- UNIVERSITY Of I -X CALIFORNIA I A TREATISE ON ELECTRICITY AND MAGNETISM MAXWELL VOL. II. Honfcon MACMILLAN AND CO. PUBLISHERS TO THE UNIVERSITY OF Clareniron A TREATISE ON ELECTRICITY AND MAGNETISM BY JAMES CLERK MAXWELL, M.A. LLD. EDIN., F.R.SS. LONDON AND EDINBURGH HONORARY FELLOW OP TRINITY COLLEGE, AND PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE VOL. II AT THE CLARENDON PRESS 1873 [All rights reserved] v. . J/VV*Wt 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 nortb, 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 FORCE 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. Vll 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 <I> exceeds that on the nearest point on the negative side by 477$ 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 PKOBLEMS. 431. Theory of a hollow spherical shell 56 432. Case when K. is large 58 433. When t = 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. Page 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. Broun's method of correction 115 463. Joule's suspension 115 464. Balance vertical force magnetometer 117 CHAPTER VIII. TERRESTRIAL 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. Grauss' 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. Keflexions on magnetic investigations 126 PART IV. ELECTROMAGNET ISM. CHAPTER I. ELECTROMAGNETIC FORCE. 475. Orsted's discovery 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. Kesolution 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 CUEEENTS. 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 CUEEENT 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, (Tp) .. .. ,. .. 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 Tp 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 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. EXPLOBATION 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, Ql .. 214 591. Its relation to the magnetic induction, 3B. 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 ELECTKIC 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 XI. 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 •$ O7T 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. CURRENT-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. xvn 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 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. PAKALLEL 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 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 CONTENTS. xix CHAPTER XVI. ELECTROMAGNETIC OBSERVATIONS. Art. Page 730. Observation of vibrations , ;. 335 731. Motion in a logarithmic spiral 336 732. Eectilinear 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^ 353 754. Determination of gl 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. Page 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 CONTENTS. xxi 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. Verdet'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 xxii CONTENTS. 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 THEOEY 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. Page 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 ERRATA. VOL. II. p. 11, 1.1, for r. dV, d2 .lx read W = m9-^— = — m, m,^- — — (-)• 2 2^' „ equation (8), insert — before each side of this equation. p. 1 3, last line but one, dele — . p. 14, 1. 8, for XVII read XIV. p. 15, equation (5), for VpdS read Vpdxdydz. p. 16, 1. 4 from bottom, after equation (3) insert of Art. 389. p. 17, equation (14), for r read r5. p. 21, 1. 1, for 386 read 385. „ 1. 7 from bottom for in read on. p. 28, last line but one, for 386 read 385. dF dH _ <W d# p. 41, equation (10), for ^--^ ttffi ^-^' p. 43, equation (14), put accents on #, ?/, z. p. 50, equation (19), for — , &c. rmc? — , &c., inverting all the differ du x cL v ential coefficients. p. 51, 1. 11, for 309 read 310. p. 61, 1. 16, for Y=Fsm0 read Z=Fsm6. „ equation (10), for TT read 7i2. p. 62, equation (13), for § read f. p. 63, 1. 3, for pdr read pdv. p. 67, right-hand side of equation should be 4 p. 120, equation (1), for downwards read upwards. „ equation (2), insert — before the right-hand member of each equation. p. 153, 1. 15, for =(3 read =/3'. p. 155, 1. 8, for A A read AP. p. 190, equation (11), for Fbq1 read Fb^. p. 192, 1. 22, for Tp read Tp. p. 193, after 1. 5 from bottom, insert, But they will be all satisfied pro vided the n determinants formed by the coefficients having the indices 1 ; 1, 2 ; 1, 2, 3, &c. ; 1, 2, 3, ..n are none of them negative. p. 197, 1. 22, for (x^ #15 &c.) read fax^&c. „ 1. 23, for (xlt 052, &c.) read (x-^x^)^ &c. p. 208, 1. 2 from bottom, for Ny£ read \Ny£. p. 222, 1. 9 from bottom, for -^~ or % read -^ or -& p. 235, equations (5), for - read ju j and in (6) for — read — • p. 245, first number of last column in the table should be 1010. p. 258, 1. 14, for perpendicular to read along. p. 265, 1. 2 after equation (9), for -~ read -=~« ay ciy ERRATA. VOL. II. 3 from bottom, for (-) read (-) - p. ;281y equation (19), for n read %. p. 282, 1. 8, for z2 read z*. p. 289, equation (22), for 4a24 read 2af ; and for 4«'24 read 2 a' p. 293, equation (17), dele — . p. 300, 1. 7, for when read where. „ 1. 17, insert — after =. „ 1. 26, for Q* read ft. p. 301, equation (4') for / read r\ „ equation (5), insert — after = . p. 302, 1. 4 from bottom, for M= \ read M=—J- „ 1. 3 from bottom, insert at the beginning M— n the denominator of the last term should be c, „ last line, before the first bracket, for c22 read c2. p. 303, 1. 1 1 from bottom, for ft' read ft', p. 306, 1. 14, for 277 read 4-77. „ 1. 15, for >fAa read 2 V~Aa. 1. 19 should be 7 Tlf „ lines 23 and 27, change the sign of --=— • p. 316, equation (3), for =My- read my. p. 317, 1. 7, for ~| read -3. p. 318, 1. 8 from bottom for 36 to 31 read ^36 to p. 320, 1. 9, for 627, read 672. „ last line, after = insert f. p. 324, equation (14) should be - ~ (1 -H—y^)=~^ = constant. TT y y p. 325, 1. 5 from bottom, should be #=| ^-2 ^ (a^-a3). p. 346, 1. 2, for 0 read 0^ p. 359, equation (2), /or ^^ read —Ex. p. 365, equation (3), last term, dele y. PART III. MAGNETISM. CHAPTEK I. ELEMENTARY THEORY OF MAGNETISM. 371.] CERTAIN bodies, as, for instance, the iron ore called load stone, the earth itself, and pieces of steel which have been sub jected to certain treatment, are found to possess the following properties, and are called Magnets. • If, near any part of the earth's surface except the Magnetic Poles, a magnet be suspended so as to turn freely about a vertical axis, it will in general tend to set itself in a certain azimuth, and if disturbed from this position it will oscillate about if. An un- magnetized body has no such tendency, but is in equilibrium in all azimuths alike. 372.] It is found that the force which acts on the body tends to cause a certain line in the body, called the Axis of the Magnet, to become parallel to a certain line in space, called the Direction of the Magnetic Force. Let us suppose the magnet suspended so as to be free to turn in all directions about a fixed point. To eliminate the action of its weight we may suppose this point to be its centre of gravity. Let it come to a position^of equilibrium. Mark two points on the magnet, and note their positions in space. Then let the magnet be placed in a new position of equilibrium, and note the positions in space of the two marked points on the magnet. Since the axis of the magnet coincides with the direction of magnetic force in both positions, we have to find that line in the magnet which occupies the same position in space before and VOL. II. B 2 ELEMENTARY THEORY OF MAGNETISM. [373- after the motion. It appears, from the theory of the motion of >•;{ ^'bodies of invariable form, that such a line always exists, and that a motion equivalent to the actual motion might have taken place by simple rotation round this line. To find the line, join the first and last positions of each of the marked points, and draw planes bisecting these lines at right angles. The intersection of these planes will be the line required, which indicates the direction of the axis of the magnet and the direction of the magnetic force in space. The method just described is not convenient for the practical determination of these directions. We shall return to this subject when we treat of Magnetic Measurements. The direction of the magnetic force is found to be different at different parts of the earth's surface. If the end of the axis of the magnet which points in a northerly direction be marked, it has been found that the direction in which it sets itself in general deviates from the true meridian to a considerable extent, and that the marked end points on the whole downwards in the northern fc hemisphere and upwards in the southern. The azimuth of the direction of the magnetic force, measured from the true north in a westerly direction, is called the Variation, or the Magnetic Declination. The angle between the direction of the magnetic force and the horizontal plane is called the Magnetic Dip. These two angles determine the direction of the magnetic force, and, when the magnetic intensity is also known, the magnetic force is completely determined. The determination of the values of these three elements at different parts of the earth's surface, the discussion of the manner in which they vary according to the place and time of observation, and the investigation of the causes of the magnetic force and its variations, constitute the science of Terrestrial Magnetism. 373.] Let us now suppose that the axes of several magnets have been determined, and the end of each which points north marked. Then, if one of these be freely suspended and another brought near it, it is found that two marked ends repel each other, that a marked and an unmarked end attract each other, and that two unmarked ends repel each other. If the magnets are in the form of long rods or wires, uniformly and longitudinally magnetized, see below, Art. 384, it is found that the greatest manifestation of force occurs when the end of one magnet is held near the end of the other, and that the 374-] LAW OF MAGNETIC FORCE. 3 phenomena can be accounted for by supposing- that like ends of the magnets repel each other, that unlike ends attract each other, and that the intermediate parts of the magnets have no sensible mutual action. The ends of a long thin magnet are commonly called its Poles. In the case of an indefinitely thin magnet, uniformly magnetized throughout its length, the extremities act as centres of force, and the rest of the magnet appears devoid of magnetic action. In all actual magnets the magnetization deviates from uniformity, so that no single points can be taken as the poles. Coulomb, how ever, by using long thin rods magnetized with care, succeeded in establishing the law of force between two magnetic poles *. The repulsion between two magnetic poles is in the straight line joining them, and is numerically equal to the product of the strengths of the poles divided by the square of the distance between them. 374.] This law, of course, assumes that the strength of each pole is measured in terms of a certain unit, the magnitude of which may be deduced from the terms of the law. The unit-pole is a pole which points north, and is such that, when placed at unit distance from another unit-pole, it repels it with unit offeree, the unit of force being defined as in Art. 6. A pole which points south is reckoned negative. If m1 and m2 are the strengths of two magnetic poles, I the distance between them, and / the force of repulsion, all expressed numerically, then . ~ But if [m], [I/I and [F] be the concrete units of magnetic pole, length and force, then whence it follows that or [m] = \Il*T-lM*\. The dimensions of the unit pole are therefore f as regards length, ( — 1) as regards time, and \ as regards mass. These dimensions are the same as those of the electrostatic unit of electricity, which is specified in exactly the same way in Arts. 41, 42. * His experiments on magnetism with the Torsion Balance are contained in the Memoirs of the Academy of Paris, 1780-9, and in Biot's Traite de Physique, torn. iii. 4 ELEMENTARY THEORY OF MAGNETISM. [375- 375.] The accuracy of this law may be considered to have been established by the experiments of Coulomb with the Torsion Balance, and confirmed by the experiments of Gauss and Weber, and of all observers in magnetic observatories, who are every day making measurements of magnetic quantities, and who obtain results which would be inconsistent with each other if the law of force had been erroneously assumed. It derives additional support from its consistency with the laws of electromagnetic phenomena. 376.] The quantity which we have hitherto called the strength of a pole may also be called a quantity of ' Magnetism,' provided we attribute no properties to ' Magnetism ' except those observed in the poles of magnets. Since the expression of the law of force between given quantities of 'Magnetism' has exactly the same mathematical form as the law of force between quantities of 'Electricity' of equal numerical value, much of the mathematical treatment of magnetism must be similar to that of electricity. There are, however, other properties of magnets which must be borne in mind, and which may throw some light on the electrical properties of bodies. Relation between the Poles of a Magnet. 377.] The quantity of magnetism at one pole of a magnet is always equal and opposite to that at the other, or more generally thus : — In every Magnet the total quantity of Magnetism (reckoned alge braically) is zero. Hence in a field of force which is uniform and parallel throughout the space occupied by the magnet, the force acting on the marked end of the magnet is exactly equal, opposite and parallel to that on the unmarked end, so that the resultant of the forces is a statical couple, tending to place the axis of the magnet in a determinate direction, but not to move the magnet as a whole in any direction. This may be easily proved by putting the magnet into a small vessel and floating it in water. The vessel will turn in a certain direction, so as to bring the axis of the magnet as near as possible to the direction of the earth's magnetic force, but there will be no motion of the vessel as a whole in any direction ; so that there can be no excess of the force towards the north over that towards the south, or the reverse. It may also be shewn from the fact that magnetizing a piece of steel does not alter its weight. It does alter the apparent position of its centre of gravity, causing it in these 380.] MAGNETIC 'MATTER/ 5 latitudes to shift along the axis towards the north. The centre of inertia, as determined by the phenomena of rotation, remains unaltered. 378.] If the middle of a long thin magnet be examined, it is found to possess no magnetic properties, but if the magnet be broken at that point, each of the pieces is found to have a magnetic pole at the place of fracture, and this new pole is exactly equal and opposite to the other pole belonging to that piece. It is impossible, either by magnetization, or by breaking magnets, or by any other means, to procure a magnet whose poles are un equal. If we break the long thin magnet into a number of short pieces we shall obtain a series of short magnets, each of which has poles of nearly the same strength as those of the original long magnet. This multiplication of poles is not necessarily a creation of energy, for we must remember that after breaking the magnet we have to do work to separate the parts, in consequence of their attraction for one another. 379.] Let us now put all the pieces of the magnet together as at first. At each point of junction there will be two poles exactly equal and of opposite kinds, placed in contact, so that their united action on any other pole will be null. The magnet, thus rebuilt, has therefore the same properties as at first, namely two poles, one at each end, equal and opposite to each other, and the part between these poles exhibits no magnetic action. Since, in this case, we know the long magnet to be made up of little short magnets, and since the phenomena are the same as in the case of the unbroken magnet, we may regard the magnet, even before being broken, as made up of small particles, each of which has two equal and opposite poles. If we suppose all magnets to be made up of such particles, it is evident that since the algebraical quantity of magnetism in each particle is zero, the quantity in the whole magnet will also be zero, or in other words, its poles will be of equal strength but of opposite kind. Theory of Magnetic ''Matter? 380.] Since the form of the law of magnetic action is identical with that of electric action, the same reasons which can be given for attributing electric phenomena to the action of one ' flu id' or two ' fluids' can also be used in favour of the existence of a magnetic matter, or of two kinds of magnetic matter, fluid or 6 ELEMENTARY THEORY OF MAGNETISM. [380. otherwise. In fact, a theory of magnetic matter, if used in a purely mathematical sense, cannot fail to explain the phenomena, provided new laws are freely introduced to account for the actual facts. One of these new laws must be that the magnetic fluids cannot pass from one molecule or particle of the magnet to another, but that the process of magnetization consists in separating to a certain extent the two fluids within each particle, and causing the one fluid to be more concentrated at one end, and the other fluid to be more concentrated at the other end of the particle. This is the theory of Poisson. A particle of a magnetizable body is, on this theory, analogous to a small insulated conductor without charge, which on the two- fluid theory contains indefinitely large but exactly equal quantities of the two electricities. When an electromotive force acts on the conductor, it separates the electricities, causing them to become manifest at opposite sides of the conductor. In a similar manner, according to this theory, the magnetizing force causes the two kinds of magnetism, which were originally in a neutralized state, to be separated, and to appear at opposite sides of the magnetized particle. In certain substances, such as soft iron and those magnetic substances which cannot be permanently magnetized, this magnetic condition, like the electrification of the conductor, disappears when the inducing force is removed. In other substances, such as hard steel, the magnetic condition is produced with difficulty, and, when produced, remains after the removal of the inducing force. This is expressed by saying that in the latter case there is a Coercive Force, tending to prevent alteration in the magnetization, which must be overcome before the power of a magnet can be either increased or diminished. In the case of the electrified body this would correspond to a kind of electric resistance, which, unlike the resistance observed in metals, would be equivalent to complete insulation for electromotive forces below a certain value. This theory of magnetism, like the corresponding theory of electricity, is evidently too large for the facts, and requires to be restricted by artificial conditions. For it not only gives no reason why one body may not differ from another on account of having more of both fluids, but it enables us to say what would be the properties of a body containing an excess of one magnetic fluid. It is true that a reason is given why such a body cannot exist, 381.] MAGNETIC POLARIZATION. 7 but this reason is only introduced as an after-thought to explain this particular fact. It does not grow out of the theory. 381.] We must therefore seek for a mode of expression which shall not be capable of expressing too much, and which shall leave room for the introduction of new ideas as these are developed from new facts. This, I think, we shall obtain if we begin by saying that the particles of a magnet are Polarized. Meaning of the term ' Polarization? When a particle of a body possesses properties related to a certain line or direction in the body, and when the body, retaining these properties, is turned so that this direction is reversed, then if as regards other bodies these properties of the particle are reversed, the particle, in reference to these properties, is said to be polarized, and the properties are said to constitute a particular kind of polarization. Thus we may say that the rotation of a body about an axis constitutes a kind of polarization, because if, while the rotation continues, the direction of the axis is turned end for end, the body will be rotating in the opposite direction as regards space. A conducting particle through which there is a current of elec tricity may be said to be polarized, because if it were turned round, and if the current continued to flow in the same direction as regards the particle, its direction in space would be reversed. In short, if any mathematical or physical quantity is of the nature of a vector, as defined in Art. 11, then any body or particle to which this directed quantity or vector belongs may be said to be Polarized *9 because it has opposite properties in the two opposite directions or poles of the directed quantity. The poles of the earth, for example, have reference to its rotation, and have accordingly different names. * The word Polarization has been used in a sense not consistent with this in Optics, where a ray of light is said to be polarized when it has properties relating to its sides, which are identical on opposite sides of the ray. This kind of polarization refers to another kind of Directed Quantity, which may be called a Dipolar Quantity, in opposition to the former kind, which may be called Unipolar. When a dipolar quantity is turned end for end it remains the same as before. Tensions and Pressures in solid bodies, Extensions, Compressions and Distortions and most of the optical, electrical, and magnetic properties of crystallized bodies are dipolar quantities. The property produced by magnetism in transparent bodies of twisting the plane of polarization of the incident light, is, like magnetism itself, a unipolar property. The rotatory property referred to in Art. 303 is also unipolar. 8 ELEMENTARY THEORY OF MAGNETISM. [382. Meaning of the term ' Magnetic Polarization.'' 382.] In speaking of the state of the particles of a magnet as magnetic polarization, we imply that each of the smallest parts into which a magnet may be divided has certain properties related to a definite direction through the particle, called its Axis of Magnetization, and that the properties related to one end of this axis are opposite to the properties related to the other end. The properties which we attribute to the particle are of the same kind as those which we observe in the complete magnet, and in assuming that the particles possess these properties, we only assert what we can prove by breaking the magnet up into small pieces, for each of these is found to be a magnet. Properties of a Magnetized Particle. 383.] Let the element dxdydz be a particle of a magnet, and let us assume that its magnetic properties are those of a magnet the strength of whose positive pole is mt and whose length is ds. Then if P is any point in space distant r from the positive pole and / from the negative pole, the magnetic potential at P will be — due to the positive pole, and -- -^ due to the negative pole, or If ds, the distance between the poles, is very small, we may put /— r = dscos e, (2) where e is the angle between the vector drawn from the magnet to P and the axis of the magnet, or , N cose. (3) Magnetic Moment. 384.] The product of the length of a* uniformly and longitud inally magnetized bar magnet into the strength of its positive pole is called its Magnetic Moment. Intensity of Magnetization. The intensity of magnetization of a magnetic particle is the ratio of its magnetic moment to its volume. We shall denote it by /. The magnetization at any point of a magnet may be defined by its intensity and its direction. Its direction may be defined by its direction-cosines A, /u,, v. 385.] COMPONENTS OF MAGNETIZATION. 9 Components of Magnetization. The magnetization at a point of a magnet (being a vector or directed quantity) may be expressed in terms of its three com ponents referred to the axes of coordinates. Calling these A, B, C, A = I\, B = Iy., C=Iv, and the numerical value of I is given by the equation (4) ja = A*+B* + C2. (5) 385.] If the portion of the magnet which we consider is the differential element of volume dxdydz, and if / denotes the intensity of magnetization of this element, its magnetic moment is Idxdydz. Substituting this for mds in equation (3), and remembering that rcose = \(£-x)+iL(ri—y) + v(C—z), (6) where £, 77, f are the coordinates of the extremity of the vector r drawn from the point (#, y, z), we find for the potential at the point (£, 77, () due to the magnetized element at (a?, y, z\ W= {A(£-x) + B(ri-y)+C({-z)}±;dxdydz. (7) To obtain the potential at the point (£. r], f) due to a magnet of finite dimensions, we must find the integral of this expression for every element of volume included within the space occupied by the magnet, or (8) Integrating by parts, this becomes dc where the double integration in the first three terms refers to the surface of the magnet, and the triple integration in the fourth to the space within it. If I, m, n denote the direction-cosines of the normal drawn outwards from the element of surface dS, we may write, as in Art. 21 j the sum of the first three terms, where the integration is to be extended over the whole surface of the magnet. 10 ELEMENTARY THEORY OF MAGNETISM. [386. If we now introduce two new symbols a and p} defined by the equations <r = (dA dB dC^ p: ~^ + ^ + ^;j the expression for the potential may be written 386.] This expression is identical with that for the electric potential due to a body on the surface of which there is an elec trification whose surface-density is o-, while throughout its substance there is a bodily electrification whose volume-density is p. Hence, if we assume cr and p to be the surface- and volume-densities of the distribution of an imaginary substance, which we have called t magnetic matter,' the potential due to this imaginary distribution will be identical with that due to the actual magnetization of every element of the magnet. The surface-density v is the resolved part of the intensity of magnetization 7 in the direction of the normal to the surface drawn outwards, and the volume-density p is the ' convergence' (see Art. 25) of the magnetization at a given point in the magnet. This method of representing the action of a magnet as due to a distribution of f magnetic matter ' is very convenient, but we must always remember that it is only an artificial method of representing the action of a system of polarized particles. On the Action of one Magnetic Molecule o 387.] If, as in the chapter on Spherical Harmonics, Art. 129, we make d , d d d ~TL = ^ T~ + m ~j — \- n r "> W dh dx dy dz where I, m, n are the direction-cosines of the axis It, then the potential due to a magnetic molecule at the origin, whose axis is parallel to klt and whose magnetic moment is mlt is y _ d ml ml ( '** ~5*77~"HAi' where A.L is the cosine of the angle between h± and r. Again, if a second magnetic molecule whose moment is m2, and whose axis is parallel to hz, is placed at the extremity of the radius vector r, the potential energy due to the action of the one magnet on the other is 387.] FORCE BETWEEN TWO MAGNETIZED PARTICLES. 11 (3) (4) where /u12 is the cosine of the angle which the axes make with each other, and Xls A2 are the cosines of the angles which they make with r. Let us next determine the moment of the couple with which the first magnet tends to turn the second round its centre. Let us suppose the second magnet turned through an angle d(f) in a plane perpendicular to a third axis &3, then the work done against the magnetic forces will be -^ — dti, and the moment of the a(f> forces on the magnet in this plane will be dW ml m2 ,dyl2 d\2^ ~~d^ = ~^~\d$~ Al3^' The actual moment acting on the second magnet may therefore be considered as the resultant of two couples, of which the first acts in a plane parallel to the axes of both magnets, and tends to increase the angle between them with a force whose moment is while the second couple acts in the plane passing through r and the axis of the second magnet, and tends to diminish the angle between these directions with a force 3 m* m9 >~^cos(r/h)siu(r/^, (7) where (f^), (?'^2); (^1^2) denote the angles between the lines r, To determine the force acting on the second magnet in a direction parallel to a line 7/3, we have to calculate dW d* ,K (9) (10) If we suppose the actual force compounded of three forces, R, H^ and H2, in the directions of r, ^ and ^2 respectively, then the force in the direction of ^3 is (11) 12 ELEMENTARY THEORY OF MAGNETISM. [388. Since the direction of h% is arbitrary, we must have 3 tYl-i tlfli\ ~\ _/L ^^ .— — vMl2 "~~ 1 2/5 (12) The force 72 is a repulsion, tending to increase r ; H^ and ZT2 act on the second magnet in the directions of the axes of the first and second magnet respectively. This analysis of the forces acting between two small magnets was first given in terms of the Quaternion Analysis by Professor Tait in the Quarterly Math. Journ. for Jan. 1860. See also his work on Quaternions, Art. 414. Particular Positions. 388.] (1) If Aj and A2 are each equal to 1, that is, if the axes of the magnets are in one straight line and in the same direction, fj.12 = 1, and the force between the magnets is a repulsion p. TT , TT Qm1m2 . . Jic-f jczi-f/ZgTs -- 4 -- (13) The negative sign indicates that the force is an attraction. (2) If A: and A2 are zero, and /*12 unity, the axes of the magnets are parallel to each other and perpendicular to /, and the force is a repulsion 3m1m2 In neither of these cases is there any couple. (3) If A! = 1 and A2 = 0, then /u12 = 0. (15) The force on the second magnet will be - — *— 2 in the direction of its axis, and the couple will be — ^— 2 t tending to turn it parallel to the first magnet. This is equivalent to a single force - ^ 2 acting parallel to the direction of the axis of the second magnet, and cutting r at a point two-thirds of its length from m2. Fig. 1. Thus in the figure (1) two magnets are made to float on water, 388.] FORCE BETWEEN TWO SMALL MAGNETS. 13 being in the direction of the axis of m1 , but having- its own axis at right angles to that of ml. If two points, A, B, rigidly connected with % and m2 respectively, are connected by means of a string T, the system will be in equilibrium,, provided T cuts the line m1m2 at right angles at a point one-third of the distance from ml to m2 . (4) If we allow the second magnet to turn freely about its centre till it comes to a position of stable equilibrium, ?Fwill then be a minimum as regards k2 , and therefore the resolved part of the force due to m2, taken in the direction of ^15 will be a maximum. Hence, if we wish to produce the greatest possible magnetic force at a given point in a given direction by means of magnets, the positions of whose centres are given, then, in order to determine the proper directions of the axes of these magnets to produce this effect, we have only to place a magnet in the given direction at the given point, and to observe the direction of stable equilibrium of the axis of a second magnet when its centre is placed at each of the other given points. The magnets must then be placed with their axes in the directions indicated by that of the second magnet. Of course, in performing this experi ment we must take account of terrestrial magnetism, if it exists. Let the second magnet be in a posi tion of stable equilibrium as regards its direction, then since the couple acting on it vanishes, the axis of the second magnet must be in the same plane with that of the first. Hence (M2) = (V)+M2), (16) and the couple being Fig. 2. m (sin (h-^ /t>2) — 3 cos (h-^ r) sin (r h2)), (17) we find when this is zero tan (^ r) = 2 tan (r 7*2) , (18) or tan^Wg-B = 2 ta,nRm2ff2. (19) When this position has been taken up by the second magnet the dV value of W becomes where h2 is in the direction of the line of force due to ml at 14 ELEMENTARY THEORY OF MAGNETISM. [389. Hence W ,-.V; T ~1 * (20) Hence the second magnet will tend to move towards places of greater resultant force. The force on the second magnet may be decomposed into a force R, which in this case is always attractive towards the first magnet, and a force ffl parallel to the axis of the first magnet, where H L = 3^ ** _ . (21) ^ 73 Ax2 + 1 In Fig. XVII, at the end of this volume, the lines of force and equipotential surfaces in two dimensions are drawn. The magnets which produce them are supposed to be two long cylindrical rods the sections of which are represented by the circular blank spaces, and these rods are magnetized transversely in the direction of the arrows. Jf we remember that there is a tension along the lines of force, it is easy to see that each magnet will tend to turn in the direction of the motion of the hands of a watch. That on the right hand will also, as a whole, tend to move towards the top, and that on the left hand towards the bottom of the page. On the Potential Energy of a Magnet placed in a Magnetic Field. 389.] Let V be the magnetic potential due to any system of magnets acting on the magnet under consideration. We shall call V the potential of the external magnetic force. If a small magnet whose strength is m, and whose length is ds, be placed so that its positive pole is at a point where the potential is T3 and its negative pole at a point where the potential is F', the potential energy of this magnet will be mCF—P'), or, if ds is measured from the negative pole to the positive, dV - ,1X m-f-ds. (1) as If / is the intensity of the magnetization, and A, p, v its direc tion-cosines, we may write, mds = dV dV dV dV and -7- = A-y--f-ju-^ — |- v^-> ds dx dy dz and, finally, if A, B, C are the components of magnetization, A=\I, B=pl, C=vl, 390.] POTENTIAL ENERGY OP A MAGNET. 15 so that the expression (1) for the potential energy of the element of the magnet becomes To obtain the potential energy of a magnet of finite size, we must integrate this expression for every element of the magnet. We thus obtain W = fff(A df + Bll^ + Cd-f) dxdydz (3) J J J ^ dx dy dz ' as the value of the potential energy of the magnet with respect to the magnetic field in which it is placed. The potential energy is here expressed in terms of the components of magnetization and of those of the magnetic force arising from external causes. By integration by parts we may express it in terms of the distribution of magnetic matter and of magnetic potential ~ + -- + -dxdydzy (4) where /, m, n are the direction-cosines of the normal at the element of surface dS. If we substitute in this equation the expressions for the surface- and volume-density of magnetic matter as given in Art. 386, the expression becomes pdS. (5) We may write equation (3) in the form + Cy}dxdydz, (6) where a, ft and y are the components of the external magnetic force. On the Magnetic Moment and Axis of a Magnet. 390.] If throughout the whole space occupied by the magnet the external magnetic force is uniform in direction and magnitude, the components a, /3, y will be constant quantities, and if we write IJJAdxdydz=lK, jjJBdxdydz=mK, [((cdxdydz = nKt (7) the integrations being extended over the whole substance of the magnet, the value of ^may be written y). (8) 16 ELEMENTAEY THEORY OF MAGNETISM. In this expression I, m, n are the direction-cosines of the axis of the magnet, and K is the magnetic moment of the magnet. If e is the angle which the axis of the magnet makes with the direction of the magnetic force «£), the value of W may be written JF = -K$cos€. (9) If the magnet is suspended so as to be free to turn about a vertical axis, as in the case of an ordinary compass needle, let the azimuth of the axis of the magnet be $, and let it be inclined 0 to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is 5 and dip £, then a = «$p cos £ cos bj (3 = «£j cos £ sin 8, y = «£) sin f; (10) I = cos 0 cos <£, m = cos 0 sin <£, n — sin 0 ; (11) whence W— — KQ (cos £ cos 6 cos ($ — 8) + sin ( sin e). (12) The moment of the force tending to increase $ by turning the magnet round a vertical axis is _ ^L=_K cos Ccos<9 sin (<J>-5). (13) On the Expansion of the Potential of a Magnet in Solid Harmonics. 391.] Let V be the potential due to a unit pole placed at the point (£, T?, f). The value of F" at the point #, y, z is r= {(f-*)2+(>/-,?o2 +(<r-*)Ti (i) This expression may be expanded in terms of spherical harmonics, with their centre at the origin. We have then (2) when FQ = - , r being the distance of (f, 77, f ) from the origin, (3) (4) _ 2~ 2r5 fee. To determine the value of the potential energy when the magnet is placed in the field of force expressed by this potential, we have to integrate the expression for W in equation (3) with respect to x, y and z, considering £, 77, (" and r as constants. If we consider only the terms introduced by F~0, Ft and V2 the result will depend on the following volume-integrals, 392.] EXPANSION OF THE POTENTIAL DUE TO A MAGNET. 17 lK = jjJAdxdydz, mK = fjfsdxdydz, nK =JJJ Cdxdydz; (6) L=jjJAxdxdydz> M = jjj Bydxdydz, N =jjJCzdxdydz', (7) P = (B* + Cy)dxdydz, Q = R = ^y + Bnyndydz- (8) We thus find for the value of the potential energy of the magnet placed in presence of the unit pole at the point (^17, Q, _ r5 This expression may also be regarded as the potential energy of the unit pole in presence of the magnet, or more simply as the potential at the point £ , 17, f due to the magnet. On ike Centre of a Magnet and its Primary and Secondary Axes. 392.] This expression may be simplified by altering the directions of the coordinates and the position of the origin. In the first place, we shall make the direction of the axis of x parallel to the axis of the magnet. This is equivalent to making l—\^ m = 0, n — 0. (10) If we change the origin of coordinates to the point (#', y', /), the directions of the axes remaining unchanged, the volume-integrals IK, mK and nK will remain unchanged, but the others will be altered as follows : L'=L-lKx', M'=M-mKy', Nf = N-nKz/-f (11) P'=P—K(mz'+ny), Q'=Q- K(nx' + lz'\ R' — R— K(ly' + mx'}. If we now make the direction of the axis of x parallel to the axis of the magnet, and put , Zl-M-N , R , Q , . x'= —^ > y = Tr> z = -^> (13) 2A A A then for the new axes M and N have their values unchanged, and the value of 1! becomes \ (M+N). P remains unchanged, and Q and R vanish. We may therefore write the potential thus, VOL. II. 18 ELEMENTARY THEOEY OF MAGNETISM. \_392- We have thus found a point, fixed with respect to the magnet, such that the second term of the potential assumes the most simple form when this point is taken as origin of coordinates. This point we therefore define as the centre of the magnet, and the axis drawn through it in the direction formerly defined as the direction of the magnetic axis may be defined as the principal axis of the magnet. We may simplify the result still more by turning the axes of y and z round that of x through half the angle whose tangent is p -=£ — — . This will cause P to become zero, and the final form of the potential may be written Kt ttf- 3 2 This is the simplest form of the first two terms of the potential of a magnet. When the axes of y and z are thus placed they may be called the Secondary axes of the magnet. We may also determine the centre of a magnet by finding the position of the origin of coordinates, for which the surface-integral of the square of the second term of the potential, extended over a sphere of unit radius, is a minimum. The quantity which is to be made a minimum is, by Art. 141, 4 (Z2 + Mz + N*-MN-NL-LM] + 3 (P2 + Q2 +^2). (16) The changes in the values of this quantity due to a change of position of the origin may be deduced from equations (11) and (12). Hence the conditions of a minimum are 21(2 L—M— N)+3nQ+3mR = 0, 2m(2M-N-L)+3lR+3nP = 0, (17) 2n (2N—Z—M)+3mP+3lQ = 0. If we assume I = I, m = 0, n = Q, these conditions become 2L-M—N=0, q = 0, R=0, (18) which are the conditions made use of in the previous invest igation. This investigation may be compared with that by which the potential of a system of gravitating matter is expanded. In the latter case, the most convenient point to assume as the origin is the centre of gravity of the system, and the most convenient axes are the principal axes of inertia through that point. In the case of the magnet, the point corresponding to the centre of gravity is at an infinite distance in the direction of the axis, 394'] CONVENTION RESPECTING SIGNS. 19 and the point which we call the centre of the magnet is a point having- different properties from those of the centre of gravity. The quantities If, M, N correspond to the moments of inertia, and P, Q, R to the products of inertia of a material body, except that Z, M and N are not necessarily positive quantities. When the centre of the magnet is taken as the origin, the spherical harmonic of the second order is of the sectorial form, having its axis coinciding with that of the magnet, and this is true of no other point. When the magnet is symmetrical on all sides of this axis, as in the case of a figure of revolution, the term involving the harmonic of the second order disappears entirely. 393.] At all parts of the earth's surface, except some parts of the Polar regions, one end of a magnet points towards the north, or at least in a northerly direction, and the other in a southerly direction. In speaking of the ends of a magnet we shall adopt the popular method of calling the end which points to the north the north end of the magnet. When, however, we speak in the language of the theory of magnetic fluids we shall use the words Boreal and Austral. Boreal magnetism is an imaginary kind of matter supposed to be most abundant in the northern, parts of the earth, and Austral magnetism is the imaginary magnetic matter which prevails in the southern regions of the earth. The magnetism of the north end of a magnet is Austral, and that of the south end is Boreal. When therefore we speak of the north and south ends of a magnet we do not compare the magnet with the earth as the great magnet, but merely express the position which the magnet endeavours to take up when free to move. When, on the other hand, we wish to compare the distribution of ima ginary magnetic fluid in the magnet with that in the earth we shall use the more grandiloquent words Boreal and Austral magnetism. 394.] In speaking of a field of magnetic force we shall use the phrase Magnetic North to indicate the direction in which the north end of a compass needle would point if placed in the field of force. In speaking of a line of magnetic force we shall always suppose it to be traced from magnetic south to magnetic north, and shall call this direction positive. In the same way the direction of magnetization of a magnet is indicated by a line drawn from the south end of the magnet towards the north end, and the end of the magnet which points north is reckoned the positive end. 20 ELEMENTARY THEORY OF MAGNETISM. \_394-- We shall consider Austral magnetism, that is, the magnetism of that end of a magnet which points north, as positive. If we denote its numerical value by m> then the magnetic potential and the positive direction of a line of force is that in which V diminishes. CHAPTER II. MAGNETIC FORCE AND MAGNETIC INDUCTION. 395.] WE have already (Art. 386) determined the magnetic potential at a given point due to a magnet, the magnetization of which is given at every point of its substance, and we have shewn that the mathematical result may be expressed either in terms of the actual magnetization of every element of the magnet, or in terms of an imaginary distribution of ' magnetic matter,' partly condensed on the surface of the magnet and partly diffused through out its substance. The magnetic potential, as thus denned, is found by the same mathematical process, whether the given point is outside the magnet or within it. The force exerted on a unit magnetic pole placed at any point outside the magnet is deduced from the potential by the same process of differentiation as in the corresponding electrical problem. If the components of this force are a, /3, y, dV dV dV m a= — > /3 = j-j y— j-- (1) dx dy dz To determine by experiment the magnetic force at a point within the magnet we must begin by removing part of the magnetized substance, so as to form a cavity within which we are to place the magnetic pole. The force acting on the pole will depend, in general, in the form of this cavity, and on the inclination of the walls of the cavity to the direction of magnetization. Hence it is necessary, in order to avoid ambiguity in speaking of the magnetic force within a magnet, to specify the form and position of the cavity within which the force is to be measured. It is manifest that when the form and position of the cavity is specified, the point within it at which the magnetic pole is placed must be regarded as 22 MAGNETIC FORCE AND MAGNETIC INDUCTION. [396. no longer within the substance of the magnet, and therefore the ordinary methods of determining the force become at once applicable. 396.] Let us now consider a portion of a magnet in which the direction and intensity of the magnetization are uniform. Within this portion let a cavity be hollowed out in the form of a cylinder, the axis of which is parallel to the direction of magnetization, and let a magnetic pole of unit strength be placed at the middle point of the axis. Since the generating lines of this cylinder are in the direction of magnetization, there will be no superficial distribution of mag netism on the curved surface, and since the circular ends of the cylinder are perpendicular to the direction of magnetization, there will be a uniform superficial distribution, of which the surface- density is /for the negative end, and —/for the positive end. Let the length of the axis of the cylinder be 2 b, and its radius a. Then the force arising from this superficial distribution on a magnetic pole placed at the middle point of the axis is that due to the attraction of the disk on the positive side, and the repulsion of the disk on the negative side. These two forces are equal and in the same direction, and their sum is ---!=. (2) From this expression it appears that the force depends, not on the absolute dimensions of the cavity, but on the ratio of the length to the diameter of the cylinder. Hence, however small we make the cavity, the force arising from the surface distribution on its walls will remain, in general, finite. 397.] We have hitherto supposed the magnetization to be uniform and in the same direction throughout the whole of the portion of the magnet from which the cylinder is hollowed out. Wlien the magnetization is not thus restricted, there will in general be a distribution of imaginary magnetic matter through the substance of the magnet. The cutting out of the cylinder will remove part of this distribution, but since in similar solid figures the forces at corresponding points are proportional to the linear dimensions of the figures, the alteration of the force on the magnetic pole due to the volume-density of magnetic matter will diminish indefinitely as the size of the cavity is diminished, while the effect due to the surface-density on the walls of the cavity remains, in general, finite. If, therefore, we assume the dimensions of the cylinder so small 399-1 MAGNETIC FORCE IN A CAVITY. 23 that the magnetization of the part removed may be regarded as everywhere parallel to the axis of the cylinder, and of constant magnitude I, the force on a magnetic pole placed at the middle point of the axis of the cylindrical hollow will be compounded of two forces. The first of these is that due to the distribution of magnetic matter on the outer surface of the magnet, and throughout its interior, exclusive of the portion hollowed out. The components of this force are a, /3 and y, derived from the potential by equations (1). The second is the force 72, acting along the axis of the cylinder in the direction of magnetization. The value of this force depends on the ratio of the length to the diameter of the cylindric cavity. 398.] Case I. Let this ratio be very great, or let the diameter of the cylinder be small compared with its length. Expanding the expression for R in terms of j- , it becomes a quantity which vanishes when the ratio of b to a is made infinite. Hence, when the cavity is a very narrow cylinder with its axis parallel to the direction of magnetization, the magnetic force within the cavity is not affected by the surface distribution on the ends of the cylinder, and the components of this force are simply a, /3, y, where dV dV dV ,,. a = -- 7-, 0 = — -=-, y= — -—. (4) dx dy dz We shall define the force within a cavity of this form as the magnetic force within the magnet. Sir William Thomson has called this the Polar definition of magnetic force. When we have occasion to consider this force as a vector we shall denote it *>7$. 399.] Case II. Let the length of the cylinder be very small compared with its diameter, so that the cylinder becomes a thin disk. Expanding the expression for R in terms of - , it becomes _ £+££-*..}, (5) a 2 #3 3 the ultimate value of which, when the ratio of a to b is made infinite, is 4 TT J. Hence, when the cavity is in the form of a thin disk, whose plane is normal to the direction of magnetization, a unit magnetic pole 24 MAGNETIC FORCE AND MAGNETIC INDUCTION. [400. placed at the middle of the axis experiences a force 4 IT I in the direction of magnetization arising from the superficial magnetism on the circular surfaces of the disk *. Since the components of J are A, B and (7, the components of this force are 4 -n A, 4 TT B and 4 TT C. This must be compounded with the force whose components are a, {3, y. 400.] Let the actual force on the unit pole be denoted by the vector 35, and its components by a, b and c, then a = a + 4 TT A, 0=/3 + 47T.£, (6) C = y -f 4 TT C. We shall define the force within a hollow disk, whose plane sides are normal to the direction of magnetization, as the Magnetic Induction within the magnet. Sir William Thomson has called this the Electromagnetic definition of magnetic force. The three vectors, the magnetization 3, the magnetic force <!fj, and the magnetic induction S3 are connected by the vector equation 47:3. (7) Line-Integral of Magnetic Force. 401.] Since the magnetic force, as denned in Art. 398, is that due to the distribution of free magnetism on the surface and through the interior of the magnet, and is not affected by the surface- magnetism of the cavity, it may be derived directly from the general expression for the potential of the magnet, and the line- integral of the magnetic force taken along any curve from the point A to the point B is where VA and V^ denote the potentials at A and B respectively. * On the force within cavities of other forms. 1. Any narrow crevasse. The force arising from the surface-magnetism is 47r/cos€ in the direction of the normal to the plane of the crevasse, where 6 is the angle between this normal and the direction of magnetization. When the crevasse is parallel to the direction of magnetization the force is the magnetic force £ ; when the crevasse is perpendicular to the direction of magnetization the force is the magnetic induction 93. 2. In an elongated cylinder, the axis of which makes an angle « with the direction of magnetization, the force arising from the surface-magnetism is 27r/sin e, perpendicular to the axis in the plane containing the axis and the direction of magnetization. 3. In a sphere the force arising from surface-magnetism is f IT I in the direction of magnetization. 402.] SURF ACE -INTEGRAL. 25 Surface-Integral of Magnetic Induction. 402.] The magnetic induction through the surface 8 is defined as the value of the integral Q = ff%cos€dS, (9) where 23 denotes the magnitude of the magnetic induction at the element of surface clS, and e the angle between the direction of the induction and the normal to the element of surface, and the integration is to be extended over the whole surface, which may be either closed or bounded by a closed curve. If a, b, c denote the components of the magnetic induction, and /, m, n the direction-cosines of the normal, the surface-integral may be written q = jj(la+mb+nG)d8. (10) If we substitute for the components of the magnetic induction their values in terms of those of the magnetic force, and the magnetization as given in Art. 400, we find Q = n(la + mp + ny)dS + 4 TT (lA + m£ + nC)dS. (11) We shall now suppose that the surface over which the integration extends is a closed one, and we shall investigate the value of the two terms on the right-hand side of this equation. Since the mathematical form of the relation between magnetic force and free magnetism is the same as that between electric force and free electricity, we may apply the result given in Art. 77 to the first term in the value of Q by substituting a, ft, y, the components of magnetic force, for X, Y, Z, the components of electric force in Art. 77, and M, the algebraic sum of the free magnetism within the closed surface, for e, the algebraic sum of the free electricity. We thus obtain the equation ny)48*x 4irM. (12) Since every magnetic particle has two poles, which are equal in numerical magnitude but of opposite signs, the algebraic sum of the magnetism of the particle is zero. Hence, those particles which are entirely within the closed surface S can contribute nothing to the algebraic sum of the magnetism within S. The 26 MAGNETIC FORCE AND MAGNETIC INDUCTION. [403. value of M must therefore depend only on those magnetic particles which are cut by the surface S. Consider a small element of the magnet of length s and trans verse section kz, magnetized in the direction of its length, so that the strength of its poles is m. The moment of this small magnet will be ms, and the intensity of its magnetization, being the ratio of the magnetic moment to the volume, will be /=£• (13) Let this small magnet be cut by the surface S, so that the direction of magnetization makes an angle e' with the normal drawn outwards from the surface, then if dS denotes the area of the section, p = ds cos e/t ( 1 4) The negative pole — m of this magnet lies within the surface S. Hence, if we denote by dM the part of the free magnetism within S whic*h is contributed by this little magnet, IS. (15) To find M, the algebraic sum of the free magnetism within the closed surface S, we must integrate this expression over the closed surface, so that M=- or writing A, .Z?, C for the components of magnetization, and I, m, n for the direction-cosines of the normal drawn outwards, (16) This gives us the value of the integral in the second term of equation (11). The value of Q in that equation may therefore be found in terms of equations (12) and (16), Q = 47r3/-47rl/= 0, (17) or, the surface-integral of the magnetic induction through any closed surface is zero. 403.] If we assume as the closed surface that of the differential element of volume dx dy dz, we obtain the equation *! + *+* = 0. (18) dx dy dz This is the solenoidal condition which is always satisfied by the components of the magnetic induction. 405.] LINES OF MAGNETIC INDUCTION. 27 Since the distribution of magnetic induction is solenoidal, the induction through any surface bounded by a closed curve depends only on the form and position of the closed curve, and not on that of the surface itself. 404.] Surfaces at every point of which la + mb + nc = 0 (19) are called Surfaces of no induction, and the intersection of two such surfaces is called a Line of induction. The conditions that a curve, Sj may be a line of induction are 1 dx 1 dy \ dz , . = 'L = . (20) a ds I ds c ds A system of lines of induction drawn through every point of a closed curve forms a tubular surface called a Tube of induction. The induction across any section of such a tube is the same. If the induction is unity the tube is called a Unit tube of in duction. All that Faraday * says about lines of magnetic force and mag netic sphondyloids is mathematically true, if understood of the lines and tubes of magnetic induction. The magnetic force and the magnetic induction are identical outside the magnet, but within the substance of the magnet they must be carefully distinguished. In a straight uniformly mag netized bar the magnetic force due to the magnet itself is from the end which points north, which we call the positive pole, towards the south end or negative pole, both within the magnet and in the space without. The magnetic induction, on the other hand, is from the positive pole to the negative outside the magnet, and from the negative pole to the positive within the magnet, so that the lines and tubes of induction are re-entering or cyclic figures. The importance of the magnetic induction as a physical quantity will be more clearly seen when we study electromagnetic phe nomena. When the magnetic field is explored by a moving wire, as in Faraday's Exp. Res. 3076, it is the magnetic induction and not the magnetic force which is directly measured. The Vector-Potential of Magnetic Induction. 405.] Since, as we have shewn in Art. 403, the magnetic in duction through a surface bounded by a closed curve depends on * Exp. Res., series xxviii. 28 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406. the closed curve, and not on the form of the surface which is bounded by it, it must be possible to determine the induction through a closed curve by a process depending only on the nature of that curve, and not involving the construction of a surface forming a diaphragm of the curve. This may be done by finding a vector 21 related to 33, the magnetic induction, in such a way that the line-integral of SI, extended round the closed curve, is equal to the surface-integral of 33, extended over a surface bounded by the closed curve. If, in Art. 24, we write F9 G, H for the components of SI, and a, b, c for the components of 33, we find for the relation between these components dH dG dF dH dG dF a= — .j 7 dz dz ax ax ay The vector SI, whose components are F, G, //, is called the vector- potential of magnetic induction. The vector-potential at a given point, due to a magnetized particle placed at the origin, is nume rically equal to the magnetic moment of the particle divided by the square of the radius vector and multiplied by the sine of the angle between the axis of magnetization and the radius vector, and the direction of the vector-potential is perpendicular to the plane of the axis of magnetization and the radius vector, and is such that to an eye looking in the positive direction along the axis of magnetization the vector-potential is drawn in the direction of rotation of the hands of a watch. Hence, for a magnet of any form in which A^ B, C are the components of magnetization at the point xyz, the components of the vector-potential at the point f 77 £ are (22) where p is put, for conciseness, for the reciprocal of the distance between the points (f, 77, Q and (#, y, z), and the integrations are extended over the space occupied by the magnet. 406.] The scalar, or ordinary, potential of magnetic force, Art. 386, becomes when expressed in the same notation, 406.] VECTOR- POTENTIAL. 29 /v /y\ t-j /v\ Kemembering that ~ = — -~, and that the integral dx u/ £ has the value — 4 TT ( A) when the point (£, 77, f) is included within the limits of integration, and is zero when it is not so included, (A) being the value of A at the point (f, 77, (*), we find for the value of the ^-component of the magnetic induction, dH _ dG_ dr] d£ f d^p dzp \ d'*p d2j) } \dydr) dzdC' dx dr] dxd^S 7> r, ^ 7 7 -ri - ~ + B -/- -f- (7 7 \dxdydz d£jJJ ( dx dy d The first term of this expression is evidently -- ^ , or a, the component of the magnetic force. The quantity under the integral sign in the second term is zero for every element of volume except that in which the point (f, ry, £) is included. If the value of A at the point (f, r/, f) is (A), the value of the second term is 4 TT (A)9 where (A) is evidently zero at all points outside the magnet. We may now write the value of the ^-component of the magnetic induction « = o+4w(^), (25) an equation which is identical with the first of those given in Art. 400. The equations for b and c will also agree with those of Art. 400. We have already seen that the magnetic force § is derived from the scalar magnetic potential V by the application of Hamilton's operator y , so that we may write, as in Art. 1 7, £=-vF, (26) and that this equation is true both without and within the magnet. It appears from the present investigation that the magnetic induction S3 is derived from the vector-potential SI by the appli cation of the same operator, and that the result is true within the magnet as well as without it. The application of this operator to a vector-function produces, 30 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406. in general, a scalar quantity as well as a vector. The scalar part, however, which we have called the convergence of the vector- function, vanishes when the vector-function satisfies the solenoidal condition dF dG dH •Jl + -J~ + -7TF = °* df; dr] d£ By differentiating the expressions for F, G, If in equations (22), we find that this equation is satisfied by these quantities. We may therefore write the relation between the magnetic induction and its vector-potential 23 = V % which may be expressed in words by saying that the magnetic induction is the curl of its vector-potential. See Art. 25. CHAPTER III MAGNETIC SOLENOIDS AND SHELLS*. On Particular Forms of Magnets. 407.] IF a long narrow filament of magnetic matter like a wire is magnetized everywhere in a longitudinal direction, then the product of any transverse section of the filament into the mean intensity of the magnetization across it is called the strength of the magnet at that section. If the filament were cut in two at the section without altering the magnetization, the two surfaces, when separated, would be found to have equal and opposite quan tities of superficial magnetization, each of which is numerically equal to the strength of the magnet at the section. A filament of magnetic matter, so magnetized that its strength is the same at every section, at whatever part of its length the section be made, is called a Magnetic Solenoid. If m is the strength of the solenoid, ds an element of its length, r the distance of that element from a given point, and e the angle which r makes with the axis of magnetization of the element, the potential at the given point due to the element is m ds cos € m dr .. — o = —s- ~r~ ds. r2 r* ds Integrating this expression with respect to s} so as to take into account all the elements of the solenoid, the potential is found to be ,11^ V = m ( ) > rl r2 T! being the distance of the positive end of the solenoid, and r^ that of the negative end from the point where V exists. Hence the potential due to a solenoid, and consequently all its magnetic effects, depend only on its strength and the position of * See Sir W. Thomson's 'Mathematical Theory of Magnetism,' Phil. Trans., 1850, or Reprint. 32 MAGNETIC SOLENOIDS AND SHELLS. [408. its ends, and not at all on its form, whether straight or curved, between these points. Hence the ends of a solenoid may be called in a strict sense its poles. If a solenoid forms a closed curve the potential due to it is zero at every point, so that such a solenoid can exert no magnetic action, nor can its magnetization be discovered without breaking it at some point and separating the ends. If a magnet can be divided into solenoids, all of which either form closed curves or have their extremities in the outer surface of the magnet, the magnetization is said to be solenoidal, and, since the action of the magnet depends entirely upon that of the ends of the solenoids, the distribution of imaginary magnetic matter will be entirely superficial. Hence the condition of the magnetization being solenoidal is dA dB dC _ dx dy dz where A, B, C are the components of the magnetization at any point of the magnet. 408.] A longitudinally magnetized filament, of which the strength varies at different parts of its length, may be conceived to be made up of a bundle of solenoids of different lengths, the sum of the strengths of all the solenoids which pass through a given section being the magnetic strength of the filament at that section. Hence any longitudinally magnetized filament may be called a Complex Solenoid. If the strength of a complex solenoid at any section is m, then the potential due to its action is ds where m is variable, Cm dr f% - m\ mi /I fll* 4* i 4* /I /*> J I l dm 7 ds This shews that besides the action of the two ends, which may in this case be of different strengths, there is an action due to the distribution of imaginary magnetic matter along the filament with a linear density dm /V. — - •"— j * ds Magnetic Shells. 409.] If a thin shell of magnetic matter is magnetized in a SHELLS. 33 direction everywhere normal to its surface, the intensity of the magnetization at any place multiplied by the thickness of the sheet at that place is called the Strength of the magnetic shell at that place. If the strength of a shell is everywhere equal, it is called a Simple magnetic shell; if it varies from point to point it may be conceived to be made up of a number of simple shells superposed and overlapping each other. It is therefore called a Complex magnetic shell. Let dS be an element of the surface of the shell at Q, and 4> the strength of the shell, then the potential at any point, P, due to the element of the shell, is d V = <J> — - dS cos €* r2 where e is the angle between the vector QP, or r and the normal drawn from the positive side of the shell. But if du> is the solid angle subtended by dS at the point P r2 da — dS cos e, whence dF = <&da>, and therefore in the case of a simple magnetic shell or, the potential due to a magnetic shell at any point is the product of its strength into the solid angle subtended by its edge at the given point*. 410.] The same result may be obtained in a different way by supposing the magnetic shell placed in any field of magnetic force, and determining the potential energy due to the position of the shell. If V is the potential at the element dS, then the energy due to this element is dy dy dy * (^ -r- +m~j- + n ~r) <*** \ da dy dz' or, the product of the strength of the shell into the part of the surface-integral of V due to the element dS of the shell. Hence, integrating with respect to all such elements, the energy due to the position of the shell in the field is equal to the product of the strength of the shell and the surf ace -integral of the magnetic induction taken over the surface of the shell. Since this surface-integral is the same for any two surfaces which * This theorem is due to Gauss, General Theory of Terrestrial Magnetism, § 38. VOL. II. D 34 MAGNETIC SOLENOIDS AND SHELLS. [411- have the same bounding- edge and do not include between them any centre of force, the action of the magnetic shell depends only on the form of its edge. Now suppose the field of force to be that due to a magnetic pole of strength m. We have seen (Art. 76, Cor.) that the surface- integral over a surface bounded by a given edge is the product of the strength of the pole and the solid angle subtended by the edge at the pole. Hence the energy due to the mutual action of the pole and the shell is and this (by Green's theorem. Art. 100) is equal to the product of the strength of the pole into the potential due to the shell at the pole. The potential due to the shell is therefore 4> co. 411.] If a magnetic pole m starts from a point on the negative surface of a magnetic shell, and travels along any path in space so as to come round the edge to a point close to where it started but on the positive side of the shell, the solid angle will vary continuously, and will increase by 4 TT during the process. The work done by the pole will be 4 TT 4> m, and the potential at any point on the positive side of the shell will exceed that at the neighbouring point on the negative side by 4 TT 4>. If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4 TT 4>, being positive when the positive side of the shell is inward. Hence such a shell exerts no action on any magnet placed either outside or inside the shell. 412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, the distribution of magnetism is called Lamellar. If <£ is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point xy z by a line drawn within the magnet, then the conditions of lamellar magnetization are ,_<Z<I> d<}> d(f> A = — =— , JD = -r— , L> = — T~ * dx dy dz The quantity, <J>, which thus completely determines the magnet ization at any point may be called the Potential of Magnetization. It must be carefully distinguished from the Magnetic Potential. 413.] A magnet which can be divided into complex magnetic shells is said to have a complex lamellar distribution of mag netism. The condition of such a distribution is that the lines of 415.] POTENTIAL DUE TO A LAMELLAE MAGNET. 35 magnetization must be such that a system of surfaces can be drawn cutting them at right angles. This condition is expressed by the well-known equation Aff__<lB} ^A_<IC ^_<U ^dy dz> ^dz dx' ^dx dy ' Forms of the Potentials of Solenoidal and Lamellar Magnets. 414.] The general expression for the scalar potential of a magnet where p denotes the potential at (#, y, z) due to a unit magnetic pole placed at f, TJ, £ or in other words, the reciprocal of the distance between (f, r;, Q, the point at which the potential is measured, and (#, y> z), the position of the element of the magnet to which it is due. This quantity may be integrated by parts, as in Arts. 96, 386. where I, m, n are the direction-cosines of the normal drawn out wards from dS, an element of the surface of the magnet. When the magnet is solenoidal the expression under the integral sign in the second term is zero for every point within the magnet, so that the triple integral is zero, and the scalar potential at any point, whether outside or inside the magnet, is given by the surface- integral in the first term. The scalar potential of a solenoidal magnet is therefore com pletely determined when the normal component of the magnet ization at every point of the surface is known, and it is independent of the form of the solenoids within the magnet. 415.] In the case of a lamellar magnet the magnetization is determined by c/>, the potential of magnetization, so that dcf) d<j> d$ •**• -— ~^ — j .£> = —7— , <-/ = — ; — • ax ay dz The expression for V may therefore be written = fff, JJJ \ dp . ' dx dx dy dy dz dz Integrating this expression by parts, we find D 2 36 MAGNETIC SOLENOIDS AND SHELLS. The second term is zero unless the point (f, r/, f) is included in the magnet, in which case it becomes 4 TT (<£) where (<£) is the value of <p at the point £, 77, f The surface-integral may be expressed in terms of rt the line drawn from (x, y, z] to (f, rj, f ), and 0 the angle which this line makes with the normal drawn outwards from dSt so that the potential may be written where the second term is of course zero when the point (f, TJ, f) is not included in the substance of the magnet. The potential, F, expressed by this equation, is continuous even at the surface of the magnet, where $ becomes suddenly zero, for if we write fit = and if £1L is the value of H at a point just within the surface, and 122 that at a point close to the first but outside the surface, fla = ^ + 477^), r2 = r,. The quantity H is not continuous at the surface of the magnet. The components of magnetic induction are related to 12 by the equations d& da da a= -- =— , 0= -- =-, c — -- -j- • dx dy dz 416.] In the case of a lamellar distribution of magnetism we may also simplify the vector-potential of magnetic induction. Its ^-component may be written By integration by parts we may put this in the form of the surface-integral or F . The other components of the vector-potential may be written down from these expressions by making the proper substitutions. On Solid Angles. 417.] We have already proved that at any point P the potential 4 1 8.] SOLID ANGLES. 37 due to a magnetic shell is equal to the solid angle subtended by the edge of the shell multiplied by the strength of the shell. As we shall have occasion to refer to solid angles in the theory of electric currents, we shall now explain how they may be measured. Definition. The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. This area is to be reckoned positive or negative according as it lies on the left or the right- hand of the path of the radius vector as seen from the given point. Let (£, r], f) be the given point, and let (#, y, z) be a point on the closed curve. The coordinates- x, y, z are functions of s, the length of the curve reckoned from a given point. They are periodic functions of s, recurring whenever s is increased by the whole length of the closed curve. We may calculate the solid angle o> directly from the definition thus. Using spherical coordinates with centre at (£, 77, Q, and putting x — f = r sin0cos$, y— rj = r sin 0 sin^, z — C=rcos0, we find the area of any curve on the sphere by integrating co = /(I— cos0) d$, or, using the rectangular coordinates, the integration being extended round the curve s. If the axis of z passes once through the closed curve the first term is 2 IT. If the axis of z does not pass through it this term is zero. 418.] This method of calculating a solid angle involves a choice of axes which is to some extent arbitrary, and it does not depend solely on the closed curve. Hence the following method, in which no surface is supposed to be constructed, may be stated for the sake of geometrical propriety. As the radius vector from the given point traces out the closed curve, let a plane passing through the given point roll on the closed curve so as to be a tangent plane at each point of the curve in succession. Let a line of unit-length be drawn from the given point perpendicular to this plane. As the plane rolls round the 38 MAGNETIC SOLENOIDS AND SHELLS. [4 1 9. closed curve the extremity of the perpendicular will trace a second closed curve. Let the length of the second closed curve be o-, then the solid angle subtended by the first closed curve is 00 = 27T — (7. This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circum ference of the polar curve, is numerically equal to the circumference of a great circle of the sphere. This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem. 419.] A closed curve s is given in space, and we have to find the solid angle subtended by s at a given point P. If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if cr is the path of the pole as it approaches the point P, the potential must be the result of a line-integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and a. When P is at an infinite distance, the solid angle is evidently zero. As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point ap proaches. As the point P moves from P to P' over the element do-, the element QQ' of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ' will sweep over an area on the spherical surface, which we may write da = Udsdcr. (I) To find FT let us suppose P fixed while the closed curve is moved parallel to itself through a distance da- equal to PPf but in the opposite direction. The relative motion of the point P will be the same as in the real case. 420.] GENERATION OF A SOLID ANGLE. 39 During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to Q Q' and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment d& which we are in search of. To determine the value of this solid angle, let 6 and tf be the angles which ds and dcr make with PQ respect ively, and let <£ be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds .dcr on a. plane per pendicular to PQ or r will be ds dcr sin Q sin 6' sin and since this is equal to r2 d<a, we find Fig. 3. Hence du> = II ds dcr = -g sin Q sin 6' sin </> ds dcr. n = — - sin 6 sin 0' sin <>. (2) (3) 420.] We may express the angles 6, 6', and $ in terms of and its differential coefficients with respect to s and o-, for cos0= -=-, /»/ cos<9'= •-=-, dcr and sin 6 sin 6' cos cp = r dsdcr (4) We thus find the following value for D2, (5) A third expression for II in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is d& and whose axis is r is J r* do) = J r* FT ds dcr. But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and dcr on the axis of #, y and zt as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of n, n = -^ -=— > -^— > -=— c— *i T\—y> <* -. l— *> -7— > dcr drj -j— > dcr T«' dx Ts* d_y_ 7 ^ ds dz ~ds" (6) 40 MAGNETIC SOLENOIDS AND SHELLS. [421. This expression gives the value of FT free from the ambiguity of sign introduced by equation (5). 421.] The value of o>, the solid angle subtended by the closed curve at the point P, may now be written a) = ndsdv-i-WQ, (7) where the integration with respect to s is to be extended completely round the closed curve, and that with respect to <r from A a fixed point on the curve to the point P. The constant <o0 is the value of the solid angle at the point A. It is zero if A is at an infinite distance from the closed curve. The value of o> at any point P is independent of the form of the curve between A and P provided that it does not pass through the magnetic shell itself. If the shell be supposed infinitely thin, and if P and Pf are two points close together, but P on the positive and P' on the negative surface of the shell, then the curves AP and AP/ must lie on opposite sides of the edge of the shell, so that PAP' is a line which with the infinitely short line PP forms a closed circuit embracing the edge. The value of o> at P exceeds that at P' by 47T, that is, by the surface of a sphere of radius unity. Hence, if a closed curve be drawn so as to pass once through the shell, or in other words, if it be linked once with the edge of the shell, the value of the integral I lUdsdv extended round both curves will be 47r. This integral therefore, considered as depending only on the closed curve s and the arbitrary curve AP, is an instance of a _ function of multiple values, since, if we pass from A to P along different paths the integral will have different values according to the number of times which the curve AP is twined round the curve s. If one form of the curve between A and P can be transformed into another by continuous motion without intersecting the curve s, the integral will have the same value for both curves, but if during the transformation it intersects the closed curve n times the values of the integral will differ by 47m. If s and a- are any two closed curves in space, then, if they are not linked together, the integral extended once round both is zero. If they are intertwined n times in the same direction, the value of the integral is 4iTn. It is possible, however, for two curves 422.] VECTOR- POTENTIAL OF A CLOSED CURVE. 41 to be intertwined alternately in opposite directions, so that they are inseparably linked together though the value of the integral is zero. See Fig. 4. It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while de scribing a closed curve in presence of a closed electric current, and indicating the geometrical connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibnitz, Euler and Vandermonde. We have now, how- Flg> 4> ever, some progress to report, chiefly due to Riemann, Helmholtz and Listing. 422.] Let us now investigate the result of integrating with respect to s round the closed curve. One of the terms of FT in equation (7) is f — x dri dz _ di) d A dz^ , . r3 da- ds ~~ da d£ W ds' If we now write for brevity ^ f 1 dx 7 „ f 1 dy .. TT f 1 dz F — I - -r- ds, G = I - -f- ds, R—\- ~ ds, (9) J r ds J r ds J r ds the integrals being taken once round the closed curve s, this term of FT may be written da- d£ds and the corresponding term of / n ds will be da- d£ Collecting all the terms of n, we may now write This quantity is evidently the rate of decrement of co, the magnetic potential, in passing along the curve a-, or in other words, it is the magnetic force in the direction of da: By assuming da- successively in the direction of the axes of x, y and z, we obtain for the values of the components of the magnetic force 42 MAGNETIC SOLENOIDS AND SHELLS. [423- do> _ dH dG Ot — ~~~ 7 f. — ~~j ~" T"T~ dt, d-r] d£ d<* _ dF dH dr] d£ d£ do> _ dG dF y =~ JT> — ,7 /• ~j (11) The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential o>, functions having a series of values, but are perfectly determinate for every point in space. The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction : Let a point Q travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When Q has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P. Potential Energy of a Magnetic Shell placed in a Magnetic Field. 423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength <£ placed in a magnetic field whose potential is T9 is rffidV d7 dY\ 70 x-tJJ ('is +*?+•*)** ^ where I, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell. Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we -+cf+^, where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell. If we now suppose that the magnetic field is that due to a 423.] POTENTIAL OF TWO CLOSED CURVES. 43 second magnetic shell whose strength is <£', the values of F, G, H will be where the integrations are extended once round the curve /, which forms the edge of this shell. Substituting these values in the expression for M we find , ff I fdx dx dy dy dz dz^ . Jf = —$$'// - (-J- -j-' + ir j' + -j--,,)dsds', (15) ^ JJ r ^ds ds ds ds ds ds' where the integration is extended once round s and once round /. This expression gives the potential energy due to the mutual action of the two shells, and is, as it ought to be, the same when s and / are interchanged. This expression with its sign reversed, when the strength of each shell is unity, is called the potential of the two closed curves s and /. It is a quantity of great importance in the theory of electric currents. If we write e for the angle between the directions of the elements ds and ds', the potential of s and / may be written (16) It is evidently a quantity of the dimension of a line. CHAPTER IV. INDUCED MAGNETIZATION. 424.] WE have hitherto considered the actual distribution of magnetization in a magnet as given explicitly among the data of the investigation. We have not made any assumption as to whether this magnetization is permanent or temporary, except in those parts of our reasoning in which we have supposed the magnet broken up into small portions, or small portions removed from the magnet in such a way as not to alter the magnetization of any part. We have now to consider the magnetization of bodies with respect to the mode in which it may be produced and changed. A bar of iron held parallel to the direction of the earth's magnetic force is found to become magnetic, with its poles turned the op posite way from those of the earth, or the same way as those of a compass needle in stable equilibrium. Any piece of soft iron placed in a magnetic field is found to exhibit magnetic properties. If it be placed in a part of the field where the magnetic force is great, as between the poles of a horse-shoe magnet, the magnetism of the iron becomes intense. If the iron is removed from the magnetic field, its magnetic properties are greatly weakened or disappear entirely. If the magnetic properties of the iron depend entirely on the magnetic force of the field in which it is placed, and vanish when it is removed from the field, it is called Soft iron. Iron which is soft in the magnetic sense is also soft in the literal sense. It is easy to bend it and give it a permanent set, and difficult to break it. Iron which retains its magnetic properties when removed from the magnetic field is called Hard iron. Such iron does not take 425.] SOFT AND HARD STEEL. 45 up the magnetic state so readily as soft iron. The operation of hammering-, or any other kind of vibration, allows hard iron under the influence of magnetic force to assume the magnetic state more readily, and to part with it more readily when the magnetizing force is removed. Iron which is magnetically hard is also more stiff to bend and more apt to break. The processes of hammering, rolling, wire-drawing, and sudden cooling tend to harden iron, and that of annealing tends to soften it. The magnetic as well as the mechanical differences between steel of hard and soft temper are much greater than those between hard and soft iron. Soft steel is almost as easily magnetized and de magnetized as iron, while the hardest steel is the best material for magnets which we wish to be permanent. Cast iron, though it contains more carbon than steel, is not so retentive of magnetization. If a magnet could be constructed so that the distribution of its magnetization is not altered by any magnetic force brought to act upon it, it might be called a rigidly magnetized body. The only known body which fulfils this condition is a conducting circuit round which a constant electric current is made to flow. Such a circuit exhibits magnetic properties, and may therefore be called an electromagnet, but these magnetic properties are not affected by the other magnetic forces in the field. We shall return to this subject in Part IV. All actual magnets, whether made of hardened steel or of load stone, are found to be affected by any magnetic force which is brought to bear upon them. It is convenient, for scientific purposes, to make a distinction between the permanent and the temporary magnetization, defining the permanent magnetization as that which exists independently of the magnetic force, and the temporary magnetization as that which depends on this force. We must observe, however, that this distinction is not founded on a knowledge of the intimate nature of magnetizable substances : it is only the expression of an hypothesis introduced for the sake of bringing calculation to bear on the phenomena. We shall return to the physical theory of magnetization in Chapter VI. 425.] At present we shall investigate the temporary magnet ization on the assumption that the magnetization of any particle of the substance depends solely on the magnetic force acting on 46 INDUCED MAGNETIZATION. [425. that particle. This magnetic force may arise partly from external causes, and partly from the temporary magnetization of neigh bouring particles. A body thus magnetized in virtue of the action of magnetic force, is said to be magnetized by induction, and the magnetization is said to be induced by the magnetizing force. The magnetization induced by a given magnetizing force differs in different substances. It is greatest in the purest and softest iron, in which the ratio of the magnetization to the magnetic force may reach the value 32, or even 45 *. Other substances, such as the metals nickel and cobalt, are capable of an inferior degree of magnetization, and all substances when subjected to a sufficiently strong magnetic force, are found to give indications of polarity. When the magnetization is in the same direction as the magnetic force, as in iron, nickel, cobalt, &c., the substance is called Para magnetic, Ferromagnetic, or more simply Magnetic. When the induced magnetization is in the direction opposite to the magnetic force, as in bismuth, &c., the substance is said to be Diamagnetic. In all these substances the ratio of the magnetization to the magnetic force which produces it is exceedingly small, being only about — 4 o (H) o Q m the case °f bismuth, which is the most highly diamagnetic substance known. In crystallized, strained, and organized substances the direction of the magnetization does not always coincide with that of the magnetic force which produces it. The relation between the com ponents of magnetization, referred to axes fixed in the body, and those of the magnetic force, may be expressed by a system of three linear equations. Of the nine coefficients involved in these equa tions we shall shew that only six are independent. The phenomena of bodies of this kind are classed under the name of Magnecrystallic phenomena. When placed in a field of magnetic force, crystals tend to set themselves so that the axis of greatest paramagnetic, or of least diamagnetic, induction is parallel to the lines of magnetic force. See Art. 435. In soft iron, the direction of the magnetization coincides with that of the magnetic force at the point, and for small values of the magnetic force the magnetization is nearly proportional to it. * Thaten, Nova Ada, Reg. Soc. Sc., Upsal., 1863. 427.] PROBLEM OF INDUCED MAGNETIZATION. 47 As the magnetic force increases, however, the magnetization in creases more slowly, and it would appear from experiments described in Chap. VI, that there is a limiting value of the magnetization, beyond which it cannot pass, whatever be the value of the magnetic force. In the following outline of the theory of induced magnetism, we shall begin by supposing the magnetization proportional to the magnetic force, and in the same line with it. Definition of the Coefficient of Induced Magnetization. 426.] Let $ be the magnetic force, defined as in Art. 398, at any point of the body, and let 3 be the magnetization at that point, then the ratio of 3 to § is called the Coefficient of Induced Magnetization. Denoting this coefficient by K, the fundamental equation of induced magnetism is The coefficient K is positive for iron and paramagnetic substances, and negative for bismuth and diamagnetic substances. It reaches the value 32 in iron, and it is said to be large in the case of nickel and cobalt, but in all other cases it is a very small quantity, not greater than 0.00001. The force <£) arises partly from the action of magnets external to the body magnetized by induction, and partly from the induced magnetization of the body itself, Both parts satisfy the condition of having a potential. 427.] Let V be the potential due to magnetism external to the body, let X2 be that due to the induced magnetization, then if U is the actual potential due to both causes u= r+a. (2) Let the components of the magnetic force «£), resolved in the directions of x, y, z, be a, /3, y, and let those of the magnetization 3 be A, B, C, then by equation (1), A = K a, *=K/3, (3) C = Ky. Multiplying these equations by dx, dy, dz respectively, and adding, we find Adx + Bdy+Cdz = K( 48 INDUCED MAGNETIZATION. [427. But since a, (3 and y are derived from the potential U, we may write the second member —KdU. Hence, if /c is constant throughout the substance, the first member must also be a complete differential of a function of #, y and z, which we shall call $, and the equation becomes i A d(b „ d(b d(b where A = -f- , B = ~- , C — — - . (5) ax dy dz The magnetization is therefore lamellar, as defined in Art. 412. It was shewn in Art. 386 that if p is the volume-density of free magnetism, (dA dB dC. P- — (-J- +-J- + T-}' x## dy dz ' which becomes in virtue of equations (3), /da d(3 dy\ \lx dy dz' But, by Art. 77, da dj3 dy _ dx dy dz ~ Hence (l+47r*)p = 0, whence p = 0 (6) throughout the substance, and the magnetization is therefore sole- noidal as well as lamellar. See Art. 407. There is therefore no free magnetism except on the bounding surface of the body. If v be the normal drawn inwards from the surface, the magnetic surface-density is d^> (-^ a- = j-- (7) dv The potential II due to this magnetization at any point may therefore be found from the surface-integral «-//= dS. (8) The value of £1 will be finite and continuous everywhere, and will satisfy Laplace's equation at every point both within and without the surface. If we distinguish by an accent the value of H outside the surface, and if v be the normal drawn outwards, we have at the surface Of =0.1 (9) 428.] POISSON'S METHOD. 49 da da' — + ^ = -4™, by Art. 78, = 4*8.^). , ., -..: : . dU = -47rKj;> bF(4)» fdV d^ , = -47rK(^+^),by(2). We may therefore write the surface-condition Hence the determination of the magnetism induced in a homo geneous isotropic body, bounded by a surface S, and acted upon by external magnetic forces whose potential is V9 may be reduced to the following mathematical problem. We must find two functions H and H' satisfying the following conditions : Within the surface S9 XI must be finite and continuous, and must satisfy Laplace's equation. Outside the surface S, Of must be finite and continuous, it must vanish at an infinite distance, and must satisfy Laplace's equation. At every point of the surface itself, H = Of, and the derivatives of H, Of and V with respect to the normal must satisfy equation (10). _ This method of treating the problem of induced magnetism is due to Poisson. The quantity k which he uses in his memoirs is not the same as *, but is related to it as follows : 47TK(£-l)+3/&= 0. (11) The coefficient K which we have here used was introduced by J. Neumann. 428.] The problem of induced magnetism may be treated in a different manner by introducing the quantity which we have called, with Faraday, the Magnetic Induction. The relation between 23, the magnetic induction, «£j, the magnetic force, and 3> the magnetization, is expressed by the equation 53 = $ + 471 3. (12) The equation which expresses the induced magnetization in terms of the magnetic force is 3 = K$. (13) VOL. IT. E 50 INDUCED MAGNETIZATION. [428. Hence, eliminating- 3, we find $ = (1+47TK)£ (14) as the relation between the magnetic induction and the magnetic force in substances whose magnetization is induced by magnetic force. In the most general case K may be a function, not only of the position of the point in the substance, but of the direction of the vector «jp, but in the case which we are now considering K is a numerical quantity. If we next write ^ = I + 4 -n K} (15) we may define /x as the ratio of the magnetic induction to the magnetic force, and we may call this ratio the magnetic inductive capacity of the substance, thus distinguishing it from K, the co efficient of induced magnetization. If we write U for the total magnetic potential compounded of T7, the potential due to external causes, and 12 for that due to the induced magnetization, we may express a, b, c, the components of magnetic induction, and a, (3, y, the components of magnetic force, as follows : dU ~} a = "0 = -M' dU e = ™=-*&'j The components #, d, c satisfy the solenoidal condition £+!+£=«• (17> Hence, the potential U must satisfy Laplace's equation at every point where /ot is constant, that is, at every point within the homogeneous substance, or in empty space. At the surface itself, if v is a normal drawn towards the magnetic substance, and v one drawn outwards, and if the symbols of quan tities outside the substance are distinguished by accents, the con dition of continuity of the magnetic induction is dv , dv dv , dv ,, dv , dv a-j- +6-j- +0-=- +a'-j- +V -r- +<f -j- = 0; (19) dx dy dz dx dy dz 429.] FARADAY'S THEORY OF MAGNETIC INDUCTION. 51 or, by equations (16), fjf, the coefficient of induction outside the magnet, will be unity unless the surrounding medium be magnetic or diamagnetic. If we substitute for U its value in terms of V and H, and for fj> its value in terms of K, we obtain the same equation (10) as we arrived at by Poisson's method. The problem of induced magnetism, when considered with respect to the relation between magnetic induction and magnetic force, corresponds exactly with the problem of the conduction of electric currents through heterogeneous media, as given in Art. 309. The magnetic force is derived from the magnetic potential, pre cisely as the electric force is derived from the electric potential. The magnetic induction is a quantity of the nature of a flux, and satisfies the same conditions of continuity as the electric current does. In isotropic media the magnetic induction depends on the mag netic force in a manner which exactly corresponds with that in which the electric current depends on the electromotive force. The specific magnetic inductive capacity in the one problem corre sponds to the specific conductivity in the other. Hence Thomson, in his Theory of Induced Magnetism (Reprint, 1872, p. 484), has called this quantity the permeability of the medium. We are now prepared to consider the theory of induced magnetism from what I conceive to be Faraday's point of view. When magnetic force acts on any medium, whether magnetic or diamagnetic, or neutral, it produces within it a phenomenon called Magnetic Induction. Magnetic induction is a directed quantity of the nature of a flux, and it satisfies the same conditions of continuity as electric currents and other fluxes do. In isotropic media the magnetic force and the magnetic induction are in the same direction, and the magnetic induction is the product of the magnetic force into a quantity called the coefficient of induction, which we have expressed by p. In empty space the coefficient of induction is unity. In bodies capable of induced magnetization the coefficient of induction is 1 + 4 TT K = /x, where K is the quantity already defined as the co efficient of induced magnetization. 429.] Let p, [k be the values of p on opposite sides of a surface E 52 INDUCED MAGNETIZATION. [4^9- separating two media, then if F, V are the potentials in the two media, the magnetic forces towards the surface in the two media dV , dV' are -7- and -3-7- • Av dv The quantities of magnetic induction through the element of dV dV surface dS are u-^-dS and u? -^-j-dS in the two media respect- r dv dv ively reckoned towards dS. Since the total flux towards dS is zero, dV ,dV But by the theory of the potential near a surface of density o-, dV dV + 4.47r(r:r= o. dv dv Hence -7- (l — —A + 4 TT or = 0. c?i> V ju, / If K! is the ratio of the superficial magnetization to the normal force in the first medium whose coefficient is jot, we have 4 77 KI = Hence K± will be positive or negative according as /ut is greater or less than //. If we put ju = 4 TT /c + 1 and p' '= 4 77 / + 1 , "47T/+1 In this expression K and K' are the coefficients of induced mag netization of the first and second medium deduced from experiments made in air, and KX is the coefficient of induced magnetization of the first medium when surrounded by the second medium. If K is greater than K, then /q is negative, or the apparent magnetization of the first medium is in the opposite direction from the magnetizing force. Thus, if a vessel containing a weak aqueous solution of a para magnetic salt of iron is suspended in a stronger solution of the same salt, and acted on by a magnet, the vessel moves as if it were magnetized in the opposite direction from that in which a magnet would set itself if suspended in the same place. This may be explained by the hypothesis that the solution in the vessel is really magnetized in the same direction as the mag netic force, but that the solution which surrounds the vessel is magnetized more strongly in the same direction. Hence the vessel is like a weak magnet placed between two strong ones all mag- 43°-] POISSON'S THEORY OP MAGNETIC INDUCTION. 53 netized in the same direction, so that opposite poles are in contact. The north pole of the weak magnet points in the same direction as those of the strong- ones, but since it is in contact with the south pole of a stronger magnet, there is an excess of south magnetism in the neighbourhood of its north pole, which causes the small magnet to appear oppositely magnetized. In some substances, however, the apparent magnetization is negative even when they are suspended in what is called a vacuum. If we assume K = 0 for a vacuum, it will be negative for these substances. No substance, however, has been discovered for which K has a negative value numerically greater than — — , and therefore for all known substances /x is positive. Substances for which K is negative, and therefore p less than unity, are called Diamagnetic substances. Those for which K is positive, and ^ greater than unity, are called Paramagnetic, Ferro magnetic, or simply magnetic, substances. We shall consider the physical theory of the diamagnetic and paramagnetic properties when we come to electromagnetism, Arts. 831-845. 430.] The mathematical theory of magnetic induction was first given by Poisson *. The physical hypothesis on which he founded his theory was that of two magnetic fluids, an hypothesis which has the same mathematical advantages and physical difficulties as the theory of two electric fluids. In order, however, to explain the fact that, though a piece of soft iron can be magnetized by induction, it cannot be charged with unequal quantities of the two kinds of magnetism, he supposes that the substance in general is a non-conductor of these fluids, and that only certain small portions of the substance contain the fluids under circumstances in which they are free to obey the forces which act on them. These small magnetic elements of the substance contain each pre cisely equal quantities of the two fluids, and within each element the fluids move with perfect freedom, but the fluids can never pass from one magnetic element to another. The problem therefore is of the same kind as that relating to a number of small conductors of electricity disseminated through a dielectric insulating medium. The conductors may be of any form provided they are small and do not touch each other. If they are elongated bodies all turned in the same general * Memoires de I'lnstitut, 1824. 54 INDUCED MAGNETIZATION. [43O. direction, or if they are crowded more in one direction than another, the medium, as Poisson himself shews, will not be isotropic. Poisson therefore, to avoid useless intricacy, examines the case in which each magnetic element is spherical, and the elements are dissem inated without regard to axes. He supposes that the whole volume of all the magnetic elements in unit of volume of the substance is k. We have already considered in Art. 314 the electric conductivity of a medium in which small spheres of another medium are dis tributed. If the conductivity of the medium is ^ , and that of the spheres ju2, we have found that the conductivity of the composite system is 2) P = f*l-j Putting fa = 1 and /ot2 = oc, this becomes _ 1 + 2/fc This quantity ju is the electric conductivity of a medium con sisting of perfectly conducting spheres disseminated through a medium of conductivity unity, the aggregate volume of the spheres in unit of volume being k. The symbol ^ also represents the coefficient of magnetic induction of a medium, consisting of spheres for which the permeability is infinite, disseminated through a medium for which it is unity. The symbol k, which we shall call Poisson's Magnetic Coefficient, represents the ratio of the volume of the magnetic elements to the whole volume of the substance. The symbol K is known as Neumann's Coefficient of Magnet ization by Induction. It is more convenient than Poisson's. The symbol ^ we shall call the Coefficient of Magnetic Induction. Its advantage is that it facilitates the transformation of magnetic problems into problems relating to electricity and heat. The relations of these three symbols are as follows : 47TK 3 * = 3* 477 If we put K = 32, the value given by Thalen's* experiments on * Recherches sur les Proprietes Magnetiques dufer, Nova Ada, Upsal, 1863. 430.] POISSON'S THEORY OF MAGNETIC INDUCTION. 55 soft iron, we find k = |f|-. This, according to Poisson's theory, is the ratio of the volume of the magnetic molecules to the whole volume of the iron. It is impossible to pack a space with equal spheres so that the ratio of their volume to the whole space shall be so nearly unity, and it is exceedingly improbable that so large a proportion of the volume of iron is occupied by solid molecules whatever be their form. This is one reason why we must abandon Poisson's hypothesis. Others will be stated in Chapter VI. Of course the value of Poisson's mathematical investigations remains unimpaired, as they do not rest on his hypothesis, but on the experimental fact of induced magnetization. CHAPTER V. PARTICULAR PROBLEMS IN MAGNETIC INDUCTION. A Hollow Spherical Shell. 431.] THE first example of the complete solution of a problem in magnetic induction was that given by Poisson for the case of a hollow spherical shell acted on by any magnetic forces whatever. For simplicity we shall suppose the origin of the magnetic forces to be in the space outside the shell. If V denotes the potential due to the external magnetic system, we may expand V in a series of solid harmonics of the form 7= CQ80 + C1S1r + to. + CiSiiA, (1) where r is the distance from the centre of the shell, #< is a surface harmonic of order i, and Ci is a coefficient. This series will be convergent provided r is less than the distance of the nearest magnet of the system which produces this potential. Hence, for the hollow spherical shell and the space within it, this expansion is convergent. Let the external radius of the shell be a2 and the inner radius alf and let the potential due to its induced magnetism be H. The form of the function H will in general be different in the hollow space, in the substance of the shell, and in the space beyond. If we expand these functions in harmonic series, then, confining our attention to those terms which involve the surface harmonic Si9 we shall find that if Q^ is that which corresponds to the hollow space within the shell, the expansion of Q^ must be in positive har monics of the form Al St r*, because the potential must not become infinite within the sphere whose radius is a^. In the substance of the shell, where r± lies between aL and a2, the series may contain both positive and negative powers of /*, of the form Outside the shell, where r is greater than a2, since the series HOLLOW SPHERICAL SHELL. 57 must be convergent however great r may be, we must have only negative powers of /, of the form The conditions which must be satisfied by the function 12, are : It must be (1) finite, and (2) continuous, and (3) must vanish at an infinite distance, and it must (4) everywhere satisfy Laplace's equation. On account of (1) Bl = 0. On account of (2) when r = a^ (4-4,H2i+1-52=0, (2) and when r = «2, (^2-J3)^2i+1 + ^2-^3 = 0. (3) On account of (3) Az = 0, and the condition (4) is satisfied everywhere, since the functions are harmonic. But, besides these, there are other conditions to be satisfied at the inner and outer surface in virtue of equation (10), Art. 427. At the inner surface where r = alt , d£l9 d&, dV ,.. <1+4*«>V-ifr+4"'* = <)' and at the outer surface where r = a2, d dV ,KN 0. From these conditions we obtain the equations iCia12i+l = <), (6) «22»+1-(^+l)^2)+(^+l)^3+47r^^22i+1=0^ (7) and if we put we find / /, 2» + l\ 4 = -(4™)^ + l)(l-Q) }NtClt (9) [I a 2t+l^-j 2^+l+477K(^+l)(l-(^) )J^Ci, (10) (11) «12i+1)^Ci. (12) These quantities being substituted in the harmonic expansions give the part of the potential due to the magnetization of the shell. The quantity Ni is always positive, since 1 -f 4 ir K can never be negative. Hence A1 is always negative, or in other words, the 58 MAGNETIC PEOBLEMS. [432. action of the magnetized shell on a point within it is always op posed to that of the external magnetic force whether the shell he paramagnetic or diamagnetic. The actual value of the resultant potential within the shell is or (l + 4wjc)(2i+ l^NiCtS.r. (13) 432.] When K is a large number, as it is in the case of soft iron, then, unless the shell is very thin, the magnetic force within it is hut a small fraction of the external force. In this way Sir W. Thomson has rendered his marine galvano meter independent of external magnetic force hy enclosing it in a tube of soft iron. 433.] The case of greatest practical importance is that in which i = 1. In this case (14) 9(l+47TK)+2(477K)2(l-0') = -477*13+ 8w«(l — (^) )UViQ, !> (15) L X dr> —I £3= 4 7TK(3 + 8 7TK)(#23 — «13)^V1 Ci. The magnetic force within the hollow shell is in this case uniform and equal to 9(1+477*) If we wish to determine K by measuring the magnetic force within a hollow shell and comparing it with the external magnetic force, the best value of the thickness of the shell may be found from the equation 1_ - 2 (4 TT K)2 The magnetic forc"e inside the shell is then half of its value outside. Since, in the case of iron, K is a number between 20 and 30, the thickness of the shell ought to be about the hundredth part of its radius. This method is applicable only when the value of K is large. When it is very small the value of A^ becomes insensible, since it depends on the square of K. 434-1 SPHERICAL SHELL. 59 For a nearly solid sphere with a very small spherical hollow, . 2(4ir«)« 1J 4 77 K The whole of this investigation might have been deduced directly from that of conduction through a spherical shell, as given in Art. 312, by putting ^ = (1 -f 47TK)/£2 in the expressions there given, remembering that A^ and A2 in the problem of conduction are equi valent to C1 + A1 and C1 + A2 in the problem of magnetic induction. 434.] The corresponding solution in two dimensions is graphically represented in Fig. XV, at the end of this volume. The lines of induction, which at a distance from the centre of the figure are nearly horizontal, are represented as disturbed by a cylindric rod magnetized transversely and placed in its position of stable equi librium. The lines which cut this system at right angles represent the equipotential surfaces, one of which is a cylinder. The large dotted circle represents the section of a cylinder of a paramagnetic substance, and the dotted horizontal straight lines within it, which are continuous with the external lines of induction, represent the lines of induction within the substance. The dotted vertical lines represent the internal equipotential surfaces, and are continuous with the external system. It will be observed that the lines of induction are drawn nearer together within the substance, and the equipotential surfaces are separated farther apart by the paramag netic cylinder, which, in the language of Faraday, conducts the lines of induction better than the surrounding medium. If we consider the system of vertical lines as lines of induction, and the horizontal system as equipotential surfaces, we have, in the first place, the case of a cylinder magnetized transversely and placed in the position of unstable equilibrium among the lines of force, which it causes to diverge. In the second place, considering the large dotted circle as the section of a diamagnetic cylinder, the dotted straight lines within it, together with the lines external to it, represent the effect of a diamagnetic substance in separating the lines of induction and drawing together the equipotential surfaces, such a substance being a worse conductor of magnetic induction than the surrounding medium. 60 MAGNETIC PROBLEMS. [435- Case of a Sphere in which the Coefficients of Magnetization are Different in Different Directions. 435.] Let a, (B, y be the components of magnetic force, and A, £, C those of the magnetization at any point, then the most general linear relation between these quantities is given by the equations A = ^0+^3/3+ q2y, \ £ = q9a+r2p+fly, { (1) C = p2a+q1h2 + 7-3 y, ) where the coefficients r,jo, q are the nine coefficients of magnet ization. Let us now suppose that these are the conditions of magnet ization within a sphere of radius a, and that the magnetization at every point of the substance is uniform and in the same direction, having the components A, 13, C. Let us also suppose that the external magnetizing force is also uniform and parallel to one direction, and has for its components X, Y, Z. The value of V is therefore and that of &' the potential of the magnetization outside the sphere is (3) The value of H, the potential of the magnetization within the sphere, is 4-n- (4) o The actual potential within the sphere is V-\- £1, so that we shall have for the components of the magnetic force within the sphere a = X — ^TtA, \ 0 = 7-J.ir-B, (5) y =Z- Hence +i*r1)^+ twftjjB + iir& C = &J+ r2Y+frZ, (6) +(1 + Solving these equations, we find A = r/^+K '' (7) 43^.] CRYSTALLINE SPHERE. 61 where I/ /•/ = r± + ^ TT ( rB rl — p2 q2 4 r-± r2 — ;-A^i)> &c., where D is the determinant of the coefficients on the right side of equations (6), and D' that of the coefficients on the left. The new system of coefficients _p' ', /_, / will be symmetrical only when the system p, q, r is symmetrical, that is, when the co efficients of the form p are equal to the corresponding ones of the form q. 436.] The moment of the couple tending to turn the sphere about the axis of x from y towards z is f. n Y\\ (Q\ — Jr2 ))* \ / If we make X = 0, Y = Fcos 0, Y = Fsin 0, this corresponds to a magnetic force F in the plane of yz, and inclined to y at an angle 0. If we now turn the sphere while this force remains constant the work done in turning the sphere will T27T be / LdQ in each complete revolution. But this is equal to 0 Hence, in order that the revolving sphere may not become an inexhaustible source of energy, j»1/= fa', and similarly j»./= q2 and These conditions shew that in the original equations the coeffi cient of B in the third equation is equal to that of C in the second, and so on. Hence, the system of equations is symmetrical, and the equations become when referred to the principal axes of mag netization, TI A = rr*"i ' C = (11) The moment of the couple tending to turn the sphere round the axis of x is 62 MAGNETIC PROBLEMS. [437- In most cases the differences between the coefficients of magnet ization in different directions are very small, so that we may put This is the force tending to turn a crystalline sphere about the axis of oo from y towards z. It always tends to place the axis of greatest magnetic coefficient (or least diamagnetic coefficient) parallel to the line of magnetic force. The corresponding case in two dimensions is represented in Fig. XVI. If we suppose the upper side of the figure to be towards the north, the figure represents the lines of force and equipotential surfaces as disturbed by a transversely magnetized cylinder placed with the north side eastwards. The resultant force tends to turn the cylinder from east to north. The large dotted circle represents a section of a cylinder of a crystalline substance which has a larger coefficient of induction along an axis from north-east to south-west than along an axis from north-west to south-east. The dotted lines within the circle represent the lines of induction and the equipotential surfaces, which in this case are not at right angles to each other. The resultant force on the cylinder is evidently to turn it from east to north. 437.] The case of an ellipsoid placed in a field of uniform and parallel magnetic force has been solved in a very ingenious manner by Poisson. If V is the potential at the point (as, y, z\ due to the gravitation dV of a body of any form of uniform density p, then — -=- is the potential of the magnetism of the same body if uniformly mag netized in the direction of x with the intensity I = p. For the value of -- =— 8# at any point is the excess of the value clx of V3 the potential of the body, above V, the value of the potential when the body is moved — §x in the direction of x. If we supposed the body shifted through the distance — 8#, and its density changed from p to — p (that is to say, made of repulsive dV instead of attractive matter,) then — y-8# would be the potential due to the two bodies. Now consider any elementary portion of the body containing a volume b v. Its quantity is pbv, and corresponding to it there is 437-] ELLIPSOID. 63 an element of the shifted body whose quantity is — pbv at a distance — 8#. The effect of these two elements is equivalent to that of a magnet of strength pbr and length 8#. The intensity of magnetization is found hy dividing the magnetic moment of an element by its volume. The result is p 8#. dV Hence -=- 8# is the magnetic potential of the body magnetized rl V with the intensity p bx in the direction of x, and — is that of ax the body magnetized with intensity p. This potential may be also considered in another light. The body was shifted through the distance — 8# and made of density —p. Throughout that part of space common to the body in its two positions the density is zero, for, as far as attraction is con cerned, the two equal and opposite densities annihilate each other. There remains therefore a shell of positive matter on one side and of negative matter on the other, and we may regard the resultant potential as due to these. The thickness of the shell at a point where the normal drawn outwards makes an angle e with the axis of a? is 8 a? cos e and its density is p. The surface-density is therefore dV p bx cos 6, and, in the case in which the potential is — , the surface-density is p cos e. In this way we can find the magnetic potential of any body uniformly magnetized parallel to a given direction. Now if this uniform magnetization is due to magnetic induction, the mag netizing force at all points within the body must also be uniform and parallel. This force consists of two parts, one due to external causes, and the other due to the magnetization of the body. If therefore the external magnetic force is uniform and parallel, the magnetic force due to the magnetization must also be uniform and parallel for all points within the body. Hence, in order that this method may lead to a solution of the clV problem of magnetic induction, -=- must be a linear function of doc the coordinates x, y> z within the body, and therefore V must be a quadratic function of the coordinates. Now the only cases with which we are acquainted in which V is a quadratic function of the coordinates within the body are those in which the body is bounded by a complete surface of the second degree, and the only case in which such a body is of finite dimen- 64 MAGNETIC PROBLEMS. [437- sions is when it is an ellipsoid. We shall therefore apply the method to the case of an ellipsoid. be the equation of the ellipsoid, and let 4>0 denote the definite integral f '0 Then if we make dfr the value of the potential within the ellipsoid will be 70 = - (L x2 + My* + Nz*} + const. (4) 2 If the ellipsoid is magnetized with uniform intensity / in a direction making angles whose cosines are I, m, n with the axes of #, y, z, so that the components of magnetization are A = II, B = Im, C = In, the potential due to this magnetization within the ellipsoid will be a = —I(Llx + Mmy + Nnz). (5) If the external magnetizing force is «§, and if its components are a, ft, y, its potential will be r=Xx + Yy + Zz. (6) The components of the actual magnetizing force at any point within the body are therefore X-AL, Y-BM, Z-CN. (7) The most general relations between the magnetization and the magnetizing force are given by three linear equations, involving nine coefficients. It is necessary, however, in order to fulfil the condition of the conservation of energy, that in the case of magnetic induction three of these should be equal respectively to other three, so that we should have A = K,(X-AL) + Kfs(Y-BM) + K'2(Z-CN}, B = K\ (X-AL) + K2i(Y-BM) + K\(Z-CN], (8) C = K'2(X-AL) + K\(Y-BM) + Kz(Z-CN}. From these equations we may determine J, B and C in terms of X, Y} Z, and this will give the most general solution of the problem. The potential outside the ellipsoid will then be that due to the * See Thomson and Tait's Natural Philosophy, § 522. 438.] ELLIPSOID. 65 magnetization of the ellipsoid together with that due to the external magnetic force. 438.] The only case of practical importance is that in which K\ = K2 = K3 = 0. (9) We have then If the ellipsoid flattened form, A — "i X 1 (10) and is of the planetary or : (ID 7? K2 T ' JJ V C = has two b= c 1+K2M~ K3 g l+K3N axes equal, a (12) l-e M = N = 2 , (±-^ sin-'*- ™) . \ e* e2 ' J If the ellipsoid is of the ovary or elongated form a — b = A/1 — e*c; (13) In the case of a sphere, when e = 0, — .«. -^- ^ j In the case of a very flattened planetoid L becomes in the limit equal to 4 TT, and M and JV become 7r2 - • In the case of a very elongated ovoid L and M approximate to the value 2 TT, while N approximates to the form a2,, 2c , and vanishes when e = 1 . It appears from these results that — (1) When K, the coefficient of magnetization, is very small, whether positive or negative, the induced magnetization is nearly equal to the magnetizing force multiplied by K, and is almost independent of the form of the body. VOL. II. F 66 MAGNETIC PROBLEMS. (2) When K is a large positive quantity, the magnetization depends principally on the form of the body,, and is almost independent of the precise value of /c, except in the case of a longitudinal force acting on an ovoid so elongated that NK is a small' quantity though K is large. (3) If the value of K could be negative and equal to — we should have an infinite value of the magnetization in the case of a magnetizing force acting normally to a flat plate or disk. The absurdity of this result confirms what we said in Art. 428. Hence, experiments to determine the value of K may be made on bodies of any form provided K is very small, as it is in the case of all diamagnetic bodies, and all magnetic bodies except iron, nickel, and cobalt. If, however, as in the case of iron, K is a large number, experi ments made on spheres or flattened figures are not suitable to determine K ; for instance, in the case of a sphere the ratio of the magnetization to the magnetizing force is as 1 to 4.22 if K = 30, as it is in some kinds of iron, and if K were infinite the ratio would be as 1 to 4.19, so that a very small error in the determination of the magnetization would introduce a very large one in the value of K. But if we make use of a piece of iron in the form of a very elongated ovoid, then, as long as NK is of moderate value com pared with unity, we may deduce the value of K from a determination of the magnetization, and the smaller the value of JV the more accurate will be the value of K. In fact, if NK be made small enough, a small error in the value of N itself will not introduce much error, so that we may use any elongated body, such as a wire or long rod, instead of an ovoid. We must remember, however, that it is only when the product JV~/c is small compared with unity that this substitution is allowable. In fact the distribution of magnetism on a long cylinder with flat ends does not resemble that on a long ovoid, for the free mag netism is very much concentrated towards the ends of the cylinder, whereas it varies directly as the distance from the equator in the case of the ovoid. The distributi6n of electricity on a cylinder, however, is really comparable with that on an ovoid, as we have already seen, Art. 152. 439-] CYLINDER. 67 These results also enable us to understand why the magnetic moment of a permanent magnet can be made so much greater when the magnet has an elongated form. If we were to magnetize a disk with intensity / in a direction normal to its surface, and then leave it to itself, the interior particles would experience a constant demagnetizing force equal to 4 TT I, and this, if not sufficient of itself to destroy part of the magnetization, would soon do so if aided by vibrations or changes of temperature. If we were to magnetize a cylinder transversely the demagnet izing force would be only 2 TT I. If the magnet were a sphere the demagnetizing force would be £*/. In a disk magnetized transversely the demagnetizing force is a 7T2 - 1) and in an elongated ovoid magnetized longitudinally it 0 a2 2c is least of all, being 4 TT -^ 7 log --- G a Hence an elongated magnet is less likely to lose its magnetism than a short thick one. The moment of the force acting on an ellipsoid having different magnetic coefficients for the three axes which tends to turn it about the axis of #, is Hence, if *2 and K3 are small, this force will depend principally on the crystalline quality of the body and not on its shape, pro vided its dimensions are not very unequal, but if K2 and *3 are considerable, as in the case of iron, the force will depend principally on the shape of the body, and it will turn so as to set its longer axis parallel to the lines of force. If a sufficiently strong, yet uniform, field of magnetic force could be obtained, an elongated isotropic diamagnetic body would also set itself with its longest dimension parallel to the lines of magnetic force. 439.] The question of the distribution of the magnetization of an ellipsoid of revolution under the action of any magnetic forces has been investigated by J. Neumann*. Kirchhofff has extended the method to the case of a cylinder of infinite length acted on by any force. * Crelle, bd. xxxvii (1848). t Crelle, bd. xlviii (1854). F 2 68 MAGNETIC PROBLEMS. [439- Green, in the 17th section of his Essay, has given an invest igation of the distribution of magnetism in a cylinder of finite length acted on by a uniform external force parallel to its axis. Though some of the steps of this investigation are not very rigorous, it is probable that the result represents roughly the actual magnetization in this most important case. It certainly expresses very fairly the transition from the case of a cylinder for which K is a large number to that in which it is very small, but it fails entirely in the case in which K is negative, as in diamagnetic substances. Green finds that the linear density of free magnetism at a distance x from the middle of a cylinder whose radius is a and whose length is 2 I, is px ea +e where p is a numerical quantity to be found from the equation 0.231863 — 2 \ogep + 2p = - -— The following are a few of the corresponding values of p and K. K K oo 0 336.4 0.01 62.02 0.02 48.416 0.03 29.475 0.04 20.185 0.05 14.794 0.06 11.802 0.07 9.137 0.08 7.517 0.09 6.319 0.10 0.1427 1.00 0.0002 10.00 0.0000 oo negative imaginary. When the length of the cylinder is great compared with its radius, the whole quantity of free magnetism on either side of the middle of the cylinder is, as it ought to be, M= v2aKX. Of this \p M is on the flat end of the cylinder, and the distance of the centre of gravity of the whole quantity M from the end a of the cylinder is - P When K is very small p is large, and nearly the whole free magnetism is on the ends of the cylinder. As K increases p diminishes, and the free magnetism is spread over a greater distance 44O-] FORCE ON PARA- AND DIA-MAGNETIC BODIES. 69 from the ends. When K is infinite the free magnetism at any point of the cylinder is simply proportional to its distance from the middle point, the distribution being similar to that of free electricity on a conductor in a field of uniform force. 440.] In all substances except iron, nickel, and cobalt, the co efficient of magnetization is so small that the induced magnetization of the body produces only a very slight alteration of the forces in the magnetic field. We may therefore assume, as a first approx imation, that the actual magnetic force within the body is the same as if the body had not been there. The superficial magnetization dV dV of the body is therefore, as a first approximation, K -j- , where -=- is the rate of increase of the magnetic potential due to the external magnet along a normal to the surface drawn inwards. If we now calculate the potential due to this superficial distribution, we may use it in proceeding to a second approximation. To find the mechanical energy due to the distribution of mag netism on this first approximation we must find the surface-integral taken over the whole surface of the body. Now we have shewn in Art. 100 that this is equal to the volume-integral /*/*/* ~^r~T7 ^ j 77" 2 taken through the whole space occupied by the body, or, if R is the resultant magnetic force, E = - Now since the work done by the magnetic force on the body during a displacement 8# is Xbos where X is the mechanical force in the direction of SB, and since / = constant, which shews that the force acting on the body is as if every part of it tended to move from places where R2 is less to places where it is greater with a force which on every unit of volume is rf.JP K dx ' 70 MAGNETIC PEOBLEMS. If K is negative, as in diamagnetic bodies, this force is, as Faraday first shewed, from stronger to weaker parts of the magnetic field. Most of the actions observed in the case of diamagnetic bodies depend on this property. Skip's Magnetism. 441.] Almost every part of magnetic science finds its use in navigation. The directive action of the earth's magnetism on the compass needle is the only method of ascertaining the ship's course when the sun and stars are hid. The declination of the needle from the true meridian seemed at first to be a hindrance to the appli cation of the compass to navigation, but after this difficulty had been overcome by the construction of magnetic charts it appeared likely that the declination itsylf would assist the mariner in de termining his ship's place. The greatest difficulty in navigation had always been to ascertain the longitude ; but since the declination is different at different points on the same parallel of latitude, an observation of the de clination together with a knowledge of the latitude would enable the mariner to find his position on the magnetic chart. But in recent times iron is so largely used in the construction of ships that it has become impossible to use the compass at all without taking into account the action of the ship, as a magnetic body, on the needle. To determine the distribution of magnetism in a mass of iron of any form under the influence of the earth's magnetic force, even though not subjected to mechanical strain or other disturb ances, is, as we have seen, a very difficult problem. In this case, however, the problem is simplified by the following considerations. The compass is supposed to be placed with its centre at a fixed point of the ship, and so far from any iron that the magnetism of the needle does not induce any perceptible magnetism in the ship. The size of the compass needle is supposed so small that we may regard the magnetic force at any point of the needle as the same. The iron of the ship is supposed to be of two kinds only. (1) Hard iron, magnetized in a constant manner. (2) Soft iron, the magnetization of which is induced by the earth or other magnets. In strictness we must admit that the hardest iron is not only SHIP'S MAGNETISM. 71 capable of induction but that it may lose part of its so-called permanent magnetization in various ways. The softest iron is capable of retaining what is called residual magnetization. The actual properties of iron cannot be accurately represented by supposing it compounded of the hard iron and the soft iron above defined. But it has been found that when a ship is acted on only by the earth's magnetic force, and not subjected to any extraordinary stress of weather, the supposition that the magnetism of the ship is due partly to permanent magnetization and partly to induction leads to sufficiently accurate results when applied to the correction of the compass. The equations on which the theory of the variation of the compass is founded were given by Poisson in the fifth volume of the Memoires de I'Institut, p. 533 (1824). The only assumption relative to induced magnetism which is involved in these equations is, that if a magnetic force X due to external magnetism produces in the iron of the ship an induced magnetization, and if this induced magnetization exerts on the compass needle a disturbing force whose components are JT', Y'9 Z', then, if the external magnetic force is altered in a given ratio, the components of the disturbing force will be altered in the same ratio. It is true that when the magnetic force acting on iron is very great the induced magnetization is no longer proportional to the external magnetic force, but this want of proportionality is quite insensible for magnetic forces of the magnitude of those due to the earth's action. Hence, in practice we may assume that if a magnetic force whose value is unity produces through the intervention of the iron of the ship a disturbing force at the compass needle whose com ponents are a in the direction of #, d in that of y, and g in that of z, the components of the disturbing force due to a force X in the direction of x will be aX, dX, and gX. If therefore we assume axes fixed in the ship, so that x is towards the ship's head, y to the starboard side, and z towards the keel, and if X, Y, Z represent the components of the earth's magnetic force in these directions, and X', Y', Z' the components of the combined magnetic force of the earth and ship on the compass needle, X' = X+aX+bY+c Z+P, ) Y' = Y+dX+eY+fZ+Q, (1) 72 MAGNETIC PROBLEMS. [44 1- In these equations #, #, c, d, e,f, g, h, Jc are nine constant co efficients depending on the amount, the arrangement, and the capacity for induction of the soft iron of the ship. P, Q, and E are constant quantities depending on the permanent magnetization of the ship. It is evident that these equations are sufficiently general if magnetic induction is a linear function of magnetic force, for they are neither more nor less than the most general expression of a vector as a linear function of another vector. It may also be shewn that they are not too general, for, by a proper arrangement of iron, any one of the coefficients may be made to vary independently of the others. Thus, a long thin rod of iron under the action of a longitudinal magnetic force acquires poles, the strength of each of which is numerically equal to the cross section of the rod multiplied by the magnetizing force and by the coefficient of induced magnet ization. A magnetic force transverse to the rod produces a much feebler magnetization, the effect of which is almost insensible at a distance of a few diameters. If a long iron rod be placed fore and aft with one end at a distance x from the compass needle, measured towards the ship's head, then, if the section of the rod is A, and its coefficient of magnetization K, the strength of the pole will be A K X, and, if A = — — , the force exerted by this pole on the compass needle will be aX. The rod may be supposed so long that the effect of the other pole on the compass may be neglected. We have thus obtained the means of giving any required value to the coefficient a. If we place another rod of section B with one extremity at the same point, distant x from the compass toward the head of the vessel, and extending to starboard to such a distance that the distant pole produces no sensible effect on the compass, the dis turbing force due to this rod will be in the direction of x, and B K.Y bx* equal to — x - , or if B = , the force will be b Y. X2 K ' This rod therefore introduces the coefficient b. A third rod extending downwards from the same point will introduce the coefficient <?. The coefficients d, e,f may be produced by three rods extending to head, to starboard, and downward from a point to starboard of 44i.] SHIP'S MAGNETISM. 73 the compass, and g, h, k by three rods in parallel directions from a point below the compass. Hence each of the nine coefficients can be separately varied by means of iron rods properly placed. The quantities P, Q, R are simply the components of the force on the compass arising from the permanent magnetization of the ship together with that part of the induced magnetization which is due to the action of this permanent magnetization. A complete discussion of the equations (1), and of the relation between the true magnetic course of the ship and the course as indicated by the compass, is given by Mr. Archibald Smith in the Admiralty Manual of the Deviation of the Compass. A valuable graphic method of investigating the problem is there given. Taking a fixed point as origin, a line is drawn from this point representing in direction and magnitude the horizontal part of the actual magnetic force on the compass-needle. As the ship is swung round so as to bring her head into different azimuths in succession, the extremity of this line describes a curve, each point of which corresponds to a particular azimuth. Such a curve, by means of which the direction and magnitude of the force on the compass is given in terms of the magnetic course of the ship, is called a Dygogram. There are two varieties of the Dygogram. In the first, the curve is traced on a plane fixed in space as the ship turns round. In the second kind, the curve is traced on a plane fixed with respect to the ship. The dygogram of the first kind is the Lima9on of Pascal, that of the second kind is an ellipse. For the construction and use of these curves, and for many theorems as interesting to the mathe matician as they are important to the navigator, the reader is referred to the Admiralty Manual of the Deviation of the Compass. CHAPTER VI. WEBER'S THEORY OF INDUCED MAGNETISM. 442.] WE have seen that Poisson supposes the magnetization of iron to consist in a separation of the magnetic fluids within each magnetic molecule. If we wish to avoid the assumption of the existence of magnetic fluids, we may state the same theory in another form, hy saying that each molecule of the iron, when the magnetizing force acts on it, becomes a magnet. Weber's theory differs from this in assuming that the molecules of the iron are always magnets, even before the application of the magnetizing force, but that in ordinary iron the magnetic axes of the molecules are turned indifferently in every direction, so that the iron as a whole exhibits no magnetic properties. When a magnetic force acts on the iron it tends to turn the axes of the molecules all in one direction, and so to cause the iron, as a whole, to become a magnet. If the axes of all the molecules were set parallel to each other, the iron would exhibit the greatest intensity of magnetization of which it is capable. Hence Weber's theory implies the existence of a limiting intensity of magnetization, and the experimental evidence that such a limit exists is therefore necessary to the theory. Experiments shewing an approach to a limiting value of magnetization have been made by Joule * and by J. Miiller f. The experiments of Beetz J on electrotype iron deposited under the action of magnetic force furnish the most complete evidence of this limit, — A silver wire was varnished, and a very narrow line on the * Annals of Electricity, iv. p. 131, 1839 ; Phil Mag. [4] ii. p. 316. t Pogg., Ann. Ixxix. p. 337, 1850. + Pogg. cxi. 1860. 443-] THE MOLECULES OF IRON ARE MAGNETS. 75 metal was laid bare by making1 a fine longitudinal scratch on the varnish. The wire was then immersed in a solution of a salt of iron, and placed in a magnetic field with the scratch in the direction of a line of magnetic force. By making the wire the cathode of an electric current through the solution, iron was deposited on the narrow exposed surface of the wire, molecule by molecule. The filament of iron thus formed was then examined magnetically. Its magnetic moment was found to be very great for so small a mass of iron, and when a powerful magnetizing force was made to act in the same direction the increase of temporary magnetization was found to be very small, and the permanent magnetization was not altered. A magnetizing force in the reverse direction at once reduced the filament to the condition of iron magnetized in the ordinary way. Weber's theory, which supposes that in this case the magnetizing force placed the axis of each molecule in the same direction during the instant of its deposition, agrees very well with what is observed. Beetz found that when the electrolysis is continued under the action of the magnetizing force the intensity of magnetization of the subsequently deposited iron diminishes. The axes of the molecules are probably deflected from the line of magnetizing force when they are being laid down side by side with the mole cules already deposited, so that an approximation to parallelism. can be obtained only in the case of a very thin filament of iron. If, as Weber supposes, the molecules of iron are already magnets, any magnetic force sufficient to render their axes parallel as they are electrolytically deposited will be sufficient to produce the highest intensity of magnetization in the deposited filament. If, on the other hand, the molecules of iron are not magnets, but are only capable of magnetization, the magnetization of the deposited filament will depend on the magnetizing force in the same way in which that of soft iron in general depends on it. The experiments of Beetz leave no room for the latter hy pothesis. 443.] We shall now assume, with Weber, that in every unit of volume of the iron there are n magnetic molecules, and that the magnetic moment of each is m. If the axes of all the molecules were placed parallel to one another, the magnetic moment of the unit of volume would be M = n m, 76 WEBER'S THEORY OF INDUCED MAGNETISM. [443. and this would be the greatest intensity of magnetization of which the iron is capable. In the unmagnetized state of ordinary iron Weber supposes the axes of its molecules to be placed indifferently in all directions. To express this, we may suppose a sphere to be described, and a radius drawn from the centre parallel to the direction of the axis of each of the n molecules. The distribution of the extremities of these radii will express that of the axes of the molecules. In the case of ordinary iron these n points are equally distributed over every part of the surface of the sphere, so that the number of molecules whose axes make an angle less than a with the axis of x is n . - (I - cos a), and the number of molecules whose axes make angles with that of ^, between a and a-f da is therefore n . j - sin a a a. 2t This is the arrangement of the molecules in a piece of iron which has never been magnetized. Let us now suppose that a magnetic force X is made to act on the iron in the direction of the axis of a?, and let us consider a molecule whose axis was originally inclined a to the axis of so. If this molecule is perfectly free to turn, it will place itself with its axis parallel to the axis of a?, and if all the molecules did so, the very slightest magnetizing force would be found sufficient to develope the very highest degree of magnetization. This, how ever, is not the case. The molecules do not turn with their axes parallel to a?, and this is either because each molecule is acted on by a force tending to preserve it in its original direction, or because an equivalent effect is produced by the mutual action of the entire system of molecules. Weber adopts the former of these suppositions as the simplest, and supposes that each molecule, when deflected, tends to return to its original position with a force which is the same as that which a magnetic force D, acting in the original direction of its axis, would produce. The position which the axis actually assumes is therefore in the direction of the resultant of X and D. Let APB represent a section of a sphere whose radius represents, on a certain scale, the force D. 443-] DEFLEXION OF AXES OF MOLECULES. 77 Let the radius OP be parallel to the axis of a particular molecule in its original position. Let SO represent on the same scale the magnetizing force X which is supposed to act from 8 towards 0. Then, if the molecule is acted on by the force X in the direction SO, and by a force D in a direction parallel to OP, the original direction of its axis, its axis will set itself in the direction SP, that of the resultant of X and D. Since the axes of the molecules are originally in all directions, P may be at any point of the sphere indifferently. In Fig. 5, in which X is less than D, SP, the final position of the axis, may be in any direction whatever, but not indifferently, for more of the molecules will have their axes turned towards A than towards JS. In Fig. 6, in which X is greater than D, the axes of the molecules will be all confined within the cone STT' touching the sphere. Fig. 5. Hence there are two different cases according as X is less or greater than D. Let a = AOP, the original inclination of the axis of a molecule to the axis of x. 0 = ASP, the inclination of the axis when deflected by the force X. (3 = SPO, the angle of deflexion. SO = X, the magnetizing force. OP = D, the force tending towards the original position. SP = R, the resultant of X and D. m = magnetic moment of the molecule. Then the moment of the statical couple due to X, tending to diminish the angle 0, is mL = mX sin#, and the moment of the couple due to D, tending to increase 6, is mL — 78 WEBER'S THEORY OF INDUCED MAGNETISM. [443. Equating these values, and remembering that /3 = a — 0, we find J)sina tan0 = - -- (1) X +D cos a to determine the direction of the axis after deflexion. We have next to find the intensity of magnetization produced in the mass by the force X, and for this purpose we must resolve the magnetic moment of every molecule in the direction of #, and add all these resolved parts. The resolved part of the moment of a molecule in the direction of x is m cos 0. The number of molecules whose original inclinations lay between a and a -{-da is % . -smaaa. 2 We have therefore to integrate /= f* — cos 6 tin a da, (2) JQ 2 'remembering that 0 is a function of a. We may express both 9 and a in terms of JR, and the expression to be integrated becomes ' (3) the general integral of which is In the first case, that in which X is less than D, the limits of integration are R = D + X and R = D— X. In the second case, in which X is greater than D, the limits are R = X+ D and R = X-D. When X is less than D, I = | ~X. (5) 2 When X is equal to D, I = -mn. (6) 3 1 712 When X is greater than D, I — mn(\ -- — ) ; (7) * o J\. I and when X becomes infinite / = mn. (8) According to this form of the theory, which is that adopted by Weber *, as the magnetizing force increases from 0 to D, the * There is some mistake in the formula given by Weber (Trans. Acad. Sax. i. p. 572 (1852), or Pogg., Ann. Ixxxvii. p. 167 (1852)) as the result of this integration, the steps of which are not given by him. His formula is 444-] L1MIT OF MAGNETIZATION. 79 magnetization increases in the same proportion. When the mag netizing force attains the value D, the magnetization is two-thirds of its limiting value. When the magnetizing force is further increased, the magnetization, instead of increasing indefinitely, tends towards a finite limit. D 2D 3D 4D Fig. 7. The law of magnetization is expressed in Fig. 7, where the mag netizing force is reckoned from 0 towards the right and the mag netization is expressed by the vertical ordinates. Weber's own experiments give results in satisfactory accordance with this law. It is probable, however, that the value of D is not the same for all the molecules of the same piece of iron, so that the transition from the straight line from 0 to E to the curve beyond E may not be so abrupt as is here represented. 444.] The theory in this form gives no account of the residual magnetization which is found to exist after the magnetizing force is removed. I have therefore thought it desirable to examine the results of making a further assumption relating to the conditions under which the position of equilibrium of a molecule may be permanently altered. Let us suppose that the axis" of a magnetic molecule, if deflected through any angle /3 less than /30, will return to its original position when the deflecting force is removed, but that if the deflexion j3 exceeds ^0, then, when the deflecting force is removed, the axis will not return to its original position, but will be per manently deflected through an angle /3 — j30, which may be called the permanent set of the molecule. This assumption with respect to the law of molecular deflexion is not to be regarded as founded on any exact knowledge of the intimate structure of bodies, but is adopted, in our ignorance of the true state of the case, as an assistance to the imagination in following out the speculation suggested by Weber. Let L = Dsin /30, (9) 80 WEBER'S THEORY OF INDUCED MAGNETISM. [444. then, if the moment of the couple acting on a molecule is less than ml/, there will be no permanent deflexion, but if it exceeds mL there will be a permanent change of the position of equilibrium. To trace the results of this supposition, describe a sphere whose centre is 0 and radius OL = L. As long as X is less than L everything will be the same as in the case already considered, but as soon as X exceeds L it will begin to produce a permanent deflexion of some of the molecules. Let us take the case of Fig. 8, in which X is greater than L but less than D. Through S as vertex draw a double cone touching the sphere L. Let this cone meet the sphere D in P and Q. Then if the axis of a molecule in its original position lies between OA and OP, or between OB and OQ, it will be deflected through an angle less than /30, and will not be permanently deflected. But if Fig. 8. Fig. 9. the axis of the molecule lies originally between OP and OQ, then a couple whose moment is greater than L will act upon it and will deflect it into the position SP, and when the force X ceases to act it will not resume its original direction, but will be per manently set in the direction OP. Let us put L = Xsin00 when 0 = PSA or QSB, then all those molecules whose axes, on the former hypotheses, would have values of 6 between 00 and TT — 00 will be made to have the value 00 during the action of the force X. During the action of the force X, therefore, those molecules whose axes when deflected lie within either sheet of the double cone whose semivertical angle is 00 will be arranged as in the former case, but all those whose axes on the former theory would lie outside of these sheets will be permanently deflected, so that their axes will form a dense fringe round that sheet of the cone which lies towards A. 445-] MODIFIED THEORY. 81 As X increases, the number of molecules belonging to the cone about B continually diminishes, and when X becomes equal to D all the molecules have been wrenched out of their former positions of equilibrium, and have been forced into the fringe of the cone round A, so that when X becomes greater than D all the molecules form part of the cone round A or of its fringe. When the force X is removed, then in the case in which X is less than L everything returns to its primitive state. When X is between L and D then there is a cone round A whose angle AOP = 00 + /30, and another cone round B whose angle BOQ = 00-/30. Within these cones the axes of the molecules are distributed uniformly. But all the molecules, the original direction of whose axes lay outside of both these cones, have been wrenched from their primitive positions and form a fringe round the cone about A. If X is greater than D, then the cone round B is completely dispersed, and all the molecules which formed it are converted into the fringe round A, and are inclined at the angle 00-f-/30. 445.] Treating this case in the same way as before, we find for the intensity of the temporary magnetization during the action of the force X, which is supposed to act on iron which has never before been magnetized, When X is less than L, I = - M -_- • 3 J-f When X is equal to It, I = - M -=j • When X is between L and 2), When X is equal to D, ' When X is greater than D> When X is infinite, I = M. When X is less than L the magnetization follows the former law, and is proportional to the magnetizing force. As soon as X exceeds L the magnetization assumes a more rapid rate of increase VOL. n. G 82 WEBER'S THEORY OF INDUCED MAGNETISM. [445. on account of the molecules beginning to be transferred from the one cone to the other. This rapid increase, however, soon conies to an end as the number of molecules forming the negative cone diminishes, and at last the magnetization reaches the limiting value M. If we were to assume that the values of L and of D are different for different molecules, we should obtain a result in which the different stages of magnetization are not so distinctly marked. The residual magnetization, /', produced by the magnetizing force X, and observed after the force has been removed, is as follows : When X is less than I/, No residual magnetization. When X is between L and D, When X is equal to D, T2 2 When X is greater than D, '-J When X is infinite, If we make M = 1000, L = 3, .# = 5, we find the following values of the temporary and the residual magnetization : — Magnetizing Temporary Residual Force. Magnetization. Magnetization. x i r 000 1 133 0 2 267 0 3 400 0 4 729 280 5 837 410 6 864 485 7 882 537 8 897 574 oo 1000 810 446.] TEMPORARY AND RESIDUAL MAGNETIZATION. 83 These results are laid down in Fig. 10. 10 I 2 3 4 5 6 7 8 J JHcufn.etizin.tp jforce Fig. 10. The curve of temporary magnetization is at first a straight line from X = 0 to X = L. It then rises more rapidly till X = 1), and as X increases it approaches its horizontal asymptote. The curve of residual magnetization begins when X = _Z/, and approaches an asymptote at a distance = .8lJf. It must be remembered that the residual magnetism thus found corresponds to the case in which, when the external force is removed, there is no demagnetizing force arising from the distribution of magnetism in the body itself. The calculations are therefore applicable only to very elongated bodies magnetized longitudinally. In the case of short, thick bodies the residual magnetism will be diminished by the reaction of the free magnetism in the same way as if an external reversed magnetizing force were made to act upon it. 446.] The scientific value of a theory of this kind, in which we make so many assumptions, and introduce so many adjustable constants, cannot be estimated merely by its numerical agreement with certain sets of experiments. If it has any value it is because it enables us to form a mental image of what takes place in a piece of iron during magnetization. To test the theory, we shall apply it to the case in which a piece of iron, after being subjected to a magnetizing force XQ> is again subjected to a magnetizing force X1. If the new force X± acts in the same direction in which X0 acted, which we shall call the positive direction, then, if X± is less than X^9 it will produce no permanent set of the molecules, and when X1 is removed the residual magnetization will be the same as G 2 84: WEBER'S THEORY OF INDUCED MAGNETISM. [446. that produced by X0 . If Xl is greater than X0 , then it will produce exactly the same effect as if X0 had not acted. But let us suppose Xl to act in the negative direction, and let us suppose XQ = L cosec 00, and Xl = — I/cosec01. As X1 increases numerically, 0: diminishes. The first molecules on which X1 will produce a permanent deflexion are those which form the fringe of the cone round A, and these have an inclination when undeflected of 00 + J30 . As soon as 61 — /30 becomes less than 00-f~/30 the process of de magnetization will commence. Since, at this instant, ^ = 00-f 2^30, X13 the force required to begin the demagnetization, is less than XQ, the force which produced the magnetization. If the value of D and of L were the same for all the molecules, the slightest increase of X1 would wrench the whole of the fringe of molecules whose axes have the inclination 00 + /30 into a position in which their axes are inclined 01 + )30 to the negative axis OB. Though the demagnetization does not take place in a manner so sudden as this, it takes place so rapidly as to afford some confirmation of this mode of explaining the process. Let us now suppose that by giving a proper value to the reverse force Xj we have exactly demagnetized the piece of iron. The axes of the molecules will not now be arranged indiffer ently in all directions, as in a piece of iron which has never been magnetized, but will form three groups. (1) Within a cone of semiangle 01— /30 surrounding the positive pole, the axes of the molecules remain in their primitive positions. (2) The same is the case within a cone of semiangle 00— /30 surrounding the negative pole. (3) The directions of the axes of all the other molecules form a conical sheet surrounding the negative pole, and are at an inclination 0l + /30 . When X0 is greater than D the second group is absent. When •Xj_ is greater than I) the first group is also absent. The state of the iron, therefore, though apparently demagnetized, is in a different state from that of a piece of iron which has never been magnetized. To shew this, let us consider the effect of a magnetizing force X2 acting in either the positive or the negative direction. The first permanent effect of such a force will be on the third group of molecules, whose axes make angles = 01 + /30 with the negative axis. 447-1 MAGNETISM AND TORSION, 85 If the force X2 acts in the negative direction it will begin to produce a permanent effect as soon as 02 + /30 becomes less than ^i + A)5 that is, as soon as X2 becomes greater than XL. But if X2 acts in the positive direction it will begin to remagnetize the iron as soon as 02 — {3 becomes less than Oi + P0, that is, when 02 = Ol -j- 2/30, or while X2 is still much less than X±. It appears therefore from our hypothesis that — When a piece of iron is magnetized by means of a force X0i its magnetism cannot be increased without the application of a force greater than X0. A reverse force, less than Jf0, is sufficient to diminish its magnetization. If the iron is exactly demagnetized by a reversed force X19 then it cannot be magnetized in the reversed direction without the application of a force greater than X1} but a positive force less than Xx is sufficient to begin to remagnetize the iron in its original direction. These results are consistent with what has been actually observed by Ritchie*. Jacobi f, Marianini J, and Joule §. A very complete account of the relations of the magnetization of iron and steel to magnetic forces and to mechanical strains is given by Wiedemann in his Galvanismus. By a detailed com parison of the effects of magnetization with those of torsion, he shews that the ideas of elasticity and plasticity which we derive from experiments on the temporary and permanent torsion of wires can be applied with equal propriety to the temporary and permanent magnetization of iron and steel. 447.] Matteucci || found that the extension of a hard iron bar during the action of the magnetizing force increases its temporary magnetism. This has been confirmed by Wertheim. In the case of soft bars the magnetism is diminished by extension. The permanent magnetism of a bar increases when it is extended, and diminishes when it is compressed. Hence, if a piece of iron is first magnetized in one direction, and then extended in another direction, the direction of magnet ization will tend to approach the direction of extension. If it be compressed, the direction of magnetization will tend to become normal to the direction of compression. This explains the result of an experiment of Wiedemann's. A * Phil. Mag., 1833. t Pog., Ann., 1834. J Ann. de Chimie d de Physique, 1846. § Phil. Trans., 1855, p. 287. || Ann. de Chimie et de Physique, 1858. 86 WEBER S THEORY OF INDUCED MAGNETISM. [448. current was passed downward through a vertical wire. If, either during the passage of the current or after it has ceased, the wire be twisted in the direction of a right-handed screw, the lower end becomes a north pole. Fi. Here the downward current magnetizes every part of the wire in a tangential direction, as indicated by the letters NS. The twisting of the wire in the direction of a right-handed screw causes the portion ABCD to be extended along the diagonal AC and compressed along the diagonal BD. The direction of magnet ization therefore tends to approach AC and to recede from BD, and thus the lower end becomes a north pole and the upper end a south pole. Effect of Magnetization on the Dimensions of the Magnet. 448.] Joule *, in 1842, found that an iron bar becomes length ened when it is rendered magnetic by an electric current in a coil which surrounds it. He afterwards f shewed, by placing the bar in water within a glass tube, that the volume of the iron is not augmented by this magnetization, and concluded that its transverse dimensions were contracted. Finally, he passed an electric current through the axis of an iron tube, and back outside the tube, so as to make the tube into a closed magnetic solenoid, the magnetization being at right angles to the axis of the tube. The length of the axis of the tube was found in this case to be shortened. He found that an iron rod under longitudinal pressure is also elongated when it is magnetized. When, however, the rod is under considerable longitudinal tension, the effect of magnetization is to shorten it. * Sturgeon's Annals of Electricity, vol. viii. p. 219. t Phil. Mag., 1847. 448-] CHANGE OF FORM. 87 This was the case with a wire of a quarter of an inch diameter when the tension exceeded 600 pounds weight. In the case of a hard steel wire the effect of the magnetizing force was in every case to shorten the wire, whether the wire was under tension or pressure. The change of length lasted only as long as the magnetizing force was in action, no alteration of length was observed due to the permanent magnetization of the steel. Joule found the elongation of iron wires to be nearly proportional to the square of the actual magnetization, so that the first effect of a demagnetizing current was to shorten the wire. On the other hand, he found that the shortening effect on wires under tension, and on steel, varied as the product of the magnet ization and the magnetizing current. Wiedemann found that if a vertical wire is magnetized with its north end uppermost, and if a current is then passed downwards through the wire, the lower end of the wire, if free, twists in the direction of the hands of a watch as seen from above, or, in other words, the wire becomes twisted like a right-handed screw. In this case the magnetization due to the action of the current on the previously existing magnetization is in the direction of a left-handed screw round the wire. Hence the twisting would indicate that when the iron is magnetized it contracts in the direction of magnetization and expands in directions at right angles to the magnetization. This, however, peems not to agree with Joule's results. For further developments of the theory of magnetization, see Arts. 832-845. CHAPTER VII. MAGNETIC MEASUREMENTS. 449.] THE principal magnetic measurements are the determination of the magnetic axis and magnetic moment of a magnet, and that of the direction and intensity of the magnetic force at a given place. Since these measurements are made near the surface of the earth, the magnets are always acted on by gravity as well as by terrestrial magnetism, and since the magnets are made of steel their mag netism is partly permanent and partly induced. The permanent magnetism is altered by changes of temperature, by strong in duction, and by violent blows ; the induced magnetism varies with every variation of the external magnetic force. The most convenient way of observing the force acting on a magnet is by making the magnet free to turn about a vertical axis. In ordinary compasses this is done by balancing the magnet on a vertical pivot. The finer the point of the pivot the smaller is the moment of the friction which interferes with the action of the magnetic force. For more refined observations the magnet is suspended by a thread composed of a silk fibre without twist, either single, or doubled on itself a sufficient number of times, and so formed into a thread of parallel fibres, each of which supports as nearly as possible an equal part of the weight. The force of torsion of such a thread is much less than that of a metal wire of equal strength, and it may be calculated in terms of the ob served azimuth of the magnet, which is not the case with the force arising from the friction of a pivot. The suspension fibre can be raised or lowered by turning a horizontal screw which works in a fixed nut. The fibre is wound round the thread of the screw, so that when the screw is turned the suspension fibre always hangs in the same vertical line. 450.] SUSPENSION". 89 The suspension fibre carries a small horizontal divided circle called the Torsion-circle, and a stirrup with an index, which can be placed so that the index coincides with any given division of the torsion circle. The stirrup is so shaped that the magnet bar can be fitted into it with its axis horizontal, and with any one of its four sides uppermost. To ascertain the zero of torsion a non-magnetic body of the same weight as the magnet is placed in the stirrup, and the position of the torsion circle when in equilibrium ascertained. The magnet itself is a piece of hard-tempered steel. According to Gauss and Weber its length ought to be at least eight times its greatest transverse dimension. This is neces sary when permanence of the direc tion of the magnetic axis within the magnet is the most important con sideration. Where promptness of motion is required the magnet should be shorter, and it may even be ad visable in observing sudden altera tions in magnetic force to use a bar magnetized transversely and sus pended with its longest dimension vertical *. 450.1 The magnet is provided with an arrangement for ascertaining its angular position. For ordinary pur poses its ends are pointed, and a divided circle is placed below the Fig. 13. ends, by which their positions are read oif by an eye placed in a plane through the suspension thread and the point of the needle. For more accurate observations a plane mirror is fixed to the magnet, so that the normal to the mirror coincides as nearly as possible with the axis of magnetization. This is the method adopted by Gauss and Weber. Another method is to attach to one end of the magnet a lens and to the other end a scale engraved on glass, the distance of the lens * Joule, Proc. Phil. Soc., Manchester, Nov. 29, 1864. 90 MAGNETIC MEASUREMENTS. [45O. from the scale being1 equal to tlie principal focal length of the lens. The straight line joining the zero of the scale with the optical centre of the lens ought to coincide as nearly as possible with the magnetic axis. As these optical methods of ascertaining the angular position of suspended apparatus are of great importance in many physical researches, we shall here consider once for all their mathematical theory. Theory of the Mirror Method. We shall suppose that the apparatus whose angular position is to be determined is capable of revolving about a vertical axis. This axis is in general a fibre or wire by which it is suspended. The mirror should be truly plane, so that a scale of millimetres may be seen distinctly by reflexion at a distance of several metres from the mirror. The normal through the middle of the mirror should pass through the axis of suspension, and should be accurately horizontal. We shall refer to this normal as the line of collimation of the ap paratus. Having roughly ascertained the mean direction of the line of collimation during the experiments which are to be made, a tele scope is erected at a convenient distance in front of the mirror, and a little above the level of the mirror. The telescope is capable of motion in a vertical plane, it is directed towards the suspension fibre just above the mirror, and a fixed mark is erected in the line of vision, at a horizontal distance from the object glass equal to twice the distance of the mirror from the object glass. The apparatus should, if possible, be so arranged that this mark is on a wall or other fixed object. In order to see the mark and the suspension fibre at the same time through the telescope, a cap may be placed over the object glass having a slit along a vertical diameter. This should be removed for the other observations. The telescope is then adjusted so that the mark is seen distinctly to coincide with the vertical wire at the focus of the telescope. A plumb-line is then adjusted so as to pass close in front of the optical centre of the object glass and to hang below the telescope. Below the telescope and just behind the plumb-line a scale of equal parts is placed so as to be bisected at right angles by the plane through the mark, the suspension-fibre, and the plumb-lino. The sum of the heights of the scale and the 450.] THE MIRROR METHOD. 91 object glass should be equal to twice the height of the mirror from the floor. The telescope being now directed towards the mirror will see in it the reflexion of the scale. If the part of the scale where the plumb-line crosses it appears to coincide with the vertical wire of the telescope, then the line of collimation of the mirror coincides with the plane through the mark and the optical centre of the object glass. If the vertical wire coincides with any other division of the scale, the angular position of the line of collimation is to be found as follows : — Let the plane of the paper be horizontal, and let the various points be projected on this plane. Let 0 be the centre of the object glass of the telescope, P the fixed mark, P and the vertical wire of the telescope are conjugate foci with respect to the object glass. Let M be the point where OP cuts the plane of the mirror. Let MN be the normal to the mirror ; then OMN = 6 is the angle which the line of collimation makes with the fixed plane. Let MS be a line in the plane of OM and MN, such that NMS = OMN, then S will be the part of the scale which will be seen by reflexion to coincide with the vertical wire of the telescope. Now, since X X xx ---'V Fig. 14. MN is horizontal, the projected angles OMN and NMS in the figure are equal, and QMS =20. Hence OS = OMtan.20. We have therefore to measure OM in terms of the divisions of the scale ; then, if s0 is the division of the scale which coincides with the plumb-line, and s the observed division, whence 6 may be found. In measuring OM we must remember that if the mirror is of glass, silvered at the back, the virtual image of the reflecting surface is at a distance behind the front surface 92 MAGNETIC MEASUREMENTS. [450. of the glass = — , where t is the thickness of the glass, and //, is the index of refraction. We must also remember that if the line of suspension does not pass through the point of reflexion, the position of M will alter with 0. Hence, when it is possible, it is advisable to make the centre of the mirror coincide with the line of suspension. It is also advisable, especially when large angular motions have to be observed, to make the scale in the form of a concave cylindric surface, whose axis is the line of suspension. The angles are then observed at once in circular measure without reference to a table of tangents. The scale should be carefully adjusted, so that the axis of the cylinder coincides with the suspension fibre. The numbers on the scale should always run from the one end to the other in the same direction so as to avoid negative readings. Fig. 1 5 Fig. 15. represents the middle portion of a scale to be used with a mirror and an inverting telescope. This method of observation is the best when the motions are slow. The observer sits at the telescope and sees the image of the scale moving to right or to left past the vertical wire of the telescope. With a clock beside him he can note the instant at which a given division of the scale passes the wire, or the division of the scale which is passing at a given tick of the clock, and he can also record the extreme limits of each oscillation. When the motion is more rapid it becomes impossible to read the divisions of the scale except at the instants of rest at the extremities of an oscillation. A conspicuous mark may be placed at a known division of the scale, and the instant of transit of this mark may be noted. When the apparatus is very light, and the forces variable, the motion is so prompt and swift that observation through a telescope METHODS OF OBSERVATION. 93 would be useless. In this case the observer looks at the scale directly, and observes the motions of the image of the vertical wire thrown on the scale by a lamp. It is manifest that since the image of the scale reflected by the mirror and refracted by the object glass coincides with the vertical wire, the image of the vertical wire, if sufficiently illuminated, will coincide with the scale. To observe this the room is darkened, and the concentrated rays of a lamp are thrown on the vertical wire towards the object glass. A bright patch of light crossed by the shadow of the wire is seen on the scale. Its motions can be followed by the eye, and the division of the scale at which it comes to rest can be fixed on by the eye and read off at leisure. If it be desired to note the instant of the passage of the bright spot past a given point on the scale, a pin or a bright metal wire may be placed there so as to flash out at the time of passage. By substituting a small hole in a diaphragm for the cross wire the image becomes a small illuminated dot moving to right or left on the scale, and by substituting for the scale a cylinder revolving by clock work about a horizontal axis and covered with photo graphic paper, the spot of light traces out a curve which can be afterwards rendered visible. Each abscissa of this curve corresponds to a particular time, and the ordinate indicates the angular position of the mirror at that time. In this way an automatic system of continuous registration of all the elements of terrestrial magnetism has been established at Kew and other observatories. In some cases the telescope is dispensed with, a vertical wire is illuminated by a lamp placed behind it, and the mirror is a concave one, which forms the image of the wire on the scale as a dark line across a patch of light. 451.] In the Kew portable apparatus, the magnet is made in the form of a tube, having at one end a lens, and at the other a glass scale, so adjusted as to be at the principal focus of the lens. Light is admitted from behind the scale, and after passing through the lens it is viewed by means of a telescope. Since the scale is at the principal focus of the lens, rays from any division of the scale emerge from the lens parallel, and if the telescope is adjusted for celestial objects, it will shew the scale in optical coincidence with the cross wires of the telescope. If a given division of the scale coincides with the intersection of the cross wires, then the line joining that division with the optical centre of the lens must be parallel to the line of collimation of 94 MAGNETIC MEASUKEMENTS. [45 2. the telescope. By fixing the magnet and moving the telescope, we may ascertain the angular value of the divisions of the scale, and then, when the magnet is suspended and the position of the tele scope known, we may determine the position of the magnet at any instant by reading off the division of the scale which coincides with the cross wires. The telescope is supported on an arm which is centred in the line of the suspension fibre, and the position of the telescope is read off by verniers on the azimuth circle of the instrument. This arrangement is suitable for a small portable magnetometer in which the whole apparatus is supported on one tripod, and in which the oscillations due to accidental disturbances rapidly subside. Determination of the Direction of the Axis of the Magnet, and of the Direction of Terrestrial Magnetism. 452.] Let a system of axes be drawn in the magnet, of which the axis of z is in the direction of the length of the bar, and x and y perpendicular to the sides of the bar supposed a parallelepiped. Let I, m, n and A, /u, v be the angles which the magnetic axis and the line of collimation make with these axes respectively. Let M be the magnetic moment of the magnet, let H be the horizontal component of terrestrial magnetism, let Z be the vertical component, and let 6 be the azimuth in which H acts, reckoned from the north towards the west. Let ( be the observed azimuth of the line of collimation, let a be the azimuth of the stirrup, and (3 the reading of the index of the torsion circle, then a — /3 is the azimuth of the lower end of the suspension fibre. Let y be the value of a — /3 when there is no torsion, then the moment of the force of torsion tending to diminish a will be T(a-/3-y), where r is a coefficient of torsion depending on the nature of the fibre. To determine A, fix the stirrup so that y is vertical and up wards, z to the north and so to the west, and observe the azimuth f of the line of collimation. Then remove the magnet, turn it through an angle TT about the axis of z and replace it in this inverted position, and observe the azimuth f of the line of col limation when y is downwards and x to the east, 452.] BISECTION OF MAGNETIC FOKCE. 95 f=a+f-A, (1) r=a-|+A. (2) Hence x = |+i(f-0. (3) Next, hang the stirrup to the suspension fibre, and place the magnet in it, adjusting it carefully so that y may be vertical and upwards, then the moment of the force tending to increase a is 1— T (a— /3 — y). (4) But if C is the observed azimuth of the line of collimation C=a+|-A, (5) so that the force may be written MHsin *» sin (d - f + J- A) -T (f + A- - — 0 - y) • (6) When the apparatus is in equilibrium this quantity is zero for a particular value of f When the apparatus never comes to rest, but must be observed in a state of vibration, the value of £ corresponding to the position of equilibrium may be calculated by a method which will be described in Art. 735. When the force of torsion is small compared with the moment of the magnetic force, we may put d — £+ 1—\ for the sine of that angle. • If we give to /3, the reading of the torsion circle, two different values, p! and /32, and if £ and £2 are the corresponding values of £ MHsinm^-Q = r (£-£_& + &), (7) or, if we put " , (8) and equation (7) becomes, dividing by Jf/Jsin m, -^-y = 0. (9) If we now reverse the magnet so that y is downwards, and adjust the apparatus till y is exactly vertical, and if f is the new value of the azimuth, and 5' the corresponding declination, /(f-X + -/3-y=0> (10) whence - = i (f+C') + i/ (C+C'-2(/3-f y)). (11) 96 MAGNETIC MEASUREMENTS. [452. The reading of the torsion circle should now be adjusted, so that the coefficient of r may be as nearly as possible zero. For this purpose we must determine y, the value of a — (3 when there is no torsion. This may be done by placing a non-magnetic bar of the same weight as the magnet in the stirrup, and determining a — /3 when there is equilibrium. Since / is small, great accuracy is not required. Another method is to use a torsion bar of the same weight as the magnet, containing within it a very small magnet whose magnetic moment is - of that of the principal magnet. Ifi Since r remains the same, / will become m'} and if (^ and f/ are the values of ( as found by the torsion bar, 6 = iCt + fiO+i*!" (£ + &'- 2 (/3 + y)). (12) Subtracting this equation from (11), 2(»-l)(/3 + y) = (» + ^)(CI + C1')-(l + ^,)tf+O. (13) Having found the value of /3-fy in this way, /3, the reading of the torsion circle, should be altered till f+f'-2(/3 + y) = 0, (14) as nearly as possible in the ordinary position of the apparatus. Then, since r' is a very small numerical quantity, and since its coefficient is very small, the value of the second term in the ex pression for 5 will not vary much for small errors in the values of T and y, which are the quantities whose values are least ac curately known. The value of 8, the magnetic declination, may be found in this way with considerable accuracy, provided it remains constant during the experiments, so that we may assume 5'= 8. When great accuracy is required it is necessary to take account of the variations of 8 during the experiment. For this purpose observations of another suspended magnet should be made at the same instants that the different values of £ are observed, and if r], if are the observed azimuths of the second magnet corresponding to f and f ', and if 8 and 8' are the corresponding values of 8, then 8'-8 = rj'-r?. (15) Hence, to find the value of 8 we must add to (11) a correction i (')-•?')• The declination at the time of the first observation is therefore 8 = 4(C+r+ ^-770 + 4/^+^-2/3-2^. (16) 453-] OBSERVATION OP DEFLEXION. 97 To find the direction of the magnetic axis within the magnet subtract (10) from (9) and add (15), ^ = A + i(f-r)-H^-^Hi^(f-r-f2A-7r). (17) By repeating the experiments with the bar on its two edges, so that the axis of OB is vertically upwards and downwards, we can find the value of m. If the axis of collimation is capable of ad justment it ought to be made to coincide with the magnetic axis as nearly as possible, so that the error arising from the magnet not being exactly inverted may be as small as possible *. On the Measurement of Magnetic Forces. 453.] The most important measurements of magnetic force are those which determine M, the magnetic moment of a magnet, and //, the intensity of the horizontal component of terrestrial magnetism. This is generally done by combining the results of two experiments, one of which determines the ratio and the other the product of these two quantities. The intensity of the magnetic force due to an infinitely small magnet whose magnetic moment is M, at a point distant r from the centre of the magnet in the positive direction of the axis of the magnet, is ^ = 2— (I) and is in the direction of r. If the magnet is of finite size but spherical, and magnetized uniformly in the direction of its axis, this value of the force will still be exact. If the magnet is a solenoidal bar magnet of length 2 It, *=2*(l + 2§ + sg + &c.). 00 If the magnet be of any kind, provided its dimensions are all small compared with r, JL)+fcc., (3) where Alt A2, &c. are coefficients depending on the distribution of the magnetization of the bar. Let H be the intensity of the horizontal part of terrestrial magnetism at any place. H is directed towards magnetic north. Let r be measured towards magnetic west, then the magnetic force at the extremity of r will be H towards the north and R towards * See a Paper on 'Imperfect Inversion,' by W. Swan. Trans. R. S. Edin., vol. xxi (1855), p. 349. VOL. TT. H 98 MAGNETIC MEASUREMENTS. [453- the west. The resultant force will make an angle 0 with the magnetic meridian, measured towards the west, and such that (4) Hence, to determine -~= we proceed as follows : — JdL The direction of the magnetic north having been ascertained, a magnet, whose dimensions should not be too great, is suspended as in the former experiments, and the deflecting magnet M is placed so that its centre is at a distance r from that of the sus pended magnet, in the same horizontal plane, and due magnetic east. The axis of M is carefully adjusted so as to be horizontal and in the direction of r. The suspended magnet is observed before M is brought near and also after it is placed in position. If 0 is the observed deflexion, we have, if we use the approximate formula ( 1 ), f=^tau*; (5) or, if we use the formula (3), •.-•••. \ JrHan^l + ^i+^+fec. (6) Here we must bear in mind that though the deflexion 0 can be observed with great accuracy, the distance r between the centres of the magnets is a quantity which cannot be precisely deter mined, unless both magnets are fixed and their centres defined by marks. This difficulty is overcome thus : The magnet M is placed on a divided scale which extends east and west on both sides of the suspended magnet. The middle point between the ends of M is reckoned the centre of the magnet. This point may be marked on the magnet and its position observed on the scale, or the positions of the ends may be observed and the arithmetic mean taken. Call this Sj, and let the line of the suspension fibre of the suspended magnet when produced cut the scale at *0, then r1 = s1 — s0) where ^ is known accurately and s0 ap proximately. Let 01 be the deflexion observed in this position of M. Now reverse M, that is, place it on the scale with its ends reversed, then ^ will be the same, but M and Alt A3, &c. will have their signs changed, so that if 02 is ^ne deflexion, - I r,»tan 9, = 1 -A, ± + J,± -&c. (7) 454-] DEFLEXION OBSERVATIONS. 99 Taking the arithmetical mean of (6) and (7), i ^(tan^-tanfy = 1+^72 +^4^ + &c. (8) Now remove M to the west side of the suspended magnet, and place it with its centre at the point marked 2<$0 — s on the scale. Let the deflexion when the axis is in the first position be 03, and when it is in the second 04, then, as before, 2 Let us suppose that the true position of the centre of the sus pended magnet is not SQ but <?0 -f or, then (10) and (V +, 2») = ,»(!. + 'l^ + &c.); (11) O and since -^ may be neglected if the measurements are .carefully made, we are sure that we may take the arithmetical mean of rLn and r2n for rn. Hence, taking the arithmetical mean of (8) and (9), --^ or, making = 1 + A2~ +&c., (12) - (tan Ol — tan 62 + tan 03 — tan 04) = D, (13) 454.] We may now regard D and r as capable of exact deter mination. The quantity A2 can in no case exceed 2^2, where L is half the length of the magnet, so that when r is considerable compared with L we may neglect the term in A2 and determine the ratio of H to M at once. We cannot, however, assume that A2 is equal to 2i/2, for it may be less, and may even be negative for a magnet whose largest dimensions are transverse to the axis. The term in A±, and all higher terms, may safely be neglected. To eliminate A2, repeat the experiment, using distances rlt ra, ?*3, &c., and let the values of D be J)19 D2, #3, &c., then -2M(l , 4 2~~iT^ + ^ &c. &c. II 2 100 MAGNETIC MEASUREMENTS. [454- If we suppose that the probable errors of these equations are equal, as they will be if they depend on the determination of D only, and if there is no uncertainty about r, then, by multiplying each equation by r~3 and adding the results, we obtain one equation, and by multiplying each equation by r~5 and adding we obtain another, according to the general rule in the theory of the com bination of fallible measures when the probable error of each equation is supposed the same. Let us write 2(Vr-*) for AT3 + -02V3 + A^f 3 + &c., and use similar expressions for the sums of other groups of symbols, then the two resultant equations may be written *} = (2 (r-&) + 4 2 O TUT 2 (J)r~5) = -g- (2 (*-«) + A2 2 whence 1 W -=- 2 /-6 2r-10~2/-82 = 2 Z>r and 4>{2 (D?-3) 2 (r~10)-2 (Dr~5) 2 (*-8)} = 2 (Dr-B) 2 (r-«)-2 (Dr~*) 2 (r-8). The value of A2 derived from these equations ought to be less than half the square of the length of the magnet M. If it is not we may suspect some error in the observations. This method of observation and reduction was given by Gauss in the ( First Report of the Magnetic Association/ When the observer can make only two series of experiments at 2M distances r± and r2, the value of -=- derived from these experi ments is - - If 5Z)X and bD2 are the actual errors of the observed deflexions ^ and _Z)2, the actual error of the calculated result Q will be If we suppose the errors 8^ and bD2 to be independent, and that the probable value of either is SD, then the probable value of the error in the calculated value of Q will be 5 Q, where 455-1 METHODS OF TANGENTS AND SINES. 101 If we suppose that one of these distances, say the sinaHar,; ijs- given, the value of the greater distance may be determined so as to make b Q a minimum. This condition leads to an equation of the fifth degree in rf^ which has only one real root greater than r22. From this the best value of ^ is found to be rx = 1.3189/2*. If one observation only is taken the best distance is when bD r-lr -— = x/3 — , D ° r where b D is the probable error of a measurement of deflexion, and br is the probable error of a measurement of distance. Method of Sines. 455.] The method which we have just described may be called the Method of Tangents, because the tangent of the deflexion is a measure of the magnetic force. If the line rl5 instead of being measured east or west, is adjusted till it is at right angles with the axis of the deflected magnet, then R is the same as before, but in order that the suspended magnet may remain perpendicular to r, the resolved part of the force H in the direction of r must be equal and opposite to R. Hence, if 0 is the deflexion, R — Hsm 0. This method is called the Method of Sines. It can be applied only when R is less than H. In the Kew portable apparatus this method is employed. The suspended magnet hangs from a part of the apparatus which revolves along with the telescope and the arm for the deflecting magnet, and the rotation of the whole is measured on the azimuth circle. The apparatus is first adjusted so that the axis of the telescope coincides with the mean position of the line of collimation of the magnet in its undisturbed state. If the magnet is vibrating, the true azimuth of magnetic north is found by observing the ex tremities of the oscillation of the transparent scale and making the proper correction of the reading of the azimuth circle. The deflecting magnet is then placed upon a straight rod which passes through the axis of the revolving apparatus at right angles to the axis of the telescope, and is adjusted so that the axis of the deflecting magnet is in a line passing through the centre of the suspended magnet. The whole of the revolving apparatus is then moved till the line * See Airy's Magnetism. 102 MAGNETIC MEASUREMENTS. [45$. of coilimation of the suspended magnet again coincides with the axis of the telescope, and the new azimuth reading is corrected, if necessary, by the mean of the scale readings at the extremities of an oscillation. The difference of the corrected azimuths gives the deflexion, after which we proceed as in the method of tangents, except that in the expression for D we put sin & instead of tan 6. In this method there is no correction for the torsion of the sus pending fibre, since the relative position of the fibre, telescope, and magnet is the same at every observation. The axes of the two magnets remain always at right angles in this method, so that the correction for length can be more ac curately made. 456.] Having thus measured the ratio of the moment of the deflecting magnet to the horizontal component of terrestrial mag netism, we have next to find the product of these quantities, by determining the moment of the couple with which terrestrial mag netism tends to turn the same magnet when its axis is deflected from the magnetic meridian. There are two methods of making this measurement, the dy namical, in which the time of vibration of the magnet under the action of terrestrial magnetism is observed, and the statical, in which the magnet is kept in equilibrium between a measurable statical couple and the magnetic force. The dynamical method requires simpler apparatus and is more accurate for absolute measurements, but takes up a considerable time, the statical method admits of almost instantaneous measure ment, and is therefore useful in tracing the changes of the intensity of the magnetic force, but it requires more delicate apparatus, and is not so accurate for absolute measurement. Method of Vibrations. The magnet is suspended with its magnetic axis horizontal, and is set in vibration in small arcs. The vibrations are observed by means of any of the methods already described. A point on the scale is chosen corresponding to the middle of the arc of vibration. The instant of passage through this point of the scale in the positive direction is observed. If there is suffi cient time before the return of the magnet to the same point, the instant of passage through the point in the negative direction is also observed, and the process is continued till n+I positive and 456.] TIME OF VIBKATION. 103 n negative passages have been observed. If the vibrations are too rapid to allow of every consecutive passage being observed, every third or every fifth passage is observed, care being taken that the observed passages are alternately positive and negative. Let the observed times of passage be T1} T2, T2n+1, then if we put I 4 + y + y 4 &c. then Tn+1 is the mean time of the positive passages, and ought to agree with T'n+v the mean time of the negative passages, if the point has been properly chosen. The mean of these results is to be taken as the mean time of the middle passage. After a large number of vibrations have taken place, but before the vibrations have ceased to be distinct and regular, the observer makes another series of observations, from which he deduces the mean time of the middle passage of the second series. By calculating the period of vibration either from the first series of observations or from the second, he ought to be able to be certain of the number of whole vibrations which have taken place in the interval between the time of middle passage in the two series. Dividing the interval between the mean times of middle passage in the two series by this number of vibrations, the mean time of vibration is obtained. The observed time of vibration is then to be reduced to the time of vibration in infinitely small arcs by a formula of the same kind as that used in pendulum observations, and if the vibrations are found to diminish rapidly in amplitude, there is another cor rection for resistance, see Art. 740. These corrections, however, are very small when the magnet hangs by a fibre, and when the arc of vibration is only a few degrees. The equation of motion of the magnet is - = 0 where 0 is the angle between the magnetic axis and the direction of the force H, A is the moment of inertia of the magnet and suspended apparatus, M is the magnetic moment of the magnet, H the intensity of the horizontal magnetic force, and MHr' the coefficient of torsion : / is determined as in Art. 452, and is a very small quantity. The value of 0 for equilibrium is T "y 00 = - - T 5 a very small angle, 104 MAGNETIC MEASUREMENTS. [457- and the solution of the equation for small values of the amplitude, C is f t \ 0 = Ccos (2 TT -^ 4- a) + 00, where T is the periodic time, and C the amplitude, and yr2 whence we find the value of MH9 Here T is the time of a complete vibration determined from observation. A, the moment of inertia, is found once for all for the magnet, either by weighing and measuring it if it is of a regular figure, or by a dynamical process of comparison with a body whose moment of inertia is known. Combining this value of Mil with that of -~ formerly obtained, we get Jp and //* 457.] We have supposed that //and M continue constant during the two series of experiments. The fluctuations of // may be ascertained by simultaneous observations of the bifilar magnet ometer to be presently described, and if the magnet has been in use for some time, and is not exposed during the experiments to changes of temperature or to concussion, the part of M which de pends on permanent magnetism may be assumed to be constant. All steel magnets, however, are capable of induced magnetism depending on the action of external magnetic force. Now the magnet when employed in the deflexion experiments is placed with its axis east and west, so that the action of ter restrial magnetism is transverse to the magnet, and does not tend to increase or diminish M. When the magnet is made to vibrate, its axis is north and south, so that the action of terrestrial mag netism tends to magnetize it in the direction of the axis, and therefore to increase its magnetic moment by a quantity Jc //, where k is a coefficient to be found by experiments on the magnet. There are two ways in which this source of error may be avoided without calculating Jc, the experiments being arranged so that the magnet shall be in the same condition when employed in deflecting another magnet and when itself swinging. 457-] ELIMINATION OF INDUCTION. 105 We may place the deflecting magnet with its axis pointing north, at a distance r from the centre of the suspended magnet, the line r making an angle whose cosine is \/J with the magnetic meridian. The action of the deflecting magnet on the suspended one is then at right angles to its own direction, and is equal to Here M is the magnetic moment when the axis points north, as in the experiment of vibration, so that no correction has to be made for induction. This method, however, is extremely difficult, owing to the large errors which would be introduced by a slight displacement of the deflecting magnet, and as the correction by reversing the deflecting magnet is not applicable here, this method is not to be followed except when the object is to determine the coefficient of induction. The following method, in which the magnet while vibrating is freed from the inductive action of terrestrial magnetism, is due to Dr. J. P. Joule *. Two magnets are prepared whose magnetic moments are as nearly equal as possible. In the deflexion experiments these mag nets are used separately, or they may be placed simultaneously on opposite sides of the suspended magnet to produce a greater deflexion. In these experiments the inductive force of terrestrial magnetism is transverse to the axis. Let one of these magnets be suspended, and let the other be placed parallel to it with its centre exactly below that of the sus pended magnet, and with its axis in the same direction. The force which the fixed magnet exerts on the suspended one is in the opposite direction from that of terrestrial magnetism. If the fixed magnet be gradually brought nearer to the suspended one the time of vibration will increase, till at a certain point the equilibrium will cease to be stable, and beyond this point the suspended magnet will make oscillations in the reverse position. By experimenting in this way a position of the fixed magnet is found at which it exactly neutralizes the effect of terrestrial magnetism on the sus pended one. The two magnets are fastened together so as to be parallel, with their axes turned the same way, and at the distance just found by experiment. They are then suspended in the usual way and made to vibrate together through small arcs. * Proc. Phil. S., Manchester, March 19, 1867. 106 MAGNETIC MEASUREMENTS. [45 8. The lower magnet exactly neutralizes the effect of terrestrial magnetism on the upper one, and since the magnets are of equal moment, the upper one neutralizes the inductive action of the earth on the lower one. The value of M is therefore the same in the experiment of vibration as in the experiment of deflexion, and no correction for induction is required. 458.] The most accurate method of ascertaining the intensity of the horizontal magnetic force is that which we have just described. The whole series of experiments, however, cannot be performed with sufficient accuracy in much less than an hour, so that any changes in the intensity which take place in periods of a few minutes would escape observation. Hence a different method is required for ob serving the intensity of the magnetic force at any instant. The statical method consists in deflecting the magnet by means of a statical couple acting in a horizontal plane. If L be the moment of this couple, M the magnetic moment of the magnet, // the horizontal component of terrestrial magnetism, and 0 the deflexion, M H sin 0 = L. Hence, if L is known in terms of 0, MH can be found. The couple L may be generated in two ways, by the torsional elasticity of a wire, as in the ordinary torsion balance, or by the weight of the suspended apparatus, as in the bifilar suspension. In the torsion balance the magnet is fastened to the end of a vertical wire, the upper end of which can be turned round, and its rotation measured by means of a torsion circle. We have then X, = r(a — a0 — 6) = Mil sin 6. Here a0 is the value of the reading of the torsion circle when the axis of the magnet coincides with the magnetic meridian, and a is the actual reading. If the torsion circle is turned so as to bring the magnet nearly perpendicular to the magnetic meridian, so that e = ~tf, then r(a-a0- + 00 or By observing 0', the deflexion of the magnet when in equilibrium, we can calculate Mil provided we know r. If we only wish to know the relative value of H at different times it is not necessary to know either M or T. We may easily determine T in absolute measure by suspending 459-] BIFILAB SUSPENSION. 107 a non-magnetic body from the same wire and observing its time of oscillation, then if A is the moment of inertia of this body, and T the time of a complete vibration, The chief objection to the use of the torsion balance is that the zero-reading a0 is liable to change. Under the constant twisting force, arising from the tendency of the magnet to turn to the north, the wire gradually acquires a permanent twist, so that it becomes necessary to determine the zero-reading of the torsion circle afresh at short intervals of time. Bifilar Suspension. 459.] The method of suspending the magnet by two wires or fibres was introduced by Gauss and Weber. As the bifilar sus pension is used in many electrical instruments, we shall investigate it more in detail. The general appearance of the suspension is shewn in Fig. 16, and Fig. 17 represents the projection of the wires on a horizontal plane. AB and A'B' are the projections of the two wires. AA and BB' are the lines joining the upper and the lower ends of the wires. a and b are the lengths of these lines. a and /3 their azimuths. TFand W the vertical components of the tensions of the wires. Q and Q' their horizontal components. h the vertical distance between AA and BB'. The forces which act on the magnet are — its weight, the couple arising from terrestrial magnetism, the torsion of the wires (if any) and their tensions. Of these the effects of magnetism and of torsion are of the nature of couples. Hence the resultant of the tensions must consist of a vertical force, equal to the weight of the magnet, together with a couple. The resultant of the vertical components of the tensions is therefore along the line whose pro jection is 0, the intersection of A A and BB', and either of these lines is divided in 0 in the ratio of W to W. The horizontal components of the tensions form a couple, and are therefore equal in magnitude and parallel in direction. Calling either of them Q, the moment of the couple which they form is L=Q.PF, (1) where PP7 is the distance between the parallel lines AB and AB'. 108 MAGNETIC MEASUREMENTS. [459- To find the value of L we have the equations of moments Qh = W. AB = Jr. AK> (2) and the geometrical equation (AB + A'ff) PPf = ab sin (a- ft), ( 3) whence we obtain, ab WW 1= W+ W r, sin(a-/3). Fig. 16. Fig. 17. (4) If m is the mass of the suspended apparatus, and g the intensity of gravity, w+ W' = mg. (5) If we also write W— W — nmg> (6) L — - (i—.n?)m.ff-jr sin (a— ft). we find L — - (1 —nz}mff — sin (a — /3V (7) The value of L is therefore a maximum with respect to n when n 459-] BIFILAR SUSPENSION. 109 is zero, that is, when the weight of the suspended mass is equally borne by the two wires. We may adjust the tensions of the wires to equality by observing1 the time of vibration, and making it a minimum, or we may obtain a self-acting adjustment by attaching the ends of the wires, as in Fig. 16, to a pulley, which turns on its axis till the tensions are equal. The distance of the upper ends of the suspension wires is re gulated by means of two other pullies. The distance between the lower ends of the wires is also capable of adjustment. By this adjustment of the tension, the couple arising from the tensions of the wires becomes T I ab . . L = - -j- mg sin (a— -/3). The moment of the couple arising from the torsion of the wires is of the form T (y—p\ where r is the sum of the coefficients of torsion of the wires. The wires ought to be without torsion when a = ft, we may then make y — a. The moment of the couple arising from the horizontal magnetic force is of the form MS BIU (3 — 0), where 8 is the magnetic declination, and 0 is the azimuth of the axis of the magnet. We shall avoid the introduction of unnecessary symbols without sacrificing generality if we assume that the axis of the magnet is parallel to £JB', or that /3 = 0. The equation of motion then becomes 4--j72= MHsw(b — 0} + - ^-^sin(a — 0) + r(a-0). (8) There are three principal positions of this apparatus. (1) When a is nearly equal to 8. If T^ is the time of a complete oscillation in this position, then 47r2^ lab -yrr- = l-fi>"ff+T + MH. (9) (2) When a is nearly equal to 8 + 77. If T2 is the time of a complete oscillation in this position, the north end of the magnet being now turned towards the south, 1 ab ^-jrWff + T-MH. (10) The quantity on the right-hand of this equation may be made 130 MAGNETIC MEASUREMENTS. [459. as small as we please by diminishing a or £, but it must not be made negative, or the equilibrium of the magnet will become un stable. The magnet in this position forms an instrument by which small variations in the direction of the magnetic force may be rendered sensible. For when 5—0 is nearly equal to TT, sin (8 — 0) is nearly equal to 6 — by and we find (8-a). (11) = a- 7 l ah 71* rr - ~j-mg-\-T —MH 4 fl By diminishing the denominator of the fraction in the last term we may make the variation of 0 very large compared with that of 8. We should notice that the coefficient of 8 in this expression is negative, so that when the direction of the magnetic force turns in one direction the magnet turns in the opposite direction. (3) In the third position the upper part of the suspension- apparatus is turned round till the axis of the magnet is nearly perpendicular to the magnetic meridian. If we make 0-8=|+0/, and a-6 = p-P, (12) the equation of motion may be written (/:J-0'). (13) If there is equilibrium when //= EQ and 0'= 0, = 0, (14) and if H is the value of the horizontal force corresponding to a small angle 0/, x ^ - -j- mg cos /3 -|- T \ '--~ - In order that the magnet may be in stable equilibrium it is necessary that the numerator of the fraction in the second member should be positive, but the more nearly it approaches zero, the more sensitive will be the instrument in indicating changes in the value of the intensity of the horizontal component of terrestrial magnetism. The statical method of estimating the intensity of the force depends upon the action of an instrument which of itself assumes 46 1. J DTP. Ill different positions of equilibrium for different values of the force. Hence, by means of a mirror attached to the magnet and throwing1 a spot of light upon a photographic surface moved by clockwork, a curve may be traced, from which the intensity of the force at any instant may be determined according to a scale, which we may for the present consider an arbitrary one. 460.] In an observatory, where a continuous system of regis tration of declination and intensity is kept up either by eye ob servation or by the automatic photographic method, the absolute values of the declination and of the intensity, as well as the position and moment of the magnetic axis of a magnet, may be determined to a greater degree of accuracy. For the declinometer gives the declination at every instant affected by a constant error, and the bifilar magnetometer gives the intensity at every instant multiplied by a constant coefficient. In the ex periments we substitute for b, 8 + 80 where 8' is the reading of the declinometer at the given instant, and 80 is the unknown but constant error, so that 8' + 80 is the true declination at that instant. In like manner for H, we substitute CH' where IF is the reading* ' " O of the magnetometer on its arbitrary scale, and C is an unknown but constant multiplier which converts these readings into absolute measure, so that CH' is the horizontal force at a given instant. The experiments to determine the absolute values of the quan tities must be conducted at a sufficient distance from the declino meter and magnetometer, so that the different magnets may not sensibly disturb each other. The time of every observation must be noted and the corresponding values of 8' and H' inserted. The equations are then to be treated so as to find 80, the constant error of the declinometer, and C the coefficient to be applied to the readings of the magnetometer. When these are found the readings of both instruments may be expressed in absolute measure. The absolute measurements, however, must be frequently repeated in order to take account of changes which may occur in the magnetic axis and magnetic moment of the magnets. 461.] The methods of determining the vertical component of the terrestrial magnetic force have not been brought to the same degree of precision. The vertical force must act on a magnet which turns about a horizontal axis. Now a body which turns about a hori zontal axis cannot be made so sensitive to the action of small forces as a body which is suspended by a fibre and turns about a vertical axis. Besides this, the weight of a magnet is so large compared 112 MAGNETIC MEASUREMENTS. [461. with the magnetic force exerted upon it that a small displace ment of the centre of inertia by unequal dilatation, &c. produces a greater effect on the position of the magnet than a considerable change of the magnetic force. Hence the measurement of the vertical force, or the comparison of the vertical and the horizontal forces, is the least perfect part of the system of magnetic measurements. The vertical part of the magnetic force is generally deduced from the horizontal force by determining the direction of the total force. If i be the angle which the total force makes with its horizontal component, i is called the magnetic Dip or Inclination, and if H is the horizontal force already found, then the vertical force is //tan i, and the total force is H sec i. The magnetic dip is found by means of the Dip Needle. The theoretical dip-needle is a magnet with an axis which passes through its centre of inertia perpendicular to the magnetic axis of the needle. The ends of this axis are made in the form of cylinders of small radius, the axes of which are coincident with the line passing through the centre of inertia. These cylindrical ends rest on two horizontal planes and are free to roll on them. When the axis is placed magnetic east and west, the needle is free to rotate in the plane of the magnetic meridian, and if the instrument is in perfect adjustment, the magnetic axis will set itself in the direction of the total magnetic force. It is, however, practically impossible to adjust a dip-needle so that its weight does not influence its position of equilibrium, because its centre of inertia, even if originally in the line joining the centres of the rolling sections of the cylindrical ends, will cease to be in this line when the needle is imperceptibly bent or un equally expanded. Besides, the determination of the true centre of inertia of a magnet is a very difficult operation, owing to the interference of the magnetic force with that of gravity. Let us suppose one end of the needle and one end of the pivot to be marked. Let a line, real or imaginary, be drawn on the needle, which we shall call the Line of Collimation. The position of this line is read off on a vertical circle. Let 6 be the angle which this line makes with the radius to zero, which we shall suppose to be horizontal. Let A. be the angle which the magnetic axis makes with the line of collimation, so that when the needle is in this position the line of collimation is inclined 0 + A. to the horizontal. 461.] DIP CIRCLE. 11.3 Let p be the perpendicular from the centre of inertia on the plane on which the axis rolls, then p will be a function of 6, whatever be the shape of the rolling surfaces. If both the rolling sections of the ends of the axis are circular, p — c — #sin(0+a) (1) where a is the distance of the centre of inertia from the line joining the centres of the rolling sections, and a is the angle which this line makes with the line of collimation. If M is the magnetic moment, m the mass of the magnet, and g the force of gravity, I the total magnetic force, and i the dip, then, by the conservation of energy, when there is stable equilibrium, MIcos(0 + \ — i) — mgjp (2) must be a maximum with respect to 0, or MIsm(0 + \-i)=-m<?d^> (3) = —mg a cos (6 + a), if the ends of the axis are cylindrical. Also, if T be the time of vibration about the position of equi librium, : /,x MI+ mga sin (6+ a) = -^- where A is the moment of inertia of the needle about its axis of rotation. In determining the dip a reading is taken with the dip circle in the magnetic meridian and with the graduation towards the west. Let 61 be this reading, then we have MIsm(01 + \—i) = —m(/acos(0l + a). (5) The instrument is now turned about a vertical axis through 180°, so that the graduation is to the east, and if 02 is the new reading, MIsm(02 + X — v+i) ——mga cos (02 + a). (6) Taking (6) from (5), and remembering that 6^ is nearly equal to i, and 02 nearly equal to TT— i, and that X is a small angle, such that mgaK may be neglected in comparison with MI, MI(0l—02-{-7f—2i') =—2mgaco$icosa. (7) Now take the magnet from its bearings and place it in the deflexion apparatus, Art. 453, so as to indicate its own magnetic moment by the deflexion of a suspended magnet, then M=\r*HD (8) where D is the tangent of the deflexion. VOL. II. I 114 MAGNETIC MEASUREMENTS. [461. Next, reverse the magnetism of the needle and determine its new magnetic moment M', by observing a new deflexion, the tan gent of which is D' ', M> = i ^ H1)^ (9) whence MD' = M'D. ( 1 0) Then place it on its bearings and take two readings, 03 and 04, in which 03 is nearly ir + i, and 04 nearly —i, 3/'/' sin (03 + A' — 77 — i) = mgaco8(0B+a), (11) M'l' sin (04 + A' + i) = m g a cos (04 + a), (1 2) whence, as before, M'I(93— 04 — 77 — 2i) — 2mgacosicosa, (13) adding (8), Jf/^-^ + Tr — 2z') + lT/(03 — 04— IT — 2 a) = 0, (14) or J9(01-02 + 7r-2;) + .Z/(03-04-7r-2*) = 0, (15) whence we find the dip -e4-Tr) , . where D and _Z/ are the tangents of the deflexions produced by the needle in its first and second magnetizations respectively. In taking observations with the dip circle the vertical axis is carefully adjusted so that the plane bearings upon which the axis of the magnet rests are horizontal in every azimuth. The magnet being magnetized so that the end A dips, is placed with its axis on the plane bearings, and observations are taken with the plane of the circle in the magnetic meridian, and with the graduated side of the circle east. Each end of the magnet is observed by means of reading microscopes carried on an arm which moves concentric with the dip circle. The cross wires of the microscope are made to coincide with the image of a mark on the magnet, and the position of the arm is then read off on the dip circle by means of a vernier. We thus obtain an observation of the end A and another of the end B when the graduations are east. It is necessary to observe both ends in order to eliminate any error arising from the axle of the magnet not being concentric with the dip circle. The graduated side is then turned west, and two more observ ations are made. The magnet is then turned round so that the ends of the axle are reversed, and four more observations are made looking at the other side of the magnet. 463.] JOULE'S SUSPENSION. 115 The magnetization of the magnet is then reversed so that the end B dips, the magnetic moment is ascertained, and eight ohserva- tions are taken in this state, and the sixteen observations combined to determine the true dip. 462.] It is found that in spite of the utmost care the dip, as thus deduced from observations made with one dip circle, differs per ceptibly from that deduced from observations with another dip circle at the same place. Mr. Broun has pointed out the effect due to ellipticity of the bearings of the axle, arid how to correct it by taking observations with the magnet magnetized to different strengths. The principle of this method may be stated thus. We shall suppose that the error of any one observation is a small quantity not exceeding a degree. We shall also suppose that some unknown but regular force acts upon the magnet, disturbing it from its true position. If L is the moment of this force, 00 the true dip, and 0 the observed dip, then L = Jf/sin(0-00), (17) = MI(0-00), (18) since 0 — 6$ is small. It is evident that the greater M becomes the nearer does the needle approach its proper position. Now let the operation of taking the dip be performed twice, first with the magnetization equal to Mlt the greatest that the needle is capable of, and next with the magnetization equal to M~29 a much smaller value but sufficient to make the readings distinct and the error still moderate. Let 01 and 62 be the dips deduced from these two sets of observ ations, and let L be the mean value of the unknown disturbing force for the eight positions of each determination, which we shall suppose the same for both determinations. Then L = M1i(01-e0) = M2i(02-00). (19) If we find that several experiments give nearly equal values for L, then we may consider that 00 must be very nearly the true value of the dip. 463.] Dr. Joule has recently constructed a new dip-circle, in which the axis of the needle, instead of rolling on horizontal agate planes, is slung on two filaments of silk or spider's thread, the ends I 2 116 MAGNETIC MEASUREMENTS. [463- of the filaments being attached to the arms of a delicate balance. The axis of the needle thus rolls on two loops of silk fibre, and Dr. Joule finds that its freedom of motion is much greater than when it rolls on agate planes. In Fig. 18, NS is the needle, CC' is its axis, consisting of a straight cylindrical wire, and PCQ, P'C'Q' are the filaments on which the axis rolls. POQ is the balance, consisting of a double bent lever supported by a wire, 0 0, stretched horizont ally between the prongs of a forked piece, and having a counterpoise It which can be screwed up or down, so that the balance is in neutral equilibrium about 0 0. In order that the needle may be in neutral equilibrium as the needle rolls on the filaments the centre of gra vity must neither rise nor fall. Hence the distance OC must remain constant as the needle rolls. This condition will be fulfilled if the arms of the balance OP and 0 Q are equal, and if the filaments are at right angles to the arms. Dr. Joule finds that the needle should not be more than five inches long. When it is eight inches long, the bending of the needle tends to diminish the apparent dip by a fraction of a minute. The axis of the needle was originally of steel wire, straightened by being brought to a red heat while stretched by a weight, but Dr. Joule found that with the new suspension it is not necessary to use steel wire, for platinum and even standard gold are hard enough. The balance is attached to a wire 00 about a foot long stretched horizontally between the prongs of a fork. This fork is turned round in azimuth by means of a circle at the top of a tripod which supports the whole,. Six complete observations of the dip can be 464.] VEETICAL FORCE. 117 obtained in one hour, and the average error of a single observation is a fraction of a minute of arc. It is proposed that the dip needle in the Cambridge Physical Laboratory shall be observed by means of a double image instru ment, consisting of two totally reflecting prisms placed as in Fig. 19 and mounted on a vertical graduated circle, so that the plane of reflexion may be turned round a horizontal axis nearly coinciding with the prolongation of the axis of the suspended dip- needle. The needle is viewed by means of a telescope placed behind the prisms, and the two ends of the needle are seen together as in Fig. 20. By turning the prisms about the axis of the vertical circle, the images of two lines drawn on the needle may be made to coincide. The inclination of the needle is thus determined from the reading of the vertical circle. Fig. 19. Fig. 20. The total intensity / of the magnetic force in the line of dip may be deduced as follows from the times of vibration in the four positions already described, T13 Tz, jP3, 5JL _L JL J_l. " 2M+2' I Zi2 + Tf "h T* h T* ) The values of M and M' must be found by the method of deflexion and vibration formerly described, and A is the moment of inertia of the magnet about its axle. The observations with a magnet suspended by a fibre are so much more accurate that it is usual to deduce the total force from the horizontal force from the equation /= H sec 6, where / is the total force, H the horizontal force, and 0 the dip. 464.] The process of determining the dip being a tedious one, is not suitable for determining the continuous variation of the magnetic 118 MAGNETIC MEASUREMENTS. [464. force. The most convenient instrument for continuous observa tions is the vertical force magnetometer, which is simply a magnet balanced on knife edges so as to be in stable equilibrium with its magnetic axis nearly horizontal. If Z is the vertical component of the magnetic force, M the magnetic moment, and 0 the small angle which the magnetic axis makes with the horizon HZ = mgacQ&(a.~6), where m is the mass of the magnet, g the force of gravity, a the distance of the centre of gravity from the axis of suspension, and a the angle which the plane through the axis and the centre of gravity makes with the magnetic axis. Hence, for the small variation of vertical force bZ, there will be a variation of the angular position of the magnet bO such that In practice this instrument is not used to determine the absolute value of the vertical force, but only to register its small variations. For this purpose it is sufficient to know the absolute value of Z when 0 = 0, and the value of -y-r • civ The value of Z, when the horizontal force and the dip are known, is found from the equation Z = ZTtan00, where 00 is the dip and H the horizontal force. To find the deflexion due to a given variation of Z, take a magnet and place it with its axis east and west, and with its centre at a known distance i\ east or west from the declinometer, as in ex periments on deflexion, and let the tangent of deflexion be Dl . Then place it with its axis vertical and with its centre at a distance rz above or below the centre of the vertical force mag netometer, and let the tangent of the deflexion produced in the magnetometer be D2. Then, if the moment of the deflecting magnet is M, jr. M^IIr^D^ = ^r^D2. clZ r^ DL Hence -7— = H -^ -~ • dO r23 D2 The actual value of the vertical force at any instant is 7 7 +fidZ & = &Q H- v -j^ > where ZQ is the value of Z when Q = 0. For continuous observations of the variations of magnetic force 464.] VERTICAL FOKCE. 119 at a fixed observatory the Unifilar Declinometer, the Bifilar Hori zontal Force Magnetometer, and the Balance Vertical Force Mag netometer are the most convenient instruments. At several observatories photographic traces are now produced on prepared paper moved by clock work, so that a continuous record of the indications of the three instruments at every instant is formed. These traces indicate the variation of the three rectangular com ponents of the force from their standard values. The declinometer gives the force towards mean magnetic west, the bifilar magnet ometer gives the variation of the force towards magnetic north, and the balance magnetometer gives the variation of the vertical force. The standard values of these forces, or their values when these instruments indicate their several zeros, are deduced by frequent observations of the absolute declination, horizontal force, and dip. CHAPTER VIII. ON TERRESTRIAL MAGNETISM. 465.] OUR knowledge of Terrestrial Magnetism is derived from the study of the distribution of magnetic force on the earth's sur face at any one time, and of the changes in that distribution at different times. The magnetic force at any one place and time is known when its three coordinates are known. These coordinates may be given in the form of the declination or azimuth of the force, the dip or inclination to the horizon, and the total intensity. The most convenient method, however, for investigating the general distribution of magnetic force on the earth's surface is to consider the magnitudes of the three components of the force, X=Hcosb, directed due north, \ Y=Hsmb, directed due west, (1) Z = If tan 0, directed vertically downwards, ) where H denotes the horizontal force, 8 the declination, and 0 the dip. If V is the magnetic potential at the earth's surface, and if we consider the earth a sphere of radius a, then Y i dr i dv dv , } A = -- ^yj Y = - j — > ^=-^-7 (*) a dl a cos I dK dr where I is the latitude, and A. the longitude, and r the distance from the centre of the earth. A knowledge of V over the surface of the earth may be obtained from the observations of horizontal force alone as follows. Let FQ be the value of V at the true north pole, then, taking the line-integral along any meridian, we find, o, (3) for the value of the potential on that meridian at latitude I. 466.] MAGNETIC SURVEY. 121 Thus the potential may be found for any point on the earth's surface provided we know the value of X, the northerly component at every point, and F0, the value of Fat the pole. Since the forces depend not on the absolute value of V but on its derivatives, it is not necessary to fix any particular value for F0. The value of V at any point may be ascertained if we know the value of X along any given meridian, and also that of T over the whole surface. Let JF»»/j:tf+7*, W where the integration is performed along the given meridian from the pole to the parallel I, then F= ^+«fVco8/#A, (5) ^AO where the integration is performed along the parallel I from the given meridian to the required point. These methods imply that a complete magnetic survey of the earth's surface has been made, so that the values of X or of Y or of both are known for every point of the earth's surface at a given epoch. What we actually know are the magnetic com ponents at a certain number of stations. In the civilized parts of the earth these stations are comparatively numerous ; in other places there are large tracts of the earth's surface about which we have no data. Magnetic Survey. 466.] Let us suppose that in a country of moderate size, whose greatest dimensions are a few hundred miles, observations of the declination and the horizontal force have been taken at a con siderable number of stations distributed fairly over the country. Within this district we may suppose the value of V to be re presented with sufficient accuracy by the formula F= Vt + a(AJ + Ai\+\BJ*+EJ\+\3iK* + ^ (6) whence X = A1 + Bl I + £2 X, (7) Ycosl = A2 + £2l + 33\. (8) Let there be n stations whose latitudes are ll} £2, ...&c. and longitudes \lt A2, &c., and let X and 7 be found for each station. Let J = 122 TERRESTRIAL MAGNETISM. [466- /0 and A0 may be called the latitude and longitude of the central station. Let X0=-i(i)- and rocosJ0=:-2(rcosJ), (10) tl ti then X0 and Y0 are the values of X and Y at the imaginary central station, then \-\0), (11) A-A0). (12) We have n equations of the form of (11) and n of the form (12). If we denote the probable error in the determination of X by £, and that of Ycos I by q, then we may calculate f and r/ on the supposition that they arise from errors of observation of H and 8. Let the probable error of H be ^, and that of 8, d, then since dX — cos 5 . dff—Hsm 8 . db, £2 = 7,2 COS2 8 + d*H* sin2 8 Similarly 7?2 = /fc2 sin2 8 + d2 //2 cos2 8. If the variations of X and T from their values as given by equa tions of the form (11) and (12) considerably exceed the probable errors of observation, we may conclude that they are due to local attractions, and then we have no reason to give the ratio of £ to r\ any other value than unity. According to the method of least squares we multiply the equa tions of the form (11) by r/, and those of the form (12) by £ to make their probable error the same. We then multiply each equation by the coefficient of one of the unknown quantities J3lt H2, or BZ and add the results, thus obtaining three equations from which to find B B and B. in which we write for conciseness, *1 = 2(^2)-»^ ^ = Pl = 2(lX)-nlQXQ, By calculating £19 J52, and J53, and substituting in equations (11) and (12), we can obtain the values of X and Y at any point within the limits of the survey free from the local disturbances 468.] MAGNETIC FEATURES OF THE EARTH. 123 which are found to exist where the rock near the station is magnetic, as most igneous rocks are. Surveys of this kind can be made only in countries where mag netic instruments can be carried about and set up in a great many stations. For other parts of the world we must be content to find the distribution of the magnetic elements by interpolation between their values at a few stations at great distances from each other. 467.] Let us now suppose that by processes of this kind, or by the equivalent graphical process of constructing charts of the lines of equal values of the magnetic elements, the values of X and Y, and thence of the potential V, are known over the whole surface of the globe. The next step is to expand V in the form of a series of spherical surface harmonics. If the earth were magnetized uniformly and in the same direction throughout its interior, V would be an harmonic of the first degree, the magnetic meridians would be great circles passing through two magnetic poles diametrically opposite, the magnetic equator would be a great circle, the horizontal force would be equal at all points of the magnetic equator, and if H0 is this constant value, the value at any other point would be H= //Ocos I', where V is the magnetic latitude. The vertical force at any point would be Z = 2 HQ sin I' , and if Q is the dip, tan 6 = 2 tan I' . In the case of the earth, the magnetic equator is defined to be the line of no dip. It is not a great circle of the sphere. The magnetic poles are defined to be the points where there is no horizontal force or where the dip is 90°. There are two such points, one in the northern and one in the southern regions, but they are not diametrically opposite, and the line joining them is not parallel to the magnetic axis of the earth. 468.] The magnetic poles are the points where the value of V on the surface of the earth is a maximum or minimum, or is stationary. At any point where the potential is a minimum the north end of the dip-needle points vertically downwards, and if a compass- needle be placed anywhere near such a point, the north end will point towards that point. At points where the potential is a maximum the south end of the dip-needle points downwards, and the south end of the compass- needle points towards the point. If there are p minima of V on the earth's surface there must be p — \ other points, where the north end of the dip-needle points 124: TERRESTRIAL MAGNETISM. [469. downwards, but where the compass-needle, when carried in a circle round the point, instead of revolving so that its north end points constantly to the centre, revolves in the opposite direction, so as to turn sometimes its north end and sometimes its south end towards the point. If we call the points where the potential is a minimum true north poles, then these other points may be called false north poles, because the compass-needle is not true to them. If there are p true north poles, there must be p — I false north poles, and in like manner, if there are q true south poles, there must be y — 1 false south poles. The number of poles of the same name must be odd, so that the opinion at one time prevalent, that there are two north poles and two south poles, is erroneous. According to Gauss there is in fact only one true north pole and one true south pole on the earth's surface, and therefore there are no false poles. The line joining these poles is not a diameter of the earth, and it is not parallel to the earth's magnetic axis. 469.] Most of the early investigators into the nature of the earth's magnetism endeavoured to express it as the result of the action of one or more bar magnets, the position of the poles of which were to be determined. Gauss was the first to express the distribution of the earth's magnetism in a perfectly general way by expanding its potential in a series of solid harmonics, the coefficients of which he determined for the first four degrees. These coeffi cients are 24 in number, 3 for the first degree, 5 for the second, 7 for the third, and 9 for the fourth. All these terms are found necessary in order to give a tolerably accurate representation of the actual state of the earth's magnetism. To find what Part of the Observed Magnetic Force is due to External and what to Internal Causes. 470.] Let us now suppose that we have obtained an expansion of the magnetic potential of the earth in spherical harmonics, consistent with the actual direction and magnitude of the hori zontal force at every point on the earth's surface, then Gauss has shewn how to determine, from the observed vertical force, "whether the magnetic forces are due to causes, such as magnetization or electric currents, within the earth's surface, or whether any part is directly due to causes exterior to the earth's surface. Let V be the actual potential expanded in a double series of spherical harmonics, 472.] SUBTERRANEAN OH CELESTIAL I 125 -2 The first series represents the part of the potential due to causes exterior to the earth,, and the second series represents the part due to causes within the earth. The observations of horizontal force give us the sum of these series when r — a, the radius of the earth. The term of the order i is The observations of vertical force give us Z=* — > dr ' and the term of the order i in aZ is Hence the part due to external causes is and the part due to causes within the earth is _ r- The expansion of V has hitherto been calculated only for the mean value of V at or near certain epochs. No appreciable part of this mean value appears to be due to causes external to the earth. 471.] We do not yet know enough of the form of the expansion of the solar and lunar parts of the variations of V to determine by tills method whether any part of these variations arises from magnetic force acting from without. It is certain, however, as the calculations of MM. Stoney and Chambers have shewn, that the principal part of these variations cannot arise from any direct magnetic action of the sun or moon, supposing these bodies to be magnetic *. 472.] The principal changes in the magnetic force to which attention has been directed are as follows. * Professor Hornstein of Prague has discovered a periodic change in the magnetic elements, the period of which is 26.33 days, almost exactly equal to that of the synodic revolution of the sun, as deduced from the observation of sun-spots near his equator. This method of discovering the time of rotation of the unseen solid body of the sun by its effects on the magnetic needle is the first instalment of the repayment by Magnetism of its debt to Astronomy. Akad., Wien, June 1,5, 1871. See Proc. R.8., Nov. 16,1871. 126 TERRESTRIAL MAGNETISM. [473- I. The more Regular Variations. (1) The Solar variations, depending on the hour of the day and the time of the year. (2) The Lunar variations, depending on the moon's hour angle and on her other elements of position. (3) These variations do not repeat themselves in different years,, but seem to be subject to a variation of longer period of about eleven years. (4) Besides this, there is a secular alteration in the state of the earth's magnetism, which has been going on ever since magnetic observations have been made, and is producing changes of the magnetic elements of far greater magnitude than any of the varia tions of small period. II. The Disturbances. 473.] Besides the more regular changes, the magnetic elements are subject to sudden disturbances of greater or less amount. It is found that these disturbances are more powerful and frequent at one time than at another, and that at times of great disturbance the laws of the regular variations are masked, though they are very distinct at times of small disturbance. Hence great attention has been paid to these disturbances, and it has been found that dis turbances of a particular kind are more likely to occur at certain times of the day, and at certain seasons and intervals of time, though each individual disturbance appears quite irregular. Besides these more ordinary disturbances, there are occasionally times of excessive disturbance, in which the magnetism is strongly disturbed for a day or two. These are called Magnetic Storms. Individual disturbances have been sometimes observed at the same instant in stations widely distant. Mr. Airy has found that a large proportion of the disturbances at Greenwich correspond with the electric currents collected by electrodes placed in the earth in the neighbourhood, and are such as would be directly produced in the magnet if the earth-current, retaining its actual direction, were conducted through a wire placed underneath the magnet. It has been found that there is an epoch of maximum disturbance every eleven years, and that this appears to coincide with the epoch of maximum number of spots in the sun. 474.] The field of investigation into which we are introduced 474-] VARIATIONS AND DISTURBANCES. 127 by the study of terrestrial magnetism is as profound as it is ex tensive, We know that the sun and moon act on the earth's magnetism. It has been proved that this action cannot be explained by sup posing these bodies magnets. The action is therefore indirect. In the case of the sun part of it may be thermal action, but in the case of the moon we cannot attribute it to this cause. Is it pos sible that the attraction of these bodies, by causing strains in the interior of the earth, produces (Art. 447) changes in the magnetism already existing in the earth, and so by a kind of tidal action causes the semidiurnal variations ? But the amount of all these changes is very small compared with the great secular changes of the earth's magnetism. What cause, whether exterior to the earth or in its inner depth s, produces such enormous changes in the earth's magnetism, that its magnetic poles move slowly from one part of the globe to another ? When we consider that the intensity of the magnetization of the great globe of the earth is quite comparable with that which we produce with much difficulty in our steel magnets, these immense changes in so large a body force us to conclude that we are not yet acquainted with one of the most powerful agents in nature,, the scene of whose activity lies in those inner depths of the earth, to the knowledge of which we have so few means of access. PART IV. ELECTROMAGNETISM. CHAPTEK I. ELECTROMAGNETIC FORCE. 475.] IT had been noticed by many different observers that in certain cases magnetism is produced or destroyed in needles by electric discharges through them or near them, and conjectures of various kinds had been made as to the relation between mag netism and electricity, but the laws of these phenomena, and the form of these relations, remained entirely unknown till Hans Christian Orsted *, at a private lecture to a few advanced students at Copenhagen, observed that a wire connecting the ends of a voltaic battery affected a magnet in its vicinity. This discovery he published in a tract entitled Experiments circa effectum Conflictus Electrici in Acum Magneticam, dated July 21, 1820. Experiments on the relation of the magnet to bodies charged with electricity had been tried without any result till Orsted endeavoured to ascertain the effect of a wire heated by an electric current. He discovered, however, that the current itself, and not the heat of the wire, was the cause of the action, and that the e electric conflict acts in a revolving manner,' that is, that a magnet placed near a wire transmitting an electric current tends to set itself perpendicular to the wire, and with the same end always pointing forwards as the magnet is moved round the wire. 476.] It appears therefore that in the space surrounding a wire * See another account of Orsted's discovery in a letter from Professor Hansteen in the Life of Faraday by Dr. Bence Jones, vol. ii. p. 395. 478.] STRAIGHT CURRENT. 129 transmitting an electric current a magnet is acted on by forces depending on the position of the wire and on the strength of the current. The space in which these forces act may therefore be considered as a magnetic field, and we may study it in the same way as we have already studied the field in the neighbourhood of ordinary magnets, by tracing the course of the lines of magnetic force, and measuring the intensity of the force at every point. 477.] Let us begin with the case of an indefinitely long straight wire carrying an electric current. If a man were to place himself in imagination in the position of the wire, so that the current should flow from his head to his feet, then a magnet suspended freely before him would set itself so that the end which points north would, under the action of the current, point to his right hand. The lines of magnetic force are everywhere at right angles to planes drawn through the wire, and are there fore circles each in a plane perpendicular to the wire, which passes through its centre. The pole of a magnet which points north, if carried round one of these circles from left to right, would experience a force acting always in the direction of its motion. The other pole of the same magnet would experience a force in the opposite direction. 478.] To compare these forces let the wire be supposed vertical, and the current a de scending one, and let a magnet be placed on an apparatus which is free to rotate about a vertical axis coinciding with the wire. It is found that under these circumstances the current has no effect in causing the rotation of the apparatus as a whole about itself as an axis. Hence the action of the vertical current on the two poles of the magnet is such that the statical moments of the two forces about the current as an axis are equal and opposite. Let % and m2 be the strengths of the two poles, rl and r2 their distances from the axis of the wire, 5\ and T2 the intensities of the magnetic force due to the current at Fig. 21. the two poles respectively, then the force on m1 is and s since it is at right angles to the axis its moment l Similarly that of the force on the other pole is m2T2r2, and since there is no motion observed, mlT1rl + m2T2r2 = 0. VOL. II. K 130 ELECTROMAGNETIC FORCE. [479- But we know that in all magnets m-L + m^ = 0. Hence T^ = T2r2, or the electro magnetic force due to a straight current of infinite length is perpendicular to the current, and varies inversely as the distance from it. 479.] Since the product Tr depends on the strength of the current it may be employed as a measure of the current. This method of measurement is different from that founded upon elec trostatic phenomena, and as it depends on the magnetic phenomena produced by electric currents it is called the Electromagnetic system of measurement. In the electromagnetic system if i is the current, Tr = 2i. 480.] If the wire be taken for the axis of z} then the rectangular components of T are Here Xdx+Ydy+Zdz is a complete differential, being that of Hence the magnetic force in the field can be deduced from a potential function, as in several former instances, but the potential is in this case a function having an infinite series of values whose common difference is 4:iri. The differential coefficients of the potential with respect to the coordinates have, however, definite and single values at every point. The existence of a potential function in the field near an electric current is not a self-evident result of the principle of the con servation of energy, for in all actual currents there is a continual expenditure of the electric energy of the battery in overcoming the resistance of the wire, so that unless the amount of this expenditure were accurately known, it might be suspected that part of the energy of the battery may be employed in causing work to be done on a magnet moving in a cycle. In fact, if a magnetic pole, m, moves round a closed curve which embraces the wire, work is actually done to the amount of 4 TT m i. It is only for closed paths which do not embrace the wire that the line-integral of the force vanishes. We must therefore for the present consider the law of force and the existence of a potential as resting on the evidence of the experiment already described. 483.] MAGNETIC POTENTIAL. 131 481.] If we consider the space surrounding an infinite straight line we shall see that it is a cyclic space, because it returns into itself. If we now conceive a plane, or any other surface, com mencing at the straight line and extending on one side of it to infinity, this surface may be regarded as a diaphragm which reduces the cyclic space to an acyclic one. If from any fixed point lines be drawn to any other point without cutting the diaphragm, and the potential be defined as the line-integral of the force taken along one of these lines, the potential at any point will then have a single definite value. The magnetic field is now identical in all respects with that due to a magnetic shell coinciding with this surface, the strength of the shell being i. This shell is bounded on one edge by the infinite straight line. Tho other parts of its boundary are at an infinite distance from the part of the field under consideration. 482.] In all actual experiments the current forms a closed circuit of finite dimensions. We shall therefore compare the magnetic action of a finite circuit with that of a magnetic shell of which the circuit is the bounding edge. It has been shewn by numerous experiments, of which the earliest are those of Ampere, and the most accurate those of Weber, that the magnetic action of a small plane circuit at distances which are great compared with the dimensions of the circuit is the same as that of a magnet whose axis is normal to the plane of the circuit, and whose magnetic moment is equal to the area of the circuit multiplied by the strength of the current. If the circuit be supposed to be filled up by a surface bounded by the circuit and thus forming a diaphragm, and if a magnetic shell of strength i coinciding with this surface be substituted for the electric current, then the magnetic action of the shell on all distant points will be identical with that of the current. 483.] Hitherto we have supposed the dimensions of the circuit to be small compared with the distance of any part of it from the part of the field examined. We shall now suppose the circuit to be of any form and size whatever, and examine its action at any point P not in the conducting wire itself. The following method, which has important geometrical applications, was introduced by Ampere for this purpose. Conceive any surface S bounded by the circuit and not passing through the point P. On this surface draw two series of lines crossing each other so as to divide it into elementary portions, the K 2 132 ELECTROMAGNETIC FORCE. [484. dimensions of which are small compared with their distance from P, and with the radii of curvature of the surface. Round each of these elements conceive a current of strength i to flow, the direction of circulation being the same in all the ele ments as it is in the original circuit. Along every line forming the division between two contiguous elements two equal currents of strength i flow in opposite direc tions. The effect of two equal and opposite currents in the same place is absolutely zero, in whatever aspect we consider the currents. Hence their magnetic effect is zero. The only portions of the elementary circuits which are not neutralized in this way are those which coincide with the original circuit. The total effect of the elementary circuits is therefore equivalent to that of the original circuit. 484.] Now since each of the elementary circuits may be con sidered as a small plane circuit whose distance from P is great compared with its dimensions, we may substitute for it an ele mentary magnetic shell of strength i whose bounding edge coincides with the elementary circuit. The magnetic effect of the elementary shell on P is equivalent to that of the elementary circuit. The whole of the elementary shells constitute a magnetic shell of strength i, coinciding with the surface 8 and bounded by the original circuit, and the magnetic action of the whole shell on P is equivalent to that of the circuit. It is manifest that the action of the circuit is independent of the form of the surface S9 which was drawn in a perfectly arbitrary manner so as to fill it up. We see from this that the action of a magnetic shell depends only on the form of its edge and not on the form of the shell itself. This result we obtained before, at Art. 410, but it is instructive to see how it may be deduced from electromagnetic considerations. The magnetic force due to the circuit at any point is therefore identical in magnitude and direction with that due to a magnetic shell bounded by the circuit and not passing through the point, the strength of the shell being numerically equal to that of the current. The direction of the current in the circuit is related to the direction of magnetization of the shell, so that if a man were to stand with his feet on that side of the shell which we call the positive side, and which tends to point to the north, the current in front of him would be from right to left. 486.] MAGNETIC POTENTIAL DUE TO A CIRCUIT. 133 485.] The magnetic potential of the circuit, however, differs from that of the magnetic shell for those points which are in the substance of the magnetic shell. If co is the solid angle subtended at the point P by the magnetic shell, reckoned positive when the positive or austral side of the shell is next to P, then the magnetic potential at any point not in the shell itself is coc/>, where $ is the strength of the shell. At any point in the substance of the shell itself we may suppose the shell divided into two parts whose strengths are ^ and c/>2, where </>! -f c/>2 = c/>, such that the point is on the positive side of c^1 and on the negative side of c/>2 . The potential at this point is On the negative side of the shell the potential becomes $ (co— • 47r). In this case therefore the potential is continuous, and at every point has a single determinate value. In the case of the electric circuit, on the other hand, the magnetic potential at every point not in the conducting wire itself is equal to ia>, where i is the strength of the current, and co is the solid angle subtended by the circuit at the point, and is reckoned positive when the current, as seen from P, circulates in the direction opposite to that of the hands of a watch. The quantity ^co is a function having an infinite series of values whose common difference is 4 TT i. The differential coefficients of id) with respect to the coordinates have, however, single and de terminate values for every point of space. 486.] If a long thin flexible solenoidal magnet were placed in the neighbourhood of an electric circuit, the north and south ends of the solenoid would tend to move in opposite directions round the wire, and if they were free to obey the magnetic force the magnet would finally become wound round the wire in a close coil. If it were possible to obtain a magnet having only one pole, or poles of unequal strength, such a magnet would be moved round and round the wire continually in one direction, but since the poles of every magnet are equal and opposite, this result can never occur. Faraday, however, has shewn how to produce the continuous rotation of one pole of a magnet round an electric current by making it possible for one pole to go round and round the current while the other pole does not. That this process may be repeated in definitely, the body of the magnet must be transferred from one side of the current to the other once in each revolution. To do this without interrupting the flow of electricity, the current is split 134 ELECTROMAGNETIC FORCE. into two branches, so that when one branch is opened to let the magnet pass the current continues to flow through the other. Faraday used for this purpose a circular trough of mercury, as shewn in Fig. 23, Art. 491. The current enters the trough through the wire AB, it is divided at B, and after flowing through the arcs £QP and BRP it unites at P, and leaves the trough through the wire PO, the cup of mercury 0, and a vertical wire beneath 0, down which the current flows. The magnet (not shewn in the figure) is mounted so as to be capable of revolving about a vertical axis through 0, and the wire OP revolves with it. The body of the magnet passes through the aperture of the trough, one pole, say the north pole, being beneath the plane of the trough, and the other above it. As the magnet and the wire OP revolve about the vertical axis, the current is gradually transferred from the branch of the trough which lies in front of the magnet to that which lies behind it, so that in every complete revolution the magnet passes from one side of the current to the other. The north pole of the magnet revolves about the descending current in the direction N.E.S.W. and if w, o>' are the solid angles (irrespective of sign) subtended by the circular trough at the two poles, the work done by the electromagnetic force in a complete revolution is mi (ITT — o> — a/), where m is the strength of either pole, and i the strength of the current. 487.] Let us now endeavour to form a notion of the state of the magnetic field near a linear electric circuit. Let the value of o>, the solid angle subtended by the circuit, be found for every point of space, and let the surfaces for which co is constant be described. These surfaces will be the equipotential surfaces. Each of these surfaces will be bounded by the circuit, and any two surfaces, o^ and o>2, will meet in the circuit at an angle i(o>1-<i)2). Figure XVIII, at the end of this volume, represents a section of the equipotential surfaces due to a circular current. The small circle represents a section of the conducting wire, and the hori zontal line at the bottom of the figure is the perpendicular to the plane of the circular current through its centre. The equipotential surfaces, 24 of which are drawn corresponding to a series of values of CD differing by — > are surfaces of revolution, having this line for 489.] ACTION OF A CIRCUIT ON A MAGNETIC SYSTEM. 135 their common axis. They are evidently oblate figures, being flat tened in the direction of the axis. They meet each other in the line of the circuit at angles of 1 5°. The force acting on a magnetic pole placed at any point of an equipotential surface is perpendicular to this surface, and varies inversely as the distance between consecutive surfaces. The closed curves surrounding the section of the wire in Fig. XVIII are the lines of force. They are copied from Sir W. Thomson's Paper on 'Vortex Motion*.' See also Art. 702. Action of an Electric Circuit on any Magnetic System. 488.] We are now able to deduce the action of an electric circuit on any magnetic system in its neighbourhood from the theory of magnetic shells. For if we construct a magnetic shell, whose strength is numerically equal to the strength of the current, and whose edge coincides in position with the circuit, while the shell itself does not pass through any part of the magnetic system, the action of the shell on the magnetic system will be identical with that of the electric circuit. Reaction of the Magnetic System on the Electric Circuit. 489.] From this, applying the principle that action and reaction are equal and opposite, we conclude that the mechanical action of the magnetic system on the electric circuit is identical with its action on a magnetic shell having the circuit for its edge. The potential energy of a magnetic shell of strength $ placed in a field of magnetic force of which the potential is T, is, by Art. 410, T- -J- > x dy dz' where I, m, n are the direction-cosines of the normal drawn from the positive side of the element dS of the shell, and the integration is extended over the surface of the shell. Now the surface-integral where #, I, c are the components of the magnetic induction, re presents the quantity of magnetic induction through the shell, or, * Trans. R. 8. Edin., vol. xxv. p. 217, (1869). 136 ELECTROMAGNETIC FORCE. [490. in the language of Faraday, the number of lines of magnetic in duction, reckoned algebraically, which pass through the shell from the negative to the positive side, lines which pass through the shell in the opposite direction being reckoned negative. Remembering that the shell does not belong to the magnetic system to which the potential V is due, and that the magnetic force is therefore equal to the magnetic induction, we have dV dV dV a= -- =-, b= -- =-, c = -- j-> dx dy dz and we may write the value of M, M=-<t>N. If bx1 represents any displacement of the shell, and X1 the force acting on the shell so as to aid the displacement, then by the principle of conservation of energy, "= 0, ^ or X = 6 -- r §x We have now determined the nature of the force which cor responds to any given displacement of the shell. It aids or resists that displacement accordingly as the displacement increases or diminishes N, the number of lines of induction which pass through the shell. The same is true of the equivalent electric circuit. Any dis placement of the circuit will be aided or resisted accordingly as it increases or diminishes the number of lines of induction which pass through the circuit in the positive direction. We must remember that the positive direction of a line of magnetic induction is the direction in which the pole of a magnet which points north tends to move along the line, and that a line of induction passes through the circuit in the positive direction, when the direction of the line of induction is related to the direction of the current of vitreous electricity in the circuit as the longitudinal to the rotational motion of a right-handed screw. See Art. 23. 490.] It is manifest that the force corresponding to any dis placement of the circuit as a whole may be deduced at once from the theory of the magnetic shell. But this is not all. If a portion of the circuit is flexible, so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into 49O.] FOKCE ACTING ON A CUKRENT. 137 a sufficient number of portions connected by flexible joints. Hence we conclude that if by the displacement of any portion of the circuit in a given direction the number of lines of induction which pass through the circuit can be increased, this displacement will be aided by the electromagnetic force acting on the circuit. Every portion of the circuit therefore is acted on by a force urging it across the lines of magnetic induction so as to include a greater number of these lines within the embrace of the circuit, and the work done by the force during this displacement is numerically equal to the number of the additional lines of in duction multiplied by the strength of the current. Let the element ds of a circuit, in which a current of strength i is flowing, be moved parallel to itself through a space §x, it will sweep out an area in the form of a parallelogram whose sides are parallel and equal to ds and bx respectively. If the magnetic induction is denoted by 33, and if its direction makes an angle e with the normal to the parallelogram, the value of the increment of N corresponding to the displacement is found by multiplying the area of the parallelogram by 33 cos e. The result of this operation is represented geometrically by the volume of a parallelepiped whose edges represent in magnitude and direction 8ar, ds, and 33, and it is to be reckoned positive if when we point in these three directions in the order here given the pointer moves round the diagonal of the parallelepiped in the direction of the hands of a watch. The volume of this parallelepiped is equal to Xb%. If 0 is the angle between ds and 33, the area of the parallelogram is ds . 33 sin 6, and if 77 is the angle which the displacement b% makes with the normal to this parallelogram, the volume of the parallelepiped is ds . 33 sin 0 . bx cos 77 — 8 N. Now X bx = i 5 N = i ds . 33 sin 0 fix cos 77, and X =. i ds . 33 sin 0 cos 77 is the force which urges ds, resolved in the direction 8#. The direction of this force is therefore perpendicular to the paral lelogram, and is equal to i . ds . 33 sin 0. This is the area of a parallelogram whose sides represent in mag nitude and direction i ds and 33. The force acting on ds is therefore represented in magnitude by the area of this parallelogram, and in direction by a normal to its plane drawn in the direction of the longitudinal motion of a right-handed screw, the handle of which 138 ELECTROMAGNETIC FORCE. [491. South East is turned from the direction of the current ids to that of the magnetic induction 33. We may express in the language of Quaternions, both the direction and West ^ J^ North the magnitude of this force by saying that it is the vector part of the result of multiplying the vector ids, the element of the current, by the vector 33, the magnetic induction. 491.] We have thus completely de termined the force which acts on any portion of an electric circuit placed in a magnetic field. If the circuit is moved in any way so that, after assuming various forms and positions, it returns to its original place, the strength of the current remaining constant during the motion, the whole amount of work done by the electromagnetic forces will be zero. Since this is true of any cycle of motions of the circuit, it follows that it is impossible to maintain by electromagnetic forces a motion of continuous rotation in any part of a linear circuit of constant strength against the resistance of friction, &c. It is possible, however, to produce continuous rotation provided that at some part of the course of the electric current it passes from one conductor to another which slides or glides over it. When in a circuit there is sliding contact of a conductor over the surface of a smooth solid or a fluid, the circuit can no longer be considered as a single linear circuit of constant strength, but must be regarded as a system of two or of some greater number of circuits of variable strength, the current being so distributed among them that those for which N is increasing have currents in the positive direc tion, while those for which N is diminishing have currents in the negative direction. Thus, in the apparatus represented in Fig. 23, OP is a moveable conductor, one end of which rests in a cup of mercury 0, while the other dips into a circular trough of mercury concentric with 0. Fig. 23. 492.] CONTINUOUS KOTATION. 139 The current i enters along AB, and divides in the circular trough into two parts, one of which, #, flows along the arc BQP, while the other, y, flows along BRP. These currents, uniting at P, flow along the moveable conductor PO and the electrode OZ to the zinc end of the battery. The strength of the current along OP and OZ is x + y or i. Here we have two circuits, ABQPOZ, the strength of the current in which is x, flowing in the positive direction, and ABRPOZ, the strength of the current in which is y> flowing in the negative direction. Let 23 be the magnetic induction, and let it be in an upward direction, normal to the plane of the circle. While OP moves through an angle 9 in the direction opposite to that of the hands of a watch, the area of the first circuit increases by i#P2. 0, and that of the second diminishes by the same quantity. Since the strength of the current in the first circuit is #, the work done by it is J x. OP2. 0.33, and since the strength of the second is — y, the work done by it is \y.OP2. 6 33. The whole work done is therefore i(tf + 3/)OP2.033 or ii.OP2.0«B, depending only on the strength of the current in PO. Hence, if i is maintained constant, the arm OP will be carried round and round the circle with a uniform force whose moment is \i .OP2 53. If, as in northern latitudes, 33 acts downwards, and if the current is inwards, the rotation will be in the negative direction, that is, in the direction PQBR. 492.] We are now able to pass from the mutual action of magnets and currents to the action of one current on another. For we know that the magnetic properties of an electric circuit C± , with respect to any magnetic system M2, are identical with those of a magnetic shell S19 whose edge coincides with the circuit, and whose strength is numerically equal to that of the electric current. Let the magnetic system M2 be a magnetic shell S2, then the mutual action between ^ and 82 is identical with that between ^ and a circuit C2, coinciding with the edge of S2 and equal in numerical strength, and this latter action is identical with that between Ct and C2. Hence the mutual action between two circuits, Cl and C2) is identical with that between the corresponding magnetic shells Sl and S2. We have already investigated, in Art. 423, the mutual action 140 ELECTROMAGNETIC FORCE. [493- of two magnetic shells whose edges are the closed curves s1 and s2 . /**2 /**! COS 6 If we make M= I - &,<&«, J0 ^0 ? where e is the angle between the directions of the elements ds1 and ds2, and r is the distance between them, the integration being extended once round s.2 and once round slf and if we call M the potential of the two closed curves ^ and <s2, then the potential energy due to the mutual action of two magnetic shells whose strengths are ^ and a'2 bounded by the two circuits is and the force X, which aids any displacement 8#, is The whole theory of the force acting on any portion of an electric circuit due to the action of another electric circuit may be deduced from this result. 493.] The method which we have followed in this chapter is that of Faraday. Instead of beginning, as we shall do, following Ampere, in the next chapter, with the direct action of a portion of one circuit on a portion of another, we shew, first, that a circuit produces the same effect on a magnet as a magnetic shell, or, in other words, we determine the nature of the magnetic field due to the circuit. We shew, secondly, that a circuit when placed in any magnetic field experiences the same force as a magnetic shell. We thus determine the force acting on the circuit placed in any magnetic field. Lastly, by supposing the magnetic field to be due to a second electric circuit we determine the action of one circuit on the whole or any portion of the other. 494.] Let us apply this method to the case of a straight current of infinite length acting on a portion of a parallel straight con ductor. Let us suppose that a current i in the first conductor is flowing vertically downwards. In this case the end of a magnet which points north will point to the right-hand of a man looking at it from the axis of the current. The lines of magnetic induction are therefore horizontal circles, having their centres in the axis of the current, and their positive direction is north, east, south, west. Let another descending vertical current be placed due west of the first. The lines of magnetic induction clue to the first current 496.] ELECTROMAGNETIC MEASURE OF A CURRENT. 141 are here directed towards the north. The direction of the force acting on the second current is to be determined by turning the handle of a right-handed screw from the nadir, the direction of the current, to the north, the direction of the magnetic induction. The screw will then move towards the east, that is, the force acting on the second current is directed towards the first current, or, in general, since the phenomenon depends only on the relative position of the currents, two parallel currents in the same direction attract each other. In the same way we may shew that two parallel currents in opposite directions repel one another. 495.] The intensity of the magnetic induction at a distance r from a straight current of strength i is, as we have shewn in Art. 479, i 2-. r Hence, a portion of a second conductor parallel to the first, and carrying a current i' in the same direction, will be attracted towards the first with a force where a is the length of the portion considered, and r is its distance from the first conductor. Since the ratio of a to r is a numerical quantity independent of the absolute value of either of these lines, the product of two currents measured in the electromagnetic system must be of the dimensions of a force, hence the dimensions of the unit current are [i] = [F*] = \_M* L* T-*]. 496.] Another method of determining the direction of the force which acts on a current is to consider the relation of the magnetic action of the current to that of other currents and magnets. If on one side of the wire which carries the current the magnetic action due to the current is in the same or nearly the same direction as that due to other currents, then, on the other side of the wire, these forces will be in opposite or nearly opposite directions, and the force acting on the wire will be from the side on which the forces strengthen each other to the side on which they oppose each other. Thus, if a descending current is placed in a field of magnetic force directed towards the north, its magnetic action will be to the north on the west side, and to the south on the east side. Hence the forces strengthen each other on the west side and oppose each 142 ELECTROMAGNETIC FORCE. [497- other on the east side, and the current will therefore be acted on by a force from west to east. See Fig. 22, p. 138. In Fig. XVII at the end of this volume the small circle represents a section of the wire carrying a descending current, and placed in a uniform field of magnetic force acting towards the left-hand of the figure. The magnetic force is greater below the wire than above it. It will therefore be urged from the bottom towards the top of the figure. 497.] If two currents are in the same plane but not parallel, we may apply this principle. Let one of the conductors be an infinite straight wire in the plane of the paper, supposed horizontal. On the right side of the current the magnetic force acts downward, and on the left side it acts upwards. The same is true of the mag netic force due to any short portion of a second current in the same plane. If the second current is on the right side of the first, the magnetic forces will strengthen each other on its right side and oppose each other on its left side. Hence the second current will be acted on by a force urging it from its right side to its left side. The magnitude of this force depends only on the position of the second current and not on its direction. If the second current is on the left side of the first it will be urged from left to right. Hence, if the second current is in the same direction as the first it is attracted, if in the opposite direction it is repelled, if it flows at right angles to the first and away from it, it is urged in the direction of the first current, and if it flows toward the first current, it is urged in the direction opposite to that in which the first current flows. In considering the mutual action of two currents it is not neces sary to bear in mind the relations between electricity and magnetism which we have endeavoured to illustrate by means of a right-handed screw. Even if we have forgotten these relations we shall arrive at correct results, provided we adhere consistently to one of the two possible forms of the relation. 498.] Let us now bring together the magnetic phenomena of the electric circuit so far as we have investigated them. We may conceive the electric circuit to consist of a voltaic battery, and a wire connecting its extremities, or of a thermoelectric arrangement, or of a charged Leyden jar with a wire connecting its positive and negative coatings, or of any other arrangement for producing an electric current along a definite path. The current produces magnetic phenomena in its neighbourhood. 499-] RECAPITULATION. 143 If any closed curve be drawn, and the line-integral of the magnetic force taken completely round it, then, if the closed curve is not linked with the circuit, the line-integral is zero, but if it is linked with the circuit, so that the current i flows through the closed curve, the line-integral is 4 IT i, and is positive if the direction of integration round the closed curve would coincide with that of the hands of a watch as seen by a person passing through it in the direction in which the electric current flows. To a person moving along the closed curve in the direction of integration, and passing through the electric circuit, the direction of the current would appear to be that of the hands of a watch. We may express this in another way by saying that the relation between the direc tions of the two closed curves may be expressed by describing a right-handed screw round the electric circuit and a right-handed screw round the closed curve. If the direction of rotation of the thread of either, as we pass along it, coincides with the positive direction in the other, then the line-integral will be positive, and in the opposite case it will be negative. Fig. 24. Relation between the electric current and the lines of magnetic induction indicated by a right-handed screw. 499.] Note. — The line-integral 4 TT i depends solely on the quan tity of the current, and not on any other thing whatever. It does not depend on the nature of the conductor through which the current is passing, as, for instance, whether it be a metal or an electrolyte, or an imperfect conductor. We have reason for believing that even when there is no proper conduction, but 144 ELECTROMAGNETIC FORCE. [5OO. merely a variation of electric displacement, as in the glass of a Leyden jar during charge or discharge, the magnetic effect of the electric movement is precisely the same. Again , the value of the line-integral 4 TT i does not depend on the nature of the medium in which the closed curve is drawn. It is the same whether the closed curve is drawn entirely through air, or passes through a magnet, or soft iron, or any other sub stance, whether paramagnetic or diamagnetic. 500.] When a circuit is placed in a magnetic field the mutual action between the current and the other constituents of the field depends on the surface-integral of the magnetic induction through any surface bounded by that circuit. If by any given motion of the circuit, or of part of it, this surface-integral can be increased, there will be a mechanical force tending to move the conductor or the portion of the conductor in the given manner. The kind of motion of the conductor which increases the surface- integral is motion of the conductor perpendicular to the direction of the current and across the lines of induction. If a parallelogram be drawn, whose sides are parallel and pro portional to the strength of the current at any point, and to the magnetic induction at the same point, then the force on unit of length of the conductor is numerically equal to the area of this parallelogram, and is perpendicular to its plane, and acts in the direction in which the motion of turning the handle of a right- handed screw from the direction of the current to the direction of the magnetic induction would cause the screw to move. Hence we have a new electromagnetic definition of a line of magnetic induction. It is that line to which the force on the conductor is always perpendicular. It may also be defined as a line along which, if an electric current be transmitted, the conductor carrying it will experience no force. 501.] It must be carefully remembered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, not on the electric current, but on the con ductor which carries it. If the conductor be a rotating disk or a fluid it will move in obedience to this force, and this motion may or may not be accompanied with a change of position of the electric current which it carries. But if the current itself be free to choose any path through a fixed solid conductor or a network of wires, then, when a constant magnetic force is made to act on the system, the path of the current through the conductors is not permanently 501-] RECAPITULATION. 145 altered, but after certain transient phenomena, called induction currents, have subsided, the distribution of the current will be found to be the same as if no magnetic force were in action. The only force which acts on electric currents is electromotive force, which must be distinguished from the mechanical force which is the subject of this chapter. Fig. 25. Relations between the positive directions of motion and of rotation indicated by three right-handed screws. VOL. II. CHAPTER II. AMPERE'S INVESTIGATION OF THE MUTUAL ACTION OF ELECTRIC CURRENTS. 502.] WE have considered in the last chapter the nature of the magnetic field produced by an electric current; and the mechanical action on a conductor carrying an electric current placed in a mag netic field. From this we went on to consider the action of one electric circuit upon another, by determining the action on the first due to the magnetic field produced by the second. But the action of one circuit upon another was originally investigated in a direct manner by Ampere almost immediately after the publication of Orsted's discovery. We shall therefore give an outline of Ampere's method, resuming the method of this treatise in the next chapter. The ideas which guided Ampere belong to the system which admits direct action at a distance, and we shall find that a remark able course of speculation and investigation founded on these ideas has been carried on by Gauss, Weber, J. Neumann, Riemann, Betti, C. Neumann, Lorenz, and others, with very remarkable results both in the discovery of new facts and in the formation of a theory of electricity. See Arts. 846-866. The ideas which I have attempted to follow out are those of action through a medium from one portion to the contiguous portion. These ideas were much employed by Faraday, and the development of them in a mathematical form, and the comparison of the results with known facts, have been my aim in several published papers. The comparison, from a philosophical point of view, of the results of two methods so completely opposed in their first prin ciples must lead to valuable data for the study of the conditions of scientific speculation. 503.] Ampere's theory of the mutual action of electric currents is founded on four experimental facts and one assumption. 505.] AMPERE'S SCIENTIFIC METHOD. 147 Ampere's fundamental experiments are all of them examples of what has been called the null method of comparing' forces. See Art. 214. Instead of measuring the force by the dynamical effect of communicating1 motion to a body, or the statical method of placing it in equilibrium with the weight of a body or the elasticity of a fibre, in the null method two forces, due to the same source, are made to act simultaneously on a body already in equilibrium, and no effect is produced, which shews that these forces are them selves in equilibrium. This method is peculiarly valuable for comparing the effects of the electric current when it passes through circuits of different forms. By connecting all the conductors in one continuous series, we ensure that the strength of the current is the same at every point of its course, and since the current begins everywhere throughout its course almost at the same instant, we may prove that the forces due to its action on a suspended body are in equilibrium by observing that the body is not at all affected by the starting or the stopping of the current. 504.] Ampere's balance consists of a light frame capable of revolving1 about a vertical axis, and carrying1 a wire which forms two circuits of equal area, in the same plane or in parallel planes, in which the current flows in opposite directions. The object of this arrangement is to get rid of the effects of terrestrial magnetism on the conducting wire. When an electric circuit is free to move it tends to place itself so as to embrace the largest possible number of the lines of induction. If these lines are due to terrestrial magnetism, this position, for a circuit in a vertical plane, will be when the plane of the circuit is east and west, and when the direction of the current is opposed to the apparent course of the sun. By rigidly connecting two circuits of equal area in parallel planes, in which equal currents run in opposite directions, a combination is formed which is unaffected by terrestrial magnetism, and is therefore called an Astatic Combination, see Fig. 26. It is acted on, however, by forces arising from currents or magnets which are so near it that they act differently on the two circuits. 505.] Ampere's first experiment is on the effect of two equal currents close together in opposite directions. A wire covered with insulating material is doubled on itself, and placed near one of the circuits of the astatic balance. When a current is made to pass through the wire and the balance, the equilibrium of the balance remains undisturbed, shewing that two equal currents close together L 2 148 AMPERES THEORY. [506. in opposite directions neutralize each other. If, instead of two wires side by side, a wire be insulated in the middle of a metal Fig. 26. tube, and if the current pass through the wire and back by the tube, the action outside the tube is not only approximately but accurately null. This principle is of great importance in the con struction of electric apparatus, as it affords the means of conveying the current to and from any galvanometer or other instrument in such a way that no electromagnetic effect is produced by the current on its passage to and from the instrument. In practice it is gene rally sufficient to bind the wires together, care being taken that they are kept perfectly insulated from each other, but where they must pass near any sensitive part of the apparatus it is better to make one of the conductors a tube and the other a wire inside it. See Art. 683. 506.] In Ampere's second experiment one of the wires is bent and crooked with a number of small sinuosities, but so that in every part of its course it remains very near the straight wire. A current, flowing through the crooked wire and back again through the straight wire, is found to be without influence on the astatic balance. This proves that the effect of the current running through any crooked part of the wire is equivalent to the same current running in the straight line joining its extremities, pro vided the crooked line is in no part of its course far from the straight one. Hence any small element of a circuit is equivalent to two or more component elements, the relation between the component elements and the resultant element being the same as that between component and resultant displacements or velocities. 507.] In the third experiment a conductor capable of moving 508.] FOUK EXPERIMENTS. 149 only in the direction of its length is substituted for the astatic balance, the current enters the conductor and leaves it at fixed points of space, and it is found that no closed circuit placed in the neighbourhood is able to move the conductor. Fig. 27. The conductor in this experiment is a wire in the form of a circular arc suspended on a frame which is capable of rotation about a vertical axis. The circular arc is horizontal, and its centre coincides with the vertical axis. Two small troughs are filled with mercury till the convex surface of the mercury rises above the level of the troughs. The troughs are placed under the circular arc and adjusted till the mercury touches the wire, which is of copper well amalgamated. The current is made to enter one of these troughs, to traverse the part of the circular arc between the troughs, and to escape by the other trough. Thus part of the circular arc is traversed by the current, and the arc is at the same time capable of moving with considerable freedom in the direc tion of its length. Any closed currents or magnets may now be made to approach the moveable conductor without producing the slightest tendency to move it in the direction of its length. 508.] In the fourth experiment with the astatic balance two circuits are employed, each similar to one of those in the balance, but one of them, C, having dimensions n times greater, and the other, A, n times less. These are placed on opposite sides of the circuit of the balance, which we shall call B, so that they are similarly placed with respect to it, the distance of C from B being n times greater than the distance of B from A. The direction and 150 AMPERES THEORY. [5o8. strength of the current is the same in A and C. Its direction in B may be the same or opposite. Under these circumstances it is found that B is in equilibrium under the action of A and C, whatever be the forms and distances of the three circuits, provided they have the relations given above. Since the actions between the complete circuits may be considered to be due to actions between the elements of the circuits, we may use the following method of determining the law of these actions. Let Alt BI} Cv Fig. 28, be corresponding elements of the three circuits, and let A2, B2, C2 be also corresponding elements in an other part of the circuits. Then the situation of B± with respect to A2 is similar to the situation of C^ with respect to B.2) but the 0 u distance and dimensions of Cl and B2 are n times the distance and dimensions of Bl and A2i respectively. If the law of electromag netic action is a function of the distance, then the action, what ever be its form or quality, between Bl and A.2, may be written and that between C1 and B2 ' where #, b, c are the strengths of the currents in A, B, C. But A — CB and a = c. Hence ^ = Clt = B and this is equal to F by experiment, so that we have or, the force varies inversely as the square of the distance. 511.] FOKCE- BETWEEN TWO ELEMENTS. 151 509.] It may be observed with reference to these experiments that every electric current forms a closed circuit. The currents used by Ampere, being produced by the voltaic battery, were of course in closed circuits. It might be supposed that in the case of the current of discharge of a conductor by a spark we might have a current forming an open finite line, but according to the views of this book even this case is that of a closed circuit. No experiments on the mutual action of unclosed currents have been made. Hence no statement about the mutual action of two ele ments of circuits can be said to rest on purely experimental grounds. It is true we may render a portion of a circuit moveable, so as to ascertain the action of the other currents upon it, but these cur rents, together with that in the moveable portion, necessarily form closed circuits, so that the ultimate result of the experiment is the action of one or more closed currents upon the whole or a part of a closed current. 510.] In the analysis of the phenomena, however, we may re gard the action of a closed circuit on an element of itself or of another circuit as the resultant of a number of separate forces, depending on the separate parts into which the first circuit may be conceived, for mathematical purposes, to be divided. This is a merely mathematical analysis of the action, and is therefore perfectly legitimate, whether these forces can really act separately or not. 511.] We shall begin by considering the purely geometrical relations between two lines in space representing the circuits, and between elementary portions of these lines. Let there be two curves in space in each of which a fixed point is taken from which the arcs are measured in a defined direction along the curve. Let A, A' be these points. Let PQ and P Q' be elements of the two curves. Let AP=s, A'P'=s and let the distance PPf be de- Fig< 29' noted by r. Let the angle P*PQ be denoted by 0, and PP'(g by Qf, and let the angle between the planes of these angles be denoted by rj. The relative position of the two elements is sufficiently defined by their distance r and the three angles 0, 6' , and r/, for if these be 152 AMPERES THEORY. given their relative position is as completely determined as if they formed part of the same rigid body. 512.] If we use rectangular coordinates and make #, y, z the coordinates of P, and of, y', z' those of P', and if we denote by I, m, n and by I', m', n' the direction-cosines of PQ, and of P'Q' re spectively, then dx j dy dz -» -J-—1) -f- = m, = n, as as as dx' ,, dy' , dz' , (2) and I \x' — x) + m (y' — y} + n (z' — z) = rcos0, I' (xf — x] -f- m' (y' — y) -f n (zf — z) = — rcos6\ (3) II' -f mm' -f nn' = cos e, where e is the angle between the directions of the elements them selves, and cos e = — cos 6 cos 6' + sin 0 sin (f cos rj. (4) r* = (af—x)* + (tf—y)* + (af-z)2, (5) Again . whence + -*, dr . , . dx , , N dy , , . dz - = -(* -*) _(y _,) -(, -z) = — rcosO. dr Similarly r= (^- . i . < -^) +(/-*) \ (6) = — r cos 6 ; and differentiating r -=- with respect to /, dr dr dx dx' dy dy dz dz' CvS CtS CvS CvS CvS CvS CtS dS (7) — — (II' -j- mm' + n n'} = — cos e. j We can therefore express the three angles 0, 6', and r;, and the auxiliary angle e in terms of the differential coefficients of r with respect to s and s' as follows, dr cos 0 = dr cose = — r dr dr d2r sin 6 sin 6' cos 77 = — r - (8) 513-] GEOMETRICAL RELATIONS OF TWO ELEMENTS. 153 513.] We shall next consider in what way it is mathematically conceivable that the elements PQ and PQ' might act on each other, and in doing so we shall not at first assume that their mutual action is necessarily in the line joining them. We have seen that we may suppose each element resolved into other elements, provided that these components, when combined according to the rule of addition of vectors, produce the original element as their resultant. We shall therefore consider ds as resolved into cos 6 ds — a in the direction of r, and sin 6 ds = /3 fl ^ in a direction perpendicular to \ / *^\/ T in the plane P'PQ. p « «'>"' We shall also consider ds' as resolved into cos Q' els' = a in the direction of r reversed, mntfoO8ri(tf=P in a direction parallel to that in which /3 was measured, and sin 0' sin 17 els' = y in a direction perpendicular to a and /3'. Let us consider the action between the components a and j3 on the one hand, and a, /3', / on the other. (1) a and a are in the same straight line. The force between them must therefore be in this line. We shall suppose it to be an attraction = Aa<xii't where A is a function of r, and i} i' are the intensities of the currents in ds and els' respectively. This expression satisfies the condition of changing sign with i and with i'm (2) /3 and (3' are parallel to each other and perpendicular to the line joining them. The action between them may be written This force is evidently in the line joining (3 and /3', for it must be in the plane in which they both lie, and if we were to measure (3 and ft in the reversed direction, the value of this expression would remain the same, which shews that, if it represents a force, that force has no component in the direction of f3, and must there fore be directed along r. Let us assume that this expression, when positive, represents an attraction. (3) /3 and y are perpendicular to each other and to the line joining them. The only action possible between elements so related is a couple whose axis is parallel to T. We are at present engaged with forces, so we shall leave this out of account. (4) The action of a and /3', if they act on each other, must be expressed by 154 AMPERE'S THEORY. The sign of this expression is reversed if we reverse the direction in which we measure j3'. It must therefore represent either a force in the direction of ft', or a couple in the plane of a and /3'. As we are not investigating couples, we shall take it as a force acting on a in the direction of ft'. There is of course an equal force acting on /3' in the opposite direction. We have for the same reason a force Cay'ii' acting on a in the direction of y', and a force acting on /3 in the opposite direction. 514.] Collecting our results, we find that the action on ds is compounded of the following forces, X = (Aaa' + B (3fi')ii' in the direction of r, Y— C(a(B' — aj3)ii' in the direction of (3, (9) and Z — C ay ii' in the direction of y'. Let us suppose that this action on ds is the resultant of three forces, Rii'dsds' acting in the direction of r, Sii'dsds' acting in the direction of ds, and S'ii'dsds' acting in the direction of ds' , then in terms of 6, d', and 77, R = A cos 0 cos 0' + J9sin0sin0'cosr7, In terms of the differential coefficients of .r o, r ^ + G-yyJ & = — G--1 ds ds J In terms of I, m, n, and I', m', n'9 R =- where f, ??, fare written for af—x, y' — y, and / — z respectively. 515.] We have next to calculate the force with which the finite current / acts on the finite current s. The current s extends from A, where s = 0, to P, where it has the value s. The current / extends from A', where s'= 0, to P', where it has the value /. 5 1 6.] ACTION OF A CLOSED CIRCUIT ON AN ELEMENT. 155 The coordinates of points on either current are functions of s or of /. If F is any function of the position of a point, then We shall use the subscript (s o) to denote the excess of its value at P over that at A, thus jr(SiQ} = FP-FA, Such functions necessarily disappear when the circuit is closed. Let the components of the total force with which A' P* acts on A A be iif Xj ii'Y, and ii'Z. Then the component parallel to X of the force with which da' acts on ds will be ii' - — 7-7 da ds'. dsds Hence -T = R+8l+8'l'. (13) r Substituting the values of R, S, and S' from (12), remembering (14) and arranging the terms with respect to lt m, n, we find ds Since A, B, and C are functions of r, we may write P = f (A + £)~dr, Q=[ Cdr, (16) jr r jr the integration being taken between r and oo because A, JB, C vanish when r = oo. Hence (A + £)-L = -~, and <? = -^. (17) 516.] Now we know, by Ampere's third case of equilibrium, that when / is a closed circuit, the force acting on ds is perpendicular to the direction of ds, or, in other words, the component of the force in the direction of ds itself is zero. Let us therefore assume the direction of the axis of x so as to be parallel to ds by making I = 1 , m — 0, n == 0. Equation (15) then becomes - To find — , the force on ds referred to unit of length, we must ds 156 AMPERE'S THEORY. [5J7- integrate this expression with respect to /. Integrating the first term by parts, we find *X=(Pp-Q)Va-£(2Pr-3-O?l-<U'. (19) When / is a closed circuit this expression must be zero. The first term will disappear of itself. The second term, however, will not in general disappear in the case of a closed circuit unless the quantity under the sign of integration is always zero. Hence, to satisfy Ampere's condition, (20) 517.] We can now eliminate P, and find the general value of When / is a closed circuit the first term of this expression vanishes, and if we make (22) & r /=r JQ Z T where the integration is extended round the closed circuit /, we may write c^ Similarly =na'_iyt (23) u/s dZ -j-=lp ds The quantities a', (3', y are sometimes called the determinants of the circuit / referred to the point P. Their resultant is called by Ampere the directrix of the electrodynamic action. It is evident from the equation, that the force whose components dX dY . dZ . are -^—> -=-, and ~ is perpendicular both to ds and to this as as ds directrix, and is represented numerically by the area of the parallel ogram whose sides are ds and the directrix. 5 1 9-] FORCE BETWEEN TWO FINITE CURRENTS. 157 In the language of quaternions, the resultant force on ds is the vector part of the product of the directrix multiplied by ds. Since we already know that the directrix is the same thing as the magnetic force due to a unit current in the circuit /, we shall henceforth speak of the directrix as the magnetic force due to the circuit. 518.] We shall now complete the calculation of the components of the force acting between two finite currents, whether closed or open. Let p be a new function of r, such that "oo P = i/ (B-C)dr, (24) then by (17) and (20) d% d and equations (11) become & — - ' ' au&(Q+P) B ** , ** \ O = — ^7-7 > O = — - • ds ds J With these values of the component forces, equation (13) becomes l_L_ I' _UL . (27} w ds ds' ds' ds 519.] Let F = I Ipds, G = I mpds, H = I npds, (28) i/O JQ JQ F = f'l'p ds', G' = f'm'pds', H'= [''n'pds'. (29) ^0 "'0 Jo These quantities have definite values for any given point of space. When the circuits are closed, they correspond to the components of the vector-potentials of the circuits. Let L be a new function of r, such that fr L — I r(Q + p)dr, (30) ^o and let M be the double integral M = I I pcosedsds', (31) «^0 * 0 158 AMPERE'S THEORY. [520. which, when the circuits are closed, becomes their mutual potential, then (27) may be written «•„•* \dM dL } dsds'~ dsds I da dx^ \ 520.] Integrating1, with respect to s and *', between the given limits, we find d_ dx dx + F'P-FfA-FP, + FAf, (33) where the subscripts of L indicate the distance, r, of which the quantity L is a function, and the subscripts of F and F' indicate the points at which their values are to be taken. The expressions for Y and Z may be written down from this. Multiplying the three components by dxt dy, and dz respectively, we obtain Xdx+Ydy + Zdz = DM-D(Lpp,—LAP,—LA,p X — — — =- (LPp> — LAP'— LA' p P,-Ay, (34) where D is the symbol of a complete differential. Since Fdx + Gdy + Hdz is not in general a complete differential of a function of #,y, £, Xdx + Ydy + Zdz is not a complete differential for currents either of which is not closed. 521.] If, however, both currents are closed, the terms in I/, F, G, H, F, G't H' disappear, and Xdx+Ydy + Zdz = DM, (35) where M is the mutual potential of two closed circuits carrying unit currents. The quantity M expresses the work done by the electro magnetic forces on either conducting circuit when it is moved parallel to itself from an infinite distance to its actual position. Any alteration of its position, by which M is increased, will be assisted by the electromagnetic forces. It may be shewn, as in Arts. 490, 596, that when the motion of the circuit is not parallel to itself the forces acting on it are still determined by the variation of M, the potential of the one circuit on the other. 522.] The only experimental fact which we have made use of in this investigation is the fact established by Ampere that the action of a closed current on any portion of another current is perpendicular to the direction of the latter. Every other part of 524.] HIS FORMULA. 159 the investigation depends on purely mathematical considerations depending on the properties of lines in space. The reasoning there fore may be presented in a much more condensed and appropriate form by the use of the ideas and language of the mathematical method specially adapted to the expression of such geometrical relations — the Quaternions of Hamilton. This has been done by Professor Tait in the Quarterly Mathe matical Journal, 1866, and in his treatise on Quaternions, § 399, for Ampere's original investigation, and the student can easily adapt the same method to the somewhat more general investigation given here. 523.] Hitherto we have made no assumption with respect to the quantities A, B, C, except that they are functions of r, the distance between the elements. We have next to ascertain the form of these functions, and for this purpose we make use of Ampere's fourth case of equilibrium, Art. 508, in which it is shewn that if all the linear dimensions and distances of a system of two circuits be altered in the same proportion, the currents remaining the same, the force between the two circuits will remain the same. Now the force between the circuits for unit currents is -= — , and dos since this is independent of the dimensions of the system, it must be a numerical quantity. Hence M itself, the coefficient of the mutual potential of the circuits, must be a quantity of the dimen sions of a line. It follows, from equation (31), that p must be the reciprocal of a line, and therefore by (24), B— (7 must be the inverse square of a line. But since B and C are both functions of r, B—C must be the inverse square of r or some numerical multiple of it. 524.] The multiple we adopt depends on our system of measure ment. If we adopt the electromagnetic system, so called because it agrees with the system already established for magnetic measure ments, the value of M ought to coincide with that of the potential of two magnetic shells of strength unity whose boundaries are the two circuits respectively. The value of M in that case is, by Art. 423, /"/"cos* , M = J I - ds ds', (36) the integration being performed round both circuits in the positive direction. Adopting this as the numerical value of M, and com paring with (31), we find p = , and S-C=~. (37) 160 AMPERE'S THEORY. [525- 525.] We may now express the components of the force on ds arising from the action of ds' in the most general form consistent with experimental facts. The force on ds is compounded of an attraction 1 /dr dr d2r \ . d2 0 . 7 , 1 R = -H- l-y- -=-, — 2r -7-77) ^^ dsds' -f r -^—j-,11 ds ds rz ^ds ds dsds' dsds in the direction of r, S = 77 i i'ds ds in the direction of ds, as and S' = —^ ii'ds ds' in the direction of ds'. ds /NO where Q — / Cdr, and since C is an unknown function of r, we J r know only that Q is some function of r. 526.] The quantity Q cannot be determined, without assump tions of some kind, from experiments in which the active current forms a closed circuit. If we suppose with Ampere that the action between the elements ds and ds' is in the line joining them, then S and 8' must disappear, and Q must be constant, or zero. The force is then reduced to an attraction whose value is (39) Ampere, who made this investigation long before the magnetic system of units had been established, uses a formula having a numerical value half of this, namely 1 A dr dr dr N . ., _ R = -2 (- -7- -T7 - r -j—r^Jjds ds'. (40) f2 \9 fix Of d*njfJ** v ' Here the strength of the current is measured in what is called electro dynamic measure. If i, i' are the strength of the currents in electromagnetic measure, and j, j' the same in electrodynamic mea sure, then it is plain that jf = 2ii', or j = ^i. (41) Hence the unit current adopted in electromagnetic measure is greater than that adopted in electrodynamic measure in the ratio of «/2 to 1. The only title of the electrodynamic unit to consideration is that it was originally adopted by Ampere, the discoverer of the law of action between currents. The continual recurrence of <s/2 in calculations founded on it is inconvenient, and the electro magnetic system has the great advantage of coinciding numerically 527.] FOUK ASSUMPTIONS. 161 with all our magnetic formulae. As it is difficult for the student to bear in mind whether he is to multiply or to divide by \/2, we shall henceforth use only the electromagnetic system, as adopted by Weber and most other writers. Since the form and value of Q have no effect on any of the experiments hitherto made, in which the active current at least is always a closed one, we may, if we please, adopt any value of Q which appears to us to simplify the formulae. Thus Ampere assumes that the force between two elements is in the line joining them. This gives Q = 0, (42) r Grassmann * assumes that two elements in the same straight line have no mutual action. This gives Q 1 R- 3 d*T 8- l dr 8'- l - (43) V= ~2~r' ~Trdsds" 2r* els'3 ~ 2r* ds ( } We might, if we pleased, assume that the attraction between two elements at a given distance is proportional to the cosine of the angle between them. In this case „ 1 _ 1 0 1 dr 0, 1 dr , . . «=--> JZ = ^«»c, * = -F5p. S'=^Ts. (44) Finally, we might assume that the attraction and the oblique forces depend only on the angles which the elements make with the line joining them, and then we should have 0- 2 R- 3ldrdr S- 2- S'-*~. (45) V* ~P * VS3?1 ~PdS> ~ r* ds ( } 527.] Of these four different assumptions that of Ampere is undoubtedly the best, since it is the only one which makes the forces on the two elements not only equal and opposite but in the straight line which joins them. * Pogg., Ann. Ixiv. p. 1 (1845). VOL. II. M CHAPTER III ON THE INDUCTION OF ELECTRIC CURRENTS. 528.] THE discovery by Orsted of the magnetic action of an electric current led by a direct process of reasoning to that of magnetization by electric currents, and of the mechanical action between electric currents. It was not, however, till 1831 that Faraday, who bad been for some time endeavouring to produce electric currents by magnetic or electric action, discovered the con ditions of magneto-electric induction. The method which Faraday employed in his researches consisted in a constant appeal to ex periment as a means of testing the truth of his ideas, and a constant cultivation of ideas under the direct influence of experiment. In his published researches we find these ideas expressed in language which is all the better fitted for a nascent science, because it is somewhat alien from the style of physicists who have been accus tomed to established mathematical forms of thought. The experimental investigation by which Ampere established the laws of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the ' Newton of elec tricity.' It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics. The method of Ampere, however, though cast into an inductive form, does not allow us to trace the formation of the ideas which guided it. We can scarcely believe that Ampere really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself*, that he * Theorie des Phenomenes Elect rodynamiqucs, p. 9. 529.] ' FARADAY'S SCIENTIFIC METHOD. 163 discovered the law by some process which he has not shewn us, and that when he had afterwards built up a perfect demon stration he removed all traces of the scaffolding by which he had raised it. Faraday, on the other hand, shews us his unsuccessful as well as his successful experiments, and his crude ideas as well as his developed ones, and the reader, however inferior to him in inductive power, feels sympathy even more than admiration, and is tempted to believe that, if he had the opportunity, he too would be a dis coverer. Every student therefore should read Ampere's research as a splendid example of scientific style in the statement of a dis covery, but he should also study Faraday for the cultivation of a scientific spirit, by means of the action and reaction which will take place between newly discovered facts and nascent ideas in his own mind. It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathe matical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural, untechnical language. It is mainly with the hope of making these ideas the basis of a mathematical method that I have undertaken this treatise. 529.] We are accustomed to consider the universe as made up of parts, and mathematicians usually begin by considering a single par ticle, and then conceiving its relation to another particle, and so on. This has generally been supposed the most natural method. To conceive of a particle, however, requires a process of abstraction, since all our perceptions are related to extended bodies, so that the idea of the all that is in our consciousness at a given instant is perhaps as primitive an idea as that of any individual thing. Hence there may be a mathematical method in which we proceed from the whole to the parts instead of from the parts to the whole. For example, Euclid, in his first book, conceives a line as traced out by a point, a surface as swept out by a line, and a solid as generated by a surface. But he also defines a surface as the M 2 164 MAGNETO-ELECTRIC INDUCTION,* [530. boundary of a solid, a line as the edge of a surface, and a point as the extremity of a line. In like manner we may conceive the potential of a material system as a function found by a certain process of integration with respect to the masses of the bodies in the field, or we may suppose these masses themselves to have no other mathematical meaning than the volume-integrals of — V2^? where ^ is the potential. In electrical investigations we may use formulae in which the quantities involved are the distances of certain bodies, and the electrifications or currents in these bodies, or we may use formulae which involve other quantities, each of which is continuous through all space. The mathematical process employed in the first method is in tegration along lines, over surfaces, and throughout finite spaces, those employed in the second method are partial differential equa tions and integrations throughout all space. The method of Faraday seems to be intimately related to the second of these modes of treatment. He never considers bodies as existing with nothing between them but their distance, and acting on one another according to some function of that distance. He conceives all space as a field of force, the lines of force being in general curved, and those due to any body extending from it on all sides, their directions being modified by the presence of other bodies. He even speaks * of the lines of force belonging to a body as in some sense part of itself, so that in its action on distant bodies it cannot be said to act where it is not. This, however, is not a dominant idea with Faraday. I think he would rather have said that the field of space is full of lines of force, whose arrangement depends on that of the bodies in the field, and that the mechanical and electrical action on each body is determined by the lines which abut on it. PHENOMENA OF MAGNETO-ELECTRIC INDUCTION f. 530.] 1. Induction by Variation of the Primary Current. Let there be two conducting circuits, the Primary and the Secondary circuit. The primary circuit is connected with a voltaic * Exp. Res., ii. p. 293 ; iii. p. 447. t Read Faraday's Experimental Researches, series i and ii. 530.] ELEMENTARY PHENOMENA. 165 battery by which the primary current may be produced, maintained, stopped, or reversed. The secondary circuit includes a galvano meter to indicate any currents which may be formed in it. This galvanometer is placed at such a distance from all parts of the primary circuit that the primary current has no sensible direct influence on its indications. Let part of the primary circuit consist of a straight wire, and part of the secondary circuit of a straight wire near, and parallel to the first, the other parts of the circuits being at a greater distance from each other. It is found that at the instant of sending a current through the straight wire of the primary circuit the galvanometer of the secondary circuit indicates a current in the secondary straight wire in the opposite direction. This is called the induced current. If the primary current is maintained constant, the induced current soon disappears, and the primary current appears to produce no effect on the secondary circuit. If now the primary current is stopped, a secondary current is observed, which is in the same direction as the primary current. Every variation of the primary current produces electromotive force in the secondary circuit. When the primary current increases, the electromotive force is in the opposite direction to the current. When it diminishes, the electromotive force is in the same direction as the current. When the primary current is constant, there is no electromotive force. These effects of induction are increased by bringing the two wires nearer together. They are also increased by forming them into two circular or spiral coils placed close together, and still more by placing an iron rod or a bundle of iron wires inside the coils. 2. Induction ~by Motion of the Primary Circuit. We have seen that when the primary current is maintained constant and at rest the secondary current rapidly disappears. Now let the primary current be maintained constant, but let the primary straight wire be made to approach the secondary straight wire. During the approach there will be a secondary current in the opposite direction from the primary. If the primary circuit be moved away from the secondary, there will be a secondary current in the same direction as the primary. 3. Induction by Motion of the Secondary Circuit. If the secondary circuit be moved, the secondary current is 166 MAGNETO-ELECTRIC INDUCTION. [S31- opposite to the primary when the secondary wire is approaching- the primary wire, and in the same direction when it is receding- from it. In all cases the direction of the secondary current is such that the mechanical action between the two conductors is opposite to the direction of motion, being a repulsion when the wires are ap proaching, and an attraction when they are receding. This very important fact was established by Lenz *. 4. Induction by the Relative Motion of a Magnet and the Secondary Circuit. If we substitute for the primary circuit a magnetic shell, whose edge coincides with the circuit, whose strength is numerically equal to that of the current in the circuit, and whose austral face cor responds to the positive face of the circuit, then the phenomena produced by the relative motion of this shell and the secondary circuit are the same as those observed in the case of the primary circuit. 531.] The whole of these phenomena may be summed up in one law. When the number of lines of magnetic induction which pass through the secondary circuit in the positive direction is altered, an electromotive force acts round the circuit, which is measured by the rate of decrease of the magnetic induction through the circuit. 532.] For instance, let the rails of a railway be insulated from the earth, but connected at one terminus through a galvanometer, and let the circuit be completed by the wheels and axle of a rail way carriage at a distance x from the terminus. Neglecting the height of the axle above the level of the rails, the induction through the secondary circuit is due to the vertical component of the earth's magnetic force, which in northern latitudes is directed downwards. Hence, if b is the gauge of the railway, the horizontal area of the circuit is bx, and the surface-integral of the magnetic induction through it is Zbxt where Z is the vertical component of the magnetic force of the earth. Since Z is downwards, the lower face of the circuit is to be reckoned positive, and the positive direction of the circuit itself is north, east, south, west, that is, in the direction of the sun's apparent diurnal course. Now let the carriage be set in motion, then x will vary, and * Pogg., Ann. xxi. 483 (1834.) 533-] DIRECTION OF THE FORCE. 167 there will be an electromotive force in the circuit whose value „, das is — Zb -=-. dt If x is increasing, that is, if the carriage is moving away from the terminus, this electromotive force is in the negative direction, or north, west, south, east. Hence the direction of this force through the axle is from right to left. If x were diminishing, the absolute direction of the force would be reversed, but since the direction of the motion of the carriage is also reversed, the electro motive force on the axle is still from right to left, the observer in the carriage being always supposed to move face forwards. In southern latitudes, where the south end of the needle dips, the electromotive force on a moving body is from left to right. Hence we have the following rule for determining the electro motive force on a wire moving through a field of magnetic force. Place, in imagination, your head and feet in the position occupied by the ends of a compass needle which point north and south respec tively ; turn your face in the forward direction of motion, then the electromotive force due to the motion will be from left to right. 533.] As these directional relations are important, let us take another illustration. Suppose a metal girdle laid round the earth at the equator, and a metal wire laid along the meridian of Green wich from the equator to the north pole. / Let a great quadrantal arch of r/A metal be constructed, of which one extremity is pivoted on the north pole, while the other is carried round the equator, sliding on the great girdle of the earth, and following the sun in his daily course. There will then be an electromotive force along the moving quadrant, acting from the pole towards the equator. The electromotive force will be the same whether we suppose the earth at rest and the quadrant moved from east to west, or whether we suppose the quadrant at rest and the earth turned from west to east. If we suppose the earth to rotate, the electromotive force will be the same whatever be the form of the part of the circuit fixed in space of which one end touches one of the pole& 168 MAGNETO-ELECTRIC INDUCTION. [534- and the other the equator. The current in this part of the circuit is from the pole to the equator. The other part of the circuit, which is fixed with respect to the earth, may also be of any form, and either within or without the earth. In this part the current is from the equator to either pole. 534.] The intensity of the electromotive force of magneto -electric induction is entirely independent of the nature of the substance of the conductor in which it acts, and also of the nature of the conductor which carries the inducing current. To shew this, Faraday * made a conductor of two wires of different metals insulated from one another by a silk covering, but twisted together, and soldered together at one end. The other ends of the wires were connected with a galvanometer. In this way the wires were similarly situated with respect to the primary circuit, but if the electromotive force were stronger in the one wire than in the other it would produce a current which would be indicated by the galvanometer. He found, however, that such a combination may be exposed to the most powerful electromotive forces due to in duction without the galvanometer being affected. He also found that whether the two branches of the compound conductor consisted of two metals, or of a metal and an electrolyte, the galvanometer was not affected f. Hence the electromotive force on any conductor depends only on the form and the motion of that conductor, together with the strength, form, and motion of the electric currents in the field. 535.] Another negative property of electromotive force is that it has of itself no tendency to cause the mechanical motion of any body, but only to cause a current of electricity within it. If it actually produces a current in the body, there will be mechanical action due to that current, but if we prevent the current from being formed, there will be no mechanical action on the body itself. If the body is electrified, however, the electro motive force will move the body, as we have described in Electro statics. 536.] The experimental investigation of the laws of the induction of electric currents in fixed circuits may be conducted with considerable accuracy by methods in which the electromotive force, and therefore the current, in the galvanometer circuit is rendered zero. For instance, if we wish to shew that the induction of the coil * Rrp. fas., 195. f Ib., 200. 536.] EXPERIMENTS OF COMPARISON. 169 A on the coil X is equal to that of B upon Y, we place the first pair of coils A and X at a sufficient distance from the second pair Fig. 32. £ and Y. We then connect A and B with a voltaic battery, so that we can make the same primary current flow through A in the positive direction and then through B in the negative direction. We also connect X and Y with a galvanometer, so that the secondary current, if it exists, shall flow in the same direction through X and Yin series. Then, if the induction of A on X is equal to that of B on Y, the galvanometer will indicate no induction current when the battery circuit is closed or broken. The accuracy of this method increases with the strength of the primary current and the sensitiveness of the galvanometer to in stantaneous currents, and the experiments are much more easily performed than those relating to electromagnetic attractions, where the conductor itself has to he delicately suspended. A very instructive series of well devised experiments of this kind is described by Professor Felici of Pisa *. I shall only indicate briefly some of the laws which may be proved in this way. (1) The electromotive force of the induction of one circuit on another is independent of the area of the section of the conductors and of the material of which they are made. For we can exchange any one of the circuits in the experiment for another of a different section and material, but of the same form, without altering the result. * Annettes dc Chimie, xxxiv. p. G6 (1852), and Nuovo Cimento, ix. p. 345 (1859). 170 MAGNETO-ELECTRIC INDUCTION. [537- (2) The induction of the circuit A on the circuit X is equal to that of X upon A. For if we put A in the galvanometer circuit, and X in the battery circuit, the equilibrium of electromotive force is not disturbed. (3) The induction is proportional to the inducing current. For if we have ascertained that the induction of A on X is equal to that of B on Y, and also to that of C on Z, we may make the battery current first flow through A, and then divide itself in any proportion between B and C. Then if we connect X reversed, Y and Z direct, all in series, with the galvanometer, the electromotive force in X will balance the sum of the electromotive for