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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

<r is now

s ;

p

is r ;

is b ; y is g ; and a is a.]

10

ELECTRICAL PAPERS.

be the products of the principal permeabilities and capacities, the theorem gives, applied to (28) and (29),

... ........... . ...... (31)
wH = V/xs/xE ....................... ........ (32)
H Putting the value of given by (32) in (28) first, and then the value of E given by (31) in (29), we have

......................... (33)

(34)

To these apply the transformation-formula (18), giving

and

-mcE = /xs(s^E)-/xE(s/xs), ..................... (33a)

= - w/*H

-
cs(scH) cH(scs),

...................... (340)

where the bracketed quantities are scalar products. Put in this form,
= {(s^s)/x-mc}E /xs(s/>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<J"

!2-'.28.80

(446)

U to the last of which I have added an additional term. The getting ot
the forms of Rt, and 2, depending upon the return, is less easy,
though only a question of long division. I shall give the formula later. At present I give their ultimate forms at very high frequencies.
Let = p resistivity, and q = frequency = n/'2ir, then

(456)

These are also Lord Rayleigh's. For the return we have

^/ = (Wsg)*

74 = m/n

(466)

I express R{ and R( in terms of the resistivity rather than the

resistance of the wire and return because their resistances have really

nothing to do with it, as we see in especial from the It?2 formula. The

Rf z

of

the

tube

depends

upon

its

inner

radius

only,

no

matter

how

thick

it may be, that is to say upon extent of conducting surface, varying

inversely as the area, which is 2;r 2 per unit length. The proof of (466)

will follow.

Now, as regards the meanings. Let us call the ratio of the impressed

force to the current in a line when electrostatic induction is ignorable

the Impedance of the line, from the verb impede. It seems as good a

term as Resistance, from resist. (Put the accent on the middle e in

impedance.) When the flow is steady, the impedance is wholly con-

ditioned by the dissipation of energy, and is then simply the resistance

Rl of the line. This is also sensibly the case when the frequency is

very low ; but with greater frequency inertia becomes sensible. Then

L + 2
(Pi'

Lrri2)^

is

the

impedance.

Here fi and

are, in the ordinary

sense, the resistance and inductance of unit length of line, including

wire and return. When, further, differences of current-density are

sensible,

the

impedance

is

f2
(R

+

U-n^l.

This is greater or less than

the former, according to the frequency, becoming ultimately less,

U especially if the wire is of iron, owing to the then large reduction in the
value of as compared with L.

Now, when we further take electrostatic induction into account we

shall have the above equations (346) and (356), in which PJ and Lf are

the same as if there were no static charge. The proof of this I must

also postpone. It is the only thing to be proved to make the above

quite complete, excepting (466), which is a mere matter of detail. The

proof arises out of the short sketch I gave in Section xxvu. of the

general electrostatic investigation, used there for illustration.

The impedance

is

made variable ;

it is no

longer

the

same

all

along

the line, simply because the current-amplitude decreases from the place

of impressed force, where it is greatest, to the far end of the line,

ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 65

where it is least. The question arises whether we shall confine imped-

ance according to the above definition to the place of impressed force,
or extend its meaning. If we confine the use, a new word must be

invented. I therefore, at least temporarily, extend the meaning to

signify the ratio of F" to 6' anywhere.

It is very convenient to express impedance in ohms, whatever may be

its ultimate structure. Thus the greatest impedance of a line is what

its resistance would have to be in order that in steady-flow the current

should equal that arriving at the far end under the given circumstances.

It will usually be far greater than the resistance. But there is this

remarkable thing about the joint action offerees of inertia and elasticity.

The impedance may be far less than in the electromagnetic theory.

F JW)k That is,

/(7

according

to

(41ft)

may be

far

less

than

+ 2
(/2'

This is clearly of great importance in connection with the future of

long-distance telegraphy ana telephony.

(In passing I will give an illustration of reduction of impedance pro-

duced by inertia. Let an oscillatory current be kept up in a submarine

cable and in the receiving coils. Insert an iron core in them. The

result is to increase the amplitude of the current-waves. More fully,

increasing the inductance of the coil continuously from zero, whilst

keeping its resistance constant, increases the amplitude up to a certain

U point, after which it decreases. The theory will follow.) To get the submarine cable formulae, ignoring inertia, take

= and

PJ 11. To get the more correct formulae, not allowing for variations

of current-density, but including inertia, take Lf = L the steady induct-

R ance, and f = R. To get the linear magnetic theory formulae, take

= 0,

arid

L' = L,

R' = R.

Finally, using R' and J7, but with

=0, we

have the complete magnetic formulae suitable for short lines. Thus
S=0 in (4:11) brings it to

R U Equations (34ft) to (36ft) are true generally, that is, with f and the
proper functions of d/dt. The solution in the case of steady impressed
force will follow, including the interior state of the wire. Also the
interior state in the oscillatory case.
A great deal may be dug out of (41ft). In the remainder of this
Section, however, we may merely notice the form it takes at very high
frequencies, so high as to bring surface conduction into play, and show how much less the impedance is than according to the magnetic theory.
Let n be so great as to make B'/L'n small. Then we may also take

Q = n/v.

Also,

if

pl f.~

is

small,

as it

will

be

on

increasing the

frequency, we

need only consider the first term under the radical sign in (41ft),

which becomes

(*

Take for R' its ultimate form

H.E.P. VOL. II.

ELECTRICAL PAPERS.
got from (455) and (46&) by supposing wire and sheath of the same
= material, and 2/a + l/a1 l/a2 .
Then the impedance is

where exp is defined by e* = exp x, convenient when x is complex. Here

L Q

is

a

numeric,

and

= 30 ohms (i.e., when we reckon the impedance

in ohms); ^ = 1600 and /*=!, if the conductors are copper; and

= / 10 5^, if ^ is the length of the line in kilom. ; therefore

= 15Z x

To see how it

works

out,

take

L Q

=

1,

a= 1

cm,

and

q = 104 ;

then

F /(7 = 1 5 x exp 4^/300 ohms.

If the line is 100 kilom., PI is made 1-J, which is too small for our
approximate formula. If 1,000 kilom., it is made 13J, which is rather large. Pl= 10 is large. If it is 500 kilom., then

jr
Q

/C

= 15

x exp 6f = 1,178

ohms.

So the impedance is only 1,178 ohms at 500 kilom. distance at the enormous frequency of 10,000 waves per second. It is of course much less at a lower frequency, but the more complete formula will have to be used if it be much lower.
Now compare this real impedance with the resistance of the line in the
steady state, its effective resistance according to the magnetic theory,
and the impedance according to the same. The resistance of the line we may take to be twice that of the wire, by choosing the return of a
proper thickness, or
Rl= 2 x 500 x 10 5 x 1600/7r = 50 ohms, say.

L will be a little more than 1J, say 1-6, therefore Lln='8x27rx 104 = 5060,

U so that the linear-theory impedance is nearly 5,100 ohms. But, owing to the high frequency, we should use R' and

instead

R of

and L ;

here take L' = L

+ P/jn,

then

This large increase of resistance is more than counterbalanced by the reduction of inductance, so that the impedance is brought down from the above 5,100 to about 3,500 ohms, the magnetic theory impedance; and this is about three times the real impedance at its greatest, viz., at the distant end of the line.
It is further to be noted that the wire and return need not be solid,
as we see from the value of R! compared with R. What is needed at
very high frequencies is two conducting sheets of small thickness, of the highest conductivity and lowest inductivity ; i.e., of copper.

ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 67
SECTION XXX. IMPEDANCE FORMULA FOR SHORT LINES.
RESISTANCE OF TUBES.
In the case of a short line, a very high frequency is needed in general
to make it necessary to take electrostatic induction into account in estimating the impedance. Keeping below such a frequency, the impedance per unit length is simply

This is greater than the common (R2 + L2n2)% at first, when the

frequency is low, equal to it at some higher frequency, and less than it

for still higher frequencies. Thus, for simplicity, let the return con-

tribute

nothing

to

the

resistance or the

inductance ;

then, using (446),

we shall have

- <"

R and L being the steady resistance and inductance of the line per unit
R U length (the latter to include L for the external medium), f and the

real values at frequency w/27r per second, p the inductivity of the wire, and g = (pn/It)*.

Thus the first increase in the square of the impedance over that of

the

linear

theory

is

2
J/^% ,

independent

of

resistance;

large in iron,

small in copper.

But

as

the

frequency

is raised,

the

2
g

term

becomes

We sensible ;

being negative,

it puts a stop to the increase.

can get a

rough idea of the frequency required to bring the impedance down to

that

of

the

linear

theory

by

ignoring

the

3
g

term.

This gives

(48i)

The real frequency required must be greater than this, and taking

the

s
g

term

into

account,

we

shall

obtain,

as

a

higher

limit,

........................ (496)

We approximately.

see that the simpler (486) is near enough.

If the wire is of copper of a resistance of 1 ohm per kilom., making

R= 104 ,

we

shall

have,

using

(486),

If the return is distant, we can easily have L = 9. Then the

frequency required is about 100 waves per second. This is a low

telephonic frequency, so that we see that telephonic signalling is

somewhat assisted by the approximation to surface conduction.

If the wire

is

of iron,

then,

on

account

of the

large value

of /x,

a

much

lower frequency is sufficient to reduce the impedance below that of the

linear theory ; that is, an iron wire is not by any means so disadvan-

68

ELECTRICAL PAPERS.

tageous, compared with a copper wire of the same diameter, as its higher resistivity and far higher inductivity would lead one to expect.
But it is not to be inferred that there is any advantage in using iron, electrically speaking, from the fact that the impedance is so easily made much less than that of the linear theory. Copper is, of course, the best to use, in general, being of the highest conductivity, and lowest inductivity. Nor is any great importance to be attached to the matter in any case, for, on a short line, to which we at present refer, it will usually happen that the telephones themselves are of more importance
than the line in retarding changes of current.
We also see that in electric-light mains with alternating currents
there may easily be a reduction of impedance if the wires be thick and the returns not too close. On the other hand, the closer they are
brought the less is the impedance, according to the ordinary formula.
It should be borne in mind that we are merely dealing with a correction,
not with the absolute value of the impedance, which is really the
important thing.
Now take the frequency midway between and the second frequency w^hich gives the linear-theory impedance. Then IF + L-ifi becomes

wherein use the value of n 2 given by (486). The increase of impedance

is not, therefore, in a copper wire, anything of a startling nature.

We Impedances are not additive, in general.

cannot say that the

impedance of a wire is so much, that of a coil so much more, and then

that their sum is the impedance when they are put in sequence, at the

same frequency.

In passing, I may as well caution the reader against the false idea

somewhere prevalent. The increased resistance of a wire is not in any

way caused or evidenced by the weakness of the current in the variable

period compared with its final strength, a result due to the back E.M.F.
of inertia. No matter how great the inertia, and how slowly it makes

the current rise, there is no change of resistance, unless there be

changed distribution of current. There must always be some change,

but it is usually negligible. When, however, as notably in the case of

iron, the central part of the wire is inoperative, of course this changed

distribution of current means a large increase of resistance, though not

of impedance, which is reduced. It is a hollow tube, not a solid wire,

that must, to a first approximation, be regarded as the conductor.

There cannot be said to be any definite resistance unless the current

distribution is definite.

Thus, in the rise of the current from zero to the steady state there is,
presuming that there is large departure from the regular final distribution, no definite resistance, and it is clearly not possible to balance a wire in which the above takes place against a thin wire, a conclusion that is easily verified. But the case of simple-harmonic impressed force is peculiar. The distribution of current, though not constant, goes through the same regular changes over and over again in such a manner

ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 69

that the total current at every moment is the same as if a true linear

circuit of definite resistance and inductance were substituted. This is

very considerably departed from when mere rapid makes and breaks

are employed.
Consider now the resistance of a tube at a given frequency. It

depends materially upon whether the return-current be within it or

^ outside it. Let there be two tubes, and the inner and outer radii

of

the

inner,

and

a.2

and

a 3

of

the

outer.

By an easy extension of

equation (286), the form quoted in the last Section, the differential

equation of the total current is

the dots indicating repetition of what is above them. The first term is for the insulator between tubes, the second for the inner tube, the
third for the outer. Or,

U where R(, 1&, L{,

2

are

functions

p of 2 ,

and

therefore

constants

when

the current is simple-harmonic. The division of the numerators by

the denominators, a simple matter in the case of a solid wire, becomes

a very complex matter in the tube case.

I

give

the

results

as

far

p as

2
.

It is not necessary to do the work separately for the two tubes, for,

if we compare the expressions carefully, we shall see that they only

differ in the exchange of the inner and outer radii, and in changed

sign of the whole.
For the inner tube we have

where 11^ is the steady resistance per unit length. This is the coefficient

of p, and is therefore nothing more than the inductance per unit length

of the

tube

in

steady flow,

the first correction to which

depends

p on

3
.

This may be immediately verified by the square- of-force method.

The resistance of the inner tube per unit length is

To obtain, from (516) and (526), the corresponding expressions for

R the

outer

tube,

change

7?
x

to

v

pl

to

pB,

/^

to /A3,

a x

to

and a
2,

to aB .

The change of $ign is not necessary, because it is involved in the
E substitution of E^ for v Or, simply, (516) and (526) holding good
when the return is outside the tube, exchange a^ and a , and we have

the corresponding formulae when the return is inside it.

70

ELECTRICAL PAPERS.

Let & = Jftj.

This removes a fourth part of the material from the

central part of a solid wire of radius ar The return being outside,

the resistance is

x '01 2.

If

solid,

the

'012

would be

*083 ;

or the

correction

is

reduced

seven

times by removing only a fourth part of the material.

But if the return is inside, all else being the same, the resistance is

^ R{ = R,+ B^nftiraHfr) x -503 = + P^n^iraf/p,) x -031,

so now the correction is reduced less than three times instead of seven

times, as when the return was outside.

This difference will be, of course, greatly magnified when the ratio

di/ctQ is large ; for instance, consider a solid wire surrounded by a very

thick tube

for return ;

the steady resistance

of the return will be

only a

small fraction of that of the wire, but the percentage increase of resistance of the outer conductor will be a large multiple of that of the wire.
Thus the earth's resistance, which, in spite of the low conductivity, is so small to a steady current, will be largely multiplied when the current

is a periodic function of the time.

Now, as regards the resistance of the tube at high frequencies. If

the return is outside it is

q being the frequency. But if the return is inside, it is

thus depending upon the inner radius when the return is inside, and on the outer when it is outside, for an obvious reason, when the position

of the magnetic field where the primary transfer of energy takes place

is considered.

Suppose we fix the outer radius, and then thin the tube from a solid wire down to a mere skin. In doing so we increase the steady resistance as much as we please. But the high-frequency formula (536)
remains the same. Now, as it would involve an absurdity for the

resistance to be less than that in steady flow, it is clear that (536)

cannot be valid until the frequency is so high as to make R{ much greater

R than lt which is itself very great when the tube is thin. That is to

say, removing the central part of a wire, when the return is outside it,

makes it become more a linear conductor, so that a much higher

frequency

is

required

to

change

its

resistance ;

and when the tube is

very thin the frequency must be enormous. Practically, then, a thin

tube is always a linear conductor, although it is only a matter of raising
the frequency to make (536) or (546) applicable.
To get them, use in (506) the appropriate J (x), etc., formulae when x

is very large. They are

J= (x)

- KI(X) = (sin x + cos x) -f

K J^x) =

= Q (x) (sin x - cos x) + (TTX)%. )

ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. 71
These, used in (50&), putting the circular functions in the exponential forms, reduce it to

where i = (- l)i Here

=
sfai ^iTfaJcflfp,

therefore

and

similarly

for

s s;

so we get

Here,

since p2 =

- n2 ,

pl = (Jw)*(l +i) ;

which brings us to

where

^'M*!1'

fM =
f!

<i)

as before given, except that the inner tube was a solid wire.
If, however, the frequency were really so high as to make these highfrequency formulae applicable when the conductors are thin tubes, it is clear that we should, by reason of the high frequency, need, at least in
general, to take electrostatic induction into account even on a short
line, and therefore not estimate the impedance by the magnetic formulae,
R but by the more general of the last Section, in which the same f and
Lr occur. As for long lines, it is imperative to consider electrostatic
induction. There is no fixed boundary between a "short" and a "long" line; we must take into account in a particular case the
circumstances which control it, and judge whether we may treat it as a short or a long-line question. To the more general formula I shall
return in the following Section, merely remarking at present that there is a curious effect arising from the to-and-fro reflection of the electro-
magnetic waves in the dielectric, which causes the impedance to have maxima and minima values as the speed continuously increases ; and that when the period of a wave is somewhere about equal to the time taken to travel to the distant end and back, the amplitude of the
received current may easily be greater than the steady current from the
same impressed force. And, in correction of the definition in Section
V xxix. of as the surface potential of the wire, substitute this defini-
tion, Q = SF, where Q is the charge and S the electrostatic capacity,
both per unit length of wire.

SECTION XXXI. THE INFLUENCE OF ELECTRIC CAPACITY. IMPEDANCE FORMULAE.
Let us now return to the more general case of Section XXIX., the
amplitude of the current due to a simple-harmonic impressed force at one end of a line. Although the formula (416) for the amplitude at the distant end is very compact, yet the exponential form of the functions does not allow us to readily perceive the nature of the change made by lengthening the line, or making any other alteration that will cause the

72

ELECTRICAL PAPERS,

effect of the electric charge to be no longer negligible, by causing the

magnetic formula to be sensibly departed from. Let us, therefore, put

(416)

in

the

form

F" /(7

= etc.,

and

then

expand

the

right

member

in

an

infinite series of which the first term shall be the magnetic impedance

itself, whilst the others depend on the electric capacity as well as on the

resistance and inductance.

On expanding the exponentials and the cosine in (416), we obtain a

P P series in which the quantities

4-
Q,\

6-

^6 ,

etc.,

occur,

all

divided by

To put these in terms of the resistance, etc., we have, by (376),

P* + Q* = SnI,

2PQ = SnR',

Q2 -pi = Sn*L', ...(586)

where

/=(' + Z'% 2

2
)i

........................... (596)

/ being the short-line impedance per unit length. Using these, we
convert (416) to the following form,

*
...... (606)

V Here we may repeat that Q and G' are the amplitudes of the impressed
force at one end and of the current in the wire at the other end of the

R double wire of length Z, whose "constants" are

f ,

Z/,

and

S,

the

R resistance, inductance, and electric capacity per unit length, f and

Lf being functions of the frequency already given. I do not give more

terms than are above expressed, owing to the complexity of the coefficients of the subsequent powers of S. To go further, it will be

desirable to modify the notation, and also to entirely separate the

terms depending upon resistance in the [ ] from the others. Let

SLf = v-\

f=(Bf/I/n)*,

= h nljv. ../. ..... (616)

Here v is a velocity, / and h numerics. The least value of the velocity

is (SL)~t, at zero frequency, L being the full steady inductance per

unit length, as before. As the frequency increases, so does v. Its

(L limiting value is

)"* or (/*2c.2 )~, the speed of undissipated waves

through the dielectric.

The

/ ratio

falls

from

infinity

at

zero

fre-

quency, to zero at infinite frequency. See equations (436) to (466).
The ratio h is such that lift* is the ratio of the time a wave travelling

at speed v takes to traverse the line, to the wave-period.
In terms of /, /, and h, our formula (416), or rather (606), when extended, becomes

From this, seeing that in the [], resistance appears in / only, we see
that the corresponding no-resistance formula is simply
Vsia-> ..................... (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.