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A Treatise on Electricity and Magnetism - Volume 2.djvu

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CONTENTS.

PART III.

MAGNETISM. CHAPTER I.

ELEMENTARY THEOEY OF MAGNETISM. Art. Page

371. Properties of a magnet when acted on by the earth .. .. 1

372. Definition of the axis of the magnet and of the direction of

magnetic force 1

373. Action of magnets on one another. Law of magnetic force .. 2

374. Definition of magnetic units and their dimensions 3

375. Nature of the evidence for the law of magnetic force .. .. 4

376. Magnetism as a mathematical quantity 4

377. The quantities of the opposite kinds of magnetism in a magnet

are always exactly equal * .. 4

378. Effects of breaking a magnet .. .. 5

379. A magnet is built up of particles each of which is a magnet .. 5

380. Theory of magnetic matter 5

381. Magnetization is of the nature of a vector 7

382. Meaning of the term Magnetic Polarization 8

383. Properties of a magnetic particle 8

384. Definitions of Magnetic Moment, Intensity of Magnetization,

and Components of Magnetization .. .. 8

385. Potential of a magnetized element of volume 9

386. Potential of a magnet of finite size. Two expressions for this

potential, corresponding respectively to the theory of polari zation, and to that of magnetic matter 9

387. Investigation of the action of one magnetic particle on another 10

388. Particular cases 12

389. Potential energy of a magnet in any field of force 14

390. On the magnetic moment and axis of a magnet 15

812246

�� � vi CONTENTS.

Art. Page

391. Expansion of the potential of a magnet in spherical harmonics 16

392. The centre of a magnet and the primary and secondary axes

through the centre 17

393. The north end of a magnet in this treatise is that which points

north, and the south end that which points south. Boreal magnetism is that which is supposed to exist near the north pole of the earth and the south end of a magnet. Austral magnetism is that which belongs to the south pole of the earth and the north end of a magnet. Austral magnetism is con sidered positive 19

394. The direction of magnetic force is that in which austral mag

netism tends to move, that is, from south to north, and this is the positive direction of magnetic lines of force. A magnet is said to be magnetized from its south end towards its north end.. 19

��CHAPTER II.

MAGNETIC FOECE AND MAGNETIC INDUCTION.

395. Magnetic force defined with reference to the magnetic potential 21

396. Magnetic force in a cylindric cavity in a magnet uniformly

magnetized parallel to the axis of the cylinder 22

397. Application to any magnet 22

398. An elongated cylinder. Magnetic force 23

399. A thin disk. Magnetic induction 23

400. Relation between magnetic force, magnetic induction, and mag

netization 24

401. Line-integral of magnetic force, or magnetic potential .. .. 24

402. Surface-integral of magnetic induction 25

403. Solenoidal distribution of magnetic induction .. .. .. .. 26

404. Surfaces and tubes of magnetic induction 27

405. Vector-potential of magnetic induction 27

406. Relations between the scalar and the vector-potential .. .. 28

��CHAPTER III.

PARTICULAR FORMS OF MAGNETS.

407. Definition of a magnetic solenoid 31

408. Definition of a complex solenoid and expression for its potential

at any point .. .. 32

�� � CONTENTS. vii

Art. Page

409. The potential of a magnetic shell at any point is the product of

its strength multiplied by the solid angle its boundary sub tends at the point 32

410. Another method of proof 33

411. The potential at a point on the positive side of a shell of

strength 4> exceeds that on the nearest point on the negative

side by 47r<J> 34

412. Lamellar distribution of magnetism 34

413. Complex lamellar distribution 34

414. Potential of a solenoidal magnet 35

415. Potential of a lamellar magnet 35

416. Vector-potential of a lamellar magnet 36

417. On the solid angle subtended at a given point by a closed curve 36

418. The solid angle expressed by the length of a curve on the sphere 37

419. Solid angle found by two line-integrations 38

420. II expressed as a determinant 39

421. The solid angle is a cyclic function 40

422. Theory of the vector-potential of a closed curve 41

423. Potential energy of a magnetic shell placed in a magnetic field 42

��CHAPTER IV.

INDUCED MAGNETIZATION.

424. When a body under the action of magnetic force becomes itself

magnetized the phenomenon is called magnetic induction .. 44

425. Magnetic induction in different substances 45

426. Definition of the coefficient of induced magnetization .. .. 47

427. Mathematical theory of magnetic induction. Poisson s method 47

428. Faraday s method .. 49

429. Case of a body surrounded by a magnetic medium 51

430. Poisson s physical theory of the cause of induced magnetism .. 53

CHAPTER V.

MAGNETIC PEOBLEMS.

431. Theory of a hollow spherical shell 56

432. Case when K is large 58

433. When i=l 58

434. Corresponding case in two dimensions. Fig. XV 59

435. Case of a solid sphere, the coefficients of magnetization being

different in different directions 60

�� � viii CONTENTS.

Art.

436. The nine coefficients reduced to six. Fig. XVI 61

437. Theory of an ellipsoid acted on by a uniform magnetic force .. 62

438. Cases of very flat and of very long ellipsoids 65

439. Statement of problems solved by Neumann, Kirchhoff and Green 67

440. Method of approximation to a solution of the general problem

when K is very small. Magnetic bodies tend towards places of most intense magnetic force, and diamagnetic bodies tend to places of weakest force 69

441. On ship s magnetism 70

��CHAPTER VI.

WEBER S THEORY OF MAGNETIC INDUCTION.

442. Experiments indicating a maximum of magnetization .. .. 74

443. Weber s mathematical theory of temporary magnetization .. 75

444. Modification of the theory to account for residual magnetization 79

445. Explanation of phenomena by the modified theory 81

446. Magnetization, demagnetization, and remagnetization .. .. 83

447. Effects of magnetization on the dimensions of the magnet .. 85

448. Experiments of Joule 86

CHAPTER VII.

MAGNETIC MEASUREMENTS.

449. Suspension of the magnet 88

450. Methods of observation by mirror and scale. Photographic

method 89

451. Principle of collimation employed in the Kew magnetometer .. 93

452. Determination of the axis of a magnet and of the direction of

the horizontal component of the magnetic force 94

453. Measurement of the moment of a magnet and of the intensity of

the horizontal component of magnetic force 97

454. Observations of deflexion 99

455. Method of tangents and method of sines 101

456. Observation of vibrations 102

457. Elimination of the effects of magnetic induction 105

458. Statical method of measuring the horizontal force 106

459. Bifilar suspension 107

460. System of observations in an observatory Ill

461. Observation of the dip-circle Ill

�� � CONTENTS. ix

Art, Page

462. J. A. BrourTs method of correction 115

463. Joule s suspension 115

464. Balance vertical force magnetometer 117

��CHAPTER VIII.

TERRESTKIAL MAGNETISM.

465. Elements of the magnetic force 120

466. Combination of the results of the magnetic survey of a country 121

467. Deduction of the expansion of the magnetic potential of the

earth in spherical harmonics 123

468. Definition of the earth s magnetic poles. They are not at the

extremities of the magnetic axis. False poles. They do not exist on the earth s surface 123

469. Gauss calculation of the 24 coefficients of the first four har

monics 124

470. Separation of external from internal causes of magnetic force .. 124

471. The solar and lunar variations 125

472. The periodic variations 125

473. The disturbances and their period of 11 years 126

474. Reflexions on magnetic investigations 126

��PART IV.

ELECTROMAGNETISM. CHAPTER I.

ELECTROMAGNETIC FORCE.

475. Orsted s disco very of the action of an electric current on a

magnet 128

476. The space near an electric current is a magnetic field .. .. 128

477. Action of a vertical current on a magnet 129

478. Proof that the force due to a straight current of indefinitely

great length varies inversely as the distance 129

479. Electromagnetic measure of the current 130

�� � X CONTENTS.

Art. Page

480. Potential function due to a straight current. It is a function

of many values 130

481. The action of this current compared with that of a magnetic

shell having an infinite straight edge and extending on one side of this edge to infinity 131

482. A small circuit acts at a great distance like a magnet .. .. 131

483. Deduction from this of the action of a closed circuit of any form

and size on any point not in the current itself 131

484. Comparison between the circuit and a magnetic shell .. .. 132

485. Magnetic potential of a closed circuit 133

486. Conditions of continuous rotation of a magnet about a current 133

487. Form of the magnetic equipotential surfaces due to a closed

circuit. Fig. XVIII 134

488. Mutual action between any system of magnets and a closed

current 135

489. Reaction on the circuit 135

490. Force acting on a wire carrying a current and placed in the

magnetic field * 136

491. Theory of electromagnetic rotations 138

492. Action of one electric circuit on the whole or any portion of

another 139

493. Our method of investigation is that of Faraday 140

494. Illustration of the method applied to parallel currents .. .. 140

495. Dimensions of the unit of current 141

496. The wire is urged from the side on which its magnetic action

strengthens the magnetic force and towards the side on which

it opposes it 141

497. Action of an infinite straight current on any current in its

plane .. 142

498. Statement of the laws of electromagnetic force. Magnetic force

due to a current 142

499. Generality of these laws .. 143

500. Force acting on a circuit placed in the magnetic field .. .. 144

501. Electromagnetic force is a mechanical force acting on the con

ductor, not on the electric current itself 144

��CHAPTER II.

MUTUAL ACTION OF ELECTRIC CURRENTS.

502. Ampere s investigation of the law of force between the elements

of electric currents .. 146

�� � CONTENTS. xi

Art. Page

503. His method of experimenting 146

504. Ampere s balance 147

505. Ampere s first experiment. Equal and opposite currents neu

tralize each other 147

506. Second experiment. A crooked conductor is equivalent to a

straight one carrying the same current ..148

507. Third experiment. The action of a closed current as an ele

ment of another current is perpendicular to that element .. 148

508. Fourth experiment. Equal currents in systems geometrically

similar produce equal forces 149

509. In all of these experiments the acting current is a closed one .. 151

510. Both circuits may, however, for mathematical purposes be con

ceived as consisting of elementary portions, and the action

of the circuits as the resultant of the action of these elements 151

511. Necessary form of the relations between two elementary portions

of lines 151

512. The geometrical quantities which determine their relative posi

tion 152

513. Form of the components of their mutual action 153

514. Resolution of these in three directions, parallel, respectively, to

the line joining them and to the elements themselves .. .. 154

515. General expression for the action of a finite current on the ele

ment of another 154

516. Condition furnished by Ampere s third case of equilibrium .. 155

517. Theory of the directrix and the determinants of electrodynamic

action 156

518. Expression of the determinants in terms of the components

of the vector-potential of the current 157

519. The part of the force which is indeterminate can be expressed

as the space-variation of a potential 157

520. Complete expression for the action between two finite currents 158

521. Mutual potential of two closed currents 158

522. Appropriateness of quaternions in this investigation .. .. 158

523. Determination of the form of the functions by Ampere s fourth

case of equilibrium 159

524. The electrodynamic and electromagnetic units of currents .. 159

525. Final expressions for electromagnetic force between two ele

ments 160

526. Four different admissible forms of the theory 160

527. Of these Ampere s is to be preferred 161

�� � xii CONTENTS.

��CHAPTER III.

INDUCTION OF ELECTRIC CURRENTS.

Art. Page

528. Faraday s discovery. Nature of his methods 162

529. The method of this treatise founded on that of Faraday .. .. 163

530. Phenomena of magneto-electric induction 164

531. General law of induction of currents 166

532. Illustrations of the direction of induced currents 166

533. Induction by the motion of the earth 167

534. The electromotive force due to induction does not depend on

the material of the conductor 168

535. It has no tendency to move the conductor 168

536. Felici s experiments on the laws of induction 168

537. Use of the galvanometer to determine the time-integral of the

electromotive force 170

538. Conjugate positions of two coils 171

539. Mathematical expression for the total current of induction .. 172

540. Faraday s conception of an electrotonic state 173

541. His method of stating the laws of induction with reference to

the lines of magnetic force 174

542. The law of Lenz, and Neumann s theory of induction .. .. 176

543. Helmholtz s deduction of induction from the mechanical action

of currents by the principle of conservation of energy .. .. 176

544. Thomson s application of the same principle 178

545. Weber s contributions to electrical science .. 178

��CHAPTER IV.

INDUCTION OF A CURRENT ON ITSELF.

546. Shock given by an electromagnet 180

547. Apparent momentum of electricity 180

548. Difference between this case and that of a tube containing a

current of water 181

549. If there is momentum it is not that of the moving electricity .. 181

550. Nevertheless the phenomena are exactly analogous to those of

momentum 181

551. An electric current has energy, which may be called electro-

kinetic energy 182

552. This leads us to form a dynamical theory of electric currents .. 182

�� � CONTENTS. xiii

CHAPTER V.

GENERAL EQUATIONS OF DYNAMICS. Art. Page

553. Lagrange s method furnishes appropriate ideas for the study of

the higher dynamical sciences , 184

554. These ideas must be translated from mathematical into dy

namical language 184

555. Degrees of freedom of a connected system 185

556. Generalized meaning of velocity 186

557. Generalized meaning of force , , .. .. 186

558. Generalized meaning of momentum and impulse ,. ,. .. 186

559. Work done by a small impulse .. ., 187

560. Kinetic energy in terms of momenta, (T p ) 188

561. Hamilton s equations of motion 189

562. Kinetic energy in terms of the velocities and momenta, (Tp,j) .. 190

563. Kinetic energy in terms of velocities, (T^) ,, 191

564. Relations between T p and T^p and q ,. 191

565. Moments and products of inertia and mobility .. *.. .. 192

566. Necessary conditions which these coefficients must satisfy .. 193

567. Relation between mathematical, dynamical, and electrical ideas 193

CHAPTER VI.

APPLICATION OF DYNAMICS TO ELECTROMAGNETISM.

568. The electric current possesses energy 195

569. The current is a kinetic phenomenon 195

570. Work done by electromotive force 196

571. The most general expression for the kinetic energy of a system

including electric currents .. 197

572. The electrical variables do not appear in this expression .. .. 198

573. Mechanical force acting on a conductor 198

574. The part depending on products of ordinary velocities and

strengths of currents does not exist 200

575. Another experimental test ,, ., .. 202

576. Discussion of the electromotive force 204

577. If terms involving products of velocities and currents existed

they would introduce electromotive forces, which are not ob served , 204

CHAPTER VII.

ELECTROKINETICS.

578. The electrokinetic energy of a system of linear circuits .. .. 206

579. Electromotive force in each circuit .. ., 207

�� � CONTENTS.

PART III.

MAGNETISM. CHAPTER I.

ELEMENTARY THEOEY OF MAGNETISM. Art. Page

371. Properties of a magnet when acted on by the earth .. .. 1

372. Definition of the axis of the magnet and of the direction of

magnetic force 1

373. Action of magnets on one another. Law of magnetic force .. 2

374. Definition of magnetic units and their dimensions 3

375. Nature of the evidence for the law of magnetic force .. .. 4

376. Magnetism as a mathematical quantity 4

377. The quantities of the opposite kinds of magnetism in a magnet

are always exactly equal * .. 4

378. Effects of breaking a magnet .. .. 5

379. A magnet is built up of particles each of which is a magnet .. 5

380. Theory of magnetic matter 5

381. Magnetization is of the nature of a vector 7

382. Meaning of the term Magnetic Polarization 8

383. Properties of a magnetic particle 8

384. Definitions of Magnetic Moment, Intensity of Magnetization,

and Components of Magnetization .. .. 8

385. Potential of a magnetized element of volume 9

386. Potential of a magnet of finite size. Two expressions for this

potential, corresponding respectively to the theory of polari zation, and to that of magnetic matter 9

387. Investigation of the action of one magnetic particle on another 10

388. Particular cases 12

389. Potential energy of a magnet in any field of force 14

390. On the magnetic moment and axis of a magnet 15

812246

�� � xiv CONTENTS.

Art. Page

580. Electromagnetic force 208

581. Case of two circuits 208

582. Theory of induced currents 209

583. Mechanical action between the circuits 210

584. All the phenomena of the mutual action of two circuits depend

on a single quantity, the potential of the two circuits .. .. 210

��CHAPTER VIII.

EXPLOEATION OF THE FIELD BY MEANS OF THE SECONDARY CIRCUIT.

585. The electrokinetic momentum of the secondary circuit .. .. 211

586. Expressed as a line-integral 211

587. Any system of contiguous circuits is equivalent to the circuit

formed by their exterior boundary 212

588. Electrokinetic momentum expressed as a surface-integral .. .212

589. A crooked portion of a circuit equivalent to a straight portion 213

590. Electrokinetic momentum at a point expressed as a vector, $1 .. 214

591. Its relation to the magnetic induction, 33. Equations (A) .. 214

592. Justification of these names 215

593. Conventions with respect to the signs of translations and rota

tions 216

594. Theory of a sliding piece 217

595. Electromotive force due to the motion of a conductor .. .. 218

596. Electromagnetic force on the sliding piece ..218

597. Four definitions of a line of magnetic induction 219

598. General equations of electromotive force, (B) 219

599. Analysis of the electromotive force 222

600. The general equations referred to moving axes 223

601. The motion of the axes changes nothing but the apparent value

of the electric potential 224

602. Electromagnetic force on a conductor 224

603. Electromagnetic force on an element of a conducting body.

Equations (C) 226

CHAPTER IX.

GENERAL EQUATIONS.

604. Recapitulation 227

605. Equations of magnetization, (D) 228

606. Relation between magnetic force and electric currents .. .. 229

607. Equations of electric currents, (E) 230

608. Equations of electric displacement, (F) 232

�� � CONTENTS. xv

Art - Page

609. Equations of electric conductivity, (G) 232

610. Equations of total currents, (H) 232

611. Currents in terms of electromotive force, (I) .. .. .. .. 233

612. Volume-density of free electricity, (J) 233

613. Surface-density of free electricity, (K) 233

614. Equations of magnetic permeability, (L) 233

615. Ampere s theory of magnets 234

616. Electric currents in terms of electrokinetic momentum .. .. 234

617. Vector-potential of electric currents 236

618. Quaternion expressions for electromagnetic quantities .. .. 236

619. Quaternion equations of the electromagnetic field 237

CHAPTER X.

DIMENSIONS OF ELECTRIC UNITS.

620. Two systems of units .. .. 239

621. The twelve primary quantities 239

622. Fifteen relations among these quantities 240

623. Dimensions in terms of [e] and [m] 241

624. Reciprocal properties of the two systems 241

625. The electrostatic and the electromagnetic systems 241

626. Dimensions of the 12 quantities in the two systems .. .. 242

627. The six derived units 243

628. The ratio of the corresponding units in the two systems .. 243

629. Practical system of electric units. Table of practical units .. 244

CHAPTER XL

ENERGY AND STRESS.

630. The electrostatic energy expressed in terms of the free electri

city and the potential 246

631. The electrostatic energy expressed in terms of the electromotive

force and the electric displacement 246

632. Magnetic energy in terms of magnetization and magnetic force 247

633. Magnetic energy in terms of the square of the magnetic force .. 247

634. Electrokinetic energy in terms of electric momentum and electric

current 248

635. Electrokinetic energy in terms of magnetic induction and mag

netic force 248

636. Method of this treatise 249

637. Magnetic energy and electrokinetic energy compared .. .. 249

638. Magnetic energy reduced to electrokinetic energy 250

�� � xvi CONTENTS.

Art. Page

639. The force acting on a particle of a substance due to its magnet

ization 251

640. Electromagnetic force due to an electric current passing through

it 252

641. Explanation of these forces by the hypothesis of stress in a

medium 253

642. General character of the stress required to produce the pheno

mena 255

643. "When there is no magnetization the stress is a tension in the

direction of the lines of magnetic force, combined with a pressure in all directions at right angles to these lines, the

magnitude of the tension and pressure being ^ 2 , where <>

oTf

is the magnetic force 256

644. Force acting on a conductor carrying a current 257

645. Theory of stress in a medium as stated by Faraday .. .. 257

646. Numerical value of magnetic tension 258

CHAPTER XII.

CURKENT-SHEETS.

647. Definition of a current-sheet 259

648. Current-function 259

649. Electric potential ,. .. 260

650. Theory of steady currents , 260

651. Case of uniform conductivity 260

652. Magnetic action of a current-sheet with closed currents .. .. 261

653. Magnetic potential due to a current-sheet 262

654. Induction of currents in a sheet of infinite conductivity .. .. 262

655. Such a sheet is impervious to magnetic action 263

656. Theory of a plane current-sheet 263

657. The magnetic functions expressed as derivatives of a single

function 264

658. Action of a variable magnetic system on the sheet 266

659. When there is no external action the currents decay, and their

magnetic action diminishes as if the sheet had moved off with constant velocity R 267

660. The currents, excited by the instantaneous introduction of a

magnetic system, produce an effect equivalent to an image of that system 267

661. This image moves away from its original position with velo

city R 268

662. Trail of images formed by a magnetic system in continuous

motion . 268

�� � CONTENTS. xvii

Art. Page

663. Mathematical expression for the effect of the induced currents 269

664. Case of the uniform motion of a magnetic pole .. ., .. 269

665. Value of the force acting on the magnetic pole r . .. ,. 270

666. Case of curvilinear motion 271

667. Case of motion near the edge of the sheet 271

668. Theory of Arago s rotating disk 271

669. Trail of images in the form of a helix 274

670. Spherical current-sheets 275

671. The vector- potential 276

672. To produce a field of constant magnetic force within a spherical

shell 277

673. To produce a constant force on a suspended coil 278

674. Currents parallel to a plane 278

675. A plane electric circuit. A spherical shell. An ellipsoidal

shell .. 279

676. A solenoid 280

677. A long solenoid 281

678. Force near the ends 282

679. A pair of induction coils 282

680. Proper thickness of wire 283

G81. An endless solenoid .. .. .. .. 284

CHAPTER XIII.

PARALLEL CURRENTS.

682. Cylindrical conductors .".."<. .. 286

683. The external magnetic action of a cylindric wire depends only

on the whole current through it .. 287

684. The vector-potential ^ .. .. .1 288

685. Kinetic energy of the current .. .. .. ... ... ... .. 288

686. Repulsion between the direct and the return current .. 289

687. Tension of the wires. Ampere s experiment 289

688. Self-induction of a wire doubled on itself 290

689. Currents of varying intensity in a cylindric wire 291

690. Relation between the electromotive force and the total current 292

691. Geometrical mean distance of two figures in a plane .. .. 294

692. Particular cases 294

693. Application of the method to a coil of insulated wires .. .. 296

CHAPTER XIV.

CIRCULAR CURRENTS.

694. Potential due to a spherical bowl *. 299

695. Solid angle subtended by a circle at any point 301

VOL. II. b

�� � xviii CONTENTS.

Art. Page

696. Potential energy of two circular currents .. 302

697. Moment of the couple acting between two coils 303

698. Values of Q? 303

699. Attraction between two parallel circular currents 304

700. Calculation of the coefficients for a coil of finite section .. .. 304

701. Potential of two parallel circles expressed by elliptic integrals 305

702. Lines of force round a circular current. Fig. XVIII .. .. 307

703. Differential equation of the potential of two circles 307

704. Approximation when the circles are very near one another .. 309

705. Further approximation 310

706. Coil of maximum self-induction . 311

��CHAPTER XV.

ELECTROMAGNETIC INSTRUMENTS.

707. Standard galvanometers and sensitive galvanometers .. .. 313

708. Construction of a standard coil 314

709. Mathematical theory of the galvanometer 315

710. Principle of the tangent galvanometer and the sine galvano

meter 316

711. Galvanometer with a single coil 316

712. Gaugain s eccentric suspension 317

713. Helmholtz s double coil. Fig. XIX 318

714. Galvanometer with four coils 319

715. Galvanometer with three coils 319

716. Proper thickness of the wire of a galvanometer 321

717. Sensitive galvanometers 322

718. Theory of the galvanometer of greatest sensibility 322

719. Law of thickness of the wire .. .. 323

720. Galvanometer with wire of uniform thickness 325

721. Suspended coils. Mode of suspension 326

722. Thomson s sensitive coil 326

723. Determination of magnetic force by means of suspended coil

and tangent galvanometer ..327

724. Thomson s suspended coil and galvanometer combined .. .. 328

725. Weber s electrodynamometer 328

726. Joule s current-weigher 332

727. Suction of solenoids 333

728. Uniform force normal to suspended coil 333

729. Electrodynamometer with torsion-arm 334

�� � CONTEXTS. xix CHAPTER XVI.

ELECTROMAGNETIC OBSERVATIONS.

Art. Page

730. Observation of vibrations 335

731. Motion in a logarithmic spiral 336

732. Rectilinear oscillations in a resisting medium 337

733. Values of successive elongations .. ... 338

734. Data and qusesita 338

735. Position of equilibrium determined from three successive elon

gations 338

736. Determination of the logarithmic decrement 339

737. When to stop the experiment - .* .. .. 339

738. Determination of the time of vibration from three transits .. 339

739. Two series of observations .. .. 340

740. Correction for amplitude and for damping .. .. .. ..341

741. Dead beat galvanometer .. .. .. .. .* ... .. .. 341

742. To measure a constant current with the galvanometer .. .. 342

743. Best angle of deflexion of a tangent galvanometer 343

744. Best method of introducing the current .. .. 343

745. Measurement of a current by the first elongation 344

746. To make a series of observations on a constant current .. .. 345

747. Method of multiplication for feeble currents 345

748. Measurement of a transient current by first elongation .. .. 346

749. Correction for damping .. * .. .. 347

750. Series of observations. Zurilckwerfungs methode 348

751. Method of multiplication ? . .. .. 350

CHAPTER XVII.

ELECTRICAL MEASUREMENT OF COEFFICIENTS OF INDUCTION.

752. Electrical measurement sometimes more accurate than direct

measurement 352

753. Determination of G l .. ..- 353

754. Determination of g l 354

755. Determination of the mutual induction of two coils .. .. 354

756. Determination of the self-induction of a coil 356

757. Comparison of the self-induction of two coils 357

CHAPTER XVIII.

DETERMINATION OF RESISTANCE IN ELECTROMAGNETIC MEASURE.

758. Definition of resistance ~ 358

759. Kirchhoff s method 358

�� � xx CONTENTS.

Art. Pa * e

760. Weber s method by transient currents .......... 360

761. His method of observation .............. 361

762. Weber s method by damping .............. 361

763. Thomson s method by a revolving coil .......... 364

764. Mathematical theory of the revolving coil ..- ...... 364

765. Calculation of the resistance .......... 365

766. Corrections .................... 366

767. Joule s calorimetric method .............. 367

��CHAPTER XIX.

COMPARISON OF ELECTROSTATIC WITH ELECTROMAGNETIC UNITS.

768. Nature and importance of the investigation ........ 368

769. The ratio of the units is a velocity ............ 369

770. Current by convection ................ 370

771. Weber and Kohlrausch s method ............ 370

772. Thomson s method by separate electrometer and electrodyna-

mometer ...................... 372

773. Maxwell s method by combined electrometer and electrodyna-

mometer ...................... 372

774. Electromagnetic measurement of the capacity of a condenser.

Jenkin s method .................. 373

775. Method by an intermittent current ............ 374

776. Condenser and "Wippe as an arm of Wheatstone s bridge .. 375

777. Correction when the action is too rapid .......... 376

778. Capacity of a condenser compared with the self-induction of a

coil ........................ 377

779. Coil and condenser combined .............. 379

780. Electrostatic measure of resistance compared with its electro

magnetic measure .................. 382

CHAPTER XX.

ELECTROMAGNETIC THEORY OF LIGHT.

781. Comparison of the properties of the electromagnetic medium

with those of the medium in the undulatory theory of light 383

782. Energy of light during its propagation .......... 384

783. Equation of propagation of an electromagnetic disturbance .. 384

784. Solution when the medium is a non-conductor ...... 386

785. Characteristics of wave-propagation ............ 386

786. Velocity of propagation of electromagnetic disturbances .. .. 387

787. Comparison of this velocity with that of light ........ 387

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Art. Page

788. The specific inductive capacity of a dielectric is the square of

its index of refraction ..... .. 388

789. Comparison of these quantities in the case of paraffin .. .. 388

790. Theory of plane waves 389

791. The electric displacement and the magnetic disturbance are in

the plane* of the wave-front, and perpendicular to each other 390

792. Energy and stress during radiation 391

793. Pressure exerted by light .. .... ..391

794. Equations of motion in a crystallized medium .. .. .. 392

795. Propagation of plane waves 393

796. Only two waves are propagated 393

797. The theory agrees with that of Fresnel .. .. 394

798. Relation between electric conductivity and opacity .. .. 394

799. Comparison with facts 395

800. Transparent metals 395

801. Solution of the equations when the medium is a conductor .. 395

802. Case of an infinite medium, the initial state being given .. 396

803. Characteristics of diffusion 397

804. Disturbance of the electromagnetic field when a current begins

to flow .. .. .. ,. .. 397

805. Rapid approximation to an ultimate state 398

��CHAPTER XXI.

MAGNETIC ACTION ON LIGHT.

806. Possible forms of the relation between magnetism and light .. 399

807. The rotation of the plane of polarization by magnetic action .. 400

808. The laws of the phenomena 400

809. Yerdet s discovery of negative rotation in ferromagnetic media 400

810. Rotation produced by quartz, turpentine, &c., independently of

magnetism 401

811. Kinematical analysis of the phenomena 402

812. The velocity of a circularly-polarized ray is different according

to its direction of rotation 402

813. Right and left-handed rays 403

814. In media which of themselves have the rotatory property the

velocity is different for right and left-handed configurations 403

815. In media acted on by magnetism the velocity is different for

opposite directions of rotation 404

816. The luminiferous disturbance, mathematically considered, is a

vector 404

817. Kinematic equations of circularly-polarized light .. .. .. 405

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Art. Page

818. Kinetic and potential energy of the medium 406

819. Condition of wave-propagation 406

820. The action of magnetism must depend on a real rotation about

the direction of the magnetic force as an axis 407

821. Statement of the results of the analysis of the phenomenon .. 407

822. Hypothesis of molecular vortices 408

823. Variation of the vortices according to Helmholtz s law .. .. 409

824. Variation of the kinetic energy in the disturbed medium .. 409

825. Expression in terms of the current and the velocity .. .. 410

826. The kinetic energy in the case of plane waves 410

827. The equations of motion 411

828. Velocity of a circularly -polarized ray 411

829. The magnetic rotation 412

830. Researches of Verdet 413

831. Note on a mechanical theory of molecular vortices 415

CHAPTER XXII.

ELECTRIC THEOKY OF MAGNETISM.

832. Magnetism is a phenomenon of molecules 418

833. The phenomena of magnetic molecules may be imitated by

electric currents 419

834. Difference between the elementary theory of continuous magnets

and the theory of molecular currents 419

835. Simplicity of the electric theory 420

836. Theory of a current in a perfectly conducting circuit .. .. 420

837. Case in which the current is entirely due to induction .. .. 421

838. Weber s theory of diamagnetism 421

839. Magnecrystallic induction 422

840. Theory of a perfect conductor 422

841. A medium containing perfectly conducting spherical molecules 423

842. Mechanical action of magnetic force on the current which it

excites 423

843. Theory of a molecule with a primitive current 424

844. Modifications of Weber s theory 425

845. Consequences of the theory 425

CHAPTER XXIII.

THEORIES OF ACTION AT A DISTANCE.

846. Quantities which enter into Ampere s formula 426

847. Relative motion of two electric particles 426

�� � CONTENTS. xxiii

��Art.

848. Relative motion of four electric particles. Fechner s theory .. 427

849. Two new forms of Ampere s formula 428

850. Two different expressions for the force between two electric

particles in motion 428

851. These are due to Gauss and to Weber respectively 429

852. All forces must be consistent with the principle of the con

servation of energy 429

853. Weber s formula is consistent with this principle but that of

Gauss is not 429

854. Helmholtz s deductions from Weber s formula 430

855. Potential of two currents 431

856. Weber s theory of the induction of electric currents .. ..431

857. Segregating force in a conductor 432

858. Case of moving conductors 433

859. The formula of Gauss leads to an erroneous result 434

860. That of Weber agrees with the phenomena 434

861. Letter of Gauss to Weber 435

862. Theory of Riemann 435

863. Theory of C. Neumann 435

864. Theory of Betti 436

865. Repugnance to the idea of a medium 437

866. The idea of a medium cannot be got rid of 437

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