Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39.48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991.QJornell Hmuersity Hibrarg 3tljara, TSm $ork THE ALEXANDER GRAY MEMORIAL LIBRARY ELECTRICAL ENGINEERING THE GIFT OF The McGraw-Hill Book Co.. Inc. 1921Cfavenbon (jJveee Irenes n V.f RECENT RESEARCHES IN ELECTRICITY AND MAGNETISM ./. •/. THOMSONJSondon HENRY FROWDE Oxford University Press Warehouse Amen Corner, E.C. (JWw 2)or8 MACMILLAN & CO., 112 FOURTH AVENUENOTES ON RECENT RESEARCHES IN ELECTRICITY AND MAGNETISM INTENDED AS A SEQUEL TO PROFESSOR CLERK-MAXWELL’S TREATISE ON ELECTRICITY AND MAGNETISM J. J. THOMSON, M.A., F.R.S. Hon. Sc. D. Dublin FELLOW OF TRINITY COLLEGE PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE Oxford AT THE CLARENDON PRESS 1893©jcf©t& PRINTED AT THE CLARENDON PRESS BY HORACE HART, PRINTER TO THE UNIVERSITYPREFACE In the twenty years which have elapsed since the first appearance of Maxwell’s Treatise on Electricity and Magnetism great progress has been made in these sciences. This progress has been largely—perhaps it would not be too much to say mainly—due to the influence of the views set forth in that Treatise, to the value of which it offers convincing testimony. In the following work I have endeavoured to give an account of some recent electrical researches, experimental as well as theoretical, in the hope that it may assist students to gain some acquaintance with the recent progress of Electricity and yet retain Maxwell’s Treatise as the source from which they learn the great principles of the science. I have adopted exclusively Maxwell’s theory, and have not attempted to discuss the con- sequences which would follow from any other view of electrical action. I have assumed throughout the equations of the Electro- magnetic Field given by Maxwell in the ninth chapter of the second volume of his Treatise. The first chapter of this work contains an account of a method of regarding the Electric Field, which is geometrical and physical rather than analytical. I have been induced to dwell on this because I have found that students, especially those who com- mence the subject after a long course of mathematical studies, have a great tendency to regard the whole of Maxwell’s theory as a matter of the solution of certain differential equations, and to dispense with any attempt to form for themselves a mental picture of the physical processes which accompany the phe- nomena they are investigating. I think that this state of things is to be regretted, since it retards the progress of the science ofVI PREFACE. Electricity and diminishes the value of the mental training afforded by the study of that science. In the first place, though no instrument of research is more powerful than Mathematical Analysis, which indeed is indispens- able in many departments of Electricity, yet analysis works to the best advantage when employed in developing the suggestions afforded by other and more physical methods. One example of such a method, and one which is very closely connected with the initiation and development of Maxwell's Theory, is that of the ‘ tubes of force ’ used by Faraday. Faraday interpreted all the laws of Electrostatics in terms of his tubes, which served him in place of the symbols of the mathematician, while in his hands the laws according to which these tubes acted on each other served instead of the differential equations satisfied by such symbols. The method of the tubes is distinctly physical, that of the symbols and differential equations is analytical. The physical method has all the advantages in vividness which arise from the use of concrete quantities instead of abstract symbols to represent the state of the electric field; it is more easily wielded, and is thus more suitable for obtaining rapidly the main features of any problem ; when, however, the problem has to be worked out in all its details, the analytical method is necessary. In a research in any of the various fields of electricity we shall be acting in accordance with Bacon's dictum that the best results are obtained when a research begins with Physics and ends with Mathematics, if we use the physical theory to, so to speak, make a general survey of the country, and when this has been done use the analytical method to lay down firm roads along the line indicated by the survey. The use of a physical theory will help to correct the tendency —which I think all who have had occasion to examine in Mathe- matical Physics will admit is by no means uncommon—to look on analytical processes as the modern equivalents of the Philosopher’s Machine in the Grand Academy of Lagado, and to regard as the normal process of investigation in this subject the manipulation of a large number of symbols in the hope that every now and then some valuable result may happen to drop out.PREFACE. Vll Then, again, I think that supplementing the mathematical theory by one of a more physical character makes the study of electricity more valuable as a mental training for the student. Analysis is undoubtedly the greatest thought-saving machine ever invented, but I confess I do not think it necessary or desir- able to use artificial means to prevent students from thinking too much. It frequently happens that more thought is required, and a more vivid idea of the essentials of a problem gained, by a rough solution by a general method, than by a complete solution arrived at by the most recent improvements in the higher analysis. The method of illustrating the properties of the electric field which I have given in Chapter I has been devised so as to lead directly to the distinctive feature in Maxwell’s Theory, that changes in the polarization in a dielectric produce magnetic effects analogous to those produced by conduction currents. Other methods of viewing the processes in the Electric Field, which would be in accordance with Maxwell’s Theory, could, I have no doubt, be devised; the question as to which particular method the student should adopt is however for many purposes of secondary importance, provided that he does adopt one, and acquires the habit of looking at the problems with which he is occupied as much as possible from a physical point of view. It is no doubt true that these physical theories are liable to imply more than is justified by the analytical theory they are used to illustrate. This however is not important if we remember that the object of such theories is suggestion and not demonstration. Either Experiment or rigorous Analysis must always be the final Court of Appeal; it is the province of these physical theories to supply cases to be tried in such a court. Chapter II is devoted to the consideration of the discharge of electricity through gases; Chapter III contains an account of the application of Schwarz’s method of transformation to the solu- tion of two-dimensional problems in Electrostatics. The rest of the book is chiefly occupied with the consideration of the pro- perties of alternating currents; the experiments of Hertz and the development of electric lighting have made the use of theseVlll PREFACE. currents, both for experimental and commercial purposes, much more general than when Maxwell’s Treatise was written; and though the principles which govern the action of these currents are clearly laid down by Maxwell, they are not developed to the extent which the present importance of the subject demands. Chapter IY contains an investigation of the theory of such currents when the conductors in which they flow are cylin- drical or spherical, while in Chapter V an account of Hertz’s experiments on Electromagnetic Waves is given. This Chapter also contains some investigations on the Electromagnetic Theory of Light, especially on the scattering of light by small metallic particles ; on reflection from metals ; and on the rotation of the plane of polarization by reflection from a magnet. I regret that it was only when this volume was passing through the press that I became acquainted with a valuable paper by Drude (Wiede- mann’s Annalen, 46, p. 353, 1892) on this subject. Chapter VI mainly consists of an account of Lord Rayleigh’s investigations on the laws according to which alternating currents distribute themselves among a network of conductors ; while the last Chapter contains a discussion of the equations which hold when a dielectric is moving in a magnetic field, and some problems on the distribution of currents in rotating conductors. I have not said anything about recent researches on Magnetic Induction, as a complete account of these in an easily accessible form is contained in Professor Ewing’s ‘Treatise on Magnetic Induction in Iron and other Metals.5 I have again to thank Mr. Chree, Fellow of King’s College, Cambridge, for many most valuable suggestions, as well as for a very careful revision of the proofs. J. J. THOMSON.CONTENTS CHAPTER I. ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE. Art. Page 1. Electric displacement ........................................... 1 2. Faraday tubes ................................................... 2 3. Unit Faraday tubes............................................... 3 4. Analogy with kinetic theory of gases ............................ 4 5. Reasons for taking tubes of electrostatic induction as the unit 4 6. Energy in the electric field .................................... 4 7. Behaviour of Faraday tubes in a conductor ....................... 5 8. Connection between electric displacement and Faraday tubes.. 6 9. Bate of change of electric polarization expressed in terms of the velocity of Faraday tubes ................................ 6 10. Momentum due to Faraday tubes .................................. 9 11. Electromotive intensity due to induction....................... 10 12. Velocity of Faraday tubes...................................... 11 13. Systems of tubes moving with different velocities.............. 12 14. Mechanical forces in the electric field ....................... 14 15. Magnetic force due to alteration in the dielectric polarization 15 16. Application of Faraday tubes to find the magnetic force due to a moving charged sphere ...................................... 16 17. Rotating electrified plates.................................... 23 18. Motion of tubes in a steady magnetic field .................... 28 19. Induction of currents due to changes in the magnetic field .. 32 20. Induction due to the motion of the circuit .................... 33 21. Effect of soft iron in the field............................... 34 22. Permanent magnets.............................................. 35 23. Steady current along a straight wire........................... 36 24. Motion of tubes when the currents are rapidly alternating .. 38 25. Discharge of a Leyden jar...................................... 38 26. Induced currents ...............................................40 27. Electromagnetic theory of light ............................... 42X CONTENTS. Art. 28-32. Behaviour of tubes in conductors ...................... 43-47 33. Galvanic cell .................................................. 48 34. Metallic and electrolytic conduction............................ 50 CHAPTER II. PASSAGE OF ELECTRICITY THROUGH GASES. 35. Introduction ................................................... 53 36. Can the molecules of a gas be electrified ? 53 37. Hot gases....................................................... 54 38. Electric properties of flames .................................. 57 39. Effect of ultra-violet light on the discharge .................. 57 40. Electrification by ultra-violet light........................... 59 41. Disintegration of the negative electrode.........................60 42. Discharge of electricity from illuminated metals ............... 60 43. Discharge of electricity by glowing bodies ..................... 62 44. Volta-potential................................................. 63 45. Electrification by sun-light ................................... 66 46. 4 Electric Strength’ of a gas .................................. 68 47. Effect of the nature of the electrodes on the spark length .. 69 48. Effect of curvature of the electrodes on the spark length.. .. 69 49. Bailie’s experiments on the connection between potential differ- ence and spark length ........................................ 70 50. Liebig’s on the same subject ................................ 72 51. Potential difference expressed in terms of spark length .. .. 74 52-53. Minimum potential difference required to produce a spark 74 54-61. Discharge when the field is not uniform ............... 77-84 62-65. Peace’s experiments on the connection between pressure and spark potential ........................................ 84-89 66-68. Critical pressure .. .. .......................... 89-90 69-71. Potential difference required to spark through various gases .. ....................................... 90-92 72-76. Methods of producing electrodeless discharges .. .. 92-95 77. Appearance of such discharges.................................. 96 78-80. Critical pressure for such discharges.................. 96-97 81. Difficulty of getting the discharge to pass across from gas to metal ........................................................ 98 82-86. High conductivity of rarefied gases ...................... 99-103 87. Discharge through a mixture of gases .........................103 88-93. Action of a magnet on the electrodeless discharges .. 104-107 94. Appearance of discharge when electrodes are used.............108 95. Crookes’ theory of the dark space ...........................109CONTENTS. xi Art. Page 96. Length of dark space .......................................110 97-98. Negative glow .. .. 110 99-103. Positive column and striations.......................... 111-114 104-107. Velocity of discharge along positive column .. .. 115-118 108-116. Negative rays............................................119-124 117. Mechanical effects produced by negative rays...............124 118-123. Shadows cast by negative rays ...........................125-128 124-125. Relative magnitudes of time quantities in the dis- charge .. .. 128-130 126-128. Action of a magnet on the discharge ..............131-132 129. Action of a magnet on the negative glow ...................132 130-133. Action of a magnet on the negative rays .. .. 134-138 134. Action of a magnet on the positive column..............138 135. Action of a magnet on the negative rays in very high vacua .. 139 136. Action of a magnet on the course of the discharge......140 137-138. Action of a magnet on the striations ...............141-142 139-147. Potential gradient along the discharge tube .. .. 142-149 148-151. Effect of the strength of the current on the cathode fall.......................................................150-153 152-155. Small potential difference sufficient to maintain current when once started.............................. .. 153-155 156—162. Warburg’s experiments on the cathode fall .. .. 155—158 163-165. Potential gradient along positive column .. .. 159-160 166-168. Discharge between electrodes placed close together 160-163 169-176. The arc discharge .......................................163-167 177-178. Heat produced by the discharge ..........................167-168 179-182. Difference between effects at positive and negative electrodes ................................................169-171 183-186. Lichtenberg’s and Kundt’s dust figures...................172-174 187-193. Mechanical effects due to the discharge...........174-177 194-201. Chemical action of the discharge ........................177-181 202. Phosphorescent glow due to the discharge ........................184 203-206. Discharge facilitated by rapid changes in the strength of the field..................................................185-189 207-229. Theory of the discharge................................. 189-207 CHAPTER in. CONJUGATE FUNCTIONS. 230-233. Schwarz and Christoffers transformation .. .. 208-211 234. Method of applying it to electrostatics..........................211Xll CONTENTS. Art Page 235. Distribution of electricity on a plate placed parallel to an infinite plate................. ........................212 236. Case of a plate between two infinite parallel plates .. ..216 237. Correction for thickness of plate ...........................218 238. Case of one cube inside another ..........................222 239-240. Cube over an infinite plate ...................... 225-227 241. Case of condenser with guard-ring when the slit is shallow .. 227 242. Correction when guard-ring is not at the same potential as the plate ...................................................231 243. Case of condenser with guard-ring when the slit is deep .. 232 244. Correction when guard-ring is not at the same potential as the plate ...................................................235 245. Application of elliptic functions to problems in electrostatics 236 246. Capacity of a pile of plates ..............................239 247. Capacity of a system of radial plates .....................241 248. Finite plate at right angles to two infinite ones ........242 249. Two sets of parallel plates .................................244 250. Two sets of radial plates....................................246 251. Finite strip placed parallel to two infinite plates.......246 252. Two sets of parallel plates .................................248 253. Two sets of radial plates....................................249 254. Limitation of problems solved................................250 CHAPTER IV. ELECTRICAL WAVES AND OSCILLATIONS. 255. Scope of the chapter ........................................251 256. General equations............................................251 257. Alternating currents in two dimensions.......................253 258. Case when rate of alternation is very rapid..................259 259-260. Periodic currents along cylindrical conductors .. .. 262 261. Value of Bessel's functions for very large or very small values of the variable ............................................262 262. Propagation of electric waves along wires ...................263 263-264. Slowly alternating currents .......................... 270-273 265. Expansion of x J0{x)/J0' (#) 274 266. Moderately rapid alternating currents .......................276 267. Very rapidly alternating currents............................278 268. Currents confined to a thin skin ............................280 269. Magnetic force in dielectric ................................282 270. Transmission of disturbances along wires ....................283CONTENTS. Xlll Art, Page 271. Relation between external electromotive force and current .. 288 272. Impedance and self-induction..................................292 273-274. Values of these when alternations are rapid .. .. 294-295 275-276. Flat conductors ............................................296 277. Mechanical force between flat conductors .....................300 278. Propagation of longitudinal magnetic waves along wires .. 302 279. Case when the alternations are very rapid ...........306 280. Poynting's theorem............................................308 281. Expression for rate of heat production in a wire...........314 282. Heat produced by slowly varying current ...................314 283-284. Heat produced by rapidly varying currents .. .. 316-317 285. Heat in a transformer due to Foucault currents when the rate of alternation is slow.......................................318 286. When the rate of alternation is rapid .....................321 287. Heat produced in a tube...................................323 288. Vibrations of electrical systems .............................328 289. Oscillations on two spheres connected by a wire...........328 290. Condition that electrical system should oscillate.........329 291. Time of oscillation of a condenser........................331 292. Experiments on electrical oscillations .......................332 293-297. General investigation of time of vibration of a condenser .............................................. 333-340 298-299. Vibrations along wires in multiple arc.............. 341-344 300. Time of oscillations on a cylindrical cavity..............344 301. On a metal cylinder surrounded by a dielectric ...............347 302. State of the field round the cylinder ........................350 303. Decay of currents in a metal cylinder.........................352 304-305. When the lines of magnetic force are parallel to the axis of the cylinder ........................................ 354-356 306-307. When the lines of force are at right angles to the axis ................................................... 357-359 308. Electrical oscillations on spheres ...........................361 309. Properties of the functions S and E ..........................363 310. General solution .............................................366 311. Equation giving the periods of vibration .....................367 312. Case of the first harmonic distribution...................368 313. Second and third harmonics .. .♦ .......... .. .. 371 314. Field round vibrating sphere..............................372 315. Vibration of two concentric spheres ..........................372 316. When the radii of the spheres are nearly equal ...............375 317. Decay of currents in spheres..............................377 318. Rate of decay when the currents flow in meridional planes .. 380xiv CONTENTS. Art Page 319-320. Effect of radial currents in the sphere.......... 382-383 321. Currents induced in a sphere by the annihilation of a uniform magnetic field ..........................................384 322. Magnetic effects of these currents when the sphere is not made of iron.............................................386 323. When the sphere is made of iron..........................387 CHAPTER V. ELECTROMAGNETIC WAVES. 324. Hertz’s experiments .......................................388 325-327. Hertz’s vibrator .................................. 388-390 328. The resonator..............................................391 329. Effect of altering the position of the air gap...........391 330-331. Explanation of these effects ...........................392 332. Resonance...................................................395 333-335. Rate of decay of the vibrations ................... 395-397 336-339. Reflection of waves from a metal plate........... 398-400 340-342. Sarasin’s and De la Rive’s experiments........... 400-404 343. Parabolic mirrors ..........................................404 344-346. Electric screening................................. 405-406 347. Refraction of electromagnetic waves .......................406 348. Angle of polarization .....................................406 349-350. Theory of reflection of electromagnetic waves by a dielectric ......................................... 407-411 351. Reflection of these waves from and transmission through a thin metal plate.........................................414 352-354. Reflection of light from metals ....................417-419 355. Table of refractive indices of metals .....................420 356. Inadequacy of the theory of metallic reflection ...........421 357. Magnetic properties of iron for light waves................422 358. Transmission of light through thin films ..................423 359-360. Reflection of electromagnetic waves from a grating 425-428 361-368. Scattering of these waves by a wire ............. 428-436 369. Scattering of light by metal spheres ..................... 437 370. Lamb’s theorem ............................................438 371. Expressions for magnetic force and electric polarization .. 440 372. Polarization in plane wave expressed in terms of spherical harmonics ...............................................441 373-376. Scattering of a plane wave by a sphere of any size 443-445 377. Scattering by a small sphere...............................447 378. Direction in which the scattered light vanishes ...........449CONTENTS. XV Art. Page 379-384. Hertz's experiments on waves along wires .. .. 451-456 385. Sarasin's and De la Rive's experiments on waves along wires 459 390-392. Comparison of specific inductive capacity with refractive index .. .. 468-471 393-401. Experiments to determine the velocity of electromagnetic waves through various dielectrics ................... 471-481 402. Effects produced by a magnetic field on light................482 403. Kerr's experiments ..........................................482 404. Oblique reflection from a magnetic pole......................484 405. Reflection from tangentially magnetized iron.................485 406. Kundt's experiments on films.................................485 407. Transverse electromotive intensity...........................486 408. Hall effect..................................................486 409-414. Theory of rotation of plane of polarization by reflection from a magnet ........................................ 490-501 415-416. Passage of light through thin films in a magnetic field ................................................ 504-508 CHAPTER VI. DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 417-418. Very rapidly alternating currents distribute themselves so as to make the Kinetic Energy a minimum...................510 419. Experiments to illustrate this.............................511 420. Distribution of alternating currents between two wires in parallel .. ............................................512 421. Self-induction and impedance of the wires .................516 422. Case of any number of wires in parallel....................517 423-426. General case of any number of circuits.............. 520-524 427-428. Case of co-axial solenoids.......................... 524-526 429. Wheatstone's bridge with alternating currents ...............527 430-432. Combination of self-induction and capacity .. .. 529-531 432*. Effect of two adjacent vibrators on each other’s periods .. 532 CHAPTER VII. ELECTROMOTIVE INTENSITY IN MOVING BODIES. 433. Equations of electromotive intensity for moving bodies .. .. 534 434-439. Sphere rotating in a symmetrical magnetic field .. 535-542 440. Propagation of light through a moving dielectric...........543 441. Currents induced in a sphere rotating in an unsymmetrical field ...................................................546Xvi CONTENTS. Art. Page 442. Special case when the field is uniform....................550 443. Case when the rotation is very rapid ................. ,.551 444. Magnetic force outside the sphere..........................553 445. Couples and Forces on the Rotating Sphere...................554 446. The magnetic force is tangential when the rotation is rapid .. 555 447. Force on the sphere.........................................556 448. Solution of the previous case gives that of a sphere at rest in an alternating field ......................................557 Appendix. The Electrolysis of Steam .............................559 ADDITIONS AND CORRECTIONS. Page 67. For further remarks on electrification by incandescent bodies see Appendix, p. 569. „ 122. E. Wiedemann and Ebert have shown (Wied. Ann. 46, p. 158, 1892) that the repulsion between two pencils of negative rays is due to the influence which the presence of one cathode exerts on the emission of rays from a neighbouring cathode. „ 174. Dewar (Proc, Soy. Soc. 33, p. 262, 1882) has shown that the interior of the gaseous envelope of the electric arc always shows a fixed pressure amounting to about that due to a millimetre of water above that of the surrounding atmosphere. „ 182. Line itfor go°C. read ioo*C.NOTES ON ELECTRICITY AND MAGNETISM. CHAPTER I. ELECTBIC DISPLACEMENT AND FABADAY TDBES OF FOECE. 1.] The influence which the notation and ideas of the fluid theory of electricity have ever since their introduction exerted over the science of Electricity and Magnetism, is a striking illustration of the benefits conferred upon this science by a concrete representation or ‘ construibar vorstellung ’ of the sym- bols, which in the Mathematical Theory of Electricity define the state of the electric field. Indeed the services which the old fluid theory has rendered to Electricity by providing a lan- guage in which the facts of the science can be clearly and briefly expressed can hardly be over-rated. A descriptive theory of this kind does more than serve as a vehicle for the clear ex- pression of well-known results, it often renders important services by suggesting the possibility of the existence of new phenomena. The descriptive hypothesis, that of displacement in a dielec- tric, used by Maxwell to illustrate his mathematical theory, seems to have been found by many readers neither so simple nor so easy of comprehension as the old fluid theory; indeed this seems to have been one of the chief reasons why his views did not sooner meet with the general acceptance they have since received. As many students find the conception of1 displacement ’ difficult, I venture to give an alternative method of regarding the pro- cesses occurring in the electric field, which I have often found useful and which is, from a mathematical point of view, equiva- lent to Maxwell’s Theory.2 ELECTRIC DISPLACEMENT AND [2. 2.] This method is based on the conception, introduced by Faraday, of tubes of electric force, or rather of electrostatic* induction. Faraday, as is well known, used these tubes as the language in which to express the phenomena of the electric field. Thus it was by their tendency to contract, and the lateral repulsion which similar tubes exert on each other, that he explained the mechanical forces between electrified bodies, while the influence of the medium on these tubes was on his view indicated by the existence of specific inductive capacity in dielectrics. Although the language which Faraday used about lines of force leaves the impression that he usually regarded them as chains of polarized particles in the dielectric, yet there seem to be indications that he occasionally regarded them from another aspect; i.e. as something having an existence apart from the molecules of the dielectric, though these were polarized by the tubes when they passed through the dielectric. Thus, for example, in § 1616 of the Experimental Researches he seems to regard these tubes as stretching across a vacuum. It is this latter view of the tubes of electrostatic induction which we shall adopt, we shall regard them as having their seat in the ether, the polarization of the particles which accompanies their passage through a dielectric being a secondary phenomenon. We shall for the sake of brevity call such tubes Faraday Tubes. In addition to the tubes which stretch from positive to nega- tive electricity, we suppose that there are, in the ether, multitudes of tubes of similar constitution but which form discrete closed curves instead of having free ends; we shall call such tubes ‘ closed ’ tubes. The difference between the two kinds of tubes is similar to that between a vortex filament with its ends on the free surface of a liquid and one forming a closed vortex ring inside it. These closed tubes which are supposed to be present in the ether whether electric forces exist or not, impart a fibrous structure to the ether. In his theory of electric and magnetic phenomena Faraday made use of tubes of magnetic as well as of electrostatic induction, we shall find however that if we keep to the con- ception of tubes of electrostatic induction we can explain the phenomena of the magnetic field as due to the motion of such tubes.3-] FARADAY TUBES OF FORCE. 3 The Faraday Tubes. 3.] As is explained in Art. 82 of Maxwell’s Electricity and Magnetism, these tubes start from places where there is posi- tive and end at places where there is negative electricity, the quantity of positive electricity at the beginning of the tube being equal to that of the negative at the end. If we assume that the tubes in the field are all of the same strength, the quantity of free positive electricity on any surface will be proportional to the number of tubes leaving the surface. In the mathematical theory of electricity there is nothing to indicate that there is any limit to the extent to which a field of electric force can be subdivided up into tubes of continually diminishing strength, the case is however different if we regard these tubes of force as being no longer merely a form of mathematical ex- pression, but as real physical quantities having definite sizes and shapes. If we take this view, we naturally regard the tubes as being all of the same strength, and we shall see reasons for believing that this strength is such that when they terminate on a conductor there is at the end of the tube a charge of negative electricity equal to that which in the theory of electrolysis we associate with an atom of a monovalent element such as chlorine. This strength of the unit tubes is adopted because the pheno- mena of electrolysis show that it is a natural unit, and that fractional parts of this unit do not exist, at any rate in elec- tricity that has passed through an electrolyte. We shall assume in this chapter that in all electrical processes, and not merely in electrolysis, fractional parts of this unit do not exist. The Faraday tubes either form closed circuits or else begin and end on atoms, all tubes that are not closed being tubes that stretch in the ether along lines either straight or curved from one atom to another. When the length of the tube connect- ing two atoms is comparable with the distance between the atoms in a molecule, the atoms are said to be in chemical com- bination ; when the tube connecting the atoms is very much longer than this, the atoms are said to be ‘ chemically free \ The property of the Faraday tubes of always forming closed circuits or else having their ends on atoms may be illustrated by the similar property possessed by tubes of vortex motion in a frictionless fluid, these tubes either form closed circuits orELECTRIC DISPLACEMENT AND 4 [6. have their ends on the boundary of the liquid in which the vortex motion takes place. The Faraday tubes may be supposed to be scattered through- out space, and not merely confined to places where there is a finite electromotive intensity, the absence of this intensity being due not to the absence of the Faraday tubes, but to the want of arrangement among such as are present: the electromotive in- tensity at any place being thus a measure, not of the whole number of tubes at that place, but of the excess of the number pointing in the direction of the electromotive intensity over the number of those pointing in the opposite direction. 4. ] In this chapter we shall endeavour to show that the various phenomena of the electromagnetic field may all be interpreted as due to the motion of the Faraday tubes, or to changes in their position or shape. Thus, from our point of view, this method of looking at electrical phenomena may be regarded as forming a kind of molecular theory of Electricity, the Faraday tubes taking the place of the molecules in the Kinetic Theoiy of Gases: the object of the method being to explain the pheno- mena of the electric field as due to the motion of these tubes, just as it is fche object of the Kinetic Theory of Gases to explain the properties of a gas as due to the motion of its molecules. These tubes also resemble the molecules of a gas in another re- spect, as we regard them as incapable of destruction or creation. 5. ] It may be asked at the outset, why we have taken the tubes of electrostatic induction as our molecules, so to speak, rather than the tubes of magnetic induction ? The answer to this question is, that the evidence afforded by the phenomena which accoim- pany the passage of electricity through liquids and gases shows that molecular structure has an exceedingly close connection with tubes of electrostatic induction, much closer than we have any reason to believe it has with tubes of magnetic induction. The choice of the tubes of electrostatic induction as our molecules seems thus to be the one which affords us the greatest facilities for explaining those electrical phenomena in which matter as well as the ether is involved. 6. ] Let us consider for a moment on this view the origin of the energy in the electrostatic and electromagnetic fields. We suppose that associated with the Faraday tubes there is a dis- tribution of velocity of the ether both in the tubes themselvesFAEADAY TUBES OF FOEOE. 5 7-] and in the space surrounding them. Thus we may have rotation in the ether inside and around the tubes even when the tubes themselves have no translatory velocity, the kinetic energy due to this motion constituting the potential energy of the electro- static field: while when the tubes themselves are in motion we have super-added to this another distribution of velocity whose energy constitutes that of the magnetic field. The energy we have considered so far is in the ether, but when a tube falls on an atom it may modify the internal motion of the atom and thus affect its energy. Thus, in addition to the kinetic energy of the ether arising from the electric field, there may also be in the atoms some energy arising from the same cause and due to the alteration of the internal motion of the atoms produced by the incidence of the Faraday tubes. If the change in the energy of an atom produced by the incidence of a Faraday tube is different for atoms of different substances, if it is not the same, for example, for an atom of hydrogen as for one of chlorine, then the energy of a number of molecules of hydrochloric acid would depend upon whether the Faraday tubes started from the hydrogen and ended on the chlorine or vice versa. Since the energy in the molecules thus depends upon the disposition of the tubes in the molecule, there will be a tendency to make all the tubes start from the hydrogen and end on the chlorine or vice versa, according as the first or second of these arrangements makes the difference between the kinetic and potential energies a maximum. In other words, there will, in the language of the ordinary theory of electricity, be a tendency for all the atoms of hydrogen to be charged with electricity of one sign, while all the atoms of chlorine are charged with equal amounts of electricity of the opposite sign. The result of the different effects on the energy of the atom produced by the incidence of a Faraday tube will be the same as if the atoms of different substances attracted elec- tricity with different degrees of intensity: this has been shown by v. Helmholtz to be sufficient to account for contact and fric- tional electricity. It also, as we shall see in Chapter II, accounts for some of the effects observed when electricity passes from a gas to a metal or vice vers&. 7.] The Faraday tubes when they reach a conductor shrink to molecular dimensions. We shall consider the processes by which6 ELECTRIC DISPLACEMENT AND [9. this is effected at the end of this chapter, and in the meantime proceed to discuss the effects produced by these tubes when moving through a dielectric. 8. ] In order to be able to fix the state of the electric field at any point of a dielectric, we shall introduce a quantity which we shall call the ‘polarization’ of the dielectric, and which while mathema- tically identical with Maxwell’s ‘ displacement ’ has a different physical interpretation. The ‘polarization ’ is defined as follows: Let A and B be two neighbouring points in the dielectric, let a plane whose area is unity be drawn between these points and at right angles to the line joining them, then the polarization in the direction AB is the excess of the number of the Faraday tubes which pass through the unit area from the side A to the side B over those which pass through the same area from the side B to the side A. In a dielectric other than air we imagine the unit area to be placed in a narrow crevasse cut out of the di- electric, the sides of the crevasse being perpendicular to AB. The polarization is evidently a vector quantity and may be resolved into components in the same way as a force or a velo- city; we shall denote the components parallel to the axes of x, 2/, z by the letters f, g> h; these are mathematically identical with the quantities which Maxwell denotes by the same letters, their physical interpretation however is different. 9. ] We shall now investigate the rate of change of the compo- nents of the polarization in a dielectric. Since the Faraday tubes in such a medium can neither be created nor destroyed, a change in the number passing through any fixed area must be due to the motion or deformation of the tubes. We shall suppose, in the first place, that the tubes at one place are all moving with the same velocity. Let u, v, w be the components of the velocities of these tubes at any point, then the change in f, the number of tubes passing at the point sc, y, 0, through unit area at right angles to the axis of x, will be due to three causes. The first of these is the motion of the tubes from another part of the field up to the area under consideration ; the second is the spreading out or concentration of the tubes due to their relative motion; and the third is the alteration in the direction of the tubes due to the same cause. Let 62/ be the change in / due to the first cause, then in consequence of the motion of the tubes, the tubes which at the9-] FARADAY TUBES OF FORCE, 7 time t + bt pass through the unit area will be those which at the time t were at the point x—ubt, y—vbt9 z—wht, hence will be given by the equation 'dyT^ In consequence of the motion of the tubes relatively to one another, those which at the time t passed through unit area at right angles to x will at the time t + bt be spread over an area \dv . dw) Idy thus b2f9 the change in / due to this cause, will be given by the equation / , ., Cdv , dw) [ dv dw n or 1+8f{#+ dzS + £} In consequence of the deflection of the tubes due to the relative motion of their parts some of those which at the time t were at right angles to the axis of x will at the time t + bt have a component along it. Thus, for example, the tubes which at the time t were parallel to y will after a time bt has elapsed be twisted towards the axis of x through an angle bt-similarly dy du those parallel to z will be twisted through an angle bt-7- (mZ towards the axis of x in the time bt; hence b3fy the change in / due to this cause, will be given by the equation du , du1 }• Hence if bf is the total change in/ in the time bt, since »/= \f+h/+hf, we have T / df , df , df\ x,dv dwx , du J81- ■which may be written as df d d df dg dh8 ELECTRIC DISPLACEMENT AND [9. If p is the density of the free electricity, then since by the definition of Art. 8 the surface integral of the normal polarization taken over any closed surface must be equal to the quantity of electricity inside that surface, it follows that __ df dg dh P dx^ dy + dz9 hence equation (1) may be written Tt+Up = Similarly = | (2) dh d , n T1 d . , x dt+wp = rJwf-uh)-^/vk-wlj)-' If p, q, r are the components of the current parallel to x, y, z respectively, a, (3, y the components of the magnetic force in the same directions, then we know 4 up — dy dp dz' __<*P \ dy . da dy ~~ dz dx* dB da 4 7rr = -=--=- • dx dy (3) Hence, if we regard the current as made up of the convection current whose components are up, vp, wp respectively, and the polarization current whose components are ^, we see by comparing equations (2) and (3) that we may regard the moving Faraday tubes as giving rise to a magnetic force whose components a, /3, y are given by the equation a = 4 7r (vh — wg), \ /3 = 4 7’(ivf-uh), > (4) y = 4tTr(ug—vf ). ) Thus a Faraday tube when in motion produces a magnetic force at right angles both to itself and to its direction of motion, whose magnitude is proportional to the component of the velocity at right angles to the direction of the tube. The magnetic forceFARADAY TUBES OF FORCE. 9 IO.] and the rotation from the direction of motion to that of the tube at any point are related like translation and rotation in a right-handed screw. 10.] The motion of these tubes involves kinetic energy, and this kinetic energy is the energy of the magnetic field. Now if /x is the magnetic permeability we know that the energy per unit volume is £(a* + JS» + y»), o 7T or substituting the values of a, /3, y from equations (4), 2 irfjL [(hv—gw)2 + (fw—Jm)2 + (g u The momentum per unit volume of the dielectric parallel to x is the differential coefficient of this expression with regard to u, hence if U, V, W are the components of the momentum parallel to x, y> 0, we have U= 4tt/x {g(gu-fv)-h(fw-hu)} = gc—hb, if ay b, c are the components of the magnetic induction parallel to a, y, 0. Similarly Thus the momentum per unit volume in the dielectric, which is due to the motion of the tubes, is at right angles to the polarization and to the magnetic induction, the magnitude of the momentum being equal to the product of the polarization and the component of the magnetic induction at right angles to it. We may regard each tube as having a momentum proportional to the intensity of the component of the magnetic induction at right angles to the direction of the tube. It is interesting to notice that the components of the momentum in the field as given by equations (5) are proportional to the amounts of energy transferred in unit time across unit planes at right angles to the axes of x} y, 0 in Poynting’s theory of the transfer of energy in the electromagnetic field (Phil. Trans. 1884, Part II. p. 343); hence the direction in which the energy in Poynting’s theory is supposed to move is the same as the direction of the momentum determined by the preceding investigation. V — ha—fc, l w = fb-aa.\ (5)10 ELECTBIC DISPLACEMENT AND [II- 11.] The electromotive intensities parallel to x, y> z due to the motion of the tubes are the differential coefficients of the kinetic energy with regard to /, g, k respectively, hence we obtain the following expressions for X, F, Z the components of the electromotive intensity, X = wb—vc, } F = uc—wa, V (6) Z — va —ub. ' Thus the direction of the electromotive intensity due to the motion of the tubes is at right angles both to the magnetic induction and to the direction of motion of the tubes. From equations (6) we get dZ dY da da (db dc\ dy dz ~~V dy^W dz U 'dy d& But since the equation ,dv dwx , du dvj da db dc _ dx + dy^ dz~~ holds, as we shall subsequently show, on the view we have taken of the magnetic force as well as on the ordinary view, we have dZ dY da da da ,dv dw\ ,du du l^-te=UdX + VTy+WTz + ) F=aSA~c2/, > (9) W=b2f-a2gj Thus when we have a number of tubes moving about in the electric field the resultant momentum at any point is per- pendicular both to the resultant magnetic induction and to the , resultant polarization, and is equal to the product of these two quantities into the sine of the angle between them. The electromotive intensities X, Yy Z parallel to the axes of x, 2/, 2? respectively are equal to the mean values of dT/df, dT/dg, dT/dh, hence we have X = bw—cvy \ Y = cu-aW, > (10) Z = av — bu; ^ where a bar placed over any quantity indicates that the mean value of that quantity is to be taken.14 ELEOTEIC DISPLACEMENT AND [14. Thus when a system of Faraday tubes is in motion, the electromotive intensity is at right angles both to the resultant magnetic induction and to the mean velocity of the tubes, and is equal in magnitude to the product of these two quantities into the sine of the angle between them. We see from the preceding equations that there may be a resultant magnetic force due to the motion of the positive tubes in one direction and the negative ones in the opposite, without either resultant momentum or electromotive intensity; for if there are as many positive as negative tubes passing through each unit area so that there is no resultant polariza- tion, there will, by equations (9), be no resultant momentum, while if the number of tubes moving in one direction is the same as the number moving in the opposite, equations (10) show that there will be no resultant electromotive intensity due to the motion of the tubes. We thus see that when the mag- netic field is steady the motion of the Faraday tubes in the field will be a kind of shearing of the positive past the negative tubes; the positive tubes moving in one direction and the negative at an equal rate in the opposite. When, however, the field is not in a steady state this ceases to be the case, and then the electromotive intensities due to in- duction are developed. Mechanical Forces in the Field. 14.] The momentum parallel to x per unit volume of the medium, due to the motion of the Faraday tubes, is by equation (9) c2g—b2h; thus the momentum parallel to x which enters a portion of the medium bounded by the closed surface S in unit time is equal to // [c'2g(lu + mv + nw)—b 2 h (lu + mv + nw)] d S> where dS is an element of the surface and l, m, n the direction- cosines of its inwardly directed normal. If the surface S is so small that the external magnetic field may be regarded as constant over it, the expression may be written as c ff 2 g {lu -f mv + nw) d 8—bff 2h {lu+mv + nw) d 8.FARADAY TUBES OF FORCE. 15 I5-] ff 2g (lu + mv + nw) d 8, + mv + nw) dS, are the number of Faraday tubes parallel to y and z respectively which enter the element in unit time, that is, they are the volume integrals of the components q and r of the current parallel to y and 0 respectively: if the medium surrounded by 8 is a dielectric this is a polarization current, if it is a con- ductor it is a conduction current. Thus the momentum parallel to x communicated in unit time to unit volume of the medium, in other words the force parallel to x acting on unit volume of the medium, is equal to eg — br; similarly the forces parallel to y and z are respectively When the medium is a conductor these are the ordinary expressions for the components of the force per unit volume of the conductor when it is carrying a current in a magnetic field. When, as in the above investigation, we regard the force on a conductor carrying a current as due to the communication to the conductor of the momentum of the Faraday tubes which enter the conductor, the origin of the force between two currents will be very much the same as that of the attraction between two bodies on Le Sage's theory of gravitation. Thus, for example, if we have two parallel currents A and B flowing in the same direction, then if A is to the left of B more tubes will enter A from the left than from the right, because some of those which would have come from the right if B had been absent will be absorbed by B, thus in unit time the momentum having the direction left to right which enters A will exceed that having the opposite direction; thus A will tend to move towards the right, that is towards B, while for a similar reason B will tend to move towards A. 15.] We have thus seen that the hypothesis of Faraday tubes in motion explains the properties and leads to the ordinary equations of the electromagnetic field. This hypothesis has the advantage of indicating very clearly why polarization and con- Now and16 ELECTRIC DISPLACEMENT AND [i6 duction currents produce similar mechanical and magnetic effects. For the mechanical effects and the magnetic forces at any point in the field are due to the motion of the Faraday tubes at that point, and any alteration in the polarization involves motion of these tubes just as much as does an ordinary conduction current. 16.] We shall now proceed to illustrate this method of re- garding electrical phenomena by applying it to the consideration of some simple cases. We shall begin with the case which suggested the method; that of a charged sphere moving uni- formly through the dielectric. Let us suppose the charge on the sphere is e and that it is moving with velocity w parallel to the axis of z. Faraday tubes start from the sphere and are carried along with it as it moves through the dielectric; since these tubes are moving they will, as we have seen, produce a magnetic field. We shall suppose that the system has settled down into a steady state, so that the sphere and its tubes are all moving with the same velocity w. Let /, g, h be the com- ponents of the polarization at any point, a, /3, y those of the magnetic force. The expressions for X, F, Z, the components of the electromotive intensity, will consist of two parts, one due to the motion of the Faraday tubes and given by equations (6), the other due to the distribution of these tubes and derivable from a potential 'P ; we thus have, if the magnetic permeability is unity, X = w/3 dx 5 v dq/ Y = — wa---7- 3 dy d w, is equal to 2 {F2-™2}*. cos20> hence so that /= F =eF{F2-'iy2}i, e F a: 4w {F2-^ }i{^ + 2/2+F^24 e 9 = TZ y F2—w2' z Thus (15) f_~L=h.. x y z The Faraday tubes are radial and the resultant polarization varies inversely as ( . r2 < 1 + F2 —w2 cos20 }*■ where r is the distance of the point from the centre, and 0 the angle which r makes with the direction of motion of the sphere. We see from this result that the polarization is greatest where 0 = 77/2, least where 0 — 0; the Faraday tubes thus leave the poles of the sphere and tend to congregate at the equator. This arises from the tendency of these tubes to set themselves at right angles to the direction in which they are moving. The surface density of the electricity on the moving sphere varies in- versely as ^ \l + W^?C08 e\ ’FARADAY TUBES OP FORCE. 19 16.] it is thus a maximum at the equator and a minimum at the poles. The components a, ft, y of the magnetic force are given by the equations a — 4wwg = eVw { F2—w>2}* 1^,2 + y2 + _y____ v* V^—w2 \ ft = 4 TSWf y = o. e Vw x } (•«) These expressions as well as (15) were abtained by Mr. Heaviside by another method in the Phil. Mag. for April, 1889. Thus the lines of magnetic force are circles with their centres in and their planes at right angles to the axis of z. When w is so small that w2/V2 may be neglected, the preceding equations take the simpler forms - x_ _ e y , _ e z 4. u r3 5 ^ 4 7r r3 5 4 tt r3 5 eivy ^>3 3 ft — ewx /y» 3 (See J. J. Thomson 4 On the Electric and Magnetic Effects produced by the Motion of Electrified Bodies’, Phil. Mag. April, 1881.) The moving sphere thus produces the same magnetic field as an element of current at the centre of the sphere parallel to z whose moment is equal to ew. When as a limiting case V = w, that is when the sphere is moving with the velocity of light, we see from equations (15) and (16) that the polarization and magnetic force vanish except when z = 0 when they are infinite. The equatorial plane is thus the seat of infinite magnetic force and polarization, while the rest of the field is absolutely devoid of either. It ought to be noticed that in this case all the Faraday tubes have arranged themselves so as to be at right angles to the direction in which they are moving. We shall now consider the momentum in the dielectric due to the motion of the Faraday tubes. Since the dielectric isELECTRIC DISPLACEMENT AND 20 [16. non-magnetic the components U, V\ W of this are by equations (9) given by the following expressions : 62 V2W U = —ph = — xz 4tt V2 — IV2 / 2 9, V2 +y +72_W2 z2) F' = a/i F2w 2/z 4 IT F2— F2 TF: I3f-ag = (x2 + y2+ yz_w2 Z2) V2w (x2+yZ) (17) 4ir F2 —-w2 F2 The resultant momentum at any point is thus at right angles to the radius and to the magnetic force; it is therefore in the plane through the radius and the direction of motion and at right angles to the former. The magnitude of the resultant momentum per unit volume at a point at a distance r from the centre of the sphere, and where the radius makes an angle 0 with the direction of motion, is V2 sin# 47r 1 + w2 V2- COS20> Thus the momentum vanishes along the line of motion of the sphere, where the Faraday tubes are moving parallel to themselves, and continually increases towards the equator as the tubes get to point more and more at right angles to their direction of motion. The resultant momentum in the whole of the dielectric is evidently parallel to the direction of motion; its magnitude I is given by the equation j__e2iu V2 r°° r* r2ir sin2$r2dr sin0d0d(f> ( Vz — iv2 ) __ &%v Y2 f1 sin2 0 d (cos 0) —iv2 .L V2-iv vr V2-iv2 cos20 w or putting {V2 — w2} cos 0 = tan \j/,FAEADAY TUBES OF FOECE. 21 16.] we see that I = or if 1= ' e2V2 a {V2—w2}^ Jo tan-1 - COS' !^(l - Z! W2 sm ’f) d'l'l w rr = S. {F2-w2}i cbTiH1_l5')+isin2H1+^cos2^)}' Thus the momentum of the sphere and dielectric parallel to 0 is miv +1, where m is the mass of the sphere; so that the effect of the charge will be to increase the apparent mass of the sphere by I/w or by e2 V2 C / F2n ^ V2 si i-----™~i~~2)+ 4 sin2$ (1 +1 -y cos 2s)l* a w {V2-n?}il v w2' K w2 ') When the velocity of the sphere is very small compared to that of light, o ^ — y (l + T y-2) approximately, and the apparent increase in the mass of the sphere is 2 ^ 3 a When in the limit w = V the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain con- stant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light. The kinetic energy per unit volume of the dielectric is 87r («2 + n and hence by equations (16) and (17) it is equal to — W- thus the total kinetic energy in the dielectric is equal to \wl,22 ELECTRIC DISPLACEMENT AND [16. that is to £_ 4 a w V2 {F2—^}* K1 ~1 +^sin (x+f ^cos 2s)| ■ We shall now proceed to investigate the mechanical forces acting on the sphere when it is moving parallel to the axis of z in a uniform magnetic field in which the magnetic force is everywhere parallel to the axis of x and equal to H. If ?7, V\ W are the components of the momentum, TJ = go —Kb, V' = ha—fc, W=fb —ga. In this case c = 0, b = /3, a = a + H, where a and /3 have the values given in equations (16). The momentum transmitted in unit time across the surface of a sphere concentric with the moving one has for components JfwUcoaOdS, JjwV' cos 0 dS, JJwW cosOdS, the integration being extended over the surface of the sphere. Substituting the values of Z7, V\ W, we see that the first and third of these expressions vanish, while the second reduces to VHw 4 7T { Y2 — W2]^ JJ oL W2 i- cos 2ddS or eVHw r* cos2 6 sin Odd — 'UptfJ'O {F2— W*}%Jo which is equal to eHwV vr cos2 e }* F*-w* |(F2-w2)t ^F2—/F+«k*j Fit;2 ~ V" yji ) i0%\v-w' f w ( w When w/V is very small this expression reduces to ieHw.FARADAY TUBES OF FORCE. 23 *7-] This is the rate at which momentum is communicated to the sphere, in other words it is the force on the sphere; hence the force on the charged sphere coincides in direction with the force on an element of current parallel to the axis of 0, but the magnitude of the force on the moving sphere is only one-third that of the force on an element of current along z whose moment is ew. By the moment of an element of current we mean the product of the intensity of the current and the length of the element. When w = F, that is when the sphere moves through the magnetic field with the velocity of light, we see from the preceding expression that the force acting upon it vanishes. We can get a general idea of the origin of the mechanical force on the moving sphere if we remember that the uniform magnetic field is (Art. 13) due to the motion of Faraday tubes, the positive tubes moving in one direction, the negative ones in the opposite, and that in their motion through the field these tubes have to traverse the sphere. The momentum due to these tubes when they enter the sphere is proportional to the magnetic force at the place where they enter the sphere, while their momentum when they leave the sphere is proportional to the magnetic force at the place of departure. Now the magnetic forces at these places will be different, because on one side of the sphere the magnetic force arising from its own motion will increase the original magnetic field, while on the other side it will diminish it. Thus by their passage across the sphere the tubes will have gained or lost a certain amount of momentum; this will have been taken from or given to the sphere, which will thus be subject to a mechanical force. Rotating Electrified Plates. 17.] The magnetic effects due to electrified bodies in motion are more conveniently examined experimentally by means of electrified rotating plates than by moving electrified spheres. The latter have, as far as I know, not been used in any experiments on electro-convection, while most interesting experiments with rotating plates have been made by Rowland (Berichte d. Perl. Acad. 1876, p. 211), Rowland and Hutchinson {Phil. Mag. 27, p. 445, 1889), Rontgen (Wied. Ann. 35, p. 264, 1888; 40, p. 93, 1890), Himstedt (Wied. Ann. 38, p. 560, 1889). The general plan of24 ELECTRIC DISPLACEMENT AND [17. these experiments is as follows: an air condenser with circular parallel plates is made to rotate about an axis through the centres of the plates and at right angles to their planes. To prevent induced currents being produced by the rotation of the plates in the earth’s magnetic field, radial divisions filled with insulating material are made in the plates. When the plates are charged and set in rotation a magnetic field is found to exist in their neighbourhood similar to that which would be produced by electric currents flowing in concentric circular paths in the plates of the condenser, the centres of these circles being the points where the axis of rotation cuts the plates. Let us now consider how these magnetic forces are produced. Faraday tubes at right angles to the plates pass from one plate to the other. We shall suppose when the condenser is rotating as a rigid body these tubes move as if they were rigidly connected with it. Then, taking the axis of rotation as the axis of 0, the component velocities of a tube at a point whose coordinates are x, y are respectively — o>y and a>x, where o> is the angular velocity with which the plates are rotating. If these were the only Faraday tubes in motion the com- ponents a, /3, y of the magnetic force would by equations (4) be given by the equations a = 4 7T(TCOX, /3 = 4770-0)2/, (18) y = 0, where a ( = h) is the surface-density of the electricity on either plate. These values for the components of the magnetic force do not however satisfy the relation cla dt3 c?y __ dx + dy dz 9 which must be satisfied since the value of gl~fff (a? + P* + y2)dxdydz must, in a medium whose magnetic permeability is unity, be stationary for all values of a, /3, y which give assigned values to the currents, that is to df3 da da dy dy d(3 dx dy9 dz dx dy dzFARADAY TUBES OF FORCE. 25 17-] For let a0, /30, y0 be any particular values of the components of the magnetic force which satisfy the assigned conditions, then the most general values of these components are expressed by the equations /3 =^o + y = y0 + d(j> d($> da' where is an arbitrary function of x, 0. Then if Iff (a* + £2 + 72) dy dz is stationary, t/yj'(aba-\-j3b/3 + yby) dxdydz = 0. (19) Let the variations in a, /3, y be due to the increment of by an arbitrary function b , then ba = dbcfr dx 8/3 = dbcf) ~dy~3 by = db$ dz Substituting these values for 8 a, 8/3, 8y, and integrating by parts, equation (19) becomes JJb is arbitrary d a j3 dy efo + dy + ” The values of a, /3, y given by equation (18) cannot therefore be the complete expressions for the magnetic force, and since we regard all magnetic force as due to the motion of Faraday tubes, it follows that the tubes which connect the positive to the nega- tive charges on the plates of the condenser cannot be the only tubes in the field which are in motion; the motion of these tubes must set in motion the closed tubes which, Art. 2, exist in their neighbourhood. The motion of the closed tubes will produce a magnetic field in which the forces can be derived from26 ELECTRIC DISPLACEMENT AND [17. a magnetic potential X2. When we include the magnetic field due to the motion of these closed tubes, we have . d£l dSl a = 4-7TfTtoX----7— = —7— 3 dx dx P . d£l dQf c?X2 11 1 t4!- if = 2 7T(T(o (x2 + y2)-Q.-, and since da ^ dfi dy dx dy dz we have d2iY d2Q' d2iY dx2 1 dy2 1 dz2 The question now arises, does the motion of the tubes which connect the positive and negative electrifications on the plates only set those closed tubes in motion which are between the plates of the condenser, or does it affect the tubes outside as well ? Let us examine the consequences of the first hypothesis. In this case, since the Faraday tubes outside the condenser are at rest, the magnetic force will vanish except between the plates of the condenser; it follows, however, from the properties of the magnetic potential that it must vanish inside as well, so that no magnetic force at all would be produced by the rotation of the plates. As this is contrary to the result of Rowland’s experi- ments, the Faraday tubes stretching between the plates must by their rotation set in motion tubes extending far away from the region between the plates. The motion of these closed tubes must however be consistent with the condition that the magnetic force parallel to the plates due to the motion of the tubes must be continuous. Let us consider for a moment the radial magnetic force due to the closed tubes: this may arise either from the rota- tion round the axis of tubes which pass through the plates, or from the motion at right angles to the plates of tubes parallel to them. In the first case, the velocity tangential to the plates of the tubes must be continuous, otherwise the tubes would break, and since the tangential velocity is continuous, the radial magnetic force due to the motion of these tubes will be continuous also. In the second case, the product of the normal velocity of the tubesFARADAY TUBES OF FORCE. 27 17-] and their number per unit volume must be the same on the two sides of a plate, otherwise there would be an accumulation of these tubes in the plate. The product of the normal velocity into the number of the tubes is, however, equal to the tangential magnetic force due to the motion of the closed tubes, so that this must be continuous. The open tubes which stretch from the positive electricity on one plate to the negative on the other will, however, by their motion produce a discontinuity in the radial magnetic force, since these tubes stop at the plates, and do not pass through them. The radial magnetic force at a point due to these tubes is 4 7r o'cor, where r is the distance of the point from the axis of rota- tion. The conditions to determine the magnetic field are thus, (1) that except in the substance of the plates there must be a magnetic potential satisfying Laplace’s equation, and (2) that at either plate the discontinuity in the radial magnetic force must be 4 7r o- co r, where the usual expression for the magnetic force outside the wire produced by a straight current. When the field is steady, there will be as many positive as negative tubes in each unit area, and therefore no electromotive intensity; if however the intensity of the current changes, this will no longer hold. To take an extreme case, let us suppose that the circuit is suddenly broken, then the inertia of the positive tubes will make them continue to move inwards ; and since as the circuit is broken they can no longer shrink to molecular dimen- sions when they enter it, the positive tubes will accumulate in the region surrounding the wire: the inertia of the negative tubes carries them out of this region, so that now there will be a pre- ponderance of positive tubes in the field around the wire. If any conductor is in this field these positive tubes will give rise to a positive current, which is the ‘direct’ induced current which occurs on breaking the circuit. When the field was steady no current would be produced in this secondary circuit, be- cause there were as many positive as negative tubes in its neighbourhood. The Faraday tubes have momentum which they give up when they enter the wire. If we consider a single wire where everything is symmetrical, the wire is bombarded by these tubes on all sides, so that there is no tendency to make it move off in any definite direction. Let us suppose, however, that we have two parallel wires conveying currents in the same direction, let a and b denote the cross-sections of these wires, b being to the right of A. Then some of the tubes which if B were absent would pass into a from the region on the right, will when b is present be absorbed by it, and so prevented from entering A. The supply of positive tubes to A will thus no longer be symmetrical; more will now come into A from the region on its left than from that on its right; hence since each of the tubes has momentum, more38 ELECTRIC DISPLACEMENT AND [25 momentum will come to a from the left than from the right; thus A will be pushed from left to right or towards B. There will thus be an attraction between the parallel currents. 24.] It will be noticed that the tubes in the preceding case move radially in towards the wire, so that the energy which is converted into heat in the circuit comes from the dielectric side- ways into the wire and is not transmitted longitudinally along it. This was first pointed out by Poynting in his paper on the Transfer of Energy in the Electromagnetic Field {Phil. Trans. 1884, Part. II. p. 343). When however the current instead of being constant is alternating very rapidly, the motion of the tubes in the dielec- tric is mainly longitudinal and not transversal. We shall show in Chapter IV that if p is the frequency of the current, or the specific resistance of the wire, a its radius, and its magnetic permeability, then when 4 ir^pa2/ it means that x sets of cells, each containing 600 elements, were connected up in parallel. Experiments with Nitrogen. Iridium electrodes at a distance of 15 mm. Number fixing the experiment Pressure of the nitrogen in 1 2 3 4 5 millimetres of mercury . . 19-65 31-9 53-1 534 524 Number of cells 600 x3 600 x3 600x3 600x4 400 x6 Strength of current in Ampbres Kick of the galvanometer due to the charge in the con- denser produced by the po- tential difference between •535 1-225 14 2-0 24 the electrodes 75-82 25-32 25-32 15-20 17-20 In the first experiment a reddish-yellow positive column stretched at first from the anode to an intensely bright patch on the cathode; the cathode however soon became white hot along the whole of its length, and then showed no trace of the negative glow, nor were any nitrogen lines detected when the region round the cathode was examined by the spectroscope. The tip of the anode was white hot. From the second experiment we see that though the density of the nitrogen was much greater, the potential difference was less than half what it was in the first experiment. This is due to the electrodes being hotter in this experiment than in the pre- ceding one. In the third experiment only half of the cathode was white hot, but the length of the anode which was incan- descent was greater than in the preceding experiment. In the fourth experiment, in which a current of 2 Amperes passed through the gas, the end of the anode was hotter than that of the cathode, in fact with this current the anode, though made of iridium, began to melt. In the ordinary arc lamp, in which we152 THE PASSAGE OE [IS©- have probably a discharge closely resembling that in this ex- periment, the anode is also hotter than the cathode when the current is intense. In this case the gas was quite dark. A very remarkable feature shown by it is the smallness of the potential difference between the electrodes, not amounting to more than 100 volts, though the gas was at the pressure of 53*1 millimetres, and the distance between the electrodes 15 mm. When the electrodes were cold, the battery power used, about 1200 volts, was not sufficient to break down the gas: the discharge had to be started by sending a spark from a Leyden jar through the tube. The conduction through the gas in this case is of the same character as that described in Art. 169. 150.] Hittorf also made experiments on hydrogen and carbonic oxide; the results for hydrogen are given in the following table (Wied. Ann. 21, p. 113, 1884):— Experiments with Hydrogen. Distance of the Iridium electrodes 15 mm. Number fixing the experiment . . 1 2 3 4 5 6 Pressure of hydro- gen in millimetres of Hg 20 33-8 47-05 47-05 47-05 68-55 Number of cells. . 400x6 400x6 400x6 600x4 800x3 800x3 Intensity of current in Ampbres . . -5465 •3415 .3074 -9222 .9905 •8197 Kick of the galvano- meter due to the charge in the con- denser produced by the potential difference between the electrodes. . 100 107-108 110 100-110 107-110 110 In experiment 1 the pressure and the current were almost the same as for experiment 1, Art. 149, in nitrogen; the potential difference between the electrodes was however much greater in hydrogen than in nitrogen, though the potential difference required to initiate a discharge in hydrogen is considerably less than in nitrogen. In these experiments the potential dif- ference between the electrodes for this dark discharge seems almost independent of the current and of the density of the gas.I52.] ELECTRICITY THROUGH GASES. 153 Experiments with Carbonic Oxide Gas. Distance between Iridium electrodes 15 mm. Number fixing the experiment Pressure of CO in millimetres 1 2 3 4 of mercury 131 22-75 517 75-85 Number of cells 800x3 800x3 800x3 800x3 Intensity of current in Amperes Kick of the galvanometer due to the charge in the con- denser produced by the po- tential difference between •8880 -9734 1-3662 1-2978 the electrodes 92-100 89-92 40 42 c_> c_:> The great fall in potential, which occurs between experiments 2 and 3 on CO, was accompanied by a loss of luminosity; in 1 and 2 there was a little positive blue light at the anode, but in 3 this had disappeared, and the discharge was quite dark and showed in the spectroscope no trace of the carbonic oxide bands. 151. ] When repeating these experiments with carbon elec- trodes instead of iridium ones, Hittorf found that with strong currents and at pressures between 10 mm. and 2 mm. the discharge through hydrogen took a very peculiar form, it consisted of ring-shaped striae, the insides of which were dark. These rings extended through the tubes and encircled both the anode and the cathode, as shown in Fig. 65. 152. ] The preceding experiments show that when the electrodes are white hot, the negative glow disap- pears, and the potential difference between the elec- trodes when a current is passing through the gas sinks to a fraction of the value it has when the electrodes are cold and the negative glow exists. Hittorf (Wied. Ann. 21, p. 133) has shown by a direct experiment that when the cathode is white hot a very small electromotive force is sufficient to maintain the discharge. The arrangement he used is shown in Fig. 66. A thin carbon filament which serves as a cathode is stretched between two conductors mn% and can be raised to a white heat by a current passing through it and these con- ductors ; the anode a is vertically below the cathode and remains cold. When the pressure was very low, Hittorf found that 1 cell of his battery, equivalent to about 2 volts, would maintain a current between the anode and cathode when they were separated Fig. 65.154 THE PASSAGE OP CI53* by 6 cm.; in this case the discharge was quite dark. When ten or more cells were used a pale bluish light spread over the anode. It should be noticed that the single cell does not start the cur- rent, it only maintains it: the cur- rent has previously to be started by the application of a much greater potential difference. Hit- torf generally started the current by discharging a Leyden jar through the tube. No current at all will pass if the poles are re- versed so that the anode is hot and the cathode cold. In these experiments it is necessary for the cathode to be at a white heat for an appreciable current to pass between the electrodes; very little effect seems to be produced on the potential difference at the cathode until the latter is hotter than a bright red heat. The current produced by a given electro- motive force is greater at higher exhaustions than at low ones, but Hittorf found he could get ap- preciable effects at pressures up to 9 or 10 mm. 153.] In considering the results of experiments in which carbon filaments or platinum wires are raised to incandescence, we must remember that, as Elster and Geitelhave shown (Art. 43), electri- fication is produced by the incandescent body, the region around which receives a charge of electricity; though whether the carrier of this charge is the disintegrated particles of the incandescent wire, or the dissociated molecules of the gas itself, is not clear. This electrification often makes the interpretation of experiments in which incandescent bodies are used ambiguous. Thus for example, Hittorf in one experiment (Wied. Ann. 21, p. 137, 1884) used a U-shaped discharge tube, in one limb of which a carbon filament was raised to incandescence; the other limb of the tube contained a small gold leaf electroscope; when the pressure of the gas in the tube was very low, Hittorf found thatELECTRICITY THROUGH GASES. 155 156.] the electroscope would retain a charge of negative electricity but immediately lost a positive charge. This experiment does not however show conclusively that positive electricity escapes more easily than negative from a metal into a gas which is in the con- dition in which it conducts electricity, because the same effect would occur if the incandescent carbon filament produced a negative electrification in the gas around it. 154. ] The way in which the passage of electricity from metal to gas, or vice versd, is facilitated by increasing the temperature of the metal to the point of incandescence is illustrated by an effect observed in the experiments on hot gases described in Art. 37. It was found that when a current was passing between electrodes immersed in a platinum tube at a bright yellow heat and containing some gas, such as iodine, which conducts well, the current was at once stopped if a large piece of cold platinum foil was lowered between the electrodes, although there was a strong up-current of gas in the tube which prevented a cold layer of gas being formed against the platinum foil: as soon, however, as the foil became incandescent the current from one or two Leclanche cells passed freely. It would appear, therefore, that even when the gas is in the condition in which it conducts electricity freely, some of the cathode potential difference will remain as long as the cathode itself is not incandescent. 155. ] The passage of electricity from a gas to a negative elec- trode seems, as we shall see later, to require something equivalent to chemical combination between the charged atoms of the metal and the atoms of the gas which carry the discharge; and the reason for the disappearance of the fall in potential at the cathode when the latter is incandescent is probably due to this combina- tion taking place under these circumstances much more easily than when the electrode is cold. 156. ] Warburg (Wied. Ann. 31, p. 545, 1887 : 40. p. 1, 1890) has made a valuable series of experiments on the circumstances which influence the fall of potential at the cathode. He has in- vestigated the effect produced on this fall by altering the gas, the size and material of the electrodes, and the amount of im- purity in the gas. Hittorf, as we have seen, had already shown that as long as there is room for the negative glow to spread over the surface of the cathode, the cathode fall in potential is approximately independent of the intensity of the current.156 THE PASSAGE OP [15 7< In Warburg’s experiments, the fall in potential at the cathode, by which is meant the potential difference between the cathode and a wire at the luminous boundary of the negative glow, was measured by a quadrant electrometer. Warburg found that, so long as the whole of the cathode was not covered by the negative glow, the fall in potential at the cathode was nearly independent of the density of the gas : this is shown by the following table (1. c. p. 579), in which E represents the potential difference be- tween the electrodes, which were made of aluminium, e the poten- tial fall at the cathode, E and e being measured in volts, p the pressure of the gas, dry hydrogen, measured in millimetres of mer- cury, i the current through the gas in millionths of an Ampere. e. E-e. t. 9.5 191 139 6140 6.4 190 103 4740 4.4 190 70 4810 3.0 189 50 2640 1.79 191 40 1730 1.20 192 39 1360 •80 191 39 508 This table shows that though the fall in potential in the positive light decreased as the pressure diminished, the fall in potential at the cathode remained almost constant. 157.] In imperfectly dried nitrogen, which contained also a trace of oxygen, the cathode potential difference depended to some extent on the metal of which the electrode was made; platinum, zinc, and iron electrodes had all practically the same potential fall; for copper electrodes the fall was about 3 per cent, and for aluminium electrodes about 15 per cent, less than for platinum. In hydrogen which contained a trace of oxygen, the potential fall for platinum, silver, copper, zinc, and steel was practically the same, about 300 volts. In the case of the last three metals, however, the value of the cathode potential fall at the beginning of the experiment was much lower than 300 volts, and it was not until after long sparking that it rose to its normal value; Warburg attributed this to the presence at the beginning of the experiment of a thin film of oxide which gradually got dissipated by the sparking; he found by direct experiment that the potential fall of a purposely oxidised steel electrode was less than the value reached by a bright steel electrode afterELECTRICITY THROUGH GASES. 157 I59-] being used far some time. The potential fall for aluminium and magnesium electrodes was about 180 volts, and was thus consider- ably smaller than for platinum electrodes (cf. Art. 47); these metals, however, are easily oxidised ; and as, unlike other metals, they do not disintegrate when used as cathodes, the film of oxide would not get removed by use. 158. ] The fact that a large number of metals give the same potential fall, while others give a varying one, seems to indicate that this potential fall depends upon whether the electrodes do or do not take part in some chemical change occurring at the cathode; and the connection between this fall in potential and the chemical changes which take place near the cathode seems still more clearly shown by the surprisingly large effects pro- duced by a small quantity of impurity in the gas. Warburg found that the fall of potential at the cathode in nitrogen which contained traces both of moisture and oxygen was 260 volts, while the same nitrogen, after being very carefully dried, gave a cathode fall of 343 volts: thus, in this case, a mere trace of moisture had diminished the cathode fall by 25 per cent., the removal of the trace of oxygen produced equally remarkable effects, see Art. 160. This points clearly to the influence exerted by chemical actions at the cathode on the fall of potential in that region; since a mere trace of a substance is often suffi- cient to start chemical reactions which would be impossible without it: thus, for example, Pringsheim ( Wied. Ann. 32, p. 384, 1887) found that unless traces of moisture were present, hydrogen and chlorine gas would not combine to form hydro-chloric acid under the action of sunlight unless it was very intense. 159. ] The fall of potential at the cathode seems to be lowered as much by a trace of moisture as by a larger quantity, as long as the total quantity of moisture in the nitrogen remains small; if, however, the amount of aqueous vapour is considerable, the fall in potential is greater than for pure nitrogen; thus in a mixture of nitrogen and aqueous vapour, in which the pressure due to the nitrogen was 3-9 mm., that due to the aqueous vapour 2-3 mm., Warburg found that the fall in potential was about 396 volts, as against about 343 volts for nitrogen containing a trace of oxygen; the increase in the fall of potential at the cathode was, however, not nearly so great comparatively as the increase in the potential differences along the positive column.158 THE PASSAGE OP [l62. In hydrogen, Warburg found that a trace of aqueous vapour increased the potential difference at the cathode instead of diminishing it as in nitrogen. 160. ] Warburg (Wied. Ann. 40, p. 1, 1890) also investigated the effects produced by removing from the nitrogen or hydrogen any trace of oxygen that might have been present. This was done by placing sodium in the discharge tube, and then after the other gas had been let into the tube, heating up the sodium, which combined with any oxygen there might be in the tube. The effect of removing the oxygen from the nitrogen was very remarkable: thus, in nitrogen free from oxygen, the fall of potential at the cathode when platinum electrodes were used was only 232 volts as against 343 volts when there was a trace of oxygen present; when magnesium electrodes were used the fall in potential was 207 volts ; in hydrogen free from oxygen the fall of potential was 300 volts with platinum electrodes, and 168 volts with magnesium electrodes; thus with platinum electrodes the potential fall in hydrogen is greater than in nitrogen, while with magnesium electrodes it is less. 161. ] Warburg also investigated a case in which the conditions for chemical change at the cathode were as simple as possible, one in which the gas was mercury vapour (with possibly a trace of air) and the cathode a mercury surface; he found that the negative dark space was present, and that the cathode fall was very considerable, amounting to about 340 volts; this, at the pressures used in these experiments between 3-5 mm. and 14*0 mm., was much greater than the potential difference in a portion of the positive light about half as long again as the piece at the cathode, for which the potential fall was measured. 162. ] In air free from carbonic acid, but containing a little moisture, Warburg (Wied. Ann. 31, p. 559, 1887) found that the potential fall was about 340 volts : this is very nearly the value found by Mr. Peace for the smallest potential difference which would send a spark between two parallel plates. When we consider the theory of the discharge we shall see that there are reasons for concluding that it is impossible to produce a spark by a smaller potential difference than the cathode fall of potential in the gas through which the spark has to pass. The researches made by Hittorf on the distribution of poten- tial along the tube show, as we have seen, Art. 140, that the165.] ELECTRICITY THROUGH GASES. 159 potential gradient is by no means constant; to produce the changes in this gradient which occur in the neighbourhood of the cathode, there must in that region be a quantity of free electricity in the tube. Schuster (Proc. Roy. Soc. 47, p. 542,1890) concludes from his measurements of the potential in the neigh- bourhood of the cathode that if p is the volume density of the free positive electricity at a distance x from the cathode, p varies as €-**. 163. ] The measurements of potential along the positive column have been less numerous than those of the negative dark space. Hittorf, De la Rue and Hugo Muller concur in finding that the potential gradient close to the anode is, though not comparable with that at the cathode, greater than that in the middle of the tube. 164. ] The potential gradient in the positive column is not like the fall in potential at the cathode approximately independent of the density, it diminishes as the pressure of the gas diminishes: but as the pressure of the gas diminishes, the distance between two consecutive striations increases, and though I can find no experiments bearing on this point, it would be a matter of great interest to know whether or not the potential difference along a length of the positive column equal to the distance between two striations, where these are regular, is approximately independent of the density of the gas. 165. ] De la Rue and Hugo Muller (Phil. Trans. 1878, Part I, p. 159) measured the potential gradients along a tube in which two wide portions were connected by a piece of capillary tubing, narrow enough to constrict the striae; they found the potential gradient much greater along the capillary portion than along the wide one. Thus the potential difference along 4*25 inches of the positive column in the wide tube, which was about yf of an inch in diameter, was, on an arbitrary scale, 75, while the potential difference along a portion of the positive column, which included 2 inches of the wide tube and 3*75 inches of the capillary tube (| of an inch in diameter), was 138 ; the potential gradients along the wide and narrow portions are thus in the proportion of 1 to 1-55. In this case the cathode was in the wide part of the tube; when the tube round the cathode is so narrow that it re- stricts the negative glow, the increase in the potential differ-160 THE PASSAGE OE [166. ence at the cathode produced by this restriction makes it very much more difficult to get a discharge to pass through the narrow tube than through a wider one. An experiment due to Hittorf (Wied. Ann. 21, p. 93, 1884) illustrates this effect in a very remarkable way; at a pressure of *03 mm. of mercury, it took 1100 of his cells to force the discharge through a tube 1 cm. in diameter, while 300 cells were sufficient to force it between similar electrodes the same distance apart in a tube 11 cm. in diameter, filled with the same kind of gas at the same pressure. 166.] When the electrodes are placed so near together that the dark space round the cathode ex- tends to the anode, the appearance of the discharge is completely changed : this is very well shown in an ex- periment due to Hittorf (Pogg. Ann. 136, p. 213, 1869) represented in Fig. 67; the electrodes were parallel to each other, and the pressure of the gas in the discharge tube was so low that the dark space round the cathode extended beyond the anode ; the posi- tive discharge in this case, instead of turning towards the cathode, started from the bend in the anode on the side furthest away from the cathode, and then crept along the surface of the glass until it reached the boundary of the negative dark space. I observed a similar effect in the course of some experiments on the discharge between large parallel plates (Proc. Camb. Philos. Soc. 5, p. 395,1886); when the pressure of the gas was very small, the positive column, instead of passing between the plates, went, as in Fig. 68, from the under side of the lower plate which was the positive electrode, and166.] ELECTRICITY THROUGH GASES. 161 after passing between the glass and the plates reached right up to the negative glow, which was above the negative plate: the space between the plates was quite dark and free from glow. a b Fig. 69. Lehmann (Molekularphysik, bd. 2, p. 295) has observed with a microscope the appearance of the discharge passing between electrodes of different shapes, placed very close together; they exhibit in a very beautiful way the same peculiarities as those just described; Lehmann’s figures are represented in Fig. 69. M162 THE PASSAGE OF [167. When the distance between the electrodes is less than the thickness of the dark space, it is very difficult to get the dis- charge to pass between them ; this is very strikingly illustrated by another experiment of Hittorf’s (WiecL Ann. 21, p. 96,1884) which is represented in Fig. 70. The two electrodes were only 1 mm. apart, but the regions surrounding them were connected by a long spiral tube 3f m. long ; in spite of the enormous difference between the lengths of the two paths, the discharge, when the pressure was very low, all went round through the spiral, and the space between the electrodes remained quite dark. 167.] In cases of this kind the potential difference required to produce discharge between two electrodes must be diminished by increasing the distance between them. For in Hittorf’s experiments, the potential difference between the electrodes was equal to the potential fall at the cathode, plus the change in potential due to the 3| m. of positive light in the spiral, while if the shortest distance between the electrodes had been increased until it was just greater than the thickness of the negative dark space, the potential difference between the electrodes when the discharge passed would only have amounted to the cathode fall, plus the potential difference due to a short positive column instead of to one 3f metres long, so that the potential difference would have been less than when the electrodes are nearer together. Peace’s experiment described in Art. 53. is a direct proof of the truth of this statement for higher pressure, and is free from the objection to which the preceding deduction from Hittorf s experiment is liable, that the cathodeELECTRICITY THROUGH GASES. 163 169.] fall may not be the same when the discharge starts in the large vessel when the negative glow is unrestricted, as it is when the discharge passes through the narrow tubes, the walls of which constrict the negative glow. 168. ] These results explain a peculiar effect which is observed when the discharge passes between slightly curved electrodes at not too great a distance apart; until the pressure is very low the discharge passes across the shortest distance between the electrodes, but after a very low pressure is reached the discharge leaves the centre of the field, and in order to get a longer spark length departs further and further from it as the pressure of the gas is reduced. The Arc Discharge. 169. ] The ; arc discharge,’ of which the well-known arc lamp is a familiar example, is characterised by the passage of a large current and the incandescence of both the terminals, as well as by the comparatively small potential difference between them ; we considered a case of this discharge in Art. 148, the gas was, how- ever, in that case, at a low pressure; the cases when the gas is at higher pressures are of special interest, on account of the exten- sive use made of this form of discharge for lighting purposes. If the current through a vacuum tube with electrodes is gradu- ally increased, the discharge, as Gassiot found in 1863, gradually changes from the ordinary type of the vacuum tube discharge with the negative space and a striated positive column to the arc discharge, in which there is comparatively little difference between the appearances at the terminals. The terminals are brilliantly incandescent while the gas remains comparatively dark, being probably in the state in which it has a large supply of dissociated molecules by means of which it can transmit the current even though the potential gradient is small. The connection between spark length, potential difference and current in the arc discharge, has been investigated by many physicists, who have all found that the potential difference V is almost independent of the current and can be expressed by the formula V = a + bl, where l is the spark length and a and h are constants. Ayrton m 2164 THE PASSAGE OF [I7I. and Perry (Phil. Mag. [5] 15, p. 346, 1883), using a formula which is identical with the preceding one if the sparks are not very short, found that for carbon electrodes a — 63 volts and h= 21*6 volts, if l is measured in centimetres. The value of a probably depends on the quality of the carbon of which the elec- trodes are made, as other observers, who have also used carbon electrodes, have found considerably smaller values for a. When more volatile substances than carbon are used the values of a are smaller, the more volatile the substance the smaller in general being the value of a. This is borne out by the following deter- minations made by Lecher (Wied. Ann. 33, p. 625, 1888); the length l in these equations is measured in centimetres, and V in volts:— Horizontal Carbon Electrodes v = 33 + 451. Vertical Carbon Electrodes V = 35-5 + 571. Platinum Electrodes, (-5 cm. in diameter) V = 28 + 4 ll. Iron Electrodes, (-55 cm. in diameter) V = 20 + 501. Silver Electrodes, (-49 cm. in diameter) . v- 8 + 601. 170.] The form of the expression for V shows that the potential required to maintain the current between two incandescent electrodes cannot fall short of a certain minimum value, however short the arc may be. The preceding measurements for a show that this potential difference, though small compared with the 4 cathode fall * when the electrodes are cold, is much greater than that which Hittorf in his experiments (see Art. 152) found necessary to maintain a constant current when the cathode was incandescent; we must remember, however, that in Lechers experiments the gas was at atmospheric pressure, while in Hittorf’s the pressure was very low. 171.] Lecher (1. c.) investigated the potential gradient in the arc by inserting a spare carbon electrode, and found that it was far from uniform : thus when the difference of potential between the anode and the cathode was 46 volts, there was a fall of 36 volts close to the anode, and a smaller fall of ten volts near the cathode. The result that the great fall of potential in the arc discharge occurs close to the anode is confirmed by an experi- ment made by Fleming (Proc. Roy. 80c. 47, p. 123, 1890), in which a spare carbon electrode was put into the arc; when this electrode was connected with the anode sufficient current went165 174.] ELECTRICITY THROUGH * GASES. round the new circuit to ring an electric bell, but when it was connected to the cathode the current which went round the circuit was not appreciable. 172. ] The term in the expression for the potential in Art. 169, which is independent of the length of the arc, and which involves an expenditure of energy when electricity travels across an infinitesimally small air space, is probably connected with the work required to disintegrate the electrodes, since the more volatile are the electrodes the smaller is this term. 173. ] The disintegration of the electrodes is a very marked feature of the arc discharge, and it is not, as in the case when small currents pass through a highly exhausted gas, con- fined to the negative elec- trode; in'fact, when carbon electrodes are used, the loss in weight of the anode is greater than that of the cathode, the anode getting hollowed out and taking a crater-like form. 174. ] Perhaps the most interesting examples of the arc discharge are those which occur when we are able by means of transformers to pro- duce a great difference of potential, say thirty or forty thousand volts between two electrodes, and also to trans- mit through the arc a very considerable current. In this case the arc presents the appearance illustrated in Fig. 71. The discharge, instead of pass- ing in a straight line between the electrodes, rises from the electrodes in two columns which unite at the top, where stria- tions are often seen though these do not appear in the photo- graph from which Fig. 71 was taken. The vertical columns are sometimes from eighteen inches to two feet in length, they flicker slowly about and are very easily blown out, a very slight puff of166 THE PASSAGE OF [X76. air being sufficient to extinguish them. The air blast apparently breaks the continuity of the belt of dissociated molecules along which the current passes, and the current is stopped just as a current through a wire would be stopped if the wire were cut. The discharge is accompanied by a crackling sound, as if a number of minute sparks were passing between portions of the arc temporarily separated by very short intervals of space. 175. ] The relation between the losses of weight of the anode and the cathode in the arc discharge depends however very much on the material of which the electrodes are made; thus Mat- teucci (Comptes Rendus, 30, p. 201, 1850) found that for copper, silver and brass electrodes the cathode lost more than the anode, while for iron the loss in weight of the anode was greater than that of the cathode. The electrodes in the arc discharge are at an exceedingly high temperature, in fact probably the highest temperatures we can produce are obtained in this way. With carbon electrodes the anode is much hotter than the cathode (compare Art. 149). Since the temperature of the electrodes is so high, it is probable that they are disintegrated partly by the direct action of the heat and not wholly by purely electrical processes such as those which occur in electrolysis; for this reason, we should not expect to find any simple relation between the loss in weight of the electrode and the quantity of electricity which has passed through the arc. Grove (Phil. Mag. [3] 16, p. 478, 1840), who used a zinc anode sufficiently large for the temperature not to rise about its melting point, came to the conclusion that the amounts of zinc lost and oxygen absorbed by the electrode were chemically equivalent to the oxygen liberated in a voltameter placed in the circuit. On the other hand, Herwig, (Pogg. Ann. 149, p. 521, 1873), who investigated the relation between the loss of weight of a silver electrode in the arc and the amount of chemical decomposition in a voltameter placed in the same circuit, was however unable to find any simple law connecting the two. The brightness of the light given by carbon electrodes is much increased by soaking them in a solution of sodium sulphate. 176. ] The particles projected from the electrodes in the arc discharge are presumably charged with electricity, since they are deflected by a magnet; thus some of the electricity passingELECTRICITY THROUGH GASES. 167 177-] between the electrodes will be carried by these particles. Com- paratively few experiments bearing on this point have, however, been made on the arc discharge, and we have not the information which would enable us to estimate how much of the current is carried by the disintegrated electrodes and how much by the gas. Fleming (Proc. Roy. Soc. 47, p. 123, 1890) has suggested that all the current is carried by particles tom off the electrodes, that these particles are projected (chiefly from the cathode) with enormous velocities, and that the incandescence of the electrodes is due to the heat developed by their bombardment by these particles; the hollowing out of the anode is on this theory supposed to be due to a kind of sand blast action exerted by the particles coming from the negative electrode. On this theory, if I understand it rightly, the gas by which the electrodes are enveloped plays no part in the discharge. I do not think that the theory is consistent with Hittorfs and Gassiot’s observations on the continuity of the arc discharge with the ordinary striated discharge produced in a vacuum tube through which only a very small current is passing, nor does it seem in accordance with what we know about the high conductivity of gases which are at a high temperature or through which an electric discharge has recently passed. The Heat produced by the Discharge. 177.] Though the electric discharge is generally accompanied by intense light, the average temperature of the molecules of the gas through which it passes is often by no means high. Thus E. Wiedemann (Wied. Ann. 6, p. 298, 1879) has found that the average temperature of a column of air at a pressure of about 3 mm. made luminous by the passage of the discharge can be under 100° C. As, however, any instrument which we may use to measure the temperature of the gas merely measures the average temperature of molecules filling a considerable space, the fact that this temperature is low does not preclude the existence of a small number of molecules moving with velocities immensely greater than the mean velocity corresponding to the temperature indicated by the thermometer. On the other hand, the fact that the gas is luminous during the discharge does not afford conclusive evidence of the168 THE PASSAGE OF [l78. existence of molecules in a state comparable with that of the majority of the molecules in a gas at a very high temperature, for mere increase of temperature unaccompanied by chemical changes seems to have little effect in increasing the luminosity of a gas ; thus in one of Hittorf s experiments already men- tioned, where the temperature of the electrodes was great enough to melt iridium, the gas surrounding them when examined by the spectroscope did not show any spectroscopic lines. It would seem that the interchange of atoms between the mole- cules which probably goes on when the discharge passes through the gas is much more effective in making it luminous than mere increase in temperature unaccompanied by chemical changes. 178.] Many experiments have been made by G. and E. Wiede- mann, Hittorf, and others on the distribution along the line of discharge of the heat produced by the spark. Hittorf s experi- ments are the easiest to interpret, since by means of a large battery he produced through the discharge tube a current which, if not absolutely continuous, was so nearly so, that no want of continuity could be detected either by a revolving mirror or by a telephone; the gas had therefore a much better chance of getting into a steady state than if intermittent discharges such as those produced by an induction coil had been used. Hittorf (Wied. Ann. 21, p. 128, 1884) inserted three thermo- meters in the discharge tube, one close to the cathode, another in the bright part of the negative glow, and the third in the positive column. He found, using small currents and low gaseous pressures, that the temperature of the thermometer next the cathode was the highest, that of the one in the negative glow the next, and that of the one in the positive column the lowest. The distribution of temperature depends very much upon the intensity of the current. Hittorf found that when the strength was increased the difference between the temperatures of his thermometers increased also. When however the increase in the current is so great that the discharge becomes an arc discharge, then, at any rate when carbon electrodes are used, the temperature at the anode is higher than that of the cathode; with weak currents we have seen that it is lower. E. Wiedemann (Wied. Ann. 10, p. 225 et seq., 1880) found that the distribution of temperature along the discharge depended on the pressure. In his experiments the temperature at the anodeELECTRICITY THROUGH GASES. 169 I79-] was slightly higher than that at the cathode when the pressure was about 26 mm. of mercury, at lower pressures the cathode was the hotter, and the difference between the temperatures of the cathode and the anode increased as the pressure diminished. Differences between the Phenomena at the Positive and Negative Electrodes. 179.] We have seen already that when the pressure of the gas is small the two electrodes present very different appearances, there are however many differences between an anode and a cathode even at atmospheric pressure. The appearance of the spark discharge at the two electrodes is different. The following figure is from a photograph of the spark in air at atmospheric pressure. It will be noticed that the sparks seem to reach a definite point on the negative elec- trode, but to spread over a considerable area of the positive. Bright dots of light are often to be seen on the positive electrode but not on the negative, these are still more striking at lower pressures. When the spark is branched as in Fig. 73, the branches point to the negative electrode. If the electrodes are not of the same size, the spark length for the same potential difference seems to depend upon whether the larger or smaller electrode is used as the cathode, though it is a disputed question whether this difference exists if the spark is not accompanied by some other form of discharge.170 THE PASSAGE OP [ 181. Thus, if for example the electrodes are spheres of different sizes, Faraday (Experimental Researches, § 1480) found that the spark length was greater when the smaller sphere was positive than when it was negative. We may express this result by saying that when the electric field is not uniform the gas does not break down so easily when the greatest electromotive intensity is at the cathode as it does when it is at the anode. Macfarlane’s measurements {Phil, Mag. [5] 10, p. 403, 1880) of the potential difference required to start a discharge between a ball and a disc are in accordance with this result, as he found that for a given length of spark the potential difference between the electrodes was smaller when the ball was positive than when it was negative. 180. ] De la Rue and Hugo Muller {Phil. Trans. 1878, Part I, p. 55) observed analogous effects in the experiments they made with their large chloride of silver battery on the sparking distance between a point and a disc. They found that for potential differ- ences between 5000 and 8000 volts the sparking distance was greatest when the point was positive and the disc negative, while for smaller potential differences they found that the opposite result was true. The appearance of the discharge at the positive point they found was different from that at the negative. The discharge at the negative point is represented in Fig. 74, that at the positive in Fig. 75. 181. ] Wesendonck (Wied. Ann. 38, p. 222, 1889), however, concludes from his experiments that there are no polar differ- ences of this kind when the discharge passes entirely as a spark, and that the differences which have been observed are due to the coexistence of other kinds of discharge such as a brush and glow. The existence of this kind of discharge would put the gas into a condition in which it is electrically weak and thus ill- Fig. 74. Fig. 75.ELECTRICITY THROUGH GASES. 171 182.] fitted to resist the passage of the spark. This explanation does not seem inconsistent with Faraday’s experiment, for, as we shall see in the next paragraph, the negative brush is formed more easily than the positive one. Thus if the sparks in his experi- ments only passed when they were preceded by the formation of brushes at both the electrodes, it might be produced if the greatest electromotive intensity was at the place where the brush was formed with the greatest difficulty—the anode—while it might not be produced if the smallest intensity was at the anode, thus the gas would be electrically weaker in the first case than in the second. 182.] Considerable polar differences seem undoubtedly to occur in the brush and glow discharges. Thus Faraday [Experimental Researches, § 1501) found that if two equal spheres were electri- fied until they discharged their electricity by a brush discharge into the air, the discharge occurred at a lower potential for the negative ball than for the positive; more electricity thus accumulates on the positive ball than on the negative before the brush occurs, so that when the positive brush does take place it is finer than the negative one. The brush discharge is also intermittent, and since the positive brush requires a greater accumulation of electricity than the negative one, the interval between consecutive discharges is greater for the positive than for the negative brush. Fig. 76. The positive and negative brushes are represented in Fig. 76, copied from a figure given by Faraday. In the brush discharge the electricity seems to be carried partly by particles of metal torn from the electrodes. Nahrwold (Wied. Ann. 31, p. 473, 1887) has confirmed the conclusion that the negative brush is more easily formed than the positive. Wesendonck (Wied. Ann. 39,p,601,1890) has shown that when the discharge passes as a glow discharge from a point into air, hydrogen, or nitrogen, the potential at which the discharge begins is less when the point is negative than when it is positive.172 THE PASSAGE OF [183. Lichtenberg’s Figures and Kundfs Dust Figures. 183.] Very tangible differences between the discharges from the positive and negative.electrodes at ordinary pressures are obtained if we allow the discharge from one or other of the electrodes to pass on to a non-conducting plate covered with Fig. 77. some badly conducting powder. If, for example, we powder a plate with a mixture of red minium and yellow sulphur and then cause a discharge from a positively electrified point to pass to the plate, the sulphur, which by friction against the minium is negatively electrified, adheres to the positively electrified parts of the plate, and will be found to be arranged in a star-like form like that represented in Fig. 77. If, on the other hand, the discharge is taken from a negatively electrified body the appearance of the minium on the plate is that represented in Fig. 78. These are known as Liohten- berg’s figures ; the positive ones are larger than the negative. Fig. 78. If the electrodes are made of very bad conductors, such as wood, there is no difference between the positive and the negative figures.ELECTRICITY THROUGH GASES. 173 185] 184.] Very beautiful figures are obtained if a plate of glass covered with a non-conducting powder, such as lycopodium* is placed on a metal plate, and two wires connected with the poles of an induction coil made to touch the powdered surface of the glass. When the discharge passes the powder arranges itself in patterns which are finely branched and have a moss-like appear- ance at the anode and a more feathery or lichenous appearance at the cathode. The accompanying figure is from a paper by Joly (Proo. Roy. Soc. 47, p. 84, 1890); the negative electrode is on the deft. Fig. 79. 185.] As Lehmann has remarked (Molekularphysik, b, 11, p. 303), the differences between the positive and negative figures are what we should expect if the discharge passed as a brush from the positive electrode and as a glow from the negative one. He has verified by direct observation that this is frequently the case. Fig. 80. A good deal of light is also, I think, thrown on the difference between the positive and negative figures by Fig. 80, which is given by De la Rue and Hugo Muller (Phil. Trans. 1878, Part I, p. 118) as the discharge produced by 11,000 of their174 THE PASSAGE OF [i87. chloride of silver cells in free air. It will be noticed that there is at the negative electrode a continuous discharge superposed on the streamers which are the only form of discharge at the positive, this continuous discharge will fully account for the comparative want of detail in the negative figure. 186. ] Kundt’s figures are obtained by scattering non-conduct- ing powders over a horizontal metal plate, instead of, as in Lichtenberg’s figures, over a non-conducting one. If the plate be shaken after a discharge has passed from a negative point to the positive plate, it will be found that the powder will fall from. every part of the plate except a small circle under the negative electrode, where the powder sticks to the plate and forms what is called Kundt’s ‘ dust figure.’ The dimensions of this circle are very variable, ranging in Kundt’s original experiments (Pogg. Ann. 136, p. 612, 1869) from 10 to 200 mm. in diameter. If the point is positive and the plate negative Kundt’s figures are only formed with great difficulty. Mechanical Effects produced by the Discharge. 187. ] We have already considered the mechanical effects pro- duced by the projection of particles from the cathode: many other such effects are however produced by the electric discharge. One of the most interesting of these is that described by De la Rue and Hugo Muller {Phil. Trans. 1880, p. 86): they found that when the discharge from their large chloride of silver battery passed through air at the pressure of 53 mm. of mercury, the pressure of the air was increased by about 30 per cent., and they proved, by measuring the temperature, that the increase in pressure could not be accounted for by the heat produced by the spark. This effect can easily be observed if a pressure gauge is attached to any ordinary discharge tube, the gas inside being most conveniently at a pressure of from 2 to 10 mm. of mercury. At the passage of each spark there is a quick movement of the liquid in the gauge as if it had been struck by a blow coming from the tube; immediately after the passage of the spark the liquid in the gauge springs back to within a short distance of its position of equilibrium, and then slowly creeps back the rest of the way. This creeping effect is probably due to the slow escape of the heat produced by the passage of the spark.ELECTRICITY THROUGH GASES. 175 190.] The gauge behaves as if a wave of high pressure rushed through the tube when the spark passed. 188. ] Meissner, Abhand. dev Konig. Gesellschaft, Gottingen, 16, p. 98 et seq., 1871 (who seems to have been the first to observe this effect, though in his experiments it was not developed to such an extent as in De la Rue’s and Muller’s), found that if a tube provided with a gauge were placed between the plates of a condenser there was an increase of pressure when the plates were charged or discharged, and no effect as long as the charge on the condenser remained constant. In this case there was no spark between the plates of the condenser, and the effect must have been due to the passage through the gas of the electricity which, when it was in equilibrium before the spark passed, was spread over the glass of the tube. Meissner observed this effect when the tube was filled with oxygen, hydrogen, carbonic acid, and nitrogen, though it was very small when the tube was filled with hydrogen. 189. ] The effect seems too great to be accounted for merely by the increased statical pressure due to the decomposition of the molecules of the gas by the discharge, for in De la Rue’s experiment, where the gas was contained in a large vessel and the discharge passed as a narrow thread between the elec- trodes, the pressure was increased by about 30 per cent. Now if this increase of pressure was due to the splitting up of the molecules into atoms it would require about one-third of the molecules to be so split up by a discharge which only occupied an infinitesimal fraction of the volume of the gas. 190. ] It would seem more probable that in this case we had something analogous to the driving off of particles from an electrified point, as in the ordinary phenomenon of the ‘ electrical wind,’ or that of the projection of particles from , the cathode which occurs when the discharge passes through a gas at a very low pressure; the difference between this case and the one we are considering being that in the latter, since the pressure is greater, the molecules shot off from the cathode communicate their momentum to the surrounding gas instead of retaining it until they strike against the walls of the discharge tube. This would have the effect of diminishing the density of the gas in the neighbourhood of the line of discharge, and would therefore increase the density and pressure in other parts of the tube.THE PASSAGE OP 176 [192. 191.] Topler (Pogrgr. Ann. 134, p. 194, 1868) has investigated by means of a stroboscopic arrangement the disturbance in the air produced by the passage of a spark. The following figures taken b c Fig. 81. from his paper show the regions when the gas is Expanded in the neighbourhood of the spark line at successive small intervals of time after the passage of the spark. It will be noticed that these regions show periodic swellings and contractions as if the centres of greatest disturbance were distributed at regular and fipite intervals along the line of discharge. A similar appear- ance was observed by Antolik (Pogg. Ann. 154, p. 14, 1875) when the discharge passed over a plate covered with fine powder; the powder placed itself in ridges at regular intervals along the line of discharge. 192.] This effect is also beautifully illustrated in an experiment made by Joly (Proc. Roy. Soc. 47, p. 78 et seq., 1890), in which the discharge passed from one strip of platinum to another between plates of glass placed so close together that they showed Newton’s rings; it was only with difficulty that the discharge could be got through this narrow space at all, it declined to go through the centre of the rings, and went out of its way to get through the places where the distance between the plates was greatest. Where it passed it made furrows on the glass at right angles to the line of discharge and separated by regularELECTRICITY THROUGH GASES. 177 I94-] intervals; a magnified representation of these is 'shown in Fig. 82, taken from Joly’s paper. When the air between the plates was replaced by hydrogen these furrows had a tendency to be more widely separated. 193.] The explosive effects produced by the spark are well illustrated by an experiment due to Hertz (Wied. Ann. 19, p, 87, 1883), in which the anode was placed at the bottom of a glass tube with a narrow mouth, while the cathode was placed Fig. 82. outside the tube and close to the open end. The tube and the electrodes were in a bell jar filled with dry air at a pressure of 40-50 mm. of mercury. When the discharge from a Leyden jar charged by an induction-coil passed, the glow accompanying it was blown out of the tube and extended several centimetres from the open end. In this experiment, as in the well-known ‘electric wind/ the explosive effects seem to be more vigorous at the anode than they are at the cathode. Chemical Action of the Electric Discharge. 194.] When the electric discharge passes through a gas, it produces in the majority of cases perceptible chemical changes, though whether these changes are due to the electrical action of the spark, or whether they are secondary effects due to a great increase of temperature occurring either at the electrodes or along the path of the discharge, is very difficult to determine when the discharge takes the form of a bright spark. N178 THE PASSAGE OP [196. 195.] For this reason we shall mainly consider the chemical changes produced by those forms of discharge in which the thermal effects are as small as possible, though even in these cases, since we can only measure the average temperature of a large number of molecules, it is always possible to account for any chemical effect by supposing that although the average temperature is not much increased by the discharge, a small number of molecules have their kinetic energy so much increased that they can enter into fresh chemical combinations. The thermal explanation of the chemical changes requires that they should be subsequent to, and not contemporaneous with the passage of the discharge; on the view adopted in this book chemical changes of some kind are necessary before the dis- charge can pass at all, though it by no means follows that the chemical changes which are instrumental in carrying the current are those which are finally apparent. When electricity passes through a liquid, electrolyte the substances liberated at the electrodes are in consequence of secondary chemical actions frequently different from the ions which carry the current. 196.] A very convenient method of producing discharges as free as possible from great heat is by using a Siemens’ ozonizer, repre- sented in Fig. 83. Two glass tubes are fused together, and the gas through which the discharge takes place circulates between them, entering by one of the side tubes and leaving by the other; the inside of the inner tube and the outside of the outer are coated with tin-foil, and are connected with the poles of anELECTRICITY THROUGH GASES. 179 x97-] induction-coil. When the coil is working a quiet discharge passes as a series of luminous threads between the surfaces of the glass opposed to each other. This form of discharge is often called the ‘silent discharge/ and by French writers Veffluve electrique. When air or oxygen is sent through a tube of this kind when the coil is working a considerable amount of ozone is produced. Ozone is not produced by the action of a steady electric field on oxygen or air unless the field is intense enough to produce a discharge through the gas (see J. J. Thomson and R. Threlfall, Proc.Roy. Soc. 40, p. 340, 1886). Meissner (Abhandlungen der Konig. Gesell. Gottingen, 16, p. 3, 1871) found that ozone was produced in tubes placed between the plates of a condenser when the condenser was charged or discharged, although no sparks passed between the plates, but that no ozone was produced when the charges on the plates of the condenser were kept constant. This was pro- bably due to the passage through the gas of electricity which had distributed itself over the walls of the tube under the induc- tive action of the charged plates of the condenser. Bichat and Guntz (.Annales de Chimie et de Physique [6], 19, p. 131, 1890) ascribe the formation of ozone, even by the silent discharge, to purely thermal causes. They regard the bright thread-like discharge surrounded by the non-luminous gas as a column of very hot oxygen surrounded by a cold atmosphere, and consider the conditions analogous to those which obtain in a St. Claire Deville ‘ chaud froid * tube, by the aid of which they state that Troost and Hautefeuille have produced ozone from oxygen without the use of the electric discharge. 197.] By the aid of the silent discharge a great many chemical changes are produced, of which the following are given by Lehmann, Molekularphysik, (bd. 2, p. 328.) Carbonic acid is split up by the discharge into carbonic oxide, oxygen, and ozone: water vapour into hydrogen and oxygen: when the discharge passes through acetylene a solid and a liquid are produced: phosphoretted hydrogen yields under similar circumstances a solid: methyl hydride gives marsh gas, hydrogen, and an acid: nitrous oxide splits up into nitrogen and oxygen: nitric oxide into nitrous oxide, nitrogen and oxygen. A mixture of carbonic acid and marsh gas gives a viscous N %180 THE PASSAGE OF [199. fluid; nitrogen partly combines with ammonia: carbonic oxide and hydrogen give a solid product: carbonic oxide and marsh gas a resinous substance: nitrogen and hydrogen ammonia. Dextrine, benzine, and sodium absorb nitrogen under the influence of the discharge, and enter into chemical combination with it. Hydrogen forms with benzine and turpentine resinous compounds. 198. ] Eerthelot (Annales de Chimie et de Physique, [5], 10, p. 55, 1877) has shown that the absorption of nitrogen by dextrine takes place under very small electromotive intensities; he showed this by connecting the inside and the outside coatings of the ozonizer to points at different heights above the surface of the ground, and found that this difference of potential, which varied in the course of the experiments from + 60 to —180 volts, was sufficient to produce in the course of a few weeks an appre- ciable absorption of nitrogen by a solution of dextrine in contact with it. The potential differences in these experiments were so small, and their rate of variation so slow, that it seems im- probable that any discharge could have passed through the nitrogen, and the experiments suggest that chemical action between a gas and a substance with which it is in contact can be produced by the action of a variable electric field without the passage of electricity through the bulk of the gas. Berthelot suggests that plants may, under the influence of atmospheric electricity, absorb nitrogen by an action of this kind. This suggestion also raises the very important question as to whether the chemical changes which accompany the growth of plants can have any influence on the development of atmospheric electricity. 199. ] We must now consider the relation between the quantity of electricity which passes through a gas and the amount of chemical action which takes place in consequence. It is neces- sary here to make a distinction, which has been too much neglected, between the part of this action which occurs at the electrodes and the part which occurs along the length of the spark. When a current of electricity passes through a liquid electrolyte the only evidence of chemical decomposition is to be found at the electrodes. When, however, the electric dis- charge passes through a gas the chemical changes are not con- fined to the electrodes but occur along the line of the discharge as well. This is proved by the fact that when the electrodelessELECTRICITY THROUGH GASES. 181 201.] discharge passes through oxygen ozone is produced, as is testified by the existence for several seconds after the discharge has passed of a beautiful phosphorescent glow: the same thing is also proved by the behaviour of the discharge when it passes through acetylene; the first two or three sparks are of a beautiful light green colour, while all subsequent discharges are a kind of whitish pink, showing that the first two or three sparks have decomposed the gas. 200. ] Since chemical decomposition is not confined to the elec- trodes its amount must depend upon the length of the spark; this has been proved by Perrot (Annales de Chimie et de Phy- sique [3], 61, p. 161, 1861), who compared the amounts of water vapour decomposed in the same time in a number of discharge tubes placed in series, the spark lengths in the tubes ranging from two millimetres to four centimetres; he found that the volumes of gas decomposed varied from 2 c.c. to 52 c.c., and that neither the longest nor the shortest spark produced the maximum effect. By placing a voltameter in the circuit Perrot found that in one of his tubes the amount of water vapour decomposed by the sparks was about 20 times the amount of water decomposed in the voltameter. It is evident from this that if we wish to arrive at any simple relation between the quantity of electricity passing through the gas and the amount of chemical decomposi- tion produced we must separate the part of the latter which occurs along the length of the spark from that which takes place at the electrodes. 201. ] This seems to have been done in a remarkable investi- gation made more than thirty years ago by Perrot (l.c.), which does not seem to have attracted the attention it merits, and which would well repay repetition. The apparatus used by Perrot in his experiments is represented in Fig. 84 from his paper. The spark passed between two platinum wires sealed into glass tubes, cfg,df g, which they did not touch except at the places where they were sealed: the open ends, c, d, of these tubes were about 2 mm. apart, and the wires terminated inside the tubes at a distance of about 2 mm. from the ends. The other ends of these tubes were inserted under test tubes e e, in which the gases which passed up the tubes were collected. The air was exhausted from the vessel A and the water vapour through which the discharge passed was obtained by heating the water in the182 THE PASSAGE OF [20 r. vessel to about 90°C.: special precautions were taken to free this water from any dissolved gas. The stream of vapour arising from this water drove up the tubes the gases produced by the passage of the spark; part of thesei gases was produced along the length of the spark, but in this case the hydrogen and Fig. 84. oxygen would be in chemically equivalent proportions ; part of the gases driven up the tubes would however be liberated at the electrodes, and it is this part only that we could expect to bear any simple relation to the quantity of electricity which had passed through the gas. When the sparking had ceased, the gases which had collected in the test tubes e and e were analysed ; in the first place they were exploded by sending a strong spark through them, this at once got rid of the hydrogen and oxygen which existed in chemically equivalent proportions and thus got rid of the gas produced along the length of the spark. After the explosion the gases left in the tubes were the hydrogen or oxygen in excess, together with a small quantity of nitrogen, due to a little air which had leaked into the vessel in the course of the experiments, or which had been absorbed by the water. The results of these analyses showed that there was always an excess of oxygen in the test tube in connection with the positive electrode, and an excess of hydrogen in the test tube connected with the negative electrode, and also that the amounts of oxygen and hydrogen in the respective tubes were very nearly chemically equivalent to the amount of copper deposited from a solution of copper sulphate in a voltameter placed in series with the discharge tube.ELECTRICITY THROUGH GASES. 183 201.] These results are so important that I shall quote one of Perrot’s experiments in full (l.a pp. 182-3). Duration of experiment 4 hours. 8-5 milligrammes of copper deposited in the voltameter from oqpper sulphate; this amount of copper is chemically equivalent to 3 c.c. of hydrogen and 1*5 e.c. of oxygen at atmospheric pressure. In the test tube over the negative electrode there were at the end of the experiment 37-5 c.c. of gas, after the explosion by the spark this was reduced to 3*1 c.c., so that by far the greater part of the gas collected consisted of hydrogen and oxygen in chemically equivalent proportions, produced not at the electrodes but along the line of the spark. 5*3 c.c. of oxygen were added to the original gas, which was again exploded and the contraction was 4*5 c.c.; in the original gas in the test tube there was there- fore an excess of 3 c.c. of hydrogen and *1 c.c. of something besides hydrogen and oxygen, probably nitrogen. In the test tube over the positive electrode there were 35-8 c.c. of gas at the end of the experiment, after the explosion by the spark this was reduced to 1*6 c.c. 1*8 c.c. of oxygen were added, but there was no ex- plosion when the spark passed; 8*7 c.c. of hydrogen were added and the mixture exploded when the spark passed; the contrac- tion produced was 9*6 c.c., showing that the excess of oxygen originally present was 1*4 c.c. and that *2 c.c. of nitrogen were mixed with it. Thus the excesses of hydrogen and oxygen in the tubes were very nearly chemically equivalent to the amount of copper deposited in the voltameter. This is also borne out by the following results of other experiments made by Perrot (l.c. p. 183). 2nd experiment. Duration of experiment 4 hours. Copper deposited in voltameter 6 milligrammes, chemically equivalent to 2*12 c.c. of hydrogen and 1*06 c.c. of oxygen. Gas in the test tube over the positive electrode 3540 c.c.; excess of oxygen *95 c.c.; nitrogen *2 c.c. Gas in the test tube over the negative electrode 32*40 c.c.; excess of hydrogen 2*10 c.c.; nitrogen *1 c.c. 4th experiment. Duration of experiment 3 hours. Copper deposited in voltameter 5*5 milligrammes, chemically equivalent to 1*94 c.c. of hydrogen and to *97 c.c. of oxygen. Gas in the test tube over the positive electrode 25*10 c.c.; excess of oxygen *85 c.c.; nitrogen 45 c.c.184 TfiE PASSAGE OP [202. Gas in the test tube over the negative electrode 27*70 c.c.; excess of hydrogen 1*8 c.c.; nitrogen *21 c.c. 6th experiment. Duration of experiment 3 ^ hours. Copper deposited in voltameter 6 milligrammes, chemically equivalent to 2*12 c.c. of hydrogen and to 1*06 c.c. of oxygen. Gas in the test tube over the positive electrode 30*20 c.c.; excess of oxygen *90 c.c.; nitrogen *2 c.c. Gas in the test tube over the negative pole 32*50 c.c.; excess of hydrogen 2*05 c.c.; nitrogen -2 c.c. These results seem to prove conclusively (assuming that the discharge passed straight between the platinum wires and did not pass through a layer of moisture on the sides of the tubes) that the conduction through water vapour is produced by chemi- cal decomposition, and also that in a molecule of water vapour the atoms of hydrogen and oxygen are associated with the same electrical charges as they are in liquid electrolytes. 202.] Another way in which the chemical changes which accom- pany the passage of the spark through a gas manifest themselves is by the production of a phosphorescent glow, which often lasts for several seconds after the discharge has ceased. In a great many gases this glow does not occur, it is however extremely bright in oxygen. A convenient way of producing the glow is to take a tube about a metre long filled with oxygen at a low pressure, and produce an electrodeless discharge at the middle of the tube. From the bright ring produced by the discharge a phospho- rescent haze will spread through the tube moving sufficiently slowly for its motion to be followed by the eye. The haze seems to come from the ozone, and the phosphorescence to be due to the gradual reconversion of the ozone into oxygen. This view is borne out by the fact that if the tube is heated the glow is not formed by the discharge, but as soon as the tube is allowed to cool down the glow is again produced: thus the glow, like ozone, cannot exist at a high temperature. The spectrum of this glow in oxygen is a continuous one, in which, however, a few bright lines can be observed if very high dispersive power is used. The glow is also formed in air, though not so brightly as in pure oxygen. When electrodes are used it seems to form most readily over the negative electrode, especially if this is formed of a flat surface of sulphuric acid. I have experimented with a large number of gases in order to203.] ELECTRICITY THROUGH GASES. 185 see whether or not the glow was formed when the electrodeless discharge passed through them. I have never detected any glow in a single gas (as distinct from a mixture) unless that gas was one which formed polymeric modifications, but all the gases I examined which do polymerize have shown the after-glow. The gases in which I have found the glow are oxygen, cyanogen (in which it is extremely persistent, though not so bright as in oxygen), acetylene, and vinyl chloride, all of which polymerize. A bulb filled with oxygen seems to retain its power of glowing unimpaired, however much it may be sparked through. In bulbs filled with the other gases, however, the glow after long sparking is not so bright as it was originally. This seems to suggest that the polymeric modification produced by the sparking does not get completely reconverted into the original form. Spark facilitated by rapid changes in the intensity of . the Electric Field. 203.] Jaumann (Sitzb. d. WienAkad. 97, p. 765, 1888) has made some interesting experiments on the effect on the spark length of small but rapid changes in the electrical condition of the elec- trodes. The arrangement used for these experiments is repre- sented in Fig. 85, which is taken from Jaumann’s paper. The main current from an electrical machine charged the con- denser B, while a neighbouring condenser c could be charged through the air-space F; C was a small condenser whose capacity was only *55 m., while B was a battery of Leyden Jars whose capa- city was 1000 times that of C. Another circuit connected with the186 THE PASSAGE OP [204. machine led to a thin wire placed about 5 mm. above a plate e which was connected to the earth. A glow discharge passed between the wire and the plate, and the difference of potential between the in- side and outside coatings of the jar b was constant and equal to about 12 electrostatic units. When the knobs of the air-break f were pushed suddenly together a spark about -5 mm. in length was produced at F, and in addition a bright spark 5 mm. long jumped across the air space at e where there was previously only a glow. The passage of the spark at F put the two condensers B and c into electrical communication, and this was equivalent to increasing the capacity of B by about one part in a thousand; this alteration in the capacity produced a correspond- ing diminution in the potential difference between its coatings. This disturbance of the electrical equilibrium would give rise to small but very rapid oscillations in the potential difference be- tween the wire and the plate e, and this variable field seemed able to send a spark across e, where when the potential was steady nothing but a glow was to be seen. 204.] It thus appears that a gas is electrically weaker under oscillating electric fields than under steady ones, for it is not apparent why the addition of the capacity of the small condenser to that of B should produce any considerable difference in the electromotive intensity at e. It is true that while the discharge is oscillating the tubes of electrostatic induction are not distri- buted in the same way as they are when the field is steady, and some concentration of these tubes may very likely take place, but it does not seem probable that the disturbance produced by so small a condenser would be sufficient to account for the large effects observed by Jaumann, unless, as he supposes, the gas is electrically weaker in variable electric fields. * Another point which might affect the electromotive intensity at e is the following: the comparatively small difference of potential between the wire and the plate is partly due to the glowing air-space at e acting as a conductor, this conductivity is due to dissociated molecules produced by the discharge, and it is likely that this would exhibit what are called ‘unipolar’ properties, that is, that its conductivity for a current in one direction would not be the same as for one in the opposite. Even when the change produced in the distribution of elec- tricity is not so great as that due to an actual reversal of theELECTRICITY THROUGH GASES. 187 205-] current it is conceivable that the conductivity of the space at e might depend upon the way the electricity was distributed over the wire and plate. Thus when this distribution of electricity was altered, the air, by becoming a worse conductor, might cause the electricity to accumulate on the wire and thus increase the electro- motive intensity at e. Since, however, there is a condenser of large capacity in electrical connection with the wire any increase in its electrification would be slow, whereas the spark observed by Jaumann seems to have followed that across f without the lapse of any appreciable interval. 205.] The observations of other physicists seem to afford con- firmatory evidence of the way in which electric discharge is facilitated by rapid alterations in the electromotive intensity. Thus Meissner (Abhand. der Konig. Gesell. Gottingen, 16, p. 3, 1871; see also Art. 196) found that ozone was produced in a tube placed between the plates of a condenser when these were suddenly charged or discharged, while none was produced when the charges on the plates were kept constant; the potential dif- ference in this experiment was not sufficient to cause a spark to pass between the plates. Again, R. v. Helmholtz and Richarz (Wied. Ann. 40, p. 161, 1890) using an induction coil that would give sparks in air about 4 inches long, found that when the electrodes were separated by about a foot and encased in wet linen bags to stop any particles of metal that might be given off from them, a steam jet some distance away from the electrodes showed very distinct signs of condensation whenever the current in the primary of the coil was broken. A steam jet is a very sensitive detector of chemical decomposition, free atoms produc- ing condensation of the steam even when no particles of dust are present. If we suppose that the electric field produces a polarized arrangement of the molecules of the gas, then considering the case when the left-hand electrode is the negative one, the right- hand the positive, there will be between the electrodes a chain of molecules arranged as in the first line in Fig. 86, the positively charged atoms being denoted by A, the negatively charged ones by £. If the field is now reversed, the molecules will be arranged as in the second line in Fig. 86. If the reversal takes place very slowly, the molecules will reverse their polarity by swinging round, but if the rate of reversal is very rapid the188 THE PASSAGE OP [206, resistance offered by the inertia of the molecules to this rotation will give rise to a tendency to produce the reversal of polarity of the molecules by chemical decomposition without rotation. A B A— —B A B A B B A B A B A B A A B A B A B A B Fig. 86. This may be done by the molecules splitting up and rearranging themselves as in the third line of Fig. 86. I have observed the effect of the reversal of the electric field when experimenting on the discharge produced in hydrogen at low pressures by a battery consisting of a large number of storage cells. I found that when the electromotive force was insufficient to produce continuous discharge, a momentary dis- charge occurred when the battery was reversed; this discharge merely flashed out for an instant, and took place when no discharge could be obtained by merely making or breaking the circuit without reversing the battery. A momentary discharge, however, occurred on making the circuit long before the electro- motive force was sufficient to maintain a permanent discharge. Fig. 87. 206.] Jaumann (1. c.) gives some examples of brushes which are formed at places where the electromotive intensity for steady charges is not a maximum. He explains these by supposing that the variations in the density of the electricity are more rapidELECTRICITY THROUGH GASES. 189 208.] at some parts of the electrodes than at others, and that ceteris paribus the discharge takes place most readily at the places where the rate of variation of the charge is greatest. Some of these brushes are represented in Fig. 87, taken from Jaumann. Theory of the Electric Discharge. 207. ] The phenomena attending the electric discharge through gases are so beautiful and varied that they have attracted the attention of numerous observers. The attention given to these phenomena is not, however, due so much to the beauty of the experiments, as to the wide-spread conviction that there is perhaps no other branch of physics which affords us so promising an opportunity of penetrating the secret of electricity; for while the passage of this agent through a metal or an elec- trolyte is invisible, that through a gas is accompanied by the most brilliantly luminous effects, which in many cases are so much influenced by changes in the conditions of the discharge as to give us many opportunities of testing any view we may take of the nature of electricity, of the electric discharge, and of the relation between electricity and matter. Though the account we have given in this chapter of the dis- charge through gases is very far from complete, it will probably have been sufficient to convince the student that the phenomena are very complex and very extensive. It is therefore desirable to find some working hypothesis by which they can be co- ordinated : the following method of regarding the discharge seems to do this to a very considerable extent. 208. ] This view is, that the passage of electricity through a gas as well as through an electrolyte, and as we hold through a metal as well, is accompanied and effected by chemical changes; also thatc chemical decomposition is not to be considered merely as an accidental attendant on the electrical discharge, but as an essential feature of the discharge without which it could not occur’ [Phil, Mag. [5], 15, p. 432, 1883). The nature of the chemical changes which accompany the discharge may be roughly described as similar to those which on Grotthus’ theory of electrolysis are supposed to occur in a Grotthus chain. The way such chemical changes effect the passage of the electricity has been already described in Art. 31, when we considered the way190 THE PASSAGE OF [209. in which a tube of electrostatic induction contracted when in a conductor. The shortening of a tube of electrostatic induc- tion is equivalent to the passage of electricity through the conductor. In conduction through electrolytes the signs of chemical change are so apparent both in the deposition on the electrodes of the constituents of the electrolyte and in the close connection, expressed by Faraday’s Laws, between the quantity of electricity transferred through the electrolyte and the amount of chemical change produced, that no one can doubt the importance of the part played in this case by chemical decomposition in the trans- mission of the electric current. 209.] When electricity passes through gases, though there is (with the possible exception of Perrot’s experiment, see Art. 200) no one phenomenon whose interpretation is so unequivocal as some in electrolysis, yet the consensus of evidence given by the very varied phenomena shown by the gaseous discharge seems to point strongly to the conclusion that here, as in electrolysis, the discharge is accomplished by chemical agency. Perrot, in 1861, seems to have been the first to suggest that the discharge through gases was of an electrolytic nature. In 1882 Giese (Wied. Ann. 17, pp. 1, 236, 519) arrived at the same conclusion from the study of the conductivity of flames. Before applying this view to explain in detail the laws govern- ing the electric discharge through gases, it seems desirable to mention one or two of the phenomena in which it is most plainly suggested. The experiments bearing most directly on this subject are those made by Perrot on the decomposition of steam by the dis- charge from a RuhmkorfFs coil (see Art. 200). Perrot found that when the discharge passed through steam there was an excess of oxygen given off at the positive pole and an excess of hydrogen at the negative, and that these excesses were chemically equiva- lent to each other and to the amount of copper deposited from a voltameter containing copper sulphate placed in series with the discharge tube. If this result should be confirmed by subsequent researches, it would be a direct and unmistakeable proof that the passage of electricity through gases, just as much as through electrolytes, is effected by chemical means. It would also show that the charge of electricity associated with an atom of anELECTRICITY THROUGH GASES. 191 211.] element in a gas is the same as that associated with the same atom in an electrolyte. 210. ] Again, Grove (Phil. Trans. 1852, Part I, p. 87) made nearly forty years ago some experiments which show that the chemical action going on at the positive electrode is not the same as that at the negative. Grove made the discharge from a Ruhmkorff’s coil pass between a steel needle and a silver plate, the distance between the point of the needle and the plate being about 2-5 mm.; the gas through which the discharge passed was a mixture of hydrogen and oxygen at pressures about 2 cm. of mercury. When the silver plate was positive and the needle negative a patch of oxide was formed on the plate, while if the plate were originally negative no oxidation occurred. When the silver plate had been oxidised while being used as a positive electrode, if the current were reversed so that the plate became the negative electrode, the oxide was reduced by the hydrogen and the plate became clean. When pure hydrogen was sub- stituted for the mixture of hydrogen and oxygen no chemical action could be observed on the plate, which was however a little roughened by the discharge; if however the plate was oxidised to begin with, it rapidly deoxidised in the hydrogen, especially when it was connected with the negative pole of the coil. Reitlinger and Wachter (Wied. Ann. 12, p. 590, 1881) found that the oxidation was very dependent upon the quantity of water vapour present; when the gas was thoroughly dried very little oxidation took place. The effect may therefore be due to the decomposition of the water vapour into hydrogen and oxygen, an excess of oxygen going to the positive and an excess of hydrogen to the negative pole. Ludeking (Phil. Mag. [5], 33, p. 521, 1892) has found that when the discharge passes through hydriodic acid gas, iodine is deposited on the positive electrode but not on the negative. 211. ] Again, chemical changes take place in many gases when the electric discharge passes through them. Perhaps the best known example of this is the formation of ozone by the silent discharge through oxygen. There are however a multitude of other instances, thus ammonia, acetylene, phosphoretted hydrogen, and indeed most gases of complex chemical constitution are decomposed by the spark. Another fact which also points to the conclusion that the dis-192 THE PASSAGE OP [213. charge is accomplished by chemical means is that mentioned in Art. 38, that the halogens chlorine, bromine, and iodine, which are dissociated at high temperatures, and which at such tem- peratures have already undergone the chemical change which we regard as preliminary to conduction, have then lost all power of insulation and allow electricity to pass through them with ease. Then, again, we have the very interesting result discovered by R. v. Helmholtz (Wied. Ann. 32, p. 1, 1887), that a gas through which electricity is passing and one in which chemical changes are known to be going on both affect a steam jet in the same way. 212. ] Again, one of the most striking features of the discharge through gases is the way in which one discharge facilitates the passage of a second; the result is true whether the discharge passes between electrodes or as an endless ring, as in the experi- ments described in Art. 77. Closely connected with this effect is Hittorf’s discovery {Wied. Ann. 7, p. 614, 1879) that a few galvanic cells are able to send a current through gas which is conveying the electric discharge. Schuster {Proceedings Royal Soc., 42, p. 371, 1887) describes a somewhat similar effect. A large discharge tube containing air at a low pressure was divided into two partitions by a metal plate with openings round the perimeter, which served to screen off from one com- partment any electrical action occurring in the other, if a vigorous discharge passed in one of these compartments, the electromotive force of about one quarter of a volt was sufficient to send a current through the air in the other. Since such electromotive forces would not produce any dis- charge through air in its normal state, these experiments suggest that the chemical state of the gas has been altered by the dis- charge. 213. ] We shall now go on to discuss more in detail the conse- quences of the view that dissociation of the molecules of a gas always accompanies electric discharge through gases. We notice, in the first place, that the separation of one atom from another in the molecule of a gas is very unlikely to be produced by the um aided agency of the external electric field. Let us take the case of a molecule of hydrogen as an example ; we suppose that the molecule consists of two atoms, one with a positive charge, the other with an equal negative one. The most obvious assumption,ELECTRICITY THROUGH GASES. 193 213-] which indeed is not an assumption if we accept Perrot’s results, to make about the magnitude of the charges on the atoms is that each is equal in magnitude to that charge which the laws of elec- trolysis show to be associated with an atom of a monovalent element. We shall denote this charge bye; it is the one molecule of electricity which Maxwell speaks about in Art. 260 of the Electricity and Magnetism. The electrostatic attraction between the atoms is the molecule where r is the distance between them. If the other molecules of hydrogen present do not help to split up the molecule, the force tending to pull the atoms apart is 2 Fe, where F is the external electromotive intensity. The ratio of the force tending to separate the atoms, to their electrostatic attraction, is thus 2 Fr2/e; now at atmospheric pres- sure discharge will certainly take place through hydrogen if F in electrostatic units is as large as 100, while at lower pressures a very much smaller value of F will be all that is required. To be on the safe side, however, we shall suppose that F= 102; then, assuming that the electrochemical equivalent of hydrogen is 10-4 and that there are 1021 molecules per cubic centimetre at atmospheric pressure, since the mass of a cubic centimetre of hydrogen is 1/11 x 103 of a gramme, e in electromagnetic units will be 104/11 x 1024, or e in electrostatic units will be about 2*7 x 10-11 and r is of the order 10~8, hence 2Fr2/e, the ratio under consideration, will be about 1/1*4 xlO3; this is so small that it shows the separation of the atoms cannot be effected by the direct action of the electric field upon them when the molecule is not colliding with other molecules. If the atoms in a molecule were almost but not quite shaken apart by a collision with another molecule, the action of the electric field might be sufficient to complete the separation. The electric field, however, by polarizing the molecules of the gas, may undoubtedly exert a much greater effect than it could produce by its direct action on a single molecule. When the gas is not polarized, the forces exerted on one molecule by its neighbours act some in one direction, others in the opposite, so o194 THE PASSAGE OP [214. that the resultant effect is very small; when, however, the medium is polarized, order is introduced into the arrangement of the molecules, and the inter-molecular forces by all tending in the same direction may produce very large effects. 214.] The arrangement of the molecules of a gas in the electric field and the tendency of the inter-molecular forces may be illus- trated to some extent by the aid of a model consisting of a large number of similar small magnets suspended by long strings attached to their centres. The positive and negative atoms in the molecules of the gas are represented by the poles of the magnets, and the forces between the molecules by those between the magnets. The way the molecules tend to arrange them- selves in the electric field is represented by the arrangement of the magnets in a magnetic field. The analogy between the model and the gas, though it may serve to illustrate the forces between the molecules, is very im- perfect, as the magnets are almost stationary, while the molecules are moving with great rapidity, and the collisions which occur in consequence introduce effects which are not represented in the model. The magnets, for example, would form long chains similar to those formed by iron filings when placed in the magnetic field ; in the gas, however, though some of the molecules would form chains, they would be broken up into short lengths by the bombardment of other molecules. The length of these chains would depend upon the intensity of the bombardment to which they were subjected, that is upon the pressure of the gas ; the greater the pressure the more intense the bombardment, and therefore the shorter the chain. We shall call these chains of molecules Grotthus* chains, because we suppose that when the discharge passes through the gas it passes by the agency of these chains, and that the same kind of interchange of atoms goes on amongst the molecules of these chains as on Grotthus* theory of electrolysis goes on between the molecules on a Grotthus* chain in an electrolyte. The molecules in such a chain tend to pull each other to pieces, and the force with which the last atom in the chain is attracted to the next atom will be much smaller than the force between two atoms in an isolated molecule ; this atom will therefore be much more easily detached from the chain than it would from a single molecule, and thus chemical change, and therefore electricELECTRICITY THROUGH GASES. 195 216.] discharge, will take place much more easily than if the chains were absent. 215. ] As far as the electrical effects go, it does not matter whether the effect of the electric field is merely to arrange chains which already exist scattered about in the gas, or whether it actually produces new chains ; we are more concerned with the presence of such chains than with their method of pro- duction. The existence of a small number of such chains (and it only requires a most insignificant fraction of the whole number of molecules to be arranged in chains to enable the gas to convey the most intense discharge) would have important chemical results, as it would greatly increase the ability of the gas to enter into chemical combination. Ax-----Bx A2---------B2 A3----------Bz A4---------B4 Fig. 88. 216. ] The way in which the electric discharge passes along such a chain of molecules is similar to the action in an ordinary Grotthus’ chain. Thus, let Ax Bv A2 B2, Az B3) &c., Fig. 88, repre- sent consecutive molecules in such a chain, the A’s being the positive atoms and the B’a the negative. Let one atom, Av at the end of the chain be close to the positive electrode. Then when the chain breaks down the atom Ax at the end of the chain goes to the positive electrode, Bx the other atom in this molecule, combining with the negative atom A2 in the next molecule, B2 combining with A3 ; the last molecule being left free and serving as a new electrode from which a new series of recombinations in a consecutive chain originates. There would thus be along the line of discharge a series of quasi-electrodes, at any of which the products of the decomposition of the gas might appear. The whole discharge between the electrodes consists on this view in a series of non-contemporaneous discharges, these dis- charges travelling consecutively from one chain to the next. The experiment described in Art. 105 shows that this discharge starts from the positive electrode and travels to the negative with a velocity comparable with that of light. The introduction of these Grotthus’ chains enables us to see how the velocity of the discharge can be so great, while the velocity of the individual molecules is comparatively small. The smallness of the velocity of these molecules has been proved by spectroscopic observations;196 THE PASSAGE OF [217. many experiments have shown that there is no appreciable dis- placement in the lines of the spectrum of the gas in the discharge tube when the discharge is observed end on, while if the mole- cules were moving with even a very small fraction of the velocity of light, Doppler s principle shows that there would be a measur- able displacement of the lines. It does not indeed require spec- troscopic analysis to prove that the molecules cannot be moving with half the velocity of light; if they did it can easily be shown that the kinetic energy of the particles carrying the discharge of a condenser would have to be greater than the potential energy in the condenser before discharge. When, however, we consider the discharge as passing along these Grotthus’ chains, since the recombinations of the different molecules in the chain go on simultaneously, the electricity will pass from one end of the chain to the other in the time required for an atom in one molecule to travel to the oppositely charged atom in the next molecule in the chain. Thus the velocity of the discharge will exceed that of the individual atoms in the pro- portion of the length of the chain to the distance between two adjacent atoms in neighbouring molecules. This ratio may be very large, and we can understand therefore why the velocity of the electric discharge transcends so enormously that of the atoms. 217.] We thus see that the consideration of the smallness of the electromotive intensity required to produce chemical change or discharge, as well as of the enormous velocity with which the discharge travels through the gas, has led us to the conclusion that a small fraction of the molecules of the gas are held together in Grotthus’ chains, while the consideration of the method by which the discharge passes along these chains indicates that the spark through the gas consists of a series of non-contemporaneous discharges, the discharge travelling along one chain, then wait- ing for a moment before it passes through the next, and so on. It is remarkable that many of the physicists, who have paid the greatest attention to the passage of electricity through gases, have been driven by their observations to the conclusion that the electric discharge is made up of a large number of separate discharges. The behaviour of striae under the action of magnetic force is one of the chief reasons for coming to this conclusion. On this point Spottiswoode and Moulton (Phil. Trans. 1879, part 1,ELECTRICITY THROUGH GASES. 197 217.] p. 205) say, ‘ If a magnet be applied to a striated column, it will be found that the column is not simply thrown up or down as a whole, as would be the case if the discharge passed in direct lines from terminal to terminal, threading the striae in its passage. On the contrary, each stria is subjected to a rotation or deformation of exactly the same character as would be caused if the stria marked the termination of flexible currents radiating from the bright head of the stria behind it and terminating in the hazy inner surface of the stria in question. An examination of several cases has led the authors of this paper to conclude that the currents do thus radiate from the bright head of a stria to the inner surface of the next, and that there is no direct passage from one terminal of the tube to the other/ With regard to the way the discharge takes place, the same authors say (Phil. Trans. 1879, part 1, p. 201)—‘ If, then, we are right in supposing that the series of artificially produced hollow shells are analogous in their structures and functions to striae, it is not difficult to deduce, from the explanation above given, the modus operandi of an ordinary striated discharge. The passage of each of the intermittent pulses from the bright surface of a stria towards the hollow surface of the next may well be supposed, by its inductive action, to drive from the next stria a similar pulse, which in its turn drives one from the next stria, and so on. . . . The passage of the discharge is due in both cases to an action consisting of an independent discharge from one stria to the next, and the idea of this action can perhaps be best illustrated by that of a line of boys crossing a brook on stepping stones, each boy stepping on the stone which the boy in front of him has left/ Goldstein (Phil. Mag. [5] 10, p. 183, 1880) expresses much the same opinion. He says : ‘ By numerous comparisons, and taking account of all apparently essential phenomena, I have been led to the following view:— ‘ The kathode-light, each bundle of secondary negative light, as well as each layer of positive light, represent each a separate current by itself, which begins at the part of each structure turned towards the kathode, and ends at the end of the negative rays or of the stratified structure, without the current flowing in one structure propagating itself into the next, without the elec- tricity which flows through one also traversing the rest in order.198 THE PASSAGE OP [220. ‘ I suspect, then, that as many new points of departure of the discharge are present in a length of gas between two electrodes as this shows of secondary negative bundles or layers—that as according to experiments repeatedly mentioned all the pro- perties and actions of the discharge at the kathode are found again at the secondary negative light and with each layer of positive light, the intimate action is the same with these as it is with those.’ 218. ] Thus, if we regard a stria as a bundle of Grotthus’ chains in parallel rendered visible, the bright parts of the stria corre- sponding to the ends of the chain, the dull parts to the middle, the conclusion of the physicists just quoted are almost identical with those we arrived at by the consideration of the chains. We therefore regard the stratification of the discharge as evidence of the existence of these chains, and suppose that a stria is in fact a bundle of Grotthus’ chains. 219. ] As far as phenomena connected with the electric discharge are concerned, the Grotthus* chain is the unit rather than the molecule ; now the length of this chain is equal to the length of a stria, which is very much greater than the diameter of a molecule, than the average distance between two molecules, or even than the mean free path of a molecule: thus the structure of a gas, as far as phenomena connected with the electric dis- charge are concerned, is on a very much coarser scale than its structure with reference to such properties as gaseous diffusion where the fundamental length is that of the mean free path of the molecules. 220. ] Peace’s discovery that the density—which we shall call the critical density—at which the ‘ electric strength * of the gas is a minimum depends upon the distance between the electrodes, proves that the gas, when in an electric field sufficiently intense to produce discharge, possesses a structure whose length scale is comparable with the distance between the electrodes when these are near enough together to influence the critical density. As this distance is very much greater than any of the lengths recognized in the ordinary Kinetic Theory of Gases, the gas when under the influence of the electric field must have a structure very much coarser than that recognized by that theory. In our view this structure consists in the formation of Grotthus* chains.ELECTRICITY THROUGH GASES. 199 222.] 221. ] The striations are only clearly marked within somewhat , narrow limits of pressure. But it is in accordance with the con- clusion which all who have studied the spark have arrived at— that there is complete continuity between the bright well-defined spark which occurs at high pressures and the diffused glow which represents the discharge at high exhaustions—to suppose that they always exist in the spark discharge, but that at high pressures they are so close together that the bright and dark parts cease to be separable by the eye. The view we have taken of the action of the Grotthus’ chains in propagating the electric discharge, and the connection between these chains and the striations, does not require that every dis- charge should be visibly striated; on the contrary, since the striations will only be visible when there is great regularity in the disposition of these chains, we should expect that it would only be under somewhat exceptional circumstances that the con- ditions would be regular enough to give rise to visible striations. 222. ] We shall now proceed to consider more in detail the application of the preceding ideas to the phenomena of the electric discharge. The first case we shall consider is the calcu- lation of the potential difference required to produce discharge under various conditions. It is perhaps advisable to begin with the caution that in com- paring the potential differences required to produce discharge through a given gas we must be alive to the fact that the con- dition of the gas is altered for a time by the passage of the discharge. Thus, when the discharges follow each other so rapidly that the interval between two discharges is not suffi- ciently long to allow the gas to return to its original condition before the second discharge passes, this discharge is in reality passing through a gas whose nature is a function of the electrical conditions. Thus, though this gas may be called hydrogen or oxygen, it is by no means identical with the gas which was called by the same name before the discharge passed through it. When the discharges follow each other with great rapidity the supply of dissociated molecules left by preceding discharges may be so large that the discharge ceases to be disruptive, and is analogous to that through a very hot gas whose molecules are dissociated by the heat. The measurements of the potential differences required to send200 THE PASSAGE OF [222. the first spark through a gas are thus more definite in their interpretation than measurements of potential gradients along the path of a nearly continuous discharge. The striations on the preceding view of the discharge may, since they are equivalent to a bundle of Grotthus’ chains, be regarded as forming a series of little electrolytic cells, the beginning and the end of a stria corresponding to the electrodes of the cell. Let F be the electromotive intensity of the field, A the length of a stria, then when unit of electricity passes through the stria the work done on it by the electric field is F\. The passage of the elec- tricity through the stria is accompanied just as in the case of the electrolytic cell, by definite chemical changes, such as the decomposition of a certain number of molecules of the gas; thus if w is the increase in the potential energy of the gas due to the changes which occur when unit of electricity passes through the stria, then neglecting the heat produced by the current we have by the Conservation of Energy Fk = tv, or the difference in potential between the beginning and end of a stria is equal to w. If the chemical and other changes which take place in the consecutive striae are the same, the potential difference due to each will be the same also. There is however one stria which is under different conditions from the others, viz. that next the negative electrode, i. e. the negative dark space. For in the body of the gas, the ions set free at an extremity of the stria, are set free in close proximity to the ions of opposite sign at the extremity of an adjacent stria. In the stria next the electrode the ions at one end are set free against a metallic surface. The experiments described in the account we have already given of the discharge show that the chemical changes which take place at the cathode are abnormal; one reason for this no doubt is the presence of the metal, which makes many chemical changes possible which could not take place if there were nothing but gas present. This stria is thus under excep- tional circumstances and may differ in size and fall of potential from the other striae. Hittorfs experiments, Art. 140, show that the fall of potential at the cathode is abnormally great. If we call this potential fall K and consider the case of discharge between two parallel metal plates; the discharge on this view, starting from the positive electrode, goes consecutively across a number n of222.] ELECTRICITY THROUGH GASES. 201 similar striae, one of which reaches up to the positive electrode, the fall of potential across each of these is w; the discharge finally crosses the stria in contact with the negative electrode in which the fall of potential is K; thus V, the total fall of potential as the discharge goes from the positive to the negative electrode, is given by the equation V=K + mv. (1) If l is the distance between the plates, A 0 the length of the stria next the cathode, A the length of the other stria, then Substituting this value for n in (l) we get r=(*-*T,)+X”’- which mav be written V=K' + al (2) According to this equation the curve representing the relation between potential difference and spark length for constant pressure is a straight line which does not pass through the origin. The curves we have given from the papers by Paschen and Peace show that this is very approximately true. The curves show that for air Kf would at atmospheric pressure be about 600 volts from Paschen’s experiments and about 400 volts from Peace’s. If R is the electromotive intensity required to produce a spark of length l between two parallel infinite plates, then since R= V'/l R = ~+a. (3) Since K' is positive, the electromotive intensity required to pro- duce discharge increases as the length of the spark diminishes; in other words, the electric strength of a thin layer of gas is greater than that of a thick layer. The electric strength will sensibly increase as soon as K'/l become appreciable in com- parison with a, this will occur as soon as l ceases to be a very large multiple of the length of a stria. Thus the thickness of the layer when the ‘ electric strength ’ begins to vary appreciably is com- parable with the length of a stria at the pressure at which the discharge takes place; this length is very large when compared with molecular distances or with the mean free path of the202 THE PASSAGE OF [223 molecules of the gas ; hence we see why the change in the ‘electric strength ’ of a gas takes place when the spark length is very large in comparison with lengths usually recognized in the Kinetic Theory of Gases. According to formula (3), the curve representing the relation between electromotive intensity and spark length is a rectangular hyperbola ; this is confirmed by the curves given by Dr. Liebig for air, carbonic acid, oxygen and coal gas (see Fig. 19), and by those given by Mr. Peace for air. 223.] The preceding formulae are not applicable when the dis- tance between the electrodes is less than A0 the length of the stria next the cathode. But if the discharge passes through the gas and is not carried by metal dust torn from the electrodes we can easily see that the electric strength must increase as the distance between the electrodes diminishes. For as we have seen, the molecules which are active in carrying the discharge are not tom in pieces by the direct action of the electric field but by the attraction of the neighbouring molecules in the Grotthus’ chain. Now when we push the electrodes so near together that the distance between them is less than the normal length of the chain, we take away some of the molecules from the chain and so make it more difficult for the molecules which remain to split up any particular molecule into atoms, so that in order to effect this splitting up we must increase the number of chains in the field, in other words, we must increase the electromotive intensity. Peace’s curves, Fig. 27, showing the relation between the potential difference and spark length are exceedingly flat in the neighbourhood of the critical spark length. This shows that the potential difference required to produce discharge increases very slowly at first as the spark length is shortened to less than the length of a Grotthus’ chain. We now proceed to consider the relation between the spark potential and the pressure. As we have already remarked, the length of a Grotthus’ chain depends upon the density of the gas ; the denser the gas the shorter the chain: this is illustrated by the way in which the striae lengthen out when the pressure is reduced. The experiments which have been made on the connection between the length of a stria and the density of the gas are not sufficiently decisive to enable us to formulate the224-] ELECTRICITY THROUGH GASES. 203 exact law connecting these two quantities, we shall assume however that it is expressed by the equation \=pp-*, where A is the length of a stria, p the density of the gas, and j3, k positive constant. Equation (1) involves K the fall of potential at the cathode and w the fall along a stria as well as A. Warburg’s experiments (Art. 160) show that the cathode fall K is almost independent of the pressure, and although no observations have been made on the influence of a change in the pressure on the value of w, it is not likely that w any more than K depends to any great extent upon the pressure. If we substitute the preceding value of A in equation (2) we get V = K'+l^w. Both Paschen’s and Peace’s experiments show that when the spark length is great enough to include several striae the curve representing the relation between the spark potential and density for a constant spark length, though very nearly straight, is slightly convex to the axis along which the densities are measured. This shows that k is slightly, but only slightly, greater than unity. 224.] It is interesting to trace the changes which take place in the conditions of discharge between two electrodes at a fixed distance apart as the pressure of the gas gradually diminishes. When the pressure is great the strise are very close together, so that if the distance between the electrodes is a millimetre or more, a large number of strise will be crowded in between them. As the pressure diminishes the strise widen out, and fewer and fewer of them can find room to squeeze in between the electrodes, and as the number of strise between the elec- trodes diminishes, the potential required to produce a spark dimin- ishes also, each stria that is squeezed out corresponding to a definite diminution in the spark potential. This diminution in potential will go on until the strise have all been eliminated with the exception of one. There can now be no further reduction in the number of strise as the pressure diminishes, and the Grotthus’ chain which is left, and which is required to split up the molecules to allow the discharge to take place,204 THE PASSAGE OF [226. gets curtailed as the pressure falls by a larger and larger fraction of its natural length, and therefore has greater and greater difficulty in effecting the decomposition of the molecules, so that the electric strength of the gas will now increase as the pressure diminishes. There will thus be a density at which the electric strength of the gas is a minimum, and that density will be the one at which the length of the stria next the cathode is equal or nearly equal to the distance between the electrodes. Thus the length of a stria at the minimum strength will have to be very much less when the electrodes are very near together than when they are far apart, and since the stria-length is less the density at which the ‘ electric strength ’ is a minimum will be very much greater when the electrodes are near together than when they are far apart. This is most strikingly exemplified in Mr. Peace’s experiments, for when the distance between the electrodes was reduced from 1/5 to 1/100 of a millimetre the critical pressure was raised from 30 to 250 mm. of mercury. The mean free path of a molecule of air at a pressure of 30 mm. is about 1/400 of a millimetre. 225. ] The existence of a critical pressure, or pressure at which the electric strength is a minimum, when the discharge passes between electrodes can thus be explained if we recognize the formation of Grotthus5 chains in the gas, and the theory leads to the conclusion which, as we have seen, is in accordance with the facts, that the critical pressure depends on the spark length. 226. ] We have seen that when the distance between the elec- trodes is less than the length of the stria next the negative electrode, the intensity of the field required to produce discharge will increase as the distance between the electrodes diminishes. Peace’s observations show that this increase is so rapid that the potential difference between the electrode when the spark passes increases when the spark length is diminished, or in other words, that the electromotive intensity increases more rapidly than the reciprocal of the length of a Grotthus’ chain. This will explain the remarkable results observed by Hittorf (Art. 170) and Lehmann (Art. 170) when the electrodes were placed very near together in a gas at a somewhat low pressure. In such cases it was found that the discharge instead of passing in the straight line between the electrodes took a very roundabout course. To explain this, suppose that in the experiment shown in Fig. 68ELECTRICITY THROUGH GASES. 205 228.] the electrodes are nearer together than the length of the chain next the electrode, i. e. the negative dark space; then if the discharge passed along the shortest path between the plates, the potential difference required would, by Peace’s experiments, con- siderably exceed Ky the normal cathode potential fall; if however the discharge passed as in the figure along a line of force, whose length is greater than the negative dark space, the potential difference required would be K plus that due to any small posi- tive column which may exist in the discharge. The latter part of the potential difference is small compared with A, so that the potential difference required to produce discharge along this path will only be a little in excess of A, while that required to produce discharge along the shortest path would, by Peace’s experiments, be considerably greater than A, the discharge will therefore pass as in the figure in preference to taking the shortest path. 227. ] Since a term in the expression (1) for the potential differ- ence required to produce a spark of given length is inversely proportional to the length of a stria, anything which diminishes the length of a stria will tend to increase this potential difference. Now the length of a stria is influenced by the size of the discharge tube as soon as the length becomes comparable with the diameter of the tube ; the narrower the tube the shorter are the striae. Hence we should expect to find that it would require a greater potential difference to produce at a given pressure a spark through a narrow tube than through a wide one. This is confirmed by the experiments made by De la Rue and Hugo Muller, described in Art. 169. 228. ] We do not at present know enough about the laws which govern the passage of electricity from a gas to a solid, or from a solid to a gas, to enable us to account for the difference between the appearances presented by the discharge at the cathode and anode of a vacuum tube; it may, however, be well to consider one or two points which must doubtless influence the behaviour of the discharge at the two electrodes. We have seen (Art. 108) that the positive column in the electric discharge starts from the positive electrodes, and that with the ex- ception of the negative rays, no part of the discharge seems to begin at the cathode; we have also seen that the potential differ- ences in the neighbourhood of the cathode are much greater than206 THE PASSAGE OP [228. those near the anode. These results might at first sight seem in- consistent with the experiments we have described (Art. 40) on the electrical effect on metal surfaces of ultra-violet light and incan- descence. In these experiments we saw that under such influences negative electricity escaped with great ease from a metallic electrode, while, on the other hand, positive electricity had great difficulty in doing so In the ordinary discharge through gases it seems, on the contrary, to be the positive electricity which escapes with ease, while the negative only escapes with great difficulty. We must remember, however, that the vehicle conveying the elec- tricity may not be the same in the two cases. When ultra-violet light is incident on a metal plate, there seems to be nothing in the phenomena inconsistent with the hypothesis that the negative electrification is carried away by the vapour or dust of the metal. In the case of vacuum tubes, however, the electricity is doubtless conveyed for the most part by the gas and not by the metal. In order to get the electricity from the gas into the metal, or from the metal into the gas, something equivalent to chemical combination must take place between the metal and the gas. Some experiments have been made on this point by Stanton (Proc. Roy. Soc. 47, p. 559, 1890), who found that a hot copper or iron rod connected to earth only discharged the electricity from a positively electrified conductor in its neigh- bourhood when chemical action was visibly going on over the surface of the rod, e.g. when it was being oxidised in an atmo- sphere of oxygen. When it was covered with a film of oxide it did not discharge the adjacent conductor; if when coated with oxide it was placed in an atmosphere of hydrogen it discharged the electricity as long as it was being deoxidised, but as soon as the deoxidation was complete the leakage of the electricity stopped. On the other hand, when the conductor was negatively electrified, it leaked even when no apparent chemical action was taking place. I have myself observed (.Proc. Roy. Soc. 49, p. 97, 1891) that the facility with which electricity passed from a gas to a metal was much increased when chemical action took place. If this is the case, the question as to the relative ease with which the electricity escapes from the two electrodes through a vacuum tube, depends upon whether a positively or negatively electrified surface more readily enters into chemical combination with the adjacent gas, while the sign of the electrification of a metal229.] ELECTRICITY THROUGH GASES. 207 surface under the influence of ultra-violet light may, on the other hand, depend upon whether the ‘Volta-potential’ (see Art. 44) for the metal in its solid state is less or greater than for the dust or vapour of the metal. 229.] In framing any theory of the difference between the positive and negative electrodes, we must remember that at the electrodes we have either two different substances or the same substance in two different states in contact, and it is in accordance with what we know of the electrical effects produced by the contact of different substances that the gas in the immediate neighbourhood of the electrodes should be polarized, that is, that the molecular tubes of induction in the gas should tend to point in a definite direction relatively to the outward drawn normals to the electrode : let us suppose that the polarization is such that the negative ends of the tubes are the nearest to the electrode: we may regard the molecules of the gas as being under the influence of a couple tending to twist them into this position. If now this electrode is the cathode, then before these molecules are avail- able for carrying the discharge, they must be twisted right round against the action of an opposing couple, so that to produce discharge at this electrode the electric field must be strong enough to twist the molecules out of their original alignment into the opposite one, it must therefore be stronger than in the body of the gas where the opposing couple does not exist: a polarization of this kind would therefore make the cathode potential gradient greater than that in the body of the gas.CHAPTER III* Conjugate Functions. 230. ] The methods given by Maxwell for solving problems in Electrostatics by means of Conjugate Functions are somewhat indirect, since there is no rule given for determining the proper transformation for any particular problem. Success in using these methods depends chiefly upon good fortune in guessing the suitable transformation. The use of a general theorem in Trans- formations given by Schwarz (Ueber einige Abbildungsauf- gaben, Crelle 70, pp. 105-120, 1869), and Christoffel (Sul prob- lema delle temperature stazionarie,Am\sX\ di Matematica, I. p. 89, 1867), enables us to find by a direct process the proper trans- formations for electrostatical problems in two dimensions when the lines over which the potential is given are straight. We shall now proceed to the discussion of this method which has been applied to Electrical problems by KirchhofF (Zur Theorie des Compensators, Gesammelte Abhandlungen, p. 101), and by Potier (Appendix to the French translation of Maxwell’s Electricity and Magnetism); it has also been applied to Hydrodynamical prob- lems by Michell {On the Theory of Free Stream Lines, Phil. Trans. 1890, A. p. 389), and Love {Theory of Discontinuous Fluid Motions in tivo dimensions, Proc. Camb. Phil. Soc. 7, p. 175, 1891). 231. ] The theorem of Schwarz and Christoffel is that any polygon bounded by straight lines in a plane, which we shall call the 0 plane, where z = x + iy,x and y being the Cartesian coordinates of a point in this plane, can be transformed into the axis of £ in a plane which we shall call the t plane, where t = £ + it), £ and rj being the Cartesian coordinates of a point in this plane; and that points inside the polygon in the z planeCONJUGATE FUNCTIONS. 209 231-] transform into points on one side of the axis of £. The trans- formation which effects this is represented by the equation where alf a2,...an are the internal angles of the polygon in the z plane; tx, t2,...tn are real quantities and are the coordinates of points on the axis of £ corresponding to the angular points of the polygon in the 0 plane. To prove this proposition, we remark that the argument of dz/dt, that is the value of 0 when dz/dt is expressed in the form ReQ where R is real, remains unchanged as long as z remains real and does not pass through any one of the values t19 t2,...tn\ in other words, the part of the real axis of t be- tween the points tr and tr+l corresponds to a straight line in the plane of 0. We must now investigate what happens when t passes through one of the points such as tr on the axis of £. With centre tr describe a small semi-circle BDC on the positive side of the axis of £, and consider the change in dz/dt as t passes round BDC from B to C. D p ___________________________________________________________________. Fig. 89. Since we suppose <0, the radius of this semi-circle, indefinitely small, if any finite change in dz/dt occurs in passing round this £-1 semi-circle it must arise from the factor (t—tr)17 Now for a point on the semi-circle BDC £ — tr = £-1 aS-i 1(^-1) — =<*>* 1e, hence, since 0 decreases from 7r to zero as the point travels ~1 round the semi-circle, the argument of (t—tr)17 , and there- fore of dz/dt, is increased by 7r—ar, that is the line correspond- ing to the portion tr tr+1 of the axis of £ makes with the line corresponding to the portion t^tr the angle 77 — ^; in other210 CONJUGATE FUNCTIONS. [23T. words, the internal angle of the polygon in the 0 plane at the point corresponding to tr is ar. If we imagine a point to travel along the axis of £ in the plane of t from £ = — 00 to £ = -f and then back again from 4- 00 to — 00 along a semi-circle of infinite radius with its centre at the origin of coordinates in the t plane, then, as long as the point is on the axis of £, the corresponding point in the plane 0 is on one of the sides of the polygon. To find the path in 0 corresponding to the semi-circle in t we put t = Ree, where R is very great and is subsequently made infinite: equa- tion (1) then becomes a=cr • « • ■ <2> since R is infinite compared with any of the quantities tl9 t2) ...£w. Since along the semi-circle dt^iR^dd, equation (2) becomes 1 -fa2+ ...an dz = l CR " € w dO, °i + «2 + -°« _(n_1) lSaL + a* + a£_(n-l)l 0 or z = GR - 1 7r Thus the path in the 0 plane corresponding to the semi-circle in the plane of 0 is a portion of a circle subtending an angle ai + a2+ ...ow —(71—1)77 at the origin, and whose radius is zero or infinite according as a1 + a2jL.._.a„ is positive or negative. If this quantity is zero, then equation (2) becomes dz _ C _ C dt~ Rtl9~ t ’ hence z = Clogt + A = C log R -f- l CO + Ay where A is the constant of integration.CONJUGATE FUNCTIONS. 211 234-] Thus as the point in the t plane moves round the semi-circle the point in the z plane will travel over a length Gtt of a straight line parallel to the axis of y at an infinite distance from the origin. 232. ] Since by equation (l) the value of dz/dt cannot vanish or become infinite for values of t inside the area bounded by the axis of £ and the infinite semi-circle, this area can be conformably transformed to the area bounded by the polygon in the 2? plane. 233. ] When we wish to transform any given polygon in the z plane into the axis of £ in the t plane we have the values of al9 a2,...att given. As regards the values of tl9t2,...tn some may be arbitrarily assumed while others will have to be determined from the dimensions of the polygon. Whatever the values of t19 the transformation(1) will transform the axis of £ into a polygon whose internal angles have the required values. In order that this polygon should be similar to the given one we require n — 3 conditions to be satisfied; hence as regards the n quantities t19 t2, the values of 3 of them may be arbi- trarily assumed, while the remaining w —3 must be determined from the dimensions of the polygon in the z plane. 234. ] The method of applying the transformation theorem to the solution of two dimensional problems in Electrostatics in which the boundaries of the conductors are planes, is to take the polygon whose sides are the boundaries of the conductors, which we shall speak of as the polygon in the z plane, and transform it by the Schwarzian transformation into the real axis in a new plane, which we shall call the t plane. If \jr represents the potential function, the stream function, and w = <£ + n/f, the condition that \j/ is constant over the conductors may be repre- sented by a diagram in the w plane consisting of lines parallel to the real axis in this plane: we must transform these lines by the Schwarzian transformation into the real axis in the t plane. Thus corresponding to a point on the real axis in the t plane we have a point in the boundary of a conductor in the 0 plane and a point along a line of constant potential in the w plane, and we make this potential correspond to the potential of the con- ductor in the electrostatical problem whose solution we require. In this way we find * + ‘2/ = /(*), + if = F(t), v zCONJUGATE FUNCTIONS. 212 [235- where / and F are known functions; eliminating t between these equations we get + = x(x + iy)> which gives us the solution of our problem. 235.] We shall now proceed to consider the application of this method to some special problems. The first case we shall consider is the one discussed by Maxwell in Art. 202 of the Electricity and Magnetism, in which a plate bounded by a straight edge and at potential V is placed above and parallel to an infinite plate at zero potential. The diagrams in the 0 an now r 47t ax d __ d(t> dt dx ~ dt dx 2 V (*+!)(*-!) T(*+l)(*_l) C(t2 — a2)* _ V 2 ttC (£2—a2)* 1 H (t2-a2)*' Hence the density of the electricity on the plate is V 1 4tt H{t2-a2)*' This is infinite at the edges C and E. When EP is a large multiple of H, t = 1 approximately, and the density is V 1 4 7T JET ^ 1— or since (1-a2)* H-h H ’ the density is uniform and equal to 1 v 9 47r H—h 238.] Condensers are sometimes made by placing one cube inside another; in order to find the capacity of a condenser of this kind we shall investigate the distribution of electricity on a system of conductors such as that represented in Fig. 95, where ABC is maintained at zero potential and FED at potential V. _D /s=i The diagram in the z plane is bounded by the lines AB, BC, DE, EF; we shall assume that t = — 00 at the point on the line AB where £/ = + <», £=0 at B, £ = 1 at the point on BC where t=o B t=a Fig. 95.CONJUGATE FUNCTIONS. 223 238.] x = + 00, and t = a at E, where a is a quantity greater than unity which has to be determined by the geometry of the system. The internal angles of the polygon in the z plane are 7r/2 at B, zero at C, 3 7t/2 at E. The transformation which turns the boundary of the z polygon into the real axis in the t plane is by equation (1) expressed by the equation dz _ G(a-t)i dt #(1 — t) The diagram in the w plane consists of the real axis and a line parallel to it. The internal angle of the polygon is at t = 1 and is equal to zero, hence the transformation which turns this diagram into the real axis of t is dw __ B dt ~ 1 — t5 V or (f) -f i\js — t F— — log (l 7r since F is the increment in \jr when t passes through the value 1. To integrate (17) put t — a u* We have then dz 1+u2 2Ca du (1 +u2) {1—(a—l)^2} = 2C1r7^ + Hence ____ /1-f Fa—1 Ux z = 2(7tan'1u+ Fa— lOlog^ Fa—1 u = 2(7sin-1 a + Fa—1 —1 ? (18) V a 6 ts/a — t— Va—lVV where the constants have been chosen so as to make x and y vanish when t = 0. When t = a, we have #+t2/= C 7T + Fa-lCi7r.224 CONJUGATE FUNCTIONS. [238. Hence if h and k are the coordinates of E referred to the axes bc, ab, we have h = On, k = CVa— l7r. We can also deduce these equations from equation (17) by the process used to determine the constants in Art. 237. We may write (18) in the form , . , /t k, i(Va—t+ Va-l^t)2) . —i(i=<5—Ll <19> The quantity of electricity on the strip BP, where p is a point on BC, is equal to 4 77 77 Now if BP is large compared with k, the value of t at P is approximately unity; from (19) we get the more accurate value - log (1 - 0 = j * - ^ sm-‘- 2 !og j 2 ^=1}, x 2h, t h . 2k = 7r------tan-1---2 log —- ---• k k k *Vh2 + k2 Hence the quantity of electricity on the strip is Vi 2 h , li 2k, Vh2 + k2) 47ik (77 k 77 8 2k ) Hence the quantity is the same as if the electricity were dis- tributed with the uniform density — 7/477k over a strip whose breadth was less than BP by 2 h ,h 2k, \/h2 + kl * * — tan-1 -=--------log - 77 k 77 & 2k In the important case when h = k, this becomes h hCONJUGATE FUNCTIONS. 225 239] The surface density of the electricity at any point on bc or ED is - y /~ + 4 the — or 4* sign being taken according as the point is on BC or ED. This expression vanishes at B and is infinite at E. At p, a point on BC at some distance from B, t is approxi- mately unity, so that the surface density is V liPCVa—X = _ 4 7jk This result is of course obvious, but it may be regarded as affording a verification of the preceding solution. E D7H ” c /—co A--------------------------------------B/»0 Fig. 96. 239.] Another case of some interest is that represented in Fig. 96, where we have an infinite plane AB at potential Fin pre- sence of a conductor at zero potential bounded by two semi-infinite planes CD, DE at right angles to each other. The diagram in the z plane is bounded by the lines AB, CD, DE and a quadrant of a circle whose radius is infinite. We shall assume t = — 00 at the point on the line AB where & = — 00, £ = Oat the point on the same line where # = + oo,£ = latD. The internal angles of the polygon in the z plane are zero at B and 3 tt/2 at D. The trans- QCONJUGATE FUNCTIONS, 226 [239- formation which turns the boundary of the z polygon into the real axis in the t plane is therefore, by equation (1), dz _ C(l-f)* dt ~~ t (20) The diagram in the w plane consists of two straight lines parallel to the real axis, the internal angle being zero at the point t = 0 ; hence we have V W = + lx// = — log t, IT since the plane AB is at potential V and CDE at potential zero. Integrating equation (20), we find when t > 0< ,1 z = x + iy = G(2 \/l —£ — log —(21) where no constant of integration is needed if the origin of coordinates is taken at D where t = +1. If h is the distance between CD and AB, then 0 increases by ih when t changes sign, hence we have by equation (20), by the process similar to that by which we deduced the constant in Art. (237), h = — Ctt ; so that (21) becomes, 0 < t < 1, = <22) The quantity of electricity on a strip dp where P is a point on DC is y if tP is the value of t at P. If dp is large compared with h, tP will be very nearly zero; the value of log tP is then readily got by writing (22) in the form x + iy = ^{2log(l + \/l— t)— logt — 2\/T^4}. So that if x = DP, we have approximately, 2h, ^ 2h) IT2 4-I-] CONJUGATE FUNCTIONS. 227 Thus the quantity of electricity on DP is We can prove in a similar way that if Q is a point on DE the charge on DQ is equal to 240. ] If the angle CDE, instead of being equal to tt/2} were equal to tt/ti, the transformation of the diagram in the 0 plane to the real axis of t could be effected by the relation 71 — 1 dz _ C(t—1) n dt ~~ t 241. ] We shall now proceed to discuss aproblem which enablesus to estimate the effect produced by the slit between the guard-ring and the plate of a condenser on the capacity of the condenser. /—00H V n^1 tm — aF JU D C/= + a IL 1 II Fig. 97. D t = +a — a /«*<* a Fig. 98. When the plate and the guard-ring are of finite thickness the integration of the differential equation between 0 and t involves the use of Elliptic Functions. In the two limiting cases when the thickness of the plate is infinitely small or infinitely great, the necessary integrations can however be effected by simpler means. We shall begin with the case where the thickness of the plate is very small, and consider the distribution of electricity on two semi-infinite plates separated by a finite interval 2 k and placed parallel to an infinite plane at the distance h from it. We shall suppose that the two semi-infinite plates are at the same potential Vt and that the infinite plate is at potential zero. The diagrams in the 0 and w planes are represented in Figs. 97 and 98.228 CONJUGATE FUNCTIONS. [241. The diagram in the z plane is bounded by the infinite straight line ED, the two sides AB and BC of the semi-infinite line on the right, the two sides FG, GH of the semi-infinite line on the left, and a semi-circle of infinite radius. A point travers- ing the straight portion of the boundary might start from A and travel to B on the upper side of the line on the right, then from B to C along the under side, from D to E along the infinite straight line, from F to G on the under side of the line on the left and from G to H on the upper side of this line. We shall suppose that t = + 00 at A, t == + 1 at B, t = +a(a< 1) at C, t — —a at F, t = — 1 at G, t = — 00 at H. The internal angles of the polygon in the 0 plane are 2 tt at B, zero at C, zero at F, and 2 7T at G; hence the transformation which turns the diagram in the z plane into the real axis of t is expressed by the relation dz _ pt2— 1 (23) The diagram in the w plane consists of two straight lines parallel to the real axis and the potential changes by V when t passes through the values + a : hence we easily find V t + a 1T 0 + o/r = — log—— + tV. TT Z —* Qj We have from equation (23) n (, (1—a2), t — a (1—a2) ) = —-log-------+ -jltt[, \ 2a st+a 2a ) (24) (25) where the constant of integration has been chosen so as to make x = 0, y = 0 when t = 0. The axis of x is ED, the axis of y the line at right angles to this passing through the middle of GB. If 2 Jc is the width of the gap and h the vertical distance between the plates, x = k, y = h, when t = 1, hence we have by(25) c (l —a2) \ 2 a A = c(i=*!,. 2 a ° 1+a) Hence a is determined by the equation 7r(l—a2 bl—a) (26)CONJUGATE FUNCTIONS. 229 241.] The quantity of electricity on the lower side of the semi- infinite plate between B and P is, since t increases from P to B, 47r { <#>jp — > or by (24) v (i tp + u 1 1+&) But by (25) if bp = x—k, we have .-i = c[(,- 1 - 1^-’jlog|=?-l»gl^}]. Hence if Q is the quantity of electricity on the lower side of the plate between B and P, x-k = C(tP-l)+ or since tP = a approximately, if P is a considerable distance from B, we have q = -<*)}• (27) The quantity of electricity Q1 on the upper side of the plate, from A to B, is equal to or since t = + 00 at A, and therefore A vanishes, we have ft = T?logiT5' <28> We can by equation (26) easily express a in terms of k/h, when this ratio is either very small or very large. We shall begin by considering the first case, which is the one that most frequently occurs in practice. We see from (26) that when k/h is very small, a is very small and is approximately equal to 7r k230 CONJUGATE FUNCTIONS. [241. The corresponding value of G is \k, hence, neglecting (k/h)zi «■= 20 V k " 4r>h 2 TT k2\ sir Hence Q + the whole quantity of electricity between A and P, is approximately equal to V $ ____7T k2} 4 7tJi\X 8 h ) Hence the quantity of electricity on the plate of the condenser is to the present degree of approximation the same as if the electricity were uniformly distributed over the plate with the density it would have if the slit were absent, provided that the area of the plate is increased by that of a strip whose width is 7 o thus the breadth of the additional strip is very approximately half that of the slit. We pass on now to the case when h/k is very small. We see from equation (26) that in this case a is very nearly equal to unity, the approximate values of a and G being given by the equations ^ 1 —a = —r ’ 7rk G = k. Hence by equations (27) and (28) we have So that the total charge Q + Qx on AP is equal to242.] CONJUGATE FUNCTIONS. 231 and thus the width of the additional strip is 242.] We have hitherto supposed that the potentials of the plates ABC and FGH are the same; we can however easily modify the investigation so as to give the solution of the case when ABC is maintained at the potential and FGH at the potential F2. The relation between z and t will not be affected by this change, but the relation between w and t will now be represented by the equation 4> + if = — log (t + a) — — log (t-a) +1 Tf. TT TT The quantity of electricity between B and P, a point on the lower side of the plate, is Now if BP is large, t at P is approximately equal to a, and V V p = —--log 2a -f ~r-(x—Ca\ TT h When h/k is large a is small and approximately equal to irk/4h, and this equation becomes ^ Vx, irk X *' = — logU + Tx- Since t = 1 at B and a is small, we see that +a)----1 log (tp>—a), 7T 7T which, since tp is large, may be written as + L\jr = —log(£ + a)-log(£ — a) + iV. 77 The quantity of electricity Q on the plate of the condenser between A and P, a point on BC at some considerable distance from is 4>a} ; since t is infinite at the point corresponding to A, we see that A is zero, hence Q = ~t~ 4>p 477 V 1 = I^10g £p + Cb 4 ^i^Ta Now the point P corresponds to a point in the t plane where t is very nearly equal to a; hence we have approximately by (29) = ^og(l-a2)) 7T 2k . = 5(*-—sm" k Vh2 + k2 * Thus k n v f 2k • -1 A-nhl * VK2 + lc2 * h. h2 ~ Zl°S h2 + k h2 + k2' }• In the case which occurs most frequently in practice, that in which k is small compared with h, we have, neglecting (k/h)2} „ VCONJUGATE FUNCTIONS. 235 244.] that is, the quantity of electricity on the plate is the same as if the distribution were uniform and the width of the plate were increased by half the breadth of the slit. The quantity of electricity on the face AB of the slit is equal to V 1 +a or, substituting the value for a previously found, 1 + k */W+k2 ) 1 - — Vh2 + k2) and this when k/h is small is equal to V 2k 4 7i h 7r Thus 2/77 of the increase in the charge on ABC, over the value it would have if the surface density were uniformly F/4 77 h on BC, is on the side AB of the slit, and (77—2)/77 is on the face of the plate of the condenser. 244.] A slight modification of the preceding solution will enable us to find the distribution of electricity on the conductors when ABC and FGH are no longer at the same potential. If Tj is the potential of ABC, that of FGH, then the relation between £ and t will remain the same as before, while the relation between w and t will now be expressed by the equation w = Q + if = flog (t + a) — -2log(tf — + or <#> +~—-^log (t + a) + —log (t + a) — — log(£ ^— a) + Hence the quantity of electricity on qbp where Q is a point on AB at some distance from B will exceed the quantity that would be found from the results of the preceding Article by 4 772 log tp -j- CL ^<2 + a Since P is a point on BC at some distance from B, tp is approxi- mately equal to a, and since a is small and tQ large we mayCONJUGATE FUNCTIONS. 236 [245- replace tQ + a by ; m pression becomes aking these substitutions the preceding ex- T£-TJ\ 2a (30) When t is large, the relation between z and t, which is given by the equation dt~ P-a? ’ is by integrating this equation found to be x — k + i (y—h) _____ = Ohg(t+ ✓(■-!)+ — sin-1 1 4-at) t + a) or substituting for G (the iD of the preceding Article) its value l 2 Jc/tt, we have x — k + i(y — h) 2k log (t + Vt1— 1) +1 -1 sin-1 - tt ( r—d —at . ,l+a£) — sin-1 ----> • t + a) Hence, when t is large we have approximately lo%2t = Yk(y~h)- Substituting this value for log tQ in the expression (30), we find that the correction to be applied on account of the difference of potential between ABC and FGH to the expression given by Art. 243 for the quantity of electricity on QBP is (¥-*p 4 7T2 Vh2 + k2 4 k + where y—h = BQ. 245.] The indirect method given by Maxwell, Electrostatics, Chap. XH, in which we begin by assuming an arbitrary relation between z and w of the form x + iy = + and then proceed to find the problems in electrostatics which can be solved by this relation, leads to some interesting results when elliptic functions are employed. Thus, let us assume x + iy = 6sn( + n/f), (31) and suppose that is the potential and \jr the stream function.CONJUGATE FUNCTIONS. 237 245-] Let k be the modulus of the elliptic functions, 2 K and 2 iK' the real and imaginary periods. Let us trace the equipotential surface for which = K; we have x + iy = bsn(K + b (32) “ dn(*,*r where dn (yj/, ¥) denotes that the modulus of the elliptic function is k\ that is Vl —k2, and not k. From equation (32) we see that V = 0, and b X= dn (\fr, k') * Now dn (yj/^k') is always positive, its greatest value is unity when = 0, or an even multiple of K\ its least value is k when \j/ is an odd multiple of K\ thus the equation h x + iy = dn (yjr, k') represents the portion of the axis of x between x = b and x = b/k. If we put = — K, we have x + iy = 6sn(—J5T + i\/r), b ~ dn(*,*'); hence the equipotential surface, —K, consists of the portion of the axis of x between x = —b and x = — b/k. Thus the transformation (31) solves the case of two infinite plane strips AB, CD, a p—b o--------------d Fig. 100, of finite and equal widths, b (1 —k)/k, in one plane placed so that their sides are parallel to each other. In the above investigation the potential difference is 2 K. The quantity of electricity on the top of the strip CD is equal to the difference in the values of \jr at C and D divided by 4 7r. Now the difference in the values of ^ at C and D is K', hence the quantity of electricity on the top of the strip is Fig. 100. 4 71 K'.CONJUGATE FUNCTIONS. 238 [245- There is an equal quantity of electricity on the bottom of the strip, so that the total charge on CD is — 2 K\ 4 TT The difference of potential between the strips is 2 K, hence the capacity of the strip per unit length measured parallel to z is 1_IC 47r K The modulus k of the elliptic functions is the ratio of BC to AD, that is the ratio of the shortest to the longest distance between points in the lines AB and CD. The values of K and K' for given values of k are tabulated in Legendre’s Traite des Fonctions Elliptiques : so that with these tables the capacity of two strips of any width can be readily found. When k is small, that is when the breadth of either of the strips is large compared with the distance between them, K and K' are given approximately by the following equations, *-!■ K'= log (4/k) = log (4 AD/BC). Hence in this case the capacity is approximately, 2^1og(4 AD/BC). Returning to the general case, if +1 \j/), if \js be taken as the potential and as the stream function. 246.] Capacity of a Pile of Plates, Fig. 101. If we put x + iy = sn(4> + tV0, (33) then when = K <“=SD(* + *)-(«) ------------------K ------------------— — K A P ~B~K k ------------------K --------—---------— K Fig. 101. Thus, since dn (f, ¥) is always real and positive, V= 0, y = 2irb, y = 4irb, &c., while x varies between the values x1, x2, where (35) When

= le'2 sn ^ cn lc’y^n2 Substituting the values of sn (\jr, k'), cn {ty, k'), dn (\j/, k')247-] CONJUGATE FUNCTIONS. 241 X in terms of e we get Pa-^i) d\lf _ 1 e 26 dx ~ b ( fo-2i) _ (a-a-Q (ga - a;) l(e 6 —e J )(e 6 —e Hence the surface density is equal to AB 1 e 26 b )/ 4irb $ AP AP BP BP U* ((e 6 —€ 6 )(e & — € * )) The distribution of electricity on any one of the plates is evidently the same as if the plate were placed midway between two infinite parallel plates at potential zero, the distance between the two infinite plates being 27*6. 247.] Capacity of a system of 2 n plates arranged radially and making equal angles with each other, the alternate plates being at the same potential, the extremities of the plates lying on two coaxial right circular cylinders. Let us put (x + iy V b sn(<£ + i\/r), or, transforming to polar coordinates r and 0, (^) *tn*= sn(<£ + o/r). Then, as before, we see that when 0 = K, n 6 = 0 or 2 tt, or 4 7r, and so on, and when = — K, nd = ir or 3 7r, or 5w, &c.; hence this transformation solves the case of 2 n plates arranged radially, making angles tt/u with each other, one set of n plates being at the potential K, the other set at the potential —K. When <#> = K> we have r « 1 W =dn fcV)' Hence if rx and r2 are the smallest and greatest distances of the edges of a plate from the line to which all the plates con- verge, we have n242 CONJUGATE FUNCTIONS. [248. The total charge on both sides of one of the plates is, as before, K'/2tt, and since the potential difference is 2 K the capacity of the plate is K'/l-nK, When r1 is small compared with r2,k is small, and we have then approximately K' = log (4/k) = log 4 + n log (r2/rx). Thus the capacity of a plate is in this case approximately ^{log4 + ™log(r2/r1)}. Returning to the general case, the surface density of the electricity on one side of a plate is equal to 4tt dr ’ but since 1 =k'2 sn ^cn (*>*w(*.k')- Substituting for the elliptic functions their values in terms of r, we find when <£=# d\j/ __ nbnrn~1 dr k {(r2w — r12n)(r22w — r2n)}* Thus the surface density is equal to 1 w2wrn_1 47r |(r2w — r12w)(r22w—r2n)}^ When n = 1, this case coincides with that discussed in Art. 245. 248.] Let us next put x+iy = 6cn($ + o/r), and take ^ for the potential, and for the stream function. Then when ^ = 0, we have x + iy = Sen <#>, hence y = 0, and x can have any value between ±b: thus the ©= dn Nr, if) ’CONJUGATE FUNCTIONS. 243 248.] equipotential surface for which \jr is zero is the portion of the axis of x between x — — b, and x = -f b> When t/t = K\ x + iy = ft cnf^ + ilf') _ ftidn<£ &sn 5 hence x = 0, and y ranges from + bJc'/Jc to + 00 and from — bk'/k to — 00. Hence the section of the equipotential surface for which yjr = K' is the portion of the axis of y included between these limits. Thus the section of the conductors over which the distribution of electricity is given by this transformation is similar to that represented in Fig. 102, where the axis of x is vertical. To find the quantity of electricity on A B we notice that (f> = 0 at A and is equal to 2 K at B, hence the quantity of A electricity on one side of AB f--------------E is equal to Kj 2n, thus the total charge on AB is K/ tt. Fig 102# The difference of potential between ab and CD or ef is K\ so that the capacity of AB is equal to 1 If vK'* The modulus k of the elliptic functions is given by the equation k' _ {1-&2}* _ EC k “ k “AB* If AB is very large compared with EC then k is very nearly unity, and in this case we have K = log (4/¥) = log (4AB/EC), so that the capacity of AB is ~log (4AB/EC). The surface density of the electricity at a point P on either244 CONJUGATE FUNCTIONS. [249- side of ab is (without any limitation as to the value of h) equal to 4 7r dx 9 and since x = b cn , = *>* + *’}* z. ____________ = - CP •/ AP. BP; hence the surfaoe density is equal to b 1 4 w* cp-Zap. bp 249.] We pass on now to consider the transformation x + iy e h = cn (<£ + ti/r), where <£ is taken as the potential and \jr as the stream function. Over the equipotential surface for which = 0, we have x 4- iy e h = cn _ 1 cn(\//-,&') Hence y = 0, ±7r&, ±27r6,...; while £ ranges from 0 to infinity. For the equipotential surface for which . x' = iog4 + j, where x' = — xv Thus the capacity of a plate in this case is approximately equal to j 4 (6 log 4 + x') The surface density at a point on one of the first set of plates at246 CONJUGATE FUNCTIONS. [25I- a distance x from the edge is easily shewn by the methods previously used to be equal, whatever be the value of k, to 1 4 7ikb j 2x 2x 2xi V («T-l)(e^ +eT") 250.] The transformation with as the potential and \j/ as the stream function, gives the solution of the case represented in Fig. 104 ; where the 2 n outer planes at potential zero are supposed to extend to infinity, the 2 n inner planes at potential K bisect the angles between the outer planes, and OA = &. We can easily prove that in this case the quantity of electricity on the outer plates is equal to nK'/n, so that the capacity of the system is equal to nlT 7T K ’ when the modulus of the elliptic functions is determined by the reiation ,OC," _ k' voa' - J' 251.] The transformation x + iy = b dn ( + where

= 0, we have x + iy = b dn i\fr _ dn(f,k') cn (*,£')’ (36)CONJUGATE FUNCTIONS. 247 25I-] hence y = 0, and x ranges from + b to +00 and from —6 to — 00, thus giving the portions EF, CD of the figure. When for the potential and \f/ for the stream function, then since, when = 0, x + iy e h = dn (L\fr) _ dn (\f/9 k') cn (\jf, hr)9 we have y = 0, y = ±itb, y = + 2Trb..., while x ranges from 0 to + oo. Thus the equipotential surfaces for which $ vanishes are a pile of parallel semi-infinite plates stretching from the axis of y to infinity along the positive direction of x9 the distance between two adjacent plates being irb. When = K are a pile of parallel semi-infinite plates stretching from — oo to a distance x1 from the previous set of plates. The distance between adjacent plates in this set is again Tib, and the planes of the plates in this set are the continuations of those of the plates in the set at potential zero. This system of conductors is represented in Fig. 106.CONJUGATE FUNCTIONS. 249 253-] The quantity of electricity on both sides of one of the plates at potential zero is —K'/2v, hence the ' —" — capacity of such a ---------------------- --------------------- plate is Pig, io6. 1 K' 2-nK ’ the modulus of the elliptic functions being given by equation (38). When the distance between the edges of the two sets of plates is large compared with the distance between two adjacent parallel plates, then xx is large compared with 6, so that k' is small; in this case we have approximately 7r 2’ log (4/ft0 log4+|; hence the capacity of a plate is equal to b _ 4 (a^+6 log 4) * The surface density of the electricity at a point P on one of the planes at potential zero is in the general case easily proved to be equal to _ _1______________^____________ 4 it b j 2k 2x 2z, U ’ 253.] The transformation (JB + IJ/) =dn + where is the potential and \jr the stream function and n a positive integer, gives the solution of the case shown in Fig. 107, when the potential of the outer radial plates is zero and that of the inner K. The 2 n outer plates make equal angles with each other and extend to infinity. The quantity of electricity on both sides of one of the outer K =250 CONJUGATE FUNCTIONS. plates is — K'/2k\ since there are 2n of these plates the capacity of the system is \ / . nK' the modulus of the Elliptic Functions being given by the equation o^A B_ Fig. 107. 254.] We have only considered those applications of elliptic function to elec- trostatics where the expression for the capacity of the electrical system proves to be such that it can be readily calculated in any special case by the aid of Legendre’s Tables. There are many other transformations which are of great interest analytically, though the want of tables of the special functions involved makes them of less interest for experimental purposes than those we have considered. Thus, for example, the transformation x + iy = Z( + i\lr)9 where Z is the function introduced by Jacobi and defined by the equation ru jg Z(u) = J &n2udu — -g} if \jr is the potential and the stream function, gives the distri- bution of electricity in the important case of a condenser formed by two parallel and equal plates of finite breadth.CHAPTER IV. ELECTRICAL WAVES AND OSCILLATIONS. 255. ] The properties of electrical systems in which the distribu- tion of electricity varies periodically and with sufficient rapidity to call into play the effects of electric inertia, are so interesting and important that they have attracted a very large amount of attention ever since the principles which govern them were set forth by Maxwell in his Electricity and Magnetism. We shall in this Chapter consider the theory of such vibrating electrical systems, while the following Chapter will contain an account of some remarkable experiments by which the properties of such systems have been exhibited in a very striking way. 256. ] We shall begin by writing down the general equations which we shall require in discussing the transmission of electric disturbances through a field in which both insulators and con- ductors are present. Let jF, G, H be the components of the vector potential parallel to the axes of x, y, z respectively, P, Q, R the components of the electromotive intensity, and ay 6, c those of the magnetic induc- tion in the same directions, let <\> be the electrostatic potential, \ dt dx ’ (i)252 ELECTRICAL WAVES AND OSCILLATIONS. [256- We have also hence so that similarly _ dH dG a ~~ dy dz * da^^ddH^d^dG dt dy dt dz dt da __ dQ dR \ dt ~~ dz dy 9 db __ dR dP dt ~~ dx dz dc __ dP dQ dt dy dx (2) If a, /3, y are the components of the magnetic force, u, v, w those of the total current, then (Maxwell’s Electricity and Mag- netmn, Art. 607) _ dy dj3 K dy dz1 4 ITU . da dy '*V ~~ dz dx ’ , dp da (») In the metal the total current is the sum of the conduction and polarization currents; the conduction current parallel to x is P/ and therefore by (2) we have, assuming dP dQ dR2 5 7-] ELECTRICAL WAVES AND OSCILLATIONS. 253 similarly **-*?%■ (4) It follows from equation (2) that a, 6, c satisfy equations of the same form. In the dielectric there is only the polarization current, the com- K' dP ponent of which parallel to x is — hence in the dielectric we have dP _ dy _ d£ _ l (dc _ db^ dt ~ dy dz~ ix 'dy d& ’ and therefore by (2) /72 p similarly V2Q = ix K‘ ,tr,d2Q dt2 (5) We shall suppose that the effects are periodic and of frequency p/2 77j so that the components of the electromotive intensity, as well as of the magnetic induction, will all vary as €ipt and will not explicitly involve the time in any other way. We shall also suppose that the electric waves are travelling parallel to the axis of z, so that the variables before enumerated will contain €mz as a factor, m being a quantity which it is one of the objects of our investigation to determine. With these assump- tions we see that d/dt may be replaced by ip, and d/dz by im. Alternating Electric Currents in Two Dimensions. 257.] The cases relating to alternating currents which are of the greatest practical importance are those in which the currents flow along metallic wires. As the analysis, however, in these cases is somewhat complicated, we shall begin by con- sidering the two dimensional problem, as this, though of com- paratively small practical importance, enables us by the aid of simple analysis to illustrate some important properties possessed by alternating currents.254 ELECTRICAL WAVES AND OSCILLATIONS. [257- The case we shall first consider is that of an infinite con- ducting plate bounded by the planes x = h, x = —h, immersed in a dielectric. We shall suppose that plane waves of electromotive intensity are advancing through the dielectric, and that these waves impinge on the plate. We shall suppose also that the waves fall on both sides of the plate and are symmetrical with respect to it. These waves when they strike against the plate will be reflected from it, so that there will on either side of the plate be systems of direct and reflected waves. Let P and R denote the components of the electromotive intensity parallel to the axes of x and z respectively, the com- ponent parallel to the axis of y vanishing since the case is one in two dimensions. Then in the dielectric the part of R due to the direct wave will be of the form (mz + lx+pt) while the part due to the reflected wave will be of the form Q^i(mz — lx+pt) Thus in the dielectric on one side of the plate R = + + Cel(mz~lx+pt\ (1) If F is the velocity with which electromagnetic disturbances are propagated through the dielectric, we have by equation (5), Art. 256, since ,x'Z'=l/F2, d2R d2R _ 1 d2R dxl + dz2 ”” V2 dt2 hence l2 + m2 = K V2' If X is the wave length of the incident wave, 9 the angle between the normal to the wave front and the axis of x9 we have, since 2tt y = Tr, , 27r 27r . . I = —cos 9, m = — sin 9. X X Since Q vanishes, we have dP dR257-] ELECTRICAL WAVES AND OSCILLATIONS. 255 Substituting the value of R from equation (1), we find jp 21? ^ (viz + lx+pt) q^ (mz — lx+pt) j l The resultant electromotive intensity in the incident wave is R i (mz + lx + pt) cos 0 9 in the reflected wave (2) G COS0' t (mz — lx+pt) Let us now consider the electromotive intensity in the con- ducting plate; in this region we have, by (4), Art. 256, if /x is the magnetic permeability and ^ d(ix) When x is very large J0{ix) = Jo'(.a) = --£=. V2ttx V2 TTX 2^; K°/(lx) = l€“*V7 Tx (See Heine, Kugelfunctionen, vol. i. p. 248). 262.] We shall now proceed to apply these results to the in- vestigation of the propagation of electric disturbances along the * Heine, Kugelfunctionen, vol. i. p. 189.264 ELECTRICAL WAVES AND OSCILLATIONS. [262. wire. The axis of the wire is taken as the axis of z ; P, Q, R are the components of the electromotive intensity parallel to the axes of xy y, z respectively; Q = S%; r r hence, since S is a function of r, 0, and t, and not of 6, we may write 7 , P_^x, 0-^X. dx V~dy’ (5) where x is a function we proceed to determine. Since P and Q satisfy equations of the form d*P d2P dx2 1 dy2 we have dx2 dy2 But c?P 1 dx dy ~k2P = 0,\ («) dz ~ ’ ' so that by equations (5) and (6) 7 2 dJi k x+ d^=0- We thus have the following expressions for P, Q, R, P=-T^JOJ0(lkr) + BK0(ikr)}e^mz+^ Q = - ! Jt = {OJ0(tIcr) + BK0(ikr)} (7) To find a, 6, c, the components of the magnetic induction, we have266 ELECTBICAL WAVES AND OSCILLATIONS. [262. From these equations we find 6 - - ir«i‘kr)} c = 0 ; thus the resultant magnetic induction is equal to (8) m2—k2 d {CJ0(ikr) + DK0(ikr)} e t (mz +pt) ipk2 dr and the lines of magnetic force are circles with their centres along the axis of z and their planes at right angles to it. We now proceed to consider the wire. The differential equation satisfied by R in the wire is d2R d2R d2R _ 477/x dRi clx2 + dy2 dz2 o- dt Transforming this equation to cylindrical coordinates it be- comes, since R is independent of 0, d2R 1 dR 2 73 H-----~jr ~n2R = 0, dr* r dr where, as usual, 4tjtup n2 = m -------— « Since r can vanish in the wire, the solution of this equation is R = AJn(Lnr)el(-mz+pt\ where A is a constant. We can deduce the expressions for P and Q from R in the same way as for the dielectric, and we find P= _ —A—-T(,m«.\*l(mz + 'Pt) \ -,ATxJ0(Lnr)<‘ rj iTYb A d y f \ 1 (mz +pt) P = and also a = ipn AJ0(tnr)e‘(mz+pt), b = — c = 0. A^J0 (cnr) {mz+pt'> V ipn2 dx 0 x 7 (9) (10)ELECTRICAL WAVES AND OSCILLATIONS. 267 262.] The resultant magnetic induction is at right angles to r and z and equal to ipn* dr In the outer conductor the differential equations are of the same form, but their solution will be expressed by the K func- tions and not by the J1 s, since r can be infinite in the outer con- ductor. We find if ± , n't = m2+ Ig/g, < (7S*0'(‘*'b)*0(‘*b) + ^jK^n'^K:^)) = (^Jo'('**)KoW*) + ^/0(‘»»)Jf0'(‘**))x (i^^0'(,«'b)J-0(.*b) + ^Z0(,«'b) J-0'(ifeb)). (15) This equation gives the relation between the wave length 2 77/m along the wire and the frequency $/2 77 of the vibration. To simplify this equation, we notice that k a, k b are both very small quantities, for, as we shall subsequently find, k9 when the electrical waves are very long, is inversely proportional to the wave length, while when the waves are short k is small compared with the reciprocal of the wave length; we may therefore assume that when the waves transmitted along the cable are long compared with its radii, ka and &b are very small. But in this case we have approximately, /0(ifca) = 1, J0(ikb) = 1, J0' (ik*) = — £ifca, J0'(ikb) = — \ikb; K0(tka) = log^» Z0(t/cb) = log^, Ko'W = -7L' *o'(^) = -Jy262.] ELECTKICAL WAVES AND OSCILLATIONS. 269 Making these substitutions, equation (15) reduces to [°n (j + :r;jSi) -‘rV(s+^bl08.-fe)^^) p 2 {, io- 1 ) . m-F2t 2ir^a2log(b/a)r we have seen however that the second term in the bracket is large compared with unity, so that we have approximately ma = P “t __________l___. V2 2na2 log (b/a) If R is the resistance and T the capacity in electromagnetic measure per unit length of the wire, then since 2l0 2y / . k “ F210g in' b log (b/a) ’ (20) or ni2 = ^ 11 + log Thus approximately 2 y K...1. ?(b/a)3 TO* “ F2l°giw' b log (b/a) iTi'b log (b/a)) 2y / and since n '2 _ 4 77 fx'ip/cr', m* 1 P2 (i o-'y2 t7T) / V*\ i/ub2p + 2) log (b/a)’ hence we have approximately ( /1 a'y2 ) * m =jlp j: V2 F ( Tt 1 +lTi )}’ <2,) log (b/a) J ^ " 4 log ( and n' b are very large; when this is the case we know by Art. 261 that J'Q (ina) = — iJ0 (ma), K'0 (m'b) = iK0 (in'b). Making these substitutions, equation (17) becomes p = |!Ja + (22) V2\no, 'Rb)log(b/a) v y or ^ approximately. Since the second term inside the bracket is small compared with unity, extracting the square root we have, This represents a vibration travelling approximately with the velocity V and dying away to 1/e of its initial value after tra- versing a distance iv\Z^w s+v Since the imaginary part of m is small compared with the real part, the vibration will travel over many wave lengths before its amplitude is appreciably reduced. From the expression for the rate of decay in this case we see that when the wire is surrounded by a very much worse conductor than itself, as is practically always the case with cables, the distance to which these very rapid oscillations will travel will be governed mainly by the outside conductor, and will be almost independent of the resistance and permeability of the wire ; no appreciable advantage therefore would in this case be derived by using a well-con- ducting but expensive material like copper for the wire. In aerial wires the decay will be governed by the conductivity of the earth rather than by that of the wire, unless the height of the wire above the ground, which we may take to be comparable with b, is so great that ), where \jr = mz + pt — (2nTfxp/a)? (a—r) + ^ • Similarly, we find by equation (9) that the radial electro- motive intensity (P2 + Q2)* is given by the equation {P* + Q*}i = - (26) V 27rvar 4 The resultant magnetic force is by equation (10) equal to -4= (a-r)cos^_ . Var 4ELECTRICAL WAVES AND OSCILLATIONS. 281 268.] Since all these expressions contain the factor e-(2w/ while the radial electromotive intensity is 2 VI —cos (mz +pt), and the resultant magnetic force 21 cos (mz+pt). We see that the maximum value of the radial electromotive intensity is very great compared with that of the tangential, so that in the dielectric the Faraday tubes are approximately radial. The momentum due to these tubes is, by Art. 12, at right angles both to the tubes and the magnetic force, so that in the dielectric it is parallel to the axis of the wire, while in the wire itself it is radial. Thus for these rapidly alternating currents the momentum in the dielectric follows the wire. The radial polarization in the dielectric is 2£/4 tt times the radial electro- motive intensity, and since K = 1/F2,270.] ELECTRICAL WAVES AND OSCILLATIONS. 283 it is equal to h 2 7rVr cos [mz +pt). If the Faraday tubes in the dielectric are moving with velocity V at right angles to their length, i. e. parallel to the wire, the magnetic force due to these moving tubes is, by Art. 9, at right angles both to the direction of motion, i.e. to the axis of the wire, and to the direction of the tubes, i.e. to the radius, and the magnitude of the magnetic force being, by (4), Art 9, 4 7t V times the polarization, is 27 —°cos (mz+pt), which is the expression we have already found. Hence we may regard the magnetic force in the field as due to the motion through it of the radial Faraday tubes, these moving parallel to the wire with the velocity with which electromagnetic disturb- ances are propagated through the dielectric. In the outer conductor when nf r is large R = — J0 e—(r-») cos (irbr ) where ' = mz +pt — (27r \xp/a')* 4- ~ • The radial electromotive intensity is 4 e-(Wi>/<04 (r-n) cos (y + 1) . 2K v Ar v 4 ’ The resultant magnetic force is perpendicular to r and equal to PjL e-(2»/p/a2/o- is no longer a small quantity; such vibra- tions however die away more rapidly than the slower ones, so that when the distance from the origin of disturbance is con- siderable the latter are the only vibrations whose effects are felt. For such vibrations, we have by Art. 263 * where a is any constant and F (a) denotes an arbitrary function of a, will satisfy the electrical conditions. By Fourier’s theorem, however, is equal to F(z) when t = 0. Hence this integral, since it satisfies the equations of the electric field, will be the expression for the disturbance on the wire at 0 at the time t of the disturb- ance, which is equal to F(z) when t = 0. When the disturbance is originally confined to a space close to the origin, F(a) vanishes unless a is very small; the expression (27) becomes in this.case have to make use of the general relation between m and p given plication arises from the vibrations whose frequencies are so Hence a term in the expression for R of the form m2 F(a) e”Er cosm(0—a), where Since (28) we see by (28) that the disturbance at time t and place z will be equal to F (29)27°-] ELECTRICAL WAVES AND OSCILLATIONS. 285 Thus at a given point on the wire the disturbance will vary as 1 -- Vt where c is a constant. The rise and fall of the disturbance with the time is represented in Fig. 108, where the ordinates represent Fig. 108. the intensity of the disturbance and the abscissae the time. It will be noticed that the disturbance remains very small until t approaches c/4, when it begins to increase with great rapidity, reaching its maximum value when t = 2 c; when t is greater than this the disturbance diminishes, but fades away from its maximum value much more slowly than it approached it. Since the disturbance rises suddenly to its maximum value we may with propriety call T, the time which elapses before this value is attained at a given point, the time taken by the disturb- ance to travel to that point. We see from (29) that T=^Br. (30) Thus the time taken by the disturbance to travel a distance z is proportional to z2, it is also proportional to the product of the resistance and capacity per unit length.286 ELECTRICAL WAVES AND OSCILLATIONS. [2 70. By dividing 0 by T we get the so-called ‘ velocity of the current along the wire;9 this by (30) is equal to 2 0RT* The velocity thus varies inversely as the length of the cable, and for short lengths it may be very great. The preceding formula would in fact, unless 0 were greater than 2/VHT, indicate a velocity of propagation greater than V. This however is im- possible, and the error arises from our using the equation ip= — m2/ltr instead of the accurate equation (18). By our approximate equation vibrations of infinite frequency travel with infinite velocity, in reality we have seen (Art. 267) that they travel with the velocity V. These very rapid vibrations however die away very quickly, and when we get to a distance equal to a small multiple of 2/Fltr they will practically have disappeared, and at such distances we may trust the ex- pressions (31). A considerable number of experiments have been made on the time required to transmit messages on both aerial and submarine cables; the results of some of these, made on aerial telegraph iron wires 4 mm. in diameter, are given in the accompanying table taken from a paper by Hagenbach (Wied. Ann. 29. p. 377):— Observer. Length of line in kilo- metres. Time taken for message to travel (T.) 1020r/ (square of length of line in centimetres). Fizeau and Gonnelle . 314 •003085 313 Walker . 885 •02943 376 Mitchel 977 •02128 223 Gould and Walker 1681 •07255 257 Guillemin . 1004 •028 278 Plantamour and Hirsch 1326 •00895 5090 Lowy and Stephan 863 •024 322 Albrecht 1230 •059 390 Hagenbach. 284-8 •00176 217 Hagenbach proved by making experiments with lines of different lengths that the time taken by a message to travel along a line was proportional to the square of the length of the line. If we apply the formula T = \z2RT27O.] ELECTRICAL WAVES AND OSCILLATIONS. 287 to Hagenbach’s experiment in the above table, where 0 = 284-8 x 105, R = 9-4 xlO4, and (by estimation) T = 10~22, we find T = *0038, whereas Hagenbach found *0017. The agree- ment is not good, but we must remember that with delicate receiving instruments it will be possible to detect the disturb- ance before it reaches its maximum value, so that we should expect the observed time to be less than that at which the effect is a maximum. In Hagenbach’s experiment the line was about 4 times the length which, according to the formula, would have made the disturbance travel with the velocity of light, so that it would seem to have been long enough to warrant the ap- plication of a formula which assumes that the shorter waves would have become so reduced in amplitude that their effects might be neglected. When the wire is of length Z, we know by Fourier’s Theorem that any initial disturbance R may be represented by the equation the value of R after a time t has elapsed will be represented by the equation For a full discussion of the transmission of signals along cables the reader is referred to a series of papers by Lord Kelvin at the beginning of Vol. II of his Collected Papers. Since tp = — m2/Rr,288 ELECTRICAL WAVES AND OSCILLATIONS. [271. Relation between the External Electromotive Intensity and the Current. 271.] We have hitherto only considered the total electromotive intensity and have not regarded it as made up of two parts, one due to external causes and the other due to the induction of the alternating currents in the conductors and dielectric. For some purposes, however, it is convenient to separate the electromotive intensity into these two parts, and to find the relation between the currents and the external electromotive intensity acting on the system. We may conveniently regard the external electromotive intensity as arising from an electrostatic potential satisfying the equation V2 = 0. We suppose that, as in the preceding investigation, all the variables contain the factor +!>*). Since (p varies as eimz, the equation V2<£ = 0 is equivalent to dr2 + 1 dj> r dr 0. The solution of this is, in the wire — LJ0 (imr) e in the dielectric = {MJ0 (imr) + NK0 (imr)} e‘(mz+2,<), in the outer conductor = SK0(imr)(‘(rnz+pt\ If, as before, a and b are the radii of the internal and external boundaries of the dielectric, we have, since <\> is continuous, iJ*0(tma) = M J0(ima) + iV7iL0(ima), SK0(imb)= + M0(imb). The excess of the normal electromotive intensity due to the electrostatic potential in the dielectric over that in the wire is equal to im {LJ0' (t ma) - (MJ0' (.ma) + NK0' (t ma)} e« («•+!*>. substituting the value for L — M in terms of N from the preceding equation, this becomes N im-j—i-----\ {J0f (ima) K0 (lma) — JQ (tma) KQ'(ima)J.271.] ELECTRICAL WAVES AND OSCILLATIONS. 289 Now Jo' (imtj K0 (ma) - J0 (ima) Z/ (tw a) = for let w = /0' (as) Z0 (as)—J0 (as) Z0' (as), then g = J"(x)K0(x)-J0(x)K"(x), but J0"(x)+ lj0'(x)-J0(x) = 0, dy K0"(x)+lK0'(x)-K0(x) = 0; substituting the values of J*0" (a?), K0" (#) from these equations, we find , ^-1-{J0'(x)K0(x)-J0(x)K0'(x)\ u X3 hence 16 = O a? ‘ where (7 is a constant. Substituting from Art. 261 the values for J0 (x\ Jq (x), K0 (x), K0' (x) when x is very small, we find that C is equal to unity. Thus when r = a, the normal electromotive intensity due to the electrostatic potential in the dielectric exceeds that in the wire by e t (mz + pt) a/0(ima) Similarly we may show that when r = b the normal electro- motive intensity in the dielectric exceeds that in the outer conductor by ,, ______"______ i (mz +pt) bK0(imb)f Now the electromotive intensities arising from the induction of the currents are continuous, so that the discontinuity in the total normal intensity must be equal to the discontinuity in the components arising from the electrostatic potential. By equa- tions (7), (9), (14) the total normal intensity in the dielectric at the surface of separation exceeds that in the wire by A n2—m2 (P-m2) >} t (mz+pt) l290 ELECTRICAL WAVES AND OSCILLATIONS. [271. hence we have A—J0' (ma) { :11 = -y- n ox 1 hi(kr—m2) ) a J0 Similarly wES)-'}=- By equations (10) and (12) N (tma) ‘ M bjK’^tmb) 2 7:a (n*~m^AJ0'(tn&) e‘(mz+2,t) 2 ffb EK> (t%'b) €<(^+yo Hpn u v 1 (32) (33) are respectively the line integrals of the magnetic force round the circumference of the wire and the inner circumference of the outer conductor, hence they are respectively 4 tt times the current through the wire, and 4 7t times the current through the wire plus that through the dielectric. Unless however the radius of the outer conductor is enormously greater than that of the wire, the current through the wire is infinite in comparison with that through the dielectric: for the electromotive intensity R is of the same order in the wire and in the dielectric ; the current density in the wire is ii/ K0Umb) = log-^-> ov 1 ° tma ox J & tmb hence we have E = -lmJVlog(b/a)e‘(,,“+**)t or by equation (35), since Jr0(tma) = 1, But by Art. 263 we have, if I0*l(mz+pt^ is the total current through the wire, & r 27rat . r// x J0 = — AJ0 (tfia),292 ELECTRICAL WAVES AND OSCILLATIONS. [272. hence, since n2—m2 = 47t/xip/v, m2—k2 = p2/V2, E = 21/) ^ log (b/a). I0 e 'mz +pt\ T~2 Eut by equation (18) m*-P*S 1 1 (nrJoi'to*) m ~ V2\ 4ir?A a hence n')s - 12 x ll X + - + L (I («»«’»>) " W>)s + 9^T6‘(4*W“W + -")]1/’ ( , b 1 1 7i2/x3n2a4 13 7r4/m5n4a8 ) _ or . = 1J,{81og; + ^-j|-i|_+55j5— + ira2l +12 cr2 180 We may write this as - E = PipI+Q,I, (39)ELECTRICAL WAVES AND OSCILLATIONS. 293 272.] or since as r dl E = P §+*I. If L is the coefficient of self-induction and R the resistance of a circuit through which a current I is flowing, we have external electromotive force = L -- -f RI. dt By the analogy of this equation with (39) we may call P the self- induction and Q the resistance of the cable per unit length for these alternating currents. Q has been called the ‘ impedance ’ of unit length of the circuit by Mr. Heaviside, and this term is preferable to resistance as it enables the latter to be used ex- clusively for steady currents. By comparing (39) with (38), we see that p = 2 log* + &*/<**)+ ^(Myn'ayo-4)-.,,, * = + i^(MV-4a8A4)+ , • (40) These results are the same as those given in equation (18), Art. 690, of Maxwell’s Electricity and Magnetism, with the exception that /i is put equal to unity in that equation and in it A is written instead of 2 log (b/a). We see from these equations that as the rate of alternation increases, the impedance increases while the self-induction diminishes ; both these effects are due to the influence of the rate of alternation on the distribution of the current. As the rate of alternation increases the current gets more and more concen- trated towards the surface of the wire; the effective area of the wire is thus diminished and the resistance therefore increased. On the other hand, the concentration of the current on the surface of the wire increases the average distance between the portions of the currents in the wire, and diminishes that between the currents in the wire and those flowing in the opposite direction in the outer conductor; both these effects diminish the self- induction of the system of currents. The expression for Q does not to our degree of approxi-294 ELECTRICAL WAVES AND OSCILLATIONS. [273, mation involve b at all, while b only enters into the first term of the expression for P, which is independent of the frequency; thus, as long as w a is very small, the presence of the outer conductor does not affect the impedance, nor the way in which the self-induction varies with the frequency. When p = 0 the self-induction per unit length is 2 log (b/a) + £ ft. Since fx for soft iron may be as great as 2000, the self-induction per unit length of straight iron wires will be enormously greater than that of wires made of the non-metallic metals. 273.] We shall now pass on to the case when na is large and n'b small, so that naJ0(ina,)/pQ,Jo'(ina) is small compared with nra KQ(in'b)/pbK0'(in'b). These conditions are com- patible if the specific resistance of the outer conductor is very much greater than that of the wire. In this case equation (37) becomes E = 2 ip jlog - + ^ n‘V K0 (in' b)) 4tjp b K0' (m'b)) Since n'b is small, we have approximately K0(m'b) = log(2y/t?i'b), K0f (in'b) = — l/m'b ; hence E = 2 ip jlog b/a -f \j! log (y/VTTfx'b2p/*i') — 13/ ^ j < Thus the coefficient of self-induction in this case is 2 log (b/a) + 2/ log (y/v^/x'^b2//), and the impedance f 7rpix'. It is worthy of remark that to our order of approximation neither the impedance nor the self-induction depends upon the resistance of the wire. This is only what we should expect for the self-induction, for since n a is large the currents will all be on the surface of the wire ; the configuration of the currents has thus reached a limit beyond which it is not affected by the resistance of the wire. It should be noticed that the conditions na large and n'b small make the impedance f 7rp\i large com- pared with the resistance a/no,2 for steady currents.ELECTRICAL WAVES AND OSCILLATIONS. 295 2 74-] Very Rapid Currents. 274] We must now consider the case where the frequency is so great that no, and n'b are very large; in this case, by Art. 261, Jq (iwa) = — iJ0(Lnoi), KJ(in'b) = LKQ(*.n'h), so that equation (37) becomes E = 2 ip hi+[( (TfJ, 4:TTp& / ^ (T [X 4iTTpb* we see from this equation that the self-induction P is given by the equation P = 2 log (b/a) 4- (o-jui/27rpa2)^ + ( -kh\ m2-n2n h Jc2—m2 \xn Ce (iru-sfH) = - v ' fxn Eliminating C and D by the aid of equations (44), we have k2—m2 m2 — n-s D/&2—m2 m2-7h2\ -kh A (—;--------------)e =B(— -----------+--------Jc , ' k fxn J ' k \j.n ' A sk2—m2 m2 —__kh -r>/k2 — m2 m2—n2\ kh A (---7-- + ------- ) C — JL> [ --7------------) c ^ k fM n ' v k fxn ' J (45) From these equations we get A2 = B2. The solution A = B corresponds to the current flowing in the same direction in the two slabs, the other solution corresponds to the case when the current flows in one direction in one slab and in the opposite direction in the other; it is this case we shall proceed to investigate. Putting A = — Bi equation (45) becomes but A/ fX7b tf-m2=-.p2/V2, n2—m2 = 4 7Tfup/ P = 4 7rm/0 (V2/p) cos (mz +pt), b = — 4 7r J0 cos (mz +_p£), where n' = { 2 n^p/a} In the metal slab we have on the side where x is positive, R = al0y/2n'z~n cos (mz+jp£ — n'(x—h) + > P = — al0me~n ^“^sin(mz+pt—n' (x — h)), b = — 47r/x/0e”w cos — —&)). WTe see from these equations that P/P is very large in the dielectric and very small in the metal slab, thus the Faraday tubes are at right angles to the conductor in the dielectric and parallel to it in the metal slab. Mechanical Force between the Slabs. 277.] This may be regarded as consisting of two parts, (1) an attractive force, due to the attraction of the positive electricity of one slab on the negative of the other, (2) a repulsive force, due to the repulsion between the positive currents in one slab and the negative in the other. To calculate the first force we notice that since V2fpcr is very large, the value of P in the con-301 277-] ELECTRICAL WAVES AND OSCILLATIONS. ductor is very small compared with the value in the dielectric, and may without appreciable error be neglected; hence if e is the surface density of the electricity on the slab and K the specific inductive capacity of the dielectric, 47re = —Kb-nm (V2/p) I0 cos (mz +pt). The force on the slab per unit area is equal to Pe/2 ; substituting the values of P and e this becomes 2 7t Km2 (V*/p2) J02 cos2 (mz +pt). The force due to the repulsion between the currents in the slabs per unit volume is equal to the product of the magnetic induction b into wy the intensity of the current parallel to z. Since db i’l‘w = -Tx’ the force per unit volume is equal to 1 db2 8 7T/X dx ’ hence the repulsive force per unit area of the surface of the slab _ 1 db2 - ~~Jh s7fjLdidX = g^(b2)x = h = cos2 (mz+pt). When the alternations are so rapid that the vibrations travel with the velocity of light V2m2 = p2y and since K = l/F2, the attraction between the slabs is equal to 2 7r/02 cos2 (mz +pt), while the repulsion is 2 7Tfjil02 cos2 (mz+pt), hence the resultant repulsion is equal to 2 77 (jx — 1) I02 cos2 (mz +pt). If the slabs are non-magnetic = 1, so that for these very rapid vibrations the electrostatic attraction just counterbalances the electromagnetic repulsion. Mr. Boys (Phil. Mag. [5], 31, p. 44, 1891) found that the mechanical forces between two conductors carrying very rapidly alternating currents was too small to be302 ELECTRICAL WAVES AND OSCILLATIONS. [278. detected, even by the marvellously sensitive methods for measur- ing small forces which he has perfected, and which would have enabled him to detect forces comparable in magnitude with those due to the electrostatic charges or to the repulsion between the currents. Propagation of Longitudinal Waves of Magnetic Induction along Wires. 278.] In the preceding investigations the current has been along the wire and the lines of magnetic force have formed a series of co-axial circles, the axis of these circles being that of the wire. Another case, however, of considerable practical im- portance is when these relations of the magnetic force and current are interchanged, the current flowing in circles round the axis of the wire while the magnetic force is mainly along it. This condition might be realized by surrounding a portion of the wire by a short co-axial solenoid, then if alternating currents are sent through this solenoid periodic magnetic forces parallel to the wire will be started. We shall in this article investigate the laws which govern the transmission of such forces along the wire. The problem has important applications to the construction of transformers ; in some of these the primary coil is wound round one part of a closed magnetic circuit, the secondary round another. This arrangement will not be efficient if there is any considerable leakage of the lines of magnetic force between the primary and the secondary. We should infer from general con- siderations that the magnetic leakage would increase with the rate of alternation of the current through the primary. For let us suppose that an alternating current passes through an insulated ring imbedded in a cylinder of soft iron surrounded by air, the straight axis of the ring coinciding with the axis of the cylinder. The variations in the intensity of the current through this ring will induce other currents in the iron in its neighbourhood; the magnetic action of these currents will, on the whole, cause the com- ponent of the magnetic force along the axis of the cylinder to be less and the radial component greater than if the current through the ring were steady; in which case there are no currents in the iron. Thus the effect of the changes in the intensity of the current through the primary will be to squeeze as it were the lines of278.] ELECTRICAL WAVES AND OSCILLATIONS. 303 magnetic force out of the iron and make them complete their circuit through the air. Thus when the field is changing quickly, the lines of magnetic force, instead of taking a long path through the medium of high permeability, will take a short path, even though the greater part of it is through a medium of low permeability such as air. The case is quite analogous to the difference between the path of a steady current and that of a rapidly alternating one. A steady current flows along the path of least resistance, a rapidly alternating one along the path with least self-induction. Thus, for example, if we have two wires in parallel, one very long but made of such highly conducting material that the total resistance is small, the other wire short but of such a nature that the resistance is large, then when the current is steady by far the greater part of it will travel along the long wire; if however the current is a rapidly alternating one, the greater part of it will travel along the short wire because the self- induction is smaller than for the long wire, and for these rapidly alternating currents the resistance is a secondary con- sideration. In the magnetic problem the iron corresponds to the good conductor, the air to the bad one. When the field is steady the lines of force prefer to take a long path through the iron rather than a short one through the air ; they will thus tend to keep within the iron; when however the magnetic field is a very rapidly alternating one, the paths of the lines of force will tend to be as short as possible, whatever the material through which they pass. The lines of force will thus in this case leave the iron and complete their circuit through the air. We shall consider the case of a right circular soft iron cylinder where the lines of magnetic force are in planes through the axis taken as that of s, the corresponding system of currents flowing round circles whose axis is that of the cylinder. The cylinder is surrounded by a dielectric which extends to infinity. Let a, b, c be the components of the magnetic induction parallel to the axes of x, y, z respectively; then, since the component of the magnetic induction in the xy plane is at right angles to the axis of the cylinder, we may put a — _dx. dx , dx ~ dy' Let us suppose that a, 6, c all vary as €* ^mz +1>t\304 ELECTRICAL WAVES AND OSCILLATIONS. [278. Now in the iron cylinder a, 6, c all satisfy differential equations of the form dx2 * dy2 n2 = m2 + 4 T:[up/(r, lx being the magnetic permeability and so that an approximate solution of the equation is x = -y/logy.306 ELECTRICAL WAVES AND OSCILLATIONS. [279. If we apply this result to equation (51), we find that the approximate solution of that equation is Now ** = - 4 1 aVlogOxy*)’ and since the value we have just found for k is in any practicable case very large compared with p2/V2^ we see that k2 = m2 ap- proximately. so that - ilP 1 m_ a l/x'ogOV*)) Thus since in the expression for c there is the factor €tvnz or € a2 1 AilogOT2)! > we see that the magnetic force will die away to 1/e of its value at a distance . f , , 9X, i ia{filog(w2)}^ from its origin, 279]. In the last case the current was uniformly dis- tributed over the cross-section. We can investigate the effect of the concentration of the current at the boundary of the cylinder by supposing that na, is large compared with unity though small compared with /x. In this case, since approximately Jq {ins) = — iJ0 (ina,), equation (49) becomes ------= ika iv/-V-t-V H K0 (ika») Since the left-hand side of this equation is small, tka, is also small, so that by Art. 261 we may write this equation as — — = &2a2log(2y/6&a). (52) This equation gives a value for k2 which is very large compared with p2/F2, so that approximately m = k. We also see that k or m is small compared with n} we may therefore put 7b = {4:TTfXLp/(T}^.279-] ELECTRICAL WAVES AND OSCILLATIONS. 307 Thuff equation (52) becomes or putting i&a/2y = q, To solve this equation put q2=wz^; equating real and ima- ginary parts, we get lv\ogWC08yjf — W\lrBm\fr = — ~2 [ ’ l . , , . 1 { 77 P 8? wlogwsm^ + w^cosy = — | 2]x ") * Since w is very small, the terms in logw are much the most important; an approximate solution of these equations is, there- fore, since the solution of x \ogx = — y, is x = — yfiogy, 1 (irpa2 y2 ( ua ) w — — / 77 * = 4' Hence, since k = m and Pa2 = —4y2we4, we find /---( IT .77/ ma = 2y v — wjcos- -f « 1/ 5tt . 5tk = 2y W*(cos — + i Sin — J ■ Thus, since in the expression for c there is the factor €imz, we see that c will fade away to 1/e of its initial value at a distance from the origin equal to a 57r ----, cosec —, 2 yW* ° or substituting the value of w just found, 5 7T ^ fX(T - cosec — s 2 8 (7Tp This distance is much shorter than the corresponding one308 ELECTRICAL WAVES AND OSCILLATIONS. [280. when the current was uniformly distributed over the cross- section of the wire, and the important factor varies as n* instead of /A Thus the leakage of the lines of magnetic force out of the iron cylinder is much greater when the alternations are rapid than when they are slow. This is in accordance with the conclusion we came to from general reasoning at the beginning of Art. 278. The result of this investigation points strongly to the ad- visability of very fine lamination of the core of a transformer, so as to get a uniform distribution of magnetic force over the iron and thus avoid magnetic leakage. There are many other advantages gained by fine lamination, of which one, more important than the effect we are considering, is the diminution in the quantity of heat dissipated by eddy currents. We shall proceed to consider in the next article the dissipation of energy by the currents in the wire. Dissipation of Energy by the Heat produced by Alternating Currents. 280.] A great deal of light is thrown on the laws which govern the decay of currents in conductors by the consideration of the circumstances which affect the amount of heat produced in unit time by these currents. As we have obtained the expressions for these currents we could determine their heating effect by direct integration; we shall however proceed by a different method for the sake of introducing a very important theorem due to Professor Poynting, and given by him in his paper ‘ On the Transfer of Energy in the Electromagnetic Field/ Phil. Trans. 1884, Part II, p. 343. The theorem is that Tifff^TS +«§ + Rd-§)**d,y K dQ 4 77 dt ~~ K dR — — Hence +Qw + Rifi)dxdydz =Jff {P(u— p) + Q(v—r))dxdydz =Jj{P u + Qv + Rvj) dxdydz ~fff + Q? + dxdydz (53)310 ELECTBICAL WAVES AND OSCILLATIONS. [280. Now (Maxwell’s Electricity and Magnetism, Yol. II, Art. 598), D dF d\lr , p = cV-k--3l--a-=cjl-u+r', R = li-ai-™-i£ = bi-ay+K, where P', Q\ R' are the parts of P, Q, R which do not contain the velocities. Thus Pu + Qv + Rw = {cy -bz)u + (az -cx)v + (bx - ay) w + P'u + Q'v + R'w, = - {(vc - w6) a + (wa-uc)y + (w& -va)z\ + P'u + Q'v + R'w, = -{Zar+Fy + ^}+P'u + Q/v + iJ'w; where X, \ ^ Z are the components of the mechanical force per unit volume (Maxwell, Yol. II, Art. 603). Substituting this value for Pu + Qv + Rw in (53) and trans- posing, we obtain +QW +R^)dxdydz + yyy*(Xa;+ Yy + Zz) dxdydz+j J J(Pp + Qq + Rr)dxdydz =JfJ+ ®'v * R’w) dxdydz. (54) Now A dy dB dy dz . da dy h=—R, 4 7T 4 7T f^JJJ (P2 + Q2 + R2)dxdydz. The electromagnetic energy inside the same surface is (Maxwell. Art. 635) JJJ(aa + bfi + cy) dxdydz, or ^ Iff ^ + /32+dxdydz. Thus the first two integrals on the left-hand side of equation (55) express the gain per second in electric and magnetic energy. The third integral expresses the work done per second by the mechanical forces. The fourth integral expresses the energy transformed per second in the conductor into heat, chemical energy, and so on. Thus the left-hand side expresses the total gain in energy per second within the closed surface, and equation (55) expresses that this gain in energy may be regarded as coming across the bounding surface, the amount crossing that surface per second being expressed by the right-hand side of that equation.313 28o.] electrical waves and oscillations. Thus we may regard the change in the energy inside the closed surface as due to the transference of energy across that surface; the energy moving at right angles both to H, the resultant magnetic force, and to E, the resultant of P\ Q\ R\ The amount of energy which in unit time crosses unit area at right angles to the direction of the energy flow is HE sin 0/4 7r, where 0 is the angle between H and E. The direction of the energy flow is related to those of H and E in such a way that the rotation of a positive screw from E to H would be accompanied by a translation in the direction of the flow of energy. Equation (55) justifies us in asserting that we shall arrive at correct results as to the changes in the distribution of energy in the field if we regard the energy as flowing in accordance with the laws just enunciated: it does not however justify us in asserting that the flow of energy at any point must be that given by these laws, for we can find an indefinite number of quantities u8i v8, w8 of the dimensions of flow of energy which satisfy the condition lu8 + mv8 + nw8)dS = 0, where the integration is extended over any closed surface. Hence, we see that if the components of the flow of energy were R'P—Q'y + 2 w8 instead of R'p — Q'y, P'y — R'a + 2v8 instead of P'y—R'a, Qfa—P'p + ^w8 instead of Q'a—P'/3, the changes in the distribution of energy would still be those which actually take place. Though Professor Poynting’s investigation does not give a unique solution of the problem of finding the flow of energy at any point in the electromagnetic field, it is yet of great value, as the solution which it does give is simple and one that readily enables us to form a consistent and vivid representation of the changes in the distribution of energy which are going on in any actual case that we may have under consideration. Several appli- cations of this theorem are given by Professor Poynting in the paper already quoted, to which we refer the reader. We shall now proceed to apply it to the determination of the rate of heat production in wires at rest traversed by alternating currents.314 electrical waves and oscillations. [282. 281.] Since the currents are periodic, P2, Q2, R2, a2, /32, y2 will be of the form a . ^ A + B cos (2pt + 0), where A and B do not involve the time; hence the first two integrals on the left-hand side of equation (55) will be multiplied by factors which, as far as they involve t, will be of the form sin (2pt 4- 6); hence, if we consider the mean value of these terms over a time involving a great many oscillations of the currents, they may be neglected: the gain or loss of energy represented by these terms is periodic, and at the end of a period the energy is the same as at the beginning. The third term on the left- hand side vanishes in our case because the wires are at rest, and since y, z vanish P/, Q', R' become identical with P, Q, R. Thus when the effects are periodic we see that equation (55) leads to the result that the mean value with respect to the time of JJJ (Pp + Qq + Rr)dxdydz is equal to that of -LfflURp-Qy) + m(Py-Ra) + n(Qa-Pp)\dS. The first of these expressions is, however, the mean rate of heat production, and in the case of a wire whose electrical state is symmetrical with respect to its axis, the value of the quantity under the sign of integration is the same at each point of the circumference of a circle whose plane is at right angles to the axis of the wire ; hence in this case we have the result: The mean rate of heat production per unit length of the wire is equal to the mean value of £a (tangential electromotive intensity) x (tangential magnetic force), (56) a, as before, being the radius of the wire. 282.] Let us apply this result to find the rate of heat pro- duction in the wire and in the outer conductor of a cable when the current is parallel to the axis of the wire. By the methods of Art. 268, we see that if the total current through the wire at the point £ is equal to the real part of r i (mz+pt) >315 282.] ELECTRICAL WAVES AND OSCILLATIONS. or if m = — a +i/3, to JqC COS (10 ■f^)j then, Art. 268, equation (24), the electromotive intensity R in the wire parallel to the axis of 0 is equal to the real part of J0 (*9ir) J i( — az+jpt) 27ta °€ € (57) If we neglect the polarization currents in the dielectric in comparison with the conduction currents through the wire, then the line integral of the magnetic force round the inner surface of the outer conductor must equal 4 7r/0€t using this principle we see that E in equation (11), Art. 262, equals — Ln', which is very large since we have assumed that n2a2, i.e. 47r/rpa2/(r, is a small quantity; in this case, therefore, by far the larger proportion of the heat is produced in the wire. This explains the result found in Art. 263 that the rate of decay of the vibrations is nearly independent of the resistance of the outer conductor and depends almost wholly upon that of the wire. 283.] When the frequency is so great that no, is large though n'b is still small, then JQ(ino) = iJJ(ma), so that by (57) the tangential electromotive intensity at the surface of the wire is equal to the real part of317 284.] ELECTRICAL WAVES AND OSCILLATIONS. which is equal to ■£— \2i:npAr}l I0t~ez {cos ( — az + pt) — sin( — az+pt)}. 2 7ra Hence by (56) and (59) the mean rate of heat production in the wire is equal to 4^(2 Since n'h is supposed to be small the rate of heat production in the outer conductor is as before 3jr T Splo 2-2 /3z hence the ratio of the amount of heat produced in unit time in the wire to that produced in the outer conductor is fx ( 2 (r H [x (97r3£>/xa2 ) * Thus, since n2a2 and so ^irpfxayais very large by hypothesis, we see that unless \xfrx' is very large this ratio will be very small; in other words the greater part of the heat is produced in the outer conductor; this is in accordance with the result obtained in Art. 266, which showed that the rate of decay of the vibrations was independent of the resistance of the wire. 284.] When the frequency is so high that both 71a and n'b are large, then the expression for the heat produced in the wire is that just found. To find the heat produced in the outer con- ductor we have, when n'h is very large, K0(m'b) =-iK0' (in'b); hence by (58) the tangential electromotive intensity in the outer conductor is equal to the real part of which is equal to — p/v)^Iq*~Px {cos( — az + pt) — sin( — az+pt)}. 27T D Hence by (56) and (60) the mean rate of heat production in the outer conductor is , ^(2318 ELECTRICAL WAVES AND OSCILLATIONS. [285. Thus the ratio of the heat produced in unit time in the wire to that produced in the same time in the outer conductor is so that if, as is generally the case in cables, 0 = real part of pt) = \fxo,pH* ^*sin(az+pt)— -L7rjot2^2a3LTe ^cos(az+pt). 4 or But by equation (56) the rate of heat production in the wire per unit length is equal to the mean value — cos (az + pt); where the minus sign has been taken because (Art. 280) 0 H is proportional to the rate of flow of energy in the direction of translation of a right-handed screw twisting from 0 to H; in this case this direction is radially outwards. Thus the rate of heat production in the wire is -i-TT^Va4#2*-2^, 162/F2, n2 = 4TTfXLp/o■, thus they represent the quan- tities represented by the same symbols in previous investigations, if in these we put m = 0. Let I denote the tangential current at right angle to r and the axis of the cylinder, then at dy dr if © is the tangential electromotive intensity in the same direction, then in the dielectric K d® so that since 1 /K = F2. In the tube 47r dt K =-.y0, Ilii lp dr3 © = - © = <71 __ cr dy 477 dr Since y is continuous, we have AJ0(ik&) = BJ0(ma,) + CK0 (ma), and since © is continuous, we have ™ AJ0'(tka) = ^ {BJ0'(ma,) + CK0'(Ln*)\. Since ik = p/V, ika will be very small, hence we may put Jq (t k a) — 1, tTq (1 k a) — — \ 1 ka. Making these substitutions and remembering that J0(ina)K0'(m8,) — J0f (tna) K0(ma,) =--------, 171a B = -A {K0' (i»a)+ ^Z0(.wa)} ma, 2^. we find325 287.] ELECTRICAL WAVES AND OSCILLATIONS. To determine A we have the condition that when r = b, y = H(,pt, hence H (twa) Kq (mb) — J0 (mb) A ft The effect we are considering is one which is observed when the rate of alternation of the current is very high, so that both na and nb are very large; but when this is the case J0 (iTia) = K0 (ma) = Jo (mb) = V27rna, J0' (ma) = 1 /jL> V 2n& V2irna, = A/jL 2na 72 b \/ 2 7m b /0'(t%b)=- 72b V2irnb K0(mb) = <-n\/-J!-, V 2 Tib Z0'(t7ib) = t€-Bb / —ZI—; 'V 2wb making these substitutions and writing h for b—a, we find — 0 = real part of nh —nh , / wA —72^\ an 2yotv 7 • +• +^<* - > ITY^. (62) Now since wa is very large, tib,/ix is also very large for the non-magnetic metals, and even for the magnetic metals if the frequency of the currents in the primary is exceedingly large;326 ELECTRICAL WAVES AND OSCILLATIONS. [287. but when this is the case, then, unless h is so small that n2aA//x is no longer large, we may write equation (62) as — © = real part of ^ nh , —nh + e 4 ir nh —nh € —€ (63) Since n = {4ir/Aip/a}2} we may write 7i — 7ix (1 + 1), where nx = and equation (63) becomes e_ ff%^M_r^+2sin 2n1hs 4tt €2n‘A + €-2“»*_2cos2'n,1A _ 2n,h —2nlh 0 • r* 7 (tti1 e 1 —e 1 — 2sm2'7i1A 477 {2»i* + e-2»i*_2cos2nxh Hsin^. (64) In calculating the part of the energy flowing into the tube which is converted into heat, we need only consider the part which flows across the outer surface of the tube, because the energy flowing across the inner surface is equal to that which flows into the dielectric inside the tube, and since there is no dissipation of energy in this region the average of the flow of energy across the inner surface of the tube must vanish. Hence the amount of heat produced in unit time in the tube is by equa- tion (36) equal to the mean value of — ib©Ifcos£tf, where the — sign has been taken because the translatory motion of a right-handed screw twisting from 0 to H is radially outwards; this by (64) is equal to onxb (e2”^-e~2"‘ft+2sin2nxh) _ 16u e —2»1A_2 cos 2nxh when 71J1 is very large this is equal to o-^b 1677 which is (Art. 286), as it ought to be, the same as for a solid cylinder of radius b. When h is small and n2a,h/fx not large we must take into account terms which we have neglected in arriving at the preceding expression.327 287.] ELECTBICAL WAVES AND OSCILLATIONS. In this case, we find from (62) that _ (Vp2a2h/t9 or mcotmZ =— ip(ip¥ + tyC. (66) 290.] Let us first consider the case when the wave length of electrical vibrations is very much longer than the wire; here ml is very small, so that equation (66) becomes j = — rp(i£>P + Q) G. (67) The values of P and Q, the self-induction and impedance of the wire, are given in equation (40) of Art. 272; they depend upon the frequency of the electrical vibrations. When this is so330 ELECTRICAL WAVES AND OSCILLATIONS. [29O. slow that 71 a is a small quantity, a being the radius of the wire, then approximately P = 2T R Q=27’ where L is the coefficient of self-induction and R the resistance of the whole wire for steady currents. Substituting these values in (67), we get (tp)2L + ipR + ^ = 0, R /~2 W or ip~~2L±l\! CL~ Tl?' ^ Since the various quantities which fix the state of the electric field contain as a factor, we see that when SL>CR2 these quantities will be proportional to R ’2 L f / 2 R2 JL ) 1*+t € COS - where a is a constant. This represents an oscillation whose period is R2 4L2\ ’ and whose amplitude dies away to 1/e of its original value after the time 2 L/R. Thus, if 2/CL is greater than R2/4 i2, that is if R2 is less than 8 L/C, the charges on the spheres will undergo oscillations like those performed by a pendulum in a resisting medium. Suppose, for example, that the electrical connection between the spheres is broken, and let one sphere A be charged with positive, the other sphere B with an equal quantity of negative electricity; if now the electrical connection between the spheres is restored, the positive charge on A and the negative on B will diminish until after a time both spheres are free from electrifica- tion. They will not however remain in this state, for negative electricity will begin to appear on A, positive on B, and these charges will increase in amount until (neglecting the resistance of the circuit connecting the spheres) the charges on A and B appear to be interchanged, there being now on A the same quantity ofELECTRICAL WAVES AND OSCILLATIONS. 331 291.] negative electricity as there was initially on B, while the charge on B is the same as that originally on A. When the negative charge on A has reached this value it begins to decrease, and after a time both spheres are again free from electrification. After this positive electricity begins to reappear on A, and increases until the charge on A is the same as it was to begin with; this positive charge then decreases, vanishes, and is replaced by a negative one as before. The system thus behaves as if the charges vibrated backwards and forwards between the spheres. The changes which take place in the electrical charges on the spheres are of course accompanied by currents in the wire, these currents flowing sometimes in one direction, sometimes in the opposite. When the circuit has a finite resistance the amplitude of the oscillations gradually diminishes, while if the resistance is greater than (8 L/C)* there will not be any vibrations at all, but the charges will subside to zero without ever changing sign ; in this case the current in the connecting wire is always in one direction. 291.] If we assume that the wave length of the electrical vibrations is so great that the current may be regarded as uniform all along the wire, and that the vibrations are so slow that the current is uniformly distributed across the wire, the discharge of a condenser can easily be investigated by the following method, which is due to Lord Kelvin {Phil. Mag. [4], 5, p. 393, 1853). Let Q be the quantity of electricity on one of the plates of a condenser whose capacity is C' and whose plates, like those of a Leyden Jar, are supposed to be close together; also let R be the resistance and L the coefficient of self-induction for steady currents of the wire connecting the plates. The electromotive force tending to increase Q is —Q/C'; of this RdQ/dt is required to overcome the resistance and LdlQ/dt2 to overcome the inertia of the circuit; hence we have + R dQ dt (69) The solution of this equation is, if 1 R2 C'L > 4 L2’ Q-Ae 2Z> cosjQ^ 4£a) < + where A and /3 are arbitrary constants.332 ELECTRICAL WAVES AND OSCILLATIONS. [292. In this case we have an oscillatory discharge whose frequency is equal to When / 1 m a 'C'Z 4 L1’ ’ 1 E2 CrL< 4i2 ’ (69) is -22 1 AU C'l ) +B where A and B are arbitrary constants. In this case the dis- charge is not oscillatory. To compare the results of this investigation with those of the previous one, we must remember that the capacities which occur in the two investigations are measured in somewhat different ways. The capacity G in the first investigation is the ratio of the charge on the condenser to <£ its potential; in the second investigation G' is the ratio of the charge to 2 0, the difference between the potentials of the plates, so that to compare the results we must put C' — (7/2; if we do this the results given by the two investigations are identical. 292.] The existence of electrical vibrations seems to have been first suspected by Dr. Joseph Henry in 1842 from some ex- periments he made on the magnetization of needles placed in a coil in circuit with a wire which connected the inside to the outside coating of a Leyden Jar. He says (Scientific Writings of Joseph Henry, Vol. I, p. 201, Washington, 1886): ‘This anomaly which has remained so long unexplained, and which at first sight appears at variance with all our theoretical ideas of the connection of electricity and magnetism, was after consider- able study satisfactorily referred by the author to an action of the discharge of the Leyden jar which had never before been recog- nised. The discharge, whatever may be its nature, is not correctly represented (employing for simplicity the theory of Franklin) by the simple transfer of an imponderable fluid from one side of the jar to the other, the phenomenon requires us to admit the existence of a principal discharge in one direction, and then several reflex actions backward and forward, each more feeble than the preceding, until the equilibrium is obtained. All the facts are shown to be in accordance with this hypothesis, and a ready explanation is afforded by it of a number of phenomena293-] ELECTRICAL WAVES AND OSCILLATIONS. 333 which are to be found in the older works on electricity but which have until this time remained unexplained.’ In 1853, Lord Kelvin published {Phil. Mag. [4], 5, p. 393, 1853) the results we have just given in Art. 291, thus proving by the laws of electrical action that electrical vibrations must be produced when a Leyden Jar is short circuited by a wire of not too great resistance. From 1857 to 1862, Feddersen (Pogg. Ann. 103, p. 69, 1858; 108, p. 497, 1859; 112, p. 452, 1861; 113, p. 437, 1861 ; 116, p. 132, 1862) published accounts of some beautiful experiments by which he demonstrated the oscillatory character of the jar discharge. His method consisted in putting an air break in the wire circuit joining the two coatings of the jar. When the current through this wire is near its maximum intensity a spark passes across the circuit, but when the current is near its minimum value the electromotive force is not sufficient to spark across the air break, which at these periods therefore is not luminous. Thus the image of the air space formed by reflection from a rotating mirror will be drawn out into a series of bright and dark spaces, the interval between two dark spaces depending of course on the speed of the mirror and the frequency of the electrical vibrations. Feddersen observed this appearance of the image of the air space, and he proved that the oscillatory character of the discharge was destroyed by putting a large re- sistance in circuit with the air space, by showing that in this case the image of the air space was a broad band of light gradually fading away in intensity instead of a series of bright and dark spaces. This experiment, which is a very beautiful one, can be repeated without difficulty. To excite the vibrations the coatings of the jar should be connected to the terminals of an induction coil or an electric machine. It is advisable to use a large jar with its coatings connected by as long a wire as possible. By connecting the coatings of the jar by a circuit with very large self-induction, Dr. Oliver Lodge {Modern Views of Electricity, p. 377) has produced such slow electrical vibrations that the sounds generated by the successive discharges form a musical note. 293.] In the course of the investigation in Art. 290 we have made two assumptions, (l) that ml is small, (2) that n& is also small, which implies that the currents are uniformly distributed334 ELECTRICAL WAVES AND OSCILLATIONS. [293. across the section of the discharging circuit. This condition is however very rarely fulfilled, as the electrical oscillations which are produced by the discharge of a condenser are in general so rapid that the currents in the discharging circuit fly to the outside of the wire instead of distributing themselves uniformly across it; when the currents do this, however, the resistance of the circuit depends on the frequency of the electrical vibrations, and the, investigation of Art. 290 has to be modified. Before proceeding to the discussion of this case we shall write down the conditions which must hold when the preceding investigation is applicable. In the first place, ml is to be small; now by Art. 263 we have when na is small, m2 = — ip (resistance of unit length of the wire) x (capacity of unit length of wire), hence m2 l2 = — J ip R l T, where, as before, R is the resistance of the whole of the discharging circuit, while T is the capacity of unit length of the wire. But by equation (68) when the discharge is oscillatory, we have R (2 R2 H tp~ 2L±1\lc 4m’ thus the modulus of ip is equal to Irof. hence, when ml is small, RTl VCL must be small. The other condition is that n a is small, which since n2 = m2 4 TTfJUp a and ml is also small, is equivalent to the condition that iTTfjup a2/o- should be small. Since the modulus of ip is equal to {2/XC}i we see that if n2a2 is small, 4-Ma2{2/XC}*A294-] ELECTEICAL WAVES AND OSCILLATIONS. 335 must be small. The capacity C which occurs in this expression is measured in electromagnetic units, its value in such measure is only 1/72 (where ‘ 75 is the ratio of the units and 72 = 9 x 1020) of its value in electrostatic measure. Thus the expression which has to be small to ensure the condition we are considering, contains the large factor 3 x 1010, so that to fulfil this condition the capacity and self-induction of the circuit must be very large when the discharging circuit consists of metal wire of customary dimensions. Thus, to take an example, suppose two spheres each one metre in radius are connected by a copper wire 1 millimetre in diameter. In this case G = 1/9 x 1018, (t = 1600, a = -05, jx = 1, substituting these values we find that to ensure na being small, the self-induction of the circuit must be comparable with the enormously large value 1011, which is comparable with the self- induction of a coil with 10,000 turns of wire, the coil being about half a metre in diameter. The result of this example is sufficient to show that it is only when the self-induction of the circuit or the capacity of the condenser is exceptionally large that a theory based on the assumption that na is a small quantity is applicable, it is there- fore important to consider the case where na is large and the currents in the discharging circuit are on the surface of the wire. 294.] The theory of this case is given in Art. 274, and we see from equations (42) and (43) of that Article that when the frequency of the vibrations is so great that na and n'h (using the notation of Art. 274, and supposing that the wire connecting the spheres is a cable whose external radius is b) are large quantities, equation (66) of Art. 289 becomes mcotml = —2tp {i^logb/a + (i£>)*(jw<7/47ra2)* + {Lp)^ (/xV/4 tt b2)^} C. Retaining the condition that ml is small, which will be the case when the wave length of the electrical vibrations is very much greater than the length of the discharging circuit, this equation becomes = -lP [*F 2log(b/a) + (t2>)i 2 {(M0/4™2)* + (/A'lnb2)1}],336 ELECTRICAL WAVES AND OSCILLATIONS. [294. which we shall write as £ = — lP {tpL'+2 (ippS}, (70) where L' is the coefficient of self-induction of the discharging circuit for infinitely rapid alternating currents, and S is written ^0r { (cr/ut/4 77a2)^ + ( (74) (%2 + Vi) («12 + y£) = I—3H2, or Xj* + x* (y* + y2) + i \{y* + y2f - (yi* -y.2)2 = I-3H2. Eliminating yx2 + y^ and 2/12—j/22 by equations (73) and (74), we get 4a14+12tfa12 + (l2tf2-I)- 0, xi or putting x2 = rj9 4 rj3+12 jffr;2 +(12jET2—-/)r; — G2 = 0. (75) Since the last term of this expression is negative there is at least one positive real root of this equation, and since the values given for H and I show that when A is positive 12 if2 — ids essen- tially negative, we see by Fourier’s rule that there is only one such root. But since xL is real the value of rj will be positive, so that the root we are seeking will be the unique positive real root of equation (75), which can easily be determined by Homer’s method. The value of xx is equal to minus the square root of this root, and knowing xx we can find y2/4 tt2, the square of the corresponding frequency uniquely from equations (73) and (74). We can in this way in any special case determine with ease the logarithmic decrement and the frequency of the vibrations. 296.] If in equation (75) we substitute the values of G9 IZ,and J, and write ^ = & cgl/£, _ ^ that equation becomes C3 + 2C2(1-f2)-C(3q2-42)-3(l-q)2 = 0. We can by successive approximations expand fin terms of q, and thus when (7$4/Z/3 is small approximate to the value of (. The first term in this expansion is or since c=(?M L'Cx* = £ xx = - 8 2 iCtL'l'339 296.] ELECTRICAL WAVES AND OSCILLATIONS. The corresponding value of y* determined by equations (73) and (74) is, retaining only the lowest power of q, approximately, 2_ 2 f, 2iS&l 2/1 ~ L'Cl1 L'\ )' Now 8 = {jtx /ds, and as this is proportional to the current it must vanish at the ends A, F of the wire if the capacity there is, as we suppose, very small. Hence along AB we may write, if p/2 it is the frequency, (j> = acosmslcos^, along BCE ^ = cos msa cos+ 5 sinms2) cos pt, along BDE ^ = cos m8^ cos + c sin ms3) cos pt, and along EF = d cos m (s4—ZJ cos pt Equating the expressions for the potential at Ey we have a cos mZ2 cos mZx + b sin mZ2 = d cos mZ4,) a cos m Z3 cos mlx -f c sin mZ3 = d cos mZ4.3 The current flowing along AB at B must equal the sum of the currents flowing along BCE, BDE, hence by (76) we have asinmL be , x —■>...=-p~p- (78) ■*1 -*3 Again, the current along at E must equal the sum of the currents flowing along BCE9 BDE, hence we have (77) d sin mZ4 6 cos mZ2 3 asinm?2cos?nZl c cos mZ3 ^ + a sin mZ3 cos mZt (79) *{ We get from equations (77) and (78) sin m Zx cot m Z2 cos m l x cot m Z3 cos m Z4 ] = — cZ cos mL cosec mL 2 ^ cosec mZ3 4 cZ From equations (77) and (79) we get sin m Z4 cot m Z2 cos 971Z4 cot mZ3 cos mZ4 3 = —a cos mZj | f cosec mZ2 cosec mZ3) 3 3 r298.] ELECTRICAL WAVES AND OSCILLATIONS. 343 Eliminating a and d from these equations, we get f tan mly cot ml.2 cot rn /3) (tan m lt cot mZ2 cotw l. I-4 • 4 4 }{ _ | cosec ml., ^ cosec mZgj2 4 4 (80) If AB and EF are equal lengths of the same kind of wire, lx = Z4, and = P±y and (80) reduces to the simple form tan m cot m 1.2 cot m Z, ^3 _ , ( 3 “ l cosec m L cosec m l. 4 4 4 taking the upper sign, we have tan mZx __ cot \ml2 ~4~~~ 4 if we take the lower sign, we have 4 4 }• cot £ mZo +---^— > tanmZx __ f tan \ml2 tan JmZ2 1 4 (81) (82) 4 «. Since m = 2 7t/A, where A is the wave length, these equations determine the wave lengths of the electrical vibrations. If all the wires have the same radius, = 12 = and equa- tions (81) and (82) become respectively and tan 2 7T = cot +cot tan 2 tt y + tan tt ^ + tan 7r ^ = 0. AAA. (81*) (82*) From these equations we can determine the effect on the period of an alteration in the length of one of the wires. Suppose that the length of BBE is increased by SZ3, and let hk be the corresponding increase in A, then from (81*) bX A \ lxsec2+ \l2cosec2 + \lzcosec2\ = Z3cosec2^-3 La A A 3 A We see from this equation that 8A and 8Z3 are of the same sign, so that an increase in Z3 increases the wave length. If we take equation (82*), we have bk\ A hence, in this case also, an increase in Z3 increases A. If Z3 is -sec2^-^-f iZ2sec2^ +^Z3sec2^| = i8Z3sec2^, (A A A J A344 ELECTRICAL WAVES AND OSCILLATIONS. [3OO. infinite the wave length is 4Z1 + 2Z2 and its submultiples, as we diminish Z3 the wave length shortens, hence we see that the effect of introducing an alternative path is to shorten the wave lengths of all the vibrations. The shortening of the wave length goes on until Z3 vanishes, when the wave length of the gravest vibration is 4 Zx. 299.] The currents through the wires BCE and BDE are at B in the proportion of cot jm/ cot \ml§ if we take the vibrations corresponding to equation (81), and in the proportion of tan tan p~ *° “T-' for the vibration given by (82). We can prove by the method of Art. 298 that if we have n wires between B and F\ and if AB = EFt tan mZx cot mZ2 cot mZ3 ~Pi iT~'" f cosec ml2 cosec ml3 cosec mZ4 ) =±\—r~+~s- + ~ir~+-y It follows from this equation that if any of the wires are shortened the wave lengths of the vibrations are also shortened. Electrical Oscillations on Cylinders. Periods of Vibration of Electricity on the Cylindrical Cavity inside a Conductor. 300.] If on the surface of a cylindrical cavity inside a conductor an irregular distribution of electricity is produced, then on the removal of the cause producing this irregularity, currents of electricity will flow from one part of the cylinder to another to restore the electrical equilibrium, electrical vibra- tions will thus be started whose periods we now proceed to investigate. Take the axis of the cylinder as the axis of 0, and suppose that initially the distribution of electricity is the same on all sections at right angles to the axis of the cylinder; it will evidently remain so, and the currents which restore the electrical345 300.] ELECTRICAL WAVES AND OSCILLATIONS. distribution to equilibrium will be at right angles to the axis of z. If c is the magnetic induction parallel to 0, then in the cavity filled with the dielectric c satisfies the differential equation d2c d2c __ 1 d2c dx2 + dtf ~ V2 dt2 ’ where V is the velocity of propagation of electrodynamic action through the dielectric. In the conductor c satisfies the equation d2c d2c^4iTiidc cS2 + dy2 a dt9 where o* is the specific resistance and jx the magnetic permea- bility of the substance. Transform these equations to polar coordinates r and 0, and suppose that c varies as cossflc4^; making these assumptions, the differential equation satisfied by c in the dielectric is d2c 1 dc /p2 s2\ __ dr2 * r dr 'V2 r2' ~~ the solution of which is c = A cos$0J, €piy where J9 denotes the internal Bessel’s function of the sth order. The differential equation satisfied by c in the conductor is d2c 1 dc C 47Ttxip s2) dr2 + a Let n2 = ^TTfiLp/a, then the solution of this equation is c = Beo&sdK8(mr) where K8 denotes the external Bessel’s function of the sth order. Since the magnetic force parallel to the surface of the cylinder is continuous, we have if a denotes the radius of the cylindrical cavity ~ t> AJ‘( fa) = 7z‘(t7ia> (83) The electromotive intensity at right angles to r is also con- tinuous. Now the current at right angles to r and z is —dc/4 7r/xcZr,346 ELECTRICAL WAVES AND OSCILLATIONS. [300. hence in the conductor the electromotive intensity perpendicular to r and 0 is — (rdc/liriidr. In the dielectric the current is equal to the rate of increase of the electric displacement, i.e. to ip times the electric displacement or to ipK/An times the electro- motive intensity; we see that in the dielectric the electromotive Now the wave length of the electrical vibrations will be corn- corresponding to this will be sufficient to make ns, exceedingly large, but when n a is very large we have (Heine, Kugelfunc- tionen, vol. i. p. 248) hence K\ (ma) = lK8 (ma); thus the right-hand side of (86) will be exceedingly small, and an approximate solution of this equation will be This signifies that the tangential electromotive intensity vanishes at the surface of the cylinder, or that the tubes of electrostatic induction cut its surface at right angles. The roots of the equation (84) Eliminating A and B from (83) and (84), we get (85) 4 7T|Lt ip , so that (85) may be written K's(mb,) ina Ks(ina) (86) parable with the diameter of the cylinder, and the value of p ----approximately, 2wa rr J'8(x) = 0,347 301.] ELECTRICAL WAVES AND OSCILLATIONS. for s = 1, 2, 3, are given in the following table taken from Lord .Rayleigh's Theory of Sound, Yol. II, p. 266:— s = 1 s = 2 8 = 3 1841 3-054 4-201 5332 6-705 8-015 8-536 11-706 14-864 9-965 11-344 18-016 Thus, when 8=1, the gravest period of the electrical vibra- tions is given by the equation f=a = 1.841, or the wave length of the vibration 2irV/p =-543 x 27ra, and is thus more than half the circumference of the cylinder. In this case, as far as our approximations go, there is no decay of the vibrations, though if we took into account the right-hand side of (86) we should find there was a small imaginary term in the expression for p, which would indicate a gradual fading away of the vibrations. If it were not for the resistance of the conductor the oscillations would last for ever, as there is no radiation of energy away from the cylinder. The magnetic force vanishes in the conductor except just in the neighbourhood of the cavity, and the magnetic waves emitted by one portion of the walls of the cavity will be reflected from another portion, so that no energy escapes. Metal Cylinder surrounded by a Dielectric. 301.] In this case the waves starting from one portion of the cylinder travel away through the dielectric and carry energy with them, so that the vibrations will die away independently of the resistance of the conductor. Using the same notation as before, we have in the conducting cylinder . c = A cos sQ J8 (inr) € **, and in the surrounding dielectric P J\ jpt348 ELECTRICAL WAVES AND OSCILLATIONS. [3OI. Since the magnetic force parallel to z is continuous, we have A Since the electromotive intensity perpendicular to r is continuous, we have A A Eliminating A and B from these equations, we get or _1_ J'.(ms) = V_ ma, J,(ma) ~ ppa R ^ ' (87) Now, as before, na will be large, and therefore . n a t* € J$ (tTia) = — approximately, V'27r/Ma hence J\(mo,) = — and the left-hand side of equation (87) is very small, so that the approximate form of (87) will be ir.(fa) = 0’ (ss) which again signifies that the electromotive intensity tangential to the cylinder vanishes at its surface. In order to calculate the approximate values of the roots of the equation K\ {x) = 0, it is most convenient to use the ex- pression for K8 (x) which proceeds by powers of l/x. This series is expressed by the equation V ' (ia)H 8ia (l2-4s2) (32 —4s2) + 1.2 (8 ttc)2 (l2 —4s2) (32—4s2) (52 —4s2) 1.2.3 (81®)8 + where C is a constant (see Lord Rayleigh, Theory of Sound, Yol. H, p. 271).349 301.] ELECTRICAL WAVES AND OSCILLATIONS. When s=l, **w-°p, + ra- Thus 15 105 (ix)% 2(8 lx)2 57 2(8ix)3 195 IX 128 (ix)2 1024 (lx)3 To approximate to the roots of the equation K{ (x) = 0, put lx = y, and equate the first four terms inside the bracket to zero; W6get 3 7 , 57 195 n ^ +82/'+1282'- 1024= 0’ a cubic equation to determine y. One root of this equation is real and positive, the other two are imaginary; if a is the positive root, /3 ±iy the two imaginary roots, then we have ci -f- 2 /3 — — 7 5’ 2/3a + /32 + 57 128 9 a (ft2 + y2)= 195 1024 * We find by the rules for the solution of numerical equations that a = *26 approximately, hence /3 = — *56, y = + *64. These roots are however not large enough for the approxima- tion to be close to the accurate values. Hence from equation (88), we see that when 8=1, -^a = --56 ±i-64, V or Lp = (—56 ± t *64) — • a This represents a vibration whose period is 3.1 ^a/F, and whose amplitude fades away to l/€ of its original value after a time 1.8a /V. The radiation of energy away from the sphere in this case is so rapid that the vibrations are practically dead beat; thus after one complete vibration the amplitude is only or about one two hundred and fiftieth part of its value at the beginning of the oscillation.350 ELECTBICAL WAVES AND OSCILLATIONS. [302. 302.] If we consider the state of the field at a considerable distance from the cylinder and only retain in each expression the lowest power of l/r, we find that the magnetic induction c, the tangential and radial components © and R of the electric polarization in the dielectric, may be consistently represented by the following equations: c = cos 6 €“’56 ("V") cos -64 (—^-) > since we have TfdO __ 1 dc dt ~~ ix dr3 COS0 1 —.56 (Xhil\ nA /Vt — 0=z^F;r )■ and since we have p __ sin 6 a 1 dR 1 dc KW~pfd0' |*56 cos *64 — -64 sin »64 Thus R vanishes at all points on a series of cylinders con- centric with the original one whose radii satisfy the equation /Vt —T\ cot *64 (------) = 1.13, v a ' the distance between the consecutive cylinders in this series is 1.5 7 77 a. The Faraday tubes between two such cylinders form closed curves, all cutting at right angles the cylinder for which , y £_ 0=0, or cos *64 (-----j = 0. The closed Faraday tubes move away from the cylinder and are the vehicles by which the energy of the cylinder radiates into space. The axes of the Faraday tubes, i.e. the lines of electromotive intensity between two cylinders at which 12 = 0, are represented in Fig. 110. The genesis of these closed endless tubes from the unclosed ones, which originally stretched from one point to another of the351 302.] ELECTRICAL WAVES AND OSCILLATIONS. cylinder, which we may suppose to have been electrified initially so that the surface density was proportional to sin 6, is shown in Fig. 111. The lines represent the changes in shape in a Faraday tube which originally stretched from a positively to a negatively electri- fied place on the cylinder. The outer line A represents the original position of the tube; when the equilibrium is disturbed some of the tubes inside this one will soon run into the cylinder, and the lateral repulsion they exerted on the tube under consideration will be removed; the outside lateral pressure on this tube will352 ELECTRICAL WAVES AND OSCILLATIONS. [303. now overpower the inside pressure and will produce the indenta- tion shown in the second position B of the tube; this indenta- tion increases until the two sides of the tube meet as in the third position C of the tube; when this takes place the tube breaks up, the outer part D travelling out into space and forming one of the closed tubes shown in Fig. Ill, while the inner part E runs into the cylinder. Decay of Magnetic Force in a Metal Cylinder. 303.] In addition to the very rapid oscillations we have just investigated there are other and slower changes which may occur in the electrical state of the cylinder. Thus, for example, a uniform magnetic field parallel to the axis of the cylinder might suddenly be removed; the alteration in the magnetic force would then induce currents in the cylinder whose magnetic action would tend to maintain the original state of the magnetic field, so that the field instead of sinking abruptly to zero would die away gradually. The rate at which the state of the system changes with the time in cases like this is exceedingly slow compared with the rate of change we have just investigated. Using the same notation as in the preceding investigation, it will be slow enough to make j^a/Fan exceedingly small quantity; when however p&/V is very small, K\ (^a/F) is exceedingly large compared with Kt (pa/V), since (Heine, Kugelfunctionen, vol. i. p. 237) Ks (0) is equal to ( 6) (do2)' ’ thus since when 0 is small KQ{6) is proportional to log *1. K,(p&/V) is proportional to (V/pa.)’, and K/(pa/V) to (V/p&),+1; hence the right-hand side of equation (87) is exceed- ingly large, so that an approximate solution of that equation Willbe J,{ma.) = 0. We notice that this condition makes the normal electromotive intensity at the surface of the cylinder vanish, while it will be remembered that for the very rapid oscillations the tangential electromotive intensity vanished. As the normal intensity vanishes there is no electrification on the surface of the cylinder in this case. The equation J8 (x) = 0 has an infinite number of roots all353 303.] ELECTRICAL WAVES AND OSCILLATIONS. real, the smaller values of which from 8 = 0 to s = 5 are given in the following table, taken from Lord Rayleigh’s Theory of Sound, Yol. I, p. 274. 8 = 0 8=1 8 = 2 8 = 3 8 = 4 8 = 5 2404 3-832 5-135 6-379 7-586 8-780 5*520 7-016 8-417 9-760 11-064 12-339 8-654 10-173 11-620 13-017 14-373 15-700 11-792 13-323 14-796 16-224 17-616 18-982 14-931 16-470 17*960 19-410 20-827 22-220 18-071 19-616 21-117 22-583 24-018 25431 21-212 22-760 24-270 25-749 27-200 28-628 24-353 25-903 27-421 28-909 30-371 31-813 37494 29-047 30-571 32-050 33-512 34.983 This table may be supplemented by the aid of the theorem that the large roots of the equation got by equating a Bessel’s function to zero form approximately an arithmetical progression whose common difference is 7r. If xq denotes a root of the equation J9 (x) = 0, then, since p is given by the equation where J9 (ma) = 0, n2=^p, we see that pq, the corresponding value of py is given by the equation a Thus, since ipq is real and negative, the system simply fades away to its position of equilibrium and does not oscillate about it. The term in c which was initially expressed by / 7*\ J.COSS0/8(#fl-)> will after the lapse of a time t have diminished to A COS S0j9 (xg iTraPfi*9 t. If we call T the time which must elapse before the term sinks to l/c of its original value, the c time modulus ’ of the term, then, 811106 y=47raV3354 ELECTRICAL WAVES AND OSCILLATIONS. [304. we see that the time modulus is inversely proportional to the resistance of unit length of the cylinder and directly propor- tional to the magnetic permeability. Since jx/o- for iron is larger than it is for copper, the magnetic force will fade away more slowly in an iron cylinder than in a copper one. 304.] A case of great interest, which can be solved without difficulty by the preceding equations, is the one where a cylinder is placed in a uniform magnetic field which is suddenly annihilated, the lines of magnetic force being originally parallel to the axis of the cylinder. We may imagine, for example, that the cylinder is placed inside a long straight solenoid, the current through which is suddenly broken. Since in this case everything is symmetrical about the axis of the cylinder, s = 0, and the values of ip are therefore -(2-404)2 47ia2/x -(5-520)2 4 7ra2/x &c. Now we know from the theory of Bessel’s functions that any function of r can for values of r between 0 and a be expanded in the form AXJ0 (x1 —) 4 A2J0 (x2 -) -f- AZJ0 (#3—) + ••• 9 where x19 x2, #3... are the roots of the equation J0 0*0= o* Thus, initially c = A1Jq(x1-^) 4 A2J0 (#2“) + A3J0(x3-^) 4- ..., hence the value of c after a time t will be given by the equa- tion 5'78<. This expression is a maximum when r = 0 and gradually dies away to zero when r = a, thus the lines of magnetic force fade away most quickly at the surface of the cylinder and linger longest at the centre. The time modulus for the first term is 47raV/5*78 hence at a dis- tance r from the axis of the cylinder the intensity is j.oo j o ' Xat 2tT[X& Jx (xq) Since at the instant the magnetic force is destroyed, c is constant over the cross-section of the cylinder, the intensity of the current when t = 0 will vanish except at the surface of the cylinder, where, as the above equation shows, it is infinite. After some time has elapsed the intensity of the current will be adequately represented by the first term of the series, i. e. by J-^2.404- J,{xPl)dr = -f*i*J0'(ccPl)dr. Integrating by parts and remembering that J0 (xp) = 0, we see that each of these integrals equals which is equal to 2a3 -Ji(xp), (95) since Again, since d dx* multiplying by we get 0/h2^^1 2ar—7— 5 ax + Ji (*) j = 2xJ*(x). Hence ^=\e\ji2 (£)+(1 - £) -a2 (f)} • Thus, since fjr/(f) + J^) = £J0(i), we have if J0 (£) = 0,361 308.] electrical waves and oscillations. Hence, when xp is a root of Jo (®) = f*r Ji (XP dr = ^ a2 J\ (xp). (96) Now by (94) Hence by (95) and (96) A - H p ito,Jl(xp) Thus by (93), the currents produced by the annihilation of a magnetic field H parallel to y are given by the equation so that w = — ITcos 0 (Xpa ) 7ra «*pt 4ira2 • Thus the currents vanish at the axis of the cylinder; when t = 0 they are infinite at the surface and zero elsewhere. When, as in the case of iron, fx is very large, the equation (92) becomes approximately (ina) = o. The solution in this case can be worked out on the same lines as the preceding one; for the results of this investigation we refer the reader to a paper by Prof. H. Lamb (Proc. Lond. Math. Soc. XV, p. 270). Electrical Oscillations on a Spherical Conductor. 308.] The equations satisfied in the electromagnetic field by the components of the magnetic induction, or of the electro- motive intensity, when these quantities vary as eipt, are, denoting any one of them by F, of the form d2F d2F dx2 dy2 + d?F dz2 = -X2JT, (97) where in an insulator A.2 = p2/V2, V being the velocity of pro- pagation of electrodynamic action through the dielectric, and in a conductor, whose specific resistance is Str- and Again, since ?£*<«•>+ (i*- =»• ^ ^s.(XV) + (A^-(XV) = 0, we have }s. (»>) <* <0 - S. (*<•) §, s. (»V) J + 2r{s,pr)±8^r)-S^r)±S.W\ = (A'S-A^r^Ar)^^).309.] ELECTRICAL WAVES AND OSCILLATIONS. 365 and hence J riSn (Ar) Sn (\'r) dr = SS.(Kr)-^S.{Kr)^S, (aV)£ , (102) so that if A, A' satisfy the equations a2{8«(k'a)^8«(Xa)-s»(Ka)^s»(k'a^ = °. i2 {(K'b) (L (\'b)—Sn(Kb) ^ (A'&) j = 0, then r2 Sn (Ar) Sn (A'r) dr = 0. Proceeding to the limit A' = A, we get from (102) /WM* = - A,[r««.(Ar) _r.^dSH(kr)l^b dr ) ja The following table of the values of the first four of the S and E functions will be found useful for the subsequent work :— o / \ sina5 S0(x) S1(x) = X cos x sin x x xA ^ , x sin a? 3 cos a; 3 sin a; ^(*)=—----— + os t \ COS X X* ^ X* 6 sin a; 15 cos a; 15 sin a? a2 x .3 a?4 K (x) — — IX j E-{x) = - — (i+i), 1 W X ' X' _ , X C“£a? , 3 L 3 \ EA®)~ x (i ■ x J), w-r \ 6 15l 15\ * ^ — x v x a? x3' The values of E+ can be got from those of E by changing the sign of i.366 ELECTRICAL WAVES AND OSCILLATIONS. [3*0. 310.] We shall now proceed to the study of the oscillations of a distribution of electricity over the surface of a sphere. Let us suppose that a distribution of electricity whose surface density is proportional to a zonal harmonic of the nth order is produced over the surface of the sphere, and that the cause pro- ducing this distribution is suddenly removed; then, since this distribution cannot be in equilibrium unless under the influence of external forces, electric currents will start off to equalize it, and electrical vibrations will be started whose period it is the object of the following investigation to determine. Since the currents obviously flow in planes through the axis of the zonal harmonic, which we shall take for the axis of z, there is no electromotive force round a circuit in a plane at right angles to this axis; and since the electromotive force round a circuit is equal to the rate of diminution in the number of lines of magnetic force passing through it, we see that in this case, since the motion is periodic, there can be no lines of magnetic force at right angles to such a circuit; in other words, the magnetic force parallel to the axis of 0 vanishes. Again, taking a small closed circuit at right angles to a radius of the sphere, we see that the electromotive force round this circuit, and therefore the magnetic force at right angles to it, vanish; hence the magnetic force has no component along the radius, and is thus at right angles to both the axis of 0 and the radius, so that the lines of magnetic force are a series of small circles with the axis of the harmonic for axis. Hence, if a, b, c denote the components of magnetic induc- tion parallel to the axes of x} y, 0 respectively, we may put « = yx (r> m), b = — XX (r, m), c = 0, where x (^3 m) denotes some function of r and /ut. Comparing this with the results of Art. 308, we see that inside the sphere a = r v J dfji b=- A-SJk'r)^^*, r K 'dp c = 0, where A'2 = — 4 7rjuu$/» KM is of the order 4t1 pip or ar since p is comparable with F/a. This, when the sphere conducts as well as iron or copper, is extremely small unless a is less than the wave length of sodium light, while for a perfect conductor it absolutely vanishes, hence equation (107) is very approximately equivalent to ~{aEn(X&)} = 0. (108) This, by the relation (101), may be written ^»+i(^)-^^1(Xa) = 0, which is the form given in my paper on ‘ Electrical Oscillations/ Proc. Lond. Math. Soc. XV, p. 197. This condition makes the tangential electromotive intensity vanish, so that the lines of electrostatic induction are always at right angles to the surface of the sphere. 312.] In order to show that the equations (103) and (104) in the preceding article correspond to a distribution of electricity over the surface of the sphere represented by a zonal harmonic Qn of the 71th order, we only need to show that the current along the radius vector varies as Qn, for the difference between the312-] ELECTRICAL WAVES AND OSCILLATIONS. 369 radial currents in the sphere and in the dielectric is proportional to the rate of variation of the surface density of the electricity on the sphere, and therefore, since the surface density varies as c1^, it will be proportional to the radial current. Consider a small area at right angles to the radius, and apply the principle that 4tt times the current through this area is equal to the line integral of the magnetic force round it, we get, if p is the current along the radius and jx = cos 0, 1’p = ;|.(>'8inS)' (109) where y, as before, is the resultant magnetic force which acts along a tangent to a parallel of latitude. By equation (103), y is proportional to j dQn sin# dix’ so that p is proportional to hence p, and therefore the surface density, is proportional to Qn. We shall now consider in more detail the case n = 1. We have — = A2 y 2 “ A • We shall take as the solution of the equation p2/V2 = A2 y~~ A, and we shall take E~ (Ar) as our solution, as this corresponds to a wave diverging from the sphere. Thus, equation (108) be- comes j £{a^r(xa)} = 0> or substituting for E~ (A a) the value given in Art. 309, Xa5 1 , * 1370 ELECTRICAL WAVES AND OSCILLATIONS. [312. Hence V (* ^zk V 3 ~2 S' taking the positive sign since the wave is divergent. Hence, the time of vibration is 47ra/V3 V, and the wave length 47ra/\/lh The amplitude of the vibration falls to 1/e of its original value after a time 2a/Tr, that is after the time taken by light to pass across a diameter of the sphere. In the time 2ir occupied by one complete vibration the amplitude falls to * Vs, or about 1/35 of its original value, thus the vibrations will hardly make a complete oscillation before they become practically extinguished. This very rapid extinction of the vibrations is independent of the resistance of the conductor and is due to the emission of radiant energy by the sphere. Whenever these electrical vibrations can radiate freely they die away with immense rapidity and are practically dead beat. If we substitute this value of X in the expressions for the magnetic force and electromotive intensity in the dielectric, we shall find that the following values satisfy the conditions of the problem. If y is the resultant magnetic force, acting at right angle to the meridional plane, sin 0a ( y=—i1 (T«-0 2a COS ((f) -f- where i * r—a, 7r tan o =-----tan - • r + a 3 If 0 is the electromotive intensity at right angles to r in the meridional plane, K the specific inductive capacity of the di- electric surrounding the sphere, then by Art. 310 K& = ^ ~ \) {l + ^ + Jr}*€ cob( 35 03.- (*»» = §i<»3r Ml a f*-3f Ml- When n = 1, this becomes tan A {b — a} = A £-£?{*+ail (110) The roots of this equation are real, so that in this case there is no decay of the vibrations apart from that arising from the resistance of the conductors. If a is very small compared with b, this equation reduces to The least root of this equation other than A = 0, I find by the method of trial and error to be Ab = 2-744. This case is that of the vibration of a spherical shell excited by some cause inside, here there is no radiation of the energy into space, the electrical waves keep passing backwards and forwards from one part of the surface of the sphere to another. The wave length in this case is 27rb/2-744 or 2-29b, and is therefore less than the wave length, 477b/V3, of the oscillations which would occur if the vibrations radiated off into space: this is an example of the general principle in the theory of vibrations that when dissipation of energy takes place either from friction,374 ELECTRICAL WAVES AND OSCILLATIONS. [3I5- electrical resistance, or radiation, the time of vibration is in- creased. In this case, since the radius of the inner sphere is made to vanish in the limit, the magnetic force inside the sphere whose radius is b must be expressed by that function of r which does not become infinite when r is zero, i.e. by Sn(\r). In the case when n = 1, the components a, 6, c of the magnetic induction are given by a = '2B%S1(Kr)t'Pt, b = -2B?S81(\r)'‘**, c = 0; where the summation extends over all values of A which satisfy the equation ^ Let us consider the case when only the gravest vibration is excited. Let e be the surface density of the electricity, then it will be given by an equation of the form e = C cos 0 cos pt; where p = Fa13 Ax being equal to 2.744/b. By equation (109) the normal displacement current P is given by the equation j ^ inT = rd^e^ine^+vl^ In this case so that 1 = 4 v P -- - -B COS0&! (Ajr) itpt. (ill) When r = b the normal displacement current = de/dt, hence — 4 7r (7 cos dp swpt = — -B cos dSy (Atb) tpt. Substituting this value of Btpt in (111), we have a = - 2 7i bp sin pt C —■ ^ r\ r 11 SL (Ajb) c = 0.375 316.] ELECTRICAL WAVES AND OSCILLATIONS. At the surface of the sphere the maximum intensity of the magnetic force is 2^bpCsin 6, or since bp = FA1bJ and Axb = 2*744, the maximum magnetic force is 2ttx2*744 VC sin 0. For air at atmospheric pressure VC may be as large as 25 without the electricity escaping; taking this value of VC, the maximum value of the magnetic force will be 431 sin0; this indicates a very intense magnetic field, which however would be difficult to detect on account of its very rapid rate of reversal. Electrical Oscillations on Two Concentric Spheres of nearly equal radius. 316.] When d, the difference between the radii a and b, is very small compared with a or b, equation (110) becomes A4a4 —A2a2+ 1 (112) There will be one root of this equation corresponding to a vibration whose wave length is comparable with a, and other roots corresponding to wave lengths comparable with d. When the wave length is comparable with a, A is comparable with l/a, so that in this case Ac? is very small; when this is the case (tan\d)/kd = 1, and equation (112) becomes approximately 1+A2a2 A4a4-A2a2+r or Aa = a/2. The wave length 2tt/\ is thus equal to it*/2 times the radius of the sphere. In this case, since the distance between the spheres is very small compared with the wave length, the tangential electro- motive intensity, since it vanishes at the surface of both spheres, will remain very small throughout the space between them; the electromotive intensity will thus be very nearly radial between the spheres, and the places nearest each other on the two spheres376 ELECTRICAL WAVES AND OSCILLATIONS. [316. will have opposite electrical charges. The tubes of electrostatic induction are radial, and moving at right angles to themselves traverse during a complete oscillation a distance comparable with the circumference of one of the spheres. When the wave length is comparable with the distance between the spheres, A is comparable with 1/ci, and Aa is there- fore very large. The denominator of the right-hand side of equation (112), since it involves (Aa)4, will be exceedingly large compared with the numerator, and this side of the equation will be exceedingly small, so that an approximate solution of it is tan Ac? = 0, or A d — mr, where n is an integer. The wave length 27r/A = 2 d/n. Hence, the length of the longest wave is 2 c?, and there are harmonics whose wave lengths are d, 2c?/3, 2 in the dielectric. Hence ^{a£„(A'a)} =*^{aJS-.(Aa)}. (H7) Eliminating A and B from equations (116) and (117), we get „ 3,(X'a) _ ff.(Xa) S (»«•(*' »)l £)•*.<*•» (118) In this ease the currents and magnetic forces change so slowly that A a or p&/V is an exceedingly small quantity, but when this is the case we have proved Art. 317, that approximately d ^-{a^(Aa)} n so that equation (118) becomes ^(A'a)+^{a5n(A'a)} = 0. But by equation (100), Art. 309, d (119)382 ELECTRICAL WAVES AND OSCILLATIONS. [319. hence (119) may be written n(fx—l)SH (A'a)—A'a Snm_t (A'a) = 0. (120) For non-magnetic metals for which jx = 1 this reduces to ^n-l(^ a) = while for iron, for which /x is very great, the equation approxi- mates very closely to g __ q The smaller roots of the equation Sn(x)=0, when n = 0, 1, 2, are given below; n = 0, x = tt, 2 tt, 3 7r,...; n= 1, ic = 1-4303ir, 2-45907r, 3-4709tt; 71 = 2, a? = 1*8346tt, 2-89507r, 3-922577. Thus for a copper sphere for which S1(AJ,a) — 0 d2 d Substituting in (124) the values of S, (Apa) and ^ S1 (\pa) given by these equations, we get f*1*8.i2 (krr)dr = Jj^Si2(kp&) (V®2 + (f*+ 2) (M-1)}. c c386 ELECTRICAL WAVES AND OSCILLATIONS. [322. Hence, multiplying both sides of (123) by r2S1 (kpr) and inte- grating from 0 to a, we get = \ SfM {A/a2 + (M+2)(M-l)}. (125) To find the integral on the left-hand side, we notice + 27*^Sl(Xpr) — 2rS1(kpr) + kp2r*Sl{\pr) = 0, dr2 or r,{r‘3fs' — c 4a2 . 7T2 T3 ^p = 1 p2 The sphere produces the same effect at an external point as a small magnet whose moment is 3Ha?^p V = 1 P2 € p*iro4 4a2 323.] When jx is very great _ 6-ga Hence, the normal magnetic force at the surface of the sphere is qH ^ —COS 02€ 4tt/* . Outside the sphere the magnetic force is the same as that due to a magnet whose moment is 3 if a3 4tr/i , placed at its centre. These results are given by Lamb {l, c.). Thus the magnetic effects,of the currents induced in a soft iron sphere are less than those which would be produced by a copper sphere of the same size placed in the same field. This is due to the changes of magnetic force proceeding more slowly in the iron sphere on account of its greater self-induction; as the changes in magnetic force are slower, the electromotive forces, and therefore the currents, will be smaller. Since (Apa) = 0 when /x is large, the currents on the surface of the sphere vanish, and the currents congregate towards the middle of the sphere. c c %CHAPTER V. Experiments on Electromagnetic Waves. 324. ] Professor Hertz has recently described a series of experi- ments which show that waves of electromotive and magnetic force are present in the dielectric medium surrounding an electrical system which is executing very rapid electrical vibrations. A complete account of these will be found in his book Ausbreitung dev elektrisehen Kraft, Leipzig, 1892. The vibrations which Hertz used in his investigations are of the type of those which occur when the inner and outer coatings of a charged Leyden jar are put in electrical connection. The time of vibration of such a system when the resistance of the discharging circuit may be neglected is, as we saw in Art. 296, approximately equal to 2 7tVLG, where L is the coefficient of self-induction of the discharging circuit for infinitely rapid vibrations and C is the capacity of the jar in electromagnetic measure. If C is the capacity of the jar in electrostatic measure, then, since C = C/F2, where V is the ratio of the electromagnetic unit of electricity to the electrostatic unit, the time of vibration is equal to 2ttVL C/F. But since F is equal to the velocity of propagation of electrodynamic action through air, the distance the disturb- ance will travel in the time occupied by a complete oscilla- tion, in other words the wave length in air of these vibrations, will be 2ttVLC. By using electrical systems which had very small capacities and coefficients of self-induction Hertz succeeded in bringing the wave length down to a few metres. 325. ] The electrical vibrator which Hertz used in his earlier experiments (Wied. Ann. 34, pp. 155, 551, 609, 1888) is repre- sented in Figure 113. A and B are square zinc plates whose sides are 40 cm. long, copper wires C and D each about 30 cm. long are soldered to the plates, these wires terminate in brass balls E and F. ToEXPERIMENTS ON ELECTROMAGNETIC WAVES. 389 ensure the success of the experiments it is necessary that these balls should be exceedingly brightly and smoothly polished, and inasmuch as the passage of the sparks from one ball to the other across the air space E F roughens the balls by tearing particles of metal from them, it is necessary to keep repolishing the balls at short intervals during the course of the experiment. It is also advisable to keep the air space E F shaded from the light from any sparks that may be passing in the neigh- bourhood. In order to ex- cite electrical vibrations in this system the extremities of an induction coil are con- nected with C and D respectively. When the coil is in action it produces so great a difference of potential between the balls E and F that the electric strength of the air is overcome, sparks pass across the air gap which thus becomes a conductor; the two plates A and B are now connected by a conducting circuit, and the charges on the plates oscillate backwards and forwards from one plate to another just as in the case of the Leyden jar. 326.] As these oscillations are exceedingly rapid they will not be excited unless the electric strength of the air gap breaks down suddenly; if it breaks down so gradually that instead of a spark suddenly rushing across the gap we have an almost continuous glow or brush discharge, hardly any vibrations will be excited. A parallel case to this is that of the vibrations of a simple pendulum, if the bob of such a pendulum is pulled out from the vertical by a string and the string is suddenly cut the pendulum will oscillate; if however the string instead of breaking suddenly gifes way gradually, the bob of the pendulum will merely sink to its position of equilibrium and no vibrations will be excited. It is this which makes it necessary to keep the balls E and F well polished, if they are rough there will in all like- lihood be sharp points upon them from which the electricity will gradually escape, the constraint of the system will then give way gradually instead of suddenly and no vibrations will be excited. The necessity of shielding the air gap from light coming from other sparks is due to a similar reason. Ultra-violet light in which these sparks abound possesses, as we saw in Art. 39, the property of producing a gradual discharge of electricity from the negative390 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [327. terminal, so that unless this light is shielded off there will be a tendency to produce a gradual and therefore non-effective dis- charge instead of an abrupt and therefore effective one. 327.] The presence of the coil does not, as the following calcu- lation of the period of the compound system shows, affect the time of vibration to more than an infinitesimal extent, if, as is practically always the case, the coefficient of self-induction of the secondary of the coil is almost infinite in comparison with that of the vibrator. Let L be the coefficient of self-induction of the vibrator AB, C its capacity, Lf the coefficient of self-induction of the secondary of the coil, M the coefficient of mutual induction between this coil and the vibrator, x the quantity of electricity at any time on either plate of the condenser, y the current in the vibrator, z that through the secondary of the coil. Then we have x = y + z or x = y + z. The Kinetic energy of the currents is The potential energy is (y+z)2 7T~ or \ • Hence, if we neglect the resistance of the circuit, we have by Lagrange’s equations , Ly" + Mz"+y-^- = 0, L'2f' + My"+y±Z = 0. Thus if x and y each vary as we have y ~LP2) + a -Mp2) = 0, 2 ~LY) + y(-$-Mp2) = 0. Eliminating y and 0 we get (5(5 2 1 i. ,L 2M) /,, M2^ p-CLl1+l7 L' )/(1—ZZ')* or329.J EXPERIMENTS ON ELECTROMAGNETIC WAVES. 391 But for a circuit as short as a Hertzian vibrator L/L' and M/L' will be exceedingly small, so that we have as before 2__ 1 P ~ CL' The Resonator. 328. ] When the electrical oscillations are taking place in the vibrator the space around it will be the seat of electric and magnetic intensities. Hertz found that he could detect these by means of an instrument which is called the Resonator. It consists of a piece of copper wire bent into a circle ; the ends of the wire, which are placed very near together, are furnished with two balls or a ball and a point, these are connected by an insu- lating screw, so that the distance between them admits of very fine adjustment. A resonator without the screw adjustment is shown in Fig. 114. With a vibrator having the dimensions of the one in Art. 325, Hertz used a resonator 35 cm. in radius. 329. ] When the resonator was held near the vibrator Hertz found that sparks passed across the air space in the resonator and that the length of the air space across which Fig. 114. the sparks would pass varied with the posi- tion of the resonator. This variation was found by Hertz to be of the following kind : Let the vibrator be placed so that its axis, the line E F, Fig. 113, is horizontal; let the horizontal line which bisects this axis at right angles, i. e. which passes through the middle point of the air space E F, be called the base line. Then, when the resonator is placed so that its centre is on the base line and its plane at right angles to that line, Hertz found that sparks pass readily in the resonator when its air space is either vertically above or vertically below its centre, but that they cease entirely when the resonator is turned in its own plane round its centre until the air space is in the horizontal plane through that point. Thus the sparks are bright when the line joining the ends of the resonator is parallel to the axis of the vibrator and vanish when it is at right angles to this axis. In intermediate positions of the air gap faint sparks pass between the terminals of the resonator.392 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [331. When the centre of the resonator is in the base line and its plane at right angles to the axis of the vibrator no sparks pass, whatever may be the position of the air space. When the centre of the resonator is in the base line and its plane horizontal the sparks are strongest when the air space is nearest to the vibrator, and as the resonator turns about its centre in its own plane the length of the sparks diminishes as the air space recedes from the vibrator and is a minimum when the air gap is at its maximum distance from the axis of the vibrator. They do not however vanish in this case for any position of the air space. 330. ] In the preceding experiments the length of the sparks changes as the resonator rotates in its own plane about its centre. Since rotation is not accompanied by any change in the number of lines of magnetic force passing through the resonator circuit, it follows that we cannot estimate the tendency to spark across the air gap by calculating by Faraday’s rule the electro- motive force round the circuit from the diminution in the number of lines of magnetic force passing through it. 331. ] The effects on the spark length are, however, easily explained if we consider the arrangement of the Faraday tubes radiating from the vibrator. The tendency to spark will be proportional to the number of tubes which stretch across the air gap; these tubes may fall directly on the air gap or they may be collected by the wire of the resonator and thrown on the air gap, the resonator acting as a kind of trap for Faraday tubes. Let us first consider the case when the centre of the resonator is on, and its plane at right angles to, the base line, then in the neighbourhood of the base line the Faraday tubes are approxi- mately parallel to the axis of the vibrator, and their direction of motion is parallel to the base line; thus the Faraday tubes are parallel to the plane of the resonator and are moving at right angles to it. When they strike against the wire of the resonator they will split up into separate pieces as in Fig. 115, which represents a tube moving up to and across the resonator, and after passing the cross-section of the wire of the resonator will join again and go on as if they had not been interrupted. The resonator will thus not catch Faraday tubes and throw them in the air gap, and therefore the tendency to spark across the gap will be due only to those tubes which fall directly upon it. When331-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 393 the air gap is parallel to the tubes, i. e. when it is at the highest or lowest point of the resonator, some of the tubes will be caught and will stretch across the gap and thus tend to produce a spark. When, however, the gap is at right angles to the tubes, i. e. when it is in the horizontal plane through the centre of the resonator, the tubes will pass right through it. None of them will stretch across the gap and there will be con- sequently no tendency to spark. When the plane of the resonator is at right angles to the axis of the vibrator, the tubes when they meet the wire of the resonator are, as in the last case, travelling at right angles Fig. 115. to it, so that the wire of the resonator will not collect the tubes and throw them into the air gap. In this case the air gap is always at right angles to the tubes, which will therefore pass right through it, and none of them will stretch across the gap. Thus in this case there is no tendency to spark whatever may be the position of the air space. Let us now consider the case when the centre of the resonator is on the base line and its plane horizontal. In this case, as we see by the figures Fig. 116, Faraday tubes will be caught by the wire of the resonator and thrown into the air gap wherever that may be; thus, whatever the position of the gap, Faraday tubes will stretch across it, and there will be a tendency to spark. When the gap is as near as possible to the vibrator the Faraday tubes which strike against the resonator will break and a portion394 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [331. of them will stretch right across the gap. When however the gap is a considerable distance from this position the tubes which stretch across it are due to the bending together of two portions of the tubes broken by previously striking against the resonator, the end of one of the portions having travelled along one side of the resonator while the end of the other has travelled along c Fig. 116. the other side, [a) ; these portions bend together across the gap, (b) and (c); then break up again, one long straight tube travelling outwards, the other shorter one running into the gap, as in (d) Fig. 116. The portion connecting the two sides of the gap diverges more from the shortest distance between the terminals than in the case where the air gap is as near to the vibrator as possible, the field in Fig. 116 will not therefore be so concentrated333-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 395 round the gap, so that there will be less tendency to spark, though this tendency will still remain finite. Resonance. 332.] Hitherto we have said nothing as to the effect produced by the size of the resonator on the brightness of the sparks, this effect is however often very great, especially when we are using condensers with fairly large capacities which can execute several vibrations before the radiation of their energy reduces the amplitude of the vibration to insignificance. The cause of this effect is that the resonator is itself an electrical system with a definite period of vibration of its own, hence if we use a resonator the period of whose free vibration is equal to that of the vibrator, the efforts of the vibrator to produce a spark in the resonator will accumulate, and we may be able as the result of this accumulation to get a spark which would not have been produced if the resonator had not been in tune with the vibrator. The case is analogous to the one in which a vibrating tuning fork sets another of the same pitch in vibration, though it does not produce any appreciable effect on another of slightly different pitch. Fig. 117. 333.] Professor Oliver Lodge {Nature, Feb. 20, 1890, vol. 41, p. 368) has described an experiment which shows very beautifully the effect of electric resonance. A and B, Fig. 117, represent two Leyden jars whose inner and outer coatings are connected by a wire bent so as to include a considerable area. The circuit connecting the coatings of one of these jars, A, contains an air break. Electrical oscillations are started in this jar by connect- ing the two coatings with the poles of an electrical machine.390 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [333. The circuit connecting the coatings of the other jar, B, is pro- vided with a sliding piece by means of which the self-induction of the discharging circuit, and therefore the time of an electrical oscillation of the jar, can be adjusted. The inner and outer coatings of this jar are put almost but not quite into electrical contact by means of a piece of tin-foil bent over the lip of the jar. The jars are placed face to face so that the circuits con- necting their coatings are parallel to each other, and approxi- mately at right angles to the line joining the centre of the circuits. When the electrical machine is in action sparks pass across the air break in the circuit in A, and by moving the slider in B about it is possible to find a position for it in which sparks pass by means of the tin-foil from one coating of the jar to the other; as soon however as the slider is moved from this position the sparks cease. Resonance effects are most clearly marked in cases of this kind, where the system which is vibrating electrically has con- siderable capacity, since in such cases several complete oscilla- tions have to take place before the radiation of energy from the system has greatly diminished the amplitude of the vibra- tions. When the capacity is small, the energy radiates so quickly that only a small number of vibrations have any appre- ciable amplitude; there are thus only a small number of im- pulses acting on the resonator, and even if the effects of these few conspire, the resonance cannot be expected to be very marked. In the case of the vibrating sphere we saw (Art. 312) that for vibrations about the distribution represented by the first harmonic the amplitude of the second vibration is only about 1/35 of that of the first, in such a case as this the system is practically dead-beat, and there can be no appreciable resonance or interference effects. The Hertzian vibrator is one in which, as we can see by considering the disposition of the Faraday tubes just before the spark passes across the air, there will be very considerable radiation of energy. Many of the tubes stretch from one plate of the vibrator to the other, and when the insulation of the air space breaks down, closed Faraday tubes will break off from these in the same way as they did from the cylinder; see Fig. 14. These closed tubes will move off from the vibrator with the velocity of light, and will carry the energy of the vibrator away335*] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 397 with them. In consequence of this radiation the decay of the oscillations in the vibrator will be very rapid, indeed we should expect the rate of decay to be comparable with its value in the case of the vibrations of electricity over the surfaces of spheres or cylinders, where the Faraday tubes which originally stretched from one part to another of the electrified conductor emit closed tubes which radiate into space in the same way as the similar tubes in the case of the Hertzian vibrator : we have seen, how- ever, that for spheres and cylinders the decay of vibration is so rapid that they may almost be regarded as dead-beat. We should expect a somewhat similar result for the oscillations of the Hertzian vibrator. 334. ] On the other hand, the disposition of the Faraday tubes shows us that the electrical vibrations of the resonator will be much more persistent. In this case the Faraday tubes will stretch from side to side across the inside of the resonator as in Fig. 118, and these tubes will oscillate backwards and forwards inside the resonator; they will have no tendency to form closed curves, and consequently there will be little or no radiation of energy. In this case the decay of the vibrations will be chiefly due to the resistance of the resonator, as in the corresponding cases of oscilla- tions in the electrical distribution over spherical or cylindrical cavities in a mass of metal, which are discussed in Arts. 315 and 300. 335. ] The rate at which the vibrations die away for a vibrator and resonator of dimensions not very different from those used by Hertz has been measured by Bjerknes (Wied. Ann. 44, p. 74, 1891), who found that in the vibrator the oscillations died away to 1/f of their original value after a time T/«26, where T is the time of oscillation of the vibrator. This rate of decay, though not so rapid as for spheres and cylinders, is still very rapid, as the amplitude of the tenth swing is about 1/14 of that of the first. The amplitudes of the successive vibrations are represented graphically in Fig. 119, which is taken from Bjerknes* paper.398 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [336. The time taken by the vibrations in the resonator to fade away to 1/e of their original value was found by Bjerknes to be T'/-002 or 500 T\ where T' is the time of the electrical oscilla- tion of the resonator; thus the resonator will make more than 1000 complete oscillations before the amplitude of the vibration falls to 1/10 of its original value. The very slow rate of decay of these oscillations confirms the conclusion we arrived at from the consideration of the Faraday tubes, that there was little or no radiation of energy in this case. The rate of decay of the vibrations in the resonator compares favourably with that of pendulums or tuning-forks, and is in striking contrast to the very rapid fading away of the oscillations of the vibrator. These experiments show that, as the theory led us to expect, we must regard the vibrator as a system having a remarkably large logarithmic decrement, the resonator as one having a remarkably small one. Reflection of Electromagnetic Waves from a Metal Plate. 336.] We shall now proceed to describe the experiments by which Hertz succeeded in demonstrating, by means of the vibrator and resonator described in Arts. 325 and 328, the existence in the dielectric of waves of electromotive intensity and magnetic force (Wied. Ann. 34, p. 610, 1888). The experiments were made in a large room about 15 metres long, 14 broad, and 6 high. The vibrator was placed 2 m. from one of the main walls, in such a position that its axis was vertical and its base line at right angles to the wall. At all points along the base line the electromotive intensity is vertical, being parallel to the axis of the vibrator. At the further end of the room a piece of sheet zinc 4 metres by 2 was placed338.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 399 vertically against the wall, its plane being thus at right angles to the base line of the vibrator. The zinc plate was connected to earth by means of the gas and water pipes. In one set of experiments the centre of the resonator was on and its plane at right angles to the base line. When it is in this position the Faraday tubes from the vibrator strike the wire of the resonator at right angles; the resonator therefore does not catch the tubes and throw them into the air gap, and the spark will be due to the tubes which fall directly upon the air gap. Thus, as might be expected, the sparks vanish when the gap is at the highest or lowest point of the resonator, when the tubes are at right angles to the direction in which the sparks would pass, and the sparks are brightest when the air gap is in the horizontal plane through the base line, when the incident tubes are parallel to the sparks. 337. ] Let the air gap be kept in this plane, and the resonator moved about, its centre remaining on the base line, and its plane at right angles to it. When the resonator is quite close to the zinc plate no sparks pass across the air space ; feeble sparks, however, begin to pass as soon as the resonator is moved a short distance away from the plate. They increase rapidly in brightness as the resonator is moved away from the plate until the distance between the two is about 1*8 m., when the brightness of the sparks is a maximum. When the distance is still further in- creased the brightness of the sparks diminishes, and vanishes again at a distance of about 4 metres from the zinc plate, after which it begins to increase, and attains another maximum, and so on. Thus the sparks exhibit a remarkable periodic character, similar to that which occurs when stationary vibra- tions are produced by the reflection of wave motion from a surface at right angles to the direction of propagation of the motion. 338. ] Let the resonator now be placed so that its plane is the vertical one through the base line, the air gap being at the highest or lowest point; in this position the Faraday tubes which fall directly on the air gap are at right angles to the sparks, so that the latter are due entirely to the Faraday tubes collected by the resonator and thrown into the air gap. When the resonator is in this position and close to the reflect- ing plate sparks pass freely. As the resonator recedes from the400 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [34O. plate the sparks diminish and vanish when its distance from the plate is about 1*8 metres, the place at which they were a maximum when the resonator was at right angles to the base line; after the resonator passes through this position the sparks increase and attain a maximum 4 metres from the plate, the place where, with the other position of the resonator, they were a minimum ; when the resonator is removed still further from the plate the sparks diminish, then vanish, and so on. The sparks in this case show a periodicity of the same wave length as when the resonator was in its former position, the places of minimum intensity for the sparks in one position of the resonator corresponding to those of maximum intensity in the other. 339. ] If the zinc reflecting plate is mounted on a movable frame work so that it can be placed behind the resonator and removed at will, its effect can be very clearly shown by the following experiments:— Hold the resonator in the position it had in the last experi- ment at some distance from the vibrator and observe the sparks, the zinc plate being placed on one side out of action : then place the reflector immediately behind the resonator, the sparks will increase in brightness; now push the reflector back, and at about 2 metres from the resonator the sparks will stop. On pushing it still further back the sparks will increase again, and when the reflector is about 4 metres away they will be a little brighter than when it was absent altogether. 340. ] Hertz only used one size of resonator, which was selected so as to be in tune with the vibrator. Sarasin and De la Rive (Comptes Rendus, March 31, 1891), who repeated this experiment with vibrators and resonators of various sizes, found however that the apparent wave length of the vibrations, that is twice the distance between two adjacent places where the sparks vanish, depended entirely upon the size of the resonator, and not at all upon that of the vibrator. The following table contains the results of their experiments ; A denotes the wave length, a ‘loop* means a place where the sparks are at their maximum brightness when the resonator is held in the first position, a ‘ node ’ a place where the brightness is a minimum. The line beginning ‘ 1/4 A wire’ relates to another series of experi- ments which we shall consider subsequently. It is included here34°-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 401 to avoid the repetition of the table. The distances of the loops and nodes are measured in metres from the reflecting surface. Diameter of resonator circle (D). 1 metre, stout wire 1 cm. in diameter. •75 m. stout wire. •50 m. stout wire. •35 m. stout wire. •35 m. fine wire 2 mm. in diameter. 1st Loop . . . 2-11 1-60 1.11 •76 •75 1st Node . . . 4-14 3-01 1.49 1-51 2nd Loop . . 2.30 237 2nd Node . . 3-04 340 3rd Loop . . . 3rd Node . . JA. air. . . . 203 1.41 Lll •76 •80 £ A. wire . . . 1.92 148 •98 •73 2D .... 2-00 1-50 LOO •70 -h -r- A', B' = —2 A (sin ifcos r) (K tan r—tan i) €~ia'h + a', C = 4AK tanitanr €iah A', A' = (K2 tan2 r + tan2 i) (c*— c “ *ah) + 2K tan i tan r (el + a'hy From these equations we see that if the incident wave is equal to (A — A') cos i = {B — B') cos r, (A + A') sin i = K (B + B') sin r, } (7) where the waves in the plate will be represented by (sin i/cos r) (K tan r + tan i) x413 350.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. respectively, while the emergent wave is 2Ktanitanr cos j^^((a: + A)cosi + i/sini-f Vi)—h-D', where D'2 = (K2 tan2 r + tan2 if sin2 (y A cos r) 2ir + 4 K2 tan2 r tan2 i cos2 (--fh cos ;■), 'A ' . . , K2 tan2r + tan2£, r, \ and tan 6 = ---------;— tan ( --7- k cos r) ■ 2 K tan r tan v A/ ' From these expressions we see that, as before, there is no reflected wave when h is very small compared with A/ and when h cos r is a multiple of A//2 ; these results are the same whether the Faraday tubes are in or at right angles to the plane of incidence. We see now, however, that in addition to this the reflected wave vanishes, whatever the thickness of the plate, when K tanr = tan i, or since Vn'K sin r = sin i where // is the magnetic permeability, the reflected wave vanishes when tan2i ylK-1 ’ if the plate is non-magnetic /x'= 1, and we have tan i = */K. When K tanr = tani the reflected wave and one of the waves in the plate vanish; the electromotive intensity in the other wave in the plate is equal to cos cosr + 2/sinr+ V't), and the emergent wave is cos 27r X ((x + h) cosi + y sini+ Vt — hX. \ -r cos r) • A ' The intensity of all these waves are independent of the thickness of the plate. If the plate is infinitely thick we must put 5'= 0 in equations (7); doing this we find from these equations that . (Ktanr—tani) ~~ K tan r + tan i 9 B = A sin 2 i sin i cos r -f K cos i sin r414 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [351. If the plate is made of a non-magnetic material K = sin2i/sin2r, and in this case we have d/=dtan(*-r) tan (i + r), 5 = 4^1-^ sm r cos % sm 2z + sm 2r Reflection from a Metal Plate. 351.] The very important case when the plate is made of a metal instead of an insulator can be solved in a similar way. The expressions for the electromotive intensities in the various media will be of the same type as before; in the case of metallic reflection however the quantity a\ which occurs in the expression for the electromotive intensity in the plate, will no longer be real. In a conductor whose specific resistance is 0} we seethat a,2 + b2=-iit^up/a-. (9) To compare the magnitude of the terms in this equation, let us suppose that we are dealing with a wave whose wave length is 10? centimetres. Then since 2n/p is the time of a vibration, if V is the velocity of propagation of electromagnetic action in air, V2tt/p = A, but V is equal to 3 x 1010, hence p = 6tt1010-*. If the plate is made of zinc xA C = B'= o. Hence in this case there is complete reflection from the metal plate, and since A* + A = 0 we see that the electromotive in- tensity vanishes at the surface of the plate, and since (7=0 there is no electromotive intensity on the far side of the plate.416 EXPERIMENTS ON ELECTROMAGNETIC WAVES, [351, The condition that the plate should act as a perfect reflector or, which is the same thing, as a perfect screen, is that {litjiph* 1 2/#} - should be large. In the case of zinc plates the value of this quantity for vibrations whose wave length is 103 * * * * * centimetres is equal to 1*5 x 104-5/2 A, so that for waves 1 metre long it is equal to 1500A; thus, if h were as great as TV of a millimetre, a'h would be equal to 10, and since e10 is very large the reflection in this case would be practically perfect. We see from this result the reason why gold-leaf and tin-foil are able to reflect these very rapid oscillations almost completely. If however the conductor is an electrolyte cr may be of the order 1010, so that ah will now be only 1-5 h for waves 1 metre in length, in this case it will require a slab of electrolyte several millimetres in thick- ness to produce complete reflection. We shall consider a little more fully the wave emergent from the metallic plate. We have by equations (10) 4 Aaa'elha C = M {(aj^+a2) (e1 ,ut' - e ~ ‘ha') + (2 aa'/p) («‘/,u' + « “‘hn') } C = If ha' is very small this may be written 2 Aaa' fji, {(a'2//x2 -f a2) ha\ + (2 aa'/}x)} ’ or, since a'2//x2 is very large compared with a2, A€lha G = 1 + ~ + 21M ha 2fxa Aelha 1 + (2 TrVh/(r)+ ^Lfjiha Thus, corresponding to the incident wave 2 7r / cos — (x+ Vt), A we have, since A a is very small, an emergent wave where A _ J . Since V is equal to 3x1010 and o(l= From these equations we find 1 — A = a pa (3*) 1 + fxa Let us for the present confine our attention to the non- magnetic metals for which n = 1, in this case the preceding equation becomes 1- ar A = a The expression given by Fresnel for the amplitude of the wave reflected from a transparent substance is of exactly the same form as this result, the only difference being that for a trans- parent substance o! is real, while in the case of metals it is complex.354*] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 419 Now for transparent substances the relation between a' and a is a ! + 62 a2 + b2 '2 = Ml where y! is the refractive index of the substance. In the case of metals however the relation between a' and a is a'2 + 62_ K a2 + b2~liK Kp# Since the magnetic force parallel to the surface is continuous, we have i + j/= g (5) The other boundary condition we shall employ is that Q, the tangential electromotive intensity parallel to the axis of y, is continuous. Now if g is the electric polarization parallel to y, and v the conduction current in the same direction, then in the dielectric above the metal . dg _ dy 77 dt dx* or since a2 + b2 4irp2 Q E e %420 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [355. by equation (2) we have * («2 + ^)q_ dy P dx In the metal or 4*! + 4„„=-!g, 4 7T\ ^ dy (^+V)9—i.. this by equation (3) becomes =-s; hence, since Q is continuous when x = 0, we have jxa' a a2+F2 (i-^) = (ct'2 + fc2) Equations (5) and (6) give A' = t a' a2 + &2 ** a ^+62 . , a' a? + b2 5 1+f* a a'2 + 62 (6) which is again, for non-magnetic metals for which fi— 1, of the same form as Fresnel’s expression for the amplitude of the reflected wave from a transparent substance. So that in this case, as in the previous one, we see that we can get the results of this theory of metallic reflection by substituting in Fresnel’s expression a complex quantity for the refractive index. 355.] This result leads to a difficulty similar to the one which was pointed out by Lord Rayleigh (Phil. Mag. [4], 43, p. 321, 1872) in the theory of metallic reflection on the elastic solid theory of light. The result of substituting in Fresnel’s expressions a com- plex quantity for the refractive index has been compared with the result of experiments on metallic reflection by Eisenlohr (Pogg. Ann. 104, p. 368, 1858) and Drude (Wied. Ann. 39, p. 481,1890). The latter writer finds that if the real part of i?2e2 hence near the grating where x/k is small jtt | Tjy . 2 7rT7., C a 2ttX a 2 7r , x Ex + E2 = sin — Fit 3 — A—------A — (x + a) A ( A A 27rar ^ _4wa- ^ -J-Clog^l —2e a cos-^ + e ' a )rj and we see by Maxwell’s result that the quantity inside the bracket has a constant value over the surface of the wires ; hence, if we make this value zero, we shall have satisfied the conditions of the problem. Let 2 c be the diameter of any one of the wires in the plane of the grating, then when x = 0 and y = c the expression inside the bracket must vanish, hence — A— a + C log 4 sm2 — = 0. A a To find another relation between A} C, and a we must consider the equation to the cross-section of the wire at the origin, viz., 2irx iirx — A -^(2aj + a) + Cflog (l — 2e “ cos-^ + e « ) = 0, or substituting for C its value in terms of A, 2ttx 4irx + l)log|4sin2^| = log(l — 2e “ cos-^ + e “ )• (1) If d is the value of x when y = 0, 4 irx 2tt a = 2 d- i « . VC log 2 sm — 5 a log 2ird a (2) 2 sin TTC When c = d, this equation becomes, since c/a is small, 2a, . ire a =-------log 2sm------ it ° a The expression for E2 consists of two parts, one of which is t2ir,T^ , — (!«-(* + <»)) — K .428 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [361. which represents a reflected wave equal in intensity to the inci- dent one, but whose phase is changed by reflection by (£A — a), where a is given by (2) and depends upon the size of the wires and their distance apart. The other part of the expression for E> is 2tzy 2itx C log (l — 2 e * cos ———p e 4 TTX a This is inappreciable at a distance from the grating 4 or 5 times the distance between the wires, hence the reflection, at some distance from the grating, is the same, except for the alteration in phase as from a continuous metallic surface. 360.] If the electromotive intensity had been at right angles to the wires the reflection would have been very small; thus a grating of this kind will act like a polariscope, changing either by reflection or transmission an unpolarised set of electrical vibrations into a polarised one. When used to produce polarisa- tion by transmission we may regard it as the electrical analogue of a plate of tourmaline crystal. Scattering of Electromagnetic Waves by a Metallic Wire. 361.] The scattering produced when a train of plane electro- magnetic waves impinges on an infinitely long metal cylinder, whose axis is at right angles to the direction of propagation of the waves and whose diameter is small compared with the wave length, can easily be found as follows:— We shall begin with the case where the electromotive intensity in the incident wave is parallel to the axis of the cylinder, which we take as the axis of 0; the axis of x being at right angles to the fronts of the incident waves. Let A be the wave length, then Ev the electromotive intensity in the incident waves, may be represented by the equation i2ir E1 = {Vt + x) where the real part of the right-hand side is to be taken. The positive direction of x is opposite to that in which the waves are travelling. In the neighbourhood of the cylinder x/X is small, so that we may put362.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 429 approximately, or if r and 0 are the polar coordinates of the point where the intensity is Ev ihvt Ex — ex * 7r (^1 + i—rcos ey Let E2 be the electromotive intensity due to the currents induced in the cylinder, then E2 satisfies the differential equa- tion d2E2 . 1 dE9 . 1 d2E2 _ 1 d2E, dr2 * r dr + 73. d$2 V2 dt2 47T2 or if Eo varies as cos nd, X2 E* d2E2 , I dE^ , (t]? __VL\jr - n dr2 r dr ' X2 r2' 2 The solution of which outside the cylinder is 9 nr l^VYt E2 = An cos n0Kn (—r) e , where Kn represents the ‘ external ’ Bessel’s function of the nxh order. Thus E* = |^0Z0 r) + A, cos 0K, i^-r) + A2cos 20K2(^r) +... j eFt. Now since the cylinder is a good conductor, the total tangential electromotive intensity must vanish over its surface, see Arts. 300 and 301. Hence if c is the radius of the cylinder, Ex + E2 = 0 when r = c; from this condition we get 362.] Let us first consider the effect of the cylinder on the lines of magnetic force in its neighbourhood. If a, /3 are the430 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [362. components of the magnetic force parallel to the axes of x and y respectively, E the total electromotive intensity, then dE d(3 2 7r -ttq dx ~~ dt ~~L k ™ dE da 21t Jr ty=z~dt=~L TFa‘ Thus the direction of the magnetic force will be tangential to the curves over which E is constant, the equations to the lines of magnetic force in the neighbourhood of the cylinder are there- fore where C is independent of r and 0. Now 2ttc/\ is by hypothesis very small, and when x is small then,by Art. 261, the values of K0 and Kx are given approximately by the equations ^{x) = iog{%y/x)t where y is Euler’s constant and log y is equal to *5772157. In the neighbourhood of the cylinder r/k is small as well as c/A, so that in this region the equations to the lines of magnetic force are, approximately, iog (Vc) log (yA/uc) 2ir17. 2tt a cos —— vt + —— COS $ k k (c2-r2) T sin — v t = C. A In this expression the coefficient of cos(2 7tF^/A) is very large compared with that of sin (2 tt Vt/k), so that unless 2 irVt/k is an odd multiple of tt/2, that is, unless the intensity in the incident wave at the axis of the cylinder vanishes, the equations to the lines of magnetic force are log (c/r) = a constant, so that these lines are circles concentric with the cylinders. When 2*rt/k is an odd multiple of tt/2, the lines of magnetic force are given by the equation ,„£=!3 = c, COS I r431 364.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. or in Cartesian coordinates x {c2—(x2 + y2)} = <7(a2 + 2/2); these curves are shown in Fig. 121. 363.] Since the direction of motion of the Faraday tubes is at right angles to themselves and to the magnetic force, when the lines of magnetic force near the cylinder are circles, these tubes will, in the neighbourhood of the cylinder, move radially, the positive tubes (i.e. those parallel to the tubes in the incident wave) moving inwards, the negative ones outwards. In the special case where the electromotive intensity vanishes at the axis of the cylinder, the incident wave throws tubes of one sign into the half of the cylinder in front, where x is positive, and tubes of opposite sign into the half in the rear, where x is negative; in this case, if the positive tubes in the neighbourhood of the cylinder are moving radially inwards in front, they are moving radially outwards in the rear and vice versa; there are in this case but few tubes near the equatorial plane, and the motion of these is no longer radial. 364.] When the distance from the cylinder is large compared with the wave length, we have *‘§ — i2irr/\ (r/A)^ i2irr/\ (r/A)*432 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [365. Thus in the wave ‘ scattered ’ by the cylinder „ .-¥(-”■?)< 1 , 2 2{r/\)* (log (y?rA/c) + X2 C0S ) Thus in this case, as we should expect, the part of the scattered wave which is independent of the azimuth is very much larger than the part which varies with 0, so that there is no direction in which the intensity of the scattered light vanishes. In this respect the metal cylinder resembles one made of a non-conductor, the effect of which on a train of waves has been investigated by Lord Rayleigh [Phil. Mag. [5], 12, p. 98, 1881): there are however some important differences between the two cases; in the first place we see that since c occurs in the leading term only as a logarithm, the amount of light scattered by the cylinder changes very slowly with the dimensions of the cylinder, while in the light scattered from a dielectric cylinder the electromotive in- tensity in the scattered wave is proportional to the area of the cross-section of the cylinder. Again, when the cylinder is a good conductor the electromotive intensity in the scattered wave, if we regard the logarithmic term as approximately constant, varies as A* and so increases with the wave length, while when the cylinder is an insulator the electromotive intensity varies as A“*, so that the scattering decreases rapidly as the length of the wave increases. The most interesting case of this kind is when the wave incident on the cylinder is a wave of light; in this case the theory indicates that the light scattered by the metallic cylinder would be slightly reddish, while that from the insulating cylinder would be distinctly blue ; the blue in the latter case would be much more decided than the red of the previous one, since the variation of the intensity of the scattered light with the wave length is much more rapid when the cylinder is an insulator than when it is a good conductor. 365.] We shall now proceed to consider the case when the electromotive intensity in the incident wave is at right angles to the axis of the cylinder. This case is of more interest than the preceding because the general features of the results obtained will apply to the scattering of light by particles limited in every direction; it is thus representative of the scattering by small particles in general, while the peculiarities of the case discussed365.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 433 in the preceding article were due to the cylindrical shape of the obstacle. The only case to which the results of this article would not be applicable without further investigation is that in which the particles are highly magnetic, and we shall find that even this case constitutes no exception since our results do not involve the magnetic permeability of the cylinder. As the electromotive intensity is at right angles to the axis of the cylinder, the magnetic force will be parallel to the axis. Let the magnetic force in the incident wave be expressed by the equation *_2rr, -CVt+x) When x which is equal to r cos 0 is small compared with A, this is approximately 12 7T. vt b-i 7 “f- l2tt x rcos0 — —r2 cos 29 k k1 s- Since H, the magnetic force, satisfies the differential equation d2H d2H _ 1 d2H dar2 + dy% ~ V2 dt2 ’ the magnetic force due to the currents induced in the cylinder may be expressed by the equation 12 tr #j = e A Vt^A()K0(^-r)+A1cos6Kl(^-r)+A.lc,o%2eK2i^Yr^> where A0i AY and A2 are arbitrary constants. The condition to be satisfied at the boundary of the cylinder is that the tangential electromotive intensity at its surface should vanish. In this case we have, however, ~(H1 + H2) = 4 7r (intensity of current at right angles to v). The current in the dielectric is a polarization current, and if E is the tangential electromotive intensity, the intensity of this current at right angles to r is which is equal to K_dE 4 7T dt K t 2 7T 4 7T k VE. Thus the condition that E should vanish at the surface is f f434 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [366. equivalent to the condition that s (*>+*»-° when r = c9 c being the radius of the cylinder. From this condition we get -2cX®+'4odcff°( A ) °’ Since 2 irc/A. is very small and therefore approximately *o(^c) = log(2y/^), „ ,2ir \ A 1 'X ' 2%c ir /2ir A _ KA\C)~ 2**<*' we get A) = - 2 ^2 3 A = 42 = ~2^. Thus the magnetic force due to the currents induced in the cylinder is given by the equation _ oC2 T- ,2 TT \ rr /2ir \ 7T2C2 /2?r \) = 2tt2^€ * |-Z0(—r)+2icos^Z1(—r)-^2-cos20Z2(Yr)j 366.] To draw the lines of electromotive intensity, we notice that if ds is an element of a curve in the dielectric, d [Hx + H2)/ds is proportional to the electromotive intensity at right angles to ds} so that the lines of electromotive intensity will be the ^nes Hx + U2 = a constant.435 366.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. When r/A is small, this condition leads to the equation ‘-^rt : A. 2w2c2 A2 Tr ,2 TT \ 2 L7T ■^°vTr) +“cos0|r+ 2 7r2c4 2 7T 7T2 — -oCOS20 X2 jr2 A2 X-*. this becomes i2ir Vt [l + ^cos«— — ^cos20( Fig. 122. i 2ir Yt/X Except when e * is wholly real, i.e. except when the rate of variation of the magnetic force in the incident wave at the axis of the cylinder vanishes, by far the most important term is that which contains cos 0, so that the equations to the lines of electro- motive intensity are > + r2 cos 0 = a constant = C\ say. The lines of electromotive intensity are represented in Fig. 122. At the times when e<27r^A £s wholly real, the lines are ap- proximately circles concentric with the cross-section of the cylinder, since in this case the term involving the logarithm is the most important of the variable terms. F f 2436 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [368. 367.] When r is large compared with A, we find by intro- ducing the values of the K functions when the argument is very large, viz. *.(») = H2 = -~t 2 2„2 ~) 8/ (1 + 2 cos 0), retaining only the lowest powers of c/\. Thus the magnetic force in the scattered wave vanishes when 2 cos 0 = — 1, or in a direction making an angle of 120° with the incident ray. When the wave is scattered by an insulating cylinder Lord Rayleigh (1. c.) found that the magnetic intensity in the scattered ray was expressed by a similar formula with the exception that the factor (1+2 cos 0) was replaced by cos 0. Thus, if we take the case where the incident wave is a luminous one, the scattered light will vanish in the direction of the electric displacement when the particles are insulators, while it will vanish in a direction making an angle of 30° with this direction if the particles are metallic. If the incident light is not polarized, then with metallic particles the scattered light will be completely polarized in a direction making 120° with the direction of propagation of the incident light, while if the par- ticles are insulators the direction in which the polarization is complete is at right angles to the direction of the incident light. The observations of Tyndall, Briicke, Stokes, and Lord Rayleigh afford abundant proof of the truth of the last statement: but no experiments seem to have been published on the results of the reflection of light from small metallic particles. 368.] The preceding results have also an important application to the consideration of the influence of the size of the reflector on the intensity of reflected electromagnetic waves. When the electromotive intensity is parallel to the axis of the cylinder, the most important term in the expression for the reflected wave only involves the radius of the cylinder as a logarithm, it will thus only vary slowly with the radius, so that in this case the size of the cylinder is of comparatively little importance: hence we may conclude that we shall get good reflection if the length369.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 437 of the reflector measured in the direction of the electromotive intensity is considerable, whatever may be the breadth of the reflector at right angles to the electromotive intensity. On the other hand, when the electromotive intensity is at right angles to the axis of the cylinder, the electromotive intensity in the scattered wave increases as the square of the radius of the cylinder, so that in this case the size of the reflector is all im- portant. These results are confirmed by Trouton’s experiments on ‘The Influence the Size of the Reflector exerts in Hertz’s Experiment/ Phil. Mag. [5], 32, p. 80,1891. (i) On the Scattering of Electric Waves by Metallic Spheres. 369.] We shall proceed to discuss in some detail the problem of the incidence of a plane electric wave upon a metal sphere *. If a, /3, y; f g, h are respectively the components of the magnetic force and of the polarization in the dielectric which are radiated from the sphere, then if \f/ stands for any one of these quantities it satisfies a differential equation of the form d2\jr d2\jr d2\jf __ 1 d2\j/ dx2 + dy2 + dz2 ~~ V2 dt2 where V is the velocity with which electric action is propagated through the dielectric surrounding the sphere. If A is the wave length of the disturbance incident upon the sphere, then the components of magnetic induction and of electric polarization 'll vt will all vary as e * ; thus V~2d2\f//dt2 may be replaced by — 4'7t2^/A2, so that writing k for 2 7r/A, equation (1) may be written ^f+tt + ^±+k2^ = 0 da? dy2 dz2 ^ ’ a solution of which is by Art. 308, f = e‘*VtI,fn(]cr)Sn, where r is the distance from the centre of the sphere. Since the waves of magnetic force and dielectric polarization are radiating outwards from the sphere '■<*>-(ca^TT- * The scattering by an insulating sphere is discussed by Lord Rayleigh {Phil. Mag. 12, p. 98,1881). The incidence of a plane wave on a sphere was the subject of a disserta- tion sent in to Trinity College, Cambridge, by Professor Michell in 1890. I do not know of any papers which discuss the special problem of the scattering by metal spheres.438 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [37O. Sn is a solid spherical harmonic of degree n. It should be noted that/n (Jcr) of this article is (kr)~nf(kr) of article 308. 370.] We shall now prove a theorem due to Professor Lamb (Proc. Lond. Math. Soc. 13, p. 189, 1881), that if a, ft, y satisfy equations of the form (1), and if ^ dS dy . dx *1" dy dz 9 then the most general solution of these equations is given by a = 2 }(»+l)/_, (JT)^g> 7 = S {(» +1 )/~.M ^ “*/« MI ^TI} where con, a/n represent arbitrary solid spherical harmonics of degree n. Since da>n d 'n vanish identically, and since v2K) = o, v2^ = o, we have dt dx "r dy —S[n^7an*zf'n+1(kr)+n(2n + 3) k2r2n+1fH+i(krJ\ x ( d d d) o>„ *-r + !/r+*j-r la dp dy ,, . ( d d , d 1 tee + &!1 + dz ~ S^+ ^i^) \xdjc + y dy + Zdz\'*n {■ dy dz) r2**1 = SW.TO+ i.~\f»-i{kr) + leir2fn+1(kr) + (2n + 3)krfni.l(kr)}a)K.370.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. NoW f (kr) = (~-d e~‘*r Jn\ ) \]er+ l)~f'n-i(kr)-nk3rf'n+1(kr)-n(2n+3)lc2f„+l(kr)^ x dz dy ) and this by equations (4) and (6) is equal to Let us now consider the term in 4 tt df/dt involving a>tt'; this equals . /7 x d / d d d \ dl , Mh’)\~2dx ^Xdx + ydy + Zdz^dx)u>n +fn(kr)k(x^-r^)/- M £ jSv,] 372.] In the plane electrical wave incident on the sphere, let us suppose that the electric polarization hQ in the wave front is parallel to 0 and expressed by the equation 12 IT A0 = eT-(Fi+a:)==€^^ + -)) where the axis of x is at right angles to the wave front. We have to expand h0 in the form t‘*rt2AnQn, where Qn is a zonal harmonic of degree n whose axis is the axis of x and An is a function of r which we have to determine. Since = 24bQb, and since it satisfies the equation ^ dV dx2 + dy2 + dz2 + * “ ’ and is finite when r = 0, we see by Art. 308 that A: = A:S,{kr) = AWry{^p”£r where An' is independent of r. Since , ___ kr 3! 5! we see that when kr is very small (kr)n sin&r „ k2r2 k*iA = 1--TT- + Jb = (-1)Mb' (2%+l)(2n-l)...l (9)442 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [372. But if x/r = fj., we have -ikrp_ y a n 1::^-+£>*,*,=^ • The lowest power of kr on the left-hand side of this equation is the nth, the coefficient of this is equal to 2tn ^/_+Vq,.^ = (271 + 1) (274—1) (274 — 3)...l ’ hence when kr is small we have 2 4W (kr)n _ 2 An (274 + 1)(2t4—l).,.l “ 2 74 +1 ’ Comparing this equation with (9) we see that _ (271+1) n ~ H » A„ 2n +1 &(H so that gt fcrfi __ 2 271+1 This expression is given by Lord Rayleigh (Theory of Sound, ii. p. 239). By equation (101) of Art. 308 we have -^»-i ^n+l _ j ^ 274 — 1 271+3 lir n* This can also be proved directly thus, (10) _A 2 71- A ^n+X _,r+1 271 + 3 */_ i r *«**•/* "1 + 1 ii mH t Sr- 'S 1 M Q»+l)J ^ ^ i&7\ f + 1ftkrp (d_Q«z l—i v <2#* d/jL The terms within square brackets vanish, and since O'Qn-l dQn+1 d/A d\x = -(27l + l)Qn373-] EXPEEIMENTS ON ELECTEOMAGNETIC WAVES. 443 we have A An—\ 2n—l 2n+3 w + l = A. ikr 373.] It will be convenient to collect together the results we have obtained In the incident wave, /o = 0, g0 = 0, A0 = €‘*rt2^+lQMkr)] and therefore by Art. 9, ao = 0, y0 = 0, /30=4vk0V=47TVf‘kVtS^^QnSn(kr). For the wave scattered by the sphere, omitting the time factor, we have since d/dt—ihV + 2(2m+ l)»f.{hr) (jfjj5- <’‘krg = 2 ^L-t j („ + !)/._, (b-)^ - »*»r ✓ d (jl>„ do&~\ + 2 (2 »+ 1) **/. (*r) (« ~ « > = 2 2-A {(»+ 1)/U(fcr) ^■~'nk2rin+3fn+i(kr)^ + 2 (2 n+ 1) **/. (hr) y~^) • « =2 {(»+ l)/«(*r)^ /. X / dto/jiV ^=s{(»+!)/._, r = s {(» + !)/«(b)^-nbr«-,)/„1(b)^444 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [374. 374.] To determine that along a parallel of latitude, then the condition is equivalent to df dg dh dx dy dz d_ drv (r*R) + d-r~(r sin<9©) + ' 7 sin 0a0y ' 1___d_ sin# d(j> (r vanish all over the sphere, this, if a is the radius of the sphere, gives the condition r2R) = 0 when r = a. Now rR = x (/+/0) + y(g + g())+z (h + K); but 4^kV(xf+yg + zk) + = —'2n.n+ 1 .fn(kr) cow', by equation (6), ®/o + m + s*o — z ^AnQn, omitting the time factor. But if r, 0, cj> are the polar coordinates of the point whose Cartesian coordinates are x, y, z, and z — r sin 6 sin <£, 0 — 1 {dQn + l 2n+ 1 ( dfx dQn-1) . dp Y hence, if u>M' = rw Ynf where Yn' is a surface harmonic of degree n, the condition r2R) = 0 when r = a becomes 4UTKV 2»• (» +1) r/^(an+lfn(ka)) = sin 6 sin 2Qn~ (a2An) = sm 0 sm 2 — <-—^ da da *2n — 1 «2^w+l] . 2th- 3) ’376.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 445 but sin 0 sin is a surface harmonic of degrees n, hence dp Y' = -*• 41 “ or by (10) F/ = 4 7T F sin 0 sin d (ctAn) da J n.n+ l.fikC ' dxV r dg. / /sinOsin i-(aSn(ha)) — 2 2%+1 8«(ha) * n \ ( d d \ . dQn\ —:—-—— • -r7T\ fn vcr) \Vj-----«-j-) (r*sm0 cos^-j—)» i" a”, ra. n +1. f,t(ha) jnK ’ ^ dz dy>\ ^ dy.J g = 2---------- \ \{n +1 )/„_! (fcr) (r” sin 0 sin ^5) J n.n + 1 1 “t&r " 1V cZ r sin ( c?/ut J | da -»^^n+3/«+i (*»•) ^ \—^+1— / j -j- 2 n +1 _ Sn(ha) f d ^ a dQn d rf- -2 + 1 ^pfn{kr) (* “ -«“) (r«sin 0 cos <*>^), tna“.%.w+l. f„(ha)J“v da: v rf/u.^ * = * i{f**+ 1Wj (**s“8»ta* , /sin 0 sin ^-(aSJha)) nlA^Zf “ “ 4 ’ F2 drTT. .4 f$5? {(»+> )/.-i(b-) s C1"s" 9 ^ , /sin 0 cos

are small, we have approxi- mately . m (1 m2 , m2x) B _ (mi~mz)m\1 + Mi + MVS z” o-£)*■* =k-^) _ 2(m1—m.2)m ~ M3h since m/M is small for metals. Substituting the value of m1—ma from equation (5) we get, putting M — Rt‘a m, B _ pK'k'c0 1 2-n mR2c2,ah’ the rotation of the plane of polarization is equal to the real part of B/A, and hence to K'k'c0 p cos 2 a 2 7r mh R2 Since this is proportional to l/h we see that the rotation increases as the thickness of the plate diminishes. The expla-508 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [416. nation of this is that while the intensities of the two components reflected light, viz. the component polarized in the same plane as the incident wave and the component polarized in the plane at right angles to this, both diminish as the thickness of the plate diminishes; the first component diminishes much more rapidly than the second; thus the ratio of the second component to the first and therefore the angle of rotation of the plane of polari- zation increases as the thickness of the plate diminishes. 416.] The effect of a magnetic field in producing rotation of the plane of polarization thus seems to afford strong evidence of the existence of a transverse electromotive intensity in a con- ductor placed in a magnetic field, this intensity being quite distinct from that discovered by Hall, inasmuch as the former is proportional to the rate of variation of the electromotive in- tensity, whereas the Hall effect is proportional to the electromotive intensity itself. We shall now endeavour to form some estimate of the magnitude of this transverse intensity revealed to us by optical phenomena. Kundt (Wied. Ann. 23, p. 238, 1884) found from his ex- periments that if 4>j the rotation of the plane of polarization produced by the passage of light of wave length A through a magnetized plate of thickness h, is given by an equation of the form -h then = 1°. 48' when A = 5.8 x 10~5, and A = 5.5xl0“6, thus n—n'= *1. But we have seen that the rotation in this case is equal to pK'k'c0 7rh # 27t A ’ hence, comparing this with Kundt’s result, we find ~PK'k'c0_ t 2tr 5 but if A = 5.8x10“5, p = 2 7r x 3 x 1010 x 105/5.8 = 3.2 x 1015. Substituting these values, we find K'k' 0 , 1A 17 — cb=8.lx10-«4i 6.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 509 Now if / is the electric polarization parallel to x, the transverse electromotive intensity is equal to = V^pc,X, where X is the electromotive intensity parallel to x. Hence kfK'pc0/4:7r is the ratio of the magnitude of the transverse intensity to that producing the current; this ratio is for iron therefore equal to 1 6 x 10 ~17p for magnetic fields of the strength used by Kundt. The factor multiplying p is so small as to make it probable that the effects of this transverse force are insensible except when the electro- motive intensity is changing with a rapidity comparable with the rate of change in light waves, in other words, that it is only in optical phenomena that this transverse electromotive intensity produces any measurable effect.CHAPTER VI. THE DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 417. ] Problems concerning alternating currents have become in recent years of much greater importance than they were at the time when Maxwell’s Treatise was published ; this is due to the extensive use of such currents for electric lighting, and to the important part which the much more rapidly oscillating currents produced by the discharge of Leyden jars now play in elec- trical researches. It is therefore desirable to consider more fully than is done in the Electricity and Magnetism the appli- cation of Maxwell’s principles to such currents. In doing this we shall follow the methods used by Lord Rayleigh in his papers on £ The Reaction upon the Driving-Point of a System executing Forced Harmonic Oscillations of various Periods, with Applica- tions to Electricity,’ Phil. Mag. [5], 21, p. 369, 1886, and on ‘ The Sensitiveness of the Bridge Method in its Application to Periodic Electric Currents,’ Proc. Roy. Soc. 49, p. 203, 1891. 418. ] When the currents are steady their distribution among a net-work of conductors is determined by the condition that the rate of heat production must be a minimum, see Maxwell’s Electricity and Magnetism, vol. i. p. 408. Thus, if F is the Dissipation Function [Electricity and Magnetism, vol. i. p. 408), xx, #2, #3... the currents flowing through the circuits, these variables being chosen so that they are sufficient but not more than sufficient to determine the currents flowing through each branch of the net-work, then xu sb2, &c. are determined by the equations dF __dF _ dF dxx dx2 dx 3 = 0.DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 511 When, however, the currents are variable these equations are no longer true ; we have instead of them the equations d_d/F_ dJ__dV _ dt dxx dxx dxx ’ where T is the Kinetic Energy due to the Self and Mutual in- duction of the circuits, F as before is the Dissipation Function, and V is the Potential Energy arising from the charges that may be in any condensers in the system. If the currents are periodic and proportional to eipt, the pre- ceding equation may be written as dT_ dF _dV _ dxx dxx dxx — ^ ’ and thus when p increases indefinitely the preceding equation approximates to dxx = 0 we have similarly d2__dT__ dx2~ dxs Thus in this case tbe distribution of currents is independent of the resistances, and is determined by the condition that the Kinetic Energy and not the Dissipation Function is a minimum. 419.] We have already considered several instances of this effect. Thus, when a rapidly alternating current travels along a wire, the currents fly to the outside of the wire, since by doing this the mean distance between the parts of the current is a maximum and the Kinetic Energy therefore a minimum. Again, when two currents in opposite directions flow through two parallel plates the currents congregate on the adjacent surfaces of the plates, since by so doing the average distance between the opposite currents, and therefore the Kinetic Energy, is a minimum. Mr. G. F. C. Searle has devised an experiment which shows this tendency of the currents in a very striking way. AB, Fig. 123, is an exhausted tube through which the periodic currents produced by the discharge of a Leyden jar are sent. When none of the wires leading from the jar to the tube passes parallel to it512 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [42O. in its neighbourhood, the glow produced by the currents fills the tube uniformly. When however one of the leads is bent, as in Fig. 137, so as to pass near the tube in such a way that the current through the lead is in the opposite direction to that through the Fig. 137. tube, the glow no longer fills the tube but concentrates itself on the side of the tube next the wire, thus getting as near as possible to the current in the opposite direction through the wire. When however the wire is bent, as in Fig. 138, so that the current through the lead is in the same direction as that through the f-G ) <— ---------------<—N Fig. 138. tube, the glow flies to the part of the tube most remote from the wire. 420.] We shall now proceed to consider the distribution of alternating currents among various systems of conductors. The first case we shall consider is the distribution of an alternating current between two conductors ACB, ADB in parallel. Let the resistance and self-induction in the arm ACB be respectively C 22, L, the corresponding quantities in the arm ADB being denoted by JST, and let M be the coefficient of mutual induction between the circuits ACB, ADB. We shall suppose that the rate of alternation of the current is not so rapid as to produce any420.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 513 appreciable variation in the intensity of the current from one end of ACB or ADB to the other, in other words, that the wave length corresponding to the rate of alternation of the current is large compared with the length ACB or ADB; the case when this wave length is comparable with the length of the circuit is considered separately in Art. 298. Let the current flowing in along OA and out along BP be denoted by x ; we shall assume that x varies as eipt. Let the current in ACB be y, that in ADB will be x-i/. Then Ty the Kinetic Energy in the branch ACDB of the circuit, is expressed by the equation 27= h{Lf + 2M(x-y)y + N(x--y)2}. The dissipation function F is given by and we have F=\{R? + S{x-y?l d_ dT dF_ dt dy ^ dy ~~ * or {L + N-2M)^+{R + S)y-{N-M)^-Sdi= 0. Let as = f‘P\ then from this equation we have _ (N-M)lP + S t y~ (L + N-2M)lP+(R + 8) ’ or, taking the real part of this, corresponding to the current cos pt along OA, we find {S(R + £r) + (L + N-2M) coapt-p {B(N-M)-S {L-M)} y (£ + iV-2Jtf)V + (.R + S)a ’ (’ . = {X(S + R) + (L + N-2M) (L-M)pt} cospt+p S (l-M)} Anpt s (L+jf-zuypi+iit+sf ‘ *-) These expressions may be written in the forms . ( 8i + {N~M)2pi )i , , , , „ / 4. t \ *=\(L+lf-2M)Vi(Ii+S)A ««(?><= (?<+<>, »y, . . ( IP+lL-Mff ll , , , ,, „ , t , ,, *-»“ {(Z'+iT-iJOV-KiB + sH + + where x_ p\R{N-M)-S{L-M)} S(R + S) + (L + N-2M)(.N-M)p2’ p{R{N-M)-S{L-M)} R(R + 8) + (L + N'-2M)(L-M)p2' L 1 and tan e'=514 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [42O. The maximum currents through AC3, ABB are proportional to A and B, and we see from the preceding equations that A B_________ {S2 + (N-M)2p2}± ~ \R2 + (L-M)2p2)±' When p is very large, this equation becomes A _ B N-M ~~ X-if ’ so that in this case the distribution of the currents is governed entirely by the induction in the circuits, and not at all by their resistances. Referring to equations (1) and (2) we see that when p is infinite N_M * = LVN-2M C0s^’ . . Z-Jf *-* = L-Hf-IM™?1' An inspection of these equations leads to the interesting result that when the alternations are very rapid the maximum current in one or both of the branches may be greater than that in the leads. Consider the case when the two circuits ACB, ABB are wound close together. Suppose, for example, that they are parts of a circular coil, and that there are m turns in the circuit ACB, and n turns in ABB, then if the coils are close together we may L = Km2, Ml = Knmt K = Kn2, where if is a constant. Substituting these values for X, if, N in equations (3) and (4) we find n* — nm y — 7z—cos^ = n z-y = (n — m)2 m2 — nm (n—m)2. n — m m cos pty n — m cos pt. (5) (6) Thus the currents are of opposite signs in the two coils, the current in the coil with the smallest number of turns flows in the same direction as the current in the leads. When n—m is very small both currents become large, being now much greater than the current in the leads whose maximum value was taken420.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 515 as unity; thus by introducing an alternating current of small intensity into a divided circuit, we can produce in the arms of this circuit currents of very much greater intensity. The reason of this becomes clear when we consider the energy in the loop, when the rate of alternation is exceedingly rapid. The effects of the inertia of the system become all important, and the distri- bution of currents is that which would result if we considered merely the Kinetic Energy of the system. In this case, in accordance with dynamical principles, the actual solution is that which makes the Kinetic Energy as small as possible consistent with the condition that the algebraical sum of the currents in ACB, ABB shall be equal to x. Thus, as the Kinetic Energy is to be as small as possible, and this energy is in the field around the loop and proportional at each place to the square of the magnetic force, the currents will distribute themselves in the wires so as to neutralize as much as possible each other s magnetic effect. Thus if the wires are wound close together the currents will flow in opposite directions, the branch having the smallest number of turns having the largest current, so as to be on equal terms as far as magnetic force is concerned with the branch with the larger number of turns. In fact we see from equations (5) and (6) that the current in each branch is inversely proportional to the number of turns. If the two branches are exactly equal in all respects the current in each will be in the same direction, but this distribution will be unstable, the slightest difference of the coefficients of induction in the two branches being sufficient to make the current in the branch of least inductance flow in the direction of that in the leads, and the current in the other branch in the opposite direc- tion, the intensity in either branch at the same time increasing largely. When the currents are distributed in accordance with equa- tions (3) and (4), the Kinetic Energy in the loop is \ LN-M2 L + N—2 M p2 cos2 pt We notice that (LN—M2) / (L + N'— 2M) is always less than L or N. L + N—2M is always positive, since it is proportional to the Kinetic Energy in the loop when the currents are equal and opposite. l 1 2516 DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. [421. We see from equations (1) and (2) that when R(N-M) = 8(L-M), * = B^SC0Spt> *-y = i£rscoapt So that in this cas§ the distribution of alternating currents of any frequency is the same as when the currents are steady. 421.] We shall now consider the self-induction and resistance of the two wires in parallel. Let L0 and r be respectively the self-induction and resistance of the leads, and suppose that there is no mutual induction between the leads and the branches ACB, ABB. Then we have (L0 + N)^-(N-M)ft +(r + S)x—Sy = external electromotive force tending to increase x. Substituting in this expression the value of y in terms of x previously obtained in Art. 420, we find z -w- Ttr\ l(K-M)ip+S}2 . , . _ . jY-\f--7nOr\ -75-ci® + (T + $) x (Z + N—2M) + R + S v J = external electromotive force tending to increase x. Remembering that ipx = dx/dt, we see that the left-hand side of this equation may be written ( NR2 + LS2 + 2MRS+p2(LN-M2)(L + N-2M)} dx lL° + (R + S)*+p*(L + N-2Mf ) dt 5 R8(R + 8) + p2{R(F-M)2 + 8(L-Mf}) . + i + (R + Sy+p2{L + N-2Mf \X' From the form of this equation we see that the self-induction of the two wires in parallel is NR2 + LS2 + 2 MRS+p1 (.IN -M2) (L+ N-2M) (R + S)2+p2(L + N-2My which may be written as NR2 + LS2 + 2MRS (R+sy P‘2{L + N-2M) (R+sy +p2 (L+N—2 My {R{R--M)-S(L-M)y422.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 517 The impedance of the loop is RS(R + S)+p2{R (N-M)2 + S(L—M)2\ (R + S)2 + p2(L + N-2Mf which is equal to RS p2{R(F-M)-S{L-M)}2 R + S + (R + 8){{R + 8)* + p2{L + N-2M)2}' We see from the expression for the self-induction of the loop that it is greatest when p = 0, when its value is NR2 + 2MRS+LS2 (.R + Sf and least when p is infinite when it is equal to LN-M2 L + N-2M* If R(1T-M) = S(L-M), the self-induction of the loop is independent of the period. From the expression for the impedance of the loop we see that it is least when p = 0 when its value is RS R + S’ and greatest when p is infinite when it is equal to R(N-M)2 + S(L-M)2 m (L + N—2M)2 ’ and if R (JJT—Jf) = S (.L-M), the impedance is independent of the period. Thus in this case the self-induction and the impedance are unaltered, whatever the frequency of the currents. In all other cases the self-induction diminishes and the impedance increases as the frequency of the currents increases. 422.] We shall now proceed to investigate the general case when there are any number of wires in parallel. Let x0 be the current in the leads, xl9 x2i... xn the currents in the n wires in parallel; we shall assume, as before, that there is no induction between these wires and the leads. Let arr be the self-induction and ?v the resistance of the wire through which the current is #r, ar8 the coefficient of mutual induction between this wire and the wire through which the current is x8. Let a0 be the self- induction, r0 the resistance of the leads, E0 the electromotive force in the external circuit; we shall suppose that this varies as518 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [422. eipt. The current through the leads and those through the wires in parallel are connected by the relation #0— (^1 d“ #2 d“ • • • ®n) == ^ 5 we shall denote this by = 0 Then T being the Kinetic Energy, F the Dissipation function, and A an arbitrary multiplier, the equations determining the currents are of the form d dT dF ^d dt dx8 dx8 dx8 = external electromotive force tending to increase x8. From these equations we get (a0ip + r0)x0 + \ = («n lP + ri) + «i2 LPsbi + • a12 ipx1 + (a22 ip + r,) x2 + . alnipx1 + a2nipx2 +... Solving equations (8) we find ____________________________^2_______ ^11 + ^12+ -^-12 "h ^-^2 d* ••• A2n -^m + -42n+ • •• Avn A where A = «n 'P+ra &12 > • .. CLlnlP ®12 ^P > &22 W d" ^*2> •' »• ®/2nlP alnip , a2Mip -am<-P + rn and Apq denotes the minor of A corresponding to the constituent apq ip. Since x0 = x1 + sb2 + ..., we have from the above equations __________________= A An + Al2 + Ann + 2 Al2 + 2 An + 2 A2 3 + ... A Substituting this value of A in equation (7) we find d- ^0 d- ^ *^o = where S is written for An + A22 + ...Ann + 2A12 + 2Alz+2A2Z+.... E0> ..-A = 0,^ ..-A = 0 -A = 0. (0 («) (10)422.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 519 The self-induction and impedance of the leads can be deduced from (10); the expressions for them are however in general very complicated, but they take comparatively simple forms when ip is either very large or very small. When ip is very large, A 0 . r1(A'u + A\2+...A\nr + r2(A'12 + A'22 + ...A'2n)* + ... s~ipW + 8 '2 where a\\ > «l-2> Cf'in D = a22 > a2 n «»n and A!pq is the minor of D corresponding to the constituent apq, while >Sr= A'n + A'22 +... A'nn + 2 A\2 + 2 A\% + 2 A'23 +.... Thus the self-induction of the wires in parallel is in this case D 8'* while the impedance is {*1 iA\\ + A'12 + • • • A\«Y + r2 {A'n + A\l+-~ A\nf + • • .}/£'2. When ip is very small, (?* + ■ + ...+ 2 a 12 2 a 8 = cp 13 ri r2 V* + ...) / 1 1 lx2 (----h — 4*... —) r2 rj + 1 1 1 -----1------h ... — r. To rn So that in this case the self-induction of the wires in parallel is + + 2 a, r,rQ 12 ^ %ai3 _j_ r,Ta / 1 1 lx2 (- + — + ...-) and the resistance is 1 1 1 1 -----1------h ... — r, r2 rn When there is no induction between the wires in parallel,520 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [423. a12, + *«,... «i2 ‘^ + r12, a22ip + r23,... (H) corresponding to the constituent ap9 + rP9; we shn.11 denote the determinant by A.423.] DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. 521 Substituting the values of 12 If an electromotive force X2 of the same period as Xr acted on the second circuit, then the current xY induced in the first circuit would be given by Comparing these results we get Lord Rayleigh’s theorem, that when a periodic electromotive force F acts on a circuit A the current induced in another circuit B is the same in amplitude and phase as the current induced in A when an electromotive force equal in amplitude and phase to F acts on the circuit B. When there are only two circuits in the field, A n an + (q12tff + r12)2. *22 LP + ^22 ’ if the circuits are not in metallic connection r]2= 0, and we have ^ / P a22a it \ ^11 ~ ^22 P2 + ^22 LP + rU + W p-r^a,4 12 22" ^22 ^ + ^22 Thus the presence of the second circuit diminishes the self- induction of the first by p2 a22 a?i2 a222p2 + r222 while it increases the impedance by P2T22^12 d222pl + r\2522 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [424. These results were given by Maxwell in his paper‘A Dynamical Theory of the Electromagnetic Field ’ (Phil, Trans, 155, p. 459, 1865). We see from these expressions that the diminution in the self-induction and the increase in the impedance increase con- tinuously as the frequency of the electromotive force increases. 424.] Lord Rayleigh has shown that this result is true what- ever may be the number of circuits. We have by (12) w-x1 = Xv -°n Now while keeping xl the same we can choose #2, x3, &c., so that the two quadratic expressions «22 *22 + a33*32 + • • • 2 Ct23*2^3 + • • • ! r22a?2 + r^3 +... 2r23«2^3 +..., i.e. the expressions got by putting ^ = 0 in 2T and 2 F respect- ively, reduce to the sums of squares of #2, #3, &c.; when #2, #3, &c. are chosen in this way, ^23 = ^24 = apq ~ when p is not equal to q and both are greater than unity. In this case A = aui^ + rn, ®J2*i> + r12> a\3<-P + rlz, ••• %» lP + Tln a12li? + r12> ®22 + ^22> 0 , 0 «13‘i> + rl3» 0 a33 ‘i> + ^33. 0 aintp + fin, 0 0 ... a„Bip + rBB = («n + ^11) («22 lP + »*) • • • («»»<-P + rnn) x ( (aK ip + r12)2_______________(ayPP + rnf \ (a-n tp + rn) (a22ip + r22) (au ip + ru) (a33 ip+r33) («i»t_p + r1n)2 ) „ , , , , K^ + rii)K»‘i> + rBM)r Bn = (a22+ r22)... (aBB ip + rB„). Hence — = a ip + r (aulP + r^2 (ai3tJ> + y1?.)2 («intj> + rlB)2 •®11 U 11 a22ti5 + r22 aS3lP + '>'S3 (lnn ip -f rnn — 5ft -L V ___^aiMrl«r«n\ v / (,n» p" {ai„ ?'nB — i’-j nV2\) + — 2 LlS + 2 f P1 (ai» rn» ~ gm» rl «)2\ r« 1 M«2»Bp2+r2BB) ;•425.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 523 The coefficient of ip in the first line is the coefficient of self- induction of the first circuit,—we see that it is diminished by any increase in p; the second line is the impedance, and we see that this is increased by any increase in p. 425.] We shall now return to the general case. The reduction of A/Bn to the form Lip + R without any limitation as to the value of p would usually lead to very complicated expressions; we can, however, obtain without difficulty the values of L and iJ, (1) when p is very large, (2) when it is very small. When ip is very large we see that where "-An an, «I2-- a22 * • • a2 n • • . • . d2n • •« • ann and Au is the minor of D corresponding to the constituent au. If Apq denotes the minor of D corresponding to the constituent apv then we have by (11) A. = A. = _ A. Al -^-12 An Substituting these values of #2, #3, &c. in terms of x19 Dissipation Function, we find that (13) in the R— A 2 {^11-^11 "b ^22^122 “h • • • ^nn-^-ln2 4* “b 4* • •• } j ^11 we might of course have deduced this value directly from that of A/Bu. When ip is very small, we see by putting ip = 0 in A/Bn that where E-Rn ) rilJ ri2 •• •Tl* ri2> ^*22 •• • r2„ ^Iwj ^*2 » • • • rnn and JRn is the minor of G corresponding to the constituent rn; if Rpq denotes the minor of G corresponding to the constituent then we have by (11)524 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [427. Substituting these values of %ly #2, i?3,... in the expression for the Kinetic Energy, we see that L = 2 + a22Itl2 +... +...}. ■**11 426.] Suppose we have a series of circuits arranged so that each circuit acts by induction only on the two adjacent ones; this is expressed by the condition that a12 is finite but that alp vanishes when p> 2 ; again, a12y a23 are finite, but a2p vanishes if p differs from 2 by more than unity. Substituting these values of alpi a2py a3p..., we easily find A - „ dA* . d?Au _ d*Au 14 ai2 a23 aU ^a^ ’ Am = (-1)”"1 ai2a23a34... aw_lw. Now T, the Kinetic Energy, is always positive, but the con- dition for this is (Maxwell's Electricity and Magnetism, vol. i. p. Ill) that D, Au, ^ d2A 11 da22 da22 da should all be positive ; hence we see if we take a12, a23..., &c. all positive, ^.n, Al2, Au will be alternately plus and minus, but when the frequency of the electromotive force is very great, xlf x2,... are by (13) respectively proportional to Alv A12.t.; hence we see that in this case the adjacent currents are flowing in opposite directions: a result given by Lord Rayleigh. Another way of stating this result is to say that the direction of the currents is such that all the terms involving the product of two currents in the expression for the Kinetic Energy of the system of currents are negative, and in this form we recognise it as a consequence of the principle that the distribution of the currents must be such as to make the Kinetic Energy a minimum. 427.] We shall now apply these results to the case when the circuits are a series of m co-axial right circular solenoids of equal length, which act inductively on each other but which are not427-] DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. 525 in metallic connection. We shall suppose that a is the radius of the first solenoid, b that of the second, c that of the third, and so on, a9 b, c being in ascending order of magnitude; and that %, n2y 713... are the numbers of turns of wire per unit length of the first, second, and third circuits. Then if l is the length of the solenoids, we have an = 47r2n12Za2, a22 = 47T2n£lb2, asz = 47r2 n2lc2, au = 47r27i17?2?(X2, a23 = 4Tt2n2n3lb2, a34 = 47r2 n3nAlc2i ai3 == 4.712n1n3la2, a24 = 47r27i2n4Z&2,..................... Hence Oil* ai2> «18 ai2> a22’ tt23 aiz> tt23> a33 . . . • and (4Tt2l)mn^n£n£... a2(b2—a2) (c2 — b2) (<22 —c2)..., _ dD 11dan = (4 7721)™-1 n2 n2... b2 (c2 - b2) {d2 - c2).... Now the coefficient of self-induction of the first circuit for very rapidly alternating current is D ■A n' Substituting the preceding expressions for D and A1V we find that the self-induction equals a2 4 «**«,*«* (l-p-) Thus the only one of the circuits which affects the self- induction of the first is the one immediately adjacent to it. We can at once see the reason for this if we notice that a!2 __ ^13 (Ha ___ ^22 ^23 and therefore A13 = A14 = A15 = ... = 0. Now when the rate of alternation is very rapid, #3, #4, #5..., the currents in the third, fourth, and fifth circuits, &c. are by equation (13) Art. (425) proportional to A13, AUi A15...; hence we see that in this case these currents all vanish, in other words526 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [428. the second solenoid forms a perfect electric screen, and screens off all induction from the solenoids outside it. 428.] Let us consider the case of three solenoids each of length l when the frequency is not infinitely rapid; we shall suppose that the primary coil is inside and has a radius a, number of turns per unit length nx, resistance r; next to this is the secondary, radius 6, turns per unit length n2, resistance s; and outside this is the tertiary, radius c, turns per unit length 7i3, resistance t. Since the circuits are not in metallic connection ri2 = ri3 = r23 “ 0* If Xl9 the electromotive force acting on the primary, is proportional to then we have by equations (11) and (12) 7i1^?i22n32(4:7r2lf a'P+TnHn*9 aip 5 aip a2ip b2ip C2ip+ 21 2 We see from this expression that as long as the radius and length of the secondary remain the same, the effect produced by it on the current in the tertiary circuit depends on the ratio s/n22, since s and n2 only enter into the expression for x3 as constituents of the factor s/n22. Thus all secondaries of radius b and length l will produce the same effect if s/n2 remains constant. We can apply this result to compare resistances in the following way: take two similar systems A and B each consist- ing of three co-axial solenoids, the primaries of A and B being exactly equal, as are also the two tertiaries, while the two secondaries are of the same size but differ as to the materials of which they are made. Let us use A and £ as a Hughes’ Induc- tion Balance, putting the two primaries in series and connecting the tertiaries so that the currents generated in them by their respective primaries tend to circulate in opposite directions; then if, by altering if necessary the resistance in one of the secondaries, we make the resultant current in the combined tertiaries vanish, we know that s/n22 is the same for A and B. Suppose that the secondary in B is a thin tube of thickness t429.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 527 and specific resistance cr, then considering the tube as a solenoid wound with wire of square section a packed close together, we see that for the tube s = 2 7Tbln9- = 2itbln( 2 a 2 r Now s/n2 for the tube is equal to s/n2 for the secondary of A, which may be an ordinary solenoid. We thus have = 27r6Zcr/r, n2 1 a relation by which we can deduce i* ©• B13 = («12 <-P + rn) («23 + ^23) - (o*s '■P + ^22) («13 + rn)- Substituting the preceding values for the a’s and the r’s, we find Bu = -p2 (Jffl-NC) + tp (Mb + B/3-Nc~ Gy) + 6/3—cy. Now if 0 vanishes Blz must vanish; hence if the Bridge is balanced for all values of p we must have MB—NC = 0, Mb + Bp-Nc-Cy = 0, b/3 — cy = 0; while if the Bridge is only balanced for a particular value of p, we have bfi—cy = p2(MB—NG), p (Mb-{- jB/3 — Nc — Cy) = 0. When the frequency is very great the most important term in the expression for £13 is —p2(MB — NG), so that the most im- portant condition to be fulfilled when the Bridge is balanced is MB — NG = 0; thus for high frequencies the Bridge tests the self-induction rather than the resistances of its arms.430.] DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. 529 Combination of Self-Induction and Capacity. 430.] We have supposed in the preceding investigations that the circuits were closed and devoid of capacity; very interesting results, however, occur when some or all of the circuits are cut and their free ends connected to condensers of suitable capacity. We can by properly adjusting the capacity inserted in a circuit in relation to the frequency of the electromotive force and the self-induction of the circuit, make the circuit behave under the action of an electromotive force of given frequency as if it pos- sessed no apparent self-induction. The explanation of this will, perhaps, be clear if we consider the behaviour of a simple mechanical system under the action of a periodic force. The system we shall take is that of the rectilinear motion of a mass attached to a spring and resisted by a frictional force proportional to its velocity. Suppose that an external periodic force X acts on the system, then at any instant X must be in equilibrium with the resultant of (1) minus the rate of change of momentum of the system, (2) the force due to the compression or extension of the spring, (3) the resistance. If the frequency of X is very great, then for a given momentum (1) will be very large, so that unless it is counterbalanced by (2) a finite force of infinite frequency would produce an infinitely small momentum. Let us, however, sup- pose that the frequency of the force is the same as that of the free vibrations of the system when the friction is zero. When the mass vibrates with this frequency (1) and (2) will balance each other, thus all the external force has to do is to balance the resistance. The system will thus behave like one without either mass or stiffness resisted by a frictional force. In the corresponding electrical system, self-induction corre- sponds to mass, the reciprocal of the capacity to the stiffness of the spring, and the electric resistance to the frictional resistance. If now we choose the capacity so that the period of the electrical vibrations, calculated on the supposition that the resistance of the circuit vanishes, is the same as that of the external electro- motive force, the system will behave as if it had neither self- induction nor capacity but only resistance. Hence, if L is the self-induction of a circuit whose ends are connected to the plates of a condenser whose capacity in electromagnetic measure is (7, m m530 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [431. the system will behave as if it had no self-induction under an electromotive force whose frequency is p/2ir if LCp2 = l. 431.] We shall now consider the case represented in the figure, where we have two circuits in parallel, one of the circuits being cut and its ends connected to the plates of a condenser. Let A be the self-induction of the leads, r their resistance; L, AT the coefficients of self-induction of ACB and the condenser circuit respectively, M the coefficient of mutual induction between these circuits. Let R, S be the resistances respectively of ACB and the condenser circuit, C the capacity of the condenser. Let x be the current in the leads, y that in the condenser circuit, then that in the circuit ACB will be x — y. Let X, the electromotive force in the leads, be proportional to eipt. If there is no mutual in- duction between the leads and the wires in parallel, the equations giving x, y are (A4Z)g-(Z-J/)§ +(r + R)i-Bj = X, (L+N-2M)ft-(L-M)<^+(S+R)#-Rx+%=0. Substituting the value of y in terms of x and remembering that d/dt = ip, we get €W-(L-Mri?}-2R(R + S)(L-M)) + ^2£2 + (i2 + /S)2 ( (R + S){R?-{L-Mfp*} + 2pHR(L-M) I p*e+(R+sf (14) where £ = {L + N-2M)-~- From the form of this equation we see that the self-induction of the two circuits in parallel is J f {R%-{L-MYp*\-2R{R + S)(L-M) ^2P + (JR + ^)243^.] DISTRIBUTION OE RAPIDLY ALTERNATING CURRENTS. 531 this will vanish if Lp2£2 + £{R2-(L-M)2p2} + (R + 8){L{R + S)-2R(L-M)} = 0. (15) If the roots of this quadratic are real, then it is possible to choose G so that the self-induction of the loop vanishes. An important special case is when 8=0, M= 0, when the quadratic reduces to Lp2£2 + £(R2-L2p2)-LR2 = 0; thus the first root gives e R2 T the second 1-={L + N)f+^, = Np2; this last value of 1/G makes x = y, so that none of the current goes through ACB. When £ satisfies (15) the self-induction of the loop vanishes. If in that equation we substitute i + A for L and M+ A for M, the values of £ which satisfy the new equation will make the self-induction of the whole circuit vanish. 432.] We shall next consider the case of an induction coil or transformer, the primary of which is cut and its free ends con- nected to the plates of a condenser whose capacity is C. Let Ly N be the self-induction of the primary and secondary respect- ively, M the coefficient of mutual induction between the two, R the resistance of the primary, 8 that of the secondary, x, y the currents in the primary and secondary respectively; then if X is the electromotive force acting on the primary, we have Tdx ^.dy ' . x ~ Lm+M dt+Rx+c=x> Hence if X varies as «,pt, we find y = -MipX {£N-M2) + RS+ip {.RN+S£)’ Mm2 where532 DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. [432. The amplitude of y for a given amplitude of X is proportional to XMp {(RS-p2{£]Sr-M2))* + (RN+S$)2p2}* ’ This vanishes when p — 0, because in this case the current in the primary is steady; it also vanishes in general when p is infinite, because in consequence of the self-induction of the primary only an indefinitely small current passes through it in this case. If however far = if2, or 1 if2 Gp2 ~ L N ' then the amplitude of the current in the secondary is finite when p is infinite, and is equal to MNX RN2 + SM*; thus when the frequency of the electromotive force is very high the amplitude of the current in the secondary may be increased enormously by cutting the primary circuit and connecting its ends to a condenser of suitable capacity. 432*.] We can apply a method similar to that of Art. 424 to determine the effect of placing a vibrating electrical system near a number of other such systems. We shall suppose that the systems are not in electrical con- nection, and neglect the resistances of the circuits. Let T be the Kinetic, V the Potential Energy of the system of currents; let xx denote the current in the first circuit, and let #2, #3? ••• > the cur- rents in the other circuits, be so chosen that when xl is put equal to zero the expressions for T and V reduce to the sums of squares of #2, #3,...; #2, #3, ... respectively. Let T be given by the same expression as in Article 424, while V = Then the equations of the type ddT dV432.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 533 give, if all the variables are proportional to «>pt, (—®nP2+ r)xi—a12pix2-a13p2x3- ... = 0 ci -OuA+ (~a22P2 + -)*>2 =0 C2 ~anPixi + (-«83jPa+ ••• he the values of p for the other vibrators when the first one is absent, then 1 2 “ =Pia 22 C2 1 - = 2>32®33- c3 Thus if hp^ denotes the increase in px2 due to the presence of the other vibrators, we have -«„W = P.‘^fcri) + *13 aiPi-Pi) ' *w(pz-pi Thus we see that if p2 is greater than px the effect of the proximity of the circuit whose period is p2 is to diminish pv while if p2 is less than px the proximity of this circuit increases pr Similar remarks apply to the other circuits. Thus the first system, if its free period is slower than that of the second, is made to vibrate still more slowly by the presence of the latter ; while if its free period is faster than that of the second the presence of the latter makes it vibrate still more quickly. In other words, the effect of putting two vibrators near together is to make the difference between their periods greater than it is when the vibrators are free from each others influence; the quicker period is accelerated, the slower one retarded.CHAPTER VII. ELECTROMOTIVE INTENSITY IN MOVING BODIES. 433.] The equations(5)giveninArt. 598 of Maxwell’s Electricity and Magnetism for the components of the electromotive intensity in a moving body involve a quantity 4*, whose physical meaning it is desirable to consider more fully. The investigation by which the equations themselves are deduced tells us nothing about it is introduced after the investigation is finished, so as to make the expressions for the electromotive intensity as general as it is possible for them to be and yet be consistent with Faraday’s Law of the induction of currents in a variable magnetic field. Let uy vy iv denote the components of the velocity of the medium; a, 6, c the components of the magnetic induction; Fy Gy H those of the vector potential; X, F, Z those of the electromotive intensity. In the course of Maxwell’s investigation of the values of Xy Yy Z due to induction, the terms respectively in the final expressions for X, F, Z are included under the 4* terms. We shall find it clearer to keep these terms separate and write the expressions for X, F, Z as *-s« £(i*+ov+nv)-i£ 0)ELECTROMOTIVE INTENSITY IN MOVING BODIES. 535 For Faraday’s law to hold, the line integral of the electro- motive intensity taken round any closed curve must be inde- pendent of (j>, hence must be a continuous function. When there is no free electricity dX dY dZ__ dx + dy + dz~~ Substituting the values of X, Y, Z just given, we find, using dF dG dH _ dx + dy dz ~~ ’ ttt—io ^ , d F du dG dv dHdw^ FV2u+GV2v + HV2w + 2 (-r- -y- + — — + — ^ dx dx dy dy dz dz 7 ,dH dG\,dw dvx ,dF dH\/du dwx * ^ dy dz ' ^ dy * dz' ' dz dx dz * dx' zdG dFx d>\Xj\ + (s+3y)(a + 3ii) = -v*- If the medium is moving like a rigid body, then u = p + u2z —uzy, v — q +to3x — (t)1z, w — r — u)2x ; where p, 5, r are the components of the velocity of the origin and o)19 a)2, co3 the rotations about the axes of x, y, 0 respectively. Substituting these values we see that whenever the system moves as a rigid body 6 J V2 = 0. 434.] In order to see the meaning of we shall take the case of a solid sphere rotating with uniform angular velocity 00 about the axis of z in a uniform magnetic field where the magnetic induction is parallel to the axis 0 and is equal to c. We may suppose that the magnetic induction is produced by a large cylindrical solenoid with the axis of z for its axis; in this case F — — \ cy9 G = \ cXy H = 0. In the rotating sphere u = —coy, v — do x} w = 0. If the system is in a steady state, dF/dt, dG/dt, dH/dt all vanish.536 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [434- Thus in the sphere 'dx z = these equations reduce to d fa'9 Y____ d dx y=- d dy’ and we have also V2<£=0. In the space outside the sphere the medium does not move as a rigid body. The process by which the equations (1) were obtained could not without further investigation be held to justify us in applying them to cases where the velocity is discontinuous, for in the investigation, see Maxwell, Art. 598, it is assumed that the variations 5%, by, hz are continuous, and that these are pro- portional to the components of the velocity. To avoid any discontinuity in the velocity at the surface of the sphere we shall suppose that the medium in contact with the sphere moves at the same rate as the sphere, but that as we recede from the surface of the sphere the velocity diminishes in the same way as it does in a viscous fluid surrounding a rotating sphere. Thus we shall suppose that the rotating sphere whose radius is a is surrounded by a fixed sphere whose radius is b, and that between the spheres the components of the velocity are given by the expressions u=-(A^\+By)' v==(atxI+Bx)’ w = 0’ where r is the distance from the centre of the rotating sphere. When r =b, u = 0, v = 0, hence A434*] electromotive intensity in moving bodies. 537 Substituting these values of u, v in equation (l), we find that when a < r < b, The boundary conditions satisfied by

= 0. Again, when r > b the medium is at rest, here we have and V20 — 0. xX + yY+zZ = 0, this is equivalent to ^ a —XI — o dr ' where fa is the value of inside the rotating sphere; hence we have where if is a constant.538 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [435. If 2> 3 are the values of in the region between the fixed and moving spheres, and in the fixed sphere respectively, then we may put M a\ 2 = L+ — +-A^2(^--3)» <*>S = PQz where Z, M, N, P are constants, and Q2 is the second zonal harmonic with z for its axis. The continuity of <\> gives M a iF(b5-a5) K = L + P = A r M 0 = Z+ —> b If Kx is the specific inductive capacity of the medium between the two spheres, K2 that of the medium beyond the outer sphere ; then, since the normal electric polarization must be continuous when r = b, we have 3 jr pQl — K ^cA 4. wn 3KzP - K'fA b2 + b2 _JV^2U2 + b4^) Solving these equations we find p= cAKx b2(b5-a5) 3 K2 (b5 — a5) + Kx (2b5 + 3a5)9 jy ____________cAKx a2b2 (2) 3K2 (b5- a5) + Kx (2b5 + 3a6) ’ M = cA, L = — cA/b, K =cA (b — a)/ab, j where A = — a> a3 b3/(b3 — a3). The surface density of the electricity on the moving sphere is KxcA $Kx(2b5 + 3a5 —5a3b2)-f 3JT2(b5 —a5)) n 47ia2 \ 3if2(b5—a5) + if1(2b5+3a5) The preceding formulae are general; we shall now consider some particular cases. 435.] The first we shall consider is when b—a=6 is small compared with either b or a. In this case we have approximately, when K2 is not infinite, P = —ic(oa5, -AT = — ^ c coa'5 Jf = -i ccoa* L=i CO)t K— — Jcwa2.436.] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 539 Thus in the outer fixed sphere the components of the electro- motive intensity are equal to the differential coefficients with respect to x, y> z of the function £ca>a5%. r3 Thus the radial electromotive intensity close to the surface of the rotating sphere is —ccoaQ2, while the tangential intensity is —ca>asin0cos0. These results show that the effects produced by rotating uncharged spheres in a strong magnetic field ought to be quite large enough to be measurable. Thus if the sphere is rotating so fast that a point on its equator moves with the velocity 3 x 103, which is about 100 feet per second, and if c-103, then the maximum radial intensity is about 1/33 of a volt per centimetre, and the maximum tangential intensity about 1/2 of this: these are quite measurable quantities, and if it were necessary to in- crease the effect both c and aQ2* 4 TT 436.] If the outer fixed sphere is a conductor, the electromotive intensity must vanish when r > b, hence P= 0, so that N= 0, while M, L, K have the same values as before. In this case the surface density of the electricity on the surface of the rotating sphere is g 4 7ra' cAQ< 2> and when b —a is small, this is equal to 127t6 co)Q,2Q2. Since this expression is proportional to 1/8, the surface density can be increased to any extent by diminishing the distance between the rotating and fixed surfaces. In the general case, when b — a is not necessarily small, the540 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [437* surface density of the electricity on the rotating sphere is 4 7i a2 cAQ2i the surface density on the fixed sphere is Ki 4 7Tb2 The electrostatic potential due to this distribution of electricity at a distance r from the centre of the rotating sphere is, when T > b> cA 0 - ^ (b2-a2)-^, 5 r6 while when r < a it is cA 5 The values of in these regions are respectively zero and a constant. Hence this example is sufficient to show us that is not equal to the electrostatic potential due to the free electricity on the surface of the conductors. 437.] We may (though there does not seem to be any ad- vantage gained by so doing) regard as the sum of two parts, one of which, cj>e, is the electrostatic potential due to the distribu- tion of free electricity over the surfaces separating the different media; the other, <£w, being regarded as peculiarly due to electro- magnetic induction. Let us consider the case of a body moving in any manner, then we must have, since there is no volume distribution of electricity, v2e is the electrostatic potential, then w=-[*(i^+ d. L_<^, dy m 438.] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 541 From these conditions we see from equations (I) that V2 (m+Fu + Gv + II w) A dy = [IT {l {cv—bw) + m(aw — cu) + 7i(bu — av)}]\ From these equations <£m is uniquely determined, for we see that m + Fu + Gv + Hw is the potential due to a distribution of electricity whose volume density is - §y(aw~A+ %(b»-av)\, together with a distribution whose surface density is 1 2 — — \K {l (cv—bw) + m(aw—cu) + n(bu—av)}'\i. Having thus determined m and deducing by the process exemplified in the preceding examples we can determine (j>e. 438.] The question as to whether or not the equations (1) are true for moving insulators as well as for moving conductors, w,, v, w being the components of the velocity of the insulator, is a very important one. The truth of these equations for con- ductors has been firmly established by experiment, but we have, so far as I am aware, no experimental verification of them for insulators. The following considerations suggest, I think, that some further evidence is required before we can feel assured of the validity of the application of these equations to insulators. We may regard a steady magnetic field as one in which Faraday tubes are moving about according to definite laws, the positive tubes moving in one direction, the negative ones in the opposite, the tubes being arranged so that as many positive as negative tubes pass through any area. When a conductor is moved about in such a magnetic field it disturbs the motion of the tubes, so that at some parts of the field the positive tubes no longer balance the negative and an electromotive intensity is produced in such regions. To assume the truth of equations (1), whatever the nature of the moving body may be, is, from this point of view, to assume that the effect on these tubes is the same whether the moving body be a conductor or an insulator of = Aicv-iw) + d (aiv—cu) + -j- {bn—av)542 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [439. large or small specific inductive capacity. Now it is quite con- ceivable that though a conductor, or a dielectric with a consider- able inductive capacity, might when in motion produce a con- siderable disturbance of the Faraday tubes in the ether in and around it, yet little or no effect might be produced by the motion of a substance of small specific inductive capacity such as a gas, and thus it might be expected that the electromotive intensity due to the motion of a conductor in a magnetic field would be much greater than that due to the motion of a gas moving with the same speed. 439.] As one of the most obvious methods of determining whether or not equations (1) are true for dielectrics is to in- vestigate the effect of rotating an insulating sphere in a magnetic field: we give the solution of the case similar to the one discussed in Art. 434, with the exception that the metallic rotating sphere of that article is replaced by an insulating one, specific inductive capacity iT0, of the same radius. Using the notation of that article, we easily find that in this case P <3K, 2b (3 K1 + 2K0)K-6 Kr (K, -K0) as/b^ ! b4 + 2(Z1-Z0)a5 + (3if1 + 2Jfir0)b5 ) _0 ir a i J;___________5a2bA1__________I 1 (b2 2 (K, -K0) a5 + (3 JT, + 2 K0) b5 J ' When b —a is small, this becomes p _ 1 *3K2 + 2K0 co) a° So that in this case the components of electromotive intensities in the region at rest are equal to the differential coefficients with respect to x, y, 0 of the function i 2Ko ccoa5 3 3K2 + 2K0 r3 ^2’ and thus, by Art. 435, bear to the intensities produced by the rotating conductor the ratio of 2K0 to %K2 + 2KQ. Thus, if equations (l) are true for insulators, a rotating sphere made of an insulating material ought to produce an electric field comparable with that due to a rotating metallic sphere of the same size. The greatest difficulty in experimenting with the insulating sphere would be that it would probably get electrified by friction, but unless this completely overpowered the effect due440.] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 543 to the rotation we ought to be able to distinguish between the two effects, since the rotational one is reversed when the direction of rotation is reversed as well as when the magnetic field is reversed. In deducing equations (2) of Art. 434, we assumed that equa- tions (1) held in the medium between the fixed and moving surfaces, the general equations will therefore only be true on this assumption. In the special case, however, when the layer of this medium is indefinitely thin, the results will be the same whether this medium is an insulator or conductor, so that the results in this special case would not throw any light on whether equations (1) do or do not hold for a moving dielectric. Propagation of Light through a Moving Dielectric. 440.] We might expect that some light would be thrown on the electromotive intensity developed in a dielectric moving in a magnetic field by the consideration of the effect which the motion of the dielectric would have on the velocity of light passing through it. We shall therefore investigate the laws of propaga- tion of light through a dielectric moving uniformly with the velocity components u, v, w. In this case, since we have only to deal with insulators, all the currents in the field are polarization currents due to altera- tions in the intensity of the polarization. When the dielectric is moving we are confronted with a question which we have not had to consider previously, and that is whether the equivalent current is to be taken as equal to the time rate of variation of the polarization at a point fixed in space or at a point fixed in the dielectric and moving with it; i.e. if / is the dielectric polarization parallel to x, is the current parallel to x df dt ’ or df df df df. -fr +u-f +v-f +w-f-1 at ax ay dt z In the first case we should have, if a, /3, y are the components of df dy the magnetic force, in the second, dt (3) df df3 ,df df df df\ dy ^^di+Udx+VWy+Wdz)~dy~dz (4)544 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [44O. This point seems one which can only be settled by experi- ment. It seems desirable, however, to look at the question from as many points of view as possible; the equation connecting the current with the magnetic force is the expression of the fact that the line integral of the magnetic force round any closed curve is equal to 4 77 times the rate of increase of the number of Faraday tubes passing through the curve. We saw in Chapter I. that this was equivalent to saying that a Faraday tube when in motion gave rise to a magnetic force at right angles to itself, and to the direction in which it is moving and proportional to its velocity at right angles to itself. When the medium is moving, the question then arises whether this velocity to which the magnetic force is proportional is the velocity of the tube relative (1) to a fixed point in the region under consideration, or (2) relative to the moving dielectric, or (3) relative to the ether in this region. If the first supposition is true we have equation (3), if the second equation (4), if the third an equation similar to (4) with the components of the velocity of the ether written for u, v, w. I am not aware of any experiments which would enable us to decide absolutely which, if any, of the assumptions (1), (2), (3) is correct; a priori (3) appears the most probable. If X, F, Z are the components of the electromotive intensity; a, by c those of magnetic induction; /, g, h those of electric polarization, and F, G, H those of the vector potential, then we ^ave 4 7r - , dF d\jf \ K J dt dx Tr 47t dG d\I/ F = -r9 = a»-«-5r-^ _ 4tt 7 7 dH d\l/ Z — h = bu —av-----------7—- K dt dz Then, since the dielectric is moving uniformly, we have (0 47t ,df dqx do do do do x(di- with similar equations for dZ dX dY dZ' dx dz dz dy If the sphere is rotating with angular velocity a> about the axis of z, U—~ oayi v = (ox, w = 0 ; so that equation (10) becomes dX dY , dc dc\ <") dy dx~ ' dy *dx> If (cc y J-va, 4 TTfX a dy ^dx ~ /da dax 7 <» -2/ ) +o>&. (13) 4ir/x y dy u dx From these equations we find by the aid of (12) 4ir/n v dy * dx> „ / dq dqx V3r = tu(ic4-— 4 ity ' dy J dx > n n a 47r/x , where r, 0, <#> are the polar coordinates of a point, 0 being measured from the axis of z. Tfre18* is a surface harmonic of degree n. Substituting this value in (14), we find dI 2F 2 dF /n(n+ 1) 4Trt/utsco \ cZr2 + r dr ' r2 + o- / “” * The solution of this is, Art. 308, F(r) = S„(kr), where k2 = — In fjusa/o-. Thus xp + yq + zr = ASn(Jcr) Tn8€lS(f>9 where A is a constant. Now xp + yq + zr is proportional to the current along the radius, and this vanishes at the surface of the sphere where r = a; hence we have ASn (ka) = 0, but since the roots of Sn (x) = 0 are real, and k is partly imaginary, Sn(k a) cannot vanish, thus A must vanish. In other words, the radial currents must vanish throughout the sphere; the currents thus flow along the surfaces of spheres concentric with the rotating one. Since xp + yq + zr = 0, we may by Art. 370 put «=/.«(* I (15) where fjkrj k2 = — 4 TTJXlS 0)/(T, and a)n is a solid spherical harmonic of degree n. By Art. 372, a, y3, y, the components of magnetic force, will be given by I <» + »/- W s (ffeOS > <*• with similar expressions for j3 and y.441-] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 549 Now the magnetic force may be regarded as made up of two parts, one due to the currents induced in the sphere, the other to the external magnetic field; the latter part will be derived from a potential. Let I2n be the value of this potential in the sphere; we may regard Qn as a solid spherical harmonic of degree n, since the most general expression for the potential is the sum of terms of this type. If av j3ls yL are the components of the magnetic force due to the currents, a0, /30, y0 those due to the magnetic field, then d „ a = c^ + flo = ai-^12w. Hence in the sphere 4 TT (2 n+ 1 )Jc2 (17) with similar expressions for ^ and yr Outside the sphere the magnetic force due to the currents will (neglecting the displacement currents in the dielectric) be de- rivable from a potential which satisfies Laplace's equation ; hence outside the sphere we may put, if n, a*n' in equations (15) and (17), we get the currents induced in the sphere and the magnetic force produced by those currents. 442.] We shall consider in detail the case when n = 1, i. e. when the sphere is rotating in a uniform magnetic field. Let the magnetic potential of the external field be equal to the real Part (7rcos0 + i?7’sin0€<<*>, where C is the force parallel to z and B that parallel to x. Then in the sphere 3 =-------- (Cr cosO + Br sinfle4^). We shall first consider the case when hr is very small, so that approximately by Art. 309 fo (kr) = 1 - fx {hr) = -f2 {hr) = rV Substituting these values in (18) and (19) and retaining only the lowest powers of h, we find 1^2 j ^ 4ira>1 (1 + Io7^T2)Fa2)5rsin^‘1'>’ r 3&2a2 „ . "■ = - ioTiTira? ^““O***- The term (7rcos0 in X2 does not give rise to any terms in o>n, a)n' since s and therefore lc vanishes for this term. Substituting these values we get by equations (15) P = q = fJUt) (iJ. + 2) or zB, \ V r = __ 3 2 fJLO) (fj,+ 2)cr xB. (20) Thus the currents flow in parallel circles, having for their common axis the line through the centre of the sphere which is at right angles both to the axis of rotation and to the direction of magnetic force in the external field. The intensity of the current at any point is proportional to the distance of the point from this axis. The components of the magnetic induction in the sphere are given by the equations443*] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 551 a = — b = 3 nB M+2 3rrfx2(oB (■+^) (M + 2)a^r ^-?f,+ 2a> (21) c = 3/x jx -f" 2 ^ 5 c Thus the magnetic force due to currents consists of a radial force proportional to yr, together with a force parallel to y proportional to 2 r2 — (/x + 4) a2/(fx + 2). Outside the sphere the total magnetic potential is fjb2(oa5 y Thus the magnetic effect of the currents at a point outside the sphere is the same as that of a small magnet at the centre, with its axis at right angles to the axis of rotation and the external magnetic field, and whose moment is 6 77 2? fx2co a5 5 (/x + 2 f cr # 443.] Let us now consider the case when &a is large, since, when s = 1 = we have where k=V2K* *, 2 TTflXt) K2 = thus the real part of i&a is positive and large; hence we have approximately ik a A(i*) = 2uEi’ /■<*•>= 2TW- eika, /*(*•) = -27p^’ Hence we find ■lira)! = — 3i&3ae~‘*a.Br sintfe1’*’, ®i' = I -2^ sin 5552 ELECTROMOTIVE INTENSITY IN MOVING BODIES. [443. P = q = r = a — 6 = so that by (15) 3\^2KzB,Ti —K(a.—r} C T/ / \ IT) r)cos {*(•-»•)+-}, { ^8injZ(a-r)+iJ, ^•ge~'g(*~f)[fl!C08{g (&~r) + ?| +2/sin}'S'(a-r)+ \\\' The total components of the magnetic induction inside the sphere are given by - ® €- ■K (*-»•) cos K (a - r) - \ y.B «“ •*(•“*■) a r2 cos JT (a -r) ~ J ' ar2 sin Z(a-r)^ J, — n-B * (a-r) sin if (a — r)— i/iBf~x^a~r^ ar2 cos K (a—r) — | ixBe--K’(a~r) ar2 sin if (a—r) ^ > r v ' dy r- -\V-B<~K(*-*) ar2 cosif(a-r) ^ J (22) V (23) — ilxBe K(a r)ar2siniT(a —r) ' 1 azr3 while the magnetic potential outside due to the currents in the sphere is 3 /u5 3 a? 2 jm + 2 a r3 (24) If we compare these results with those we obtained when &a was small, we see that they differ in the same way as the distribution of rapidly varying currents in a conductor differs from that of steady or slowly varying ones. When ka, is small the currents spread through the whole of the sphere, while when &a is large they are, as equations (22) show, confined to a thin shell. The currents flow along the surfaces of spheres concentric with the rotating one, and the intensity of the currents diminishes in Geometrical Progression as the distance from the surface of the sphere increases in Arithmetical Pro- gression. The magnetic field due to these currents annuls in the interior of the sphere, as equation (23) shows, that part of the ex-444-] ELECTROMOTIVE INTENSITY IN MOVING BODIES. 553 fcemal magnetic field which is not symmetrical about the axis of rotation. Thus the rotating sphere screens its interior from all but symmetrical distributions of magnetic force if {47r/xco/cr}^a is large. A very interesting case of the rotating sphere is that of the earth; in this case a = 6-37 x 108, (o = 2tt/(24 x 60 x 60), so that approximately {4 7r/mci)/(ra = 2 x 107 + 2 r3 Thus the effect of the rotating sphere on the part of the external magnetic field which is unsym metrical about the axis of rotation, i. e. upon the term Br sintfe4^, is exactly the same as if this sphere were replaced by a sphere of diamagnetic substance for which fx = 0 ; in other words, the rotating sphere behaves like a diamagnetic body. Thus we could make a model which would exhibit the properties of a feebly diamagnetic body in a steady field, by having a large number of rotating conductors arranged so that the distance between their centres was large compared with their linear dimensions. Couples and Forces on the Rotating Sphere. 445.] We shall now proceed to investigate the couples and forces on the sphere caused by the action of the magnetic field on the currents induced in the sphere. If X, F, Z are the components of the mechanical force per unit volume, then (Maxwells Electricity and Magnetism, vol. ii. Art. 603, equations C) X = cq—bv, Y = ar — cp, Z = 6p —aq. The couple on the sphere round the axis of z is JJJ (Yx—X y) dxdydz, the integration extending throughout the sphere. Substituting the preceding values for Y and X, we see that this may be written JJ!*(r (ax + by + cz) — c(px + qy + r z)) dxdydz. But since the radial current vanishes, VX + qy + vz = 0; thus the couple round z reduces to vRr dxdydzy in■ where R is the magnetic induction along the radius.446.] ELECTBOMOTIVE INTENSITY IN MOVING BODIES. 555 Similarly the couple round x is equal to ffjpU rdxdydz, while that round y is n r r / / / qRrdxdydz. From equation (16) we see that 4 7T/Ut Rr~ (2n + XjW n(n+l) If— 1 (kr) + fc2,-V«+1 (kr)’> Now by (4), Art. 370, so that or by (15) fn-i(kr) + k2r*fn+1(kr) = ~(2n + l)fn(kr), Rr = ~ ~wn (n+o/» (*r) “»> Br =------9n(n+ l)r. 60S2 v J Thus the couple around 0 is ---~^n*(71 + !)fJJ*2dxdydz. (25) When co is small we find, by substituting the value of r given in equation (20), that when the sphere is rotating in a uniform magnetic field the couple tending to stop it is 6u2 jy, co . ----rrs 5- — 7ra5. 5 (jtx 4- 2)2 among a net-work of conductors, 510. -----distribution of between two circuits in parallel, 512. -----expression for self-induction of a single wire, 298-295. -----expression for * impedance * of a single wire, 293-295. — — expression for self-induction of systems of wires, 517. -----expression for ‘ impedance * of sys- tems of wire, 517. -----flow to surface of conductors, 260, 281. -----heat produced in wire traversed by, 818. -----in flat conductors, 296. -----in two dimensions, 253. -----in wires, 262. -----motion of Faraday tubes round a wire carrying, 38. -----of long period, rate of decay along a wire, 272. -----of long period, velocity along a wire, 271. -----of moderate period, rate of decay along a wire, 277. -----of moderate period, velocity along a wire, 277. -----of short period, rate of decay along a wire, 279. -----of short period, velocity of along a wire, 279. — magnetic force, behaviour of iron under, 323. Antolik’s figures, 176. Arc discharge, 163. -----with large potential differences, 165. -----connection between loss of weight of electrodes and quantity of electricity passing, 166. Arc discharge, connection between chemi- cal change and quantity of electricity passing, 563. ----electrification in, 566. Arons, electromagnetic waves, 461. Arons and Cohn, specific inductive capa- city of water, 48, 469. Arons and Rubens, velocity of electro- magnetic waves, 472. Arrhenius, conductivity of flames, 57. Attraction between flat conductors con- veying variable currents, 300. Aurora, 107. Ayrton and Perry, specific inductive capacity of a ‘vacuum,’ 98, 471. — arc discharge, 163. Bailie, spark discharge, 69-71, 74, 85,91. Balance, Induction, 526. Beccaria, phosphorescence, 119. Becquerel, conductivity of hot gases, 54. Berthelot, chemical action of electric dis- charge, 180. Bessel’s functions, values of, when vari- able, is small or large, 263. ----roots of, 353. v. Bezold, velocity of electromagnetic waves, 480. Bichat and Guntz, formation of ozone, 179. Bjerknes, decay of vibrations, 897. Blake, experiment with mercury vapour, 54. Blondlot, conductivity of hot gases, 55. — velocity of electromagnetic waves, 480. Boltzmann, specific inductive capacity, 468. Brush discharge, 171,188. Capacity of a semi-infinite plate parallel to an infinite one, 212. — of a plate between two infinite plates, 216, 218. — of one cube inside another, 222. — of two infinite strips, 237. — of a pile of plates, 239.572 INDEX. Capacity of a series of radial plates, 241. — of two piles of plates, 245-249. — of two series of radial plates, 246- 249. — of a strip between two plates, 246. — specific inductive, 468 et seq. — electrostatic neutralizes self-induction, 529. Cardani, effect of temperature on electric strength of gases, 92. Cathode, potential fall at, 150,153. Chemical action of electric discharge, 177. Chree on negative dark space, 110. Christoffers theorem in conjugate func- tions, 208. .Chrystal on spark discharge, 74, 84. ‘ Closed * Faraday tubes, 2. Cohn and Arons, specific inductive capa- city, 48, 469. Column, negative, 110. Concentration of alternating current on the outside of a conductor, 260. Condenser, discharge of, 331, 335. Conduction of electricity through metals and electrolytes, 50. Conductivity of rarefied gases, 99. Continuity of current through discharge- tube, 143. Contraction in discharge-tube produces effects similar to a cathode, 124. Coulomb, leakage of electricity through air, 53. Couple on a sphere rotating in a magnetic field, 555. * Critical ’ pressure, 84. ----effect of spark length on, 88. ----for electrodeless discharges, 96. Crookes on discharge through gases, 59, 104, 109, 110, 120-122, 124, 139. Crookes’ space, 108. Curie, specific inductive capacity, 469. Current, connection between and ex- ternal E.M.F., 288. — force between two parallel currents, 37. — mechanical force on conductor con- veying, 14. — motion of Faraday tubes in neighbour- hood of steady, 36. Cylinder, electrical oscillations on, 344, 347. — field of force round oscillating, 350. — scattering of electromagnetic waves by, 428. Damp air, potential required to spark through, 92. Dark space, 108. ----Crookes* theory of, 109. Decay, rate of, of slowly alternating currents along a wire, 272. Decay, rate of, of moderately rapid currents along a wire, 277. ------of very rapid currents along a wire, 279. — of currents and magnetic force in cylin- ders, 852. — of currents and magnetic force in spheres, 377, 380. — of vibrations in Hertz’s vibrator, 397. — of electrical oscillations on spheres, 37°. — of electrical oscillations on cylinders, 349. De la Rive, rotation of electric discharge, 138. De la Rive and Sarasin, experiments on electromagnetic waves, 400. De la Rue and Muller, discharge through gases, 69, 80, 90, 98, 109, 111, 114, 159, 170, 173, 174. Dewar and Liveing, effects of metallic dust in discharge, 103. Dielectric, electromotive forces in a moving, 544. — velocity of light through a moving, 545. Difference between positive and negative discharge, 169. Discharge between electrodes near to- gether, 160-162. — electrodeless, 92 et seq. ------critical pressure for, 94, 97. ------difficulty of passing from one medium to another, 98. ------action of magnet on, 105. — electric, difference between positive and negative, 169. — heat produced by, 167. — mechanical effects produced by, 174. — chemical action of, 177. — furrows made by, 176. — of a condenser, 381. ‘ Displacement,’ electric, 1, 6. Distance alternating currents travel along a wire, 272, 277, 279. Disturbance, electric, transmission of along a wire, 283. Drude on metallic reflection, 420. Du Bois, reflection of light from a magnet, 483. Dust figures, 174. — given off from electrified metals, 54. Earth’s magnetism, 553. Ebert and E. Wiedemann, effect of ultra- violet light, 58. Eisenlohr, metallic reflection, 420. Electric currents, decay of, in cylinders, 352. ------decay of, in spheres, 377, 880. Electric discharge, passage of across junc- tion of a metal and a gas, 98. ------action of magnet on, 131.INDEX. 573 Electric discharge, effect of air blast on, 132. -----heat produced by, 167. -----mechanical effects produced by, 174. -----expansion due to, 175. -----furrows made by, 176. -----chemical action of, 177. -----facilitated by rapid changes in the electric field, 185. Electric displacement, 1, 6. — screening, 405. — «skin/ 260, 281. — strength, 68. Electrical vibrations, 328 et seq. -----Feddersen on, 333. -----Lord Kelvin on, 333. -----Lodge on, 333. -----on cylinders, 344. -----on spheres, 361. Electrification of a metal plate by light, 59. — produced near glowing bodies, 63. — effect of on surface tension, 65. — in arc discharge, 566. Electrified plates, rotating, 22-28. — sphere, moving, 16. -----moving, magnetic force due to, 19. -----moving, momentum of, 20. -----moving, kinetic energy, 21. -----moving, force on in a magnetic field, 22. Electrode effect of magnet on distribution of negative glow over, 138. Electrodes, discharge between two when close together, 160-162. — difference between positive and nega- tive, 169. — spluttering of, 60. Electrodeless discharge, 92 et seq. -----existence of critical pressure for, 97. -----difficulty of passing from one me- dium to another, 98. -----action of magnet on, 105. Electrolytes, conduction of electricity through, 50. — conductivity of, 100. — under rapidly alternating currents, 4!7. Electromagnetic theory of light, 42. — repulsion, 557. — waves, 388. -----reflection of, 398. -----reflection of from grating, 406. -----refraction of, 406. -----angle of polarization of, 406. -----theory of reflection of from insula- tors, 407. -----theory of reflection of from metals, 414. -----scattering of by a cylinder, 428. -----scattering of by a metal sphere, 437. -----along wires, 451. Electromotive intensity, 10, 13. ----required to produce a spark in a variable field, 82. ----required to produce a spark across a thin layer of gas, 72. ----relation between and current for alternating currents, 288. Elster and Geitel, electrification pro- duced by glowing bodies, 61, 62, 569. Energy, transfer of, 9, 308. Equations for a moving dielectric, 544. Faraday, lines of force, 2. — spark potential through different gases, 90. — difference between positive and nega- tive discharge, 170. — rotation of plane of polarization of light, 482. — space, 111. — tubes, 2-5. ----momentum of, 9. ----velocity of, 11. ----disposition in a steady magnetic field, 28. ----effect of soft iron on their motion, 34. ----round a wire carrying a steady current, 36. ----round a wire carrying an alter- nating current, 38. ----motion of during the discharge of a Leyden jar, 38. ----shortening of in a conductor, 45- 47. ----duration of in terms of resistance, 47. ----disposition of round vibrating cy- linder, 350. — — disposition of round vibrating sphere, 370. Feddersen, effect of air blast on spark, 132. — electrical vibrations, 333. Feussner, temperature coefficient of elec- trical resistance for alloys, 51. Films, transmission of light through, 423. — transmission of light through when in magnetic field, 504. First dark space, 108. Fitzgerald, auroras, 107. — rotation of plane of polarization of light, 494. Flames, electrical properties of, 57. Fleming, arc discharge, 164, 167. — electromagnetic repulsion, 557- Force acting on a current, 15. — between two parallel currents, 37. — between flat conductors conveying alternating currents, 300. — relation between external electromo- tive force and alternating current, 288. Foster and Pryson, spark potential, 74.574 INDEX. Foucault currents, heat produced by, in a transformer, 818. Functions, Bessel’s, 263, 848, 853. —‘S’and‘E,’ 364. Galvanic cell, 48. Gases, passage of electricity through, 53 et seq. — passage of electricity through hot gases, 54-56. — high conductivity of rarefied, 99-102. Gassiot on electric discharge, 163. Gaugain, spark discharge, 69, 82. Geitel and Elster, electrification caused by glowing bodies, 62, 63. — escape of electricity from illuminated surfaces, 61. Giese, electrical properties of flames, 57. — conduction of electricity through gases, 190. Glazebrook, Report on Optical Theories, 421. Glow, discharge, 171. — produced by electrodeless discharge, 180-184. Glowing bodies, discharge of electricity by, 63. — electrification caused by, 63. Goldstein, discharge of electricity through gases, 110-114, 120, 123-125, 140- 142, 197. Gordon, reflection of light from a magnet, 483. Gradient of potential in discharge tube, 144. Grating, reflection of electromagnetic waves from, 406, 425. Grotthus* chains, 189, 195. Grove, chemical action of the discharge, 44, 191. — on the arc discharge, 166. Guard-ring, distribution of electricity on, 227, 231, 232, 235. Guntz and Bichat on the formation of ozone, 179. Hagenbach, transmission of signals along wires, 286. ‘Hall effect,’ 484, 486. Hallwachs, electrification by light, 59. Heat produced by electric discharge, 167. — — in wires carrying alternating currents, 315, 317, 318. ------by Foucault currents in a trans- former, 318. ------by currents induced in a tube, 323. Heaviside, moving electrified sphere, 19. — concentration of current, 260. — impedance, 293. Heine, Kiigelfunctionen, 263, 363. Helmholtz, v. H., attraction of electricity by different substances, 5, 64. ------on the functions ‘ S * and ‘ E,’ 364. Helmholtz, v. R., effect of electrification on a steam jet, 59, 187. Henry, on electrical vibrations, 832. Hertz, effect of ultra-violet light on the discharge, 58. — negative rays, 122, 126. — explosive effects due to spark, 177. — electromagnetic waves, 388 et seq. Herwig, arc discharge, 166. Himstedt, rotating disc, 23. — currents induced in rotating sphere, 546. Hittorf, discharge through gases, 76, 94, 98, 134, 144, 152, 153, 160, 162, 168. Hoor, effect of light on charged metals, 6L. Hopkinson, specific inductive capacity, 468. Hot gases, passage of electricity through, 54-56. Hughes, concentration of alternating current, 260. — induction balance, 526. Hutchinson and Rowland, rotating elec** trifled disc, 23, 27. Impedance, 293. — expression for, 293-295. — for flat conductors, 296. — for two wires in parallel, 517. — for a network of wire, 520. Incandescent bodies, discharge of elec- tricity by, 62. ----production of electrification by, 62. Induction balance, 526. — of currents due to changes in the magnetic field, 32. ----due to motion of the circuit, 33. ----due to alternations in the primary circuit, 41. — self, expressions for, 293-296. ----for flat conductors, 296. ----for two wires in parallel, 516. ----for a network of wires, 520. ----and capacity, 529. Inductive capacity, specific, 468. ----specific in rapidly varying fields, 472. Intensity, electromotive, 10-13. Iron, effect of, on motion of Faraday tubes, 84. — magnetic properties of, under rapidly alternating currents, 323. — decay of electromagnetic waves in, 340. Jaumann, discharge facilitated by rapid changes in the potential, 69, 185. Joly, discharge figures, 173. — furrows made by discharge, 176. Kelvin, Lord, spark discharge, 70, 73.INDEX. 575 Kelvin, Lord, transmission of an electric disturbance along a wire, 287. ----oscillatory discharge, 331, 333. Kerr, reflection of light from the pole of a magnet, 483. Kinetic energy, due to motion of Faraday tubes, 13. ----due to moving charged sphere, 21. ----a minimum for rapidly alternating currents, 511. Kinnersley, electrification by evaporation, 54. Kirchhoff, on conjugate functions, 208. Klemenfiig, specific inductive capacity, 469. Kundt, dust figures, 174. — transmission of light through thin films, 423, 508. — reflection of light from a magnet, 483. Lamb, decay of currents in cylinders, 356, 361. — decay of currents in spheres, 378, 382, 384. — on the functions * S ’ and * E ’, 363. Lamb’s theorem, 438. Larmor, currents in a rotating sphere, 547. Lebedew, specific inductive capacity, 470. Lecher, on the arc discharge, 164. — on electromagnetic waves, 463, 465. Lehmann, discharge between electrodes close together, 161. — difference between positive and nega- tive discharge, 173. — chemical action of the discharge, 179. Lenard and Wolf, dust given off under ultra-violet light, 54, 58. Leyden jar, motion of Faraday tubes during discharge of, 38. — — oscillatory discharge of, 331 et seq. Lichtenberg’s figures, 172. Liebig, spark potential, 72, 91. Light, electromagnetic theory of, 42. — effect of ultra-violet on electric dis- charge, 58. — effect of ultra-violet on electrified metals, 58. — electrification of a metal plate, 59. — reflection of from metals, 417. — transmission of through thin films, 423. — effect of magnetic field on, 482 et seq. — action of magnet on light through thin films, 504. — velocity of through moving dielectric, 546. — scattering of by cylinders, 428. ----of by metallic spheres, 437. Liveing and Dewar, dust in electric dis- charge, 103. Lodge, electrical vibration, 333. — electrical resonance, 395. Longitudinal waves of magnetic induction along wires, 802. Love on conjugate functions, 208. Ludeking, passage of electricity through steam, 191. Macfarlane, spark potential, 85, 170. Magnet, permanent, 35. Magnetic force due to the motion of Faraday tubes, 8, 13, 14. — field due to a moving charged sphere, 19. ----due to rotating electrified plates, 23, 28. ----steady, 28. ----induction of current due to change of, 32. — induction, longitudinal waves of, along wires, 302. — properties of iron in rapidly alterna- ting fields, 323. — force, decay of in cylinders, 352. ----of in spheres, 377, 380. — field, effect of on light, 482 et seq. Magnets, action of, on electrodeless dis- charge, 105. ----on negative rays, 121, 134. ----on discharge with electrodes, 131. ----on negative glow, 132. ----on positive column, 138. ----distribution of negative glow over electrodes, 138. ----striations, 141. — reflection of light from, 483 et seq. — action of light passing through thin films, 504. Matteuchi arc discharge, 166. Mechanical effects due to negative rays, 124. ----produced by electric discharge, 174. — force on current, 14. ----a moving charged sphere, 22. ----between flat conductors conveying alternating currents, 300. Meissner, expansion due to discharge, 175, 179, 187. Mercury vapour, discharge through, 110. Metallic vapours, conductivity of, 56. Metals, opacity of, 48, 51. — conduction through, 50. — reflection of light from, 417. — transmissions of light through thin films of, 423. Michell, plane electromagnetic waves, 437. — conjugate functions, 208. Mirrors, parabolic, for electromagnetic waves, 404. Molecular streams, 119. Molecule, electric field required to decom- pose, 193.576 INDEX. Momentum of Faraday tubes, 9, 261, 282. -----a moving electrified sphere, 20. Moulton and Spottiswoode, electric dis- charge, 118, 119, 124, 128. Moving dielectrics, electromotive inten- sity in, 544. -----velocity of light through, 546. Muller and de la Rue, electric discharge, 69, 80, 90, 98, 111, 114, 170, 178, 174. Multiple arc, electrical vibrations along wires in, 341. -----impedance of wires in, 517. -----self-induction of wires in, 516. N a hr wold, leakage of electricity through air, 53, 171. Negative column, 110. — dark space, second, 111. — electrode, quasi, produced by con- traction of tube, 124. -----potential fall at, 155. — glow, 110. -----action of magnet on, 182. -----distribution over electrode, 138. — and positive discharges, difference be- tween, 169. — rays, 119. -----shadows cast by, 120. -----phosphorescence due to, 121, 134. -----action of a magnet on, 121. -----repulsion of, 122, 129. -----mechanical effects produced by, 124. -----opacity of substances to, 125. Negreano, specific inductive capacity, 469. Niven, C., on the functions* S’ and ‘E ’,369. Nowak and Romich, specific inductive capacity, 469. Opacity of metals, 48. — of substances to the negative rays, 125. Oscillations, electrical, on cylinders, 344. -----on spheres, 361. Oscillatory discharge, 331. Oxygen, glow produced by discharge in, 184. Ozone, production of, 179. Ozonizer, 178. Paalzow, electromagnetic waves, 461. Parabolic mirrors for electromagnetic waves, 404. Paschen, spark discharge, 69, 85, 91. Passage of electricity across junction of a metal and a gas, 98. Peace, spark potential, 69-75, 84 et seq., 162. Permanent magnet, 35. Perrot, decomposition of steam, 44, 181 et seq., 190. Phosphorescence, due to magnetic rays, m. -----to positive column, 124. Phosphorescent glow, 180, 184. Plates, rotating electrified, 23-28. Plucker, effect of magnet on discharge, 118, 132. Polarization, 6, 38. — angle of for electromagnetic waves, 406. Positive column, 111. -----velocity of, 115. -----effect of magnet on, 138. -----potential gradient in, 145. -----striations in, 112. Positive and negative discharge, differ- ence between, 169. Potential difference at cathode, 150, 154 et seq. -----required to produce a spark in different gases, 90. — distribution of along discharge tub?, 142. — gradient in positive column, 145, 159. at low pressures, 146. Potier, conjugate functions, 208. Poynting, transfer of energy in electric field, 9. Poynting’s theorem, 308. Pressure, connection between and spark potential, 84 et seq. — critical, 84. Priestley’s History of Electricity, 54, H9. Pringsheim, combination of hydrogen and chlorine, 157. Propagation of light through moving die- lectrics, 544. — velocity of, of slowly alternating currents along a wire, 271. -----of moderately rapid currents along a wire, 277. -----of very rapid currents along a wire, 279. -----of electromagnetic waves along a wire, 451. Pryson and Foster, spark potential, 74. Puluj, dark space, 108. Quincke, transmission of light through thin films, 423. Radiant matter, 121. Rate of decay of slowly alternating currents along a wire, 272. -----of moderately rapid currents, 277. -----of very rapid currents, 279. -----of currents in cylinders, 352. -----of currents in spheres, 377, 380. -----of oscillation on cylinders, 349. -----of oscillation on spheres, 370. -----of oscillation in Hertz’s vibrator, 397.INDEX. 577 Rayleigh, Lord, Theory of Sound, 353, 356, 363, 442. ----concentration of alternating cur- rent, 260. ----metallic reflection, 420. ----scattering of light by fine particles, 432, 449. ----distribution of alternating currents, 510 et seq. Reflection of electromagnetic waves, 398. — — electromagnetic waves from a grating, 406, 425. ----light from metals, 417. ----light from a magnet, 483. Refraction of electromagnetic waves, 406. * Refractive Indices ’ of metals, 421. Repulsion, electromagnetic, 557. — of negative rays, 122, 129. Resistance of a conductor, 47. Resonance, 395. Resonator, 391. Richarz and R. v. Helmholtz, steam jet, 187. Righi, electrification by light, 59, 69. — reflection from a magnet, 483. Ritter, electromagnetic waves, 461. Roberts-Austen, conduction through al- loys, 51. Romich and Nowak, specific inductive capacity, 469. Rontgen, rotating disc, 23, 27. — discharge through gases, 91. Rosa, specific inductive capacity, 469. Rotating electrified plates, 23. — sphere in a symmetrical magnetic field, 535 et seq. ----in an unsymmetrical field, 546 et seq. Rotation of plane of polarization of light, 482. ----of polarization by a thin film, 504. Ronth’s Rigid Dynamics, 479. Rowland, rotating disc, 23. Rowland and Hutchinson, rotating disc, 23-27. Rubens, metallic reflection, 422. — electromagnetic waves, 461. Rubens and Arons, velocity of electro- magnetic waves, 472. Sack, temperature coefficients of electro- lytes, 51. Sage, Le, theory of gravitation, 15. Sarasin and De la Rive, reflection of electromagnetic waves, 400. —- — electromagnetic waves along wires, 459. Scattering of electromagnetic waves by a cylinder, 428. — of electromagnetic waves by a sphere, 437. Schuster, discharge through gases, 69, 80, 108-110, 159, 192. P Schwarz, conjugate functions, 208. Schwarz’s transformation, 208. Screening, electric, 405. Searle, experiment on alternating currents, 512. Self-induction, expression for, for variable currents, 293. ----for flat conductors, 296. ----of two wires in parallel, 516. ----of a net-work of wires, 520. ----and capacity, 529. Shadows cast by negative rays, 120. Siemens, ozonizer, 178. Sissingh, reflection of light from a magnet, 483. 1 Skin,’ electrical, 260, 281. Sohncke, electrification by evaporation, 54. Spark, discharge, 68 et seq. — length, effect of nature of electrodes • on, 69. — length, effect of size of electrodes on, 69. — length, connection between and poten- tial difference, 70 et seq. — length, effect of on critical pressure, 88. — potential difference required to pro- duce in a variable field, 82. — potential effect of pressure on, 84 et seq. — potential in different gases, 90. ----effect of temperature on, 92. — effects of rapid alternations in field on, 185. Specific inductive capacity, 468. * Spectroscopy, Radiant Matter,’ 121. Sphere, charged moving, magnetic force due to, 19. — charged moving, momentum of, 20. — charged moving, kinetic energy of, 21. — charged moving, force acting on, 22. — rotating in a symmetrical magnetic field, 535 et seq. — rotating in an unsymmetrical field, 546 et seq. — electrical oscillations on, 361. — period of these oscillations, 368. — field of force round vibrating, 370. — vibrations of concentric spheres, 372. — decay of electric currents in, 377, 380. — scattering of light by, 437. * Spluttering ’ of electrodes, 60. Spottiswoode, on striations, 113. Spottiswoode and Moulton, electric dis- charge, 118, 119, 124, 129, 130, 141, 143, 144, 196. Stanton, escape of electricity from hot metals, 206. Steady current, motion of Faraday tubes in neighbourhood of, 37. Steam, decomposition of by spark, 181. Steam jet, use of, to detect electrifica- tions, 66. Stokes, theorem, 10. — on the functions * S ’ and ( E ’, 363. p578 INDEX. Stoletow, electrification by light, 59. Striations, 111. — variation of, with density of gas, 111. — effect of magnetic force on, 141. Surface tension, effect of electrification on, 66. Temperature, effect of, on spark potential, 92. — effect of, on conductivity, 51. Thompson, Elihu, electromagnetic repul- sion, 557. Theory of electric discharge, 189. Time of 4 relaxation/ 47. -----vibration of two spheres connected by a wire, 828. -----vibration of electricity on a cylinder, 844, 847. -----vibration of electricity on a sphere, 868. -----vibration of adjacent electrical systems, 582. Times involved in electric discharge, 130. Topler, disturbance produced by spark, 176. Transfer of energy, 9, 308. Transformation, Schwarz’s, 208. Transformer, heat produced in, 318. Trouton, angle of polarization for electro- magnetic waves, 407. — influence of size of reflector on Hertz’s experiments, 437. Trowbridge, decay of vibrations along iron wires, 340. Tube, heat produced in under variable magnetic field, 323. Tubes of electric force, 2. Ultra-violet light, effect of, on electric discharge, 58. Vacuum an insulator, 98. Velocity of Faraday tube, 11. -----positive column, 115. -----propagation of slowly alternating currents along wires, 271. -----propagation of moderately rapid currents along wires, 277. -----propagation of very rapid currents along wires, 279. -----electromagnetic waves along wires, 451. -----light through moving dielectrics, 546. Vibrations along wires in multiple arc, 341. Vibrations, decay of, in Hertz’s vibrator, 897. ------on cylinders, 849. ------on spheres, 370. Vibrations of electrical systems, 828 et seq. ------electrical systems, Henry on, 382. ------electrical systems, Feddersen on, 333. ------electrical systems, Lodge on, 883. ------electrical systems, Lord Kelvin on, 333. Vibrator, Electrical, 388. 4 Volta, potential/ 64. Walker, electromagnetic repulsion, 557. Warburg, leakage of electricity through air, 53. — potential fall at cathode, 155 et seq. Waves, electromagnetic, 388. ------production of, 388. ------reflection of, 398. ------Sarasin’s and de la Hive’s experi- ments on, 400. ------reflection of from grating, 406. ------refraction of from grating, 406. ------theory of reflection of from insula- tors, 407. — — theory of reflection of from metals, 414. ------scattering of from cylinders, 427. ------scattering of from spheres, 487. ------along wires, 451. Wesendonck, positive and negative dis- charge, 170, 171. Wheatstone, velocity of discharge, 115. Wheatstone’s Bridge with alternating current, 527. Wiedemann’s JElehtricitdt, 57. Wiedemann, E., on electric discharge, 108, 167, 168. Wiedemann, E., and Ebert, effect of ultra-violet light, 58. Wiedemann, G. and E., heat produced by electric discharge, 168. Wires, electromagnetic waves along, 451. — Sarasin's and de la Rive’s experiments on, 459. Wolf, effect of pressure on spark poten- tial, 85. Wolt and Lenard, action of ultra-violet light, 54, 58. 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t, the electromotive intensity, as we proceed along the wire, will be a harmonic function of the distance from the end of the wire; if the wave length of this harmonic distribution is A, the velocity of propagation of the disturbance along the wire is defined to be \p/ 2 7r. This velocity ought, if Maxwell’s theory is true, to be equal to F, the velocity with which electrodynamic disturb- ances are propagated through air (see Art. 267). Indeed on this theory the effects observed do in reality travel through the air even though the wire is present, so that the introduction of the wire does not materially alter the physical conditions. The electrical vibrations considered in this chapter are all of very high frequency, being produced by the discharge of condensers through short discharging circuits. In this case (see Art. 269) the electromotive intensity in the region around the wire is at right angles to it, and we may suppose that the phenomena near the wire are due to radial Faraday tubes, with their ends on the wire travelling along it with the velocity of light. 380. ] Considerable interest attaches to some experiments made by Hertz, which seemed to indicate that the velocity along the wire was considerably less than that through the air; and though later experiments have shown that this conclusion is Gg 2452 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [380. erroneous, and that, as Maxwell’s theory indicates, the two velocities are identical, Hertz’s experiments are of great interest both from the methods used and the points they illustrate. In these experiments Hertz (Wied. Ann. 34, p. 551, 1888) used the vibrator described in Art. 325. This was placed in a vertical plane ; behind and parallel to one of the metal plates A, and insulated from it, was a metal plate B of equal area (see Fig. 123). A long wire was soldered to B and bent round so as to come in front of the vibrator and lie in the vertical plane of symmetry of the vibrator about a foot above the base line. The wire, which was above sixty metres long, was taken through a window, and was kept as far as possible from walls, &c., so as to avoid disturbances arising from reflected waves. In the first set of experiments the free end of the wire was insulated. The resonator used was the circular coil of wire 35 cm. in radius pre- viously described. When the plane of the resonator was at right angles to the axis of the vibrator, the electromotive intensity due to the vibrator (apart from the action of the wire) did not (Art. 331) produce any tendency to spark in the resonator, so that the sparks in this position of the resonator must have been entirely due to the disturbance produced by the wire. To observe the effects due to the wire, the resonator was turned round in its own plane until the air gap was at the highest point, and therefore parallel to the wire. When the resonator was moved along the wire the following effects were observed. At the free end of the wire (which was insulated) the sparks in the resonator were extremely small, as the resonator was moved towards the vibrator the sparks increased and attained a maximum; they then381.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 453 decreased again until they almost vanished. If we call such a place a node, then, as the resonator moved along the wire, such nodes were found to occur at approximately equal intervals. 381.] Similar periodic effects were observed when the plane of the resonator was at right angles to the wire, the air gap being vertical; in such a position there would have been no sparks unless the wire had been present. On moving the resonator along the wire the brightness of the sparks changed in a periodic way: the positions however in which the sparks were brightest with the resonator in this position were those in which they had been dullest when the resonator was in its previous position. This result is what we should expect from theoretical con- siderations. For when the resonator is in the first position, with its plane passing through the wire, the air gap is placed parallel to the wire. Now the Faraday tubes travelling along the wire are, as we saw Art. 269, at right angles to it and therefore to the air gap: thus the tubes which fall directly on the air gap do not tend to produce a spark ; the sparks must be due to the tubes collected by the resonator and thrown by it into the air gap. The tubes which travel with their ends on the wire will be reflected from the insulated extremity of it, so that there will be tubes travelling in opposite directions along the wire ; incident tubes travelling from the vibrator to the free end of the wire, and reflected tubes travelling back from the free end to the vibrator. Let us now consider what will happen when the vibrator is in such a position as that represented in Fig. 124. The tube thrown into the air gap by a positive tube, such as CD proceeding from the vibrator, will be of opposite sign to that thrown by a positive tube, such as AB proceeding from the free end : thus in this position of the vibrator the positive tubes454 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [382. moving in opposite directions will neutralize each other’s effects in producing sparks, though they increase the resultant electro- motive intensity: thus, in this case, at the places where the electromotive intensity is greatest there will be no sparks in the resonator, for this maximum intensity will be due to two sets of tubes of the same sign, one set moving in one direction, the other in the opposite. Since the free end of the wire has little or no capacity, no electricity can accumulate there, so that when one set of positive tubes arrives at the free end from the vibrator an equal number of positive tubes must start from the free end and move towards the vibrator ; thus at the free end we have equal numbers of positive (or negative) tubes travelling in opposite directions. We should expect therefore that no sparks would be produced when the resonator was placed close to the free end ; this, as we have seen, was found by Hertz to be the case. When however the resonator is placed in the second position, with its plane at right angles to the wire, the conditions are very different; for the tubes which though they strike the resonator yet miss the air gap, are not hampered by the resonator in their passage through it; thus the resonator does not in this case collect tubes and throw them into the air gap. The sparks are now en- tirely due to the tubes which strike the air gap itself, and thus will be brightest at those points on the wire where the electromotive intensity is a maximum, while at such places, as we have seen, the sparks vanish when the resonator is in the former position. 382.] Hertz found that when the wire was cut at a node the nodes in the portion of the wire which remained were not altered in position, but that they were displaced when the wire was cut at any place other than a node. Hertz also found that the distance between the nodes was in- dependent of the diameter of the wire and of the material of which it was made, and that in particular the positions of the nodes were not affected by substituting an iron wire for a copper one. The distance between the nodes is half the wave length along the wire; thus, if we know the period of the electrical vibrations of the system we can determine the velocity of propagation along the wire. Hertz, by using the formula 2ir\/LC for the wave length of the vibrations emitted by a condenser of capacity (7, whose plates are connected by a discharging circuit whose co-383.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 455 efficient of self-induction is Z, came to the conclusion that the velocity of propagation along the wire was only about 2/3 of that through the dielectric ; there are however many difficulties and doubtful points in the theoretical calculation of the period of vibration of such a system as Hertz’s. 383.] Before discussing these we shall consider another method which Hertz used to compare directly the velocity of propagation along a wire with that through the air. In this method interference was produced in the following way between the waves travelling out from the vibrator through the air and those travelling along the wire. The free end of the wire was put to earth so as to get rid of reflected waves along the wire, and as there were no metallic reflectors in the way of the waves proceeding directly through the air from the vibrator, the only reflected waves of this kind must have come from the floors or walls of the room ; we shall assume for the present that there were no reflected air waves. The resonator was placed so that the air gap was at the highest point and vertically under the wire, and the plane of the resonator could rotate about a vertical axis passing through the middle of the air gap. When the plane of the resonator was at right angles to the wire, the waves proceeding along the latter had no tendency to pro- duce a spark ; any sparks that passed across the resonator must have been entirely due to the waves travelling from the vibrator through the air independently of the wire. In Hertz’s experi- ments when the resonator was in this position the sparks were about 2 mm. long. On the other hand, when the resonator was twisted about the axis so that its plane passed through the wire and was at right angles to the axis of the vibrator, the direct waves through the air from the vibrator would have no tendency to pro- duce sparks; which in this case must have been entirely due to the waves travelling along the wire. In Hertz’s experiments when the resonator was in this position the sparks were again about 2 mm. long. When the resonator was in a position intermediate between these two, the sparks were due to the combined action of the waves travelling along the wire and those coming directly through the air. In such a case the brightness of the sparks would, in general, change when the plane of the vibrator was twisted through a considerable angle. If now the fronts of the two sets of waves were parallel and moving forward with the same456 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [384. velocity, then the effect of turning the plane of the vibrator through a definite angle in a definite direction would be the same at all points on the wire: if however the two waves were travelling at different rates the effect of turning the resonator would vary as it is moved from place to place along the wire. 384.] To prove this, let the electromotive intensity in the air gap due to the wave travelling along the wire be A cos A when the plane of the resonator passes through the wire; here the wire is taken as the axis of z, and A is the wave length of the waves travelling along it. Then, when the plane of the resonator is twisted through an angle <\> from this position, the electromotive intensity in the air gap due to the wire waves will be 2 TT A cos cos — (Vt—z), since the electromotive intensity is approximately proportional to the projection of the resonator on the plane through the wire and the base line of the vibrator. Let the electromotive intensity in the air gap due to the waves coming from the vibrator independently of the wire be, when the plane of the resonator is at right angles to the wire, Bcos^{V't-(z-a)), where A' is the wave length and F the velocity of the air waves; then, if the plane of the resonator is turned until it makes an angle <£ with the plane through the wire and the base line, the electromotive intensity resolved parallel to the air gap is equal to 2?r B sin<£ cos (Vft — (z—a)). Thus, considering both the air waves and those along the wire, the electromotive intensity when the resonator is in this position is equal to A cos <£ cos ^(Vt—0) + B sin<£ cos ^(F'£—(z—a)), which may, since V/\ is equal to F'/A', be written as R cos j~ V(t + e)j,384.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 457 where ii2 = A2 cos2 + If2 sin2 + 2AB cos sm $ cos U—— z -i--^7- av • Now R is the maximum electromotive intensity acting on the air gap, and will be measured by the brightness of the spark. We see from the preceding expression that if X = X', that is, if the velocity of the waves along the wire is the same as that of the air waves which are not affected by the wire, the last term in the expression for R2 will cease to be a periodic function of z, so that in this case there will be no periodic change in the effect produced by a given rotation as we move the resonator along the wire. When however X is not equal to X', the effect on the spark length of a given rotation of the resonator will vary harmonically along the wire. Since in Hertz’s experiments the sparks were about equally long in the two extreme positions, = 0 and <£ = 7j/2, we may in discussing these experiments put A and therefore R2 = A2 (l + 2 cos sin <\> cos j(y- — 2 + ^ aj) 5 thus, if the resonator is rotated so that changes from + to — /3, R2 is diminished by A9 . . _ (/27t 2 7Tx 27r ) 2 A2 sm 2/3 cos |(— - z + ^ ar Thus when 2 7T 7T\ 7-)* + 2 7r X^ (2n+l)|, that is, at places separated by the intervals along the wire the rotation of the resonator will produce no effect upon the sparks, while on one side of one of these posi- tions it will increase, on the other side diminish the brightness of the sparks. If X' were very large compared with X, that is, if the velocity of the waves travelling freely through the air were very much greater than that of those travelling along the wire, the distance between the places where rotation produces no effect would be ^X, which is the distance between the nodes observed in the experiments described in Art. 380. Hertz, how-458 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [384. ever, came to the conclusion that the places where rotation pro- duced no effect were separated by a much greater interval than the nodes. These he had determined to be about 2-8 metres apart, whereas the places where rotation produced no effect seemed to be separated by about 7*5 metres. Assuming these numbers we have X — 5-6, hence A' = 8*94. Thus from these experiments the velocity of the free air waves would appear to be greater than those along the wire in the proportion of 8*94 to 5-6 or 1-6 to 1; or the velocity of the air waves is about half as large again as that of the wire waves. We have, however, in the preceding investigations made several assumptions which it would be difficult to realise in practice; we have assumed, for example, that in the neighbour- hood of the resonator the front of the air waves was at right angles to the wire. Since the resonator was close to the axis of the vibrator this assumption would be justifiable if there had been no reflection of the air waves from the walls or floors of the room. Since the thickness of the walls was small compared with the wave length it is not likely, unless they were very damp, that there would be much reflection from them ; the case of the floor is however very different, and it is difficult to see how reflection from it could have been entirely avoided. Ke- flection from the floor would however introduce waves, the normals to whose fronts would make a finite angle with the wire. The electromotive intensity in the spark gap due to such waves would no longer be represented by a term of the form cos(2tT(V't-z)/X'\ but by one of the form COS (2 Tt{V't — Z cos 0)/A'), where 0 is the angle between the normal to the wave front and the wire. Thus in the preceding investigation we must, for such waves, replace A' by A'sec0, and their apparent wave length along the wire would be A' sec 0 and not A', so that the reflec- tion would have the effect of increasing the apparent wave length of the air waves. The result then of Hertz’s experiments that the wave length of the air waves, measured parallel to the459 385.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. wire, was greater than that of the wire waves, may perhaps be explained by the reflection of the waves from the floor of the room, without supposing that the velocity of the free air waves is different from that of those guided by the wire. 385.] The experiments of Sarasin and De la Rive (Archives des Sciences Physiques et Naturelles Geneve, 1890, t. xxiii, p. 113) on the distance between the nodes (1) along a wire, (2) when pro- duced by interference between direct air waves and waves re- flected from a large metallic plate, seem to prove conclusively that the velocity of the waves guided by a wire is the same as that of free air waves. The experiments on the air waves have already been described in Art. 339; those on the wire waves were made in a slightly different way from Hertz’s experiments. The method used by Sarasin and De la Rive is indicated in Fig. 125. Two metallic plates placed in front of the plates of the vibrator have parallel wires F> F soldered to them, the wires being of equal length and insulated. The plane of the resonator is at right angles to the wires, and the air gap is at the highest point, so that the air gap is parallel to the shortest distance between the wires. The resonator is mounted on a wagon by means of which it can be moved to and fro along the wires, while a scale on the bench along which the wagon slides enables the position of the latter to be determined. The resonator with its mounting is shown in Fig. 126. Sarasin and De la Rive found that as long as the same resonator was used the distance between the nodes as determined by this apparatus was the460 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [385. same as when the nodes were produced by the interference of direct air waves and those reflected from a metallic plate. The relative distances are given in the table in Art. 340, where ‘ A for wire’ indicates twice the distance between the nodes measured along the wire. They found with the wires, as later on they found for the air waves, that the distance between the nodes depended entirely upon the size of the resonator and not upon that of the vibrator; in fact the distance between the nodes was directly proportional to the diameter of the resonator; while it did not seem to depend to any appreciable extent upon the size of the vibrator. These peculiarities can be ex- plained in the same way as the corresponding ones for the air waves, see Art. 341. When the extremities of the wires remote from the vibrator are attached to large metallic plates, instead of being free, the electromotive intensity parallel to the plates at the ends must vanish; hence, whenever a bundle of positive Faraday tubes from the vibrator arrives at a plate an equal number of nega- tive tubes must start from the plate and travel towards the vibrator, while, when the end of the wire is free, the tubes start- ing from the end of the wire in response to those coming from461 386.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. the vibrator are of the same sign as those arriving. Thus, when the end is free, the current vanishes and the electromotive in- tensity is a maximum, while when the end is attached to a large plate the electromotive intensity vanishes and the current is a maximum. Since the sparks in the resonator, when used as in Sarasin and De la Rive’s experiments, are due to the tubes falling directly on the air gap, the sparks will be brightest when the electromotive intensity is a maximum, and will vanish when it vanishes; thus the loops when the ends are free will coincide with the nodes when the wires are attached to large plates. This was found by Sarasin and De la Rive to be the case. A similar point arises in connection with the experiments with wires to that which was mentioned in Art. 342 in connec- tion with the experiments on the air waves. The distance between the nodes, which is half the wave length of the vibra- tion of the resonator, is, as is seen from the table in Art. 340, very approximately four times the diameter; if the resonator were a straight wire the half wave length would be equal to the length of the wire, and we should expect that bending the wire into a circle would tend to shorten the period, we should there- fore have expected the distance between the nodes to have been a little less than the circumference of the resonator. Sarasin and De la Rive’s experiments show however that it was 80 per cent, greater than this: it is remarkable however that the dis- tance of the first node from the end of the wire, which is a loop, was always equal to half the circumference of the resonator, which is the value it would have had if the wave length of the vibration emitted by the resonator had been equal to twice its circumference. 386.] The experiments of Sarasin and De la Rive show that when vibrators of the kind shown in Fig. 113 are used, the oscillations which are detected by a circular resonator are those in the resonator rather than the vibrator. Rubens, Paalzow, Ritter, and Arons (Wied. Ann. 37, p. 529, 1889 ; 40, p. 55, 1890 ; 42, pp. 154, 581, 1891) have used another method of measuring wave lengths, which though it certainly requires great care and labour, yet when used in a particular way would seem to give very accurate results. The method depends upon the change which takes place in the resistance of a wire when it is heated by the passage of a current through462 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [387. it. Rubens finds that the rapidly alternating currents induced by the vibrator can produce heat sufficient to increase the re- sistance of a fine wire by an amount which can be made to cause a considerable deflection in a delicate galvanometer. 387.] Rubens’ apparatus, which is really a bolometer, is arranged as follows. Rapidly alternating currents pass through a very fine iron wire Z. This wire forms one of the arms of a Wheat- stone’s Bridge provided with a battery and a galvanometer. When the rapidly alternating currents do not pass through L this bridge is balanced, and there is no deflection of the galvano- meter. When however a rapidly alternating discharge passes Fig. 127. / through the fine wire it heats it and so alters its resistance, and as the Bridge is no longer balanced the galvanometer is de- flected. This arrangement is so sensitive that it is not neces- sary to place L in series with the wires connected with the plates of the vibrating system. Rubens found if a wire in series with L encircled, without touching, one of the wires EJ, DH in the experiment figured in Fig. 127 (Rubens, Wied. Ann. 42, p. 154, 1871), the deflection of the galvanometer was large enough to be easily measured. The apparatus was so delicate that a rise in temperature of 1/10,000 of a degree in the wire produced a deflection of a millimetre on the gal- vanometer scale. In one of his experiments the wire joined463 388.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. in series with L was bent round two pieces of glass tubing through which the wires EJ, BH passed, the plane of the turns round the glass tube being at right angles to the wires. In this case each turn of the wire and the wire it surrounds acted like a little Leyden jar, and the electricity which flowed through the wire L and disturbed the balance in the Bridge was due to the charging and discharging of these jars. The pieces of glass tube were attached to a frame work, see Fig. 128, which was moved along the wire, and the deflection of the galvanometer observed as it moved along the wire. The 300 250 200 150 100 50 i < /°N qf / \ \ ? / Ox \ \ \ <1 i 1 / l / / t V \ \° \ ~i / i ! \ . V \c c \ / / / d Fig. 129. relation between the galvanometer deflection and the position of the tubes is shown in Fig. 129, where the ordinates represent the deflection of the galvanometer and the abscissae, the distance of the turns in the bolometer circuit from the point F in the wire. The curve shows very clearly the harmonic character of the disturbance along the wire. 388.] The results however of experiments of this kind were not very accordant, and in the majority of his experiments Rubens used another method which had previously been used by Lecher, who instead of a bolometer employed the brightness of the discharge through an exhausted tube as a measure of the intensity of the waves.464 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [388. In these experiments the turns l, m (Fig. 128) in the bolometer circuit were kept at the ends J and H of the main wire (Fig. 127), while a metallic wire forming a bridge between the two parallel wires was moved along from one end of the wires to the other. The deflection of the bolometer depended on the position of the bridge, in the manner represented in Fig. 130, where the ordinates represent the deflection of the galvanometer, the abscissae the position of the bridge. _§JL Kj i\ i i !) i i ! L ! A [ A i i r i j TTT SI f 1 A G * H i 1 —9 ll \ \ 1, l I r c 1 11 l\ Li rf i! i i r i i i \ \ —! l ■> » L A / i l c D ... 1 5 —rrn— jil V, ii T\ i i p i i Wj ^Jl± \ 1 •— a St rnv vf \ \ V \ A li ! IVi • \J V V "V 50 100 150 200 250 300 350 400 450 500 550 Fig. 130. Rubens found that the positions of the bridge, in which the deflection of the galvanometer was a maximum, were independent of the length of the wire connecting the plates of the vibrator to the balls between which the sparks passed, and therefore of the period of vibration of the vibrator. This result shows that the vibrations in the wires which are detected by the bolometer cannot be 4 forced * by the vibrator ; for though, if this were the case, the deflection of the bolometer would vary with the position of the bridge, the places where the bridge produced a maximum deflection would depend upon the period of the vibrator. We can see this in the following way, if the bridge was at a place where the electromotive intensity at right angles to the wire389-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 465 vanished—which, if there were no capacity at the ends J,, H, would be an odd number of quarter wave lengths from these ends—the introduction of the bridge would, since no current would flow through it, produce no diminution in the electro- motive intensity at the ends J, H; in other positions of the bridge some of the current, which in its absence would go to the ends, would be diverted by the bridge, so that the electromotive intensity at the ends would be weakened. Thus, when the deflection of the bolometer was a maximum, the distances of the bridge from the ends J, H would be an odd multiple of a quarter of the wave length of the vibration travelling along the wire; thus, if these vibrations were ‘ forced ’ by the vibrator, the positions of the bridge which give a maximum deflection in the bolometer would depend upon the period of the vibrator. Rubens’ experiments show that this was not the case. We may therefore, as the result of these experiments, assume that the effect of the sparks in the vibrator is to give an electrical impulse to the wires and start the ‘ free ’ vibrations proper to them. The capacity of the plates at the ends of the wire makes the investigation of the free periods troublesome; we may how- ever avail ourselves of the results of some experiments of Lecher’s (Wied. Ann. 41, p. 850, 1890), who found that the addition of capacity to the ends might be represented by supposing the wires prolonged to an extent depending upon this additional capacity. 389.] Let AS, CD, Fig. 131, be the original wires, A a, B/3, Cy, Db the amount by which they have to be prolonged to represent the capacity at the ends, we shall call the wires a/3, yb the ‘equivalent’ wires. Let PQ represent the position of the bridge. APB a----------------------------------------0 Fig. 181. The electrical disturbance produced by the coil may start several systems of currents in the wires a/3, yb. Then there may be a system of longitudinal currents along a/3, yb determined by the condition that the currents must vanish at a, /3, and at y, b. Another system might flow round aPQy, their wave length being determined by the condition that the currents along the wire must vanish at a and y, and that by symmetry the electrification h h466 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [389. at these points must be equal and opposite. A third system of currents might flow round fiPQb, the flow vanishing at and 6. If the bridge PQ were near the ends a, y, we might expect, a priori, that- the current in the circuit aPQy would be the most intense. Since the currents induced in the wires by the coil would tend to distribute themselves so that their self-induction should be as small as possible they would therefore tend to take the shortest course, i. e. that round the circuit aPQy: these cur- rents would induce currents round the circuit (SPQb. Lecher’s experiments (Wied. Ann. 41, p. 850,1890) show that the currents circulating round aPQy, fiPQb are much more efficacious in producing the electrical disturbance at the ends than the longitudinal ones along a/3, y8. As a test of the magnitude of the disturbance at the ends, Lecher used an exhausted tube containing nitrogen and a little turpentine vapour; this was placed across the wires at the ends, and the brilliancy of the luminosity in the tube served as an indication of the magni- tude of the electromotive intensity across /38. In one of his experiments Lecher used a bridge formed of two wires, PQ, P'Q' in parallel, and moved this about until the luminosity in the tube was a maximum; he then cut the wires a/3, yd between PQ and P'Q', so that the two circuits aPQy, ft P'Q'b were no longer in metallic connection. Lecher found that this division of the circuit produced very little diminution in the brilliancy of the luminosity in the tube, though the longitudinal flow of the currents from a to /3 and from y to b must have been almost entirely destroyed by it. Lecher also found that the position of the bridge in which the luminosity of the tube was a maximum depended upon the length of the bridge; if the bridge were lengthened it had to be pushed towards, and if shortened away from the coil, to maintain the luminosity of the tube at its maximum value. He also found that, as might be expected, if the bridge were very short the tube at the end remained dark wherever the bridge was placed, while if the bridge were very long the tube was always bright whatever the position of the bridge. These experiments show that it is the currents round the circuits aPQy, fiPQb which chiefly cause the luminosity in the tube. Since the currents in the circuit fiPQb are in- duced by those in the circuit aPQy, they will be greatest when the time of the electrical vibration of the system aPQy is389.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 467 the same as that of /3PQb. The periods of vibration of these circuits are determined by the conditions that the current must vanish at their extremities and that these must be in opposite electrical conditions ; these conditions entail that the wave lengths must be odd submultiples of the lengths of the circuit. If the two circuits are in unison the wave lengths must be the same, hence the ratio of the lengths of the two circuits must be of the form (2n— l)/(2m — 1), where n and m are integers. This conclusion is verified in a remarkable way by Rubens’ experiments with the bolometer. The relation between the deflections of the bolometer (the ordinates) and the distances of the bridge from G in Fig. 127 (the abscissae) is represented in Fig. 130. The length of the bridge in these experiments was 14 cm., that of the curved piece of the wire EG was 83 cm., and that of the straight portion GJ was 570 cm. The lengths Aa, Bfi which had to be added to the wires to represent the effects of the capacity at the ends were assumed to be 55 cm. for the end of the wire next the coil, and 60 cm. for the end next the bolometer. These two lengths were chosen so as best to fit in with the observations, and were thus really determined by the measurements given in the following table ; in spite of this, so many maxima were observed that the observations furnish satisfactory evidence of the truth of the theory just described. m. n. 2m—1. 2n—1. Distance t maximuni fror Calculated. )/point of deflection nG. Observed. Corresponding point in Fig. 130. 2 1 3 1 51 50 A 4 2 7 3 89 86 B 3 2 5 3 148 143 c 4 8 7 5 181 182 D 2 2 8 3 246 245 E 3 4 5 7 311 305 F 2 8 3 5 343 334 G 2 4 3 7 402 386 H 1 2 1 3 441 443 J 1 3 1 5 506 503 K 1 4 1 7 529 523 L Hha468 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [391. Specific Inductive Capacity of Dielectrics in rapidly alternating Electric Fields. 390. ] Methods analogous to those we have just described have been applied to determine the specific inductive capacities of dielectrics when transmitting electrical waves a few metres long. One of the most striking results of Maxwell's Electromagnetic Theory of Light is the connection which it entails between the specific inductive capacity and the refractive index of a trans- parent body. On this theory the refractive index for infinitely long waves is (Maxwell’s Electricity and Magnetism, vol. ii, Art. 786) equal to the square root of the specific inductive capacity of the dielectric under a steady electric field. 391. ] Some determinations of K, the specific inductive capacity of various dielectrics in slowly varying fields, are given in the following table, which also contains the value of jx2, the square of the refractive index for such dielectrics as are transparent. The letter following the value of jx2 denotes the Frauenhofer line for which the refractive index is measured; when oo is affixed to the value of jx2 the number denotes the square of the refractive index for infinitely long waves deduced from Cauchy’s formula. When /x is given by the observer of the specific inductive capacity this value has been used, in other cases jx has been taken from Landolt’s and Bornstein’s ‘ Physicalisch-Chemische Tabellen.’ Substance. Observer. K. Tempera- ture. Glass, very light flint.... Hopkinson1 6-57 2.375 B „ light flint 5* 6-85 2-478 D ,, dense flint 59 7-4 2631 D „ extra dense flint . . . 99 10-1 2-924 D ,, hard crown 9 9 6-96 „ plate 99 8-45 Paraffin 9J 2-29 2-022 00 Sulphur, along greatest axis Boltzmann2 4-73 4.89 B „ ,, mean axis . y f 8.970 4-154 B „ ,, least axis . 3-811 3-748 B „ non-crystalline . . >? Romich & Nowak3 3-84 Calcite, perpendicular to axis ; 7-7 2-734 A „ along axis 7*5 2*19 7 A 1 Hopkinson, Phil. Trans. 1878, Part I, p. 17, and Phil. Trans. 1881, Part II, p. 855. a Boltzmann, Wien. Beriohte 70, 2nd abth. p. 842, 1874. 3 Romich and Nowak, Wien. Berichte 70, 2nd abth. p. 380, 1874.391-3 EXPERIMENTS on electromagnetic waves, 469 Substance. Observer. K. Tempera- ture. m2 Fluor Spar Romich & Nowak 6-7 2-050 B Mica KlemenSiS1 6-64 2-526 D Ebonite Boltzmann2 3-15 Resin 2-55 Quartz along optic axis . . . Curie3 4-55 241 D ,, perpendicular to axis a 4-49 2-38 D Tourmaline along axis . . . ,, 6-05 2-63 D „ perpendicular to axis 7-10 2-70 D Beryl along axis 6-24 2.48 D ,, perpendicular to axis >5 7-58 2-50 D Topaz „ 6-56 2-61 D Gypsum JJ 6-33 2 32 D Alum 6-4 22 D Rock Salt if 5*85 2 36 D Petroleum Spirit Hopkinson4 1-92 1-922 00 Petroleum Oil, Field’s . . . ft 2*07 2-075 00 „ „ Common . . a 2-10 2-078 00 Ozokerite V 2-13 2-086 00 Turpentine, commercial . . >> 223 2*12S 00 Castor Oil ?? 4-78 2-153 00 Sperm Oil > i 3-02 2-135 00 Olive Oil J > 3-16 2-131 00 Neat’s-foot Oil 1} 3-07 2-125 00 Benzene C6H6 Hopkinson 5 2-38 2-2614 D 99 99 Negreano6 2-2988 25 2-2434 D 99 99 99 2-2921 14 2*2686 D Toluene C7H8 9f 2-242 27 2-224 D a a 99 2-3013 14 2-245 D Toluene Hopkinson5 2-42 2.2470 D Xylene C8H10 a 2-39 ... 2-2238 D }> f> Negreano6 2-2679 27 2-219 B Metaxylene G8H10 r 2-3781 12 2.243 B Pseudocumene C9H12 .... 19 2-4310 14 2.201 B Cymene C10HU 99 2-4706 19 2-201 D a >> Hopkinson5 2-25 2-2254 D Terebenthine C10H16 .... Negreano6 2-2618 20 2-168 D Carbon bisulphide Hopkinson5 2-67 ... 2-673 D fat lO0! Ether ,> 4-75 1 CvU Xv J 1.8055 00 Amylene a 2-05 1.9044 D Distilled Water Cohn and Arons7 76- 15° ? 1*779 D a a Rosa8 75-7 25° 1 KlemenSiS, Wien. Berichte 96, 2nd abth. p. 807, 1887. 2 Boltzmann, Wien. Berichte 70, 2nd abth. p. 342, 1874. *■ Curie, Annales de Chimie et de Physique, 6, 17, p. 385, 1889. 4 Hopkinson, Phil. Trans. 1878, Part I, p. 17, and Phil. Trans. 1881, Part II. p. 355. 5 Hopkinson, Proc. Boy. Soc. 43, p. 161, 1887. 6 Negreano, Compt. rend. 104, p. 425, 1887. 7 Cohn and Arons, Wied. Ann. 33, p. 13, 1888. 8 Rosa, Phil. Mag. [5], 31, p. 188, 1891.470 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [391. Substance. Observer. K. Tempera- ture. Ethyl alcohol (98%) .... Cohn and Arons1 26-5 1-831 00 Amyl alcohol Mixture of Xylene and Ethyl yy » 15. ... 1.951 00 alcohol containing x parts of alcohol in unit volume x = -00 >> 236 = -09 y> yy 3-08 = -17 yy j) 3-98 = -30 yy yy 7-08 = .40 yy yy 9-53 = -50 yy j* 130 = 1. yy yy 26-5 The values of K for the following gases at the pressure of 760 mm. of mercury are expressed in terms of that for a vacuum. In deducing them it has been assumed that for air at different pressures the changes in K are proportional to the changes in the pressure. Gas. Observer. K. Tempera- ture. n2- Air Boltzmann2 1-000590 0° 1.000588 2) j Klemen5i23 1.000586 0° Hydrogen .... Boltzmann2 1-000264 0° 1-000278 JD yy KlemendiS3 1-000264 0° Carbonic acid . . . Boltzmann 2 1*000946 0° 1-000908 D y♦ yy Klemenfiid3 1-000984 0° Carbonic oxide . . Boltzmann 2 1-00069 0° 1.00067 D yy yy Klemen&S3 1.000694 0° Nitrous oxide . . Boltzmann2 1.000994 0° 1-001032 D >y yy KlemenCiS3 1-001158 0° Olefiant gas Boltzmann 2 1 001312 0° 1.001356 JD Marsh gas Boltzmann 2 1.000944 0° 1.000886 Methyl alcohol . . Lebedew4 1.0057 0 0 0 r-H Ethyl alcohol . . . yy 1.001745 D 1-0065 100° (at 0°) Methyl formate. . yy 1.0069 100° Ethyl formate . . 1.0083 100° Methyl acetate . . yy 1.0073 100° Ethyl ether .... KlemendiS3 1.0045 100° >y yy 1.0074 0° 1.003048 2) Carbon bisulphide yy 1.0029 0° 1.00296 D Toluene Lebedew4 1.0043 126° Benzene 1 f> 1.0027 100° 1 Cohn and Arons, Wied. Ann. 33, p. 13, 1888. 2 Boltzmann, JPoyg. Ann. 155, p. 403, 1875. 3 Klemen&S, Wien. Berichte 91, 2nd abth. p. 712, 1885. 4 Lebedew, Wied. Ann. 44, p. 288, 1891.393-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 471 Ayrton and Perry (Practical Electricity, p. 310) found that the specific inductive capacity of a vacuum in which they estimated the pressure to be -001 mm. was about -994. This would make K for air referred to this vacuum as the unit about 1-006, while /ut2 from a vacuum to air is about 1-000588, there is thus a serious discrepancy between these values. 392. ] We see from the above table that for some substances, such as sulphur, paraffin, liquid hydrocarbons, and the permanent gases, the relation K = /x2 is very approximately fulfilled; while for most other substances the divergence between K and \j? is con- siderable. When, however, we remember (1) that even when fx is estimated for infinitely long waves this is done by Cauchy’s formula, and that the values so deduced would be completely invalidated if there were any anomalous dispersion below the visible rays, (2) that Maxwell’s equations do not profess to con- tain any terms which would account for dispersion, the marvel is not that there should be substances for which the relation K=fx2 does not hold, but that there should be any for which it does. To give the theory a fair trial we ought to measure the specific inductive capacity for electrical waves whose wave length is the same as the luminous waves we use to determine the refractive index. 393. ] Though we are as yet unable to construct an electrical system which emits electrical waves whose lengths approach those of the luminous rays, it is still interesting to measure the values of the specific inductive capacity for the shortest electrical waves we can produce. We can do this by a method used by von Bezold (Fogg. Ann. 140, p. 541, 1870) twenty years ago to prove that the velocity with which an electric pulse travels along a wire is independent of the material of wire, it was also used by Hertz in his experi- ments on electric waves. This method is as follows. Let ABGD be a rectangle of wires with an air space at EF in the middle of CD; this rectangle is connected to one of the poles of an induction coil by a wire attached to a point K in AB, then if K is at the middle of AB the pulse coming along the wire from the induction coil will divide at K and will travel to E and F, reaching these points simultaneously; thus E and F will be in similar electric states and there will be no tendency to spark across the air gap EF.472 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [395. If now we move K to a position which is not symmetrical with respect to E and F\ then, when a pulse travels along the rect- angle, it will reach one of these points before the other; their electric states will therefore be different and there will be a tendency to spark. Suppose that with K at the middle point of AB, we insert BC into a di- electric through which electromagnetic disturbances travel more slowly than they do through air, then the pulse which goes round AJD will arrive at E before the pulse which goes round BC arrives at F\ thus E and F will not be in the same electrical state and sparks will therefore pass across the air space. To get rid of the sparks we must either move K towards B or else keep K fixed and, as the waves travel more slowly through the dielectric than through air, lengthen the side AD of the figure. If we do this until the sparks disappear we may conclude that E and F are in similar electric states, and therefore that the time taken by the pulse to travel round one arm of the circuit is the same as that round the other. By seeing how much the length of the one arm exceeds that of the other we can com- pare the velocity of electromagnetic action through the dielectric in which BC is immersed with that through air. 394. ] I have used {Phil. Mag. [5], 30, p. 129,1890) this method to determine the velocity of propagation of electromagnetic action through paraffin and sulphur. This was done by leading one of the wires, say BC, through a long metal tube filled with either paraffin or sulphur, the wire being insulated from the tube which was connected to earth. By measuring the length of wire it was necessary to insert in AD to stop the sparks, I found that the velocities with which electromagnetic action travels through sulphur and paraffin are respectively 1/1-7 and 1/1-35 of the velocity through air. The corresponding values of the specific inductive capacities would be about 2-9 and 1-8. 395. ] Rubens and Arons (Wied. Ann. 42, p. 581; 44, p. 206). while employing a method based on the same principles, have395-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 473 made it very much more sensitive by using a bolometer in- stead of observing the sparks and by using two quadrilaterals instead of one. The arrangement they used is represented in Fig. 133 (Wiecl. Ann. 42, p. 584). The poles P and Q of an induction coil are connected to the balls of a spark gap 8, to each of these balls a metal plate, 40 cm. square, was attached by vertical brass rods 15 cm. long. Two small tin plates x, y, 8 cm. square, were placed at a dis- tance of between 3 and 4 cm. from the large plates. Then wires connected to these plates made sliding contacts at u and v with the wire rectangles ABCD,EFGH *230 cm. by 35 cm. Oneof these rect- angles was placed vertically over the other, the distance between them being 8 cm. The points uy v were connected with each other by a vertical wooden rod, ending in a pointer which moved over a millimetre scale. The direct action of the coil on the rectangles was screened off by interposing a wire grating through which the474 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [395* wires u x, v y were led. The wires CD, GH were cut in the middle and the free ends were attached to small metal plates 5.5 cm. square ; metal pieces attached to these plates went between the plates of the little condensers J, K, L, M, the plates of these condensers were attached cross-wise to each other as in the figure. The two wires connecting the plates were attached to a bolo- meter circuit similar to that described in Art. 387. By means of a sliding coil attached to the bolometer circuit, Arons and Rubens investigated the electrical condition of the circuits uADJ, uBCK, &c., and found that approximately there was a node in the middle and a loop at each end; these circuits then may be regarded as executing electrical vibrations whose wave lengths are twice the lengths of the circuits. If the times of vibrations of the circuits on the left of u, v are the same as those on the right, the plates J and K will be in similar electrical states, as will also L and M, and there will be no deflection of the galvanometer in the bolometer circuit. When the wires are surrounded by air this will be when u, v are at the middle points of AB, EF. In practice Arons and Rubens found that the deflection of the galvanometer never actually vanished, but attained a very decided minimum when u, v were in the middle, and that the effect produced by sliding u, v through 1 cm. could easily be detected. To determine the velocity of propagation of electromagnetic action through different dielectrics, one of the short sides of the rectangles was made so that the wires passed through a zinc box, 18 cm. long, 13 cm. broad, and 14 cm. high; the wires were care- fully insulated from the box; the wires outside the box were straight, but the part inside was sometimes straight and some- times zigzag. This box could be filled with the dielectric under observation, and the velocity of propagation of the electro- magnetic action through the dielectric was deduced from the alteration made in the null position (i. e. the position in which the deflection of the galvanometer in the bolometer circuit was a minimum) of uv by filling the box with the dielectric. Let px and p2 be the readings of the pointer attached to uv when a straight wire of length Dg and a zigzag of length Dh are respectively inserted in the box, the box in this case being empty. Then since in each case the lengths of the circuits on the right and left of uv must be the same, the difference in the396.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 475 lengths of the circuits on the left,, when the straight wire and the zigzag respectively are inserted, must be equal to the differ- ence in the lengths of the circuits on the right. The length of the circuit on the left when the zigzag is in exceeds that when the straight wire is in by iPk ~P%) (Po .Pi)’ while the difference in the length of the circuits on the right is P2~Pl> hence Dk-Dg~(p.1 -Pl) = p2 ~Pl, or Dk—Dg = 2 (p^-px). When the wires are surrounded by the dielectric, Arons and Rubens regard them as equivalent to wires in air, whose lengths are nDg and nDk, where n is the ratio of the velocity of trans- mission of electromagnetic action through air to that through the dielectric; for the time taken by a pulse to travel over a wire of length nDg in air, is the same as that required for the pulse to travel over the length Dg in the dielectric. We shall return to this point after describing the results of these experiments. If p3 and _p4 are the readings for the null positions of uv when the box is filled with the dielectric, then we have, on Arons and Rubens’ hypothesis, n(Dk-Dv) = 2(pi-p.i)-, or, eliminating Dk-Dg,______P±-Pi. 71 — j P2-P1 hence, if pl9 p2, p3, p4 are determined, the value of n follows immediately. In this way Arons and Rubens found as the values of n for the following substances:— _ n. VK. Castor Oil. . . 2-05 2-16 Olive Oil . . . 1*71 1-75 Xylol .... 1-50 1-53 Petroleum . . . 1-40 1-44 The values of if, the specific inductive capacity in a slowly varying field, were determined by Arons and Rubens for the same samples as they used in their bolometer experiments. 396.] The method used by Arons and Rubens to reduce their observations leads to values of the specific inductive capacity476 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [397. which are in accordance with those found by other methods. It is however very difficult to see, using any theory of the action of the divided rectangle that has been suggested, why the values of the specific inductive capacity should be accurately deduced from the observations by this method, except in the particular case when the wires outside the box are very short compared with the wave length of the electrical vibrations. Considering the case of the single divided rectangle, there seem to be three ways in which it might be supposed to act. We may suppose that a single electrical impulse comes to K (Fig. 132), and there splits up into two equal parts, one travelling round AD to E, the other round BG to F. If these impulses arrived at E and F simultaneously they would, if they were of equal intensity, cause the electric states of E and F to be similar, so that there would be no tendency to spark across the gap EF' Thus, if the pulses arrived at 2£and F undiminished in intensity, the condition for there to be no spark would be that the time taken by a pulse to travel from KtoE should be equal to that from K to F. This reasoning is not applicable however when the pulse in its way round one side of the circuit passes through regions in which its velocity is not the same as when passing through air. because in this case the pulse will be partly reflected as it passes from one medium to another, and will therefore proceed with diminished intensity. Thus, though this pulse may arrive at the air gap at the same time as the pulse which has travelled round the other side of the rectangle, it will not have the same intensity as that pulse ; the electrical conditions of the knobs will there- fore be different, and there will therefore be a tendency to spark. When the pulse has to travel through media of high specific inductive capacity the reflection must be very considerable, and the inequality in the pulses on the two sides of the air gap so great that we should not expect to get under any circum- stances such a diminution in the intensity of the sparks as we know from experience actually takes place. We conclude there- fore that this method of regarding the action of the rectangle is not tenable. 397.] Another method of regarding the action is to look on the rectangle as the seat of vibrations, whose period is deter- mined by the electrical system with which it is connected. Thus we may regard the potential at K as expressed by <£0 cos pt\397-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 477 then the condition that there should be no sparks is that the potentials at E and F should be the same. We can deduce the expressions for the potentials at E and F from that at K when E and jPare nodes or loops. Let us consider the case when the capacity of the knobs E and F is so small that the current at E and F vanishes. Then we can easily show by the method of Art. 298 that if there is no discontinuity in the current along the wire, and if the self-induction per unit length of the wire is the same at all points in KADE, and if the portions AK, JDF are in air while AD is immersed in a dielectric in which the velocity of propagation of electromagnetic action is V\ that through air being V> then if the potential at K is <£0 cos pt, that at F is e(lualt0 4>0 cos pt A where A = cos (AD) cos jt (KA + DF) — sin (p AD ) x |Msin(^DF) cos {^KA) + i sin (£KA) cos (^DF)J. and ft = V/V'. The potential at E is ^C0Bpt qob^KE if KE represents the total length KB + BC -f CE, the whole of which is supposed to be surrounded by air. Hence, if the potentials at E and F are the same, we have cos AD) cos y: (KA + DF) — sin ( ^ AD) x jft sin i^DF) cos (£ KA) + ± sin(|, KA) cos (£DF) j = cos ^KE. (1) To make the interpretation of this equation as simple as possible, suppose KA = DF, equation (1) then becomes cos (^r AD) cos (-y KA) - (/x + ~) $ sin AD) sin (~ KA) = cos (fKE). (2)478 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [398. Let us now consider one or two special cases of this equation. Let us suppose that AD is so small that (jx + -) sin AD) is a small quantity, then equation (2) may be written approximately cos ^-KA + \{y. + = cos (^KE); hence 2 KA 4* i (/x2 + 1) AD = KE, hTTW therefore = £(>2+l), so that in this case the process which Arons and Rubens applied to their measurements would give (jx2 +1)/2 and not /x. If, on the other hand, KA is so small that (u -f -) sin KA v jxy V is small, equation (2) may be written approximately cos ^~7AD+ (h + j-) -~EAj = cos (£ EE), or y.AD + U + i) KA = EE, r . , SKE so that h AD = and in this case Arons and Rubens’ process gives the correct result. 398.] A third view of the action of the rectangle, which seems to be that taken by Arons and Rubens, is that the vibrations are not forced, but that each side of the rectangle executes its natural vibrations independently of the other. If the extremities are to keep in the same electrical states, then the times of vibration of the two sides must be equal. Arons and Rubens’ measurements with the bolometer show that there is a loop at K and nodes at E and F. Now if 2 7t/p is the time of vibration of a wire such as KADF with a node at F and a loop at K, surrounded by air along KA, DF, and along AD by a medium through which electromagnetic action travels with the velocity V\ then we can show by a process similar to that in Art. 298 that p is given by the equation i cot AD) - ~ cot (<£■ AD) cot (£ KA) cot (£ DF) + lcot(|^)+cot(fD^) = 0. (3)398.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 479 Let us take the case when KA = DF, then this equation becomes 60 ^>) = (m + J) cot {^KA) or ^ cot2(|,Z^)-l cot KA) = 4 (V + ~) tan ^7 AD. (4) Let us consider the special case when p. A D/V' is small, the solution of (4) is then or If p' is the time of vibration of KBCE with a loop at K and a node at E, this wire being entirely surrounded by air, then hence if p' = p> so that 2 KA + m2 + 1 AD = KE, hKE_____fx2+ 1 OD = ~~2 Arons and Kubens when reducing their observations took the ratio hKE/h AD to be always equal to /x. The above investigation shows that this is not the case when pAD/Vr is small. We might show that hKE/h AD is equal to fx when KA/AD is small. The results given on the third view of the electrical vi- brations of the compound wire seem parallel to those which hold for vibrating strings and bars. Thus if we have three strings of different materials stretched in series between two points, the time of longitudinal vibration of this system is not proportional to the sum of the times a pulse would take to travel over the strings separately (see Eouth’s Advanced Rigid Dy- namics, p. 397), but is given by an equation somewhat re- sembling (3).480 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [4OO, 399. ] The discrepancy between the results of the preceding theory of the action of the divided rectangle and the method employed by Arons and Rubens to reduce their observations, may perhaps explain to some extent the difference between the values of the specific inductive capacity of glass in rapidly alternating electric fields obtained by these observers and those obtained by M. Blondlot and myself for the same quantity. Arons and Rubens (Wied. Ann. 44, p. 206, 1891) determined the ratio of the velocity of electromagnetic action through air to that through glass by filling with glass blocks a box through which the wires on one side of their rectangle passed. Employing the same method of reduction as for liquid dielectrics, they found jx (the ratio of the velocities) to be 2-33, whence K — ^ is 5*43; while the value of K for the same glass, in slowly varying fields, was 5*37, which is practically identical with the preceding value. If, how- ever, we adopted the method of reduction indicated by the pre- ceding theory we should get a considerably smaller value of K. In order to see what kind of diminution we might expect, let us suppose that the circuit through the glass is so short that the relation expressed by (4) holds. This gives the same value for (if+l)/2 as Arons and Rubens get for /x; hence we find K = 3-66, a value considerably less than under steady fields. 400. ] Arons and Rubens checked their method by finding by means of it the specific inductive capacity of paraffin. This substance happens to be one for which either method of reduce tion leads to very much the same result. For example, for fluid paraffin their method of reduction gave jx = VK = 1*47, K = 2-16 ; if we suppose that we ought to have (A + l)/2 instead of /x we get K = 1-94, while the value in slowly varying fields is 1’98 ; so that the result for this substance is not decisive between the methods of reduction. Both M. Blondlot and myself found that the specific inductive capacity of glass was smaller under rapidly changing fields than in steady ones. The following is the method used by M. Blond- lot (Comptes Eendus, May 11, 1891, p. 1058 ; Phil. Mag. [5], 32, p. 230, 1891). A large rectangular plate of copper AA\ Fig. 134, is fixed vertically, and a second parallel and smaller plate BB' forms a condenser with the first. This condenser discharges itself by means of the knobs a, b ; a is connected with the gas pipes, b with one pole of an induction-coil, the other pole of401.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 481 which is connected to the gas pipes. When the coil is working, electrical oscillations take place in the condenser, the period of which is of the order 1/25,000,000 of a second. There is thus on the side of A A' a periodic electric field which has xx as the plane of symmetry. Two square plates, CD, C'D', are placed in this field parallel to A A' and sym- metrical with respect to xx; two wires terminating in EE' are soldered at DD' to the middle points of the sides of these plates. The wires are connected at EE' to two carbon points kept facing each other at a very small distance apart. When the coil is working no sparks are observed between E and E', this is due to the sym- metry of the apparatus. When, however, a glass plate is placed between A A' and CD sparks im- mediately pass between E and E'\ these are caused by the induction received by CD differing from that received by C'D'. By inter- posing between AA' and C' ' a sheet of sulphur of suitable thick- ness the sparks can be made to disappear again. We can thus find the relative thicknesses of plates of glass and sulphur which produce the same effect on the electromagnetic waves passing through them, and we can therefore compare the specific inductive capacity of glass and sulphur under similar electrical conditions. M. Blondlot found the specific inductive capacity of the sulphur he employed by Curie’s method (Annales de Chimie et de Phy- sique, [6], 17, p. 385, 1889), and assuming that its inductive capacity was the same in rapidly alternating fields as in steady ones, he found the specific inductive capacity of the glass to be 2*84, which is considerably less than its value in steady fields. 401.] I had previously (Proc. Roy. Soc. 46, p. 292) arrived at the same conclusion by measuring the lengths of the electrical x , E T ! tE c- A- "Tr -B' -C -A' X Fig. 134.482 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [403. waves emitted by a parallel plate condenser, (1) when the plates were separated by air, (2) when they were separated by glass. The period of vibration of the condenser depends upon its capacity, and this again upon the dielectric between the plates, so that the determination of the periods gives us the means of determining the specific inductive capacity of the glass. The parallel plate condenser loses its energy by radiation slowly, and will thus force the vibration of its own period upon any electrical system under its influence. It differs in this respect from the condenser in Fig. 113, which radiates its energy away so rapidly that its action on neighbouring electrical conductors approximates to an impulse which starts the free vibrations of such systems. The wave lengths in those observations were determined by observations on sparks. This is not comparable in delicacy with the bolometric method of Arons and Rubens; the method was however sufficiently sensitive to show a considerable falling off in the specific inductive capacity of the glass, for which I obtained the value 2*7, almost coincident with that obtained by M. Blondlot. Sulphur and ebonite on the other hand, when tested in the same way, showed no appreciable change in their specific inductive capacity. The Effects produced by a Magnetic Field on Light. 402. ] The connection between optical and electromagnetic phenomena is illustrated by the effects produced by a magnetic field on light passing through it. Faraday was the first to dis- cover the action of magnetism on light; he found {Exper imental Researches, vol. 3, p. 1) that when plane polarized light passes through certain substances, such as bisulphide of carbon or heavy glass, placed in a magnetic field where the lines of force are parallel to the direction of propagation of the light, the plane of polarization is twisted round the direction of the magnetic force. The laws of this phenomenon are described in Maxwell’s Electricity and Magnetism, Chapter XXI. 403. ] Subsequent investigations have shown that a magnetic field produces other effects upon light, which, though they probably have their origin in the same cause as that which pro- duces the rotation of the plane of polarization in the magnetic field, manifest themselves in a different way.403.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 483 Thus Kerr {Phil. Mag. [5], 3, p. 321,1877), whose experiments have been verified and extended by Righi (Annales de Chimie et de Physique, [6], 4, p. 433, 1885; 9, p. 65, 1886; 10, p. 200, 1887), Kundt (Wied. Ann. 23, p. 228, 1884), Du Bois (Wied. Ann. 39, p. 25,1890),and Sissingh {Wied. Ann. 42, p. 115,1891) found that when plane polarized light is incident on the pole of an electromagnet, polished so as to act like a mirror, the plane of polarization of the reflected light is not the same when the magnet is ‘ on ’ as when it is ‘ off/ The simplest case is when the incident plane polarized light falls normally on the pole of an electromagnet. In this case, when the magnet is not excited, the reflected ray is plane polarized, and can be completely stopped by an analyser placed in a suitable position. If the analyser is kept in this position and the electromagnet excited, the field, as seen through the analyser, is no longer quite dark, but becomes so, or very nearly so, when the analyser is turned through a small angle, showing that the plane of polarization has been twisted through a small angle by reflection from the magnetized iron. Righi (1. c.) has shown that the reflected light is not quite plane polarized, but that it is elliptically polarized, the axes of the ellipse being of very unequal magnitude. These axes are not respectively in and at right angles to the plane of incidence. If we regard for a moment the reflected elliptically polarised light as approxi- mately plane polarized, the plane of polarization being that through the major axis of the ellipse, the direction of rotation of the plane of polarization depends upon whether the pole from which the light is reflected is a north or south pole. Kerr found that the direction of rotation was opposite to that of the currents exciting the pole from which the light was reflected. The rotation produced is small. Kerr, who used a small electromagnet, had to concentrate the lines of magnetic force in the neighbourhood of the mirror by placing near to this a large mass of soft iron, before he could get any appreciable effects. By the use of more powerful magnets Gordon and Righi have succeeded in getting a difference of about half a degree between the positions of the analyser for maximum darkness with the magnetizing current flowing first in one direction and then in the opposite. A piece of gold-leaf placed over the pole entirely stops the 1 i 2484 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [404. magnetic rotation, thus proving that the rotation of the plane of polarization is not produced in the air. Hall (Phil. Mag. [5], 12, p. 157, 1881) found that the rotation takes place when the light is reflected from nickel or cobalt, instead of from iron, and is in the same direction as for iron. Righi (h c.) showed that the amount of rotation depends on the nature of the light; the longer the wave length the greater (at least within the limits of the luminous spectrum) the rotation. Oblique Incidence on a Magnetic Pole. 404.] When the light is incident obliquely and not normally on the polished pole of an electromagnet it is necessary, in order to be able to measure the rotation, that the incident light should be polarized either in or at right angles to the plane of incidence, since it is only in these two cases that plane polarized light remains plane polarized after reflection from a metallic surface, even though this is not in a magnetic field. When light polarized in either of these planes is incident on the polished pole of an electromagnet, the light, when the magnet is on, is ellipti- cally polarized after reflection, and the major and minor axes of the ellipse are not respectively in and at right angles to the plane of incidence. The ellipticity of the reflected light is very small. If we regard the light as consisting of two plane polarized waves of unequal amplitudes and complementary phases, then the rotation from the plane of polarization of the incident wave to that of the plane in which the amplitude of the reflected wave is greatest is in the direction opposite to that of the currents which circulate round the poles of the electromagnet. According to Righi the amount of this rotation when the incident light is polarized in a plane perpendicular to that of incidence reaches a maximum when the angle of incidence is between 44° and 68°; while when the light is polarized in the plane of incidence the rotation steadily decreases as the angle of incidence is increased. The rotation when the light is polarized in the plane of incidence is always less than when it is polarized at right angles to that plane, except when the incidence is normal, when of course the two rotations are equal. These results of Righi’s differ in some respects from those of some preceding investigations by Kundt, who, when the light4o6.] experiments on electromagnetic waves. 485 was polarized at right angles to the plane of incidence, obtained a reversal of the sign of the rotation of the plane of polarization near grazing incidence. Reflection from Tangentially Magnetized Iron. 405. ] In the preceding experiments the lines of magnetic force were at right angles to the reflecting surface ; somewhat similar effects are however produced when the mirror is magnetized tangentially. In this case Kerr [Phil. Mag. [5], 5, p. 161, 1878) found:— i. That when the plane of incidence is perpendicular to the lines of magnetic force no change is produced by the magnetiza- tion on the reflected light. 3. No change is produced at normal incidence. 3. When the incidence is oblique, the lines of magnetic force being in the plane of incidence, the reflected light is elliptically polarized after reflection from the magnetized surface, and the axes of the ellipse are not in and at right angles to the plane of incidence. When the light is polarized in the plane of incidence, the rotation of the plane of polarization (that is the rotation from the original plane to the plane through the major axis of the ellipse) is for all angles of incidence in the opposite direction to that of currents which would produce a magnetic field of the same sign as the magnet. When the light is polarized at right angles to the plane of incidence, the rotation is in the same direction as these currents when the angle of incidence is between 0° and 75° according to Kerr, between 0° and 80° according to Kundt, and between 0° and 78° 54'according to Righi. When the incidence is more oblique than this, the rotation of the plane of polarization is in the opposite direction to the electric currents which would produce a magnetic field of the same sign. 406. ] Kerr’s experiments were confined to the case of light reflected from metallic surfaces. Kundt (Phil. Mag. [5], 18, p. 308, 1884) has made a most interesting series of observations of the effect of thin plates of the magnetic metals iron, nickel and cobalt, on the plane of polarization of light passing through these plates in a strong magnetic field where the lines of force are at right angles to the surface of the plates. Kundt found that in these circumstances the magnetic metals possess to an extraordinary degree the power of rotating the486 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [408. plane of polarization of the light. The rotation due to an iron plate is for the mean rays of the spectrum more than 30,000 times that of a glass plate of the same thickness in the same magnetic field, and nearly 1,500 times the natural rotation (i. e. the rotation independent of magnetic force) due to a plate of quartz of the same thickness. The rotation of the plane of polarization is with all three substances in the direction of the currents which would produce a magnetic field of the same sign as the one producing the rotation. The rotation under similar circumstances is nearly the same for iron and cobalt, while for nickel it is decidedly weaker. The rotation is greater for the red rays than for the blue. 407.] The phenomena discovered by Kerr show that when the rapidly alternating currents which accompany light waves are flowing through iron, nickel, or cobalt in a magnetic field, electromotive intensities are produced which are at right angles both to the current and the magnetic force. Let us take, for example, the simple case when light is incident normally on the pole of an electromagnet. Let us suppose that the incident light is polarized in the plane of zx, where 0 = 0 is the equation to the reflecting surface, so that in the incident wave the electro- motive intensities and the currents are at right angles to this plane; Kerr found, however, that the reflected wave had a com- ponent polarized in the plane of yz; thus after reflection there are electromotive intensities and currents parallel to x, that is at right angles to both the direction of the external magnetic field which is parallel to 0 and to the intensities in the incident wave which are parallel to y. The Hall Effect. 408.] In the Philosophical Magazine for November, 1880, Hall published an account of some experiments, which show that when a steady current is flowing in a steady magnetic field electro- motive intensities are developed which are at right angles both to the magnetic force and to the current, and are proportional to the product of the intensity of the current, the magnetic force and the sine of the angle between the directions of these quantities. The nature of the experiments by which this effect was demonstrated was as follows: A thin film of metal was deposited on a glass plate; this plate was placed over the pole of an electromagnet408.] experiments on electromagnetic waves. 487 and a steady current sent through the film from two electrodes. The distribution of the current was indicated by finding two places in the film which were at the same potential; this was done by finding two points such that if they were placed in electrical connection with the terminals of a delicate galvanometer (G) they pro- duced no current through it when the electromagnet was ‘ off.’ If now the current was sent through an electro- magnet a deflection of the galvanometer (0) was pro- Fig. 135. duced, and this continued as long as the electromagnet was ‘ on/ showing that the distribution of current in the film was altered by the magnetic field. The method used by Hall to measure this effect is described in the following extract taken from one of his papers on this subject (Phil. Mag. [5], 19, p. 419, 1885). ‘In most cases, when possible, the metal was used in the form of a thin strip about 1*1 centim. wide and about 3 centim. long between the two pieces of brass B,B (Fig. 135), which, soldered to the ends of the strip, served as electrodes for the entrance and escape of the main cur- rent. To the arms a, a, about 2 millim. wide and perhaps 7 millim. long, were soldered the wires w, w, which led to a Thomson galvanometer. The notches c, c show how adjust- ment was secured. The strip thus prepared was fastened to a plate of glass by means of a cement of beeswax and rosin, all the parts shown in the figure being imbedded in and covered by this cement, which was so hard and stiff as to be quite brittle at the ordinary temperature of the air. c The plate of glass bearing the strip of metal so embedded was, when about to be tested, placed with B, B vertical in the narrow488 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [408. part of a tank whose horizontal section is shown in Fig. 136. This tank, TT, containing the plate of glass with the metal strip was placed between the poles PP of the electromagnet. The tank was filled with water which was sometimes at rest and sometimes flowing. By this means the temperature of the strip of metal was under tolerable control, and the inconvenience from thermoelectric effects at a and a considerably lessened. The diameter of the plane circular ends of the pole pieces PP were about 3*7 centim.’ By means of experiments of this kind Hall arrived at the con- clusion that if a, /3, y; u, v, tv denote respectively the com- ponents of the magnetic force and the intensity of the current, electromotive intensities are set up whose components parallel to the axes of x, y> z are respectively G (fiw — yv), C (yu — aw), C(av — (3u). The values of C in electromagnetic units for some metals at 20° C, as determined by Hall (Phil. Mag. [5], 19, p. 419, 1885), are given in the following table (1. c. p. 436):— Metal. C x 1015. Copper -520 Zinc +820 Iron . . . . +7850 Steel, soft . . . . +12060 „ tempered . . . . . . . +33000 Cobalt +2460 Nickel Bismuth Antimony . . . . +114000 Gold With regard to the magnetic metals, it is not certain that the quantity primarily involved in the Hall effect is the magnetic force rather than the magnetic induction, or the intensity of magnetization. Hall’s experiments with nickel seem to point to its being the last of these three, as he found, using strong magnetic fields, that the effect ceased to be proportional t.o the external magnetic field, and fell off in a way similar to that in which the magnetization falls off when the field is in- creased. We must remember, if we use HalFs value of C for iron and the other magnetic metals, to use in the expression for4o8.] experiments on electromagnetic waves. 489 the electromotive intensities the magnetic induction instead of the magnetic force. For in Hall’s experiments the magnetic force measured was the normal magnetic force outside the iron. Since the plate was very thin the normal magnetic force outside the iron would be large compared with that inside; the normal magnetic induction inside would however be equal to the normal magnetic force outside, so that Hall in this case measured the relation between the electromotive intensity produced and the magnetic induction producing it. Hall has thus established for steady currents the existence of an effect of the same nature as that which Kerr’s experiments proved (assuming the electromagnetic theory of light) to exist for the rapidly alternating currents which constitute light. Here however the resemblance ends ; the values of the coefficient C deduced by Hall from his experiments on steady currents do not apply to rapidly alternating light currents. Thus Hall found that for steady currents the sign of C was positive for iron, negative for nickel; the magneto-optical properties of these bodies are however quite similar. Again, both Hall and Righi found that the G for bismuth was enormously larger than that for iron or nickel. Righi, however, was unable to find any traces of magneto-optical effects in bismuth. The optical experiments previously described show that there is an electromotive intensity at right angles both to the magnetic force and to the electromotive intensity; they do not however show without further investigation on what function of the electromotive intensity the magnitude of the transverse intensity depends. Thus, for example, the complete current in the metal is the sum of the polarization and conduction currents. Thus, if the electromotive intensity is X, the total current v, is given by the equation or if the effects are periodic and proportional to eipt, where K' is the specific inductive capacity of the metal and 4 tt u = K —rr + —P. dt (t We shall confine our attention to periodic currents and suppose that the variables are proportional to €tpt; in this case the pre- ceding equation becomes 4 ttu = (K'tp + 47r/o*) P, dy d^ # but hence we have similarly 4 7tu = dy dz 9 (r.?+w„)P = %-% dy dx (K'ip + 4^)Q = d£. (K^ + i,MR = d£-^; and therefore since da dp dy dx **" dy dz 9 r Trr a / \ dR>\ d2a d a (JT.J, + 4xA)(^-^) = ^ + ^ + (Pa dz2 ’492 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [409. and hence equation (l) becomes . . .da d2a d2a d2 a (K\P + <./>)/-5=er+¥t3? k /T„ , / \ / d 1 d t d\/dy d3\ ~4^(Kip + **^(a°fa+body + C°d^(dv~dzy Similarly we have „ , . ,dp d2p d2p d2$ k / Tr, / \ / d 7 d d\/do. dy^ - Ii(ir‘J’ + i*A)(«.,5 + i.^ + «.a;)((C-%)■ nrr a / \ fdy d2 y d2y d2y (Kip + 4v/0 m 0 5 where l2 + m2 ~ Y2 9 and A0 and B0 are constants. V is the velocity of propagation of electromagnetic action through the air and so is equal to 1 /K, where K is the electro- magnetic measure of the specific inductive capacity of the air, whose magnetic permeability is taken as unity. Waves will be reflected from the surface of the metal, and the amplitudes of these waves will be proportional to et(lx-raz+pt)^ gQ a^ ^ ^ the components of the total magnetic force in the air, will, since409.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 493 it is due to both the incident and reflected waves, be represented by the equations a = j^^€i(lx + mz+pt) €i(lx — mz+pt) P __ p^€i(lx + mz+pt) _j_ j£^i(lx—mz+pt) y =____L A gi(lx + mz+pt) _j^^i(lx—mz +pt) y m 0 m ’ where A and B are constants. We shall suppose that the metal is so thick that there is no reflection except from the face 2=0; in this case the waves in the metal will travel in the negative direction of 2. Thus in the metal we may put a = A'$* + **’*+&) p = ^}'€ 1 (fo + m'z + pt) = ^ 1 (lx + m'z + pt) 7 m' 5 where if m' is complex the real part must be positive in order that the equations should represent a wave travelling in the negative direction of 2; the imaginary part of m' must be negative, otherwise the amplitude of the wave of magnetic force 'W’ould increase indefinitely as the wave travelled along. Substituting these values of a, /3, y in equations (2), we get A' (— p2 ^'+4* pip /22. Thus in the metal we have a _ A^tilx + miZ+pt) u42Cl(^ + wa*+jpOj ta)i €t(lx + m1z +pt) La)2 ^(Ix + m^z +pt) mx 1 m2 2 ’ — A 6*(k + m1*+.pO__^_^j[ ei(Za? + «i2« + pO mj 1 m2 2 We thus see that the original plane wave is in the metal split up into two plane waves travelling with the velocities p/^, p/oo2 respectively. We also see from the equations for a, /3, y that the waves are two circularly polarized ones travelling with different velocities. Starting from this result Prof. G. F. Fitzgerald (Phil. Trans. 1880, p. 691) has calculated the rotation of the plane of polarization produced by reflection from the surface of a transparent medium which under the action of magnetic force splits up a plane wave into two circularly polarized ones; some of the results which he has arrived at are not in accordance with the results of Kerr’s and Rights experiments on the reflec- tion from metallic surfaces placed in a magnetic field, proving that in them we must take into account the opacity of the medium if we wish to completely explain the results of these experiments. 410.] In order to determine the reflected and transmitted waves we must introduce the boundary conditions. We assume (l) that a and /3, the tangential components of the magnetic force, are continuous; (2) that the normal magnetic induction is con- tinuous ; and (3) that the part of the tangential electromotive intensity which is due to magnetic induction is continuous. It should be noticed that condition (3) makes the total tangential J3 = -410.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 495 electromotive intensity discontinuous, for the total electromotive intensity is made up of two parts, one due to electromagnetic induction, the other due to the causes which produce the Hall effect; it is only the first of these parts which we assume to be continuous. If P is the component parallel to x of the total electromotive intensity, P' the part of it due to electromagnetic induction, then P=P' + k(b0w-c0v); but P= - 1 ■- (jr-jr) K ip + 4.-n/ Kip m (Tt ^2^2) lo K'ip+ 4*/! Ax — 0)2 A2). iJ2€2tomv 11 2 353, the last The rotation observed is small, we shall therefore neglect the squares and higher powers of (mx—m2) ; doing this we find from the preceding equations that A _ B “ mm I vm, mJ m) (10) where ilf is the value of mA or m2, when k = 0, and w2 = £2 + AT2. From equation (5) we have, when c0 = 0, -//r+4V.|)/ff + ^+mi2 = ~(K\p + 4T:/a)(l2 + m12)UaQ, —p2f/K' + 4 Ttj/ip/a + P+m2 = -^ (K'lp + 4 */, where 6 and are real quantities, then if the reflected light polarized perpendicularly to the plane of incidence is represented by /3 = cos (pt + Ix—mz), the reflected light polarized in the plane of incidence will be represented by a = 0 cos (pt + lx—mz) — sin (pt + lx—mz)\ Kk498 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [411. thus, unless <#> vanishes, the reflected light will be elliptically polarized. If however 0 and <£ are small, then the angle between the major axis of the ellipse for the reflected light and that of the incident light (regarding this which is plane polarized as the limit of elliptically polarized light when the minor axis of the ellipse vanishes) will be approximately 0. Hence if the ana- lysing prism is set so as to extinguish the light reflected from the mirror when it is not magnetized, the field after magnetiza- tion will be darkest when the analyser is turned through an angle 0, though even in this case it will not be absolutely dark. We proceed now to find 0 from equation (11). We have by Art. (353) l2 + M2 = R2e2ia(l2 + m2), or M2 = (R2c2ia-l)l2 + R2€2iam2. Now for metals the modulus of R2e2La is large, the table in Art. 355 showing that for steel it is about 17; hence we have approximately M2 = R2e2ia(l2 + m2) We shall put //= 1 in the denominator of the right-hand side of equation (11), since there is no evidence that iron and steel retain their magnetic properties for magnetic forces alternating as rapidly as those in the light waves. Making this substitution and putting m = (p/VQ) cos where i is the angle of incidence, we find M \R2< ‘m -l)(Jf+m)=f ( P / 1 Vn ^COS i -R e‘aH------ — cosi) R‘ia ' = ~T7 ~—:(1 — cos2i—jRe‘°cosi) cos %x ' approximately, since the modulus of Rzl° is large. Hence we see A __ ik pa0VQ~~2R€iasmicos2i B 4?r sin2i—Reia cosz so that if k is real, k_____pa0Fp-2 sin3 i cos2 i R sin a 4 7T sin4 i - 2 sin2 i cos i R cos a + R2 cos2 i *412.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 499 This does not change sign for any value of i between 0 and 7r/2 ; this result is therefore inconsistent with Kerr’s and Kundt’s experiments, and we may conclude that the hypothesis on which it is founded—that the transverse intensity is proportional to the total current—is erroneous. As Kerr’s and Kundt’s experiments were made with magnetic metals it seems desirable to consider the results of supposing these metals to retain their magnetic properties. When // is not put equal to unity, 0 is proportional to 2 a'2 cos2 i sin i sin a sin2 i + —^ - cos a cos i) ; this does not change sign for any value of i between 0 and tt/2, so that the preceding hypothesis cannot be made to agree with the facts by supposing the metals to retain their magnetic properties. 412.] Let us now consider the consequence of supposing that the transverse electromotive intensity is proportional not to the total current but to the polarization current; we can do this by potting Jr'tp/4* K\p/i*+ l/o- ' where k' is a real quantity. This equation may be written 4 = K'V2 A Ko fc' R2t2ia Substituting this value of k in equation (11) we find A __ l k'K'p a0 sin i cos2 i B 4 ttR€', where 6' and <£' are real, we find k'K'pa0 sinicos2i(sina sin2i—Usin 2acosi) 4 ttR sin4 i — 2R sin2 i cos i cos a + R2 cos2 i ' ' The angle through which the analyser has to be twisted in order to produce the greatest darkness is, as we have seen, equal to 0' the real part of A/B. Equation (12) shows that this k k %500 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [412. changes sign when i passes through the value given by the equation sin a sin2 i—R sin 2 a cos i = 0, or sin2i =2R cos a cos i; with the notation of the table in Art. 355 this is sin2i = 2 n cos i. If fj! is not equal to unity the corresponding equation may easily be shown to be // sin2 i = 2 n cos i. From the table in Art. 355 we see that for steel 71=2-41, the corresponding value of i when \x = 1 is about 78°, which agrees well with the results of Kerr’s experiments. Hence we see that the consequences of the hypotheses, that the transverse electro- motive intensity is proportional to the polarization current, and that = 1, agree with the results of experiments. We shall now consider the consequences of supposing that the transverse electromotive intensity is proportional to the con- duction current. We can do this by putting 1 _ *" K'lp/lir+l/v where is a constant real quantity. This equation may be written 7 4 IT k"V* ~ ipjPtM' Substituting this value of h in equation (11) we find A __ ¥' sin i cos2 i B Reta (sin2 i — Rt1** cos i) the real part of which is ¥' sin i cos2 i (cos a sin2 i—R cos 2 a cos i) R sin4 i — 2 sin2 i cos iR cos a + R? cos2 i This is the angle through which the analyser must be twisted in order to quench the reflected light as much as possible. The rotation of the analyser will change sign when i passes through the value given by the equation cos a sin2 i = R cos 2 a cos i, R cos a sin2 i = R2 cos 2 a cos i. or414*] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 501 With the notation of Art. 355 this may be written n sin21 = n2 (1 — Jc2) cos i. From the table in Art. 355 we see that 1 — k2 is negative, hence, since n is positive there is no real value of i less than 77/2 which satisfies this equation, so that if this hypothesis were correct there would be no reversal of the direction of rotation of the analyser. Hence of the three hypotheses, (1) that the transverse electro- motive intensity concerned in these magnetic optical effects is proportional to the total current, (2) that it is proportional to the polarization current, (3) that it is proportional to the conduction current, we see that (1) and (3) are inconsistent with Kerrs experiments on the reflection from tangentially magnetized mirrors, while (2) is completely in accordance with them. 413. ] The transverse electromotive intensity indicated by hypothesis (2) is of a totally different character from that discovered by Hall. In Hall’s experiments the electromotive intensities, and therefore the currents through the metallic plates, were constant; when however this is the case the ‘ polarization ’ current vanishes. Thus in Hall’s experiments there could have been no electromotive intensity of the kind assumed in hypo- thesis (2); there is therefore no reason to expect that the order of the metals with respect to Kerr’s effect should be the same as that with respect to Hall’s. It is worth noting that reflection from a transparent body placed in a magnetic field can be deduced from the preceding equations by putting a = 0, since this makes the refractive index real. In this case we see, by equation (12), that the real part of A/B vanishes, so that the reflected light is elliptically polarized, with the major axis of the ellipse in the plane of incidence; any small rotation of the analyser would therefore in this case increase the brightness of the field. 414. ] We now proceed to consider the case of reflection from a normally magnetized mirror. We shall confine ourselves to the case of normal incidence. If the incident light is plane polarized we may (using the notation of Art. 409) put B0 = 0; we have also l = 0, a>x = mx, a>2 = m2, and since the mirror is magnetized normally, a0 = 0,502 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [4H. b0 = 0. Making these substitutions, equations (6), (7), (8), and (9) of Art. 410 become, putting //= 1, AQ + A=:A1 + A2i (13) £=-1^-^), (14) A0-A=m(-1- + M, m9 (15) 7/1 g,yJ = ir.yM,// + <•«) where K is the specific inductive capacity of air. The last equa- tion by means of (5) reduces to m lTip B = p (— — —) or since Ap2 = 7712 (17) B = tm (— — —) • vmt m2' Solving these equations we find B _ tm (tt/j—?/i2) A ~' m1m2"“m2 Now 771! — m2 is small, and we may therefore, if we neglect the squares of small quantities, in the denominator of the expression for B/A, put M for either mx or m2, where M is the value of these quantities when the magnetic field vanishes. We have by equation (5) ik —P2IJ.'K' + 4 Tryfip /t\415.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 505 The four boundary conditions at the surface z = 0 give, if */= 1, A0 + A = Ax + Ax+A2 + A2i \ A0-A = m(^±-^ (18) Km1 m1 m2 m2' J B = -i(A1 + A1'-Az-A2'), B = im(-I _ Al _ :4s + 44) . j (19) vmx m1 "m2 m2 / J The boundary conditions when z = give, writing 6 and a: for — tm-Ji, —im2h respectively, Ct-imik = A1f + 6 + A2 e* + A2'e~t (Ax / C(-imh = m (: m. ^/e~9 [ ^c-^. (20) m. to, TO, De~‘mh = tm (4lf! _ A'e~0 _ Af! V TOj TOj TO2 From equations (19), (20), and (21) we get */«-*), ) + 4AAVf <2I! m0 ) J + $ + ^(l -£) (> + ~J--*<-* (1 ■- *) - 0, mr v m2' 2 ^ m2' The solution of these equations may be expressed in the form At ^(l-=.)(l + _^L)_,-*(,+ “)(!. m27 m1m2>' v m2' v m \ m m, m, ) + 2€g(^l — —) / v m/ m2 /-0 rrt2 v m.mJ x rrioJ K mxm2' v m2 m \ m m2 a2<* "6r mxm2J v v mxm2' v m/ mx -S1) (1 + ^) (1 + —) + 2<-»(> + -) " "*i v m/ v mxm2J v m.y506 EXPERIMENTS ON ELECTROMAGNETIC WAVES. [41 5. {A (m, Now by equations (20) and (21) we have c A^ + A^-o + A^ + AJt-* Substituting the ratios of Alt A/, Ajj, A2 just found, we get _D_ G ~ A (, + 4) (<•-<-»)- A (i + 4) («♦_.-♦,+ AfL v ______ m2 v m2V v_________} m1 m2 v_______ m.} v msJ x ' m, v n%2'v ' m We notice that the numerator vanishes when m1 = m2, which case 0 = : it therefore contains the factor m1—m2; hence, if we neglect the squares and higher powers of (ml — m2)i we may in the denominator put ml = m2 = M and = 6. If the thickness of the film is so small that 0 and are small quantities, then neglecting powers of h higher than the second, we find 2) 77 = 2 -i75-¥ (t —WA). G M2—m2 v J Substituting the value of m22—mt2 from equation (5), and putting M=R*tam, we see that D __pK']c'c0 (t — mh) C 47T € — 2ia 1--------- R2 Since R2 is large for metals we may, as a first approximation, Put D pK'k'c0 . The angle through which the plane of polarization is twisted is equal to the real part of D/C, and is therefore equal to -pK'Vc0mh/4:TT; it is thus to our order of approximation independent of the opacity of the plate. We see from Art. 414 that when light is incident normally on a magnetized mirror the rotation of the plane of polarization of the reflected light is proportional to sin a, and thus depends primarily on the opacity of the mirror, vanishing when the mirror is transparent. The imaginary part of D/G remains finite though h is made415*] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 507 indefinitely small, we therefore infer that the transmitted light is elliptically polarized, and that the ratio of the axes of the ellipse is approximately independent of the thickness of the plate. Let us now consider the light reflected from the plate. We have by equations (18) and (19) B_ _ 2 A ~~ l (At -f A/— A2—A2') A± (1---) 4- A^ f 1 H-^ + A2 (1-) + A2 (1 H-) Substituting the values of Alt J./, A2, A/ previously given, we find, neglecting squares and higher powers of m1 — m2, i {< €»-