GIFT OF MICHAEL REESE ELECTEICAL PAPERS. VOL. II. ELECTRICAL PAPERS BY OLIVER HEAVISIDE IN TWO VOLUMES VOL. II. ifieto gorfe MACMILLAN AND CO. AND LONDON 1894 [All rights reserved] ~06 CONTENTS OF VOL. II. A ART. 31. ON THE ELECTEOMAGNETIC WAVE-SUEFACE. l Scalars and Vectors. 4 Scalar Product. - 4 Vector Product. 5 Hamilton's V. 5 Linear Vector Operators. 6 Inverse Operators. 6 Conjugate Property. 6 Theorem. - 7 ... Transformation-Formula. The Equations of Induction. 7 8 Plane Wave. 3 Index- Surface. The Wave-Surface. ..... 9 11 Some Cartesian Expansions. 16 Directions of B, H, D, and B. 19 Note on Linear Operators and Hamilton's Cubic. - 19 Note on Modification of Index-Equation when c and n are Eotational. 22 ART. 32. NOTES ON NOMENCLATUEE. NOTE 1. Ideas, Words, and Symbols. 23 NOTE 2. On the Eise and Progress of Nomenclature. 25 ABT. 33. NOTES ON THE SELF-INDUCTION OF WIEES. 28 ART. 34. ON THE USE OF THE BEIDGE AS AN INDUCTION BALANCE. 33 ART. 35. ELECTEOMAGNETIC INDUCTION AND ITS PEOPAGA- TION. (SECOND HALF.) SECTION 25. Some Notes on Magnetization. 39 SECTION 26. The Transient State in a Eound Wire with a close- fitting Tube for the Eeturn Current. - 44 vi ELECTRICAL PAPERS. SECTION 27. The Variable Period in a Round Wire with a Concentric Tube at any Distance for the Return Current. - SECTION 28. Some Special Results relating to the Rise of the Current in a Wire. SECTION 29. SECTION 30. Oscillatory Impressed Force at one End of a Line. Its Effect. Application to Long-Distance Telephony and Telegraphy. Impedance Formulas for Short Lines. Resistance of Tubes. - SECTION 31. The Influence of Electric Capacity. Formulae. Impedance SECTION 32. The Equations of Propagation along Wires. Ele- mentary. SECTION 33. The Equations of Propagation. Self-induction. - - Introduction of SECTION 34. Extension of the Preceding to Include the Propagation of Current into a Wire from its Boundary. - SECTION 35. The Transfer of Energy and its Application to Wires. Energy- Current. SECTION 36. Resistance and Self-induction of a Round Wire with Current Longitudinal. Ditto, with Induction Longitudinal. Their Observation and Measurement. SECTION 37. SECTION 38. General Theory of the Christie Balance. Differential Equation of a Branch. Balancing by means of Reduced Copies. Theory of the Christie as a Balance of Self and -..- Mutual Electromagnetic Induction. Felici's In- duction Balance. SECTION 39a. Felici's Balance Disturbed, and the Disturbance Equilibrated. SECTION 39&. Theory of the Balance of Thick Wires, both in the Christie and Felici Arrangements. Transformer with Conducting Core. SECTION 40. Preliminary to Investigations concerning Long- Distance Telephony and Connected Matters. - - SECTION 41. Nomenclature Scheme. Simple Properties of the Ideally Perfect Telegraph Circuit. - SECTION 42. Speed of the Current. Effect of Resistance at the Sending End of the Line. Oscillatory Establishment of the Steady State when both Ends are short-circuited. PAGE 50 55 61 67 71 76 81 86 91 97 102 106 112 115 119 124 128 CONTENTS. VII SECTION 43. Reflection due to any Terminal Resistance, and Establishment of the Steady State. Insulation. Reservational Remarks. Effect of varying the Inductance. Maximum Current. - SECTION 44. Any Number of Distortionless Circuits radiating from a Centre, operated upon simultaneously. Effect of Intermediate Resistance: Transmitted and Reflected Waves. Effect of a Continuous Distribution of Resistance. Perfectly Insulated Circuit of no Resistance. Genesis and Development of a Tail due to Resistance. Equation of a Tail in a Perfectly Insulated Circuit. SECTION 45. Effect of a Single Conducting Bridge on an Isolated Wave. Conservation of Current at the Bridge. Maximum Loss of Energy in Bridge-Coil, with Maximum Magnetic Force. Effect of any Number of Bridges, and of Uniformly Distributed Leakage. The Negative Tail. The Property of the Persistence of Momentum. SECTION 46. Cancelling of Reflection by combined Resistance and Bridge. General Remarks. True Nature of the Problem of Long-Distance Telephony. How not to do it. Non-necessity of Leakage to remoye Distortion under Good Circumstances, and the Reason. Tails in a Distortional Circuit. Complete Solutions. SECTION 47. Two Distortionless Circuits of Different Types in Sequence. Persistence of Electrification, Momen- tum, and Energy. Abolition of Reflection by Equality of Impedances. Division of a Disturbance between several Circuits. Circuit in which the Speed of the Current and the Rate of Attenuation are Variable, without any Tailing or Distortion in Reception. PAGE 132 137 141 146 151 ART. 36. SOME NOTES ON THE THEORY OF THE TELEPHONE, AND ON HYSTERESIS. 155 ART. 37. ELECTROSTATIC CAPACITY OF OVERGROUND WIRES. 159 ART. 38. MR. W. H. PREECE ON THE SELF-INDUCTION OF WIRES. 160 ART. 39. NOTES ON NOMENCLATURE. NOTE 4. Magnetic Resistance, etc. - - - - 165 NOTE 5. Magnetic Reluctance. 168 vfii ELECTRICAL PAPERS. ART. 40. ON THE SELF-INDUCTION OF WIRES. PART 1. Remarks on the Propagation of Electromagnetic Waves along Wires outside them, and the Penetration of Current into Wires. Tendency to Surface Concen- tration. Professor Hughes' s Experiments. - - New (Duplex) Method of Treating the Electromagnetic Equations. The Flux of Energy. Application of the General Equations to a Round Wire with Coaxial Return-Tube. The Differential Equations and Normal Solutions. Arbitrary Initial State. Simplifications. Thin Return Tube of Constant Resis- tance. Also Return of no Resistance. - - - Ignored Dielectric Displacement. Magnetic Theory of Establishment of Current in a Wire. Viscous Fluid Analogy. Magnetic Theory of S.H. Variations of Impressed Voltage and resulting Current. PAGE 168 172 175 178 181 183 PART 2. Extension of General Theory to two Coaxial Conducting Tubes. 185 Electrical Interpretation of the Differential Equations. V Practical Simplification in terms of Voltage and Current C. - 186 Previous Ways of treating the subject of Propagation along Wires. - 190 The Effective Resistance and Inductance of Tubes. - 192 Train of Waves due to S.H. Impressed Voltage. Practi- cal Solution. 194 Effects of Quasi -Resonance. Fluctuations in the Im- pedance. - - - 195 Derivation of Details from the Solution for the Total Current. 197 Note on the Investigation of Simple-Harmonic States. - 198 PART 3. Remarks on the Expansion of Arbitrary Functions in Series. 201 The Conjugate Property U-^- T 1Z in a Dynamical System with Linear Connections. 202 Application to the General Electromagnetic Equations. - 203 Application to any Electromagnetic Arrangements sub- ject to V = ZC. 204 CONTENTS. ix Determination of Size of Normal Systems of V and G to express Initial State. Complete Solutions obtainable L with any Terminal Arrangements provided 7?, S, are Constants. Complete Solutions obtainable when R, S, L are Func- tions of z, though not of p. Effect of Energy in Terminal Arrangements. Case of Coaxial Tubes when the Current is Longitudinal. Also when the Electric Displacement is Negligible. - Coaxial Tubes with Displacement allowed for. Failure to obtain Solutions in Terms of Fand (7, except when Terminal Conditions are F<7 = 0, or when there are no Terminals, on account of the Longitudinal EnergyFlux in the Conductors. Verification by Direct Integrations. State. A Special Initial - The Effect of Longitudinal Impressed Electric Force in the Circuit. The Condenser Method. - - Special Cases of Impressed Force. How to make a Practical Working System of V and G Connections. PART 4. Practical Working System in terms of V and G admitting of Terminal Conditions of the Form V- ZG. Extension to a Pair of Parallel Wires, or to a Single Wire. Effect of Perfect Conductivity of Parallel Straight Con- ductors. Lines of Electric and Magnetic Force strictly Orthogonal, irrespective of Form of Section of Conductors. Constant Speed of Propagation. - Extension of the Practical System to Heterogeneous Circuits, with "Constants" varying from place to place. Examination of Energy Properties. - V The Solution for and G due to an Arbitrary Distribu- tion of e, subject to any Terminal Conditions. - - Explicit Example of a Circuit of Varying Resistance, etc. Bessel Functions. Homogeneous Circuit. Fourier Functions. Expansion of Initial State to suit the Terminal Conditions. - Transition from the Case of Eesistance, Inertia, and Elastic Yielding to the same without Inertia. - - Transition from the Case of Eesistance, Inertia, and Elastic Yielding to the same without Elastic Yielding. PAGE 206 207 208 210 212 215 217 218 219 220 221 222 225 229 231 234 235 ELECTRICAL PAPERS. On Telephony by Magnetic Influence between Distant Circuits. - - PART 5. St. Venant's Solutions relating to the Torsion of Prisms applied to the Problem of Magnetic Induction in Metal Rods, with the Electric Current longitudinal, and with close-fitting Return-Current. - - - Subsidence of initially Uniform Current in a Rod of Rectangular Section, with close-fitting Return-Current. Effect of a Periodic Impressed Force acting at one end of a Telegraph Circuit with any Terminal Conditions. The General Solution. Derivation of the General Formula for the Amplitude of Current at the End remote from the Impressed Force. The Effective Resistance and Inductance of the Terminal Arrangements. Special Details concerning the above. Quickening Effect of Leakage. The Long-Cable Solution, with Magnetic Induction ignored. Some Properties of the Terminal Functions. - PART 6. General Remarks on the Christie considered as an Induction Balance. Full- Sized and Reduced Copies. - Conjugacy of Two Conductors in a Connected System. The Characteristic Function and its Properties. - Theory of the Christie Balance of Self-induction. - Remarks on the Practical Use of Induction Balances, and the Calibration of an Induetometer. - - - Some Peculiarities of Self-induction Balances. Inad- equacy of S.H. Variations to represent Intermittences. Disturbances produced by Metal, Magnetic and Non- magnetic. The Diffusion - Effect. Equivalence of Nonconducting Iron to Self-induction. - - - Inductance of a Solenoid. The Effective Resistance and Inductance pf Round Wires at a given Frequency, with the Current Longitudinal ; and the Corresponding Formulae when the Induction is Longitudinal. - The Christie Balance of Resistance, Permittance, and Inductance. General Theory of the Christie Balance with Self and Mutual Induction all over. Examination of Special Cases. Reduction of the Three Conditions of Balance to Two. PAGE 237 240 243 245 248 250 252 254 256 258 262 265 269 273 277 280 281 284 CONTENTS. xi ... Miscellaneous Arrangements. Effects of Mutual Induc- tion between the Branches. . PAGE 286 PART 7. Some Notes on Part VI. (1). Condenser and Coil Balance. 289 (2). Similar Systems. 290 (3). The Christie Balance of Kesistance, Self and Mutual Induction. 291 (4). Reduction of Coils in Parallel to a Single Coil. - 292 (5). Impressed Voltage in the Quadrilateral. General Property of a Linear Network. - - - 294 Note on Part III. Example of Treatment of Terminal Conditions. Induction-Coil and Condenser. - - 297 Some Notes on Part IV. Looped Metallic Circuits. ... Interferences due to Inequalities, and consequent Limitations of Application. - 302 PART 8. The Transmission of Electromagnetic Waves along Wires without Distortion. - - 307 Properties of the Distortionless Circuit itself, and Effect of Terminal Reflection and Absorption. - - - Effect of Resistance and Conducting Bridges Inter- mediately Inserted. - - Approximate Method of following the Growth of Tails, and the Transmission of Distorted Waves. - - 311 315 318 Conditions Regulating the Improvement of Transmission. 322 ART. 41. ON TELEGRAPH AND TELEPHONE CIRCUITS. APP. A. On the Measure of the Permittance and Retardation of Closed Metallic Circuits. - - 323 APP. B. On Telephone Lines (Metallic Circuits) considered as Induction-Balances. - 334 APP. C. On the Propagation of Signals along Wires of Low Resistance, especially in reference to Long-Distance Telephony. 339 ART. 42. ON RESISTANCE AND CONDUCTANCE OPERATORS, AND THEIR DERIVATIVES, INDUCTANCE AND PERMITTANCE, ESPECIALLY IN CONNECTION WITH ELECTRIC AND MAGNETIC ENERGY. General Nature of the Operators. - - 355 S.H. Vibrations, and the effective R', L', Kf, and #'. 356 ... Impulsive Inductance and Permittance. General Theorem relating to the Electric and Magnetic Energies. - 359 xii ELECTRICAL PAPERS. General Theorem of Dependence of Disturbances solely on the Curl of the Impressed Forcive. - Examples of the Forced Vibrations of Electromagnetic Systems. - .... Induction-Balances General, Sinusoidal, and Impulsive. - - The Resistance Operator of a Telegraph Circuit. The Distortionless Telegraph Circuit. - The Use of the Resistance-Operator in Normal Solutions. - - PAGE 361 363 366 367 369 371 ART. 43. ON ELECTROMAGNETIC WAVES, ESPECIALLY IN RELATION TO THE VORTICITY OF THE IMPRESSED FORCES AND THE FORCED VIBRATIONS OF ELEC- ; TROMAGNETIC SYSTEMS. PART 1. Summary of Electromagnetic Connections. - - - 375 Plane Sheets of Impressed Force in a Nonconducting Dielectric. 376 .... Waves in a Conducting Dielectric. How to remove the Distortion due to the Conductivity. 378 Undistorted Plane Waves in a Conducting Dielectric. - 379 Practical Application. Imitation of this Effect. - - 379 Distorted Plane Waves in a Conducting Dielectric. - - 381 Effect of Impressed Force. True Nature of Diffusion in Conductors. - ... 384 385 Infinite Series of Reflected Waves. Remarkable Identi- ties. Realized Example. 387 Modifications made by Terminal Apparatus. Certain Cases easily brought to Full Realization. - - - 390 Note A. The Electromagnetic Theory of Light. - - 392 NoteB. The Beneficial Effect of Self-Induction. - 393 Note C. The Velocity of Electricity. - - - - 393 PART 2. Note on Part 1. The Function of Self-induction in the Propagation of Waves along Wires. - - 396 PART 3. Spherical Electromagnetic Waves. - - - 402 The Simplest Spherical Waves. - - 403 Construction of the Differential Equations connected with a Spherical Sheet of Vorticity of Impressed Force. - 406 Practical Problem. Uniform Impressed Force in the Sphere. 409 Spherical Sheet of Radial Impressed Force. - - - 414 CONTENTS. xiii Single Circular Vortex Line. An Electromotive Impulse. wi = l. PAGE 414 - - - - 417 Alternating Impressed Forces. - m Conducting Medium, = 1. m A Conducting Dielectric, = 1. - 418 420 422 Current in Sphere constrained to be uniform. - - - 423 PART 4. Spherical "Waves (with Diffusion) in a Conducting Dielectric. The Steady Magnetic Field due to /Constant. - - Variable State when pj p ;= 2 . First Case. Subsiding/. - Second Case. / Constant. Unequal p1 and p2. General Case. - - - - - Fuller Development in a Special Case. involving Irrational Operators. Theorems The Electric Force at the Origin due iofv at r = a. - Effect of uniformly magnetizing a Conducting Sphere surrounded by a Nonconducting Dielectric. - - ....... Diffusion of Waves from a Centre of Impressed Force in a Conducting Medium. Conducting Sphere in a Nonconducting Dielectric. Circular Vorticity of e. Complex Keflexion. Special very Simple Case. Same Case with Finite Conductivity. Sinusoidal Solution. Resistance at the Front of a Wave sent along a Wire. - Reflecting Barriers. Construction of the Operators y1 and y . ... Thin Metal Screens. K Solution with Outer Screen ; x = oo ; /constant. - - PART 5. Alternating/ with Reflecting Barriers. Forced Vibrations, Cylindrical Electromagnetic Waves. .... Mathematical Preliminary. Longitudinal Impressed E.M.F. in a Thin Conducting Tube. - - Vanishing of External Field. J0a = 0. - - - - Case of Two Coaxial Tubes. - - - - - - 424 425 425 425 426 427 429 430 432 433 435 436 438 439 440 441 442 443 444 447 448 449 xiv ELECTRICAL PAPERS. Perfectly Reflecting Barrier. Conduction Current. Its Effects. Vanishing of - # = and #=oo. - E F H = ,9 0. Vanishing of all over, and of and also internally. = .s and #, = 0. Separate Actions of the Two Surfaces of curl e. Circular Impressed Force in Conducting-Tube. - Cylinder of Longitudinal curl of e in a Dielectric. - - Filament of curl e. Calculation of Wave. - - - PART 6. Cylindrical Surface of Circular curl e in a Dielectric. - Jla = 0. Vanishing of External Field. - y = i. Unbounded Medium. s=0. Vanishing of External E. Effect of suddenly Starting a Filament of e. - - - Sudden Starting of e longitudinal in a Cylinder. - - Cylindrical Surface of Longitudinal /, a Function of and*. Conducting Tube, e Circular, a Function of and t. - ART. 44. THE GENERAL SOLUTION OF MAXWELL'S ELECTRO- MAGNETIC EQUATIONS IN A HOMOGENEOUS ISOTROPIC MEDIUM, ESPECIALLY IN REGARD TO THE DERIVATION OF SPECIAL SOLUTIONS, AND THE FORMULAE FOR PLANE WAVES. .... Equations of the Field. General Solutions. Persistence or Subsidence of Polar Fields. - Circuital Distributions. Distortionless Cases. First Special Case. Second Special Case. - Impressed Forces. Primitive Solutions for Plane Waves. - - - Fourier-Integrals. Transformation of the Primitive Solutions (17). . - - - Special Initial States. PAGE 451 451 452 452 453 454 455 456 457 458 459 459 460 461 466 467 468 469 469 470 470 471 472 473 473 474 475 476 CONTENT. xv Arbitrary Initial States. - Evaluation of Fourier-Integrals. - ... Interpretation of Results. POSTCRIPT. On the Metaphysical Nature of the Propagation of the Potentials. PAGE 477 478 479 483 ART. 45. LIGHTNING DISCHARGES, ETC. - 486 ART. 4G. PRACTICE VERSUS THEORY. ELECTROMAGNETIC WAVES. - 488 ART. 47. ELECTROMAGNETIC WAVES, THE PROPAGATION OF POTENTIAL, AND THE ELECTROMAGNETIC EFFECTS OF A MOVING CHARGE. PART 1. The Propagation of Potential. PART 2. Convection Currents. Plane Wave. .... 490 492 PART 3. A Charge moving at any Speed < v. - 494 PART 4. Eolotropic Analogy. - 496 ART. 48. THE MUTUAL ACTION OF A PAIR OF RATIONAL CURRENT-ELEMENTS. 500 ART. 49. THE INDUCTANCE OF UNCLOSED CONDUCTIVE CIR- CUITS. 502 ART. 50. ON THE ELECTROMAGNETIC EFFECTS DUE TO THE MOTION OF ELECTRIFICATION THROUGH A DI- ELECTRIC. Theory of the Slow Motion of a Charge. 504 The Energy and Forces in the Case of Slow Motion. - - - 505 General Theory of Convection Currents. 508 Complete Solution in the Case of Steady Rectilinear Motion. Physical Inanity of ^. - - 510 Limiting Case of Motion at the Speed of Light. Application to a Telegraph Circuit. - - - 511 Special Tests. The Connecting Equations. - - 513 The Motion of a Charged Sphere. The Condition at a Surface of Equilibrium (Footnote). - - - 514 The State when the Speed of Light is exceeded. - 515 A Charged Straight Line moving in its own Line. - - - 516 A Charged Straight Line moving Transversely. - - ..... A Charged Plane moving Tranversely. ... A Charged Plane moving in its own Plane. - 517 517 519 xvi ELECTRICAL PAPERS. PAGE ART. 51. DEFLECTION OF AN ELECTROMAGNETIC WAVE BY MOTION OF THE MEDIUM. - 519 ABT. 52. ON THE FORCES, STRESSES, AND FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. - (Abstract). - General Remarks, especially on the Flux of Energy. - - On the Algebra and Analysis of Vectors without Quaternions. Outline of Author's System. - On Stresses, irrotational and rotational, and their Activities. - The Electromagnetic Equations in a Moving Medium. The Electromagnetic Flux of Energy in a Stationary Medium. Examination of the Flux of Energy in a Moving Medium, and Establishment of the Measure of " True " Current. - - Derivation of the Electric and Magnetic Stresses and Forces from the Flux of Energy. Shorter Way of going from the Circuital Equations to the Flux of Energy, Stresses, and Forces. Some Remarks on Hertz's Investigation relating to the Stresses. Modified Form of Stress-Vector, and Application to the Surface separating two Regions. Quaternionic Form of Stress-Vector. Remarks on the Translational Force in Free Ether. ... Static Consideration of the Stresses. Indeterminateness. . - Special Kinds of Stress Formulae statically suggested. - - Remarks on Maxwell's General Stress A worked-out Example to exhibit the Forcives contained in Different Stresses. - A Definite Stress only obtainable by Kinetic Consideration of the Circuital Equations and Storage and Flux of Energy. - APPENDIX. Extension of the Kinetic Method of arriving at the Stresses to cases of Non-linear Connection between the .... Electric and Magnetic Forces and the Fluxes. Preservation of Type of the Flux of Energy Formula. Example of the above, and Remarks on Intrinsic Magnetization when there is Hysteresis. ABT. 53. THE POSITION OF 4?r IN ELECTROMAGNETIC UNITS. - 521 524 528 533 539 541 543 548 550 552 554 556 557 558 561 563 565 568 570 573 575 INDEX, 579 ELECTRICAL PAPERS. XXXI. ON THE ELECTROMAGNETIC WAVE-SURFACE. [PhU. May., June, 1885, p. 397, S. 5, vol. 19.] MAXWELL showed (Electricity and Magnetism, vol. ii., art. 794) that his equations of electromagnetic disturbances, on the assumption that the electric capacity varies in different directions in a crystal, lead to the Fresnel form of wave-surface. There is no obscurity arising from the ignored wave of normal disturbance, because the very existence of a plane wave requires that there be none. In fact, the electric displace- ment and the magnetic induction are both in the wave-front, and are perpendicular to one another. The magnetic force and induction are parallel, on account of the constant permeability; whilst the electric force, though not parallel to the displacement, is yet perpendicular to the magnetic induction (and force) ; the normal to the wave-front, the electric force, and the displacement being in one plane. The ray is also in this plane, perpendicular to the electric force. There are of course two rays for (in general) every direction of wave-normal, each with separate electromagnetic variables to which the above remarks apply. It is easily proved, and it may be legitimately inferred without a formal demonstration, from a consideration of the equations of induction, that if we consider the dielectric to be isotropic as regards capacity, but eolotropic as regards permeability, the same general results will follow, if we translate capacity to permeability, electric to magnetic force, and electric displacement to magnetic induction. The three principal velocities will be (c/Xj)-*, (c/*2 )-i, and (cfi3 )-t, if c is the constant value ^ of the capacity, and /xp 2 fj.. , are the three principal permeabilities. The wave-surface will be of the same character, only differing in the constants. But a dielectric may be eolotropic both as regards capacity and permeability. The electric displacement is then a linear function of the electric force, and the magnetic induction another linear function of the magnetic force. The principal axes of capacity, or lines of parallelism of electric force and displacement, cannot, in the general case, be assumed to have any necessary relation to the principal axes of permeability, or lines of parallelism of magnetic force and induction. H.E.P. VOL. ii. A 2 ELECTRICAL PAPERS. Disconnecting the matter altogether from the hypothesis that light consists of electromagnetic vibrations, we shall inquire into the conditions of propagation of plane electromagnetic waves in a dielectric which is eolotropic as regards both capacity and permeability, and determine the equation to the wave-surface. For any direction of the normal (to the wave-front, understood) there are in general two normal velocities, i.e., there are two rays differently inclined to the normal whose ray-velocities and normal wave-velocities are different. And for any direction of ray there are in general two ray-velocities, i.e., two parallel rays having different velocities and wave-fronts. In any wave (plane) the electric displacement and the magnetic induction must be always in the wave-front, i.e., perpendicular to the normal. But they are only exceptionally perpendicular to one another. In any ray the electric force and the magnetic force are both perpendicular to the direction of the ray. But they are only exceptionally perpendicular to one another. The magnetic force is always perpendicular to the electric displace- ment, and the electric force perpendicular to the magnetic induction. This of course applies to either wave. If we have to rotate the plane through the normal and the magnetic force through an angle to bring it to coincide with the magnetic induction, we must rotate the plane through the normal and the electric displacement through the same angle in the same direction to bring it to coincide with the electric force, the axis of rotation being the normal itself. In the two waves having a common wave-normal, the displacement of either is parallel to the induction of the other. And in the two rays having a common direction, the magnetic force of either is parallel to the electric force of the other. Nearly all our equations are symmetrical with respect to capacity and permeability ; so that for every equation containing some electric variables there is a corresponding one to be got by exchanging electric force and magnetic force, etc. And when the forces, inductions, etc., are eliminated, leaving only capacities and permeabilities, these may be exchanged in any formula without altering its meaning, although its immediate Cartesian expansion after the exchange may be entirely different, and only convertible to the former expression by long processes. If either /* or c be constant, we have the Fresnel wave-surface. Perhaps the most important case besides these is that in which the principal axes of permeability are parallel to those of capacity. There are then six principal velocities instead of only three, for the velocity of a wave depends upon the capacity in the direction of displacement as well as upon the permeability in the direction of induction. For ^ ^ instance, if /x15 2, and c 1? c 2, c3, are the principal permeabilities and ^ capacities, arid the wave-normal be parallel to the common axis of and Cj, the other principal axes are the directions of induction and dis- placement, and the two normal velocities are (^3)'* and (c^.2}~^ The principal sections of the wave-surface in this case are all ellipses ON THE ELECTROMAGNETIC WAVE-SURFACE. 3 (instead of ellipses and circles, as in the one-sided Fresnel-wave) ; and two of these ellipses always cross, giving two axes of single-ray velocity. But should the ratio of the capacity to the permeability be the same for all the axes (/V^ = = /*2/c2 />i3/c3 ), the wave-surface reduces to a single ellipsoid, and any line is an optic axis. There is but one velocity, and no particular polarization. If the ratio is the same for two of the axes, the third is an optic axis. Owing to the extraordinary complexity of the investigation when written out in Cartesian form (which I began doing, but gave up aghast), some abbreviated method of expression becomes desirable. I may also add, nearly indispensable, owing to the great difficulty in making out the meaning and mutual connections of very complex formulae. In fact the transition from the velocity-equation to the wave-surface by proper elimination would, I think, baffle any ordinary algebraist, unassisted by some higher method, or at any rate by some kind of shorthand algebra, I therefore adopt, with some simplification, the method of vectors, which seems indeed the only proper method. But some of the principal results will be fully expanded in Cartesian form, which is easily done. And since all our equations will be either wholly scalar or wholly vector, the investigation is made independent of quaternions by simply defining a scalar product to be so and so, and a vector product so and so. The investigation is thus a Cartesian one modified by certain simple abbre- viated modes of expression. I have long been of opinion that the sooner the much needed intro- duction of quaternion methods into practical mathematical investigations in Physics takes place the better. In fact every analyst to a certain extent adopts them : first, by writing only one of the three Cartesian scalar equations corresponding to the single vector equation, leaving the others to be inferred ; and next, by writing the first only of the three products which occur in the scalar product of two vectors. This, systematized, is I think the proper and natural way in which quaternion methods should be gradually brought in. If to this we further add the use of the vector product of two vectors, immensely increased power is given, and we have just what is wanted in the tridimensional analytical investigations of electromagnetism, with its numerous vector magni- tudes. It is a matter of great practical importance that the notation should be such as to harmonize with Cartesian formulse, so that we can pass from one to the other readily, as is often required in mixed investiga- tions, without changing notation. This condition does not appear to me to be attained by Professor Tait's notation, with its numerous letter -S prefixes, and especially by the before every scalar product, the negative sign being the cause of the greatest inconvenience in transitions. I further think that Quaternions, as applied to Physics, should be established more by definition than at present ; that scalar and vector products should be defined to mean such or such combinations, thus avoiding some extremely obscure and quasi-metaphysical reasoning, which is quite unnecessary. The first three sections of the following preliminary contain all we 4 ELECTRICAL PAPERS. want as regards definitions ; most of the rest of the preliminary consists of developments and reference-formula?, which, were they given later, in the electromagnetic problem, would inconveniently interrupt the argument, and much lengthen the work. Scalars and Vectors. In a scalar equation every term is a scalar, or algebraic quantity, a mere magnitude ; and + and - have the ordinary signification. But in a vector equation eve :y term stands for a vector, or directed magnitude, and + and -- are to be understood as com- pounding like velocities, forces, etc. Putting all vectors upon one side, we have the general form A + B + C + D+ ... =0; where A, B, .. . , are any vectors, which, if n in number, may be repre- A sented, since their sum is zero, by the n sides of a polygon. Let v A A Aft 3 be the three ordinary scalar components of referred to any set of three rectangular axes, and similarly for the other vectors. This notation saves multiplication of letters. Then the above equation stands for the three scalar equations - ... =o; - ... =o, The - sign before a vector simply reverses its direction that is, negatives its three components. According to the above, if i, j, k, be rectangular vectors of unit length, we have A = +JA + \A^ 2 1s.A 3, ........................... (1) A A A A etc. if ; v 2, 3 be the components of referred to the axes of A kA i, j, k. That is, is the sum of the three vectors iA lt JA 2, of 3, A A A lengths lt 2, 3, parallel to i, j, k respectively. We AB tiralar Product. define thus, p ....................... (2) A and call it the scalar product of the vectors and B. Its magnitude is A that of x that of B x the cosine of the angle between them. Thus. ^ by (1) and (2), = Ai, ^= 2 Aj, ^ 3 = Ak; AN N and in general, being any unit vector, is the scalar component of A N parallel to N, or, briefly, the component of A. Similarly, 2 i =l, = J2 l, k* = l, because i and i are parallel and of length unity, etc. = ij 0, jk = 0, ki = 0, And AB because i and j, for instance, are perpendicular. Notice that = BA. We have also A* A3 and .- A or A"1 - A~ A ON THE ELECTROMAGNETIC WAVE-SURFACE. 5 Thus A" 1 has the same direction as A; its length is the reciprocal of that of A. We VAB Vector Product, define thus, XA& VAB - i(AJBs - AB 3 2) + A&) - + k(^ 2 - AJ}^ ..... (3) VAB A and call the vector product of and B. Its magnitude is that of A x that of B x the sine of the angle between them. Its direction is A perpendicular to and to B with the usual conventional relation between positive directions of translation and of rotation (the vine system). Thus, Vrj=k, Vjk = i, Vki=j. Notice that VAB = - VBA, the direction being reversed by reversing the A order of the letters ; for, by exchanging and B in (3), we negative each term. Hamilton's V- The operator may, since the differentiations are scalar, be treated as a vector, of "* course with either a scalar or a vector to follow it. P scalar we have the vector If it operate on a <> ......................... whose three components are dP/dx, etc. If it operate on a vector A, we have, by (2), the scalar product and, by (3), the vector product - dy ti dz J \ dz dx J \dx dy VA The scalar product is the divergence of the vector A, the amount leaving the unit volume, if it be a flux. The vector product (7) is the curl of A, which will occur below. There are three remarkable theorems relating to V, viz., ....................... (8) (9) (10) Starting with P, a single-valued scalar function of position, the rise in its value from any point to another is expressed in (8) as the lineintegral, along any line joining the points, of VPda, the scalar product of VP and da, the vector element of the curve. A Then passing from an unclosed to a closed curve, let be any vector function of position (single-valued, of course). Its line-integral round 6 ELECTRICAL PAPERS. the closed curve is expressed in (9) as the surface-integral over any surface bounded by the curve of another vector B, which = VvA. Bc?S is the scalar product of B and the vector element of surface f/S, whose direction is defined by its unit normal. Finally, passing from an unclosed to a closed surface, (10) expresses the surface-integral of any vector C over the closed surface (normal positive outward), as the volume-integral of its divergence within the included space. H Linear Vector Operators. If be the magnetic force at a point, B the induction, E the electric force, and D the displacement, all vectors, then B--=/*H, and D = c;E/47r .................... (11) H express the relation of B to and of D to E in a dielectric medium. If it be isotropic as regards displacement, c is the electric capacity ; and if it be isotropic as regards induction, /x is the magnetic permeability ; c and /x are then constants, if the medium be homogeneous, or scalar functions of position if it be heterogeneous. We shall not alter the form of the above equations in the case of eolotropy, when c and p become linear operators. For instance, the induction will always be /xH, to be understood as a definite vector, got H from another vector, in a manner fully defined by (in case we want H H the developments) the following equations (not otherwise needed). Let lt ..., and J5j, ..., be the components of and B referred to any rectangular axes. Then B l (12) where u /* , etc., are constants; which may have any values not making HB = negative; with the identities /x 12 /x21 , etc. Or, when the components are those referred to the principal axes of per- meability, /Xj, /x2, /x3 being the principal permeabilities, all positive. H Inverse Operators. Since B = /xH, we have =1 /x~ B, where 1 /x" is the operator inverse to /x. When referred to the principal axes, we have *'--, **i = ri , /* /-,'-th But when referred to any rectangular axes, we have x by solution of (12). The accents belong to the inverse coefficients. The rest may be written down symmetrically, by cyclical changes of the figures. In the index-surface the operators are inverse to those in the wave surface. A Conjugate Property. The following property will occur frequently. and B being any vectors, ................................ (16) ON THE ELECTROMAGNETIC WAVE-SURFACE. 7 A or the scalar product of and pJB equals that of B and /*A. It only requires writing out the full scalar products to see its truth, which = results from the identities /x12 /x91 , etc. Similarly, A/xcB = /zAcB = c/xAB, etc. , AB = A/z/x^B = /zA/^B, etc. , where in the first line c is another self-conjugate operator. E D is expressed in terms of similarly to (12) by coefficients n c , c la> etc. ; or, as in (13), by the principal capacities cv c 2, c3 . A Theorem. The following important theorem will be required. and B being any vectors, /y^VAB-^V/iA/iB .......................... (17) For completeness a proof is now inserted, adapted from that given by VAB A Tait. Since is perpendicular to and B, by definition of a vector product, therefore AVAB = 0, and BVAB = 0, by definition of a scalar product. Therefore A/x/*-iVAB = 0, and B/^VAB = 0, w~ by introducing = l 1. Hence 0, and /xB/x-!VAB = 0, VAB by the conjugate property ; that is, 1 /*~ is perpendicular to /xA and to /*B. Or where h is a scalar. Or by operating by /x. To find h, multiply by any third vector C (not to A be in the same plane as and B), giving therefore >' by the conjugate property. Now expand this quotient of two scalar products, arid it will be found to be independent of what vectors A, B, C may be. Choose them then to be i, j, k, three unit vectors parallel to the principal axes of /x. Then by the i, j, k properties before mentioned. This proves (17). Transformation-Formula. The following is very useful. A, B, C being any vectors, VAVBC = B(CA)-C(AB) ........................ (18) Here CA and AB are scalar products, merely set in brackets to separate distinctly from the vectors B and C they multiply. This formula is evident on expansion. ELECTRICAL PAPERS. H E The Equations of Induction. and being the electric and magnetic forces at a point in a dielectric, the two equations of induction are [vol. I., p. 449, equations (22), (23)] />iH; ................... ........ ,.(20) c and /z being the capacity and permeability operators, and curl standing VV for as defined in equation (7). Let T and G be the electric and the magnetic current, then r = cE/47r, G = /uH/47r............... ....... (21) The dot, as usual, signifies differentiation to the time. The electric energy is EcE/87r per unit volume, and the magnetic energy H/xH/87r A per unit volume. If is Maxwell's vector potential of the electric current, we have also curlA = /*H, E=-A..................... (21o) Similarly, we may make a vector Z the vector potential of the magnetic current, such that [vol. I., p. 467] -curlZ = cE, H= -Z...................... (22) The complete magnetic energy of any current system may, by a well-known transformation, be expressed in the two ways the 2 indicating summation through all space. Similarly, the electric energy, if there be no electrification, may be written in the two ways If there be electrification, we have also another term to add, the real electrostatic energy, in terms of the scalar potential and electrification. And if there be impressed electric force in the dielectric, part of G will be imaginary magnetic current, analogous to the imaginary electric current which may replace a system of intrinsic magnetization. Plane Wave. Let there be a plane wave in the medium. Its direction N is defined by its normal. Let then be the vector normal of unit length, and z be distance measured along the normal. If v be the velocity of the wave-front, the rate the disturbance travels along the normal, or the component parallel to the normal of the actual velocity of propagation of the disturbance, we have K=f(z-vt), if the wave be a positive one, as we shall suppose, giving < 23 > H applied to or E. Next, examine what the operator VV or curl becomes when, as at present, the disturbance is assumed not to change direction, but only ON THE ELECTROMAGNETIC WAVE-SURFACE. 9 magnitude, as we pass along the normal. Apply the theorem of Version (9) to the elementary rectangular area bounded by two sides parallel to E of length a, and two sides of length b perpendicular to E and in the same plane as E and the normal N. Since its area is ab, and b = dz sin 6, Wave front and the two sides b contribute nothing to the line-integral, we find that curl = VN-^, (24) H applied to E or or other vectors, in the case of a plane wave. Using this, and (23), in the equations of induction (19), (20), they become VN = dz dz Here, since the ^-differentiation is scalar, and occurs on both sides, it VNH= may be dropped, giving us -wE, .............................. (25) VNE= ^H ............................... (26) The induction and the displacement are therefore necessarily in the wave front, by the definition of a vector product, being perpendicular to N. Also the displacement is perpendicular to the magnetic force, and the induction is perpendicular to the electric force. Index-Surface. Let * be a vector parallel to the normal, whose length is the reciprocal of the normal velocity v. It is the vector of the index-surface. By (25) and (26) we have cE = - VsH, therefore - E = c~ lVsH ; (28) and /xH= VsE, -therefore (29) Now use the theorem (17). Then, if * [In order to secure the advantage of black letters for vectors, I have changed the notation thus : The original tE), ........................ (35) {(scs)c-7i/t*}H=es(scH), ........................ (36) and perform on them the inverse operations to those contained in the {}'s, dividing also by the scalar products on the right sides. Then E s/uE (s/iS)/x - me (37) JL = __^_ scH - (scs)c np. (38) Operate by c on (37) and by //, on (38), and transfer all operators to the denominators on the right. Then X-** say> ................... (39) (40) (It should be noted that, in thus transferring operators, care should be taken to do it properly, otherwise it had better not be done at all. Thus, we have by (37), and the left c and the right //, are to go inside the {}. Operate by and then again by +1 {} , thus cancelling the 1 j}" , giving =- l /xs {(s//.s)//. mc}c~ r lo Here we can move c" 1 inside, giving and now operating by p~ l t it may be moved inside, giving as in (39).) ON THE ELECTROMAGNETIC WAVE-SURFACE. tl We can now, by (39) and (40), get as many forms of the index- We equation as we please. know that the displacement is perpendicular to the normal, and so is the induction. Hence 8b 1 = 0, = sb 2 0, ......................... (41) where bj and b 2 are the above vectors, in (39) and (40), are two equivalent equations of the index-surface. Also, operate on (39) by 1 s/wr , and on (40) by 1 Sty*" , and the left members become unity, by the conjugate property ; hence \l = f*&c- l, =l d3/x-ib 2 .................... (42) are two other forms of the index-equation. (41) and (42) are the simplest forms. More complex forms are created with that surprising ease which is characteristic of these operators ; but we do not want any more. When expanded, the different forms look very different, and no one would think they represented the same surface. This is also true of the corresponding Fresnel surface, which is comparatively simple in expression. In any equation we may exchange the operators /x and c. Put = s Ni'" 1 in any form of index-equation, and we have the velocity- equation, a quadratic in v2 giving the two velocities of the wave-front. And if we pub N# = p, thus making p a vector parallel to the normal of length equal to the velocity, it will be the vector of the surface which is the locus of the foot of the perpendicular from the origin upon the tangent-plane to the wave-surface. By (33ft), remembering that s is parallel to the normal, we see that 12 ELECTRICAL PAPERS. But s the vector of the index-surface being = Nv" 1 = 2 p#~ , we have, by (47), dividing it by ^, sr=l .................................... (48) To find the wave-surface, we must therefore let 8 be variable and eliminate it between (48) and any one of the index-equations. This is not so easy as it may appear. General considerations may lead us to the conclusion that the equation to the wave-surface and that to the index-surface may be turned one into the other by the simple process of inverting the operators, turning c into c~ l and ^ into p~ l. Although this will be verified later, any form of index-equation giving a corresponding form of wave by inversion of operators, yet it must be admitted that this requires proof. That it is true when one of the operators c or p is a constant does not prove that it is also true when we have the inverse compound operator 1 {(scs)/*" -nc~ l }~ 1 containing both c and /z, neither being constant. I have not found an easy proof. This will not be wondered at when the similar investigations of the Fresnel surface are referred to. Professor Tait, in his "Quaternions," gives two methods of finding the wave- surface ; one from the velocity-equation, the other from the index- equation. The latter is rather the easier, but cannot be said to be very obvious, nor does either of them admit of much simplification. The difficulty is of course considerably multiplied when we have the two operators to reckon with. I believe the following transition from index to wave cannot be made more direct, or shorter, except of course by omission of steps, which is not a real shortening. Given * 1 = --^1 ........... (49) (39) to 8^ = 0, ............................ (50) -(41) to = rs=l ............................. (51) (48) bis We Eliminate s and get an equation in r. have also which will assist later. = 1 /KSe- !)^!, .................... (52) (42) bis By (49) we have = S (s/>ts)c- 1 b 1 -w/>i- 1 b1 ......................... (53) Multiply by bj and use (50) ; then - = )-m(b sXb t (s/> b 1 lC 1 t-ib 1/J 1 ) ..................... (54) By differentiation, s being variable, and therefore b x also, = 20s/>ts)(b1c- 1 b1 )4-2(s/xs)0b1 c- 1 b1 ) - 2m(db1/*- 1b1 ) ....... (55) Also, differentiating (53), = + ^ da 2(ds/zs)c~ 1 b 1 (BftB)rfc~ 1 b 1 - 1 mdfjL' ; and multiplying this by 2b x gives bl ). ...(56) ON THE ELECTROMAGNETIC WAVE-SURFACE. 13 Subtract (55) from (56) and halve the result ; thus obtaining or = {b 1 -(b 1 c- 1 b1 )/xs}^s ....................... (57) In the last five equations it will be understood that da and db, are differential vectors, and that da^a is the scalar product of da and /xs, etc. ; also in getting (56) from the preceding equation we have \l \dc~ = bjC-^bj = l d\c- \> lt etc. Equation (57) is the expression of the result of differentiating (50), with db eliminated. x Now (57) shows that the vector in the {} is perpendicular to da, the variation of a. But by (51) we also have, on differentiation, = rds ................................... (58) Hence r and the { } vector in (57) must be parallel. This gives hr^-^c-^a, ........................... (59) \ where k is a scalar. If we multiply this by c~ l and use (52), we obtain rc^b^O; ............................... (60) \ or, by (49), giving in terms of cE, rE = 0, ................................... (61) a very important landmark. The ray is perpendicular to the electric force. Similarly, if we had started from instead of (49), (50), and (52) H the corresponding equations, viz., with of course the same equation (51) connecting r and a, we should have arrived at = &'r b2 -(b2/M-ib2 )cs; .......................... (62) h' being a constant, corresponding to (59) ; of this no separate proof is needed, as it amounts to exchanging /x and c and turning E into H, to make (39) become (40). And from (62), multiplying it by ft~ 1 b2, we arrive at = r/A -ib 2 0, or rH = 0, ..................... (63) corresponding to (61). The ray is thus perpendicular both to the electric and to the magnetic force. The first half of the demonstration is now completed, but before giving the second half we may notice some other properties. Thus, to determine the values of the scalar constants h and h'. Multiply (59) by a, and use (50) and (51) ; then h= - - 14 ELECTRICAL PAPERS. the second form following from (54). Insert in (59), then (64) gives r explicitly in terms of /*s and b 1? the latter of which is known in terms of the former by (49). Multiply this by 1 fi" ^, using (50) ; then -m- r/^-ib^ 1 ............................... (65) Similarly we shall find h'= - ^(bjje-ng, ............................. (66) ' ........................ (67) and, corresponding to (65), we shall have = -IT Tc- l b 2 1 ............................... (68) Now to resume the argument, stopped at equation (63). Up to equation (59) the work is plain and straightforward, according to rule in fact, being merely the elimination of the differentials, and the getting of an equation between r and s. What to do next is not at all obvious. From (59), or from (64), the same with h eliminated, we may obtain all sorts of scalar products containing r and b 1? and if we could put b x explicitly in terms of r, (60) or (65) would be forms of the wave-surface equation. From the purely mathematical point of view no direct way presents itself; but (61) and (63), considered physically as well as mathematically, guide us at once to the second half of the transforma- tion from the index- to the wave-equation. As, at the commencement, we found the induction and the displacement to be perpendicular to the normal, so now we find that the corresponding forces are perpendicular to the ray. There was no difficulty in reaching the index-equation before, when we had a single normal with two values of v the normal velocity, and two rays differently inclined to the normal. There should then be no difficulty, by parallel reasoning, in arriving at the wave- surface equation from analogous equations which express that the ray is perpendicular to the magnetic and electric forces, considering two parallel rays travelling with different ray-velocities with two differently inclined wave-fronts. Now, as we got the index-equation from VNH= --roE, ........................... (25) bis VNE = p/*H, ........................... (26) Us Mwe must have two corresponding equations for one ray-direction. Let be a unit vector defining the direction of the ray, and w be the ray- velocity, so that = wM r ................................... (69) Operate on (25) and (26) by VM, giving VMVNH= - VMVNE= ON THE ELECTROMAGNETIC WAVE-SURFACE. 15 Now use the formula of transformation (18), giving N(HM) - H(MN) = - N(EM) - E(MN) = v HM EM But = and = 0, as proved before. Also v = w(NN), or the wave-velocity is the normal component of the ray-velocity. Hence .............................. (70) .............................. (71) which are the required analogues of (25) and (26). Or, by (69), H = VrcE, -E = Vr/*H .................................. (72) .................................. (73) are the analogues of (28) and (29). The rest of the work is plain. H Eliminating E and successively, we obtain = E + Vr/zVrcE, = H + VrcVr/xH; and, using the theorem (17), these give =E+ which, using the transformation-formula (18), become = E + mfi- l T( H = + wc- 1 r(c or, rearranging, after operating by ^ and c respectively, i)mc - /x}E = 1 mr(ft~ rcE), Or = _^ r r =g l , say, (74) TT _, = & = = -~ c- lifjR =-r 7 1 (rcr !)/* - n~ rl c say 75 () These give us the four simplest forms of equation to the wave. For, since rE = = rH, we have = rg! 0, rg2 = (76) Also, operating on (74) by p~ l rc and on (75) by c" 1!/* we get two other forms. = 1 /x~ rcg1 l, c -1r/>tg2 =l, (77) gl and g2 differ from b x and b 2 merely in the change from a to r, and in the inversion of the operators. The two forms of wave (76) are analogous to (41), and the two forms (77) analogous to (42), inverting operators and putting r for s. Similarly,, if the wave-surface equation be given and we require that 16 ELECTRICAL PAPERS. of the index-surface, we must impose the same condition rs = 1 as before, and eliminate r. This will lead us to scg^O, *P8i=-> ....................... (78) corresponding to (60) and (65) ; and = s/*g2 0, scg2 =-rc, ............. .......... (79) corresponding to (63) and (68); and the first of (78) and (79) are equivalent to or the displacement and the induction are perpendicular to the normal. This completes the first half of the process ; the second part would be the repetition of the already given investigation of the index-equation. The vector rate of transfer of energy being VEH/4?r in general, when a ray is solitary, its direction is that of the transfer of energy. It seems reasonable, then, to define the direction of a ray, whether the wave is plane or not, as perpendicular to the electric and the magnetic forces. On this understanding, we do not need the preliminary investigation of the index-surface, but may proceed at once to the wave-surface by the investigation (69) to (77), following equations (25) and (26). The following additional useful relations are easily deducible : (25) and (26) we get From ^ ............................... ( and from (72) and (73), -s.................................... <"> Also, from either set, EcE = H/xH, ..................... .....-..-....:. (8-2) expressing the equality of the electric to the magnetic energy per unit volume (strictly, at a point). Some Cartesian Expansions. In the important case of parallelism of the principal axes of capacity and permeability, the full expressions for the index- or the wave-surface equations may be written down at once from the scalar product abbreviated expressions. Thus, taking any equation to the wave, as the first of (76), for example, igl = t gl being given in (74), take the axes of coordinates parallel to the common principal axes of c and /*; so that we can employ cv c 2, c a, the principal capacities, and pv p.2, /*3 the principal permeabilities in the three com- We ponents of gr then have, x t y^z being the coordinates of r, X2 2 I/ + Z2 ~l -~ ~ ' where = r/x l i h H--- A*i f*a /*3 In (83) we may exchange the c's and /x's, getting the second of (76). Similarly the first of (77) gives ON THE ELECTROMAGNETIC WAVE-SURFACE. 17 as another form, in which, again, the yu's and c's may be exchanged (not m forgetting to change into n) to give a fourth form. These reduce = = C i c 2 cy to the Fresnel surface if either == /z x /x 2 /*3 or Let x = to find the sections in the plane y, z. The first denominator in (83) gives -r - representing an ellipse, semiaxes % = (%)-* and The other terms give Or % an ellipse, semiaxes v 31 = (c^) ~ * and = (ca/^1 )~*. Similarly, in the plane 2, a; the sections are ellipses whose semiaxes are # 21 , fl23 , and v 12 , = v 32 , where for brevity vrs ~* (crfj,s ) ; and in the plane #, y, the ellipses have semiaxes %, and 32 , 13 , #12. In one of the principal planes two of the ellipses intersect, giving four places where the two members of the double surface unite. = = If GJ//ZJ c 2 //x 2 c 3 //A3 , we have a single ellipsoidal wave-surface whose equation is ++=1 .............................. 85) Now, of course, = y< %> 12 etc. When the p and c axes are not parallel, we cannot immediately write down the full expansion of the wave-surface equation. Proceed thus : Taking Tgl = as the equation, let R= m(i^- l i), and a = m~ g1 1 ; then, by (74) and (76), r r He- p = 0, or ra = 0, where = r (Jfc-/x)a.................................. (86) R is a scalar. If 15 a 2, a s are the three components of a referred to any rectangular axes, and x, y, z the components of r, we have, by (86) and (12), x = (Ecu - y = (Ec21 - = z - (jBc81 % from which a lt ft 2 , may be solved in terms of x, y, z \ thus H.E.P. VOL. II. 18 ELECTRICAL PAPERS. where, by using (15), and the rest by symmetry. Then, since ra = xa 1 + ya.2 + za B = 0, we get the full expansion. A need not be written fully, as it goes out. The equation may be written symmetrically, thus, = 1+ + + I I mn(in~ I)(TG~ T) p - < x2 (c22 BB c33/x22 2c23/x23) ... where the coefficients of 2 y, z2 , yz, and zx are omitted. = and n c-,c9Co whilst Here m = where c^, ..., are the inverse coefficients. See equation (15). The expansion of Tp~ lT is exactly similar, using the inverse /* coefficients. If in (87) we for every c or /x write the reciprocal coefficients, we obtain the equation to the index-surface ; that is, supposing x, y, z then to be the components of s instead of r. And, since sy = N, the unit wave-normal, we have the velocity-equation as follows, in the general case, ^Va - - 3 cf8 /4, + ...... (88) ...}, N N in which JV lf 2, B are the components of N, or the direction-cosines ^ N of the normal. To show the dependence of v2 upon the capacity and permeability perpendicular to N, take = 1, = 2 Q, ^3 = 0, which does not destroy generality, because in (88) the axes of reference are arbitrary. Then (88) reduces to - = o. When the /x and c axes are parallel, and their principal axes are those of reference, we have K+ 4)}, (89) where %=(c with a similar expression for NcN, and 2 /*3 )~^, etc., as before. The solution is ...... (90) NM W where X= Nfuf + Nfu* + - 2(JV1 2 2 w 1 w2 + NiNfu^ + Nf in which =- ^= - M= - ON THE ELECTROMAGNETIC WAVE-SURFACE. 19 = = Take U L 0, or c2//x2 c3//*3 ; the two velocities (squared) are then M4 Mi N^ Nfvj, + + Nffi* and Nfv^ + + N reducing to one velocity v23 when = 1. t If, further, u 2 = 0, or w 3 = 0, making always, and = = CJ//AJ 2//u 2 c 3//x3 , Jf=0 ........................ (92) is the single value of the square of velocity of wave-front. We Directions of E, H, D, and B. may expand (45) to obtain an equation for the two directions of the induction and displacement. Thus, since A - = i(c'n t/ the determinant of the coefficients of i, j, k equated to zero gives the When required equation. the principal axes of //, and c are parallel, the equation greatly simplifies, being then (93) where u v ..., are the same differences of squares of principal velocities D as in (91). For v etc., write v etc. ; and we have the same equa- tion for the induction directions. For A, etc., write c^, etc., and the D resulting equation gives the directions of E. For lt etc., write etc., and the resulting equation gives the directions of H. Note on Linear Operators and Hamilton's Cubic. (June 12th, 1892.) [The reason of the ease with which the transformations concerned in the above can usually be effected is, it will be observed, the symmetrical property AcB = BcA of the scalar products. But when a linear operator, say c, is not its own conjugate, some change of treatment is required. Thus, let D E # = u + + E^ c l c12 2 c 13 3, D{ D ^ E ^ = 3 c 81 + + c32 2 c33 3, ^ E # = + + Di c 13 c2B 2 c33 3, We where the nine c's are arbitrary. may then write D = cE, D^c'E, where the operator c' only differs from c in the exchange of c 12 and c 21 , 20 ELECTRICAL PAPERS. etc. It is now D' that is conjugate to D, whilst c' is the operator conjugate to c. It may be readily seen that D'=/E-VeE, where / is the self-conjugate operator obtained by replacing c 12 and c 21 , etc., in c by half their sums, and e is a certain vector whose components are half their differences. Thus, = %) + + e - Ji(c82 - - ij(c13 c sl ) Jk(c21 c12 ). The conjugate property of scalar products is now = Bc'A. That is, in transferring the operator from B to A, we must simultaneously change it to its conjugate. Another way of regarding the matter is as follows : If we put C= 2 we see, by the above, that D = cE = E+ i.c1 D' = c'E = Cj.iE + c2 .jE + c3 .kE = (Cj.i + c2 .j + c3 .k)E, from which we see that c'E is the same as EC, and cE the same as EC'. In the case of AcB, therefore, we may regard it either as the scalar A product of and cB, or as the scalar product of Ac and B. This is equivalent to Professor Gibbs's way of regarding linear operators. That is (converted to my notation), is the type of a linear operator. It assumes the utmost generality when i, j, k stand for any three independent vectors, instead of a unit rectangular system. Professor Gibbs has considerably developed the theory of linear operators in his Vector Analysis. w The generalised form of (17) is got thus: Let v and be any vectors, then, as before, we have 0= vVvw= vcc~ 1Vvw, = wVvw = wcc^Vvw, Vvw where the last forms assert that c~ 1 is perpendicular to vc and we, or parallel to Vvcwc ; that is, mVvw = cVc'vc'w ', ........................... (A) from which, by multiplying by a third vector u, we find m - -- = uVvw c'uWvc'w == , ............................. ,T>\ (-D) which is an invariant. Hamilton's cubic equation in c is obtained by observing that since (A) is an identity, c being any linear operator, it remains an identity ON THE ELECTROMAGNETIC WAVE-SURFACE. 21 when c is changed to c - g, which changes c' to c' - #, where g is a scalar constant. For c - g is also a linear operator. Making this substitution in (A) and expanding, we obtain (m - m^ + m2 2g - g*) Vvw jc'uVc'vc'w - ^(uVc'vc'w + vVc'wc'u + wVc'uc'v) + 2 # (c'uVvw + c'vVwu + c'wVuv) - 3 # uVvw j = cVc'vc'w - ^(Vc'vc'w + cVvc'w + cVc'vw) + 2 # (cVvw + Vvc'w + Vc'vw) - 3 # Vvw, m m where m, v 2 are the coefficients of #, g, and 2 g in the expansion of m the left member of given by (B). Comparing coefficients we see that # and 3 g go out. The others give (remembering that we are dealing with an identity), Vc'vc'w + c(Vvc'w + Vc'vw) = ?%Vvw, cVvw + (Vvc'w + Vc r vw) = w Vvw. 2 Operate on the first by c and second by 2 c , and subtract. This eliminates the vector in the brackets, and leaves cVc'vc'w - c 3 Vvw = m cVvw : - m2c2 Vvw, where the first term on the left is mVvw. So we have m + m = m^c - c2 2 c3 0, ......................... (C) which is Hamilton's cubic. If we start instead with the conjugate operator f c we shall arrive at m'Vvw = c'Vcvcw, where W' = ^Lu^Vvvcww and then, later, to the cubic + m = ra' - m(c' - 2'c' 2 c/3 0, where m', etc., come from m, etc., by exchanging c and c'. But it may m m be easily proved that = m', and we may infer from this that 1 = m{ m m and = 2 rti2, on account of the invariantic character of being pre- served when c becomes c - g. In fact, putting c =/+ Ve and cf =/- Ve, Vw where /is self-conjugate, we may independently show that m- cuVcvcw c'uVc'vc'w uVvw uVvw uVvw m - m' - w/u + wV/u/v uVvw uVcvcw + vVcwcu + wVcucv uVvw = / = /nVvw +/vVwu +/wVuv = game with c = same with , c uVvw So in Hamilton's cubic (C) we may change c to c', leaving the ra's 22 ELECTRICAL PAPERS. unchanged ; or else in the m's only ; or make the change in both the c's and the m's, without affecting its truth. If the passage from (A) to (C) above be compared with the corre- sponding transition in Tait's Quaternions (3rd edition, 158 to 160) it will be seen that that rather difficult proof is simplified (as done above) by omitting altogether the inverse operations $~ l and -l (^ g)~ and the auxiliary operator x ', especially x> perhaps. One is led to think from Professor Tait's proof that the object of the investigation is to solve the problem of inverting <. But the mere inversion can be done by elementary methods. In Gibbs's language, if a, b, c is one set of vectors, the reciprocal set is a', b', c', given by a/_Vbc ~aVbc' b/~_VbcaW ,_ Vab ~cVa On this understanding, we may expand any vector d in terms of a, b, c thus : = + + d a . a'd b . b'd c . c'd. Similarly, if 1', m', n' is the set reciprocal to 1, m, n, we have r = 1'. lr + m'. mr + n'. nr. If, then, it be given that d = <(r) = a.lr + b.mr + c.nr, we see that = lr a'd, etc., so that r = ^(d) = 1'. a'd + m'. b'd + n'. c'd inverts <. (This is equivalent to Tait, 173.) We W see by (A) and (B) that the inverts of u, v, are c' x inverts of cu, cv, cw; or c x inverts of c'u, c'v, c'w. The cubic (C) may be written W wcw cu cvcw / c_! _ , _ lu/ vc -i y/ + WC -I A I _ c / c _ / ucu/ + vcv/ + I ') uVvw I 'J I W if u', v', w' are the inverts of u, v, (or the reciprocal set). In this identity the operators c and c" 1 may be inverted. When that is done m m we see that the of c is the reciprocal of the of c" 1 .] Note on Modification of Index-equation when c and //. are Rotational. [Let c' and /*' be the conjugates to c and /*. Then, by (A), (B), in last note, mVvw = p! V^VfjiW = />tV//v//w, where ^^ m = /^ + e/* e, if /*!, /x 2 , /AS are the principal permeabilities of yu, , the self-conjugate = + operator such that />t />t Ye. With this extension of meaning, we shall have (treating c and n similarly), - E = c- 1VsH, - ?iE = Vc'sc'H, - mE = c - 1VsV/s/E, H = /x-i VsE, mH = V/s/E, - nE = ^- 1V&Vc'sc'H, NOTES ON NOMENCLATURE. 23 where the first pair replace (28), (29), the second pair (31), (32), and the third pair (33), (34). Then = ?fyuH - c's(sc'H) c / / H(sc s) replace (33) and (34), and E~ p'B s//E (s/x's)// - me' H C?B ~~ sc'H (sc's)c' - np replace (37a), (38a) ; from which two forms of index-equation corresponding to (41) are S 8 1- 1 (s/x'sjc" lap.'- (sc's)/^" 1 - nc'~ l We obtain impossible values of the velocity for certain directions of the normal. That is, there could not be a plane wave under the circum- stances.] XXXII. NOTES ON NOMENCLATURE. [The Electrician, Note 1, Sep. 4, 1885, p. 311 ; Note 2, Jan. 26, 1886, p. 227 ; Note 3, Feb. 12, 1886, p. 271.] NOTE 1. IDEAS, WORDS, AND SYMBOLS. HOWEVER desirable it may be that writers on electrotechnics should use a common notation, at least as regards the frequently recurring magnitudes concerned which notation should not be a difficult matter to arrange, provided it be kept within practical limits it is perhaps more desirable that they should adopt a common language, within the same practical limits, of course. For whilst the use of certain letters for certain magnitudes requires no more explanation than, for instance, "Let us call the currents (7r (7 2, etc.," it is otherwise with the language used when speaking of the magnitudes, as more elaborate explanations are needed to identify the ideas meant to be expressed. As regards electric conduction currents, there is a tolerably uniform usage, and a fairly good terminology. It is seldom that any doubt can arise as to a writer's meaning, unless he be an ignoramus or a paradoxist, or have unfortunately an indistinct manner of expressing him- self. I would, however, like to see the word "intensity," as applied to the electric current, wholly abolished. It was formerly very commonly used, and there was an equally common vagueness of ideas prevalent. It is sufficient to speak of the current in a wire (total) as " the current," or "the strength of current," and when referred to unit area, the current-density. (In three dimensions, on the other hand, when every- thing is referred to the unit volume, and the current-density is meant as a matter of course, it is equally sufficient to call it the current.) 24 ELECTRICAL PAPERS. It is a matter of considerable practical advantage to have single words for names, instead of groups of words, and it is fortunate that the exist- ing this cwoanyd.uctiTohnu-sc,ur"resnptecitfeircmriensoilsotganycead"mmitasy of very be well practical called " adaptation resistivity," and specific conductance " conductivity," referring to the unit volume. Resistivity is the reciprocal of conductivity, and resistance of conduct- ance. When wires are in parallel, their conductances may be more We easy to manage than their resistances. have also the convenient adjectives "conductive" and "resistive," to save circumlocution. Passing to the subject of magnetic induction, there is considerable looseness prevailing. There is a definite magnitude called by Maxwell "the magnetic induction," which may well be called simply "the induction." It is related to the magnetic force in the same manner as c"umrargennte-tdiecnspietrymeatboiltihtey.e"lectTrhiics force. may be (B = ^H.) The ratio p is the simply called the permeability, since the word is not used in any other electrical sense. Induction and permeability may not be the best names, but (apart from their being understood by mathematical electricians) they are infinitely better than the long-winded "number of lines of force" (meaning magnetic) and " conductivity for lines of force," the use of which, though defensible enough in merely popular explanations, becomes almost absurd when the electrotechnical user actually goes so far as to give them quantita- tive expression. Conductivity should not be used at all, save in point- ing out an analogy. It has its own definite meaning. " Permeability," however, does not admit of such easy adaptation to different circumstances as conductivity. Permeability referring to the unit volume, the word permeance is suggested for a mass, analogous to We conductance. have also the adjective "permeable." By adding, maonrdeo"viemrp,erthmeeapnrceef,i"x "im," we get "impermeable," "impermeability," for the reciprocal ideas, sometimes wanted. Thus impermeability, the reciprocal of /A, would stand for the long-winded " specific resistance to lines of magnetic force." (The permeance of a L coil would be Z/47T, if is its coefficient of self-induction. In the expression T=^LC* for the magnetic energy of current C in the coil, 4?r T= does not appear, whilst it does in the form \ magnetomotive force x total induction through the circuit -f 47r. It is kirC that is the magneto- motive force, and LC the induction through the circuit. Thus we have oppositely acting 47r's. I may here remark that it would be not only a theoretical but a great practical improvement to have the electric and magnetic units recast on a rational basis. But I suppose there is no chance of such an extensive change.) It must be confessed, however, that these various words are not so good as the corresponding con- duction-current words. But now, if, thirdly, we pass to electric displacement, the analogue of magnetic induction (noting by the way that it had better not be called the electric induction, on account of our already appropriating the word induction, but be called the displacement), the existing terminology is extremely unsatisfactory; and, moreover, does not readily admit of adaptation and extension. Corresponding to conductivity and perme- NOTES ON NOMENCLATURE. 25 ability we have "specific inductive capacity," or "dielectric constant," or whatever it may be called. I usually call it the electric capacity, or the capacity. It refers to the unit volume. But here it is very unfor- tunate that it is not this specific capacity c (say), but c/4?r, that is the capacity of a unit cube condenser (such that charge = difference of potential x capacity). D, the displacement, is the charge + or - ( , D E according to the end), and we have = cE/4:7r, being the electric We D force. may get over this trouble by putting it thus, = sE, and calling s (or c/4?r) the specific capacity. Then the capacity in bulk is got in the same manner as conductance from conductivity. Supposing we have done this, there is still the trouble that capacity gives the extremely awkward inverse " incapacity," and the adjectives "capacious" and "incapacious," besides not giving us any words for use in bulk, like conductance and resistance. And, in addition, the word capacity is itself rather objectionable, as likely to give beginners entirely erroneous notions as to the physical quality involved. It is not that one dielectric absorbs electricity more readily than another. Electric displacement is an elastic phenomenon : one dielectric is more yielding (electrically) than another. The reciprocal of s above is the electric elasticity, measuring the electric force required to produce the unit displacement. Thus s should have a name to express the idea of elastic yielding or distortion, and its reciprocal also a name (not strings of words), and they should be readily adaptable, like conductivity, etc. (Perhaps also a better word than permeability might be introduced, although, as we see, it is tolerably accommodative.) Displacement itself might also be replaced by another word less suggestive of bodily translation; although, on the other hand, it harmonises well with " current," the displacement being the accumulated current, or the current the time-variation of the displacement. All these things will get right in time, perhaps. Ideas are of primary importance, scientifically. Next, suitable language. As for the nota- tion, it is an important enough matter, but still only takes the third place. NOTE 2. ON THE RISE AND PROGRESS OF NOMENCLATURE. In the beginning was the word. The importance of nomenclature was recognised in the earliest times. One of the first duties that devolved upon Adam on his installation as gardener and keeper of the zoological collection was the naming of the beasts. The history of the race is repeated in that of the individual. This grand modern generalisation explains in the most scientific manner the fondness for calling names displayed by little children. Passing over the patriarchal period, the fall of the Tower of Babel and its important effects on nomenclature, the Egyptian sojourn, the wanderings in the desert, the times of the Kings, of the Babylonian captivity, of the minor prophets, of early Christianity, of those dreadful middle ages of monkish learning and ignorance, when evolution worked backwards, and of the Elizabethan revival, and coming at once to the middle of the 19th century, we find that Mrs. Gamp was much im- 26 ELECTRICAL PAPERS. pressed by the importance of nomenclature. " Give it a name, I beg. Sairey, give it a name ! " cried that esteemed lady. She even went so far as to give a name to an entirely fictitious personage Mrs. Harris, to wit who has many scientific representatives. Having thus fortified ourselves by quoting both ancient and modern instances, let us consider the names of the electrical units. A really practical name should be short, preferably monosyllabic, pronounced in nearly the same way by all civilised peoples, and not mistakable for any other scientific unit. If, in addition, it be the name, or a part of the name, of an eminent scientist, so much the better. This is quite a sentimental matter ; but if it does no harm, it is needless to object to it. But we should never put the sentiment in the first place, and give an unpractical name to a unit on account of the sentiment. Ohm and volt are admirable; farad is nearly as good (but surely it was unpractical to make it a million times too big the present microfarad should be the farad) ; erg and dyne please me ; watt is not quite so good, but is tolerable. But what about those remarkable results of the Paris Congress, the ampere and the coulomb 1 Speaking entirely for myself, they are very unpractical. Coulomb may be turned into coul, and is then endurable ; this unit is, however, little used. But ampere shortened to am or amp is not nice. Better make it pere ; then it will do. Now an additional bit of sentiment comes in to support us. Was not Ampere the father of electrodynamics ? It seems rather unpractical for the B.A. Committee to have selected 108 c.g.s. as the practical unit of E.M.F., instead of 10 9. This will hardly be appreciated except by those who make theoretical calculations; the awkward thing is that the pere is one tenth of the c.g.s. unit of current. I suppose it was because the present volt was an approxima- tion to the E.M.F. of a Daniell ; that is, however, a very strong reason for making the practical unit much smaller; because the E.M.F. of a cell has now to be given in volts and tenths, or hundredths also. How awkward it would have been if the ohm had been made 1010 c.g.s., so as to approximate to the resistance of a mile of iron telegraph wire. The ohm and volt should be the same multiple of the c.g.s. units, both 10 9 for example. Then use the millivolt or centivolt when speaking of the E.M.F. of cells. The present 1-12 volt would be 112 millivolts. Speaking from memory, Sir W. Thomson did object to the 10s volt at the Paris Congress. Mac, torn, bob, and dick are all good names for units. Tom and mac (plural, max), have sentimental reasons for adoption ; bob and dick may also at some future time. I have used torn myself (no offence, I hope) for six years past to denote 109 c.g.s. units of self or mutual electro- magnetic induction coefficient. (Some reform is wanted here. Co- efficient of self-induction, or of electromagnetic capacity, is too lengthy.) K The advantage is that L toms divided by ohms gives L/E, seconds of time. But it is too big a unit for little coils ; then use the millitom : or even the microtom for very small coils. This applies to fine-wire coils. The c.g.s. unit itself would be most suitable for coils of a few NOTES ON NOMENCLATURE. 27 turns of thick wire. If it is called the torn, then the kilotom or megatom will come in useful for fine-wire coils. A name should certainly be given to a unit of this quantity, whether it be torn, or mac, or any other practical name. Also, names to a unit of magnetic force (intensity of), and of magnetic induction. There is also the question of the names, not of the units, but of the physical magnitudes of which they are the units, but it is too large a question to discuss here except in the most superficial manner. It is engrained in the British nature to abbreviate, to make one word do for We two or three, or a short for a long word. And quite right too. have much to be thankful for ; in the application of this general remark, consider what frightful names might have been given to the electrical units by the Germans. But, on account of this national, and also rational tendency to cut and clip, it is in the highest degree desirable that as many as possible of the most important physical magnitudes should be known, not by a long string of words, but by a single word, or the smallest number possible. Thus, I find myself frequently saying force, when I mean magnetic force, and even then, I mean the intensity of magnetic force. The context will generally make the meaning plain. But it is necessary to be very careful when there are more forces than one in question. (This use of force as an abbreviation is, of course, quite distinct from the frequent positive misuse of the word force, to indicate it may be momentum, or energy, or activity, or, very often, nothing in particular, the misuser not being able to say exactly what he means ; nor does it much matter.) It would be decidedly better if such a quantity as "intensity of magnetic force" had a one-word name, for people will abbreviate, and sometimes confusion may step in. This remark applies to most of the electromagnetic magnitudes. There is an important magnitude termed the magnetic induction. I call it often simply "the induction"; but in doing so, carefully avoid calling any other quantity " the induction " (sometimes the electric dis- placement is called the electric induction). But there is an unfortunate thing here, which somewhat militates against "the induction," or even " the magnetic induction " being a thoroughly good name for the magnitude in question. This is, that besides being a name of a physical magnitude, the word induction has a widespread use, in a rather vague manner, in connection with transient states in general, whether of the electric or of the magnetic field, exemplified, to take an extreme example, when a man explains something complex by saying it is caused by "induction," and so settling the matter. If this vague qualitative use of induction were got rid of, then as a name for a physical magnitude it would be unobjectionable. As it is, it is a question whether the physical magnitude should not have a name for itself alone. " Resistivity" for specific resistance, and "conductance" for what is sometimes called the conductibility of a wire, i.e., not its conductivity (specific conductance), but the reciprocal of its resistance, are, I think, as I have remarked before, quite practical names. 28 ELECTRICAL PAPERS. NOTE 3. THE INDUCTANCE OF A CIRCUIT. IN my first note, amongst other things, I remarked that whilst the conduction-current terminology admitted of the words resistivity and conductance being coined to make it more complete, the terminology in the allied cases of magnetic induction and electric displacement was unsatisfactory. As regards the former, the following appears to me to be practical. We First, abolish the word permeability, and substitute Inductivity. B B then have = pH, when is the Induction, and /x the Inductivity, H showing how the Induction is related to the magnetic force by the specific quality of the medium at the place, its inductivity. Now conductivity and conductance are mathematically related in the same manner (except as regards a 4?r) as inductivity and what it is naturally suggested to call Inductance. The Inductance of a circuit is what is now called its coefficient of self-induction, or of electromagnetic capacity. Thus the quantities induction, inductivity, and inductance are happily connected in a manner which is at once concise and does justice to their real relationship. When the mutual coefficient of induction of two circuits is to be referred to, it will of course be the mutual inductance. XXXIII. NOTES ON THE SELF-INDUCTION OF WIRES. [The Electrician, 1886 ; Note 1, April 23, p. 471 ; Note 2, May 7, p. 510.] We NOTE 1. read in the pages of history of a monarch who was " supra grammaticam." All truly great men are like that monarch. They have their own grammars, syntaxes, and dictionaries. They cannot be judged by ordinary standards, but require interpretation. Fortunately the liberty of private interpretation is conserved. No man has a more peculiar grammar than Prof. Hughes. Hence, he is liable, in a most unusual degree, to be misunderstood, as I venture to think he has been by many, including Mr. W. Smith, whose interesting letter appears in The Electrician, April 16, 1886, p. 455, and Prof. H. Weber, p. 451. The very first step to the understanding of a writer is to find out what he means. Before that is done there cannot possibly be a clear com- prehension of his utterances. One may, by taking his language in its ordinary significance, hastily conclude that he has either revolutionised the science of induction, or that he is talking nonsense. But to do this We would not be fair. must not judge by what a man says if we have good reason to know that what he means is quite different. To be quite fair, we must conscientiously endeavour to translate his language and ideas into those we are ourselves accustomed to use. Then, and then only, shall we see what is to be seen. When Prof. Hughes speaks of the resistance of a wire, he does not NOTES ON THE SELF-INDUCTION OF WIRES. 29 always mean what common men, men of ohms, volts, and farads, mean by the resistance of a wire only sometimes. He does not exactly define what it is to be when the accepted meaning is departed from. But by a study of the context we may arrive at some notion of its new meaning. It is not a definite quantity, and must be varied to suit circumstances. Again, there is his " inductive " capacity of a wire. We can only find roughly what that means by putting together this, that, and the other. It, too, is not a definite quantity, but must be varied to suit circumstances. It is not the coefficient of self-induction, nor is it any quantity defining a specific quality of the wire, like conductivity, or inductivity. It is a complex quantity, depending on a great many things, but which may, to a first rough approximation, be taken to be proportional to the time-constant of the wire, the quotient of its coefficient of self-induction by its resistance. Bearing these two things in mind, we shall be able to approximate to Prof. Hughes's meaning. Owing to the mention of discoveries, apparently of the most revolu- tionary kind, I took great pains in translating Prof. Hughes's language into my own, trying to imagine that I had made the same experiments in the same manner (which could not have happened), and then asking what are their interpretations ? The discoveries I looked for vanished for the most part into thin air. They became well known facts when put into common language. The satisfaction of getting verifications, however, even in so roundabout and rough a manner, is some compen- sation for the disappointment felt. I venture to think that Prof. Hughes does not do himself justice in thus deceiving us, however unwittingly, and that possibly there has been also some misapprehension on his part as to what the laws of self-induction are generally supposed to be. I have failed to find any departure from the known laws of electro- magnetism. In saying this, however, I should make a reservational remark. There may be lying latent in Prof. Hughes's results dozens of discoveries, but it is impossible to get at them. For consider what the mere existence of ohms, volts, and farads means ? It means that, even before they were made, the laws of induction in linear circuits were known, and very precisely. To get, then, at new discoveries requires very accurate comparison of experiment with theory, by methods which enable us to see what we are doing and measuring, in terms of the known electromagnetic quantities. This is practically impossible, on We the basis of Prof. Hughes's papers. can only make very rough verifications. I have had myself, for many years past, occasional experience with induction balances of an exact nature true balances of resistance and induction and always found them work properly. But, in the modification made by Prof. Hughes, the balance is generally of a mixed nature, neither a true resistance nor a true induction balance, and has to be set right by a foreign impressed force, viz., induction between the battery and telephone branches. By using a strictly simple harmonic E.M.F., as of a rotating coil, we may exactly formulate the conditions of the false balance, and then, noting all the resistances, 30 ELECTRICAL PAPERS. etc., concerned, derive, though in a complex manner, exact information. Or, if we use true balances, any kind of E.M.F. will answer. To illustrate the falsity of Prof. Hughes's balances and the difficulty of getting at exact information, he finds the comparative force of the extra-currents in two similar coils in series to be 1'74 times that of a single coil. From the context it would appear that this " comparative force of the " extra-currents is the same thing as the former " inductive " capacity of wires. Now, the coefficient of self-induction of two similar coils in series, not too near one another, is double that of either, whilst the time-constant of the two is the same as of either. This can be easily verified by true balances. The most interesting of the experiments are those relating to the effect of increased diameter on what Prof. Hughes terms the "inductive My " capacity of wires. own interpretation is roughly this. That the time-constant of a wire first increases with the diameter, and then later decreases rapidly ; and that the decrease sets in the sooner the higher the conductivity and the higher the inductivity (or magnetic perme- ability) of the wires. If this be correct, it is exactly what I should have expected and predicted. In fact, I have already described the pheno- menon substantially in The Electrician ; or, rather, the phenomenon I described contains in itself the above interpretation. In The Electrician for January 10, 1885, I described how the current starts in a wire. It begins on its boundary and is propagated inward. Thus, during the rise of the current it is less strong at the centre than at the boundary. As regards the manner of inward propagation, it takes place according to the same laws as the propagation of magnetic force and current into cores from an enveloping coil, which I have described in considerable detail in The Electrician [Reprint, vol. 1, Art. 28. See especially 20]. The retardation depends on the conductivity, on the inductivity, and on the section, under similar boundary conditions. If the conductivity be high enough, or the inductivity or the section be large enough, to make the central current appreciably less than the boundary current during the greater part of the time of rise of the current, there will be an apparent reduction in the time-constant. Go to an extreme case. Very rapid short currents, and large retardation to inward transmission. Here we have the current in layers, strong on the boundary, weak in the middle. Clearly, then, if we wish to regard the wire as a mere linear circuit, which it is not, and as we can only do to a first approximation, we should remove the central part of the wire that is, increase its resistance, regarded as a line, or reduce its time-constant. This will happen the sooner the greater the inductivity and the conductivity, as the section is continuously increased. It is only thin wires that can be treated as mere lines, and even they, if the speed be only great enough, must be treated as solid conductors. I ought also to mention that the influence of external conductors, as of the return conductor, is of importance, sometimes of very great importance, in modifying the distribution of current in the transient state. I have had for years in MS. some solutions relating to round wires, and hope to publish them soon. NOTES ON THE SELF-INDUCTION OF WIRES. 31 As a general assistance to those who go by old methods a rising current inducing an opposite current in itself and in parallel conductors this may be useful. Parallel currents are said to attract or repel, according as the currents are together or opposed. This is, however, mechanical force on the conductors. The distribution of current is not affected by it. But when currents are increasing or decreasing, there is an apparent attraction or repulsion between them. Oppositely going currents repel when they are decreasing, and attract when they are increasing. Thus, send a current into a loop, one wire the return to the other, both being close together. During the rise of the current it will be denser on the sides of the wires nearest one another than on the remote sides. It is an apparent force, not between currents (on the distance-action and real motion of electricity views), but between their accelerations. NOTE 2. I did not expect to return to the subject, and do so because Prof. Hughes has apparently misunderstood my statements. On p. 495 of The Electrician for April 30, 1886, he says : "Mr. Oliver Heaviside points out that upon a close examination it will be found that all the effects which I have described are well known to mathematicians, and A consequently old." regard for accuracy compels me to point out that I did not make the statement he credits me with ; nor, to avoid any hypercriticism, is the above a correct summary of the many things that I pointed out. I said, "The discoveries I looked for vanished, for the most part, into thin air. They became well-known facts when put into common language." Observe here my "for the most part" as against Prof. Hughes's "" all ; and that I said not a word about mathematicians in the whole letter. An immediate consequence of my statement is another, namely, that some, although a minority, of the results were not well known. There is a material difference between what I said and what Prof. Hughes makes me say. In another place I said that I had "failed to find any departure from the known laws of electro- magnetism," and then proceeded to give my reasons for it. This statement includes the well-known facts as well as those which are not well known. It may be as well that I should illustrate the difference between wellknown facts and those that are less known, or only known theoretically. The influence of the form of a thin wire (a linear conductor), and of its length, diameter, conductivity, and inductivity on the phenomena of self-induction is well known. The various relations involved form the A B C of the subject. So are the effects of concentration of the current, and of dividing it, or spreading it out in strips, well known. There is another influence that is well known, that is scarcely touched upon by Prof. Hughes. The self-induction depends upon the distribution of inductivity, that is, in another form, of inductively magnetisable matter, outside the current, as well as in it, in a manner which is quite definite when the magnetic properties of the matter are known. It is not to be inferred that verifications of well-known facts are of no 32 ELECTRICAL PAPERS. value that depends upon circumstances. To be of any use, we must know what we are measuring and verifying. The theory of self and mutual induction in linear circuits is almost a branch of pure mathe- matics, so simply are the quantities related, and so exactly. It furnishes a most remarkable example of the dependence of complex phenomena on a very small number of independent variables, by ignoring minute dielectric phenomena. In getting verifications, then, it is first necessary to employ a correct method. I have elsewhere [The Electrician, April 30, 1886, p. 489; the next Art. 33] shown the approximate character of Prof. Hughes's method of balancing, and pointed out exact methods. Next, it is necessary to put results in terms of the quantities in the electro- magnetic theory which is founded upon the well-known facts; how else can we know what we are doing, and see how near our verifica- tions go? Coming now to results that are not well known, there is the thick- wire effect, depending on size, conductivity, inductivity, place of return my current, etc. This is, in opinion, the really important part of Prof. Hughes's researches, as it, in some respects, goes beyond what was already experimentally known. Having been, so far as I know, the first to correctly describe (The Electrician, Jan. 10, 1885, p. 180) [Reprint, vol. I. pp. 439, 440] the way the current rises in a wire, viz., by diffusion from its boundary, and the consequent approximation, under certain circumstances, to mere surface conduction ; and believing my Prof. Hughes's researches to furnish experimental verifications of views, it will be readily understood that I am specially interested in this effect ; and I can (in anticipation) return thanks to Prof. Hughes for accurate measures of the same, expressed in an intelligible form, to render a comparison with theory possible if it be practicable. I send my with this a first instalment of old core investigations applied to a round wire with the current longitudinal. [Section 26 of "Electro- magnetic Induction," later.] There are also intermediate matters where one can hardly be said to be either making verifications, except roughly, or discoveries; for instance, the self-induction of an iron-wire coil. Theory indicates in the plainest manner that the self-induction coefficient will be a much smaller multiple of that of a similar copper-wire coil than if the wires were straightened. Magnetic circuits are now getting quite popularly understood, by reason of the commercial importance of the dynamo. But there is really no practical way of carrying out the theory completely, as the mathematical difficulties are so great. Hence, actual measurements of the precise amounts in various cases of magnetic circuits are of value, if they be accompanied by the data necessary for comparisons. There is, however, this little difficulty in the way when transient currents are employed. Iron, by reason of its high iuductivity, is pre- We eminently suited for showing the thick-wire effect. may not, therefore, be always measuring what we want, but something else. USE OF THE BRIDGE AS AN INDUCTION BALANCE. 33 XXXIV. ON THE USE OF THE BRIDGE AS AN INDUCTION BALANCE. [The Electrician, April 30, 1886, p. 489.] IN connection with a paper "On Electromagnets, etc.," that I wrote about six years ago [Reprint, Art. xvii., vol. 1, p. 95], which paper dealt mainly with the question of the influence of the electromagnetic induction of the lines and instruments on the magnitude of the signalling currents, an influence which is of the greatest importance on short lines, and which (of the instruments) is, even on long lines, where electrostatic induction is prominent, of importance as a retarding factor, I made a great many experiments on self-induction, amongst which were measurements of the inductances of various telegraph instruments, with a view to ascertaining their practical values, and also the multiplying powers of the iron cores. It was my intention to write a supplementary paper giving the results and also further investigations; but, having got involved, in the course of the experiments, in the difficult subject of magnetic inductivity, it was postponed, and then dropped out of mind. I used, first of all, the Bridge and condenser method described by Maxwell, with reversals, and a telephone for current indicator. This was to get results at once, or by simple calculations, in electromagnetic units. Next, I discarded the condenser, and used the simple Bridge, balancing coils against standard coils. Thirdly, I have used a differential telephone with the same object, in a similar manner. The two last are very sensitive methods, and the verifications of the theory of induction in linear conductors that I have made by them are numerous. The whole of this journal would be required to give anything like a full investigation of the various ways of using the Bridge as an induction balance. I can, therefore, only touch lightly on the subject of exact balances, especially as I have to remark upon faulty methods, approximate balances, and absolutely false balances. Prof. Hughes's balance is sometimes fairly approximate, sometimes quite false. Put a telephone in the branch 5, battery and interrupter in 6. Then, r standing for resistance, I for inductance (coefficient of self-induction), and x for l/r, the time-constant of a branch, the conditions of a true and perfect balance, however the impressed force in 6 vary, are three in number, namely, Their interpretations are as follows : If the first condition is fulfilled there will be no final current in 5 when a steady impressed force is put in 6. This is the condition for a true resistance balance. H.E.P. VOL. IT. c 34 ELECTRICAL PAPERS. If, in addition to this, the second condition be also satisfied, the integral extra-current in 5 on making or breaking 6 is zero, besides the steady current being zero. (1) and (2) together therefore give an approximate induction balance with a true resistance balance. If, in addition to (1) and (2), the third condition is satisfied, the extra-current is zero at every moment during the transient state, and the balance is exact, however the impressed force in 6 vary. Practically, take r 1 = r& and J 1= =/ 2 , (4) that is, let branches 1 and 2 be of equal resistance and inductance. Then the second and third conditions become identical; and, to get perfect balances, we need only make r =r 3 4, and = Z 8 Z4 (5) This is the method I have generally used, reducing the three con- ditions to two, whilst preserving exactness. It is also the simplest method. The mutual induction, if smy, of 1 and 2, or of 3 and 4, does not influence the balance when this ratio of equality, = i\ r 2, is employed (whether = / / x 2 or not). So branches 1 and 2 may consist of two similar wires wound together on the same bobbin to keep their temper- atures equal. The sensitiveness of the telephone has been greatly exaggerated. Altogether apart from the question of referring the sensitiveness to the human ear rather than to the telephone, it is certainly, under ordinary circumstances, often unable to appreciate the differences of the second order, which vanish when the third condition is satisfied. Thus (1) and (2) satisfied, but with (3) unsatisfied, will give silence. Take, for instance, r l =r% and r 3 = r# but ^ different from / 2 and /3 from / 4, then silence is given by ft-y/r.-ft-y/r.j (6) that is, by making the differences of the inductances on the two sides We of 5 proportional to the resistances. can therefore get silence by varying the inductance of any one or more of the four branches 1, 2, 3, 4, to suit equation (6). It is certain that we do get silence this way, but it does not follow that silence is given by exactly satisfying (6), (and (1) of course), because it is only a balance of integral extra-currents, and other balances of this kind are certainly quite false sometimes. To avoid any doubt, it is of course best to keep to the legitimate and simpler previously-described method. There are some other ways of using the Bridge as an induction balance in an exact manner, but they are less practically useful than theoretically interesting. Pass, therefore, to other approximate, and to false balances. Suppose we start with a true balance, and then upset it by increasing the inductance of the branch 4. It is clear that we should never alter the already truly established resistance balance. Now, besides by the exact ways, we can get approximate silence by allowing mutual induction between 5 and any of the other five branches, or between 6 and any of the other five branches, that is nine ways, not USE OF THE BRIDGE AS AN INDUCTION BALANCE. 35 counting combinations. (Put test coils in 5 and 6 with long leading wires, so that they may be carried about from one branch to another.) These approximate balances are all of the integral extra-current only, and therefore imperfect, however nearly there may be silence. But the silences are of very different values. I find, using h'ne-wire coils, that mutual induction between 6 and 4 or between 6 and 3 gives silence (to my ear) with the true resistance balance, just like the approximate balance of equation (6) in which no mutual induction is allowed. These are only two out of the nine ways. All the rest are bad. If the difference in the inductance of 3 and 4 be small, there is very nearly silence on using any of the other seven ways; but, the larger this difference is made, the louder becomes the " silence," and sometimes it is even a very loud noise, quite comparable with the original sound that was to be destroyed, even when the combinations 6 and 4 or 6 and 3, and the formerly-mentioned method give a silence that can be felt, with the true resistance balance. It is certainly a rather remarkable thing that the one method out of these seven faulty ways which gave the very loudest sound was the 5 and 6 combination, which is Professor Hughes's method. I do not say that it is always the worst, although it was markedly so in my experiments to test the trustworthiness of the method. And sometimes it is quite fair. In fact, when the sound to be destroyed is itself weak, all the seven faulty methods are apparently alike, nearly true. But when we exaggerate the inequality of inductance between 3 and 4, whilst the 6 to 4 and 6 to 3 combinations keep good, the others get rapidly worse, and differences appear between them. I found that by increasing the resistance of the branch whose inductance was the smaller, the sound was diminished greatly, i.e., in the seven faulty methods. The coil of greater inductance had apparently the higher resistance. That is, with a false resistance balance we may approximate to silence. Such a balance is condemned for scientific purposes. Although mutual induction between 6 and 4 or 6 and 3 gave silence, with true resistance balances, the experiments were not sufficiently extended to prove their general trustworthiness. There is, however, some reason to be given for their superiority. For, since the dis- turbance in the telephone arises from the inequality of the momenta of the currents in the branches 3 and 4, and of the electric impulses arising in them when contact is broken in branch 6 (considering the break only for simplicity), we go nearest to the root of the evil by generating an additional impulse in 3 or 4 themselves from the battery branch, of the right amount. The following is an outline of the theory of these approximate = balances. Let r^r^ r r first 23 ; so that, C standing for current, we have, in the steady state, ...... (7) The momentum of the current in branch 1 is If^ that in 2 is 1C 2 2, and 36 ELECTRICAL PAPERS. so on. Consider the break, and the integral extra-current that then arises from / (7 . It is 11 1& + + + + + {^ -f r 2 r 6 (rs r4)/(rs r4 r- 5 ) }, and + + + (r3 r4 )/(rs r4 r 5 ) is the fraction of this that goes through 5 ; so that the integral current in 5 due to ^Cj is h G + * + + + + + + r i( * r*) r {( i ?< r 2)(' 3 ?>4 ) ? ? 'o( 'i ? 2 r s r 4> } > or (7^ - {r3 + r 4 + r5 + ^r^}, by making use of equations (1) and (7). Treat the others similarly. The total extra-current in 5 is r&fa + -x + + x + - 2 x 3) 4- {r3 r4 r 5 r3r.jrj, ............ (8) without any mutual induction. So *C| ~F WA == ^/O "l **^Q gives approximate balance. This was mentioned before, and becomes an exact balance with makes and breaks when a ratio of equality is taken. Now let there be mutual induction between 6 and 4, 5 and 4, and 5 and 6, the mutual inductances being Af 64 , etc. Treating these similarly to before, we shall find the total extra-current in 5 on the break taking place to be M fato + z 4 - x 2 - x s) + M Jlf (l + rjr3] + M(l + rjrj M + l 56 (l+r B / r l )(l+rJr2 }}C\^(rs + r^ + r 5 + r 3 rJrl ) .......... (9) The theory of the make leads to the same result that is, as regards Mthe integral extra-current. Otherwise they are different. So, using 56 (Hughes's method) the zero integral current is when r4 (x1 + x4 -x 2 -x 3) + M,6 (l+r3 lr1 )(l+rJr3) = .......... (10) Using Jf 45 we have M Using 6i we have }+rJr3 ) = Q................ (12) Practically employ a ratio of equality = r^ r^ ^i = ^> ' branches 1 and 2 equal fixtures. Then these three equations become = 0, .................... (10a) ^4-^3+ JfwO+ra/r^O, .................... (lla) 1,-1B + 2M,6 = ..................... (12a) M Thus the 46 system has the simplest formula, as well as being M practically perfect. It is the same with 6y Either of these must equal half the difference of the inductances of 3 and 4. As (10a), or, more generally, (10) contains resistances, we cannot get any definite results from Prof. Hughes's numbers without a knowledge of the resistances concerned. Note, also, that (10) and (11) are faulty balances ; to improve them, destroy the resistance balance ; of course then the formula will change, and is likely to .become very complex. It will be understood that when I speak of false resistance balances in this paper I do not in any way refer to the thick-wire phenomenon, USE OF THE BRIDGE AS AN INDUCTION BALANCE. 37 my mentioned in letter [p. 30], which, from its very nature, requires the resistance balance to be upset, or be different from what it would be if the wire were thin, but of the same real [i.e., steady] resistance. The resistance balance must be upset in a perfect arrangement. Nor can there be a true balance got, but only an approximate one, unless a similar thick wire be employed to produce balance. What I refer to here is the upsetting of the true resistance balance when there is no perceptible departure whatever from the linear theory. The two effects may be mixed. To use the Bridge to speedily and accurately measure the inductance of "a coil, we should have a set of proper standard coils, of known inductance and resistance, together with a coil of variable inductance, i.e., two coils in sequence, one of which can be turned round, so as to vary the inductance from a minimum to a maximum. (The scale of this variable coil could be calibrated by (12a), first taking care that the resistance balance did not require to be upset.) This set of coils, in or out of circuit according to plugs, to form say branch 3, the coil to be measured to be in branch 4. Ratio of equality. Branches 1 and 2 equal. Of course inductionless, or practically inductionless resistances are also required, to get and keep the resistance balance. The only step to this I have made (this was some years ago) in my experiments, was to have a number of little equal unit coils, and two or three multiples ; and get exact balance by allowing induction between two little ones, with no exact measurement of the fraction of a unit. So long as we keep to coils we can swamp all the irregularities due to leading wires, etc., or easily neutralise them, and therefore easily obtain considerable accuracy. With short wires, however, it is a different matter. The inductance of a circuit is a definite quantity. So is the mutual inductance of two circuits. Also, when coils are connected together, each forms so nearly a closed circuit that it can be taken as such, so that we can add and subtract inductances, and localise them definitely as "belonging to this or that part of a circuit. But this simplicity is, to a great extent, lost when we deal with short wires, We unless they are bent round so as to make nearly closed circuits. cannot fix the inductance of a straight wire, taken by itself. It has no meaning, strictly speaking. The return current has to be considered. Balances can always be got, but as regards the interpretation, that will depend upon the configuration of the apparatus. [See Section xxxviii. of " Electromagnetic Induction," later.] Speaking with diffidence, having little experience with short wires, I should recommend 1 and 2 to be two equal wires, of any convenient length, twisted together, joined at one end, of course slightly separated at the other, where they join the telephone wires, also twisted. The exact arrangement of 3 and 4 will depend on circumstances. But always use a long wire rather than a short one (experimental wire). If this is in branch 4, let branch 3 consist of the standard coils (of appropriate size), and adjust them, inserting if necessary, coils in series with 4 also. Of course I regard the matter from the point of view of getting easily interpretable results. 38 ELECTRICAL PAPERS. The exact balance (1), (2), (3) above is quite special. If the branches 1 and 3 consist of any combination of conductors and condensers, with induction in masses of metal allowed, and branches 2 and 4 consist of an exactly equal combination, in every respect, there will never be any current in 5 due to impressed force in 6. And, more generally, 2 + 4 may be only a copy of 1 + 3, on a reduced scale, so to speak. M P.S. (April 27, 1886.) The great exactness with which, when a ratio of equality is used, the 1T 64 and methods conform to the true 6S M resistance balance, as above mentioned, together with the almost per- sistent departure of the 65 (Hughes's) method from the true resistance balance, led me to suspect that, as in the use of the simple Bridge method, with no mutual induction, the three conditions of a true balance M M M are reduced to two by a ratio of equality, the same thing happens in the and 64: 6B methods, but not in the 65. This I have verified. In Hughes's system the three conditions are ............................... (13) +y=o /1+ /2 +/3 ............... ....(15) Now = = take 1-^ 1^ r l r 2> = r 3 ?4 ; then the second and third are equivalent to 2M + - /4 13 + = 56 (l r3 /rj 0, 2^/33 = 1 + ljly The second of these is a special relation that must hold before the first is true. Hence the sound with a true resistance balance, and the M necessity of a false balance to get rid of it. But in the method the conditions are = 0, ............... (17) (18) = Take = Z 1 x 2, i\ r 2, = i 3 r 4, as before, and now the second and third conditions become identical, viz., agreeing with the previously obtained equation M M Thus, whilst Hughes's method is inaccurate, sometimes greatly so, we may employ the 64 and 63 methods without any hesitation, provided a ratio of equality be kept to. They will be as accurate as the simple Bridge method, and the choice of the methods will be purely a matter of convenience. I have verified experimentally that the Hughes system requires a false resistance balance when, instead of coils, short wires are used, the branch of greater inductance having apparently the greater resistance. M I have also verified that this effect is mixed with the thick-wire effect, which last is completely isolated by using the proper M method or the simple Bridge. Its magnitude can now be exactly measured, free from the errors of a faulty method. That is, it can be estimated for any particular speed of intermittences or reversals, for it is not a constant effect. Balance a very thin against a very thick wire, so that the effect occurs only on me side. ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 39 XXXV. ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. (SECOND HALF.) [The Electrician, 1886-7. Section XXV., April 23, 1886, p. 469 ; XXVI., May 14, p. 8 (vol. 17) ; XXVII., June 11, p. 88; XXVIJL, June 25, p. 128; XXIX., July 23, p. 212; XXX., August 6, p. 252 ; XXXI., August 20, p. 296; XXXII., August 27, p. 316; XXXIII., November 12, p. 10 (vol. 18) ; XXXIV., December 24, 1886, p. 143; XXXV., January 14, 1887, p. 211; XXXVL, February 4, p. 281 ; XXXVIL, March 11, p. 390; XXXVIIL, April 1, p. 457; XXXIXa., May 13, p. 5 (vol. 19); XXX1X6., May 27, p. 50; XL., June 3, p. 79; XLL, June 17, p. 124 ; XLII., July 1, p. 163 ; XLIIL, July 15, p. 206 ; XLIV., August 12, p. 295; XLV., August 26, p. 340; XLVL, October 7, p. 459; XLVII., December 30, 1887, p. 189 (vol. 20).] SECTION XXV. SOME NOTES ON MAGNETISATION. ALTHOUGH it is generally believed that magnetism is molecular, yet it is well to bear in mind that all our knowledge of magnetism is derived from experiments on masses, not on single molecules, or We molecular structures. may break up a magnet into the smallest pieces, and find that they, too. are little magnets. Still, they are not molecular magnets, but magnets of the same nature as the original ; solid bodies showing magnetic properties, or intrinsic- We ally magnetised. are nearly as far away as ever from a mole- cular magnet. To conclude that molecules are magnets because dividing a magnet always produces fresh magnets, would clearly be unsound reasoning. For it involves the assumption that a molecule has the same magnetic property as a mass, i.e., a large collection of molecules, having, by reason of their connection, properties not possessed by the molecules separately. (Of course, I do not define a molecule to be the smallest part of a substance that has all the properties of the mass.) If we got down to a mass of iron so small that it contained few molecules, and therefore certainly not possess- ing all the properties of a larger mass, what security have we that its magnetic property would not have begun to disappear, and that their complete separation would not leave us without any magnetic field at all surrounding them of the kind we attribute to intrinsic magnetisation. That there would be magnetic disturbances round an isolated molecule in motion through a medium, and with its parts in relative motion, it is difficult not to believe in view of the partial co-ordination of radiation and electromagnetism made by Maxwell. But it might be quite different from the magnetic field of a so-called magnetic molecule that is, the field of any small magnet. This evident magnetisation might be essentially conditioned by structure, not of single molecules, but of a collection, together with relative motions connected with the structure, this structure and relative motions conditioning that peculiar state of the medium in which they are immersed, which, when existent, implies intrinsic magnet- isation of the collection of molecules, or the little mass. However this be, two things are deserving of constant remembrance. First, that the molecular theory of magnetism is a speculation which it is 40 ELECTRICAL PAPERS. desirable to keep well separated from theoretical embodiments of known facts, apart from hypothesis. And next, that as the act of exposing a solid to magnetising influence is, it is scarcely to be doubted, always accompanied by a changed structure, we should take into account and endeavour to utilise in theoretical reasoning on magnetism which is meant to contain the least amount of hypothesis, the elastic properties of the body, speaking generally, and without knowing the exact connection between them and the magnetic property. Hooke's law, Ut tensio, sic vis, or strain is proportional to stress, implies perfect elasticity, and is the first approximate law on which to found the theory of elasticity. But beyond that, we have im- perfect elasticity, elastic fatigue, imperfect restitution, permanent set. When we expose an unmagnetised body to the action of a magnetic field of unit inductivity, it either draws in the lines of induction, in which case it is a paramagnetic, is positively magnetised inductively, and its inductivity is greater than unity ; or it wards oft' induction, in which case it is a diamagnetic, is negatively magnetised inductively, and its inductivity is less than unity; or, lastly, it may not alter the field at all, when it is not magnetised, and its induc- tivity is unity. Regarding, as I do, the force and the induction not the force and the induced magnetisation as the most significant quantities, it is clear that the language in which we describe these effects is somewhat imperfect, and decidedly misleading in so prominently directing attention to the induced magnetisation, especially in the case of no induced magnetisation, when the body is still subject to the magnetic influence, and is as much the seat of magnetic stress and energy as the surrounding medium. We may, by coining a new word pro- visionally, put the matter thus. All bodies known, as well as the so-called vacuum, can be inductized. According to whether the inductization (which is the same as "the induction," in fact) is greater or less than in vacuum (the universal magnetic medium) for the same magnetic force (the other factor of the magnetic energy product), we have positive or negative induced magnetisation. To the universal medium, which is the primary seat of the magnetic energy, we attribute properties implying the absence of dissipation of energy, or, on the elastic solid theory, perfect elasticity. (Dissipation in space is scarcely within a measurable distance of measurement.) But that the ether, resembling an elastic solid in some of its properties, is one, is not material here. Inductization in it is of the elastic or quasi-elastic character, and there can be no intrinsic magnetisation. Nor evidently can there be intrinsic magnetisation in gases, by reason of their mobility, nor in liquids, except of the most transient description. But when we come to solids the case is different. If we admit that the act of inductization produces a structural change in a body (this includes the case of no induced magnetisation), ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 41 and if, on removal of the inducing force, the structural change disappears, the body behaves like ether, so far, or has no inductive retentiveness. Here we see the advantage of speaking of inductive rather than of magnetic retentiveness. But if, by reason of im- perfect elasticity, a portion of the changed structure remains, the body has inductive retentiveness, and has become an intrinsic magnet. As for the precise nature of the magnetic structure, that is an independent question. If we can do without assuming any particular structure, as for instance, the Weber structure, which is nothing more than an alignment of the axes of molecules, a structure which I believe to be, if true at all, only a part of the magnetic structure, so much the better. It is the danger of a too special hypothesis, that as, from its definiteness, we can follow up its consequences, if the latter are partially verified experimentally we seem to prove its truth (as if there could be no other explanation), and so rest on the solid ground of nature. The next thing is to predict unobserved or unobservable phenomena whose only reason may be the hypothesis itself, one out of many which, within limits, could explain the same phenomena, though, beyond those limits, of widely diverging natures. The retentiveness may be of the most unstable nature, as in soft iron, a knock being sufficient to greatly upset the intrinsic magnetisa- tion existing on first removing the magnetising force, and completely alter its distribution in the iron ; or of a more or less permanent character, as in steel. But, whether the body be para- or dia- magnetic, or neutral, the residual or intrinsic magnetisation, if there be any, must be always of the same character as the inducing force. That is, any solid, if it have retentiveness, is made into a magnet, magnetised parallel to the inducing force, like iron. Until lately only the magnetic metals were known to show reten- tiveness. Though we should theoretically expect retentiveness in all solids, the extraordinary feebleness of diamagnetic phenomena might be expected to be sufficient to prevent its observation. But, first, Dr. Tumlirz has shown that quartz is inductively retentive, and next, Dr. Lodge (Nature, March 25th, 1886) has published some results of his experiments on the retentiveness of a great many other substances, following up an observation of his assistant, Mr. Davies. The mathematical statement of the connections between intrinsic magnetisation and the state of the magnetic field is just the same whether the magnet be iron or copper, para- or dia-magnetic, or is icutral. In fact, it would equally serve for a water or a gas magnet, rere they possible. That is, divB = being the magnetic force according to the equation B = /xH, where is the induction and //. the inductivity, F the electric current, if y, and h. the magnetic force of the intrinsic magnetisation, or the ipressed magnetic force, as I have usually called it in previous -i2 ELECTRICAL PAPERS. sections where it has occurred, because it enters into all equations as an impressed force, distinct from the force of the field, whose rotation measures the electric current. It is h and that are the two data //. concerned in intrinsic magnetisation and its field ; the quantity I, the intensity of intrinsic magnetisation, only gives the product, viz., = I fjih/^TT. It would not be without some advantage to make h and /A the objects of attention instead of I and /x, as it simplifies ideas as well as the formulae. The induced magnetisation, an extremely artificial and rather unnecessary quantity, is - (/* 1) (H - h)/4*r. It will be understood that this system, when united with the corresponding electric equations, so as to completely determine transient states, requires h to be given, whether constant or variable with the time. The act of transition of elastic induction into intrinsic magnetisation, when a body is exposed to a strong field, cannot be traced in any way by our equations. It is not formulated, and it would naturally be a matter of considerably difficulty to do it. In a similar manner, we may expect all solid dielectrics to be capable of being intrinsically electrized by electric force, as described in a previous section. I do not know, however, whether any dielectric has been found whose dielectric capacity is less than that of vacuum, or whether such a body is, in the nature of things, possible. As everyone knows nowadays, the old-fashioned rigid magnet is a myth. Only one datum was required, the intensity of magnetisation I, assuming /x to be unity in as well as outside the magnet. It is a great pity, regarded from the point of view of mathematical theory, which is rendered far more difficult, that the inductivity of intrinsic magnets is not unity. But we must take nature as we find her, and although Prof. Bottomley has lately experimented on some very unmagnetisable steel, which may approximate to /*= 1, yet it is perfectly easy to show that the inductivity of steel magnets in general is not 1, but a large number, though much less than the inductivity of soft iron, and we may use a hard steel bar, whether magnetised intrinsically or not, as the core of an electromagnet with nearly the same effects, as regards induced magnetisation, except as regards the amount, as if it were of soft iron. Regarding the measure of inductivity, especially in soft iron, this is really not an easy matter, when we pass beyond the feeble forces of telegraphy. For all practical purposes ^ is a constant when the magnetic force is small, and Poisson's assumption of a linear relation between the induced magnetisation and the magnetic force is abundantly verified. It is almost mathematically true. But go to larger forces, and suppose for simplicity we have a closed solenoid with a soft iron core, and we magnetise it. Let F be the magnetic force of the current. Then, if the induction were completely elastic, we should have the induction B = /*F. But in reality we have B = />t(F + h) = /xH. If we assume the former of these equations, that is, take the magnetic force of the current as the magnetic force, we shall obtain too large an H estimate of the inductivity, in reckoning which should be taken as the magnetic force. This may be several times as large as F. For, the ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 43 softer the iron the more imperfect is its inductive elasticity, and the more easily is intrinsic magnetisation made by large forces ; although the retentiveness may be of a very infirm nature, yet whilst the force F is on, there is h on also. This over-estimate of the inductivity may be partially corrected by separately measuring h after the original magnetising force has been removed, by then destroying h. But this h may be considerably less than the former. For one reason, when we take off F by stopping the coil-current, the molecular agitation of the heat of the induced currents in the core, although they are in such a direction as to keep up the induced magnetisation whilst they last, is sufficient to partially destroy the intrinsic magnetisation, owing to We the infirm retentiveness. should take off F by small instalments, or slowly and continuously, if we want h to be left. Another quantity of some importance is the ratio of the increment in the elastic induction to the increment in the magnetic force of the current. This ratio is the same as /x when the magnetic force is small, but is, of course, quite different when it is large. As regards another connected matter, the possible existence of magnetic friction, I have been examining the matter experimentally. Although the results are not yet quite decisive, yet there does appear to be something of the kind in steel. That is, during the act of in- ductively magnetising steel by weak magnetic force, there is a reaction on the magnetising current very closely resembling that arising from eddy currents in the steel, but produced under circumstances which would render the real eddy currents of quite insensible significance. In soft iron, on the other hand, I have failed to observe the effect. It has nothing to do with the intrinsic magnetisation, if any, of the steel. But as no hard and fast line can be drawn between one kind of iron and another, it is likely, if there be such an effect in steel, where, by the way, we should naturally most expect to find it, that it would be, in a smaller degree, also existent in soft iron. Its existence, however, will not alter the fact materially that the dissipation of energy in iron when it is being weakly magnetised is to be wholly ascribed to the electric currents induced in it. P.S. (April 13, 1886.) As the last paragraph, owing to the hypothesis involved in magnetic friction, may be somewhat obscure, I add this in explanation. The law, long and generally accepted, that the induced magnetisation is simply proportional to the magnetic force, when small, is of such importance in the theory of electromagnetism, that I wished to see whether it was minutely accurate. That is, that the curve of magnetisation is, at the origin, a straight line inclined at a definite angle to the axis of abscissae, along which magnetic force is reckoned. I employed a differential arrangement (differential telephone) admitting of being made, by proper means, of considerable sensitiveness. The law is easily verified roughly. When, however, we increase the sensi- tiveness, its accuracy becomes, at first sight, doubtful ; and besides, differences appear between iron and steel, differences of kind, not of mere magnitude. But as the sensitiveness to disturbing influences 44 ELECTRICAL PAPERS. is also increased, it is necessary to carefully study and eliminate them. The principal disturbances are due to eddy currents, and to the variation in the resistance of the experimental coil with temperature. For instance, as regards the latter, the approach of the hand to the coil may produce an effect larger than that under examination. The general result is that the law is very closely true in iron and steel, it being doubtful whether there is any effect that can be really traced to a departure from the law, when rapidly intermittent currents are employed, and that the supposed difference between iron and steel is unverified. Of course it will be understood by scientific electricians that it is necessary, if we are to get results of scientific definiteness, to have true balances, both of resistance and of induction, and not to employ an arrangement giving neither one nor the other. He will also understand that, quite apart from the question of experimental ability, the theorist sometimes labours under great disadvantages from which the pure experimentalist is free. For whereas the latter may not be bound by theoretical requirements, and can employ himself in making discoveries, and can put down numbers, really standing for complex quantities, as representing the specific this or that, the former is hampered by his theoretical restrictions, and is employed, in the best part of his time, in the poor work of making mere verifications. SECTION XXVI. THE TRANSIENT STATE IN A ROUND WIRE WITH A CLOSE-FITTING TUBE FOR THE RETURN CURRENT. The propagation of magnetic force and of electric current (a function of the former) in conductors takes place according to the mathematical laws of diffusion, as of heat by conduction, allowing for the fact of the electric quantities being vectors. This conclusion may perhaps be considered very doubtful, as depending upon some hypothesis. Since, however, it is what we arrive at immediately by the application of the laws for linear conductors to infinitely small circuits (with a tacit assumption to be presently mentioned), it seems to me more necessary for an objector to show that the laws are not those of diffusion, rather than for me to prove that they are. We may pass continuously, without any break, from transient states in linear circuits to those in masses of metal, by multiplying the number of, whilst diminishing the section of, the " linear " conductors indefinitely, and packing them closely. Thus we may pass from linear circuits to a hollow, core ; from ordinary linear differential equations to a partial differential equation ; from a set of constants, one for each circuit, to a continuous function, viz., a compound of the J function and its complementary function containing the logarithm. This I have worked out. Though very interesting mathematically, it would occupy some space, as it is rather lengthy. I therefore start from the partial differential equation itself. Our fundamental equations are, in the form I give to them, ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 45 H E and being the electric and magnetic forces, C the conduction current, k and p. the conductivity and the inductivity. The assumption I referred to is that the conductor has no dielectric capacity. Bad conductors have. We are concerned with good conductors, whose dielectric capacity is quite unknown. We are concerned with a special application, and therefore choose the suitable coordinates. All equations referring to this matter will be marked b. The investigations are almost identical with those given in my paper on " The Induction of Currents in Cores," in The Electrician for 1884. [Reprint, vol. I., p. 353, art. XXVIIL] The magnetic force was then longitudinal, the current circular ; now it is the current that is longitudinal, and the magnetic force circular. The distribution of current in a wire in the transient state depends materially upon the position of the return conductor, when it is near. The nature of the transient state is also dependent thereon. Now, if the return conductor be a wire, the distributions in the two wires are rendered unsymmetrical, and are thereby made difficult of treatment. We, therefore, distribute the return current equally all round the wire, by employing a tube, with the wire along its axis. This makes the distribution symmetrical, and renders a comparatively easy mathematical analysis possible. At the same time we may take the tube near the wire or far away, and so investigate the effect of proximity. The present example is a comparatively elementary one, the tube being supposed to be close-fitting. As I entered into some detail on the method of obtaining the solutions in " Induction in Cores," I shall not enter into much detail now. The application to round wires with the current longitudinal was made by me in The Electrician for Jan. 10, 1885, p. 180, so far as a general description of the phenomenon is concerned. my See also letter of April 23, 1886. [Reprint, vol. I., p. 440; vol. IL, p. 30.] Let there be a wire of radius a, surrounded by a tube of outer radius b, and thickness b a. In the steady state, if the current-density is F in the wire, it is - Ta 2/(b 2 - a2 ) in the tube, if both be of uniform con- H ductivity, and the tube or sheath be the return conductor of the wire. Let HI be the intensity of magnetic force in the wire, and in the 2 tube. The direction of the magnetic force is circular about the axis in We both, and the current is longitudinal. shall have H! = 2uTr, H = 2 - 2*rIV(^ - b 2 2 )/r(b' - a2 ), (2b) where r is the distance of the point considered from the axis. Test by We the first of equations (15). have curl = i ir, r dr when applied to H. Now let this steady current be left to itself, without impressed force to keep it up, so that the " extra-current " phenomena set in, and the magnetic field subsides, the circuit being left closed. At the time t later, if the current-density be 7 at distance r from the axis, it will be represented by AJ y = Z (nr)<>* () 46 ELECTRICAL PAPERS. where 2 is the sign of summation. The actual current is the sum of an infinite series of little current distributions of the type represented, in which A, n, and p are constants, and JQ (nr) is the Fourier cylinder We function. have - r dr dr ...(45) Let d/dt=p, a constant, then n is given in terms of p by = ri2 - ^TTjjikp ............................... (5b) We suppose that k and /x are the same in the wire as in the sheath. Differences will be brought in in the subsequent investigation with the sheath at any distance. In (3b) there are two sets of constants, the A's fixing the size of the normal systems, and the ris or p's, since these are connected by (5&). To find the ris, we ignore dielectric displacement, since it is electro- magnetic induction that is in question. This gives the condition #= 2 0, at = r 6; ........................ (66) i.e., no magnetic force outside the tube. This gives us (U) as the determinantal equation of the ris, which are therefore known by __ _ inspection of a Table of values of the J^ function. Find the A's by the conjugate property. Thus, - - A = rfV Jo (r)rrfr J[.YaV^arJnfr/^ a?) =~ L J na i( ) o The full solution is, therefore, 2aT giving the current at time t anywhere. The equation of the magnetic force is obtained by applying the second of equations (15) ; it is . SiraT JmJnr4* and the expression for the vector-potential of the current (for its scalar A^ magnitude that is to say, as its direction, parallel to the current, does not vary, and need not be considered), is This may be tested by /xH; .............................. (126) ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 47 curl being now= -d/dr. = In the steady state (initial), 0, in the wire, and ^/ A - 2/ - V* + r* + 262 log\ (136) in the sheath. Test by (126) applied to (136) to obtain (26). The magnetic energy being puBP/Qir per unit volume, the amount in length / of wire and sheath is, by (106), 2d n*Jt(nb) To verify, this should equal the space-integral of ^A y, using (116) and (96). This need not be written. They are identical because [j!(nr) rdr = {*J*(nr)rdr = J6V>6), Jo Jo so that we may write the expression for T thus, The dissipativity being 2 y /& per unit volume, the total heat in length I of wire and sheath is, = if p k~ l , the resistivity, and the complete variable period be included, 1 Q-PiWIV-W^ff** .......... When = t Q, either by (146) or by easy direct investigation, the initial magnetic energy in length I is giving the inductance of length I as <"" which may be got in other ways. This refers to the steady state. In the transient state there cannot be said to be a definite inductance, as the distribution varies with the time. The expression in (156) for the total heat may be shown to be equivalent to that in (166) for the initial magnetic energy, thus verifying the conservation of energy in our system. I should remark that it is the same formula (96) that gives us the current both in the wire and tube, and the same formula (106) that gives us the magnetic force. They are distributed continuously in the variable period. It is at the first moment only that they are discontinuous, requiring then separate formulae for the wire and tube, i.e., separate finite formulae, although only a single infinite series. The first term of (96) is, of course, the most important, representing 48 ELECTRICAL PAPERS. the normal system of slowest subsidence. In fact, there is an extremely rapid subsidence of the higher normal systems ; only three or four need be considered to obtain almost a complete curve ; and, at a compara- tively early stage of the subsidence, the first normal system has become far greater than the rest. In fact, on leaving the current without impressed force, there is at first a rapid change in the distribution of the current (and magnetic force), besides a rapid subsidence. It tends to settle down to be represented by the first normal system ; a certain nearly fixed distribution, subsiding according to the exponential law of a linear circuit. To see the nature of the rapid change, and of the first normal system, refer to The Electrician of Aug. 23, 1884 [vol. L, p. 387], where is a representation of the / and Jj curves. In Fig. 1, take the distance OC.2 to be the outer radius of the tube, being on the axis. Then the curve marked J^ is the curve of the magnetic force, showing its com- w parative strength from the centre of the wire to the outside of the tube, in the first normal system. And, to correspond, the curve from up to C 2 is the curve of the current, showing its distribution in the first normal system. We see that the position of the point J5 with : respect to the inner radius of the sheath determines whether the current is transferred from the wire to the sheath, or vice versa, in the early part of the subsidence. If the sheath is very thin, so that the radius of the wire extends nearly up to (7 2, there is transfer of the sheath -current (initial) from the sheath a long way into the wire. On the other hand, if the wire be of small radius compared with the outer radius of the tube, so that the tube's depth extends from C 2 nearly up to 0, there is a transfer of the original wire-current a long way into the thick sheath. In Fig. 2 [vol. I., p. 388] are shown the first four normal systems, all on the same scale as regards the vertical ordinate, but we are not concerned with them at present. Since - v~ l by (56), and -p~ l is the time-constant of subsidence of a normal system, we have, for the value of the time-constant of the first system, because the value of the first nb, say n-^b, is 3-83. Compare this with E the linear-theory time-constant L/B, where L is given by (\lb\ and is the resistance of length / of the wire and sheath (sum of resistances, as the current is oppositely directed in them). Let a = \b. Then L = M28 id. ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 49 We have also E= 1 2 Ql/3-n-kb' , therefore L/R ='211 irfjjtf, so that the time-constant of the first normal system is to that of the current in wire and tube on the linear theory as *27 to '21. But it is only after the first stage of the subsidence is over that this larger timeconstant is valid. We may write the expression for L thus. Let x = b/a, then (M r fda* log X ,\ "rfTiiVS^r nearly the same as 2/zZ log x when x is large. The minimum is when b = a; then L = ^l. This is the least value of the inductance of a round wire, viz., when it has a very thin and close-fitting sheath for the return current, so that the magnetic energy is confined to the wire. When b/a is only a little over unity, _ 3fr2 - a* - 2ab b*-a*' +2 (b a) We have also R = WjvMl)? - a2 ), and therefore L/R - dfc^ Irrespective of b/a being only a little over unity, we have, with a/b = ^, = L/R "009 2 (47r/*&& ), 55 > 10 55 TT 55 " -090 whilst the time-constant of the first normal system in all three cases is 068 2 (47r^6 ). The maximum of L/R with b/a variable is when x being b/a. This value of x is not much different from the ratio of the nodes in the first normal system, or the ratio of the value of nr making J^nr) = for the first time, to that making J (nr) = 0. For the latter value makes log# = -4'65, and makes the other side of the last equation be -486. In the subsidence from the steady state, the central part of the wire is the last to get rid of its current. But the steady state has to be first set up. Then it is the central part of the wire that is the last to get its full current. To obtain the equations showing the rise of the current and of the magnetic force in the wire and the tube, we have to reverse or negative the preceding solutions, and superpose the final steady states. As these are discontinuous, there are two solutions, one for the wire, the other for the sheath ; but the transient part of them, which ultimately disappears, is the same in both. There is no occasion to write these out. If the steady state is not fully set up before the impressed force is removed, we see that the central part of the wire is less useful as a con- II. E. P. VOL. TT. D 50 ELECTRICAL PAPERS. ductor than the outer part, as the current is there the least. If there are short contacts, as sufficiently rapid reversals, or intermittences, the central part of the wire is practically inoperative, and might be removed, so far as conducting the current is concerned. Immediately after the impressed force is put on, there is set up a positive current on the outside of the wire, and a negative on the inside of the sheath, which are then propagated inward and outward respectively. If the sheath be thin, the initial (surface) wire-current is of greater and the initial sheath-current of less density than the values finally reached by keeping on the impressed force; whilst if it be the sheath that is thick the reverse behaviour obtains. This case of a close-fitting tube is rather an extreme example of departure from the linear theory ; the return current is as close as possible and wholly envelops the wire-current. Except as regards duration, the distributions of current and magnetic force are independent of the dimensions, i.e., in the smallest possible round wire closely surrounded by the return current the phenomena are the same as in a big wire similarly surrounded, except as regards the duration of the variable period. The retardation is proportional to the conductivity, to the inductivity, and to the square of the outer radius of the tube. When, as in our next Section, we remove the tube to a distance, we shall find great changes. SECTION XXVII. THE VARIABLE PERIOD IN A EOUND WIRE WITH A CONCENTRIC, TUBE AT ANY DISTANCE FOR THE RETURN CURRENT. The case considered in the last Section was an extreme one of departure from the linear theory. This arose, not from mere size, but from the closeness of the return to the main conductor, and to its completely enclosing it. Practically we must separate the two conductors by a thickness of dielectric. The departure from the linear theory is then less pronounced ; and when we widely separate the conductors it tends to be confined to a small portion only of the variable period. The size of the wire is then also of importance. Let there be a straight round wire of radius av conductivity k lt and inductivity /Zj, surrounded by a non-conducting dielectric of specific capacity c and inductivity /x2 to radius a 2, beyond which is a tube of conductivity ky and inductivity /*3, inner radius a.2 and outer 3. The object of taking c into account, temporarily, will appear later. Let the current be longitudinal and the magnetic force circular. Then, by (1&), if y is the current-density at distance r from the axis, we shall have in the conductors, and in the dielectric respectively ; the latter form being got by taking y = cj/47r, the rate of increase of the elastic displacement. ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 51 A normal system of longitudinal current-density may therefore be represented by 7j = -^j^o(wir)j from r = to yi = AJt(njr) + BsKt(nj), r = a to l 7s = ^s^o(V) + #A(V)> r * "2 to in the wire, in the insulator, and in the sheath, respectively, at a given moment. In subsiding, free from impressed force, each of these expressions, when multiplied by the time-factor t pt , gives the state at the time t later. K /"(/) is the Fourier cylinder function, and Q (nr) the complementary function. [For their expansions see vol. L, p. 387, equations (70) and (71)]. The ^4's and H's are constants, fixing the size of the normal functions ; the n's are constants showing the nature of the distributions, and p determines the rapidity of the subsidence. By applying (186) to (196) we find n? = - lirnfap, nl = - n#p\ - //,r 4*17*3^ : ...... (206) expressing all the n's in terms of the p. Corresponding to the expressions (196) for the current, we have the following for the magnetic force : J where, as is usual, the negative of the differential coefficient of Q(z) K with respect to z is denoted by J^z) ; and, in addition, the negative of the differential coefficient of Q(z) with respect to z is denoted by K^(z). H These equations (216) are got by the second and third equations (16), in the case of H^ and H^ and in the case of 3; by using, instead of Ohm's law, the dielectric equation, giving E in the dielectric, being the electric force. Of course d/dt=p, in a normal system. We have next to find the relations between the five A's and ^'s, to make the three solutions fit one another, or harmonize. This we must do by means of the boundary conditions. These are nothing more than the surface interpretations of the ordinary equations referring to space distributions. In the present case the appropriate conditions are con- tinuity of the magnetic and of the electric force at the boundaries, because the two forces are tangential ; the conditions of continuity of the normal components of the electric current and of magnetic induc- tion are not applicable, because there are no normal components in question. If the magnetic or the electric force were discontinuous, we H H should have electric or magnetic current-sheets. ?' Thus = . 2 HI and 2 are equal These give, by (216), at r=a lt and 2 and H^ are equal at 52 ELECTRICAL PAPERS. and 2 (4:Trn 2/n.2" cp l (n^} ....... (236. E E Similarly, and l are equal at r = a v and E.2 and E* are equal at a. 2 . These give, by (196), and / + =^ / % + K (47r/cp){^ 2 (n2a2 ) jB2JST ( 2fl2 )} 1 {^ 3 ( 2) s Q (i t 2 )}. (256) ^ ^ Thus, starting with given, (225) and (245) give 2 and 7> in terms A A A of v and then (236) and (256) give z and J53 in terms of Y Similarly we might carry the system further, by putting more con- centric tubes of conductors and dielectrics, or both, outside the first tube, using similar expressions for the magnetic and electric forces; every fresh boundary giving us two boundary conditions of continuity to connect the solution in one tube with that in the next. But at present we may stop at the first tube. Ignore the dielectric displace- ment beyond it, i.e., put c = beyond = r 3 cf, because our tube is to We be the return conductor to the wire inside it. may merely remark in passing that although when such is the case, there is, in the steady state, absolutely no magnetic force outside the tube, yet this is not exactly true in a transient state. To make it true, take e = beyond r=a 3; requiring -T 3 = at r-a . s This gives, by (216), J = gJr1( 8os)+l?3Jri( 3a8) ...................... (266) A A Now % and J2 3 are, by the previous, known in terms of r Make A the substitution, and we find, first, that i is arbitrary, so that it, when given, fixes the size of the whole normal system of electric and magnetic force; and next, that the n'a are subject to the following equation : 13213 13312 OV( n 2 a 2 ~ / _ n 1 ( a\ 2 2/ where, on the left side, to save trouble, the dots represent the same fraction that appears in the numerator immediately over them. Now, the w's are known in terms of p, hence (276) is the deter- minantal equation of the j?'s, determining the rates of subsidence of We all the possible normal systems. have, therefore, all the informa- tion required in order to solve the problem of finding how any initially given state of circular magnetic force and longitudinal electric force in We the wire, insulator, and sheath subsides when left to itself. merely require to decompose the initial states into normal systems of the above types, and then multiply each term by its proper time-factor tpt to let it subside at its proper rate. To effect the decomposition, make use of the universal conjugate property of the equality of the mutual potential ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 53 U and the mutual kinetic energy of two complete normal systems, 12 = Tu We [vol. I., p. 523], which results from the equation of activity. start with a given amount of electric energy in the dielectric, and of magnetic energy in the wire, dielectric and sheath, which are finally used up in heating the wire and sheath, according to Joule's law. It would be useless to write out the expressions, for I have no intention of discussing them in the above general form, especially as regards the influence of c. Knowing from experience in other similar cases that I have examined, that the effect of the dielectric displace- ment on the wire and sheath phenomena is very minute, we may put We c = at once between the wire and the sheath. might have done this at the beginning; but it happens that although the results are more complex, yet the reasoning is simpler, by taking c into account. The question may be asked, how set up a state of purely longitudinal electric force in the tube, sheath, and intermediate dielectric? As regards the wire and sheath, it is simple enough ; a steady impressed force in any part of the circuit will do it (acting equally over a complete section). But it is not so easy as regards the dielectric. It requires the impressed force to be so distributed in the conductors as to support the current on the spot without causing difference of potential. There will then be no dielectric displacement either (unless there be impressed force in the dielectric to cause it). Now, if we remove the impressed force in the conductors, the subsequent electric force will be purely longitudinal in the dielectric as well as in the conductors. But practical^ we do not set up currents in this way, but by means of localised impressed forces. Then, although the steady state is one of longitudinal electric force in the wire and sheath, in the dielectric there is normal or outward electric force as well as tangential or longitudinal, and the normal component is, in general, far greater than the tangential. In fact, the electrostatic retardation depends upon the normal displacement. But electrostatic retardation, which is of such immense import- ance on long lines, is quite insignificant in comparison with electromagnetic on short lines, and in ordinary laboratory experiments with We closed circuits (no condensers allowed) is usually quite insensible. see, therefore, that when we put = c 0, and have purely longitudinal electric force, we get the proper solutions suitable for such cases where the influence of electrostatic charge is negligible, irrespective of the distribution of the original impressed force. Our use of the longitudinal displacement in the dielectric, then, was merely to establish a connection in time between the wire and the sheath, and to simplify the conditions. (In passing, I may give a little bit of another investigation. Take both electric and magnetic induction into consideration in this wire and sheath problem, treating them as solids in which the current distribution varies with the time. The magnetic force is circular, so is fully specified by its intensity, say H, at distance r from the axis. Its equation is, if z be measured along the axis, 54 ELECTRICAL PAPERS. in which discard the last term when the wire or sheath is in question ; H or retain it and discard the previous when the dielectric is considered. The form of the normal solution is H= B Ji(sr)(A sin + cos)mz e*', m = + for the wire, where s2 - (iirfdsp 2 ). The current has a longitudinal and a radial component, say T and y, given by F= L + sJ (sr) (A sin cos)mz e pt , y= mJ^sr^A cos - B sm)mz *". K In the dielectric and sheath the Q and K-^ functions have, of course, to be counted with the 7 and Jr ) We Now put c = in (27 b). shall have J = Q (n2r) 1 ; - %7i(v) = ; K = Q (n2r) log (n.2r) ; - n./K^r) = 1 ; which will bring : ( 27b) down to the determinantal equation in the case of ignored dielectric displace- ment. To obtain this directly, establish a rigid connection between the magnetic and electric forces at r = a^ and at r=a 2, thus. Since there is no current in the insulating space, the magnetic force varies inversely as the distance from the axis of the wire. Therefore, instead of the second of (216), we shall have #2 = - ( n i/^Ap)^i J i( n i a i)( a i/ r\ H H by the first of (216). Thus 2 at r=a 2 is known, and, equated to 3 A A B at r=a 2, gives us one equation between lt and 3, 3. Next we have H^ meaning, temporarily, the value of H^ at r = ar This, when multiplied by /*2, is the amount of induction through a rectangular portion of a plane through the axis, bounded by straight lines of unit length parallel to the axis at distances a-^ and r from it; or the line- integral of the vector-potential round the rectangle ; or the excess of the vector-potential at distance r over that at distance a x; so, when multiplied by p, it is the excess of the electric force at a^ over that at r. Thus the electric force is known in the insulating space in terms E of that at the boundary of the wire. Its value at r=a 2 equated to 5 A B at r a 2 gives us a second equation between lt A%, and . z The third is equation (266) over again, and the union of the three gives us (286) again. We now have, if y1 and y3 are the actual current-densities at time t in the wire and the sheath respectively, ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 55 where . in which only the ^4 requires to be found, so that when = t Q, the initial state may be expressed. The decomposition of the initial state into normal systems may be effected by the conjugate property of the vanishing of the mutual kinetic energy, or of the mutual dissipativity of a pair of normal systems. Thus, in the latter case, writing (296) thus, y = ^Au, = 2^r, we shall have ; + u1tf^tfr/&1 p<. Iv fa, J <2 We K ^ Wj, -fr'j, and u 2, v 2 being a pair of normal solutions. can only get rid of those disagreeable customers, the and functions, by taking the sheath so thin that it can be regarded as a linear conductor i.e., neglect variations of current-density in it, and consider instead the integral current. (Except when the sheath and wire are in contact and of the same material, as in the last section.) Let a 4 be the very small thickness of the sheath, and evaluate (286) on the supposition that a 4 is infinitely small, so that a 2 and a s are equal ultimately. The result is /oKi) = i^iK a i)(( ?h/x x 2// i) lo aa - ( 2/ i) the determinantal equation in the case of a round wire of radius c^ with a return conductor in the form of a very thin concentric sheath, radius a.2. Notice that /*3, the inductivity of the sheath itself, has gone out altogether ; that is, an iron sheath for the return, if it be thin enough, does not alter the retardation as compared with a copper sheath, provided the difference of conductivity be allowed for. We may get (306) directly, easily enough, by considering that the total sheath-current must be the negative of the total wire-current, which last is, by integrating the first of (296) throughout the wire, 27 = (^/n1)2ro1 a 1(n1 1 ) t pt . This, divided by the volume of the sheath per unit length, that is, by 27r 2r/4, gives us the sheath current-density, and this, again, divided = by & 3 gives us the electric force at r a. 2 Another expression for the electric force at the sheath is given by the previous method (the rectangle business). Equate them, and (306) results. We have now got the heavy work over, and some results of special cases will follow, in which we shall be materially assisted by the analogy of the eddy currents in long cores inserted in long solenoidal coils. SECTION XXVIII. SOME SPECIAL RESULTS RELATING TO THE RISE OF THE CURRENT IN A WIRE. Premising that the wire is of radius a lt conductivity kv inductivity f4 ; that the dielectric displacement outside is ignored ; and that the sheath for the return current is at distance and is so thin that 2, 56 ELECTRICAL PAPERS. variations of current-density in it may be ignored, so that merely the total return current need be considered ; that a4 is the small thickness of the sheath, and k 3 its conductivity, we have the determinantal equa- tion (306). Let now L = Q 2/x2 log(fl2/i), A ViT = 1 ( , -#2 = O^/'A)" 1 - L Q is the external inductance per unit length, i.e., the inductance per R unit length of surface-ciiTTent, ignoring the internal magnetic field. l and E. 2 are the resistances per unit length of the wire and sheath |^ respectively, and is the internal inductance per unit length, i.e., the inductance per unit length of uniformly distributed wire-current when the return current is on its surface, thus cancelling the external We magnetic field. can now write (306) thus : and, in this, we have = 4vpikjpOi 4&l*i/Ri, ...(326) From (316) we see that the two important quantities are the ratio of the external to the internal inductance, and the ratio of the external to LJ^ the internal resistance, i.e., the ratios and R^jR^. Suppose, first, the return has no resistance. Draw the curves = yi Jo(z)/Ji(x) and */2 = KA)//*iK < the ordinates y, abscissae x, which stands for n^. Their intersections show the required values of x. The JJJl curve is something like the LJ^ curve of cotangent. If is large, the first intersection occurs with J a small value of x, so small that (x) is very little less than unity, so that a uniform distribution of current is nearly represented by the first normal distribution, whose time-constant is a little greater than that of the linear theory. J^x) = 0. On the The remaining intersections will be nearly given by other hand, decreasing LQ/^ increases the value of = the first x ; in the limit it will be the first root of J^x) 0. Thus, if the wire be of copper, and the return distant (compared with radius of wire), the linear theory is approximated to. If of iron, on the other hand, it is not practicable to have the return sufficiently distant, on account of the large value of /x15 unless the wire be exceedingly fine. Even if of copper, bringing the return closer has the same effect of rendering the first normal system widely different from representing a uniform distribution of current. It is the external magnetic field that gives stability, and reduces differences of current-density. Next, let the return have resistance. The curve y2 must now be The effect of increasing R.2 from zero is the opposite of that of increas- ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 57 ing LQ. It increases the first x, and tends to increase it up to that given by J^x) = (not counting the zero root of this equation). Thus there is a double effect produced. Whilst on the one hand the rapidity of subsidence is increased by the resistance of the sheath, on the other the wire-current in subsiding is made to depart more from the uniform distribution of the linear theory. The physical explanation is, that as the external field in the case of sheath of no resistance cannot dissipate its energy in the sheath it must go to the wire. But when the sheath has great resistance the external field is killed by it ; then the internal J field is self-contained, or the wire-current subsides as if = l (x) Q, with a wide departure from uniform distribution. This must be marked when the wire-circuit is suddenly interrupted, making the return-resistance infinite. Now, let there be no current at the time = 0, when, put on, and keep on, a steady impressed force, of such strength that the final current-density in the wire is F . At time t the current-density F at distance r from the axis is given by __ i,t l\ ^n a ll ' I - Kjptf + {/ (Vi)AA(Vi) } 2 where the n^'a are the roots of equation (316). And the total current in the wire, say C l} and with it the equal and opposite sheath-current, will rise thus to the final value G' , ^ C 4 (1 It will give remarkably different results according as we take the resistance of the wire very small and that of the sheath great, or conversely, or as we vary the ratio LJfj^. Infinite conductivity shuts out the current from the wire altogether, and so does infinite inductivity; the retardation to the inward transmission of the current being proportional to the product fij^af. Similarly, if the sheath has no resistance, the return current is shut out from it. In either of these shuttingout cases the current becomes a mere surface-current, what it always is in the initial stage, or when we cannot get beyond the initial stage, by reason of rapidly reversing the impressed force, when the current will be oppositely directed in concentric layers, decreasing in strength with great rapidity as we pass inward from the boundary. But if both the sheath and the wire have no resistance, there will be no current at all, except the dielectric current, which is here ignored, and the two surface-currents. The way the current rises in the wire, at its boundary, and at its centre, is illustrated in " Induction in Cores." For the characteristic equation of the longitudinal magnetic force in a core placed within a long solenoid, and that of the longitudinal current in our present case, are identical. The boundary equations are also identical. That is, (316) is the boundary equation of the magnetic force in the core, except- B ing that the constants LQ/^ and 2/E1 have entirely different meanings, depending upon the number of turns of wire in the coil, and its 58 ELECTRICAL PAPERS. dimensions, and resistance. If, then, we adjust the constants to be equal in both cases, it follows that when any varying impressed force acts in the circuit of the wire and sheath, the current in the wire will be made to vary in identically the same manner as the magnetic force in the core, at a corresponding distance from the axis, when a similarly varying impressed force acts in the coil-circuit (which, however, must have only resistance in circuit with it. not external self-induction as well). Thus, we can translate our core-solutions into round-straight- wire solutions, and save the trouble of independent investigation, in case a detailed solution has been already arrived at in either case. Refer to Fig. 3 [p. 398, vol. I., here reproduced]. It represents the curves of subsidence from the steady state. The "arrival" curves are got by perversion and inversion, i.e., turn the figure upside down and look at it from behind. The case we now refer to is when the sheath has negligible resistance, and when we take the constant Z= 2/>i1 , which requires a near return when the wire is of copper, but a very distant one if it is iron. 6 -8 1-0 1-2 1-4 Regarding them as arrival-curves, the curve h-Ji^ is the linear-theory curve, showing how the current-density would rise in all parts of the wire if it followed the ordinarily assumed law (so nearly true in common H H fine-wire coils). The curve a a shows what it really becomes, at the boundary, and near to it. The current rises much more rapidly there in the first part of the variable period, and much more slowly in the later part. From this we may conclude that, when very rapid reversals are sent, the amplitude of the boundary current-density will be far greater than according to the linear theory ; whereas if they be made much slower it may become weaker. This is also verified by the separate calculation ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 59 in " Induction in Cores " of the reaction on the coil-current of the core- currents when the impressed force is simple-harmonic, the amplitude of the coil-current being lowered at a low frequency, and greatly increased at high frequencies [p. 370, vol. I.]. The curve ff ff shows how the current rises at the axis of the wire. It is very far more slowly than at the boundary. But the important characteristic is the preliminary retardation. For an appreciable interval of time, whilst the boundary-current has reached a considerable fraction of its final strength, the central current is infinitesimal. In fact the theory is similar to that of the submarine cable ; when a battery is put on at one end, there is only infinitesimal current at the far end for a certain time, after which comes a rapid rise. H H H H Between the axis and the boundary the curves are intermediate between a a at the boundary and at the axis, there being pre- liminary retardation in all, which is zero at the boundary, a maximum at the axis. It is easy to understand, from the existence of this practi- cally dead period, how infinitesimally small the axial current can be, compared with the boundary current, when very rapid reversals are sent. The formulae will follow. The fourth curve liJiQ shows the way the current rises at the axis when the return has no resistance, but when at the same time there is LJ^ no external magnetic field, or 0. The return must fit closely We over the wire. may approximate to this by using an iron wire and a close-fitting copper sheath of much lower resistance. There is pre- liminary retardation, after which the current rises far more rapidly Z than when //x1 is finite. LJ^ That is, the effect of changing from the value 2 to the value H H is to change the axial arrival-curve from to h h . Suppose it is a copper wire. Then L = 2 means Iog(a2/a1 ) = 1, or a2/a^ = 2-718. Thus, removing the sheath from contact to a distance equal to 2-7 times the radius of the wire alters the axial arrival-curve from hh Q to H^H^ Now this great alteration does not signify an increased departure from the linear theory (equal current-density over all the wire). It is We exactly the reverse. have increased the magnetic energy by adding the external field, and, therefore, make the current rise more slowly. But the shape of the curve H^H^ if the horizontal (time) scale be suit- ably altered, will approximate more closely to the linear-theory curve h-Ji^ By taking the sheath further and further away, continuously increasing the slowness of rise of the current, we (altering the scale) approximate as nearly as we please to the linear-theory curve, and gradually wipe out the preliminary axial retardation, and make the current rise nearly uniformly all over the section of the wire, except at the first moment. In fact, we have to distinguish between the absolute and the relative. When the sheath is most distant the current rises the most slowly, but also the most regularly. On the other hand, when the sheath is nearest, and the current rises most rapidly, it does so with the greatest possible departure from uniformity of distribution. ^ If the wire is of iron, say = 200, the distance to which the sheath would have to be moved would be impracticably great, so that, except GO ELECTRICAL PAPERS. in an iron wire of very low inductivity, or of exceedingly small radius, we cannot get the current to rise according to the linear theory. We The simple-harmonic solutions I must leave to another Section. may, however, here notice the water-pipe analogy [p. 384, vol. i.]. The current starts in the wire in the same manner as water starts into motion in a pipe, when it is acted upon hy a longitudinal dragging force applied to its boundary. Let the water be at rest in the first place. Then, by applying tangential force of uniform amount per unit area of the boundary we drag the outermost layer into motion instantly ; it, by the internal friction, sets the next layer moving, and so on, up to the centre. The final state will be one of steady motion resisted by surface friction, and kept up by surface force. The analogy is useful in two ways. First, because any one can form an idea of this communication of motion into the mass of water from its boundary, as it takes place so slowly, and is an everyday fact in one form or another ; also, it enables us to readily perceive the manner of propagation of waves of current into wires when a rapidly varying im- pressed force acts in the circuit, and the rapid decrease in the amplitude of these waves from the boundary inward. Next, it is useful in illustrating how radically wrong the analogy really is which compares the electric current in a wire to the current of water in a pipe, and impressed E.M.F. to bodily acting impressed force on the water. For we have to apply the force to the boundary of the water, not to the water itself in mass, to make it start into motion so that its velocity can be compared with the electric current-density. The inertia, in the electromagnetic case, is that of the magnetic field, not of the electricity, which, the more it is searched for, the more unsubstantial it becomes. It may perhaps be abolished altogether when we have a really good mechanical theory to work with, of a sufficiently simple nature to be generally understood and appreciated. In our fundamental equations of motion curl (e - E) = /xH, curl H = suppose we have, in the first place, no electric or magnetic energy, so that E = 0, H = 0, everywhere, and then suddenly start an impressed force e. The initial state is E = 0, H = 0, Thus the first effect of e is to set up, not electric current (for that requires there to be magnetic force), but magnetic current, or the rate of increase of the magnetic induction, and this is done, not by e, but by its rotation, and at the places of its rotation. [A general demonstration will be given later that disturbances due to impressed e or h always have curl e and curl h for sources.] Now, imagine e to be uniformly distributed throughout a wire. Its rotation is zero, except on the boundary, where it is numerically e, directed perpendicularly to the axis of the wire. Thus the first effect is magnetic current on the boundary of the wire, and this is propagated inward and outward through the conductor and the dielectric respec- ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 61 lively. Magnetic current, of course, leads to magnetic induction and electric current. Now, in purely electromagnetic investigations relating to wires, in which we ignore dielectric displacement, we may, for purposes of calculation, transfer our impressed forces from wherever they may be in the circuit to any other part of the circuit, or distribute them uniformly, so as to get rid of difference of potential, which is much the best plan. It is well, however, to remember that this is only a device, similar in reason and in effect to the devices employed in the statics and dynamics of supposed rigid bodies, shifting applied forces from their points of application to other points, completely ignoring how forces are really transmitted. The effect of an impressed force in one part of the circuit is assumed to be the same as if it were spread all round the circuit. It would be identically the same were there no dielectric displacement, but only the magnetic force in question. When, however, we enlarge the field of view, and allow the dielectric displacement, it is not permissible to shift the impressed forces in the above manner, for every special arrangement has its own special distribution of electric energy. The transfer of energy is, of course, always from the source, wherever it may be. The first effect of starting a current in a wire is the dielectric disturbance, directed in space by the wire, because it is a sink of energy where it can be dissipated. But the dielectric disturbance travels with such great speed that we may, unless the line is long, regard it as affecting the wire at a given moment equally in every part of its length ; and this is substantially what we do when we ignore dielectric displacement in our electromagnetic investigations, distribute the impressed force as we please, and regard a long wire in which a current is being set up from outside as similar to a long core in a magnetising helix, when we ignore any difference in action at different distances along the core. SECTION XXIX. OSCILLATORY IMPRESSED FORCE AT ONE END OF A LINE. ITS EFFECT. APPLICATION TO LONG-DISTANCE TELEPHONY AND TELEGRAPHY. Given that there is an oscillatory impressed force in a circuit, if this question be asked what is the effect produced 1 the answer will vary greatly according to the conditions assumed to prevail. I therefore make the conditions very comprehensive, taking into account frictional resistance, forces of inertia, forces of elasticity, and also the approxima- tion to surface conduction that the great frequency of telephonic currents makes of importance. Space does not permit a detailed proof from beginning to end. The results may, however, be tested for accuracy by their satisfying all the conditions laid down, most of which I have given in the last three Sections. The electrical system consists of a round wire of radius a lt conduc- ^ tivity kv and inductivity ; surrounded by an insulator of inductivity 62 ELECTRICAL PAPERS. /x2 and specific dielectric capacity c, to radius a.2 ; surrounded by the return of conductivity ky inoluctivity /*3, and outer radius ay The wire and return to be each of length I, and to be joined at the ends to make a closed conductive circuit. Let S be the electrostatic capacity, and L Q the inductance of the dielectric per unit length of the line. That is, L Q = 2^log(a.2/al \ S = c{2 .og^A)}" 1......... (335) We L have = S = Cfj, 2 v~'2 :, if v is the speed of undissipated waves through the dielectric. V Let be the surface-potential of the wire, and C the wire-current, or total current in the wire, at distance x from one end, at time /. The differential equation of F'is where R' and 1J are certain even functions of p, whose structure will be explained later, and p stands for d/dt. That of C is the same. The connection between G and V is given by ........................... (356) Both (346) and (356) assume that there is no impressed force at the place considered. If there be impressed force e per unit length, add e to the left side of (356), and make the necessary change in (346), which is connected with (356) through the equation of continuity . ax ...(366) But as we shall only have e at one end of the line, we shall not require to consider e elsewhere. Now, given (346) and (356), and that there is an impressed force R F sin nt at the x = end, find V and C everywhere. Owing to f and Lf containing only even powers of p, and to the property p2 = - n2 possessed by p in simple-harmonic arrangements, Rf and Lf become constants. The solution is therefore got readily enough. Let U Q - (4^)1 { (R'* + lW)i + L'n}**.. ' These are very important constants concerned. Let also tan X = (UnP - B'Q)/(B'P + I/n tan = 2 sin n 2Ql/(c-* - cos 2QI). ' These make O and l 2 angles less than 90. Then the potential V at distance x at time t is . (396) ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 63 and the current C is -Qx-0 + + + _ p* e sin (nt Qx - 6 l <9 2 ) e~ px sin (nt " l + * e -* I 2 cos H 2 J Each of these consists of the sum of three waves, two positive, or from x = Q to ./ and one /, negative, or the reverse way. If the line were infinitely long, we should have only the first wave. But this wave is reflected at x = l and the result is the second term. Reflection at the t x = end produces the third and least important term. The wave-speed is n/Q, and the wave-length 2ir/Q. As the waves travel their amplitudes diminish at a rate depending upon the magni- tude of P. The angles O l and 2 merely settle the phase-differences. The limiting case is wave-speed = i\ and no dissipation. The amplitude of the current (half its range) is important. It is c r o (Sn)* ryt'-*) + c-^1-" + 2 cos 2Q(l - x)~\* (R* + LV}*\_~~ + ?pl -* - 2 cos 2QI ' J at any distance .r. At the extreme end x = l it is + 1" '"" g - 2 cos As it is only the current at the distant end that can be utilised there, it is clear that (416) is the equation from which valuable information is to be drawn. E It must now be explained how to get f and Lf , and their meanings. Go back to equation (286), Section xxvii. [p. 54], which is the deter- minantal or differential equation when dielectric displacement is ignored. We may write it When p is d/dt it is the differential equation of the boundary magnetic force, or of C, since they are proportional. Separating into even and odd powers of p it will take the form, if we operate on (7, where R* and Lf are functions p of 2 . To suit the oscillatory state, put R - 2 ??. for 2 p* , making f and Lf constants. They will be of the form I4; ............... (436) where R{ depends on the wire, R( on the return ; L{ on the wire, L( on the return, and L on the intermediate insulator. The forms of R{ and L{ have been given by Lord Rayleigh. They are, if 2 g where R l = steady resistance of the wire per unit length, 64 ELECTRICAL PAPERS. 3 11# 77 = i f i _ JoL^t+-ioioavo~n to ..................... (636) where, of course, v is the speed corresponding to L , or the speed of un- ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 73 dissipated waves. The sine must be reckoned positive always. To We check (636), derive it immediately from (416) by taking = 11 0. shall find the following form of (416) in terms of/ and h useful later : F = /C' \Vv(\ 2/v +/)<{< + *-'-"'' - 2 cos 2$}*, (646) where Pl = h(W{(l +/)*- 1}*, 1 W-MJWO +/)*+!}*- /" Let us now dig something out of the above formulae. This arith- metical digging is dreadful work, only suited for very robust intellects. I shall therefore be glad to receive any corrections the following may require, if they are of any importance. It will be as well to commence with the unreal, but easily imaginable case of no resistance. Let the wire and return be of infinite conductivity. We have then merely wave propagation through the dielectric, without any dissipation of energy, at the wave-speed = (/*2c2)~* which is, in air, that of light-waves. Any disturbances originating at one end travel unchanged in form ; but owing to reflection at the other end, and then again at the first end, and the consequent coexistence of oppositely travelling waves, the result is rather complex in general. Now, if we introduce a simple-harmonic impressed force at one end, and adjust its frequency until the wave-period is nearly equal to the time taken by a wave to travel to the other end and back again at the speed r, it is clear that the amplitude of the disturbance will be enormously augmented by the to-and-fro reflections nearly timing with the impressed force. This will explain (636), according to which the distant-end impedance falls to zero when = nl/v IT, or 27r, or STT, etc. Here '27r/n is the wave-period, and 21/v the time of a to-and-fro journey. The current-amplitude goes up to infinity. If, next, we introduce only a very small amount of resistance, we may easily conclude that, although the impedance can never fall to zero, yet, at particular frequencies, it will fall to a minimum, and, at others, go up to a maximum ; and that the range between the consecutive maxi- mum and minimum impedance will be very large, if only the resistance be low enough. Increasing the resistance will tend to reduce the range between the maximum and minimum, but cannot altogether obliterate the fluctua- tions in the value of the impedance as the frequency continuously increases. In practical cases, starting from frequency zero, and raising it continuously, the impedance, which is simply M, the resistance of the line, in the first place, rises to a maximum, then falls to a minimum, then rises to a second maximum greater than the first, and falls to a second minimum greater than the first, and so on, there being a regular increase in the impedance on the whole, if we disregard the fluctuations, whilst the fluctuations themselves get smaller and smaller, so that the real maxima and minima ultimately become false, or only tendencies towards maxima and minima at certain frequencies. By this to-and-fro reflection, or electrical reverberation or resonance, 74 ELECTRICAL PAPERS, the amplitude of the received current may be made far greater than the strength of the steady current from the same impressed force, even when the electrical data are not remote from, but coincide with, or resemble, what may occur in practice. To show this, let us work out some results numerically. As this matter has no particular concern with variations of current- density in the conductors, ignore them altogether; or, what comes to R the same thing, let the conductors be sheets, so that f = R, the steady L resistance, and Lf = Q very nearly, the dielectric inductance, both per unit length. Then, in (646), let /=!, = QI TT, v = 30ohms.............. (665) Then, by the second of (656), we find that h = 2-85; r iV- ? and, by (646), that o/ao = + " = 2i '8284ir [ - 8284T 2 60 ' 6 ^o ohms ....... (676) The ratio of the distant-end impedance to the resistance is therefore 60-6 x 109 _60-6 x 1 Q9 _= 20-2 _ 202 . ~1T~ nl 107* ~285' by making use of the data (666). That is, the amplitude of the received current is 42 per cent, greater than the steady current, when = (666) is enforced. But let 6Z j7r, then To/Co = JV.21^' + -*]J = 28 L Q ohms ; -- -* and the ratio of impedance to resistance is or the amplitude of current is only 3/4 of the steady current. And if = Ql JTT, we shall find F /<7 = 43-5 ohms, and that the impedance is slightly greater than the resistance. r if = Ql ITT, we shall have o/ao = 47-8 ohms, Whilst, and find the ratio of impedance to resistance to be 63/85, making the received current 35 per cent, stronger than the steady current. The above data of/= 1, and = Ql JTT, JTT, JTT, and TT, have been chosen in order to get near the first maximum and minimum of impedance. The range, it will be seen, is very great. Let us next see how these data resemble practical data in respect to resistance, etc. Remember that 1 ohm per kilom. makes .R=10 4 , (resistance per cm. of double R conductor). Also, that/= 1 means = nl= 10 5 nlv if /j is in kilometres. Then, in the case to which (666) to (686) refer, we shall have, first assuming a given value of fi, then varying Z- , and deducing the values of n and /,, the following results : ^=10 3 , =1, n= 103 , Z = 10, = /z 10 2 , ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 75 Tin's is an excessively low resistance, T\j- ohm per kilom. ; the frequencies are rather low, and the lengths great. Next, 1 ohm per kilom. : R = W, =1, n=W\ / = 85. 1 Z = 10, = ?t 10 3 , / = 856. 1 =100, n = 102 , ^ = 8568. The L = 100 case is extravagant, requiring such a very distant return current (therefore very low electric capacity). Next, 10 ohms per kilom. : =1, L =10, Lastly, very high resistance of 100 ohms per kilom. : E=IW, L =10, n=l&, = / 8-5. 1 In all these cases the amplitude of received current is 42 per cent. greater than the steady current. In the next case, = Ql JTT, the quantity nl/v has a value one-fourth of R that assumed in the above ; hence, with the same and jL and same , frequency, the above values of ^ require to be quartered. Then, in all cases, the current-amplitude will be three-fourths of the steady current. Similarly, to meet the = ()/ i?r .case, use the above figures, with the l^s halved ; and in the Ql = JTT case, with the l^a multiplied by f. A consideration of the above figures will show that there must be, in telephony, a good deal of this reinforcement of current strength some- times ; not merely that the electrostatic influence tends to increase the amplitude all round, from what it would be were only magnetic induction concerned, but that there must be special reinforcement of certain tones, and weakening of others. It will be remembered that good reproduction of human speech is not a mere question of getting the lower tones transmitted well, but also the upper tones, through a long range ; the preservation of the latter is required for good articula- tion. The ultimate effect of electrostatic retardation, when the line is long enough, is to kill the upper tones, and convert human speech into mere murmuring. The formula (625) is the most useful if we wish to see readily to what extent the magnetic formula is departed from. In this, two quantities only are concerned, / and h, or 2 (IV/ L'n) and nl/v; and if both /and h are small, it is readily seen that the first form of (636) applies, the factor by which the magnetic impedance is multiplied being (amh)/h. Even when h is not small the /terms in (626) may be negli- gible, and the first form of (636) apply. For example, suppose h= -y t and /small, then = (sin h) /h 3 x '3272 = -9816, showing a reduction of 2 per cent, from the magnetic impedance. Now, this = i /i means nl = l 10 5 , or the high frequency of 10 5 /2?r on a line of one kilom., 10 4 /2;r on 10 kilom., and so on, down to 10/2?r on 10,000 kilom., always provided the / terms are still negligible. This may easily be the case when the line is short, but will cease to be true 76 ELECTRICAL PAPERS. as the line is lengthened, owing to the n in / getting smaller and smaller. Thus, in the just-used example, if the resistance is 10 ohms L= per kilom., and /= 10, we shall have TJT on the line of 1 kilom., /= and 1 on 10 kiloms. So far, the / terms are negligible, and the first form of (636) applies. But / becomes 100 on 100 kiloms., which /= will make an appreciable, though not large, difference ; and 10,000 on 1,000 kilom. will make a large difference and cause the first (636) formula to fail. It is remarkable, however, that this formula should have so wide a range of validity. In the above we have always referred to the distant-end impedance. But at the seat of impressed force there is a large increase of current on account of the "charge." Thus, at # = 0, by the formula preceding (416), we have - The term impedance is of course strictly applicable at the seat of impressed force. As the frequency is raised, this impedance tends to be represented by and, ultimately, by if the dielectric be air. L ^o/^o = A>v = 30 ohms, .................. (706) L is usually a small number. SECTION XXXII. THE EQUATIONS OF PROPAGATION ALONG WIRES. ELEMENTARY. In another place (Phil. Mag., Aug., 1886, and later) the method adopted by me in establishing the equations of Fand C, Section xxix., was to work down from a system exactly fulfilling the conditions involved in Maxwell's scheme, to simpler systems nearly equivalent, but more easily worked. Remembering that Maxwell's is the only complete scheme in existence that will work, there is some advantage in this ; also, wre can see the degree of approximation when a change is made. In the following I adopt the reverse plan of rising from the first rough representation of fact up to the more complete. This plan has, of course, the advantage of greater intelligibility to those who have not studied Maxwell's scheme in its complete form ; besides being, from an educational point of view, the more natural plan. Whenever the solution of a so-called physical problem has been obtained, according to which, under such or such conditions, such or such effects must happen, what has really been done has been to solve another problem, which resembles the real one more or less in those features we wish to study, which we regard as essential, whilst it is of such a greatly simplified nature that its solution is, in comparison with that of the real problem, quite elementary. This remark, which is of rather an obvious nature, conveys a lesson that is not always remem- bered ; that the difference between theory and empiricism is only one of degree, even when the word theory is used in its highest sense, and ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 77 is applied to legitimate deductions from laws which are known to be very true indeed, within wide limits. It is quite possible to imagine the solution of the general problem of the universe. There does not seem to be anything against it except its possible infinite extent. Stop the extension of the universe somewhere ; then, if its laws be fully known, and be either invariable or known to vary in some definite manner, and if its state be known at a given moment, it is difficult to see how it can be indefinite at any later time, even in the minutest particulars in the history of nations or of animal- cule, or in the development of a human soul (which is certainly im- mortal, for the good and evil worked by a soul in this life live for ever, in the permanent impress they make on the future course of events). But if this be imagined to be all done, and the universe made a machine, no one would be a bit the wiser as to the reason why of it. (Even if we ask what we mean by the reason why, we shall in all pro- bability get into a vicious circle of reasoning, from which there is no escape.) All that would be done would be the formulation of facts in a complete manner. This naturally brings us to the subject of the equations of propagation, for they are merely the instruments used in attempts to formulate facts in a more or less complete manner. The first to solve a problem in the propagation of signals was Ohm, whose investigation is a very curious chapter in the history of electricity, as he arrived at results which are, under certain conditions, nearly Ohm correct, by entirely erroneous reasoning. followed the theory of the conduction of heat in wires, as developed by Fourier. Up to a certain point there is a resemblance between the flow of heat and the electric conduction current, but after that a wide dissimilarity. Let a wire be surrounded by a non-conductor of heat, in imagination ; let the heat it contains be indestructible when in the wire, and be in a state of steady flow along it. If C is the heat-current across a given V section, and the temperature there, C will be proportional to the rate V of decrease of alon the wire. Or R if ./ be length measured along the wire. The ratio of the fall of temperature per unit length, to the current, is the "resistance" per unit length, and is, more or less, a constant. Or, the current is proportional to the difference of temperature between any two sections, and is the same all the way between. The law which Ohm discovered and correctly applied to steady con- duction currents in wires is similar to this. Make C the electric current in the wire, and Fthe potential at a certain place. The current, which is the same all the way between any two sections, is proportional to their difference of potential. The ratio of the fall of potential to the current is the electrical resistance, and is constant (at the same tem- perature). But Kis, in Ohm's memoir, an indistinctly defined quantity, V called electroscopic force, I believe. Even using the modern equivalent potential, there is not a perfect parallel between the temperature and the potential V. For a given temperature appears to involve a definite 78 ELECTRICAL PAPERS. physical state of the conductor at the place considered, whereas potential has no such meaning. The real parallel is between the tem- perature gradient, or slope, and the potential slope. Now, returning to the conduction of heat, suppose that the heatcurrent is not uniform, or that the temperature-gradient changes as we pass along the wire. If the current entering a given portion of the wire at one end be greater than that leaving it at the other, then, since the heat cannot escape laterally, it must accumulate. Applying this to the unit length of wire, we have the equation of continuity, t being the time, and q the quantity of heat in the unit length. But the temperature is a function of j, say i-sr, where S is the capacity for heat per unit length of wire, here regarded, for simplicity of reasoning, as a constant, independent of the temperature. This makes the equation of continuity become Between this and the former equation between C and the variation of F, we may eliminate C and obtain the characteristic equation of the temperature, which, when the initial state of temperature along the wire is known, enables us to find how it changes as time goes on, under the influence of given conditions of temperature and supply of heat at its ends. Ohm applied this theory to electricity in a manner which is sub- stantially equivalent to supposing that electricity (when prevented from leaving the wire) flows like heat, and so must accumulate in a given portion of the wire if the current entering at one end exceeds that leaving at the other. The quantity q is the amount of electricity in the unit length, and is proportional to F", their ratio S being the capacity per unit length. With the same formal relations we arrive, of course, at the same characteristic equation, now of the potential, so that elec- tricity diffuses itself along a wire, by difference of potential, in the same way as heat by difference of temperature. A generation later, Sir W. Thomson arrived at a system which is formally the same, but having a quite different physical significance. Between the times of Ohm and Thomson great advances had been made in electrical science, both in electrostatics and electromagnetism, and We the quantities in the system of the latter are quite distinct. have (mnbb]) where on the left appear the elementary relations, and on the right the resultant characteristic equation of V.