on account of this velocity ; then the terms depending on the velocity = I (Rn'dS' - RndS). Consider the volume enclosed by dS and dS f . The total surface integral over this surface is fda t db dc\ , , , U- + 5~ + 5- dxdydz. \dx dy dzj But this must be * = R n 'dS' + R n dS + surface integral over the tubular surface generated by the contour of dS. Now, if ds = an element of contour of dS (at x, y, 2), U = velocity of a tube, so that Udt = length of the line joining corresponding points of dS and dS' t and e = inclination of ds to the direction of the dis- placement of a tube ; we have ds Udt sin e = area of the portion of the tubular surface generated by ds. Then, the surface integral over the tubular surface = I R n ds dt U sin e dx. dy dz \ a b c dt, V X Vy V z where v x , v y , v z are the component velocities of a tube corre- sponding to the velocity U, = j(Ldx + Mdy + Ndz) dt (say), and L, M, N (bv z cvy), (cv x av z ), (av y bv x ). But I (Ldx + Mdy + Ndz) = | dS I / (|^- + ... 4- ... Hence - (+ ... + ... UdtdS cos ... . .. dy dzj * Walker, Outlines of the Theory of Electromaynetism. Ill] ELECTRO-MAGNETIC THEORY 121 where /t = angle between the direction of motion of a tube and the normal to dS, and, moreover, U cos fj, = lv x + mv y + nv z . Thus, we have (lv x + mv y + nv z ) U^ + ... + ...}dSdt . '. YT. \ (la + mb + 'nc) dS = I (Id + mb + nc) dS + (lv x + mv y + nv z ) f r- + ... + . . . J f ^(.idN dM\ ) + dfl-U - -_- ) + ... + .-4 ............ (!) J ( \ 3y 8* / J 219. Thus, we get da \ fiN dM\ - - -- -- since - ^ ] (la + 7716 + nc) dS = I (Xdx + Ydy + Zdz). The equations given by Hertz are d + Vx ^ + ... + \ + (_\D^ dt \dx J \ty dz J Dt Similarly, etc .......... (4) where L' = K (v z Y - v y Z), etc. 220. Again, since we have - + = 4?r (p +/), etc, 122 OPTICAL THEORIES [CH. When the medium is in motion / should be changed into -y , . ' . the total current will be p + ~ -Lst J-Jt , fif \ fiN" 3M"\ = P +/+ % (i x +...+...) + ( Ty - - d -) , etc., where L" = (gv z v y h), etc. c>N" dM" Thus the total current will be the sum of (1) Conduction current (p), (2) Displacement current (/), (3) Convection current (v x p) [Art. 227], (4) Current which may be identified with that dis- covered by Rontgen*. 221. Let us proceed now to apply the above equations to aberration. Let us suppose the wave to be propagated along x with velocity v x , so that a = 0, X = 0, and let us further suppose 7 = 0, F=0 (since the magnetic and electric forces should be at right angles to each other). Then, putting /JL = 1 , 0*(~7'*), and Z=$(x-V't\ since p = - F>', Z = - F'f , a/3 ., a? ., 8^=*' 8* = < ^ Substituting in the equation (2) [Art. 219], we get where (v x , v y , v z ) is the velocity of the medium If Q V = = v z , we have - V'ty' + Vafr' =' . J/ap. May, 1888. Ill] ELECTRO-MAGNETIC THEORY 12S Therefore, from the equation 3-<*,r7M ,/*<*?>,. Poincare, Electricite et Optique. where T^ = ( ^" + ^r~) ^Trr, from Art. 220, Dt \ dy dxJ we get k or V =-T^+V X . Thus, the modified equation still fails to give the correct result*. CHAPTER IV ELECTRON THEORY SECTION I. THEORETICAL 222. From the investigations of the last chapter, we conclude that the electro-magnetic theory, as expressed by the equations of Maxwell and Hertz, cannot account for aberration, dispersion and allied phenomena. In analysing the reason for this, we note that the theory is based on the following postulates : (1) The energy of the electro-magnetic field is that of the dielectric medium alone, arising from a certain strained condition of the medium. (2) The conductors having static charges serve only to limit the dielectric region, so that no part of the energy resides on it. (3) The strained condition of a dielectric is due to electric displacement or polarisation (/, g, h), subject to the condition This displacement is apparently held to involve motion of the ether in the medium, subject to a property akin to elasticity (due to interaction of matter and ether) defined by the so-called specific inductive capacity of the medium which thus appears as a constant of the medium. (4) Conduction as well as convection currents (as in the case of an electrical discharge) involve a transference ( ; 'a procession and not an arrangement" as Faraday put it) dependent on a certain property of the conductor (called its CH. IV] ELECTRON THEORY 125 conductivity). This transference is that of electric charge, but what this charge is whether it is material or ethereal is not further specified or is rather left entirely open*. 223. If we limit ourselves to two physical entities, matter and ether, the electric charge whose motion constitutes electric current must be regarded as a mode of manifestation of the ether. If, on the other hand, we agree to regard a unit of electric charge as an actual physical entity, distinct from matter and ether (but related to them and partaking the nature of both in a manner that will require further investi- gation), we are able to give an account of the various phe- nomena which are left unexplained on the above postulates. 224. We are led to this additional postulate of a unit of electric charge as a physical entity not merely on theoretical grounds, but as a result of direct experiment. For during electrolysis, each monovalent atom is known to carry with it to the anode a determined quantity of what has hitherto been called a negative electric charge which can be measured and which is independent of the nature of the transported atom. If we assume that this also is the unit charge, which takes part in electric conduction or convection, and if we call it an 'electron,' we must conceive a monovalent material atom showing no electrical properties as the result of combination of a single electron with what may fittingly be called one or more units of ' positive electricity.' This would amount then to the statement that the electrical properties of bodies (as well as those of a dielectric medium) are due to the presence of ' electrons ' associated with atoms of matter, forming systems of various complexity, and such a theory is found to be consistent with observed facts. * Maxwell's pronouncement on this point, indeed, clearly sets forth the position : Electricity and Magnetism, Vol. i. ' ' It appears to me that while we derive great advantage from the recognition of the many analogies between electric current and current of material fluid, we must carefully avoid making any assumption, not warranted by experimental evidence, and that there is as yet no experimental evidence to show whether the electric current is really a current of a material substance or a double current, or whether its velocity is great or small as measured in feet per second." 126 OPTICAL THEORIES [CH. 225. Maxwell's remarks [Art. 222 (4) note] had reference to the negative results of the experiments whereby he proposed to detect in a direct manner the inertia of an electric charge. Experimental determination of the mass of an electron as well as its velocity under certain conditions has supplied the data sought by Maxwell on which to build up the further development of the electro-magnetic theory. 226. We have thus justification for regarding the pheno- menon of electric conduction in gases as due to ' ionisation ' or generation of charged particles or ' ions ' which are carriers of electricity. 227. An electric current would then, on this view, consist of two parts one due to electric displacement which we may still regard as ethereal and the other due to motion of electrons ; or if u is the total current, in the direction of x, we shall have (ignoring Rontgen current) u=f+pd; ........................... (2), where p is the volume density of (free) electrification whose transfer or procession gives rise to convection or conduction currents, and x its velocity in the direction of x, while 228. Now, when there is free electrification, we must have This must therefore be the further condition satisfied by/ g, h, while in the case of conduction there must, in addition, be a viscous decay which has to be taken account of in a suitable dissipation function. 229. When we proceed to interpret these equations, coupled with the observed phenomena of metallic conduction and dielectric polarisation, in terms of the electron theory, we are naturally led to the conclusion that : 230. A dielectric medium must be conceived to have electrons interspersed in it, giving rise to a constrained (ethereal) motion in the medium defined by the above relation. IV] ELECTRON THEORY 127 This also necessarily imposes a constraint on the motion of electrons. On the other hand, motion of electrons in a con- ductor must be free (subject however to certain dissipation of energy), so that the static charge in them always resides on the surface. 231. A conception of this kind naturally suggests that the dielectric property of a medium may be explained as arising from the constrained motion imposed on it. We proceed to show that this is possible and that in this way we can explain dispersion and allied phenomena. 232. On comparing the equations (1) and (3), we readily see that these cannot be, obviously, satisfied at the same time if /, g, h are to have the same meaning in both. 233. Now if p = 0, that is in free ether, y+? + ?* = o. dx 'by dz Thus, we may say that in free ether that in a material medium a/ % m_ ^4-^-4-5 = p, ox oy 02 while in all cases df^ dg, dh. 'ox + d + fa where / , g Q , h are defined by the equation (4) and are equal to /, g, h, when the medium is free ether. 234. In order to specify these quantities further, we observe that from the equations and dx dy dz df dg dh _ \V6 may write dx dy dz f n - f+ A etc. . ...(5), dA dB dC _ (6) ox . oy dz 128 OPTICAL THEORIES [CH. This expression suggests that A, B, C are components of a quantity / which corresponds to the coefficient of magneti- sation on the usual theory of magnetism, so that, just as dM is magnetic moment of an elementary magnet = Idr (where dr is an element of volume), similarly, we may take dM = electric moment of an element of volume containing electric charges arising from the presence of electrons and where of course I = (A,B, C) ........................ (7). 235. Again, from (5), we get And if we agree that f = u= total current (polarisation and convection), we conclude that A=px ........................... (9), (x, y, z) being the velocity of electrons, as before, and it is easily seen that (7) and (9) are consistent, on the understanding that p(x #o) = A. (say) or e (x # ) = Adr, e being the unvarying charge in volume dr, and x X Q = the displacement of e. in the direction of x. Comparing the equations (3), (4), (5) and (6), we conclude that total polarisation in any medium may be regarded as made up of two parts one involving ethereal (/, g, h) and the other corpuscular (A, B, C) displacement. Again, comparing (8) and (3), we observe that the total current is to be regarded as similarly made up and that it is the total current and total polarisation that are subject to the solenoidal condition. From (6) since, in stationary media, we get the equation of continuity for electrons in motion, viz. IV] ELECTRON THEORY 129 Again, from (6), we can obviously derive A = X , etc. ox where \ is defined by X V 2 < -f p = 0. The equation states that < is the potential of a distribution p, provided -x _ Lfe -47/ ' where & is the specific inductive capacity of the medium. Since the medium here is the ethereal medium (whose property is modified by the presence of electrons), & is an absolute constant defining the property of the free ethereal medium. Thus =/ + . )e t, ...(10), in the neighbourhood of an electric charge e. 236. On analysing the electro-magnetic theory of mag- netism, we find that : (1) Maxwell takes magnetic force, though invariably associated with and definitely related to electricity in motion, as a separate physical entity ; (2) and distinguishes between magnetic force and mag- netic induction by means of a quantity ^ which defines the magnetic property of the medium and is taken to be practically constant, though not specifically so. But inasmuch as, on experimental grounds, //,, as thus defined, depends on the magnetic force itself, one can hardly regard such a course justi- fiable. Hertz takes it as a variable in his equations, but one finds it difficult to follow the consequences of such a hypothesis. It will be seen, however, that an immediate simplification results, as soon as the intimate relation between electricity and magnetism is recognized, and an attempt made to describe the effect of one in terms of the other. The first attempt in this direction was eminently fruitful in giving a more coherent view of the electro-magnetic field than would otherwise have been possible. This was to regard an elementary magnet as M. o. T. 9 130 OPTICAL THEORIES [CH. identical with a molecular electric current and a magnetic shell, the same as an electric circuit coinciding with its periphery, being distinguished from the latter only in this that while the first is a simply-connected region, the second is doubly- connected. The next step in simplification would obviously be not to consider magnetism as a separate entity at all, but to regard it as a condition of the medium, in which electric action is taking place defined by the equations of Maxwell, involving magnetic force. Since an electric current, i.e. electrons in motion, gives rise to a magnetic field, magnetism would, on this view, correspond to a motion of the ethereal medium, due to the motion of electric charges. Finally, the optical medium being admitted to be the same as the electro- magnetic field, optical phenomena must be held to be associated with the (ethereal) motion which constitutes magnetism. 237. On this understanding, the so-called electrostatic energy of the ether will be W l} where while the potential energy of the msrterial medium will be W, W here W=*J (ff + fff + hf) ch- ............... (12), while the kinetic energy (T) of a material medium is to be taken equal to (13). 238. From (10) we observe, what is a priori evident, that the total force producing the strained condition of the medium is made up of the force associated with the free ethereal motion and that due to electronic disturbance. From the expressions for W and T, (12) and (13), applying S I (T W) cr = 0, where 8 is the operator of the calculus of variation, and remembering that IV] i ELECTRON THEORY 131 we get /=FV/ , ^=-lJ... ............ (15). But % f,=f+A=f+px ..................... (16), ^ ...... (17), Also from Helmholtz's principle we get - ;g f(to + & + we) dS = \(Xda> + Fcfy + Zdz) . . .(19), where (a, 6, c) = magnetic induction, (X, Y, Z) = electric force. Hence we have, if X=^f [Art. 235], (on the understanding that JJL = 1), so that F is the velocity of propagation in free ether. From (14), (16) and (20), we get in stationary media The equations (18) and (22) are the fundamental equations of Lorentz's theory. 239. Let, now, 2\ = the kinetic energy of an electron (e), TV = the kinetic energy of the ether, F 2 = the potential energy of the ether, F!= the potential energy of extraneous forces. 92 132 OPTICAL THEORIES [CH. Then, since j ($h - jg) dr = momentum of the field (JJL = 1) in the direction of x [ J. J. Thomson, Recent Researches], [Art. 2481 x, y, z defining the position of an electron. Applying Lagrange's equations of motion for the system consisting of the ether and the electron, viz. d /32\ 8TA _ dt \dx dx dx dx d Ti\ dV, dT 2 d . 9F 2 r since - = 0, obviously; -?- ...... (23), since -j3 = -a.ndw = h + pz, etc. [Art. 227]. KQ \(jZ UXj Integrating by parts the first integral of (23) and sub- stituting d{3 = ?r ~ r) ' etc ' j in the second integral and again integrating by parts, we get d/dK\ + w^far f (+ty + M} dT __ (24), which may be stated in words as follows : An electron in motion is subjected to a force which is the sum of an electrostatic force and an electrodynamic force, as the result of the action of the field. [Poincar6, filectricite et Optique] IV] ELECTRON THEORY 133 240. Again, T 2 being the kinetic energy of free ether, we have w here u=f+ A, etc., while W = the potential energy = ^ f ( f* + g* + 7i 2 ) dr, where (it should be remembered) k = ^- ( F being the velocity V o of light in free ether) and /, g, h are connected by the equation of condition V + d 2 + = dx dy dz Introducing (in the manner of Larmor) an undetermined multiplier $ and writing we have ......... (25), since $ is not to vary. Now integrating by parts into surface (S) integrals and volume integrals, we get Now, from Lagrange's equation (since /, g, h may be taken as generalized coordinates) we have _ 2 _ 2 dt a/ "a/ a/ = *+T'*fi-' (26 >< ft'O U = = t ; . c v ( c _ v y . * . S = total energy-density = S l + S 2 = 1 (c + v) (c- v)' 2 ~~ c v ' c 2 - v* when v is small compared with c. [Larmor, Encyc. Brit. ' Radiation.'] 246. The system of momenta of Art. 243 is found to be " proportional to amounts of energy transferred in unit time across unit planes at right angles to the axes of x, y, z on Poynting's theory of the transfer of energy in the electro- magnetic field." Hence the direction of momentum* coincides with the direction of flow of energy on this theory. * Note that we get the same system of momenta on the electron theory. 138 OPTICAL THEORIES [CH. Again, since v i 47T/ we have JT = ^6 z> v c = ^i , etc., A if electromotive intensity (X, Y, Z) is due to the motion of ~\ rp Faraday tubes only, so that ^ = X, etc. And since b we have / = fiK [f(v x * + V + V) - tfe (/^ + gv y + H) ) , etc. Therefore multiplying in order by v x , v y , v z , we have fox + #?V + hv z = 0, and therefore ^ 2 + v y 2 + v/ = ^ = F 2 , where V is the velocity of light in the medium. 247. That is, the tubes move at right angles to the direction of polarisation and also at right angles to the direction of the magnetic force with the velocity of light. The energy of the electro-magnetic field is seen, therefore, on this view to be entirely kinetic. 248. We have seen that electrons are not hypothetical entities. The phenomena of electrolysis furnish evidence of the atomic nature of electricity, while the phenomena of ionisation prove the existence of electrified units. We have, therefore, to take account of them, apart from any simplification that their introduction brings about in our theoretical discussion : this simplification is, however, immediately noticeable, for, on this theory, magnetism is explained simply as the condition of the medium, due to the motion of electrons [Art. 236]. 249. On Lorentz's theory, (a) there is no magnetism, (6) Amperean current is current due to motion of electrons, (c) in a conductor there is free motion of electrons, (d) in a dielectric, electrons cannot move far away from the position of equilibrium. IV] ELECTRON THEORY 139 On the electron theory, if u,v,w = the total current, where pf = current of convection, f , rj, f being the velocity of an electron, and +&+*.;, a* ay a* ^' Then t/, v, w will satisfy the relation a + a j , + a, = ..................... 9# 9^ dz for the left-hand expression is the same as ' or + + = dt + *dx + ~ Dt' since ^ + . . . + . . . = 0, dx if the dilatation of the particles is zero ; but we may take if we suppose electrons to undergo no change in time, in respect of their charges. Thus, the relation (27) is satisfied. Again, V 2 F= - ^n [Art. 161], since //, is to be taken equal to unity ; dF dG dH Idu dv or oy F dG dH [I + ^- 4- -v - = - ^c d?/ 9^ J r \9^ oy This is the further relation to which the electron theory necessarily leads. 250. A suitable hypothesis in terms of moving electrons may be made to fit in with the Amperean theory of magnetism. For this, we need only conceive magnetic molecules as consisting of doublets of electrons circulating rapidly in the Amperean channels, and in order to account for the neutralization of their attraction we must conceive them revolving*. If e = ionic charge, n number of ions per unit length, * Larmor, loc. cit. 140 OPTICAL THEORIES [CH. A = the area embraced by the channels, / = intensity of mag- netisation, I = length of the orbit, q = velocity of an ion, then I A I = magnetic moment = current x A = qn A e x A ; qne " I = ~T A ' But from electro-chemical data en 10 3 , and from molecular dimensions -j = ^-10 ~ 8 . I Thus an intensity of magnetisation equal to 1700 C.G.S. units will mean a velocity of about 3 x 10 8 , that is about T J^ of the velocity of light. Measurement of the velocities of cathode rays yields the same order of quantities *. 251. A magnetic molecule therefore will be, on this view, a vortex ring, the core of the vortex being made up of revolving doublets of electrons, while these may be regarded, as Larmor has suggested, as electric centres or nuclei of radial rotational strain. 252. A conception of electrons is thus seen to be capable of helping us at least to make a mental picture of the processes connected with optical and electric phenomena. We now proceed to show that it enables us to explain all those pheno- mena which the electro-magnetic theory of Maxwell and Hertz fails to explain. SECTION II. THEORY OF ELECTRIC DISCHARGE 253. Admitting that a moving electric charge will produce a magnetic field, if we further suppose that an electric current is in all cases a procession of electrons and positive electricity, we are able to construct a theory of electric discharge and to explain in a fairly simple manner the electro-dynamic action between two currents. 254. Taking one special case in which an electric discharge is passed through a De la Rive's tube at different pressures, it is found * : That there are several stages of the discharge : at a high pressure, the discharge is in the form of a shower or spray, consisting of an infinite number of rays. These, gradually, as * Phil. Mag. Oct. 1908. IV] ELECTRON THEORY 141 the pressure is diminished form into a single ban.d or stream ; as the pressure is further reduced, the band broadens and ulti- mately fills the whole tube as a glow-discharge. 255. Now, an electric discharge is a procession of corpuscles, shot off from the negative electrode under the influence of the electric field and ions, positive and negative, produced by colli- sion of these corpuscles with the molecules of the enclosed gas. The ions and the corpuscles exert electric force on one another, and as they are in motion, they exert magnetic force as well. Moreover, these masses moving through the fluid medium must exert an additional apparent force on one an- other, besides experiencing a viscous retardation in the direction of motion. The resolved parts of these forces along a line of discharge affect motion and collision along this line, and therefore need not be considered further. 256. Consider now any two (parallel) streams. The re- sultant action due to electric and magnetic forces between two charges e, e, moving with equal velocities g, was calculated by J. J. Thomson [Phil. Mag. April 1881], and shown to be op f / f& \ a repulsion of magnitude -^- f 1 - o^S/ where K is the s.l.c. of the medium, r the distance between the charges, and V the velocity of light (the moving charges being assumed to be spherical). Suppose, now, there are n positive and N negative charges in the first stream and n, N' in the second (per unit length of each stream), and let q be the velocity of a positive ion, and q, that of a corpuscle. Then, the mutual repulsion between the elements ds, ds' will be (if e, e be the charges of the particles in the two streams) 142 OPTICAL THEORIES [CH. In the case of a discharge, since all the streams are similar, n ri y N N', and e = e. Therefore the repulsion e*dsds' 2 _ 3^ J the ions and corpuscles in any one stream being a self- equilibrating system. There will certainly be repulsion, therefore, when n is large or small compared with N, for in this case the expression within square brackets becomes = 0) .................. (2), JV 2 l-3(n = 0) .................. (3), while the resultant action will certainly be an attraction of n = N. On this view, we may evidently argue that in the case of a wire carrying current) n = N. 257. We have already found the resultant action due to electric and magnetic forces. We have next to consider the effect of masses moving through the fluid, supposed perfect. If p = density of the fluid, a the radius of a positive ion supposed spherical, and r the distance between them ; then it can be shown that the repulsion between them, when moving with velocities q, is a 6 ppsr- If, then, there are n positive and N negative charges in either stream per unit length, and if b is the radius of a negative sphere and q, its velocity, the repulsion per unit length of either stream will be jf' 2 - 2nNa s b*qq] t That is, the effect is always a repulsion, which decreases as the pressure of the gaseous medium decreases. The resultant IV] ELECTRON THEORY 143 repulsion per unit length of either stream will, therefore, be equal to the sum of (1) and (4). On taking account of these two sets of forces, it is easy to see that the observed phenomena must be due to repulsion and attraction between various streams of ions issuing from the various points of the cathode. While, of course, the resultant effect must depend on the relative values of N, n, q, q', the simplest supposition that we can make is that whenever these streams carry mostly one kind of electricity, they will spread all round the tube, but when the streams contain an equal number of both kinds of electrified particles, and the pressure is not too high, the various streams will be attracted together and form a single stream. Although we cannot, in the present state of our knowledge, trace in detail the change in the value of the above expression, it seems to be possible to follow the course of events in a general manner. Stated broadly, there will be repulsion be- tween consecutive streams, both when the inside pressure is high, and also when it is low. We have thus three main stages of the discharge*. 258. It is only during the second stage that there is a steady rotation on the application of the magnetic field [in a De la Rive tube], while during the third stage the effect is that of a twist [Appendix IV]. We have already seen that the variation in the number of corpuscles in the discharge at any moment must depend on the number of collisions and recombinations per second. Thus [Con- duction of Electricity through Gases (2nd edition), Art. 229] let \ = mean free path, X = electric field, e = charge on an ion. Then, the mean kinetic energy of an ion = X\e. Now, if X\e > a certain limiting value P, ionisation takes place. Let f(X\e) = fraction of collisions that result in ionisation, i.e. f(x) = 0, when x < P. * Recent experiments have shown that all vacuum tubes exhibit these phenomena. 144 . OPTICAL THEORIES [CH. Also, let q = average velocity of translation of corpuscles, N no. of negative ions per c.c., then ~ = no. of collisions per second, A, and no. of ions produced = ~f(Xe\). A Let 7 . ~Y- no. of ions that disappear through recom- bination. Then -% [f(Xe\) 7] = the resultant no. of ions produced, neglecting those due to positive ions. Therefore, the equation of continuity is dN dNu Nu where u = average velocity of translation due to electric field = q nearly. Now in order that there should be steady rotation of the discharge, as a whole, we must have dN dNu 8T =0 ' ar= and ' /(^>-7=o. But this is also the condition that the result of collision should be annulled by recombination. We conclude, therefore, that throughout the stage that this condition holds, n and N will continue to remain equal, if they are so, as we have argued [Art. 257] that they should be, at the beginning. The quantity 7 would obviously depend on the nature of the gas. The same is true of the -function f. 7 may also depend on the conditions of the experiment for example, whether the discharge is intermittent or continuous. It may also depend on the pressure. We shall presently see what experimental evidence we have on these points. 259. When the tube contains air, the curve connecting pres- sure and potential difference between the electrodes consists of four portions. The first portion, which is very nearly straight, IV] ELECTRON THEORY 145 corresponds to the 'spray' discharge. As the 'spray' discharge changes into the 'band' or single stream, there is a bend in the curve towards the line of pressure. The next portion is also a straight line. Gradually, the curve bends away from the line of pressure, meeting asymptotically the line of potential difference [Appendix II]. 260. We have just seen that / ( Xe\) 7 = during the band or rotatory stage, that is fi J 7 = 0, since X oc - , p being the pressure. If now 7 is constant, we have / 1 -- J = constant, so that the curve connecting p and potential difference will, in this case, be a straight line, if the spark-length is constant (as is known to be the case, both on Paschen's law and experimental grounds). If 7 is not constant, the curve will not be a straight line. This may explain why, in the case of complex gases and vapours, the curve is not a straight line. / ~V \ 261. If f(Xe\) 7 is negative, so that /( ) 7 = a, say, this will correspond to the (first) stage, during which corpuscles are decreasing as the result of collision and recom- bination and tending to equality with positive ions [Art. 257]. Assuming a to be constant and less than 7, we deduce that the curve connecting p and potential' difference corresponding to this stage will also be a straight line, inclined at a greater angle to the line of potential difference, than the line corresponding to the rotatory stage [Art. 260]. This would explain the first bend in the curve [Art. 259]. In the third stage of the discharge, the form of the curve is consistent with the supposition that collisions are then few, and 7 a rapidly decreases with pressure [Appendix III]. 262. It has been found from experiment that the rotatory stage is entirely absent under certain conditions depending on the tube and coil. This is obviously due to the fact that for such a discharge / ( Xe\} 7 is never zero, or what is the same thing, a steady state of ionisation throughout the length of the discharge tube cannot be established under the conditions of the experiment. M. o. T. 10 146 OPTICAL THEORIES [CH. This is especially the case with hydrogen, and the peculiar behaviour of hydrogen in a vacuum tube, generally, would seem to be connected with this property. That the intermittent nature of the discharge from an induction coil materially determines the value off(Xe\) 7 is proved by the fact that when the discharge is passed from storage cells into a tube (of small spark length), the first stage that of spray discharge is absent. 263. It follows from experiments that during the rotatory discharge, the product of pressure and angular velocity is a con- stant, which is proportional to the density of the gas operated on. In order to explain this peculiarity we observe that in the case of steady rotation, the moment of the electro-magnetic couple on the current due to the magnet is equal to the moment of the retarding forces. The first is proportional to the current i pi (say), the second = J (^i + A 2 ) nur ds, where n = no. of +, ions per unit length, r = distance of an ion from the axis and u the velocity, ds = an element of length of the discharge, A l} AZ = the retardations of +, - ions per unit velocity. Also, if q, q be the velocities per unit electric intensity of the ions along the line of discharge, and X the electric intensity, then for the steady state P P Xe = qXA lt i.e. A l = - ; similarly, A z - -, . Therefore * pi = cone - + - r *ds [Appendix IV]. But Xe (q + q')n = i', *** - ........................... (6) - Now, since (p being the pressure of the gas in the discharge tube) 1 1 1 p oc - ; . ' . p oc - , and also oc ; IV] ELECTRON THEORY 147 But fr 2 ds is found to be practically constant when the distance between the electrodes remains unchanged, V Therefore oc pa*. P But / so long as there is rotation. Therefore f(epco) = 7, i.e., pc* = , say. If 7 is constant, as is practically the case for air, N 2 O, etc., pa> = constant. If 7 is not constant but increases with pressure, pco will increase with pressure also. Comparing this with experimental results, we observe that S is proportional to molecular weight in either case. Again, since V = f X ds, where V is the potential difference between the electrodes, if we write p = a V + b (for a simple gas during the rotatory stage), a and 6 will be constants depending on the induction coil and the nature of the gas. Substituting in (7), we have (assuming X to be constant along the discharge) V 1 PCO OC f-Tj- rj "* pjr 2 ds .fds Therefore, for any given discharge tube, b is obviously small, since pw = constant nearly, as we have just seen, fr*ds and fds being taken to be constant (as we are justified in doing). We notice, moreover, that since experiment yields the result that pa is proportional to the E.M.F. of the induction coil, a must be proportional to this quantity also. Finally, if the distance between the electrodes is changed, we have, taking fr 2 ds oc fds (= I), very nearly, where I is the distance between the electrodes, pcol z = constant, very nearly. This is also verified by experiment. 102 148 OPTICAL THEORIES [CH. SECTION III. THEORIES OF DISPERSION AND ABERRATION 264. Returning to the result of Art. 239, we observe that - = the force due to the system of electrons in volume fdr, ex = the force on the electrons other than that arising from the action of the medium -f- the force brought into play on account of the displacement of electrons. If we take a small sphere of volume fdr, the system of , , , dA dB dC electrons denned by -= \- - \- = p may be replaced by a surface distribution / cos 6 per unit area of spherical surface. And thus, as in the corresponding magnetic theory, the first part 4 A of the force in the direction of x will be ^ TT j- per unit charge. O KQ For the second we may, obviously, assume an elastic force in the direction of x due to displacement of the electron, and this will be -/*(0-ff )(say) = -/a-[Art. 235] (1), while the frictional force will be of the form bA. Hence, for equilibrium, Putting //, = j /i , jAdr = A, fpfdr = ef, etc., we have ^ A = #(/+$!) + ^ (fy -T&) (3), or for simplicity /i, A = e 2 (/+ i^l ) + -^- e* (yy z(3) (4), all the quantities having their mean values taken over a small sphere enclosing a charge. For motion, we have px = A , or fxpdr = A (5), where for A t we take its mean value, as before. IV] ELECTRON THEORY 149 Accordingly, since for a single electron of mass m, charge e and self-inductance L, T = i(m + e 2 )( 2 + 2 + i 2 ) (6), we have, neglecting the viscous term, J Mo 4 + (L# + m) A^f (f + y) + 2 (yy - iff) = ^^(f+f) + e(yB-80) ...(7). Writing (8), Le z + m = A, 4>7T 4?r e 2 47T . we get, if the magnetic field is weak, A +p 2 A = oo/, etc (9) and /+ A - F 2 V 2 /= F 2 ^ from (18) [Art. 238]. . .(10). From (9) we deduce, since dA dB dC -^ f- o H ^r~ ~~ P d^c dy d^ 3/" 3gr 3/i, and also ^- + ~ + = p, 3^ dy dz P \Po ' "O/ I v I ,'. p =p a cos //v ^^ ' - x ' ' 265. *If ^- = 0, or po is independent of coordinates, the 000 equations will be A+pTiCAL THEORIES [CH. and the solutions are which yield where n is the index of refraction and = the periodic time P of vibrations of electrons unaffected by the field, while 2 refers to the several groups of electrons that are set vibrating on account of the impressed disturbance. 266. Returning to equation (9), and introducing a viscous term b Q A, we get A + b Q A + p 2 A = oo/, etc ............. (14). Now, assuming the solutions n ^-t we get n 2 = 1 + 2 showing that there is absorption in this case. If PQ is very nearly equal to p (confining ourselves to a single electron), 6 cannot be neglected ; this further indicates that there is always absorption under these circumstances. If p is very small, then jo 2 - ipb ' and the real part of n is fi I j + j where a = 2ft 2 c. This shows that as p increases, n diminishes, and so explains anomalous dispersion. IV] ELECTRON THEORY 151 267. It is not without interest to compare the above with the various elastic solid theories that have been proposed for the explanation of dispersion. 268. For this, let us recall the fact that in an elastic medium there is, associated with an elastic displacement, molecular rotation, and if the properties of the medium are to be capable of being expressed in terms of quantities that enter into the statement of either theory, electric displacement and magnetic force must correspond in some way with the velocity of vibration and molecular rotation. Now in the electrical theory we have two quantities, ft and K, defining the property of the medium as well as the quantities f y g, h (polarisation) and H (magnetic force), while in the theory of elasticity we have the constants 0, 7 = magnetic force, o) x , coy, (o z = molecular rotation. 270. From (15) we derive, provided - = constant, Cf 27r/o a = fl), 51 [cf. Art. 191]. And this leads to the conclusion that the resultant twist is made up of an ethereal motion in addition to an electronic displace- ment, neither of which is however of the nature of a pure rotation by itself. We are now in a position to consider the equations of motion that have been proposed* to explain dispersion on a modified elastic solid theory. Boussinesq's formula is mu + MU = | n] t^ + n 3 / 3# where m is the mass, u the velocity of the ether, M, U those of ' matter ' and K volume-elasticity, and u = f . 271. From these we may easily derive the corresponding rotational equations, viz. where a) x , etc. are the curls of f, 77, f and fl^ corresponds to 1& X . Boussinesq's theory is thus seen to be capable of being inter- preted as based on the postulate of twists, defining the disturbed state of the medium. * Glazebrook, B. A. Report, 1885. IV] ELECTRON THEORY 153 For, if we write tl x =f+A =/ , we get, putting = 7 V (16), provided V*(Q X -A) = ..................... (17). Now VU = 1 V= - = 0, from (11) and Art. 241. Further, the equation of equilibrium of a material medium regarded as an elastic body would be V 2 !^ = 0, etc., so that (17) amounts to the statement that in forming the equation of motion, we must regard the material medium to be at rest. Again the material displacement is assumed by Boussinesq to be a function (/) of the ethereal displacement (f , 77, f), and in particular, for dispersion, /() is taken by him (18), (JUG where X, C, D are constants. Hence, on our notation, & x = \a> x + DV*a) x , etc (19) or A=f+Dy (20), where 2/= a) x (1 + X) + l x ( - 1 V If we admit that /is a harmonic function, the equation (20) can obviously be written in the form A .+ p A = a f (21), the constants being suitably adjusted. 272. The equations of Helmholtz with the same notation as in Art. 270 are u) \ MU = -&(U-u)-tfU-y*U\ whence V 2 a> - a> - O x - x ......... ( 154 OPTICAL THEORIES [CH. On transformation, for purposes of comparison with the electron theory, i.e., putting to x = \f+vA, l x = \'f+v'A ............ (24), we get m(X/+ vA) = tfV* (X/+ vA) + 0* {(\ - X')/+0 - v)A] ...(25) and M(\'f+ v'A) = 2 [(V - \)f+(v - v - a 2 (X/ + i/4) - 7 2 The equation (26) is the same as the equation (21) provided X' = 0, while from (25) we get (if X' = 0) an equation of the form , kf+ A = a'*V 2 /+ &*f+ y*A. Remembering that / and A must vary as cos_p (say), we can obviously adjust the constants and variables so as to put the above equation in the form which is the equation (16). [A and / differing in value from the same quantities occurring in (25) and (26), each, by a constant factor.] Ketteler's equations are of the form MCu -f M U= -tfU- p U, etc, which yield as before MC'H X = a?t' 2 cD x , + Mti x = - a*fi x - &il xt etc., which are the same equations as (25) and (26), if we put m<* x + MC'l x =f+A and MG< X + Mfl x =X/+ vA. On eliminating/ and putting Xm MC = 0, these yield /+ A = a' 2 V 2 / and KA +/= ot^A + a z *A. If f varies as cospt, this equation is of the same form as (22). IV] ELECTRON THEORY 155 273. Inasmuch, however, as dA dB dc * 5 h ~ I- D~ ~ P ox dy oz df dg dh and ^- 4- ^ + ~- = p, dx dy dz- (A, B, C) and (/, g, h) cannot be interpreted as rotations, each by itself, and to that extent these theories are less general than the electron theory. The main difference however consists only in the fact that the elastic solid theories deal with ether and matter, while the electron theory replaces matter by electrons. 274. In attempting a comparison between the elastic solid theory and the electron theory, we have identified the electro- static energy with the energy of (rotational) strain of the elastic medium, and the electro-magnetic energy with its kinetic energy. But the identification is not unique, as we do not know which of the two expressions (in either system) is kinetic or which potential, or in fact whether both the energies are not (as they most likely are) kinetic. We may, therefore, if we like*, regard electric force as identical with the rate of elastic displacement and magnetic induction with molecular rotation. On this scheme, the electro-magnetic energy is to be identified with the energy of strain provided we neglect surface integrals and take which yields the solenoidal condition for magnetic induction, viz., da db dc _ dx dy dz * Glazebrook's Address as President of the Physical Section, B.A., 1893. . 156 OPTICAL THEORIES [CH. since -^ + -^ + -^ ox oy oz I provided n varies as J . When we proceed to identify the kinetic energy - l(u 2 + v 2 + iv*) dr of the elastic medium with the electrostatic energy, we observe that if the elastic medium is ethereal, the equation of condition should be 2 ) dr = ~ J(/* + f + /i 2 ) dr ; while if the medium is a material medium, we must take dr = ^T J(/ 2 + # 2 + V) dr. These yield the following results : , = = - , where FO is the velocity of light in the ether, n V o = - and T^ the velocity of light in a medium (or, n or Also ,. V where A is the dilatation of the ethereal medium, while i.e., the total displacement (ethereal and electronic) is sole- noidal, while the volume density of electricity is proportional to ethereal expansion. 275. The various equations appropriate to an elastic medium are then found to have their exact analogues in the electron theory. Thus the equation 4 IV] ELECTRON THEORY 157 yields the electrostatic equation for a material medium, viz., on properly choosing the signs. The equation of motion of such a medium, viz., I = F 2 V 2 f, yields f = F 2 V*f or /o=FV/ , i.e, /+ j.' = F 2 V 2 (/+.4) = F 2 A 2 / if V 2 A = ; while for the free ethereal medium we have /= VfVf. Again, Boussinesq's equation, viz. , - = F 2 , " yields, if we put The other equation of condition derivable from Boussinesq's assumed relation, viz., gives A = V/+ and, similarly, for the equations of Helmholtz and Ketteler. 276. A third method of identification will be to take n (co x , (*y, u z ) = (a, & 7), o- ( , ^o, So) = ( /", g, h) and o- (f, ^, ?) = (/ , ^o, ho), da d0 fa as in the electron theory, 7rp = 0. This is allowable, as Maxwell shows (Treatise, Chap, v, Art. ee 103, Vol. I), on the ordinary law of electric action without APPENDIX I 173 any special hypothesis as to the intimate nature of electric density in relation to the ether regarded as a material medium [Art. 120]. The solution obtained by Maxwell, however, being only a particular solution, fails to satisfy all the conditions of the problem. APPENDIX II Nearly all the salient points with regard to discharges in vacuum tubes of De la Rive pattern, containing air can be derived from the curves (fig. 1, I and II). In these, pressures in millimetres of mercury are ordinates, and the correspond- ing potential-differences between the electrodes in electrostatic units are abscissae. 80 1 70 60 50 40 30 20 10 Fig. 1. I and II. 10 20 30 40 50 60 70 80 90 Diff. of potential between electrodes in electrostatic units. I. E.M.F. of discharge due to 3 cells in the primary of the induction coil. II. B.M.F. of discharge due to 4 cells in the primary of the induction coil. 174 OPTICAL THEORIES These curves were obtained by measuring the lengths of sparks between two brass spheres of 3 cm. diameter in parallel with the discharge tube and deducing the corresponding potential-difference from the table given at p. 461 of J. J. Thomson's Conduction of Electricity through Gases, 2nd ed., by interpolation. It will be seen that we may roughly distinguish four portions in each curve, AB, BC, CD, DE. The first portion (AB), which is very nearly straight, corresponds to the 'spray' discharge. As the pressure decreases the character of the discharge changes; it forms into a band by the confluence of most of the discrete streams. This is indicated by the bend in the curve at B. After this, the relation between pressure and potential- difference is given by a straight line BC. Throughout this stage,' the discharge is in the form of a band of light which rotates according to the law pco = constant. Gradually the curve bends away from the straight line BC, and at this point (somewhere about C) it appears that the above law ceases to hold. Ultimately it bends round, as is also a priori evident from the fact that at a very low pressure the resistance to the passage of discharge is very great *. APPENDIX III The peculiarities of the curves, fig. 1, seem to be capable of explanation on such considerations as the following : Let F be the voltage of the induction coil ; then the energy supplied per unit of time by the coil will be proportional to F , say i FO, where i is the current in the circuit. Let V be the potential difference between the electrodes ; then the energy supplied to the electrodes per unit of time will be proportional to V = i' V, say. Therefore i V i'V-\- energy carried away by the positive and negative ions, thrown off from the electrodes, less the energy carried to the electrodes by positive and negative ions reaching them (per unit of time). * Phil Mag. Oct. 1908. Recent experiments have shown that similar curves are obtained with all vacuum tubes. APPENDIX III 175 But the energy carried off by an ion = Xe\. Therefore i F = i' V + Xe (Nq'\ f + nq\) - E, where n and N are the numbers of positive and negative ions thrown off from the electrodes and occupying unit length of the discharge, and X, X' their mean free paths. In order to find E, we may proceed as follows : It can be shown * that the equations of continuity in a discharge-tube can be written, in the steady state, dNq' TT-* = zNq + ynq d(nq) where a= X (2), and n, N the number of positive and negative ions per unit length of discharge, as being measured along the line of dis- charge. Therefore, we have Nq' + nq = const. = - , where i is the current carried by the discharge. Again, the energy carried to the cathode by the positive ions may be written equal to where & is a coefficient determining the dissipation of energy during the passage of these ions. Also, the energy carried to the anode by the negative ions may similarly be written equal to eVx' _ e -k'x- ( a j\y + y n( fi * J. J. Thomson's Conduction of Electricity through Gases, 2nd ed. p. 490. 176 OPTICAL THEORIES where d = distance between the electrodes ; V difference of potential between the electrodes assumed to vary uniformly. f>V C .'. E = ~\ (aNq 1 + ynq) x (e~ kx + er**) dx. But from (1), if a, 7 be regarded as constant, l )e(--y^ ......... (3), if N = N lt n = w a at the cathode. Hence -* (*-** + er**) dx o Ve 4- similar terms in &' = Xe (aN 1 qS + 7^ ^j) P, say. If a. = 7 = 0, the above equation reduces to E = 0. This we may suppose to be the case during the rotatory stage in air [Phil Mag. Oct. 1912]. Therefore, since in this case N=n, and the pressure varies inversely as the mean free path, we get the equation But i = i' + i = i' + ne (q + q') ; X i' + ne(q + q) ' As, moreover, during this stage V is small compared with F , and i' should be small compared with i , we get the simple equation (putting X' = -J Y I ^ "*" $ V \ Y F = \ , / or = const, nearly. p \ q+q / p APPENDIX III 177 This, as we have seen, is the case in air (curves I and II, fig. 1) ; when, however, the pressure is sufficiently reduced, a, 7 are no longer zero. In fact, the terms in E become sufficiently effective in making V large, for a, 7, k, k' are all proportional to pressure, and it is reasonable to suppose a < 7, since 1 1 a oc , and 7 cc - . A A, This is experimentally verified. , Although it is not possible to work out completely the theory of this variation of pressure without a knowledge of a, 7, k, k', we may get some insight into its nature in special cases by proceeding as follows: From (3), we have (otf tf/ + yn l q 1 ) e <-*>* = (*&&' + yn z q,) (4), if N=N 2 , n = n z at the anode, q=q* and q = q'lq*\ but e (N z q* + n^q 2 ) = i = e (N^' + % qj (5). If, now, n 2 = 0, a 7 a 7 We have also and as - is a small quantity, we get X as an exponential func- tion of p. It is obvious, however, that the above investigation is not capable of giving a complete account of the variation of the potential difference, for we have assumed (1) that the potential varies uniformly from cathode to anode, and (2) that w, 7 are constant. As neither of these suppositions can be true always, it is not surprising that the curves obtained* are more com- plicated than those given by theory. * Phil. Mag. July, 1916. M. O. T. 12 178 OPTICAL THEORIES APPENDIX IV An approximate theory of the magnetic action of the excited magnetic field on electric discharge through a De la Rive tube can be worked out as follows : Using cylindrical coordinates, z, p, 6 (where z is measured from the Faraday dark space), the equations of motion of an ion may be written, if m is its mass, (1), (2), ............ (3), p Cut where A coeff. of viscosity, B a coeff. to be determined, Z = electric force in the direction of z and R in the direction of p, while H = magnetic force, which we know is mainly in the direction of p (Phil. Mag. Jan. 1908). Now, considering the equation (3) (to which, alone, we shall confine ourselves), if we have N negative and n positive ions per unit length, in any stream (masses m l and ra 2 respectively), we have, taking moment about the axis and summing \(mjr + m*n)d8^(p*B)+ {(A^N + A,ri) n)dsz ...... (4) as the equation of motion of any stream of discharge. Now we may assume, as in Phil. Mag. Oct. 1912, the action between two streams of lengths ds, ds to be a repulsion where q, q' are the velocities of positive and negative ions, a, 6 their radii (assumed spherical), r the distance between ds, ds', K the s.i.c. and V the velocity of light. The third term of (4) will then be of the form APPENDIX IV 179 the other terms (depending on the velocities) being neglected. Here G is a constant depending on the form of the various streams of discharge, and a the angular coordinate defining the position of the stream, whose equation of motion is given by (4), provided n and N are constant throughout the discharge, for in this case, alone, will be the same for all points. In any case, if n = N, the equation of motion is of the form = the couple acting on the discharge due to the magnetic action of the electromagnet, where / = \(m^ + m 2 ) ndsp 2 , since p = 0, in the steady state. But this couple = f Mi, where M is the total magnetic strength of induced magnetism (Phil. Mag. Oct. 1908). Therefore we have ds. where /*, = (-4j + A 2 ) n Ip This is the same equation as was obtained in Art. 263 by identifying the discharge with an electric current. If the number of positive particles is small in comparison with that of negative particles, the number of the latter will not necessarily be constant throughout any stream of discharge. In this case, putting n = and considering the motion of a small element of a discharge, we have, when the steady stage is reached, or = ...(5), Jo where B is a function of p, a, defining the position of the element of the discharge considered. This completely explains the twist referred to in Art. 258*. * Phil. Mag. July, 1916. INDEX Aberration, 59 electron theory of, 115, 148 - Fresnel's theory of, 118 Aristotle, undulatory theory, 4 Bartika, commentary of Nyaya Vasya, 3 Boussinesq, 152 Cauchy, 55 Characteristic function, 8, 9 Corpuscles, 142 Current," 74, 83 Descartes, theory of refraction, 4, 5, 6 theory of light, 6 Diffraction, 23, 92 Dispersion, 69, 60 electron theory of, 148 Double refraction, 16, 24-30, 48 Elastic solid theory, 33 modified, 55 Elasticity, theory of, 35 Electric displacement, 66 physical meaning of, 73 Electro-kinetic energy, 85 : momentum, 84 Electro-magnetic theory, 61 Electro-magnetism, 80 Electron theory, 124 Electrons, 102 Electrostatic energy, 65, 130 Faraday, 71 JFermat, 6 Fresnel, 21, 26, 118 Glazebrook, 55, 58, 96, 115 Green, 45, 51 Hall's phenomena, 114 Helmholtz, 153 Hertz, 88, 121, 124 Hindus, 2 Hooke, 13 Huyghens, 21, 23 Hydrodynamic analogy, 107 Induction, magnetic, 78 normal, 63 Ions, 142 Karl Pearson, 1 Kelvin, theory of discharge of a Ley den jar, 88 labile ether, 40 vortex sponge, 161 Ketteler, 154 Kinetic, all energy is, 12 Kirchhoff, 94 Labile ether, theory of, 40 Larmor, 48, 101, 103, 106, 110, 133, 137 Lodge, 110 Lorentz, 104, 138 MacCullagh, 28, 43, 51 Magnetic energy, 80 permeability, 79 shell, 81 Magnetism, 76, 129 Magneto-optic energy, 114 Magneto-optic rotation, 109 INDEX 181 Malus' law, 30 Mean free path, 143 Michelson-Morley experiment, 160 Molecular vortex theory, 111 Newton, 13-20 Nyaya Kandali, 3 Sutra, 3 - Vasya, 2 Optical energy, 91 Poincare, 70, 132 Poisson, 38 Poisson's solution, 93 Polarisation, 24 rotatory, 69 Polarising angle, 31 Potential electrostatic, 62 magnetic, 76, 81 Poynting, 91 Pythagoras, 4 Raman, 97 Rayleigh, 54, 55, 93 Reflection and Refraction, 14, 31, 46, 98 metallic, 99 Resonator, 89 Rontgen, 122 Rowland, 95 Saint-Venant, 68 Specific inductive capacity, 62 Stokes, 39, 43, 95 Strain-energy function, 38 Stresses in the electrostatic field, 67 Swiftest propagation of light, 6 Thomson, Sir J. 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