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Full text of "A treatise on electricity and magnetism"

- 

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, (T p ) .. .. ,. .. 188 

561. Hamilton s equations of motion .. .. , 189 

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

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

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

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

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

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

CHAPTER VI. 

APPLICATION OF DYNAMICS TO ELECTROMAGNETISM. 

568. The electric current possesses energy 195 

569. The current is a kinetic phenomenon 195 

570. Work done by electromotive force 196 

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

including electric currents ., .. .. 197 

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

573. Mechanical force acting on a conductor 198 

574. The part depending on products of ordinary velocities and 

strengths of currents does not exist 200 

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

576. Discussion of the electromotive force 204 

577. If terms involving products of velocities and currents existed 

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

CHAPTER VII. 

ELECTROKINETICS. 

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

579. Electromotive force in each circuit . . 207 



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 g l 354 

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

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

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

CHAPTER XVIII. 

DETERMINATION OF RESISTANCE IN ELECTROMAGNETIC MEASURE. 

758. Definition of resistance 358 

759. Kirchhoff s method 358 



XX CONTENTS. 

Art. 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, d 2 .l x 

read W = m 9 -^ = 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 r 5 . 
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 7i 2 . 
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 Fbq 1 read Fb^. 
p. 192, 1. 22, for Tp read T p . 
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 (x lt 05 2 , &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 10 10 . 
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 z 2 read z*. 

p. 289, equation (22), for 4a 2 4 read 2af ; and for 4 2 4 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 c 2 2 read c 2 . 
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 -Hy^)=~^ = constant. 

TT y y 

p. 325, 1. 5 from bottom, should be #=| ^- 2 ^ (a^-a 3 ). 

p. 346, 1. 2, for 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 m 1 and m 2 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- l M*\. 

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 m t 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* + C 2 . (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(riy) + v(Cz), (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 k lt and whose magnetic moment is m lt is 

y _ d m l m l ( 

** ~5*77~"H Ai 

where A. L is the cosine of the angle between h and r. 

Again, if a second magnetic molecule whose moment is m 2 , and 
whose axis is parallel to h z , 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 /u 12 is the cosine of the angle which the axes make with each 
other, and X ls A 2 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 m l m 2 ,dy l2 d\ 2 ^ 

~~d^ = ~^~\d$~ Al 3^ 

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* m 9 

>~^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 H 2 , 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 ZT 2 
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 A 2 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 Qm 1 m 2 . . 

Jic-f jczi-f/ZgTs -- 4 -- (13) 

The negative sign indicates that the force is an attraction. 

(2) If A : and A 2 are zero, and /* 12 unity, the axes of the magnets 
are parallel to each other and perpendicular to /, and the force 
is a repulsion 3m 1 m 2 



In neither of these cases is there any couple. 

(3) If A! = 1 and A 2 = 0, then /u 12 = 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 m 2 . 




Fig. 1. 
Thus in the figure (1) two magnets are made to float on water, 



3 88.] 



FORCE BETWEEN TWO SMALL MAGNETS. 



13 



being in the direction of the axis of m 1 , but having- its own axis 
at right angles to that of m l . If two points, A, B, rigidly connected 
with % and m 2 respectively, are connected by means of a string T, 
the system will be in equilibrium,, provided T cuts the line m 1 m 2 
at right angles at a point one-third of the distance from m l to m 2 . 

(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 k 2 , and therefore the resolved part of the force 
due to m 2 , 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 

(M 2 ) = (V)+M 2 ), (16) 

and the couple being 




Fig. 2. 



m 



(sin (h-^ /t> 2 ) 3 cos (h-^ r) sin (r h 2 )), 



(17) 



we find when this is zero 

tan (^ r) = 2 tan (r 7* 2 ) , 



(18) 

or tan^Wg-B = 2 ta,nRm 2 ff 2 . (19) 

When this position has been taken up by the second magnet the 

dV 



value of W becomes 



where h 2 is in the direction of the line of force due to m l 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 ff l parallel to the axis of the first magnet, where 

H L = 3^ ** _ . (21) 

^ 73 A x 2 + 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 T 3 and its negative pole at a point where the potential is F , the 
potential energy of this magnet will be mCFP ), 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 d f + B ll ^ + C d -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 



~ + -- + -dxdydz y (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 = nK t (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 
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 cos <, m = cos sin <, n sin ; (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 +(>/-,?o 2 +(<r-*)Ti (i) 

This expression may be expanded in terms of spherical harmonics, 
with their centre at the origin. We have then 

(2) 



when F Q = - , r being the distance of (f, 77, f ) from the origin, (3) 

(4) 



_ 

2 ~ 2r 5 

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~ , F t and V 2 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, 
_ 



r 5 

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 , N f = N-nKz / - f (11) 

P =PK(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 

K t t tf- 

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 (Z 2 + M z + N*-MN-NL-LM] + 3 (P 2 + Q 2 +^ 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 LM N)+3nQ+3mR = 0, 
2m(2M-N-L)+3lR+3nP = 0, (17) 

2n (2NZM)+3mP+3lQ = 0. 
If we assume I = I, m = 0, n = Q, these conditions become 

2L-MN=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-, = -=-, 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 V A 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%cosdS, (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 k z , 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 = (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 F 9 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 d z p \ d *p d 2 j) } 

\dydr) dzdC dx dr] dxd^S 



7> r, ^ 7 7 

-ri - ~ + B -/- -f- (7 7 \dxdydz 
djJJ ( 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. 

r 2 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 ( ) > 

r l r 2 

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 \ m i /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 d m 

/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 * 
r 2 

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 

r 2 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 d y d y d y 

* (^ -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. [4 11 - 

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 

A ff__<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 r t the line drawn from (x, y, z] to (f, rj, f ), and the angle 
which this line makes with the normal drawn outwards from dS t 
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 1 L is the value of H at a point just within the surface, and 
12 2 that at a point close to the first but outside the surface, 

fla = ^ + 477^), 

r 2 = 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 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 PP f 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 r 2 d<a, we find 




Fig. 3. 



Hence 



du> = II ds dcr = -g sin Q sin 6 sin </> ds dcr. 
n = - sin 6 sin 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 D 2 , 



(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 z t 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 <o 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 P f 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^ , . 

r 3 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- dds 
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. [4 2 3- 



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 T 9 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 

- +c f + ^, 

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 f dx 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 = K y. 

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 = (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 
= - 47rK j;> b F( 4 ) 

f dV d^ , 
= - 47 rK(^+^),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 V 9 may be reduced to 
the following mathematical problem. 

We must find two functions H and H satisfying the following 
conditions : 

Within the surface S 9 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 T 7 , 
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 = " = - 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 K X 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 = 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 
ju 2 , we have found that the conductivity of the composite system is 

2) 
P = f*l-j 

Putting fa = 1 and /ot 2 = 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= C Q 8 + C 1 S 1 r + to. + C i S i i A , (1) 

where r is the distance from the centre of the shell, #< is a surface 
harmonic of order i, and C i 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 a 2 and the inner radius a lf 
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 S i9 
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 A l S t 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 a L and a 2 , 
the series may contain both positive and negative powers of /*, 
of the form 



Outside the shell, where r is greater than a 2 , 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) B l = 0. 

On account of (2) when r = a^ 

(4-4,H 2i+1 -5 2 =0, (2) 

and when r = 2 , 

(^ 2 -J 3 )^ 2i+1 + ^ 2 -^ 3 = 0. (3) 

On account of (3) A z = 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 = a lt 

, dl 9 d&, dV ,.. 

< 1+4 *>V-ifr +4 " * = <) 

and at the outer surface where r = a 2 , 

d dV 



, KN 
0. 

From these conditions we obtain the equations 



iC i a 1 2i+l = <), (6) 

2 2 +1 -(^+l)^ 2 )+(^+l)^3+ 47r ^^2 2i+1 = ^ ( 7 ) 
and if we put 



we find 

/ /, 2 + l\ 

4 = -(4)^ + l)(l-Q) }N t C lt (9) 

[I a 2t+l^-j 

2^ + l + 477K(^+l)(l-(^) )J^Ci, (10) 

(11) 
1 2i+1 )^C i . (12) 

These quantities being substituted in the harmonic expansions 
give the part of the potential due to the magnetization of the shell. 
The quantity N i is always positive, since 1 -f 4 ir K can never be 
negative. Hence A 1 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)(# 2 3 1 3 )^V 1 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 A 2 in the problem of conduction are equi 
valent to C 1 + A 1 and C 1 + A 2 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+ q 2 y, \ 

= q 9 a+r 2 p+ fl y, { (1) 

C = p 2 a+q 1 h 2 + 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, \ 
= 7-J.ir-B, (5) 

y =Z- 



Hence 

+i*r 1 )^+ twftjjB + iir& 

C = &J+ r 2 Y+frZ, (6) 



+(1 + 

Solving these equations, we find 
A = r/^+K 

(7) 



43^.] CRYSTALLINE SPHERE. 61 

where I/ // = r + ^ TT ( r B r l p 2 q 2 4 r- r 2 

;-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 



Hence, in order that the revolving sphere may not become an 
inexhaustible source of energy, j 1 / = fa , and similarly j./= q 2 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 V 3 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> denote the definite integral 



f 





Then if we make 

dfr 



the value of the potential within the ellipsoid will be 



7 = - (L x 2 + 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) + K f s(Y-BM) + K 2 (Z-CN}, 
B = K\ (X-AL) + K 2i (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 \ = K 2 = K 3 = 0. (9) 



We have then 

If the ellipsoid 
flattened form, 


A 


"i 


X 1 


(10) 

and is of the planetary or 
: (ID 


7? 


K 2 


T 
JJ 

V 


C = 
has two 
b= c 


1+K 2 M~ 

K 3 g 

l+K 3 N 

axes equal, 
a 



(12) 
l-e 



M = N = 2 , (-^ sin- *- ) . 
\ e* e 2 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 7r 2 - 

In the case of a very elongated ovoid L and M approximate 
to the value 2 TT, while N approximates to the form 

a 2 ,, 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 



7T 2 - 1) and in an elongated ovoid magnetized longitudinally it 



a 2 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 K 3 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 K 2 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 



e a + e 

where p is a numerical quantity to be found from the equation 

0.231863 2 \og e p + 2p = - - 
The following are a few of the corresponding values of p and K. 

K K 



oo 

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= v 2 a K X. 

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 R 2 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. 

X 2 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 making 1 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. 
= 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 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 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 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 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 /3 , will return to its original 
position when the deflecting force is removed, but that if the 
deflexion j3 exceeds ^ , 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 j3 , 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 /3 , (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 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 /3 , 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 = Xsin0 when = PSA or QSB, 

then all those molecules whose axes, on the former hypotheses, 
would have values of 6 between and TT will be made to have 
the value 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 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 = + /3 , 

and another cone round B whose angle 
BOQ = -/3 . 

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 -f-/3 . 

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 

2 267 

3 400 

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 = 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 X Q> is again subjected to a magnetizing 
force X 1 . 

If the new force X acts in the same direction in which X 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 
X 1 is removed the residual magnetization will be the same as 

G 2 



84: WEBER S THEORY OF INDUCED MAGNETISM. [446. 

that produced by X . If X l is greater than X , then it will produce 
exactly the same effect as if X had not acted. 

But let us suppose X l to act in the negative direction, and let us 
suppose XQ = L cosec , and X l = I/cosec0 1 . 

As X 1 increases numerically, : diminishes. The first molecules 
on which X 1 will produce a permanent deflexion are those which 
form the fringe of the cone round A, and these have an inclination 
when undeflected of + J3 . 

As soon as 6 1 /3 becomes less than -f~/3 the process of de 
magnetization will commence. Since, at this instant, ^ = -f 2^3 , 
X 13 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 X 1 would wrench the whole of the fringe 
of molecules whose axes have the inclination + /3 into a position 
in which their axes are inclined 1 + )3 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 1 /3 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 /3 
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 l + /3 . 

When X 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 
X 2 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 = 1 + /3 with the negative 
axis. 



447-1 MAGNETISM AND TORSION, 85 

If the force X 2 acts in the negative direction it will begin to 
produce a permanent effect as soon as 2 + /3 becomes less than 
^i + A)5 that is, as soon as X 2 becomes greater than X L . But if 
X 2 acts in the positive direction it will begin to remagnetize the 
iron as soon as 2 {3 becomes less than Oi + P , that is, when 
2 = O l -j- 2/3 , or while X 2 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 X 0i its 
magnetism cannot be increased without the application of a force 
greater than X . A reverse force, less than Jf , is sufficient to 
diminish its magnetization. 

If the iron is exactly demagnetized by a reversed force X 19 then 
it cannot be magnetized in the reversed direction without the 
application of a force greater than X 1} but a positive force less than 
X x 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. 



[ 44 8. 



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 being 1 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 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 



x x --- 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 s 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- - - 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 /3 2 , 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) = ( + ^)(C I + C 1 )-(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 A lt A 2 , &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 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 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 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 * , then r 1 = s 1 s 0) where ^ is known accurately and s ap 
proximately. Let 1 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 A lt A 3 , &c. will 
have their signs changed, so that if 2 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<$ s on the scale. 
Let the deflexion when the axis is in the first position be 3 , and 
when it is in the second 4 , then, as before, 



2 

Let us suppose that the true position of the centre of the sus 
pended magnet is not S Q but <? -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 r L n 
and r 2 n for r n . 

Hence, taking the arithmetical mean of (8) and (9), 

--^ 
or, making 



= 1 + A 2 ~ +&c., (12) 



- (tan O l tan 6 2 + tan 3 tan 4 ) = D, (13) 



454.] We may now regard D and r as capable of exact deter 
mination. 

The quantity A 2 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 A 2 and determine the ratio 
of H to M at once. We cannot, however, assume that A 2 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 A 2 , repeat the experiment, using distances r lt r a , ?* 3 , 
&c., and let the values of D be J) 19 D 2 , # 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 AT 3 + -0 2 V 3 + 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 (*-) + A 2 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 A 2 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 r 2 , the value of -=- derived from these experi 

ments is 






- - 

If 5Z) X and bD 2 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 bD 2 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 
r 2 2 . From this the best value of ^ is found to be r x = 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 r l5 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 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 T 1} T 2 , T 2n+1 , then if 
we put I 4 + y + y 4 &c . 



then T n+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 

- = 



where 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 for equilibrium is 

T "y 
= - - 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 \ 

= Ccos (2 TT -^ 4- a) + , 

where T is the periodic time, and C the amplitude, and 

yr2 

whence we find the value of MH 9 



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 the 
deflexion, M H sin = 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 a 6) = Mil sin 6. 

Here a 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-a - + 00 



or 



By observing , 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 a 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 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 PP 7 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) PP f = 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 n z }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 observing 1 
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 (yp\ 

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 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 

47r 2 ^ lab 

-yrr- = l-fi>"ff+T + MH. (9) 

(2) When a is nearly equal to 8 + 77. If T 2 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 50 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 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 //= E Q and = 0, 

= 0, (14) 



and if H is the value of the horizontal force corresponding to a 
small angle / , 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 throwing 1 
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 + 8 where 8 is the reading of 
the declinometer at the given instant, and 8 is the unknown but 
constant error, so that 8 + 8 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 8 , 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 + 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 6 1 be this reading, then we have 

MIsm(0 1 + \i) = m(/acos(0 l + a). (5) 

The instrument is now turned about a vertical axis through 180, 
so that the graduation is to the east, and if 2 is the new reading, 
MIsm(0 2 + X v+i) mga cos (0 2 + a). (6) 

Taking (6) from (5), and remembering that 6^ is nearly equal to 
i, and 2 nearly equal to TT i, and that X is a small angle, such 
that mgaK may be neglected in comparison with MI, 

MI(0 l 2 -{-7f2i ) =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, 3 and 4 , 
in which 3 is nearly ir + i, and 4 nearly i, 

3/ / sin (0 3 + A 77 i) = mgaco8(0 B +a), (11) 

M l sin (0 4 + A + i) = m g a cos (0 4 + a), (1 2) 

whence, as before, 

M I(9 3 4 77 2i) 2mgacosicosa, (13) 

adding (8), 

Jf/^-^ + Tr 2z ) + lT/(0 3 4 IT 2 a) = 0, (14) 
or J9(0 1 -0 2 + 7 r-2;) + .Z/(0 3 -0 4 -7r-2*) = 0, (15) 

whence we find the dip 

-e 4 -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, the true dip, and the 
observed dip, then 

L = Jf/sin(0-0 ), (17) 

= MI(0-0 ), (18) 

since 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 M lt 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 1 and 6 2 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 = M 1 i(0 1 -e ) = M 2 i(0 2 -0 ). (19) 



If we find that several experiments give nearly equal values for 
L, then we may consider that 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, 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. 

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 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, 



T 13 T z , jP 3 , 



5JL _L JL J_l. 

" 2M+2 I Zi 2 + 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 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 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, 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 = ZTtan0 , where 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 D l . 

Then place it with its axis vertical and with its centre at a 
distance r z above or below the centre of the vertical force mag 
netometer, and let the tangent of the deflexion produced in the 
magnetometer be D 2 . Then, if the moment of the deflecting 

magnet is M, jr. 

M^IIr^D^ = ^r^D 2 . 

clZ r^ D L 

Hence -7 = H -^ -~ 

dO r 2 3 D 2 

The actual value of the vertical force at any instant is 
7 7 +fi dZ 

& = &Q H- v -j^ > 

where Z Q 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 
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 F , 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 F . 

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 = A 1 + B l I + 2 X, (7) 

Ycosl = A 2 + 2 l + 3 3 \. (8) 



Let there be n stations whose latitudes are l l} 2 , ...&c. and 
longitudes \ lt A 2 , &c., and let X and 7 be found for each station. 

Let J = 



122 TERRESTRIAL MAGNETISM. [466- 

/ and A may be called the latitude and longitude of the central 
station. Let 

X =-i(i)- and r o cosJ =:-2(rcosJ), (10) 

tl ti 

then X and Y are the values of X and Y at the imaginary central 
station, then 

\-\ ), (11) 

A-A ). (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 . dffHsm 8 . db, 
2 = 7,2 COS 2 8 + d * H * sin 2 8 

Similarly 7? 2 = /fc 2 sin 2 8 + d 2 // 2 cos 2 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 J3 lt 
H 2 , 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 )-^ ^ = 
P l = 2(lX)-nl Q X Q , 



By calculating 19 J5 2 , and J5 3 , 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 H is this constant value, the value 
at any other point would be H= // O cos I , where V is the magnetic 
latitude. The vertical force at any point would be Z = 2 H Q 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. 



47 8.] 



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 m 2 be the strengths 
of the two poles, r l and r 2 their distances from the axis of the wire, 
5\ and T 2 the intensities of the magnetic force due to the current at 




Fig. 21. 



the two poles respectively, then the force on m 1 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 m 2 T 2 r 2 , and since 
there is no motion observed, 

m l T 1 r l + m 2 T 2 r 2 = 0. 

VOL. II. K 



130 ELECTROMAGNETIC FORCE. [479- 

But we know that in all magnets 

m-L + m^ = 0. 

Hence T^ = T 2 r 2 , 

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 S 9 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 bx 1 represents any displacement of the shell, and X 1 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 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 . bx cos 77 8 N. 

Now X bx = i 5 N = i ds . 33 sin fix cos 77, 

and X =. i ds . 33 sin 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#P 2 . 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. OP 2 . 0.33, and since the strength of the second 
is y, the work done by it is \y.OP 2 . 6 33. The whole work done 
is therefore 

i(tf + 3/)OP 2 .033 or ii.OP 2 .0B, 

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 .OP 2 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 M 2 , are identical with those 
of a magnetic shell S 19 whose edge coincides with the circuit, and 
whose strength is numerically equal to that of the electric current. 
Let the magnetic system M 2 be a magnetic shell S 2 , then the 
mutual action between ^ and 8 2 is identical with that between ^ 
and a circuit C 2 , coinciding with the edge of S 2 and equal in 
numerical strength, and this latter action is identical with that 
between C t and C 2 . 

Hence the mutual action between two circuits, C l and C 2) is 
identical with that between the corresponding magnetic shells S l 
and S 2 . 

We have already investigated, in Art. 423, the mutual action 



140 ELECTROMAGNETIC FORCE. [493- 

of two magnetic shells whose edges are the closed curves s 1 and s 2 . 

/**2 /**! COS 6 

If we make M= I - &,<&, 

J ^0 ? 

where e is the angle between the directions of the elements ds 1 and 
ds 2 , and r is the distance between them, the integration being 
extended once round s. 2 and once round s lf and if we call M the 
potential of the two closed curves ^ and <s 2 , 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 



5 01 -] 



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 communicating 1 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 
revolving 1 about a vertical axis, and carrying 1 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. 



[ 5 o8. 



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 A lt B I} C v Fig. 28, be corresponding elements of the three 
circuits, and let A 2 , B 2 , C 2 be also corresponding elements in an 
other part of the circuits. Then the situation of B with respect 
to A 2 is similar to the situation of C^ with respect to B. 2) but the 








u 



distance and dimensions of C l and B 2 are n times the distance and 
dimensions of B l and A 2i 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 B l and A. 2 , may be written 



and that between C 1 and B 2 





where #, b, c are the strengths of the currents in A, B, C. But 

A CB and a = c. Hence 



^ = C lt 



= 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 PP f be de- Fig< 29 

noted by r. Let the angle P*PQ be denoted by 0, and PP (g 
by Q f , 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 (x f x] -f- m (y y) -f n (z f 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 sin (f cos rj. (4) 

r* = (af x )* + (tfy)* + (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 = 



dr 



cose = r 



dr dr 



d 2 r 
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 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 cos + 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 afx, 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} = F P -F A , 

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 ii f 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 l t m, n, we find 



ds 



Since A, B, and C are functions of r, we may write 

P = f (A + )~dr, Q=[ Cdr, (16) 

j r r j r 

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. [5 J 7- 

integrate this expression with respect to /. Integrating the first 
term by parts, we find 

*X=(Pp-Q) V a-(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 _iy t ( 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 " 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 * 



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.] Integrating 1 , with respect to s and * , between the given 
limits, we find 

d_ 
dx dx 

+ F P -F f A-F P , + F Af , (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 dx t dy, and dz respectively, 
we obtain 

Xdx+Ydy + Zdz = DM-D(Lpp,L AP ,L A ,p 



X =- (L P p> LAP LA p 



P ,- A y, (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, BC 
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. [5 2 5- 

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 d 2 r \ . d 2 . 7 , 1 

R = -H- l-y- -=-, 2r -7-77) ^^ dsds -f r -^j-,11 ds ds 
r z ^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 -jr^Jjds ds . (40) 

f 2 \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 1 dr , 1 dr , . . 

=--> JZ = ^c, * = - F5 p. 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- 3 ldrdr 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 V 2 ^? 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. [S3 1 - 

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 Zbx t 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 forces in Y 



(4) In pairs of circuits forming systems geometrically similar 
the induction is proportional to their linear dimensions. 

For if the three pairs of circuits above mentioned are all similar, 
but if the linear dimension of the first pair is the sum of the 
corresponding linear dimensions of the second and third pairs, then, 
if A, B, and C are connected in series with the battery, and X 
reversed, Y and Z also in series with the galvanometer, there will 
be equilibrium. 

(5) The electromotive force produced in a coil of n windings by 
a current in a coil of m windings is proportional to the product mn. 

537.] For experiments of the kind we have been considering the 
galvanometer should be as sensitive as possible, and its needle as 
light as possible, so as to give a sensible indication of a very 
small transient current. The experiments on induction due to 
motion require the needle to have a somewhat longer period of 
vibration, so that there may be time to effect certain motions 
of the conductors while the needle is not far from its position 
of equilibrium. In the former experiments, the electromotive 
forces in the galvanometer circuit were in equilibrium during 
the whole time, so that no current passed through the galvano 
meter coil. In those now to be described, the electromotive forces 
act first in one direction and then in the other, so as to produce 
in succession two currents in opposite directions through the gal 
vanometer, and we have to shew that the impulses on the galvano 
meter needle due to these successive currents are in certain cases 
equal and opposite. 

The theory of the application of the galvanometer to the 
measurement of transient currents will be considered more at length 
in Art. 748. At present it is sufficient for our purpose to ob- 



538-J FELICl s EXPERIMENTS. 171 

serve that as long- as the galvanometer needle is near its position 
of equilibrium the deflecting force of the current is proportional 
to the current itself, and if the whole time of action of the current 
is small compared with the period of vibration of the needle, the 
final velocity of the magnet will be proportional to the total 
quantity of electricity in the current. Hence, if two currents pass 
in rapid succession, conveying equal quantities of electricity in 
opposite directions, the needle will be left without any final 
velocity. 

Thus, to shew that the induction-currents in the secondary circuit, 
due to the closing and the breaking of the primary circuit, are 
equal in total quantity but opposite in direction, we may arrange 
the primary circuit in connexion with the battery, so that by 
touching a key the current may be sent through the primary circuit, 
or by removing the finger the contact may be broken at pleasure. 
If the key is pressed down for some time, the galvanometer in 
the secondary circuit indicates, at the time of making contact, a 
transient current in the opposite direction to the primary current. 
If contact be maintained, the induction current simply passes and 
disappears. If we now break contact, another transient current 
passes in the opposite direction through the secondary circuit, 
and the galvanometer needle receives an impulse in the opposite 
direction. 

But if we make contact only for an instant, and then break 
contact, the two induced currents pass through the galvanometer 
in such rapid succession that the needle, when acted on by the first 
current, has not time to move a sensible distance from its position 
of equilibrium before it is stopped by the second, and, on account 
of the exact equality between the quantities of these transient 
currents, the needle is stopped dead. 

If the needle is watched carefully, it appears to be jerked suddenly 
from one position of rest to another position of rest very near 
the first. 

In this way we prove that the quantity of electricity in the 
induction current, when contact is broken, is exactly equal and 
opposite to that in the induction current when contact is made. 

538.] Another application of this method is the following, which 
is given by Felici in the second series of his Researches. 

It is always possible to find many different positions of the 
secondary coil I>, such that the making or the breaking of contact 
in the primary coil A produces no induction current in 7?. The 



172 MAGNETO-ELECTKIC INDUCTION. [539- 

positions of the two coils are in such cases said to be conjugate to 
each other. 

Let BI and B 2 be two of these positions. If the coil B be sud 
denly moved from the position B to the position J3 2 , the algebraical 
sum of the transient currents in the coil B is exactly zero, so 
that the galvanometer needle is left at rest when the motion of B is 
completed. 

This is true in whatever way the coil B is moved from B l to B 2 ^ 
and also whether the current in the primary coil A be continued 
constant, or made to vary during the motion. 

Again, let B be any other position of B not conjugate to A, 
so that the making or breaking of contact in A produces an in 
duction current when B is in the position B . 

Let the contact be made when B is in the conjugate position _Z? 1? 
there will be no induction current. Move B to B > there will be 
an induction current due to the motion, but if B is moved rapidly 
to B , and the primary contact then broken, the induction current 
due to breaking contact will exactly annul the effect of that due to 
the motion, so that the galvanometer needle will be left at rest. 
Hence the current due to the motion from a conjugate position 
to any other position is equal and opposite to the current due to 
breaking contact in the latter position. 

Since the effect of making contact is equal and opposite to that 
of breaking it, it follows that the effect of making contact when the 
coil B is in any position B is equal to that of bringing the coil 
from any conjugate position B l to B while the current is flowing 
through A. 

If the change of the relative position of the coils is made by 
moving the primary circuit instead of the secondary, the result is 
found to be the same. 

539.] It follows from these experiments that the total induction 
current in B during the simultaneous motion of A from A l to A 2J and 
of B from B l to B. 2 , while the current in A changes from ^ to y 2 , 
depends only on the initial state A I} B l , y l5 and the final state 
A 2 , B 2 , y 2 , and not at all on the nature of the intermediate states 
through which the system may pass. 

Hence the value of the total induction current must be of the 
form F(A 2 , B 2 , y 2 ) - F(A lf 19 7l ), 

where F is a function of A, B, and y. 

With respect to the form of this function, we know, by Art. 536, 
that when there is no motion, and therefore A l = A 2 and B l = B 2 , 



540.] ELECTROTONIC STATE. 173 

the induction current is proportional to the primary current. 
Hence y enters simply as a factor, the other factor being a func 
tion of the form and position of the circuits A and J9. 

We also know that the value of this function depends on the 
relative and not on the absolute positions of A and B, so that 
it must be capable of being 1 expressed as a function of the distances 
of the different elements of which the circuits are composed, and 
of the angles which these elements make with each other. 

Let M be this function, then the total induction current may be 
written C {M l7l -M 2 y. 2 }, 

where C is the conductivity of the secondary circuit, and M^ y 1 
are the original, and M 2 , y 2 the final values of M and y. 

These experiments, therefore, shew that the total current of 
induction depends on the change which takes place in a certain 
quantity, My, and that this change may arise either from variation 
of the primary current y, or from any motion of the primary or 
secondary circuit which alters M. 

540.] The conception of such a quantity, on the changes of which, 
and not on its absolute magnitude, the induction current depends, 
occurred to Faraday at an early stage of his researches*. He 
observed that the secondary circuit, when at rest in an electro 
magnetic field which remains of constant intensity, does not shew 
any electrical effect, whereas, if the same state of the field had been 
suddenly produced, there would have been a current. Again, if the 
primary circuit is removed from the field, or the magnetic forces 
abolished, there is a current of the opposite kind. He therefore 
recognised in the secondary circuit, when in the electromagnetic 
field, a peculiar electrical condition of matter, to which he gave 
the name of the Electrotonic State. He afterwards found that he 
could dispense with this idea by means of considerations founded on 
the lines of magnetic force f, but even in his latest researches J, 
he says, ( Again and again the idea of an electrotonic state has 
been forced upon my mind. 

The whole history of this idea in the mind of Faraday, as shewn 
in his published researches, is well worthy of study. By a course 
of experiments, guided by intense application of thought, but 
without the aid of mathematical calculations, he was led to recog 
nise the existence of something which we now know to be a mathe 
matical quantity, and which may even be called the fundamental 

* Exp. Res., series i. 60. % Ib., 3269. 

t Ib., series ii. (242). Ib., 60, 1114, 1661, 1729, 1733. 



174 MAGNETO-ELECTRIC INDUCTION. [54 1 * 

quantity in the theory of electromagnetism. But as he was led 
up to this conception by a purely experimental path, he ascribed 
to it a physical existence, and supposed it to be a peculiar con 
dition of matter, though he was ready to abandon this theory as 
soon as he could explain the phenomena by any more familiar forms 
of thought. 

Other investigators were long afterwards led up to the same 
idea by a purely mathematical path, but, so far as I know, none 
of them recognised, in the refined mathematical idea of the potential 
of two circuits, Faraday s bold hypothesis of an electrotonic state. 
Those, therefore, who have approached this subject in the way 
pointed out by those eminent investigators who first reduced its 
laws to a mathematical form, have sometimes found it difficult 
to appreciate the scientific accuracy of the statements of laws which 
Faraday, in the first two series of his Researches, has given with 
such wonderful completeness. 

The scientific value of Faraday s conception of an electrotonic 
state consists in its directing the mind to lay hold of a certain 
quantity, on the changes of which the actual phenomena depend. 
Without a much greater degree of development than Faraday gave 
it, this conception does not easily lend itself to the explanation of the 
phenomena. We shall return to this subject again in Art. 584. 

541.] A method which, in Faraday s hands, was far more powerful 
is that in which he makes use of those lines of magnetic force 
which were always in his mind s eye when contemplating his 
magnets or electric currents, and the delineation of which by 
means of iron filings he rightly regarded * as a most valuable aid 
to the experimentalist. 

Faraday looked on these lines as expressing, not only by their 
direction that of the magnetic force, but by their number and 
concentration the intensity of that force, and in his later re 
searches f he shews how to conceive of unit lines of force. I have 
explained in various parts of this treatise the relation between the 
properties which Faraday recognised in the lines of force and the 
mathematical conditions of electric and magnetic forces, and how 
Faraday s notion of unit lines and of the number of lines within 
certain limits may be made mathematically precise. See Arts. 82, 
404, 490. 

In the first series of his Researches J he shews clearly how the 
direction of the current in a conducting circuit, part of which is 
* Exp. lies., 3234. t Ib., 3122. $ Ib., 114. 



LINES OF MAGNETIC INDUCTION. 175 

moveable, depends on the mode in which the moving 1 part cuts 
through the lines of magnetic force. 

In the second series* he shews how the phenomena produced 
by variation of the strength of a current or a magnet may be 
explained, by supposing the system of lines of force to expand from 
or contract towards the wire or magnet as its power rises or falls. 

I am not certain with what degree of clearness he then held the 
doctrine afterwards so distinctly laid down by him f, that the 
moving conductor, as it cuts the lines of force, sums up the action 
due to an area or section of the lines of force. This, however, 
appears no new view of the case after the investigations of the 
second series J have been taken into account. 

The conception which Faraday had of the continuity of the lines 
of force precludes the possibility of their suddenly starting into 
existence in a place where there were none before. If, therefore, 
the number of lines which pass through a conducting circuit is 
made to vary, it can only be by the circuit moving across the lines 
of force, or else by the lines of force moving across the circuit. 
In either case a current is generated in the circuit. 

The number of the lines of force which at any instant pass through 
the circuit is mathematically equivalent to Faraday s earlier con 
ception of the electrotonic state of that circuit, and it is represented 
by the quantity My. 

It is only since the definitions of electromotive force, Arts. 69, 
274, and its measurement have been made more precise, that we 
can enunciate completely the true law of magneto -electric induction 
in the following terms : 

The total electromotive force acting round a circuit at any 
instant is measured by the rate of decrease of the number of lines 
of magnetic force which pass through it. 

When integrated with respect to the time this statement be 
comes : 

The time-integral of the total electromotive force acting round 
any circuit, together with the number of lines of magnetic force 
which pass through the circuit, is a constant quantity. 

Instead of speaking of the number of lines of magnetic force, we 
may speak of the magnetic induction through the circuit, or the 
surface-integral of magnetic induction extended over any surface 
bounded by the circuit. 

* Exp. Res., 238. t Ib., 3082, 3087, 3113. 

Ib., 217, &c. 



176 MAGNETO-ELECTRIC INDUCTION. [542. 

We shall return again to this method of Faraday. In the mean 
time we must enumerate the theories of induction which are 
founded on other considerations. 

Lenz s Law. 

542.] In 1834, Lenz* enunciated the following remarkable 
relation between the phenomena of the mechanical action of electric 
currents, as defined by Ampere s formula, and the induction of 
electric currents by the relative motion of conductors. An earlier 
attempt at a statement of such a relation was given by Ritchie in 
the Philosophical Magazine for January of the same year, but the 
direction of the induced current was in every case stated wrongly. 
Lenz s law is as follows. 

If a constant current flows in the primary circuit A, and if, by the 
motion of A, or of the secondary circuit B, a current is induced in B, the 
direction of this induced current wilt be such that, by its electromagnetic 
action on A, it tends to oppose the relative motion of the circuits. 

On this law J. Neumann f founded his mathematical theory of 
induction, in which he established the mathematical laws of the 
induced currents due to the motion of the primary or secondary 
conductor. He shewed that the quantity M, which we have called 
the potential of the one circuit on the other, is the same as the 
electromagnetic potential of the one circuit on the other, which 
we have already investigated in connexion with Ampere s formula. 
We may regard J. Neumann, therefore, as having completed for 
the induction of currents the mathematical treatment which Ampere 
had applied to their mechanical action. 

543.] A step of still greater scientific importance was soon after 
made by Helmholtz in his Essay on the Conservation of Force J, and 
by Sir W. Thomson , working somewhat later, but independently 
of Helmholtz. They shewed that the induction of electric currents 
discovered by Faraday could be mathematically deduced from the 
electromagnetic actions discovered by Orsted and Ampere by the 
application of the principle of the Conservation of Energy. 

Helmholtz takes the case of a conducting circuit of resistance R, 
in which an electromotive force A, arising from a voltaic or thermo- 

* Pogg., Ann. xxxi. 483 (1834). 

t Berlin Acad., 1845 and 1847. 

Kead before the Physical Society of Berlin, July 23, 1847. Translated in 
Taylor s Scientific Memoirs, part ii. p. 114. 

Trans. Brit. Ass., 1848, and Phil. Mag., Dec. 1851. See also his paper on 
Transient Electric Currents, Phil. Mag., 1853. . 



543-1 HELMHOLTZ AND THOMSON. 177 

electric arrangement, acts. The current in the circuit at any 
instant is /. He supposes that a magnet is in motion in the 
neighbourhood of the circuit, and that its potential with respect to 
the conductor is F, so that, during any small interval of time dt, the 
energy communicated to the magnet by the electromagnetic action 



is 



The work done in generating heat in the circuit is, by Joule s 
law, Art. 242, I 2 Belt, and the work spent by the electromotive 
force A, in maintaining the current / during the time dt, is A Idt. 
Hence, since the total work done must be equal to the work spent, 






at 
whence we find the intensity of the current 




Now the value of A may be what we please. Let, therefore, 
A = 0, and then 1 



or, there will be a current due to the motion of the magnet, equal 

dV 

to that due to an electromotive force =- 

dt 

The whole induced current during the motion of the magnet 
from a place where its potential is V^ to a place where its potential 
is Fo, is 



and therefore the total current is independent of the velocity or 
the path of the magnet, and depends only on its initial and final 
positions. 

In Helmholtz s original investigation he adopted a system of 
units founded on the measurement of the heat generated in the 
conductor by the current. Considering the unit of current as 
arbitrary, the unit of resistance is that of a conductor in which this 
unit current generates unit of heat in unit of time. The unit of 
electromotive force in this system is that required to produce the 
unit of current in the conductor of unit resistance. The adoption 
of this system of units necessitates the introduction into the equa 
tions of a quantity , which is the mechanical equivalent of the 
unit of heat. As we invariably adopt either the electrostatic or 

VOL. II. N 



178 MAGNETO-ELECTRIC INDUCTION. [544. 

the electromagnetic system of units, this factor does not occur in 
the equations here given. 

544.] Helmholtz also deduces the current of induction when a 
conducting circuit and a circuit carrying a constant current are 
made to move relatively to one another. 

Let R lt R 2 be the resistances, I 19 I 2 the currents, A lt A 2 the 
external electromotive forces, and V the potential of the one circuit 
on the other due to unit current in each, then we have, as before, 

4 /! + A, I 2 = I^R, + L?R.> + /, 7, ~ 

If we suppose 7 X to be the primary current, and 7 2 so much less 

than /u that it does not by its induction produce any sensible 

^ 
alteration in 7 15 so that we may put 7 X = - , then 



a result which may be interpreted exactly as in the case of the 
magnet. 

If we suppose J 2 to be the primary current, and I to be very 
much smaller than / 2 , we get for I lt 

A-I^ 
T AI L * dt 

This shews that for equal currents the electromotive force of the 
first circuit on the second is equal to that of the second on the first, 
whatever be the forms of the circuits. 

Helmholtz does not in this memoir discuss the case of induction 
due to the strengthening or weakening of the primary current, or 
the induction of a current on itself. Thomson * applied the same 
principle to the determination of the mechanical value of a current, 
and pointed out that when work is done by the mutual action of 
two constant currents, their mechanical value is increased by the 
same amount, so that the battery has to supply double that amount 
of work, in addition to that required to maintain the currents 
against the resistance of the circuits f. 

545.] The introduction, by W. Weber, of a system of absolute 



* Mechanical Theory of Electrolysis, Phil. Mag., Dec., 1851. 

t Nichol s Cyclopaedia of Physical Science, ed. 1860, Article Magnetism, Dy 
namical Relations of, and Reprint, 571. 



545-1 WEBER. 179 

units for the measurement of electrical quantities is one of the most 
important steps in the progress of the science. Having already, in 
conjunction with Gauss, placed the measurement of magnetic quan 
tities in the first rank of methods of precision, Weher proceeded 
in his Electrodynamic Measurements not only to lay down sound 
principles for fixing the units to be employed, but to make de 
terminations of particular electrical quantities in terms of these 
units, with a degree of accuracy previously unattempted. Both the 
electromagnetic and the electrostatic systems of units owe their 
development and practical application to these researches. 

Weber has also formed a general theory of electric action from 
which he deduces both electrostatic and electromagnetic force, and 
also the induction of electric currents. We shall consider this 
theory, with some of its more recent developments, in a separate 
chapter. See Art. 846. 



N 2 



CHAPTER IV. 



ON THE INDUCTION OF A CURRENT ON ITSELF. 



546.] FARADAY has devoted the ninth series of his Researches to 
the investigation of a class of phenomena exhibited by the current 
in a wire which forms the coil of an electromagnet. 

Mr. Jenkin had observed that, although it is impossible to pro 
duce a sensible shock by the direct action of a voltaic system 
consisting of only one pair of plates, yet, if the current is made 
to pass through the coil of an electromagnet, and if contact is 
then broken between the extremities of two wires held one in each 
hand, a smart shock will be felt. No such shock is felt on making 
contact. 

Faraday shewed that this and other phenomena, which he de 
scribes, are due to the same inductive action which he had already 
observed the current to exert on neighbouring conductors. In this 
case, however, the inductive action is exerted on the same conductor 
which carries the current, and it is so much the more powerful 
as the wire itself is nearer to the different elements of the current 
than any other wire can be. 

547.] He observes, however *, that the first thought that arises 
in the mind is that the electricity circulates with something like 
momentum or inertia in the wire. Indeed, when we consider one 
particular wire only, the phenomena are exactly analogous to those 
of a pipe full of water flowing in a continued stream. If while 
the stream is flowing we suddenly close the end of the tube, the 
momentum of the water produces a sudden pressure, which is much 
greater than that due to the head of water, and may be sufficient 
to burst the pipe. 

If the water has the means of escaping through a narrow jet 

* Exp. Res., 1077- 



55O.] ELECTRIC INERTIA. 181 

when the principal aperture is closed, it will be projected with a 
velocity much greater than that due to the head of water, and 
if it can escape through a valve into a chamber, it will do so, 
even when the pressure in the chamber is greater than that due 
to the head of water. 

It is on this principle that the hydraulic ram is constructed, 
by which a small quantity of water may be raised to a great height 
by means of a large quantity flowing down from a much lower 
level. 

548.] These effects of the inertia of the fluid in the tube depend 
solely on the quantity of fluid running through the tube, on its 
length, and on its section in different parts of its length. They 
do not depend on anything outside the tube, nor on the form into 
which the tube may be bent, provided its length remains the 
same. 

In the case of the wire conveying a current this is not the case, 
for if a long wire is doubled on itself the effect is very small, if 
the two parts are separated from each other it is greater, if it 
is coiled up into a helix it is still greater, and greatest of all if, 
when so coiled, a piece of soft iron is placed inside the coil. 

Again, if a second wire is coiled up with the first, but insulated 
from it, then, if the second wire does not form a closed circuit, 
the phenomena are as before, but if the second wire forms a closed 
circuit, an induction current is formed in the second wire, and 
the effects of self-induction in the first wire are retarded. 

549.] These results shew clearly that, if the phenomena are due 
to momentum, the momentum is certainly not that of the electricity 
in the wire, because the same wire, conveying the same current, 
exhibits effects which differ according to its form ; and even when 
its form remains the same, the presence of other bodies, such as 
a piece of iron or a closed metallic circuit, affects the result. 

550.] It is difficult, however, for the mind which has once 
recognised the analogy between the phenomena of self-induction 
and those of the motion of material bodies, to abandon altogether 
the help of this analogy, or to admit that it is entirely superficial 
and misleading. The fundamental dynamical idea of matter, as 
capable by its motion of becoming the recipient of momentum and 
of energy, is so interwoven with our forms of thought that, when 
ever we catch a glimpse of it in any part of nature, we feel that 
a path is before us leading, sooner or later, to the complete under 
standing of the subject. 



182 SELF-INDUCTION. [55 1 - 

551.] In the case of the electric current, we find that, when the 
electromotive force begins to act, it does not at once produce the 
full current, but that the current rises gradually. What is the 
electromotive force doing during the time that the opposing re 
sistance is not able to balance it ? It is increasing the electric 
current. 

Now an ordinary force, acting on a body in the direction of its 
motion, increases its momentum, and communicates to it kinetic 
energy, or the power of doing work on account of its motion. 

In like manner the unresisted part of the electromotive force has 
been employed in increasing the electric current. Has the electric 
current, when thus produced, either momentum or kinetic energy ? 

We have already shewn that it has something very like mo 
mentum, that it resists being suddenly stopped, and that it can 
exert, for a short time, a great electromotive force. 

But a conducting circuit in which a current has been set up 
has the power of doing work in virtue of this current, and this 
power cannot be said to be something very like energy, for it 
is really and truly energy. 

Thus, if the current be left to itself, it will continue to circulate 
till it is stopped by the resistance of the circuit. Before it is 
stopped, however, it will have generated a certain quantity of 
heat, and the amount of this heat in dynamical measure is equal 
to the energy originally existing in the current. 

Again, when the current is left to itself, it may be made to 
do mechanical work by moving magnets, and the inductive effect 
of these motions will, by Lenz s law, stop the current sooner than 
the resistance of the circuit alone would have stopped it. In this 
way part of the energy of the current may be transformed into 
mechanical work instead of heat. 

552.] It appears, therefore, that a system containing an electric 
current is a seat of energy of some kind ; and since we can form 
no conception of an electric current except as a kinetic pheno 
menon *, its energy must be kinetic energy, that is to say, the 
energy which a moving body has in virtue of its motion. 

We have already shewn that the electricity in the wire cannot 
be considered as the moving body in which we are to find this 
energy, for the energy of a moving body does not depend on 
anything external to itself, whereas the presence of other bodies 
near the current alters its energy. 

* Faraday, Eocp. Res. (283.) 



552.] ELECTROKINETIC ENEKGY. 183 

We are therefore led to enquire whether there may not be some 
motion going 1 on in the space outside the wire, which is not occupied 
by the electric current, but in which the electromagnetic effects of 
the current are manifested. 

I shall not at present enter on the reasons for looking in one 
place rather than another for such motions, or for regarding these 
motions as of one kind rather than another. 

What I propose now to do is to examine the consequences of 
the assumption that the phenomena of the electric current are those 
of a moving system, the motion being communicated from one part 
of the system to another by forces, the nature and laws of which 
we do not yet even attempt to define, because we can eliminate 
these forces from the equations of motion by the method given 
by Lagrange for any connected system. 

In the next five chapters of this treatise I propose to deduce 
the main structure of the theory of electricity from a dynamical 
hypothesis of this kind, instead of following the path which has 
led Weber and other investigators to many remarkable discoveries 
and experiments, and to conceptions, some of which are as beautiful 
as they are bold. I have chosen this method because I wish to 
shew that there are other ways of viewing the phenomena which 
appear to me more satisfactory, and at the same time are more 
consistent with the methods followed in the preceding parts of this 
book than those which proceed on the hypothesis of direct action 
at a distance. 



CHAPTER V. 

ON THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM. 



553.] IN the fourth section of the second part of his Mecanique 
Analytique, Lagrange has given a method of reducing the ordinary 
dynamical equations of the motion of the parts of a connected 
system to a number equal to that of the degrees of freedom of 
the system. 

The equations of motion of a connected system have been given 
in a different form by Hamilton, and have led to a great extension 
of the higher part of pure dynamics *. 

As we shall find it necessary, in our endeavours to bring electrical 
phenomena within the province of dynamics, to have our dynamical 
ideas in a state fit for direct application to physical questions, we 
shall devote this chapter to an exposition of these dynamical ideas 
from a physical point of view. 

554.] The aim of Lagrange was to bring dynamics under the 
power of the calculus. He began by expressing the elementary 
dynamical relations in terms of the corresponding relations of pure 
algebraical quantities, and from the equations thus obtained he 
deduced his final equations by a purely algebraical process. Certain 
quantities (expressing the reactions between the parts of the system 
called into play by its physical connexions) appear in the equations 
of motion of the component parts of the system, and Lagrange s 
investigation, as seen from a mathematical point of view, is a 
method of eliminating these quantities from the final equations. 

In following the steps of this elimination the mind is exercised 
in calculation, and should therefore be kept free from the intrusion 
of dynamical ideas. Our aim, on the other hand, is to cultivate 

* See Professor Cayley s Report on Theoretical Dynamics, British Association, 
3 857 ; and Thomson and Tait s Natural Philosophy. 



555-] GENERALIZED COORDINATES. 185 

our dynamical ideas. We therefore avail ourselves of the labours 
of the mathematicians, and retranslate their results from the lan 
guage of the calculus into the language of dynamics, so that our 
words may call up the mental image, not of some algebraical 
process, but of some property of moving bodies. 

The language of dynamics has been considerably extended by 
those who have expounded in popular terms the doctrine of the 
Conservation of Energy, and it will be seen that much of the 
following statement is suggested by the investigation in Thomson 
and Tait^s Natural Philosophy, especially the method of beginning 
with the theory of impulsive forces. 

I have applied this method so as to avoid the explicit con 
sideration of the motion of any part of the system except the 
coordinates or variables, on which the motion of the whole depends. 
It is doubtless important that the student should be able to trace 
the connexion of the motion of each part of the system with that 
of the variables, but it is by no means necessary to do this in 
the process of obtaining the final equations, which are independent 
of the particular form of these connexions. 

The Variables. 

555.] The number of degrees of freedom of a system is the 
number of data which must be given in order completely to 
determine its position. Different forms may be given to these 
data, but their number depends on the nature of the system itself, 
and cannot be altered. 

To fix our ideas we may conceive the system connected by means 
of suitable mechanism with a number of moveable pieces, each 
capable of motion along a straight line, and of no other kind of 
motion. The imaginary mechanism which connects each of these 
pieces with the system must be conceived to be free from friction, 
destitute of inertia, and incapable of being strained by the action 
of the applied forces. The use of this mechanism is merely to 
assist the imagination in ascribing position, velocity, and momentum 
to what appear, in Lagrange s investigation, as pure algebraical 
quantities. 

Let q denote the position of one of the moveable pieces as defined 
by its distance from a fixed point in its line of motion. We shall 
distinguish the values of q corresponding to the different pieces 
by the suffixes u 2 , &c. When we are dealing with a set of 
quantities belonging to one piece only we may omit the suffix. 



186 KINETICS. [556. 

When the values of all the variables (q) are given, the position 
of each of the moveable pieces is known, and, in virtue of the 
imaginary mechanism, the configuration of the entire system is 
determined. 

The Velocities. 

556.] During the motion of the system the configuration changes 
in some definite manner, and since the configuration at each instant 
is fully defined by the values of the variables (q), the velocity of 
every part of the system, as well as its configuration, will be com 
pletely defined if we know the values of the variables (q), together 

with their velocities (- , or, according to Newton s notation, q) 

The Forces. 

557.] By a proper regulation of the motion of the variables, any 
motion of the system, consistent with the nature of the connexions, 
may be produced. In order to produce this motion by moving 
the variable pieces, forces must be applied to these pieces. 

We shall denote the force which must be applied to any variable 
q r by F r . The system of forces (F) is mechanically equivalent (in 
virtue of the connexions of the system) to the system of forces, 
whatever it may be, which really produces the motion. 

The Momenta. 

558.] When a body moves in such a way that its configuration, 
with respect to the force which acts on it, remains always the same, 
(as, for instance, in the case of a force acting on a single particle in 
the line of its motion,) the moving force is measured by the rate 
of increase of .the momentum. If F is the moving force, and p the 
momentum, 



whence p = / Fdt. 



The time-integral of a force is called the Impulse of the force ; 
so that we may assert that the momentum is the impulse of the 
force which would bring the body from a state of rest into the given 
state of motion. 

In the case of a connected system in motion, the configuration is 
continually changing at a rate depending on the velocities (q\ so 



559-] IMPULSE AND MOMENTUM. 187 

that we can no longer assume that the momentum is the time- 
intesral of the force which acts on it. 

o 

But the increment bq of any variable cannot be greater than 
qbt, where 8^ is the time during which the increment takes place, 
and q is the greatest value of the velocity during that time. In the 
case of a system moving from rest under the action of forces always 
in the same direction, this is evidently the final velocity. 

If the final velocity and configuration of the system are given, 
we may conceive the velocity to be communicated to the system 
in a very small time t, the original configuration differing from 
the final configuration by quantities bq lt 2 , &c., which are less 
than q^btj ^ 2 5^, &c., respectively. 

The smaller we suppose the increment of time 8, the greater 
must be the impressed forces, but the time-integral, or impulse, 
of each force will remain finite. The limiting value of the impulse, 
when the time is diminished and ultimately vanishes, is defined 
as the instantaneous impulse, and the momentum p, corresponding 
to any variable q, is defined as the impulse corresponding to that 
variable, when the system is brought instantaneously from a state 
of rest into the given state of motion. 

This conception, that the momenta are capable of being produced 
by instantaneous impulses on the system at rest, is introduced only 
as a method of defining the magnitude of the momenta, for the 
momenta of the system depend only on the instantaneous state 
of motion of the system, and not on the process by which that state 
was produced. 

In a connected system the momentum corresponding to any 
variable is in general a linear function of the velocities of all the 
variables, instead of being, as in the dynamics of a particle, simply 
proportional to the velocity. 

The impulses required to change the velocities of the system 
suddenly from y l9 q. 2 , &c. to /, q 2 , &c, are evidently equal to 
Pi p\, Pz J2> ^ ne cbaBgcs of momentum of the several variables. 



Work done by a Small Impulse. 

559.] The work done by the force F l during the impulse is the 
space-integral of the force, or 



W 



=j 



188 KINETICS. [560. 

If fa is the greatest and q" the least value of tlie velocity q-^ 
during the action of the force, W must be less than 



2i< Fdt 



or 



and greater than q"\Fdt or q\(p\p\)> 

If we now suppose the impulse / Fdt to be diminished without 

limit, the values of q{ and q" will approach and ultimately coincide 
with that of q lt and we may write p{p^ = pi, so that the work 
done is ultimately 7ir 



or, the work done by a very small impulse is ultimately the product 
of the impulse and the velocity. 

Increment of the Kinetic Energy. 

560.] When work is done in setting a conservative system in 
motion, energy is communicated to it, and the system becomes 
capable of doing an equal amount of work against resistances 
before it is reduced to rest. 

The energy which a system possesses in virtue of its motion 
is called its Kinetic Energy, and is communicated to it in the form 
of the work done by the forces which set it in motion. 

If T be the kinetic energy of the system, and if it becomes 
T 4- 8 T } on account of the action of an infinitesimal impulse whose 
components are 8^ 15 5j0 2 , &c., the increment 8 T must be the sum 
of the quantities of work done by the components of the impulse, 
or in symbols, IT = &*& + j s 8 A + &c., 

= 2&8j). (1) 

The instantaneous state of the system is completely defined if 
the variables and the momenta are given. Hence the kinetic 
energy, which depends on the instantaneous state of the system, 
can be expressed in terms of the variables (q), and the momenta (/>). 
This is the mode of expressing T introduced by Hamilton. When 
T is expressed in this way we shall distinguish it by the suffix p) 
thus, T p . 

The complete variation of T p is 

^=2^+Ss ? . (2) 



561.] HAMILTON S EQUATIONS. 189 

The last term may be written 



which diminishes with 8, and ultimately vanishes with it when the 
impulse becomes instantaneous. 

Hence, equating- the coefficients of bp in equations (1) and (2), 
we obtain . = ^ (s) 

or, the velocity corresponding to the variable q is the differential 
coefficient of T p with respect to the corresponding momentum p. 

We have arrived at this result by the consideration of impulsive 
forces. By this method we have avoided the consideration of the 
change of configuration during the action of the forces. But the 
instantaneous state of the system is in all respects the same, whether 
the system was brought from a state of rest to the given state 
of motion by the transient application of impulsive forces, or 
whether it arrived at that state in any manner, however gradual. 

In other words, the variables, and the corresponding velocities 
and momenta, depend on the actual state of motion of the system 
at the given instant, and not on its previous history. 

Hence, the equation (3) is equally valid, whether the state of 
motion of the system is supposed due to impulsive forces, or to 
forces acting in any manner whatever. 

We may now therefore dismiss the consideration of impulsive 
forces, together with the limitations imposed on their time of 
action, and on the changes of configuration during their action. 



Hamilton s Equations of Motion. 
561.] We have already shewn that 

dT 



(4) 



Let the system move in any arbitrary way, subject to the con 
ditions imposed by its connexions, then the variations of p and q are 

(5) 



190 KINETICS. [562. 

and the complete variation of T p is 



But the increment of the kinetic energy arises from the work 
done by the impressed forces, or 

IT, = 2 (Fig). (8) 

In these two expressions the variations bq are all independent of 
each other, so that we are entitled to equate the coefficients of each 
of them in the two expressions (7) and (8). We thus obtain 



where the momentum^ and the force F r belong to the variable q r . 
There are as many equations of this form as there are variables. 
These equations were given by Hamilton They shew that the 
force corresponding to any variable is the sum of two parts. The 
first part is the rate of increase of the momentum of that variable 
with respect to the time. The second part is the rate of increase 
of the kinetic energy per unit of increment of the variable, the 
other variables and all the momenta being constant. 

The Kinetic Energy expressed in Terms of the Momenta and 

Velocities. 

562.] Let p l9 p 2 , &c. be the momenta, and q l} q 2 , &c. the 
velocities at a given instant, and let p x , p 2 , &c., q x , q 2 , &c. be 
another system of momenta and velocities, such that 

Pi = *Pi> 4i = 0n &c - ( 10 ) 

It is manifest that the systems p, q will be consistent with each 
other if the systems p, q are so. 

Now let n vary by bn. The work done by the force F l is 

F i*h = 4i 8 Pi = Jiftntn. (11) 

Let n increase from to 1, then the system is brought from 
a state of rest into the state of motion (qp), and the whole work 
expended in producing this motion is 



-)/ 



But 



ri 

/ ndn = \, 

Jn 



564.] LAGRANGE S EQUATIONS. 191 

and the work spent in producing the motion is equivalent to the 
kinetic energy. Hence 

TP*= iC^ift + ^fc + fcC ). (13) 

where T p $ denotes the kinetic energy expressed in terms of the 
momenta and velocities. The variables q l , q% , &c. do not enter into 
this expression. 

The kinetic energy is therefore half the sum of the products of 
the momenta into their corresponding velocities. 

When the kinetic energy is expressed in this way we shall denote 
it by the symbol T p ^ . It is a function of the momenta and velo 
cities only, and does not involve the variables themselves. 

563.] There is a third method of expressing the kinetic energy, 
which is generally, indeed, regarded as the fundamental one. By 
solving the equations (3) we may express the momenta in terms 
of the velocities, and then, introducing these values in (13), we shall 
have an expression for T involving only the velocities and the 
variables. When T is expressed in .this form we shall indicate it 
by the symbol T^ . This is the form in which the kinetic energy is 
expressed in the equations of Lagrange. 

564.] It is manifest that, since T p , T$ 9 and T p ^ are three different 
expressions for the same thing, 

T p +Tt-2T p(l = 0, 
or Tp + Tt-Piii-toto-ke. = - (14) 

Hence, if all the quantities jo, q, and q vary, 



The variations 8jt? are not independent of the variations bq and 
bq, so that we cannot at once assert that the coefficient of each 
variation in this equation is zero. But we know, from equations 

(3) that g-ft = o,fa, do) 

so that the terms involving the variations bp vanish of themselves. 
The remaining variations bq and bq are now all independent, 
so that we find, by equating to zero the coefficients of bq lt &c , 



192 KINETICS. [565. 

or, the components of momentum are the differential coefficients of T^ 
with respect to the corresponding velocities. 

Again, by equating to zero the coefficients of 8^ 15 &c., 

^+^ = 0; (.8) 

dc h d^ 

or, the differential coefficient of the kinetic energy with respect to any 
variable q l is equal in magnitude but opposite in sign when T is 
expressed as a function of the velocities instead of as a function of 
the momenta. 

In virtue of equation (18) we may write the equation of motion (9), 

p djiw, 

dt dq l 

p i VtW (20) 

at dq l dq l 

which is the form in which the equations of motion were given by 
Lagrange. 

565.] In the preceding investigation we have avoided the con 
sideration of the form of the function which expresses the kinetic 
energy in terms either of the velocities or of the momenta. The 
only explicit form which we have assigned to it is 

TP* = 4 (PiJi + J 2 ? + &c.), (21) 

in which it is expressed as half the sum of the products of the 
momenta each into its corresponding velocity. 

We may express the velocities in terms of the differential co 
efficients of Tp with respect to the momenta, as in equation (3), 



This shews that T p is a homogeneous function of the second 
degree of the momenta p l} p 2 , &c. 

We may also express the momenta in terms of T$ , and we find 

*-*&+* + *") < 23 > 

which shews that T$ is a homogeneous function of the second degree 
with respect to the velocities <? 15 q 2 , &c. 
If we write 



P n for ^, P 12 for ^- &c. 

and Q n for - ? , Q 12 for -^ /- , &c. ; 



567.] MOMENTS AND PRODUCTS OF INERTIA. 193 

then, since both T ( j and T p are functions of the second degree of 
q and of p respectively, both the P s and the Q s will be functions 
of the variables q only, and independent of the velocities and the 
momenta. We thus obtain the expressions for I\ 

2 TI = P n tf + 2P 12 q, q 2 + &c., (24) 

2T p = QuPi 2 + 2 Qi2PiP2 + & c - ( 25 ) 

The momenta are expressed in terms of the velocities by the 

linear equations ^ = p n ^ + P 12 ^ + &c., (26) 

and the velocities are expressed in terms of the momenta by the 
linear equations ^ = Q n p + Q 12 p 2 + &c. (27) 

In treatises on the dynamics of a rigid body, the coefficients 
corresponding to P n , in which the suffixes are the same, are called 
Moments of Inertia, and those corresponding to P 12 , in which 
the suffixes are different, are called Products of Inertia. We may 
extend these names to the more general problem which is now 
before us, in which these quantities are not, as in the case of a 
rigid body, absolute constants, but are functions of the variables 

In like manner we may call the coefficients of the form Q n 
Moments of Mobility, and those of the form Q 12 , Products of 
Mobility. It is not often, however, that we shall have occasion 
to speak of the coefficients of mobility. 

566.] The kinetic energy of the system is a quantity essentially 
positive or zero. Hence, whether it be expressed in terms of the 
velocities, or in terms of the momenta, the coefficients must be 
such that no real values of the variables can make T negative. 

We thus obtain a set of necessary conditions which the values of 
the coefficients P must satisfy. 

The quantities P n , P 22 , &c., and all determinants of the sym 
metrical form 

P P P 

12 22 

p p p 

* 13 * 23 * q 

which can be formed from the system of coefficients must be positive 
or zero. The number of such conditions for n variables is 2 n 1. 

The coefficients Q are subject to conditions of the same kind. 

567.] In this outline of the fundamental principles of the dy 
namics of a connected system, we have kept out of view the 
mechanism by which the parts of the system are connected. We 

VOL. n. o 



194 KINETICS. [567. 

have not even written down a set of equations to indicate how 
the motion of any part of the system depends on the variation 
of the variables. We have confined our attention to the variables, 
their velocities and momenta, and the forces which act on the 
pieces representing- the variables. Our only assumptions are, that 
the connexions of the system are such that the time is not explicitly 
contained in the equations of condition, and that the principle of 
the conservation of energy is applicable to the system. 

Such a description of the methods of pure dynamics is not un 
necessary, because Lag-range and most of his followers, to whom 
we are indebted for these methods, have in general confined them 
selves to a demonstration of them, and, in order to devote their 
attention to the symbols before them, they have endeavoured to 
banish all ideas except those of pure quantity, so as not only to 
dispense with diagrams, but even to get rid of the ideas of velocity, 
momentum, and energy, after they have been once for all sup 
planted by symbols in the original equations. In order to be able 
to refer to the results of this analysis in ordinary dynamical lan 
guage, we have endeavoured to retranslate the principal equations 
of the method into language which may be intelligible without the 
use of symbols. 

As the development of the ideas and methods of pure mathe 
matics has rendered it possible, by forming a mathematical theory 
of dynamics, to bring to light many truths which could not have 
been discovered without mathematical training, so, if we are to 
form dynamical theories of other sciences, we must have our minds 
imbued with these dynamical truths as well as with mathematical 
methods. 

In forming the ideas and words relating to any science, which, 
like electricity, deals with forces and their effects, we must keep 
constantly in mind the ideas appropriate to the fundamental science 
of dynamics, so that we may, during the first development of the 
science, avoid inconsistency with what is already established, and 
also that when our views become clearer, the language we have 
adopted may be a help to us and not a hindrance. 



CHAPTER VI. 



DYNAMICAL THEORY OF ELECTROMAGNETISM. 



568.] WE have shewn, in Art. 552, that, when an electric current 
exists in a conducting circuit, it has a capacity for doing a certain 
amount of mechanical work, and this independently of any external 
electromotive force maintaining the current. Now capacity for 
performing work is nothing else than energy, in whatever way 
it arises, and all energy is the same in kind, however it may differ 
in form. The energy of an electric current is either of that form 
which consists in the actual motion of matter, or of that which 
consists in the capacity for being set in motion, arising from forces 
acting between bodies placed in certain positions relative to each 
other. 

The first kind of energy, that of motion, is called Kinetic energy, 
and when once understood it appears so fundamental a fact of 
nature that we can hardly conceive the possibility of resolving 
it into anything else. The second kind of energy, that depending 
on position, is called Potential energy, and is due to the action 
of what we call forces, that is to say, tendencies towards change 
of relative position. With respect to these forces, though we may 
accept their existence as a demonstrated fact, yet we always feel 
that every explanation of the mechanism by which bodies are set 
in motion forms a real addition to our knowledge. 

569.] The electric current cannot be conceived except as a kinetic 
phenomenon. Even Faraday, who constantly endeavoured to 
emancipate his mind from the influence of those suggestions which 
the words electric current and electric fluid are too apt to carry 
with them, speaks of the electric current as something progressive, 
and not a mere arrangement *. 

* Exp. Res., 283. 

O 2 



196 ELECTROKINETICS. \_S7- 

The effects of the current, such as electrolysis, and the transfer 
of electrification from one body to another, are all progressive 
actions which require time for their accomplishment, and are there 
fore of the nature of motions. 

As to the velocity of the current, we have shewn that we know 
nothing about it, it may be the tenth of an inch in an hour, or 
a hundred thousand miles in a second *. So far are we from 
knowing its absolute value in any case, that we do not even know 
whether what we call the positive direction is the actual direction 
of the motion or the reverse. 

But all that we assume here is that the electric current involves 
motion of some kind. That which is the cause of electric currents 
has been called Electromotive Force. This name has long been 
used with great advantage, and has never led to any inconsistency 
in the language of science. Electromotive force is always to be 
understood to act on electricity only, not on the bodies in which 
the electricity resides. It is never to be confounded with ordinary 
mechanical force, which acts on bodies only, not on the electricity 
in them. If we ever come to know the formal relation between 
electricity and ordinary matter, we shall probably also know the 
relation between electromotive force and ordinary force. 

570.] When ordinary force acts on a body, and when the body 
yields to the force, the work done by the force is measured by the 
product of the force into the amount by which the body yields. 
Thus, in the case of water forced through a pipe, the work done 
at any section is measured by the fluid pressure at the section 
multiplied into the quantity of water which crosses the section. 

In the same way the work done by an electromotive force is 
measured by the product of the electromotive force into the quantity 
of electricity which crosses a section of the conductor under the 
action of the electromotive force. 

The work done by an electromotive force is of exactly the same 
kind as the work done by an ordinary force, and both are measured 
by the same standards or units. 

Part of the work done by an electromotive force acting on a 
conducting circuit is spent in overcoming the resistance of the 
circuit, and this part of the work is thereby converted into heat. 
Another part of the work is spent in producing the electromag 
netic phenomena observed by Ampere, in which conductors are 
made to move by electromagnetic forces. The rest of the work 

* Exp. Res., 1648. 



KINETIC ENEEGY. 197 

is spent in increasing the kinetic energy of the current, and the 
effects of this part of the action are shewn in the phenomena of the 
induction of currents observed by Faraday. 

We therefore know enough about electric currents to recognise, 
in a system of material conductors carrying currents, a dynamical 
system which is the seat of energy, part of which may be kinetic 
and part potential. 

The nature of the connexions of the parts of this system is 
unknown to us, but as we have dynamical methods of investigation 
which do not require a knowledge of the mechanism of the system, 
we shall apply them to this case. 

We shall first examine the consequences of assuming the most 
general form for the function which expresses the kinetic energy of 
the system. 

571.] Let the system consist of a number of conducting circuits, 
the form and position of which are determined by the values of 
a system of variables # 15 x 9) &c., the number of which is equal 
to the number of degrees of freedom of the system. 

If the whole kinetic energy of the system were that due to the 
motion of these conductors, it would be expressed in the form 

T = i (#! ffj a?! 2 -f &c. + (^ a? 2 ) ^ x 2 -f &c., 

where the symbols (^ 15 a: lf &c.) denote the quantities which we have 
called moments of inertia, and (# 1} sc 29 &c.) denote the products of 
inertia. 

If X is the impressed force, tending to increase the coordinate x, 
which is required to produce the actual motion, then, by Lagrange s 
equation, d dT dT _ 

dt dx dx ~ 

When T denotes the energy due to the visible motion only, we 
shall indicate it by the suffix TO , thus, T m . 

But in a system of conductors carrying electric currents, part of 
the kinetic energy is due to the existence of these currents. Let 
the motion of the electricity, and of anything whose motion is 
governed by that of the electricity, be determined by another set 
of coordinates y^ y 2 , &c., then T will be a homogeneous function 
of squares and products of all the velocities of the two sets of 
coordinates. We may therefore divide T into three portions, in the 
first of which, T m , the velocities of the coordinates x only occur, 
while in the second, T e , the velocities of the coordinates y only 
occur, and in the third, T me , each term contains the product of the 
velocities of two coordinates of which one is as and the other y. 



198 ELECTROKINETICS. 

We have therefore T T _L T -4- T 

* - L m-T-- L e ^ r L me) 

where T m = | (^ ^) ^ 2 -f &c. + (^ # 2 ) ^ # 2 + &c -> 



572.] In the general dynamical theory, the coefficients of every 
term may be functions of all the coordinates, both x and y. In 
the case of electric currents, however, it is easy to see that the 
coordinates of the class y do not enter into the coefficients. 

For, if all the electric currents are maintained constant, and the 
conductors at rest, the whole state of the field will remain constant. 
But in this case the coordinates y are variable, though the velocities 
y are constant. Hence the coordinates y cannot enter into the 
expression for T, or into any other expression of what actually takes 
place. 

Besides this, in virtue of the equation of continuity, if the con 
ductors are of the nature of linear circuits, only one variable is 
required to express the strength of the current in each conductor. 
Let the velocities y^y z , &c. represent the strengths of the currents 
in the several conductors. 

All this would be true, if, instead of electric currents, we had 
currents of an incompressible fluid running in flexible tubes. In 
this case the velocities of these currents would enter into the 
expression for T, but the coefficients would depend only on the 
variables x, which determine the form and position of the tubes. 

In the case of the fluid, the motion of the fluid in one tube does 
not directly affect that of any other tube, or of the fluid in it. 
Hence, in the value of T 6 , only the squares of the velocities y, and 
not their products, occur, and in T^ any velocity y is associated 
only with those velocities of the form x which belong to its own 
tube. 

In the case of electrical currents we know that this restriction 
does not hold, for the currents in different circuits act on each other. 
Hence we must admit the existence of terms involving products 
of the form yy^ and this involves the existence of something in 
motion, whose motion depends on the strength of both electric 
currents y^ and y 2 . This moving matter, whatever it is, is not 
confined to the interior of the conductors carrying the two currents, 
but probably extends throughout the whole space surrounding them. 
573.] Let us next consider the form which Lagrange s equations 
of motion assume in this case. Let X be the impressed force 



573-] ELECTROMAGNETIC FORCE. 199 

corresponding- to the coordinate a?, one of those which determine 
the form and position of the conducting- circuits. This is a force 
in the ordinary sense, a tendency towards change of position. It 
is given by the equation 

x/ _ cl_dT^_dT_ 
dt dx dx 

We may consider this force as the sum of three parts, corre 
sponding to the three parts into which we divided the kinetic 
energy of the system, and we may distinguish them by the same 
suffixes. Thus -% _ T 



The part X m is that which depends on ordinary dynamical con 
siderations, and we need not attend to it. 

Since T does not contain x, the first term of the expression 
for X e is zero, and its value is reduced to 

J dT * 

~ ~ dx 

This is the expression for the mechanical force which must be 
applied to a conductor to balance the electromagnetic force, and it 
asserts that it is measured by the rate of diminution of the purely 
electrokinetic energy due to the variation of the coordinate x. The 
electromagnetic force, X e , which brings this external mechanical 
force into play, is equal and opposite to it, and is therefore measured 
by the rate of increase of the electrokinetic energy corresponding 
to an increase of the coordinate x. The value of X e , since it depends 
on squares and products of the currents, remains the same if we 
reverse the directions of all the currents. 

The third part of X is 

d dT me dT^ 



_ 
me ~ dt dx dx 

The quantity T me contains only products of the form xy, so that 

dT 

me is a linear function of the strengths of the currents i/. The 

first term, therefore, depends on the rate of variation of the 
strengths of the currents, and indicates a mechanical force on 
the conductor, which is zero when the currents are constant, and 
which is positive or negative according as the currents are in 
creasing or decreasing in strength. 

The second term depends, not on the variation of the currents, 
but on their actual strength. As it is a linear function with 
respect to these currents, it changes sign when the currents change 



200 



ELECTROKINETICS. 



[574. 



sign. Since every term involves a velocity x, it is zero when the 
conductors are at rest. 

We may therefore investigate these terms separately. If the 
conductors are at rest, we have only the first term to deal with. 
If the currents are constant, we have only the second. 

574.] As it is of great importance to determine whether any 
part of the kinetic energy is of the form T me , consisting of products 
of ordinary velocities and strengths of electric currents, it is de 
sirable that experiments should be made on this subject with great 
care. 

The determination of the forces acting on bodies in rapid motion 
is difficult. Let us therefore attend to the first term, which depends 
on the variation of the strength of the current. 

If any part of the kinetic energy depends on the product of 
an ordinary velocity and the strength of a 
current, it will probably be most easily ob 
served when the velocity and the current are 
in the same or in opposite directions. We 
therefore take a circular coil of a great many 
windings, and suspend it by a fine vertical wire, 
so that its windings are horizontal, and the 
coil is capable of rotating about a vertical axis, 
either in the same direction as the current in 
the coil, or in the opposite direction. 

We shall suppose the current to be conveyed 
into the coil by means of the suspending wire, 
and, after passing round the windings, to com 
plete its circuit by passing downwards through 
a wire in the same line with the suspending 
wire and dipping into a cup of mercury. 

Since the action of the horizontal component 
pj 33 of terrestrial magnetism would tend to turn 

this coil round a horizontal axis when the 
current flows through it, we shall suppose that the horizontal com 
ponent of terrestrial magnetism is exactly neutralized by means 
of fixed magnets, or that the experiment is made at the magnetic 
pole. A vertical mirror is attached to the coil to detect any motion 
in azimuth. 

Now let a current be made to pass through the coil in the 
direction N.E.S.W. If electricity were a fluid like water, flowing 
along the wire, then, at the moment of starting the current, and as 




574-1 HAS AN " ELECTRIC CURRENT TRUE MOMENTUM. 7 ? 201 

long as its velocity is increasing, a force would require to be 
supplied to produce the angular momentum of the fluid in passing 
round the coil, and as this must be supplied by the elasticity of 
the suspending wire, the coil would at first rotate in the opposite 
direction or W.S.E.N., and this would be detected by means of 
the mirror. On stopping the current there would be another 
movement of the mirror, this time in the same direction as that 
of the current. 

No phenomenon of this kind has yet been observed. Such an 
action, if it existed, might be easily distinguished from the already 
known actions of the current by the following peculiarities. 

(1) It would occur only when the strength of the current varies, 
as when contact is made or broken, and not when the current is 
constant. 

All the known mechanical actions of the current depend on the 
strength of the currents, and not on the rate of variation. The 
electromotive action in the case of induced currents cannot be 
confounded with this electromagnetic action. 

(2) The direction of this action would be reversed when that 
of all the currents in the field is reversed. 

All the known mechanical actions of the current remain the same 
when all the currents are reversed, since they depend on squares 
and products of these currents. 

If any action of this kind were discovered, we should be able 
to regard one of the so-called kinds of electricity, either the positive 
or the negative kind, as a real substance, and we should be able 
to describe the electric current as a true motion of this substance 
in a particular direction. In fact, if electrical motions were in any 
way comparable with the motions of ordinary matter, terms of the 
form T me would exist, and their existence would be manifested by 
the mechanical force X m , . 

According to Fechner s hypothesis, that an electric current con 
sists of two equal currents of positive and negative electricity, 
flowing in opposite directions through the same conductor, the 
terms of the second class T me would vanish, each term belonging 
to the positive current being accompanied by an equal term of 
opposite sign belonging to the negative current, and the phe 
nomena depending on these terms would have no existence. 

It appears to me, however, that while we derive great advantage 
from the recognition of the many analogies between the electric 
current and a current of a material fluid, we must carefully avoid 



202 



ELECTROKINETICS. 



[575- 



making any assumption not warranted by experimental evidence, 
and that there is, as yet, no experimental evidence to shew whether 
the electric current is really a current of a material substance, or 
a double current, or whether its velocity is great or small as mea 
sured in feet per second. 

A knowledge of these things would amount to at least the begin 
nings of a complete dynamical theory of electricity, in which we 
should regard electrical action, not, as in this treatise, as a phe 
nomenon due to an unknown cause, subject only to the general 
laws of dynamics, but as the result of known motions of known 
portions of matter, in which not only the total effects and final 
results, but the whole intermediate mechanism and details of the 
motion, are taken as the objects of study. 

575.] The experimental investigation of the second term of X me , 

dT 

namely -- r , is more difficult, as it involves the observation of 
ax 

the effect of forces on a body in rapid motion. 




Fig. 34. 

The apparatus shewn in Fig. 34, which I had constructed in 
1861, is intended to test the existence of a force of this kind. 



575-] EXPERIMENT OF ROTATION. 203 

The electromagnet A is capable of rotating about the horizontal 
axis BB , within a ring which itself revolves about a vertical 
axis. 

Let A, J5, C be the moments of inertia of the electromagnet 
about the axis of the coil, the horizontal axis BB , and a third axis 
CC respectively. 

Let 6 be the angle which CG makes with the vertical, </> the 
azimuth of the axis BB , and \f/ a variable on which the motion of 
electricity in the coil depends. 

Then the kinetic energy of the electromagnet may be written 

2 T = A & sin 2 + B 6 2 + <7<j> 2 cos 2 + E (< sin 6 + ^) 2 , 

where E is a quantity which may be called the moment of inertia 
of the electricity in the coil. 

If is the moment of the impressed force tending to increase 0, 
we have, by the equations of dynamics, 

d 2 Q . . . 

= B -r^ {(A C)0 2 sm0cos0 + ^(cos0((/>sm<9 + \//)}. 
(It 

By making % the impressed force tending to increase \j/ t equal 
to zero, we obtain 

< sin -f x//- = y, 

a constant, which we may consider as representing the strength of 
the current in the coil. 

If C is somewhat greater than A, will be zero, and the equi 
librium about the axis BB will be stable when 

Ey 
sin = - r 



This value of depends on that of y, the electric current, and 
is positive or negative according to the direction of the current. 

The current is passed through the coil by its bearings at B 
and B , which are connected with the battery by means of springs 
rubbing on metal rings placed on the vertical axis. 

To determine the value of 0, a disk of paper is placed at C, 
divided by a diameter parallel to BB into two parts, one of which 
is painted red and the other green. 

When the instrument is in motion a red circle is seen at C 
when is positive, the radius of which indicates roughly the value 
of 0. When is negative, a green circle is seen at C. 

By means of nuts working on screws attached to the electro 
magnet, the axis CC is adjusted to be a principal axis having 
its moment of inertia just exceeding that round the axis A, so as 



204 ELECTROKINETICS. [5?6. 

to make the instrument very sensible to the action of the force 
if it exists. 

The chief difficulty in the experiments arose from the disturbing 
action of the earth s magnetic force, which caused the electro 
magnet to act like a dip-needle. The results obtained were on this 
account very rough, but no evidence of any change in 6 could be 
obtained even when an iron core was inserted in the coil, so as 
to make it a powerful electromagnet. 

If, therefore, a magnet contains matter in rapid rotation, the 
ang ular momentum of this rotation must be very small compared 
with any quantities which we can measure, and we have as yet no 
evidence of the existence of the terms T me derived from their me 
chanical action. 

576.] Let us next consider the forces acting on the currents 
of electricity, that is, the electromotive forces. 

Let Y be the effective electromotive force due to induction, the 
electromotive force which must act on the circuit from without 
to balance it is Y = Y t and, by Lagrange s equation, 

Y= -r= . 

dt dy dy 

Since there are no terms in T involving the coordinate ^, the 
second term is zero, and Y is reduced to its first term. Hence, 
electromotive force cannot exist in a system at rest, and with con 
stant currents. 

Again, if we divide Y into three parts, Y m , Y e , and Y me , cor 
responding to the three parts of T, we find that, since T m does not 
contain^, Y m = 0. 

W -C A V d dT e 

We also find F, = - , : -=-* 

dt dy 

dT 

Here -^- ? is a linear function of the currents, and this part of 
dy 

the electromotive force is equal to the rate of change of this 
function. This is the electromotive force of induction discovered 
by Faraday. We shall consider it more at length afterwards. 
577.] From the part of T, depending on velocities multiplied by 

currents, we find Y mc = ^- 

dt du 

dT 

Now -j^ is a linear function of the velocities of the conductors. 
dy 

If, therefore, any terms of T me have an actual existence, it would 
be possible to produce an electromotive force independently of all 
existing currents by simply altering the velocities of the conductors. 



577-] ELECTROMOTIVE FORCE. 205 

For instance, in the case of the suspended coil at Art. 559, if, when 
the coil is at rest, we suddenly set it in rotation about the vertical 
axis, an electromotive force would be called into action proportional 
to the acceleration of this motion. It would vanish when the 
motion became uniform, and be reversed when the motion was 
retarded. 

Now few scientific observations can be made with greater pre 
cision than that which determines the existence or non-existence of 
a current by means of a galvanometer. The delicacy of this method 
far exceeds that of most of the arrangements for measuring the 
mechanical force acting on a body. If, therefore, any currents could 
be produced in this way they would be detected, even if they were 
very feeble. They would be distinguished from ordinary currents 
of induction by the following characteristics. 

(1) They would depend entirely on the motions of the conductors, 
and in no degree on the strength of currents or magnetic forces 
already in the field. 

(2) They would depend not on the absolute velocities of the con 
ductors, but on their accelerations, and on squares and products of 
velocities, and they would change sign when the acceleration be 
comes a retardation, though the absolute velocity is the same. 

Now in all the cases actually observed, the induced currents 
depend altogether on the strength and the variation of currents in 
the field, and cannot be excited in a field devoid of magnetic force 
and of currents. In so far as they depend on the motion of con 
ductors, they depend on the absolute velocity, and not on the change 
of velocity of these motions. 

We have thus three methods of detecting the existence of the 
terms of the form T tne , none of which have hitherto led to any 
positive result. I have pointed them out with the greater care 
because it appears to me important that we should attain the 
greatest amount of certitude within our reach on a point bearing 
so strongly on the true theory of electricity. 

Since, however, no evidence has yet been obtained of such terms, 
I shall now proceed on the assumption that they do not exist, 
or at least that they produce no sensible effect, an assumption which 
will considerably simplify our dynamical theory. We shall have 
occasion, however, in discussing the relation of magnetism to light, 
to shew that the motion which constitutes light may enter as a 
factor into terms involving the motion which constitutes mag 
netism. 



CHAPTER VII. 



THEORY OF ELECTRIC CIRCUITS. 



578.] WE may now confine our attention to that part of the 
kinetic energy of the system which depends on squares and products 
of the strengths of the electric currents. We may call this the 
Electrokinetic Energy of the system. The part depending on the 
motion of the conductors belongs to ordinary dynamics, and we 
have shewn that the part depending on products of velocities and 
currents does not exist. 

Let A l , AD &c. denote the different conducting circuits. Let 
their form and relative position be expressed in terms of the variables 
a?!, # 2 , &c., the number of which is equal to the number of degrees 
of freedom of the mechanical system. We shall call these the 
Geometrical Variables. 

Let j/ x denote the quantity of electricity which has crossed a given 
section of the conductor A 1 since the beginning of the time t. The 
strength of the current will be denoted by y^, the fluxion of this 
quantity. 

We shall call y^ the actual current, and y^ the integral current. 
There is one variable of this kind for each circuit in the system. 

Let T denote the electrokinetic energy of the system. It is 
a homogeneous function of the second degree with respect to the 
strengths of the currents, and is of the form 

T=L l y l * + L 2 ^+&c. + M l2 y l y 2 + &c. ) (1) 

where the coefficients L, M, &c. are functions of the geometrical 
variables # 15 # 2 , &c. The electrical variables y l} y 2 do not enter 
into the expression. 

We may call L lt I/ 2 , &c. the electric moments of inertia of the 
circuits A lt A 2 , &c., and M 12 the electric product of inertia of the 
two circuits A^ and A 2 , When we wish to avoid the language of 



579-] ELECTROKINETIC MOMENTUM. 207 

the dynamical theory, we shall call L^ the coefficient of self-induction 
of the circuit A lt and M 12 the coefficient of mutual induction of the 
circuits A 1 and A 2 . M lZ is also called the potential of the circuit 
A^ with respect to A z . These quantities depend only on the form 
and relative position of the circuits. We shall find that in the 
electromagnetic system of measurement they are quantities of the 
dimension of a line. See Art. 627. 

By differentiating T with respect to y we obtain the quantity _p 1 , 
which, in the dynamical theory, may be called the momentum 
corresponding to y. In the electric theory we shall call p the 
electrokinetic momentum of the circuit A 1 . Its value is 

Pl = A ^1 + ^12^2 + &C " 

The electrokinetic momentum of the circuit A 1 is therefore made 
up of the product of its own current into its coefficient of self- 
induction, together with the sum of the products of the currents 
in the other circuits, each into the coefficient of mutual induction 
of A 1 and that other circuit. 

Electromotive Force. 

579.] Let E be the impressed electromotive force in the circuit A, 
arising from some cause, such as a voltaic or thermoelectric battery, 
which would produce a current independently of magneto-electric 
induction. 

Let R be the resistance of the circuit, then, by Ohm s law, an 
electromotive force Ey is required to overcome the resistance, 
leaving an electromotive force E Ry available for changing the 
momentum of the circuit. Calling this force Y 9 we have, by the 
general equations, dp dT 

JL = -j- -- ^ > 

at ay 

but since T does not involve y, the last term disappears. 
Hence, the equation of electromotive force is 



or - =,+ 

The impressed electromotive force E is therefore the sum of two 
parts. The first, JRy, is required to maintain the current y against 
the resistance R. The second part is required to increase the elec 
tromagnetic momentum p. This is the electromotive force which 
must be supplied from sources independent of magneto-electric 



208 LINEAR CIRCUITS. [580. 

induction. The electromotive force arising from magneto -electric 
induction alone is evidently -j-, or, the rate of decrease of the 

(A v 

electrokinetic momentum of the circuit. 

Electromagnetic Force. 

580.] Let X be the impressed mechanical force arising from 
external causes, and tending to increase the variable x. By the 
general equations ^ d dT dT 

dt dx dx 

Since the expression for the electrokinetic energy does not contain 
the velocity (#), the first term of the second member disappears, 

and we find ^y 

Ji. = -- 7 

dx 

Here X is the external force required to balance the forces arising 
from electrical causes. It is usual to consider this force as the 
reaction against the electromagnetic force, which we shall call X, 
and which is equal and opposite to X . 

v AT 

Hence X = - T - > 

dx 

or, the electromagnetic force tending to increase any variable is equal 
to the rate of increase of the electrokinetic energy per unit increase of 
that variable, the currents being maintained constant. 

If the currents are maintained constant by a battery during a 
displacement in which a quantity, W, of work is done by electro 
motive force, the electrokinetic energy of the system will be at the 
same time increased by W. Hence the battery will be drawn upon 
for a double quantity of energy, or 2 W, in addition to that which is 
spent in generating heat in the circuit. This was first pointed out 
by Sir W. Thomson*. Compare this result with the electrostatic 
property in Art. 93. 

Case of Two Circuits. 

581.] Let AI be called the Primary Circuit, and A 2 the Secondary 
Circuit. The electrokinetic energy of the system may be written 



where L and N are the coefficients of self-induction of the primary 

* Nichol s Cyclopaedia of Physical Science, ed. 1860, Article, Magnetism, Dy 
namical Relations of. 



582.] TWO CIRCUITS. 209 

and secondary circuits respectively, and M is the coefficient of their 
mutual induction. 

Let us suppose that no electromotive force acts on the secondary 
circuit except that due to the induction of the primary current. 

We have then c i 

E 2 = B 2 fa+ (My, + Ny 2 ] = 0. 



Integrating this equation with respect to t, we have 

Ry 2 + Hjfi + Ny 2 = C, a constant, 
where y.^ is the integral current in the secondary circuit. 

The method of measuring an integral current of short duration 
will be described in Art. 748, and it is easy in most cases to ensure 
that the duration of the secondary current shall be very short. 

Let the values of the variable quantities in the equation at the 
end of the time t be accented, then, if y^ is the integral current, 
or the whole quantity of electricity which flows through a section 
of the secondary circuit during the time t, 



If the secondary current arises entirely from induction, its initial 
value jr. 2 must be zero if the primary current is constant, and the 
conductors at rest before the beginning of the time t. 

If the time t is sufficient to allow the secondary current to die 
away, y y its final value, is also zero, so that the equation becomes 



The integral current of the secondary circuit depends in this case 
on the initial and final values 



Induced Currents. 

582.] Let us begin by supposing the primary circuit broken, 
or y^ = 0, and let a current y{ be established in it when contact 
is made. 

The equation which determines the secondary integral current is 



When the circuits are placed side by side, and in the same direc 
tion, M is a positive quantity. Hence, when contact is made in 
the primary circuit, a negative current is induced in the secondary 
circuit. 

When the contact is broken in the primary circuit, the primary 
current ceases, and the induced current is y^ where 



The secondary current is in this case positive. 

VOL. II. P 



210 LINEAR CIRCUITS. 

If the primary current is maintained constant, and the form or 
relative position of the circuits altered so that M becomes M , the 
integral secondary current is y 2 , where 



In the case of two circuits placed side by side and in the same 
direction M diminishes as the distance between the circuits in 
creases. Hence, the induced current is positive when this distance 
is increased and negative when it is diminished. 

These are the elementary cases of induced currents described in 
Art. 530. 

Mechanical Action between the Two Circuits. 

583.] Let x be any one of the geometrical variables on which 
the form and relative position of the circuits depend, the electro 
magnetic force tending to increase x is 

dL . dM . dN 



If the motion of the system corresponding to the variation of x 
is such that each circuit moves as a rigid body, L and N will be 
independent of %, and the equation will be reduced to the form 



dx 

Hence, if the primary and secondary currents are of the same 
sign, the force X, which acts between the circuits, will tend to 
move them so as to increase M. 

If the circuits are placed side by side, and the currents flow in 
the same direction, M will be increased by their being brought 
nearer together. Hence the force X is in this case an attraction. 

584.] The whole of the phenomena of the mutual action of two 
circuits, whother the induction of currents or the mechanical force 
between them, depend on the quantity Jf, which we have called the 
coefficient of mutual induction. The method of calculating this 
quantity from the geometrical relations of the circuits is given in 
Art. 524, but in the investigations of the next chapter we shall not 
assume a knowledge of the mathematical form of this quantity. 
We shall consider it as deduced from experiments on induction, 
as, for instance, by observing the integral current when the 
secondary circuit is suddenly moved from a given position to an 
infinite distance, or to any position in which we know that M= 0. 



CHAPTER VIII. 

EXPLORATION OF THE FIELD BY MEANS OF THE SECONDARY 

CIRCUIT. 



585.] We have proved in Arts. 582, 583, 584 that the electro 
magnetic action between the primary and the secondary circuit 
depends on the quantity denoted by M, which is a function of the 
form and relative position of the two circuits. 

Although this quantity M is in fact the same as the potential 
of the two circuits, the mathematical form and properties of which 
we deduced in Arts. 423, 492, 521, 539 from magnetic and electro 
magnetic phenomena, we shall here make no reference to these 
results, but begin again from a new foundation, without any 
assumptions except those of the dynamical theory as stated in 
Chapter VII. 

The electrokinetic momentum of the secondary circuit consists 
of two parts (Art. 578), one, Mi lt depending on the primary current 
i lt while the other, Ni z , depends on the secondary current i 2 . We 
are now to investigate the first of these parts, which we shall 
denote by j?, where n _ 



We shall also suppose the primary circuit fixed, and the primary 
current constant. The quantity jt?, the electrokinetic momentum of 
the secondary circuit, will in this case depend only on the form 
and position of the secondary circuit, so that if any closed curve 
be taken for the secondary circuit, and if the direction along this 
curve, which is to be reckoned positive, be chosen, the value of p 
for this closed curve is determinate. If the opposite direction along 
the curve had been chosen as the positive direction, the sign of 
the quantity jo would have been reversed. 

586.] Since the quantity p depends on the form and position 
of the circuit, we may suppose that each portion of the circuit 






212 ELECTROMAGNETIC FIELD. 

contributes something 1 to the value of p, and that the part con 
tributed by each portion of the circuit depends on the form and 
position of that portion only, and not on the position of other parts 
of the circuit. 

This assumption is legitimate, because we are not now considering 
a current, the parts of which may, and indeed do, act on one an 
other, but a mere circuit, that is, a closed curve along which a 
current may flow, and this is a purely geometrical figure, the parts 
of which cannot be conceived to have any physical action on each 
other. 

We may therefore assume that the part contributed by the 
element ds of the circuit is Jds, where J is a quantity depending 
on the position and direction of the element ds. Hence, the value 
of p may be expressed as a line-integral 

(2) 



where the integration is to be extended once round the circuit. 
587.] We have next to determine the form of the quantity 7~. 

In the first place, if ds is reversed in direction, / is reversed in 
sign. Hence, if two circuits ABCE and AECD 
have the arc AEG common, but reckoned in 
opposite directions in the two circuits, the sum 
of the values of p for the two circuits 



Fl *g- 35 - and AECD will be equal to the value of p for 

the circuit AJBCD, which is made up of the two circuits. 

For the parts of the line-integral depending on the arc AEG are 
equal but of opposite sign in the two partial circuits, so that they 
destroy each other when the sum is taken, leaving only those parts of 
the line- integral which depend on the external boundary of ABCD. 

In the same way we may shew that if a surface bounded by a 
closed curve be divided into any number of parts, and if the 
boundary of each of these parts be considered as a circuit, the 
positive direction round every circuit being the same as that round 
the external closed curve, then the value of p for the closed curve is 
equal to the sum of the values of p for all the circuits. See Art. 483. 

588.] Let us now consider a portion of a surface, the dimensions 
of which are so small with respect to the principal radii of curvature 
of the surface that the variation of the direction of the normal 
within this portion may be neglected. We shall also suppose that 
if any very small circuit be carried parallel to itself from one part 
of this surface to another, the value of p for the small circuit is 



589.] ADDITION OF CIRCUITS. 213 

not sensibly altered. This will evidently be the case if the dimen 
sions of the portion of surface are small enough compared with 
its distance from the primary circuit. 

If any closed curve be drawn on this portion of the surface, the 
value of p will be proportional to its area. 

For the areas of any two circuits may be divided into small 
elements all of the same dimensions, and having the same value 
of p. The areas of the two circuits are as the numbers of these 
elements which they contain, and the values of p for the two circuits 
are also in the same proportion. 

Hence, the value of p for the circuit which bounds any element 
dS of a surface is of the form IdS, 

where / is a quantity depending on the position of dS and on the 
direction of its normal. We have therefore a new expression for p, 

(3) 

where the double integral is extended over any surface bounded by 
the circuit. 

589.] Let ABCD be a circuit, of which AC is an elementary 
portion, so small that it may be considered straight. 
Let APB and CQB be small equal areas in the 
same plane, then the value of p will be the same 
for the small circuits APB and CQB, or 

p (APB) = p (CQB). 

Hence p (APBQCD) = p (ABQCD) + p (APB), 
= p (ABQCD) + 1 



= p (ABCD), Fig. 36. 

or the value of p is not altered by the substitution of the crooked 
line APQCfor the straight line AC, provided the area of the circuit 
is not sensibly altered. This, in fact, is the principle established 
by Ampere s second experiment (Art. 506), in which a crooked 
portion of a circuit is shewn to be equivalent to a straight portion 
provided no part of the crooked portion is at a sensible distance 
from the straight portion. 

If therefore we substitute for the element ds three small elements, 
dx, dy, and dz, drawn in succession, so as to form a continuous 
path from the beginning to the end of the element ds, and if 
Fdx, G dy, and II dz denote the elements of the line-integral cor 
responding to dx, dy, and dz respectively, then 

Jds = Fdse+ Gdy + Hdz. (4) 



214 ELECTROMAGNETIC FIELD. [59- 

590.] We are now able to determine the mode in which the 
quantity / dep3nds on the direction of the element ds. For, 

by (4), f=P %. + 0* +H %. (5) 

ds ds ds 

This is the expression for the resolved part, in the direction of ds, 
of a vector, the components of which, resolved in the directions of 
the axes of x, y^ and z, are F, G, and H respectively. 

If this vector be denoted by 51, and the vector from the origin 
to a point of the circuit by p, the element of the circuit will be dp, 
and the quaternion expression for / will be 



We may now write equation (2) in the form 





(7) 

The vector 51 and its constituents F, G, H depend on the position 
of ds in the field, and not on the direction in which it is drawn. 
They are therefore functions of x, y, z, the coordinates of ds, and 
not of I, m } n, its direction-cosines. 

The vector 51 represents in direction and magnitude the time- 
integral of the electromotive force which a particle placed at the 
point (x, y, z) would experience if the primary current were sud 
denly stopped. We shall therefore call it the Electrokinetic Mo 
mentum at the point (x, ?/, z}. It is identical with the quantity 
which we investigated in Art. 405 under the name of the vector- 
potential of magnetic induction. 

The electrokinetic momentum of any finite line or circuit is the 
line-integral, extended along the line or circuit, of the resolved 
part of the electrokinetic momentum at each point of the same. 

591.] Let us next determine the value of 
p for the elementary rectangle ABCD, of 
which the sides are dy and dz, the positive 
direction being from the direction of the 
axis of y to that of z. 

Let the coordinates of 0, the centre of 
gravity of the element, be a? , y Q , Z Q , and let 
-p. 37 G Q > H Q be the values of G and of H at this 

point. 
The coordinates of A, the middle point of the first side of the 




MAGNETIC INDUCTION. 215 

rectangle, are y Q and Z Q - dz. The corresponding value of G is 

(8) 



and the part of the value of p which arises from the side A is 
approximately i dG 



1 rlTT 
Similarly, for B, H dz+--^- Ay dz. 

For (7, -G,dy-\ d ^dydz. 

For D, H dz + - Ay dz. 

2 cly 

Adding these four quantities, we find the value of p for the 
rectangle m d a 



If we now assume three new quantities, #, b, c, such that 



dH dG i 
> 

(A) 



a -= -- -=-9 

d dz 



dF dH 

-j --- j- 
dz dx 

dG dF 



7 ~~ 7 

dx dy J 

and consider these as the constituents of a new vector 33, then, by 
Theorem IV, Art. 24, we may express the line-integral of 51 round 
any circuit in the form of the surface-integral of 33 over a surface 
bounded by the circuit, thus 



p = F~-^G +H~ds=(la + mb + nc}dS, (11) 
J ^ ds ds ds JJ 

or p = JT 2t cose ds = f j T<& cos TJ d8, (12) 

where e is the angle between 5( and ds, and rj that between 33 and 
the normal to dS, whose direction-cosines are I, m, n, and T 51, T 33 
denote the numerical values of 51 and 33. 

Comparing this result with equation (3), it is evident that the 
quantity / in that equation is equal to 33 cos r;, or the resolved part 
of 33 normal to dS. 

592.] We have already seen (Arts. 490, 541) that, according to 
Faraday s theory, the phenomena of electromagnetic force and 



216 ELECTROMAGNETIC FIELD. [593- 

induction in a circuit depend on the variation of the number of 
lines of magnetic induction which pass through the circuit. Now 
the number of these lines is expressed mathematically by the 
surface-integral of the magnetic induction through any surface 
bounded by the circuit. Hence, we must regard the vector 23 
and its components a, b, c as representing what we are already 
acquainted with as the magnetic induction and its components. 

In the present investigation we propose to deduce the properties 
of this vector from the dynamical principles stated in the last 
chapter, with as few appeals to experiment as possible. 

In identifying this vector, which has appeared as the result of 
a mathematical investigation, with the magnetic induction, the 
properties of which we learned from experiments on magnets, we 
do not depart from this method, for we introduce no new fact into 
the theory, we only give a name to a mathematical quantity, and 
the propriety of so doing is to be judged by the agreement of the 
relations of the mathematical quantity with those of the physical 
quantity indicated by the name. 

The vector 33, since it occurs in a surface-integral, belongs 
evidently to the category of fluxes described in Art. 13. The 
vector 51, on the other hand, belongs to the category of forces, 
since it appears in a line-integral. 

593.] We must here recall to mind the conventions about positive 
and negative quantities and directions, some of which were stated 
in Art. 23. We adopt the right-handed system of axes, so that if 
a right-handed screw is placed in the direction of the axis of x, 
and a nut on this screw is turned in the positive direction of 
rotation, that is, from the direction of y to that of z, it will move 
along the screw in the positive direction of x. 

We also consider vitreous electricity and austral magnetism as 
positive. The positive direction of an electric current, or of a line 
of electric induction, is the direction in which positive electricity 
moves or tends to move, and the positive direction of a line of 
magnetic induction is the direction in which a compass needle 
points with the end which turns to the north. See Fig. 24, Art. 
498, and Fig. 25, Art. 501. 

The student is recommended to select whatever method appears 
to him most effectual in order to fix these conventions securely in 
his memory, for it is far more difficult to remember a rule which 
determines in which of two previously indifferent ways a statement 
is to be made, than a rule which selects one way out of many. 



594-] THEORY OF A SLIDING PIECE. 217 

594.] We have next to deduce from dynamical principles the 
expressions for the electromagnetic force acting on a conductor 
carrying an electric current through the magnetic field, and for 
the electromotive force acting on the electricity within a body 
moving in the magnetic field. The mathematical method which 
we shall adopt may be compared with the experimental method 
used by Faraday * in exploring the field by means of a wire, and 
with what we have already done at Art. 490, by a method founded 
on experiments. What we have now to do is to determine the 
effect on the value of ji, the electroldnetic momentum of the 
secondary circuit, due to given alterations of the form of that 
circuit. 

Let AA , BB be two parallel straight conductors connected by 
the conducting arc (7, which may be of any form, and by a straight 




Fig. 38. 

conductor AB, which is capable of sliding parallel to itself along 
the conducting rails AA and BB . 

Let the circuit thus formed be considered as the secondary cir 
cuit, and let the direction ABC be assumed as the positive direction 
round it. 

Let the sliding piece move parallel to itself from the position AB 
to the position AB . We have to determine the variation of _p, the 
electrokinetic momentum of the circuit, due to this displacement 
of the sliding piece. 

The secondary circuit is changed from ABC to A IfC, hence, by 
Art. 587, p (AB C)-p (ABC) = p (AA B B). (13) 

We have therefore to determine the value of p for the parallel 
ogram AA BB. If this parallelogram is so small that we may 
neglect the variations of the direction and magnitude of the mag 
netic induction at different points of its plane, the value of p is, 
by Art. 591, 33 cos r\ . AA ffBj where 33 is the magnetic induction, 
* Exp. Res., 3082, 3087, 3113. 



218 ELECTROMAGNETIC FIELD. [595- 

and 77 the angle which it makes with the positive direction of the 
normal to the parallelogram AA B B. 

We may represent the result geometrically by the volume of the 
parallelepiped, whose base is the parallelogram AA B B, and one of 
whose edges is the line AM, which represents in direction and 
magnitude the magnetic induction 33. If the parallelogram is in 
the plane of the paper, and if AM is drawn upwards from the paper, 
the volume of the parallelepiped is to be taken positively, or more 
generally, if the directions of the circuit AB, of the magnetic in 
duction AM, and of the displacement AA , form a right-handed 
system when taken in this cyclical order. 

The volume of this parallelepiped represents the increment of 
the value of p for the secondary circuit due to the displacement 
of the sliding piece from AB to A B . 

Electromotive Force acting on the Sliding Piece. 

595.] The electromotive force produced in the secondary circuit 
by the motion of the sliding piece is, by Art. 579, 



If we suppose AA to be the displacement in unit of time, then 
AA will represent the velocity, and the parallelepiped will represent 

~, and therefore, by equation (14), the electromotive force in the 

Ctu 

negative direction B A. 

Hence, the electromotive force acting on the sliding piece AB, 
in consequence of its motion through the magnetic field, is repre 
sented by the volume of the parallelepiped, whose edges represent 
in direction and magnitude the velocity, the magnetic induction, 
and the sliding piece itself, and is positive when these three direc 
tions are in right-handed cyclical order. 

Electromagnetic Force acting on the Sliding Piece. 

596.] Let i 2 denote the current in the secondary circuit in the 
positive direction ABC, then the work done by the electromagnetic 
force on AB while it slides from the position AB to the position 
A B is (M M)i l i 2 , where M and M are the values of M 12 in 
the initial and final positions of AB. But (M M)^ is equal 
to// p, and this is represented by the volume of the parallelepiped 
on AB, AM, and AA . Hence, it we draw a line parallel to AB 



598.] LINES OF MAGNETIC INDUCTION. 219 

to represent the quantity AB.i 2 , the parallelepiped contained by 
this line, by AM, the magnetic induction, and by A A, the displace 
ment, will represent the work done during- this displacement. 

For a given distance of displacement this will be greatest when 
the displacement is perpendicular to the parallelogram whose sides 
are AB and AM. The electromagnetic force is therefore represented 
by the area of the parallelogram on AB and AM multiplied by ?/ 2 , 
and is in the direction of the normal to this parallelogram, drawn so 
that AB, AM, and the normal are in right-handed cyclical order. 

Four Definitions of a Line of Magnetic Induction. 

597.] If the direction AA , in which the motion of the sliding 
piece takes place, coincides with AM, the direction of the magnetic 
induction, the motion of the sliding piece will not call electromotive 
force into action, whatever be the direction of AB, and if AB carries 
an electric current there will be no tendency to slide along AA. 

Again,, if AB } the sliding piece, coincides in direction with AM, 
the direction of magnetic induction, there will be no electromotive 
force called into action by any motion of AB, and a current through 
AB will not cause AB to be acted on by mechanical force. 

We may therefore define a line of magnetic induction in four 
different ways. It is a line such that 

(1) If a conductor be moved along it parallel to itself it will 
experience no electromotive force. 

(2) If a conductor carrying a current be free to move along a 
line of magnetic induction it will experience no tendency to do so. 

(3) If a linear conductor coincide in direction with a line of 
magnetic induction, and be moved parallel to itself in any direction, 
it will experience no electromotive force in the direction of its 
length. 

(4) If a linear conductor carrying an electric current coincide 
in direction with a line of magnetic induction it will not experience 
any mechanical force. 

General Equations of Electromotive Force. 

598.] We have seen that E, the electromotive force due to in 
duction acting on the secondary circuit, is equal to j- , where 



220 ELECTROMAGNETIC FIELD. 

To determine the value of E, let us differentiate the quantity 
under the integral sign with respect to ^, remembering that if the 
secondary circuit is in motion, as, y, and z are functions of the time. 
We obtain 

f ( dF tfa dG_dy dH dz. 
J^dt ds + dt r& + <fc fr 



C,dF dx dG dy dH dz^ dx 

J ^ dx ds dx ds dx ds dt 

dF dx dG dy dHdz^ dy 

dy ds dy ds dy ds dt 

dF dx_ dG_dy dH dz. dz 

ds dz ds dz ds dt 



-/< 



ds dt ds dt 



,2, 



Now consider the second term of the integral, and substitute 

from equations (A), Art. 591, the values of and -7- . This term 

dx dx 

then becomes, 

[( ^ 7)^ z dF dx dF dy dF dz^dx 
J\ C di" ds " f das ds + ^7 Ts + Hz ds di 

which we may write 

f f ( ty 7 dz dF^ dx _ 

/ ( C / ^ 7- + T-J -T7 ^- 

J ^ ds ds ds dt 

Treating the third and fourth terms in the same way, and col- 

i ,. .-, . dx dy - dz 

lectmg the terms m - , ^ , and , remembering that 



dx 



^ 
= F~ 7 - , (3) 



dt ~ dsdt>" L dt 

and therefore that the integral, when taken round the closed 
curve, vanishes, 



f ( dz dx dG. dy 

/ ( a ^7 ~ c ^7 7-^ 7 

J ^ dt dt dt ) ds 

dx d dH dz 



598.] ELECTROMOTIVE FORCE. 221 

We may write this expression in the form 



Equations of 
Electromotive (-B) 

Force. 



dy -.dz dF d^ 
where P = c -~ o-j = =- 

dz dx dG d^J 

dt dt dt dy 

_ dx dy dH d^ 

The terms involving the new quantity ^ are introduced for the 
sake of giving generality to the expressions for P, Q, R. They 
disappear from the integral when extended round the closed circuit. 
The quantity ^ is therefore indeterminate as far as regards the 
problem now before us, in which the total electromotive force round 
the circuit is to be determined. We shall find, however, that when 
we know all the circumstances of the problem, we can assign a 
definite value to ^, and that it represents, according to a certain 
definition, the electric potential at the point x, y, z. 

The quantity under the integral sign in equation (5) represents 
the electromotive force acting on the element ds of the circuit. 

If we denote by T @, the numerical value of the resultant of P, 
Q, and R, and by e, the angle between the direction of this re 
sultant and that of the element ds, we may write equation (5), 



JT<$ cost els. (6) 



fi =JT<$ cost els. 

The vector @ is the electromotive force at the moving element 
ds. Its direction and magnitude depend on the position and 
motion of ds, and on the variation of the magnetic field, but not 
on the direction of ds. Hence we may now disregard the circum 
stance that ds forms part of a circuit, and consider it simply as a 
portion of a moving body, acted on by the electromotive force Q. 
The electromotive force at a point has already been defined in 
Art. 68. It is also called the resultant electrical force, being the 
force which would be experienced by a unit of positive electricity 
placed at that point. We have now obtained the most general 
value of this quantity in the case of a body moving in a magnetic 
field due to a variable electric system. 

If the body is a conductor, the electromotive force will produce a 
current ; if it is a dielectric, the electromotive force will produce 
only electric displacement. 



222 ELECTROMAGNETIC FIELD. [599- 

The electromotive force at a point, or on a particle, must be 
carefully distinguished from the electromotive force along an arc 
of a curve, the latter quantity being the line-integral of the former. 
See Art, 69. 

599.] The electromotive force, the components of which are 
defined by equations (B), depends on three circumstances. The first 
of these is the motion of the particle through the magnetic field. 
The part of the force depending on this motion is expressed by the 
first two terms on the right of each equation. It depends on the 
velocity of the particle transverse to the lines of magnetic induction. 
If is a vector representing the velocity, and 33 another repre 
senting the magnetic induction, then if (^ is the part of the elec 
tromotive force depending on the motion, 

^ = V. 33, (7) 

or, the electromotive force is the vector part of the product of the 
magnetic induction multiplied by the velocity, that is to say, the 
magnitude of the electromotive force is represented by the area 
of the parallelogram, whose sides represent the velocity and the 
magnetic induction, and its direction is the normal to this parallel 
ogram, drawn so that the velocity, the magnetic induction, and the 
electromotive force are in right-handed cyclical order. 

The third term in each of the equations (B) depends on the time- 
variation of the magnetic field. This may be due either to the 
time-variation of the electric current in the primary circuit, or to 
motion of the primary circuit. Let ( 2 be the part of the electro 
motive force which depends on these terms. Its components are 
dF dG dH 

-w ~w and -w 

and these are the components of the vector, or 21. Hence, 

dt 

6, = -& (8) 

The last term of each equation (B) is due to the variation of the 
function ^ in different parts of the field. We may write the third 
part of the electromotive force, which is due to this cause, 

@ 3 = - V*. (9) 

The electromotive force, as defined by equations (B), may therefore 
be written in the quaternion form, 

@= r. 33-21- V*. (10) 



600.] MOVING AXES. 223 

On the Modification of the Equations of Electromotive Force when the 
Axes to which they are referred are moving in Space. 

600.] Let # , y , / be the coordinates of a point referred to a 
system of rectangular axes moving- in space, and let #, ?/, z be the 
coordinates of the same point referred to fixed axes. 

Let the components of the velocity of the origin of the moving 
system be u, v, w, and those of its angular velocity w^ o> 2 , co 3 
referred to the fixed system of axes, and let us choose the fixed 
axes so as to coincide at the given instant with the moving ones, 
then the only quantities which will be different for the two systems 
of axes will be those differentiated with respect to the time. If 

bx 

denotes a component velocity of a point moving in rigid con- 

o t 

nexion with the moving axes, and - - and -j- that of any moving 

ci/t civ 

point, having the same instantaneous position, referred to the fixed 
and the moving axes respectively, then 

dx __ x duo , ^ 

~di = bi + ~di 
with similar equations for the other components. 

By the theory of the motion of a body of invariable form, 

bx 

= + w a * 

}> (2) 



Since F is a component of a directed quantity parallel to x, 
if r be the value of -=- referred to the moving axes, 

dl" (ZFbv dFby clFbz dF 



Substituting for -=- and -y- their values as deduced from the 
dy dz 

equations (A) of magnetic induction, and remembering that, by (2), 
d bx d ly d bz 

= = a>3 = ~^ 



_b_x d^b^ d_by dffbz d bz 
dt ~ dx U dx bt + dx U fy bt + ~dx~ ~U + dx *i 

b , bz dF 



224 ELECTKOMAGNETIC FIELD. [6OI. 

Ifweaowput 

dF dV * z dF 



_^ =j H 

Of 01 Of 



-. 

The equation for P, the component of the electromotive force 
parallel to a?, is, by (B), 



referred to the fixed axes. Substituting the values of the quanti 
ties as referred to the moving axes, we have 

dy> dz> dF d(* + V) (9) 

C dt~^Tt"~dt~ dx 
for the value of P referred to the moving axes. 

601.] It appears from this that the electromotive force is ex 
pressed by a formula of the same type, whether the motions of the 
conductors be referred to fixed axes or to axes moving in space, the 
only difference between the formulae being that in the case of 
moving axes the electric potential # must be changed into v I / + 4 // . 

In all cases in which a current is produced in a conducting cir 
cuit, the electromotive force is the line-integral 



taken round the curve. The value of * disappears from this 
integral, so that the introduction of SP has no influence on its 
value. In all phenomena, therefore, relating to closed circuits and 
the currents in them, it is indifferent whether the axes to which we 
refer the system be at rest or in motion. See Art. 668. 

On the Electromagnetic Force acting on a Conductor which carries 
an Electric Current through a Magnetic Field. 

602.] We have seen in the general investigation, Art. 583, that if 
a? x is one of the variables which determine the position and form of 
the secondary circuit, and if X L is the force acting on the secondary 
circuit tending to increase this variable, then 



. ,-v 

Since ^ is independent of x lf we may write 



602.] ELECTROMAGNETIC FORCE. 225 

(3) 



and we have for the value of X lf 



ds 

Now let us suppose that the displacement consists in moving 
every point of the circuit through a distance b% in the direction 
of #, b% being any continuous function of s, so that the different 
parts of the circuit move independently of each other, while the 
circuit remains continuous and closed. 

Also let X be the total force in the direction of x acting on 
the part of the circuit from s = to s = s, then the part corre- 

7 ~V 

spending to the element ds will be -=- ds. We shall then have the 

following expression for the work done by the force during the 
displacement, 

/dX ^ f d / ~.dx ~dy -r T dz\ 

r batk s= LI -j ( F- 7 ~ + G -f + J2V) 6# ds, (4) 

ds 2 J dbsn^ ds ds ds 

where the integration is to be extended round the closed curve, 
remembering that 80? is an arbitrary function of s. We may there 
fore perform the differentiation with respect to b x in the same 
way that we differentiated with respect to t in Art. 598, remem 

bering that dx dy dz 

-= - = 1, -y- = 0. and -= = 0. (5) 

dbx 

We thus find 



The last term vanishes when the integration is extended round 
the closed curve, and since the equation must hold for all forms 
of the function bas, we must have 

dX . / dy -, dz\ /P , N 

= | ((?__._), (7) 

ds 2V ds ds 

an equation which gives the force parallel to x on any element of 
the circuit. The forces parallel to y and z are 

dT . f dz dx\ . 

= lAa -- C-=-)* (8) 
d* 2V ds ds 

dZ . ^dx dy^ , . 

j- = 4f^-3 /! ( 9 ) 

ds 2 \ ds dx 

The resultant force on the element is given in direction and mag 
nitude by the quaternion expression i 2 Vdp$$, where i 2 is the 
numerical measure of the current, and dp and 53 are vectors 

VOL. II. Q 



226 ELECTROMAGNETIC FIELD. [603. 

representing the element of the circuit and the magnetic in 
duction, and the multiplication is to be understood in the Hamil- 
tonian sense. 

603.] If the conductor is to be treated not as a line but as a 
body, we must express the force on the element of length, and the 
current through the complete section, in terms of symbols denoting 
the force per unit of volume, and the current per unit of area. 

Let X, Y, Z now represent the components of the force referred to 
unit of volume, and u, v, w those of the current referred to unit of 
area. Then, if S represents the section of the conductor, which we 
shall suppose small, the volume of the element ds will be Sds, and 

n = -^ - - . Hence, equation (7) will become 

S(vc-w6), (10) 



(Equations of 
Electromagnetic (C) 



or X = vc wb. 

Similarly Y= wa uc, 

, r/ 7 Force. 

and Z ub va. 

Here X, J", Z are the components of the electromagnetic force on 
an element of a conductor divided by the volume of that element ; 
n, v, w are the components of the electric current through the 
element referred to unit of area, and #, b, c are the components 
of the magnetic induction at the element, which are also referred 
to unit of area. 

If the vector represents in magnitude and direction the force 
acting on unit of volume of the conductor, and if ( represents the 
electric current flowing through it, 

en) 



CHAPTER IX. 



GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. 



604.] IN our theoretical discussion of electrodynamics we began 
by assuming- that a system of circuits carrying electric currents 
is a dynamical system, in which the currents may be regarded as 
velocities, and in which the coordinates corresponding to these 
velocities do not themselves appear in the equations. It follows 
from this that the kinetic energy of the system, so far as it depends 
on the currents, is a homogeneous quadratic function of the currents, 
in which the coefficients depend only on the form and relative 
position of the circuits. Assuming these coefficients to be known, 
by experiment or otherwise, we deduced, by purely dynamical rea 
soning, the laws of the induction of currents, and of electromagnetic 
attraction. In this investigation we introduced the conceptions 
of the electrokinetic energy of a system of currents, of the electro 
magnetic momentum of a circuit, and of the mutual potential of 
two circuits. 

We then proceeded to explore the field by means of various con 
figurations of the secondary circuit, and were thus led to the 
conception of a vector 2[, having a determinate magnitude and 
direction at any given point of the field. We called this vector the 
electromagnetic momentum at that point. This quantity may be 
considered as the time-integral of the electromotive force which 
would be produced at that point by the sudden removal of all the 
currents from the field. It is identical with the quantity already 
investigated in Art. 405 as the vector-potential of magnetic in 
duction. Its components parallel to x, y, and z are F, G, and H. 
The electromagnetic momentum of a circuit is the line-integral 
of $1 round the circuit. 

We then, by means of Theorem IV, Art. 24, transformed the 

Q 2 



228 GENERAL EQUATIONS. [605. 

line-integral of 1 into the surface-integral of another vector, 53, 
whose components are a, d, c, and we found that the phenomena 
of induction due to motion of a conductor, and those of electro 
magnetic force can be expressed in terms of 53. We gave to 53 
the name of the Magnetic induction, since its properties are iden 
tical with those of the lines of magnetic induction as investigated 
by Faraday. 

We also established three sets of equations : the first set, (A), 
are those of magnetic induction, expressing it in terms of the elec 
tromagnetic momentum. The second set, (B), are those of electro 
motive force, expressing it in terms of the motion of the conductor 
across the lines of magnetic induction, and of the rate of variation 
of the electromagnetic momentum. The third set, (C), are the 
equations of electromagnetic force,, expressing it in terms of the 
current and the magnetic induction. 

The current in all these cases is to be understood as the actual 
current, which includes not only the current of conduction, but the 
current due to variation of the electric displacement. 

The magnetic induction 53 is the quantity which we have already 
considered in Art. 400. In an unmagnetized body it is identical 
with the force on a unit magnetic pole, but if the body is mag 
netized, either permanently or by induction, it is the force which 
would be exerted on a unit pole, if placed in a narrow crevasse in 
the body, the walls of which are perpendicular to the direction of 
magnetization. The components of 53 are #, #, c. 

It follows from the equations (A), by which a, b, c are defined, 
that da M (i^^ 

dx dy dz 

This was shewn at Art. 403 to be a property of the magnetic 
induction. 

605.] We have defined the magnetic force within a magnet, as 
distinguished from the magnetic induction, to be the force on a 
unit pole placed in a narrow crevasse cut parallel to the direction of 
magnetization. This quantity is denoted by ), and its components 
by a, /3, y. See Art. 398. 

If 3 is the intensity of magnetization, and A, B, C its com 
ponents, then, by Art. 400, 



a = a -f 4 TT A, 
c = y+4-n C. 



(Equations of Magnetization.) (D) 



6o6.] MAGNETIC EQUATIONS. 229 

We may call these the equations of magnetization, and they 
indicate that in the electromagnetic system the magnetic induction 
33, considered as a vector, is the sum, in the Hamiltonian sense, of 
two vectors, the magnetic force .), and the magnetization 3 multi 
plied by 47T, or 33 = + 4?r3. 

In certain substances, the magnetization depends on the magnetic 
force, and this is expressed by the system of equations of induced 
magnetism given at Arts. 426 and 435. 

606.] Up to this point- of our investigation we have deduced 
everything from purely dynamical considerations, without any 
reference to quantitative experiments in electricity or magnetism. 
The only use we have made of experimental knowledge is to re 
cognise, in the abstract quantities deduced from the theory, the 
concrete quantities discovered by experiment, and to denote them 
by names which indicate their physical relations rather than their 
mathematical generation. 

In this way we have pointed out the existence of the electro 
magnetic momentum 1 as a vector whose direction and magnitude 
vary from one part of space to another, and from this we have 
deduced, by a mathematical process, the magnetic induction, 33, as 
a derived vector. We have not, however, obtained any data for 
determining either 51 or 33 from the distribution of currents in the 
field. For this purpose we must find the mathematical connexion 
between these quantities and the currents. 

We begin by admitting the existence of permanent magnets, 
the mutual action of which satisfies the principle of the conservation 
of energy. We make no assumption with respect to the laws of 
magnetic force except that which follows from this principle, 
namely, that the force acting on a magnetic pole must be capable 
of being derived from a potential. 

We then observe the action between currents and magnets, and 
we find that a current acts on a magnet in a manner apparently the 
same as another magnet would act if its strength, form, and position 
were properly adjusted, and that the magnet acts on the current 
in the same way as another current. These observations need not 
be supposed to be accompanied with actual measurements of the 
forces. They are not therefore to be considered as furnishing 
numerical data, but are useful only in suggesting questions for 
our consideration. 

The question these observations suggest is, whether the magnetic 
field produced by electric currents, as it is similar to that produced 



230 GENERAL EQUATIONS. [607. 

by permanent magnets in many respects, resembles it also in being- 
related to a potential ? 

The evidence that an electric circuit produces, in the space sur 
rounding it, magnetic effects precisely the same as those produced 
by a magnetic shell bounded by the circuit, has been stated in 
Arts. 482-485. 

We know that in the case of the magnetic shell there is a 
potential, which has a determinate value for all points outside the 
substance of the shell, but that the values of the potential at two 
neighbouring points, on opposite sides of the shell,, differ by a finite 
quantity. 

If the magnetic field in the neighbourhood of an electric current 
resembles that in the neighbourhood of a magnetic shell, the 
magnetic potential, as found by a line-integration of the magnetic 
force, will be the same for any two lines of integration, provided 
one of these lines can be transformed into the other by continuous 
motion without cutting the electric current. 

If, however, one line of integration cannot be transformed into 
the other without cutting the current, the line-integral of the 
magnetic force along the one line will differ from that along the 
other by a quantity depending on the strength of the current. The 
magnetic potential due to an electric current is therefore a function 
having an infinite series of values with a common difference, the 
particular value depending on the course of the line of integration. 
Within the substance of the conductor, there is no such thing as 
a magnetic potential. 

607.] Assuming that the magnetic action of a current has a 
magnetic potential of this kind, we proceed to express this result 
mathematically. 

In the first place, the line-integral of the magnetic force round 
any closed curve is zero, provided the closed curve does not surround 
the electric current. 

In the next place, if the current passes once, and only once, 
through the closed curve in the positive direction, the line-integral 
has a determinate value, which may be used as a measure of the 
strength of the current. For if the closed curve alters its form 
in any continuous mariner without cutting the current, the line- 
integral will remain the same. 

In electromagnetic measure, the line-integral of the magnetic 
force round a closed curve is numerically equal to the current 
through the closed curve multiplied by 4 TT. 



607.] ELECTRIC CURRENTS. 231 

If we take for the closed curve the parallelogram whose sides 

are dy and dz, the line-integral of the magnetic force round the 

parallelogram is ^y dp 

^dy dz 

and if u, v f w are the components of the flow of electricity, the 
current through the parallelogram is 

u dy dz. 

Multiplying this by 47r, and equating the result to the line- 
integral, we obtain the equation 

dy dz 
with the similar equations 

do, dy ( (Equations of /-\ 

4 7T V = -= ~- ) Electric Currents.) W 

dz dx 

dp da 

dx dy J 

which determine the magnitude and direction of the electric currents 
when the magnetic force at every point is given. 

When there is no current, these equations are equivalent to the 
condition that adx + fi dy + y dz = Dl, 

or that the magnetic force is derivable from a magnetic potential 
in all points of the field where there are no currents. 

By differentiating the equations (E) with respect to x, y, and z 
respectively, and adding the results, we obtain the equation 
du dv dw 

. I I . Q 

dx dy dz 

which indicates that the current whose components are u, v, w is 
subject to the condition of motion of an incompressible fluid, and 
that it must necessarily flow in closed circuits. 

This equation is true only if we take #, v, and w as the com 
ponents of that electric flow which is due to the variation of electric 
displacement as well as to true conduction. 

We have very little experimental evidence relating to the direct 
electromagnetic action of currents due to the variation of electric 
displacement in dielectrics, but the extreme difficulty of reconciling 
the laws of electromagnet ism with the existence of electric currents 
which are not closed is one reason among many why we must admit 
the existence of transient currents due to the variation of displace 
ment. Their importance will be seen when we come to the electro 
magnetic theory of light. 



232 GENERAL EQUATIONS. [6o8. 

608.] We have now determined the relations of the principal 
quantities concerned in the phenomena discovered by Orsted, Am 
pere, and Faraday. To connect these with the phenomena described 
in the former parts of this treatise, some additional relations are 
necessary. 

When electromotive force acts on a material body, it produces 
in it two electrical effects, called by Faraday Induction and Con 
duction, the first being most conspicuous in dielectrics, and the 
second in conductors. 

In this treatise, static electric induction is measured by what we 
have called the electric displacement, a directed quantity or vector 
which we have denoted by ), and its components by/*, #, k. 

In isotropic substances, the displacement is in the same direction 
as the electromotive force which produces it, and is proportional 
to it, at least for small values of this force. This may be expressed 
by the equation i 

<T\ -IT- rr, (Equation of Electric /-pry 

4 IT Displacement.) 

where ^is the dielectric capacity of the substance. See Art. 69. 

In substances which are not isotropic, the components /, #, h of 
the electric displacement 2) are linear functions of the components 
P, Q, -K of the electromotive force (. 

The form of the equations of electric displacement is similar to 
that of the equations of conduction as given in Art. 298. 

These relations may be expressed by saying that K is, in isotropic 
bodies, a scalar quantity, but in other bodies it is a linear and vector 
function, operating on the vector (. 

609.] The other effect of electromotive force is conduction. The 
laws of conduction as the result of electromotive force were esta 
blished by Ohm, and are explained in the second part of this 
treatise, Art. 241. They may be summed up in the equation 

ft = C (, (Equation of Conductivity.) (G) 

where ( is the intensity of the electromotive force at the point, 
$ is the density of the current of conduction, the components of 
which are p, q, r, and C is the conductivity of the substance, which, 
in the case of isotropic substances, is a simple scalar quantity, but 
in other substances becomes a linear and vector function operating 
on the vector ($. The form of this function is given in Cartesian 
coordinates in Art. 298. 

610.] One of the chief peculiarities of this treatise is the doctrine 
which it asserts, that the true electric current (, that on which the 



614.] CURRENTS OF DISPLACEMENT. 233 

electromagnetic phenomena depend, is not the same thing as $, the 
current of conduction, but that the time- variation of 2), the electric 
displacement, must be taken into account in estimating the total 
movement of electricity, so that we must write, 

( = +2), (Equation of True Currents.) (H) 

or, in terms of the components, 



dt 
dg 

j V 

dk 



(H*) 



611.] Since both $ and 2) depend on the electromotive force ($, 
we may express the true current ( in terms of the electromotive 
force, thus 



or, in the case in which C and K are constants, 



w = CR+ - K C -j- 

47T dt 

612.] The volume-density of the free electricity at any point 
is found from the components of electric displacement by the 
equation ^f dg dk 

613.] The surface-density of electricity is 

where /, m, n are the direction-cosines of the normal drawn from 
the surface into the medium in which f, g, li are the components of 
the displacement, and / , m , n are those of the normal drawn from 
the surface into the medium in which they are f , /, //. 

614.] When the magnetization of the medium is entirely induced 
by the magnetic force acting on it, we may write the equation of 
induced magnetization, $$ = /*), (L) 

where p is the coefficient of magnetic permeability, which may 
be considered a scalar quantity, or a linear and vector function 
operating on j, according as the medium is isotropic or not. 



234 



GENEKAL EQUATIONS. 



615.] These may be regarded as the principal relations among 
the quantities we have been considering. They may be combined 
so as to eliminate some of these quantities, but our object at present 
is not to obtain compactness in the mathematical formulae, but 
to express every relation of which we have any knowledge. To 
eliminate a quantity which expresses a useful idea would be rather 
a loss than a gain in this stage of our enquiry. 

There is one result, however, which we may obtain by combining 
equations (A) and (E), and which is of very great importance. 

If we suppose that no magnets exist in the field except in the 
form of electric circuits, the distinction which we have hitherto 
maintained between the magnetic force and the magnetic induction 
vanishes, because it is only in magnetized matter that these quan 
tities differ from each other. 

According to Ampere s hypothesis, which will be explained in 
Art. 833, the properties of what we call magnetized matter are due 
to molecular electric circuits, so that it is only when we regard the 
substance in large masses that our theory of magnetization is 
applicable, and if our mathematical methods are supposed capable 
of taking account of what goes on within the individual molecules, 
they will discover nothing but electric circuits, and we shall find 
the magnetic force and the magnetic induction everywhere identical. 
In order, however, to be able to make use of the electrostatic or of 
the electromagnetic system of measurement at pleasure we shall 
retain the coefficient //, remembering that its value is unity in the 
electromagnetic system. 

616.] The components of the magnetic induction are by equa 
tions (A), Art. 591, dH dG 

n 

a/ -y- 

dy dz 

dF dH 
o --- 
dz dx 

dF 

dx dy 
The components of the electric current are by equations (E), 






Art. 607, 



dy aft 

4 77 U V- 7- > 

0* & 



da 

- 

dz 
d(B 

~ 

dx 



dy 

= 

dx 
da 

~~ 

dy 



6l6.] 



VECTOR-POTENTIAL OP CURRENTS. 



According to our hypothesis a, b, c are identical with 
respectively. We therefore obtain 



If we write 



235 
i, fift /uy 



tffo? dy dy 2 dz 2 

dF dG dH 

J = -j- + -r + ~r > 

ax dy dz 



dzdx 



we may write equation (1), 



Similarly, 



dJ 



4 TT ja v = -- + V 2 # 



If we write F =- fff U - dx dy dz, ~| 



-, j 



where r is the distance of the given point from the element xy z, 
and the integrations are to be extended over all space, then 



(7) 



The quantity x. disappears from the equations (A), and it is not 
related to any physical phenomenon. If we suppose it to be zero 
everywhere, / will also be zero everywhere, and equations (5), 
omitting the accents, will give the true values of the components 
of 51. 



* The negative sign is employed here in order to make our expressions consistent 
with those in which Quaternions are employed. 



236 GENERAL EQUATIONS. [617. 

617.] We may therefore adopt, as a definition of 2[, that it 
is the vector-potential of the electric current, standing 1 in the same 
relation to the electric current that the scalar potential stands to 
the matter of which it is the potential, and obtained by a similar 
process of integration, which may be thus described. 

From a given point let a vector be drawn, representing 1 in mag 
nitude and direction a given element of an electric current, divided 
by the numerical value of the distance of the element from the 
given point. Let this be done for every element of the electric 
current. The resultant of all the vectors thus found is the poten 
tial of the whole current. Since the current is a vector quantity, 
its potential is also a vector. See Art. 422. 

When the distribution of electric currents is given, there is one, 
and only one, distribution of the values of 31, such that 31 is every 
where finite and continuous, and satisfies the equations 
V21= 47Tf*<, fl.VSl = 0, 

and vanishes at an infinite distance from the electric system. This 
value is that given by equations (5), which may be written 



Quaternion Expressions for tJie Electromagnetic Equations. 

618.] In this treatise we have endeavoured to avoid any process 
demanding from the reader a knowledge of the Calculus of Qua 
ternions. At the same time we have not scrupled to introduce the 
idea of a vector when it was necessary to do so. When we have 
had occasion to denote a vector by a symbol, we have used a 
German letter, the number of different vectors being so great that 
Hamilton s favourite symbols would have been exhausted at once. 
Whenever therefore, a German letter is used it denotes a Hamil- 
tonian vector, and indicates not only its magnitude but its direction. 
The constituents of a vector are denoted by Roman or Greek letters. 

The principal vectors which we have to consider are : 



Constituents. 

The radius vector of a point .................. p x y z 

The electromagnetic momentum at a point 2[ F G H 

The magnetic induction ..................... 53 a I c 

The (total) electric current .................. ( u v w 

The electric displacement ..................... 2) f g h 



6 1 9.] QUATEKNION EXPRESSIONS. 237 



Constituents. 

The electromotive force ..................... ( P Q R 

The mechanical force ........................ g XYZ 

The velocity of a point ........................ or p so y z 

The magnetic force ........................... ) a /3 y 

The intensity of magnetization ............ 3 ABC 

The current of conduction .................. ft p q r 

We have also the following scalar functions : 
,The electric potential ^. 
The magnetic potential (where it exists) 12. 
The electric density e. 
The density of magnetic matter m. 

Besides these we have the following quantities, indicating physical 
properties of the medium at each point : 

(7, the conductivity for electric currents. 
K, the dielectric inductive capacity. 
fji, the magnetic inductive capacity. 

These quantities are, in isotropic media, mere scalar functions 
of p, but in general they are linear and vector operators on the 
vector functions to which they are applied. K and JJL are certainly 
always self- conjugate, and C is probably so also. 

619.] The equations (A) of magnetic induction, of which the 

first is > dH dG 

a = -= --- r- 
dy dz 

may now be written sg _ yyty 

where V is the operator 

. d . d -, d 
%-j- +7-7- + -7-1 

dx * dy dz 

and Vindicates that the vector part of the result of this operation 
is to be taken. 

Since 21 is subject to the condition $ V 2[ = 0, V[ is a pure 
vector, and the symbol V is unnecessary. 

The equations (B) of electromotive force, of which the first is 

, . dF d* 
P = cyoz -- - --- r- , 

dt dx 

become @= F33 $ V*. 

The equations (C) of mechanical force, of which the first is 

v , d^> dil 

JL = cv mv e -- m -7 j 
dx dx 

become = 7 $ 33 



238 GENERAL EQUATIONS. [619. 

The equations (D) of magnetization, of which the first is 

a a 4- 4 TT A, 
become 33 <$ 4- 4 TT 3. 

The equations (E) of electric currents, of which the first is 

dy d(3 

4 TT u -/ -- fi 
dy dz 



become 4 -n & = 

The equation of the current of conduction is, by Ohm s Law, 

= <7<g. 
That of electric displacement is 

3) = -?-K. 

4 7T 

The equation of the total current, arising from the variation of 
the electric displacement as well as from conduction, is 

< - S + 2X 
When the magnetization arises from magnetic induction, 

SB = M . 

We have also, to determine the electric volume-density, 

e = V$). 
To determine the magnetic volume-density, 

m = S V 3. 

When the magnetic force can be derived from a potential 

= - V 12. 



CHAPTER X. 



DIMENSIONS OF ELECTRIC UNITS. 

620.] EVERY electromagnetic quantity may be defined with 
reference to the fundamental units of Length, Mass, and Time. 
If we begin with the definition of the unit of electricity, as given 
in Art. 65, we may obtain definitions of the units of every other 
electromagnetic quantity, in virtue of the equations into which 
they enter along with quantities of electricity. The system of 
units thus obtained is called the Electrostatic System. 

If, on the other hand, we begin with the definition of the unit 
magnetic pole, as given in Art. 374, we obtain a different system 
of units of the same set of quantities. This system of units is 
not consistent with the former system, and is called the Electro 
magnetic System. 

We shall begin by stating those relations between the different 
units which are common to both systems, and we shall then form 
a table of the dimensions of the units according to each system. 

621.] We shall arrange the primary quantities which we have 
to consider in pairs. In the first three pairs, the product of the 
two quantities in each pair is a quantity of energy or work. In 
the second three pairs, the product of each pair is a quantity of 
energy referred to unit of volume. 

FIRST THREE PAIRS. 

Electrostatic Pair. 

Symbol. 

( 1 ) Quantity of electricity . . . . e 

(2) Line-integral of electromotive force, or electric po 

tential E 



240 DIMENSIONS OF UNITS. [622. 

Magnetic Pair. 

Symbol. 

(3) Quantity of free magnetism, or strength of a pole . m 

(4) Magnetic potential ...... H 

ElectroJcinetic Pair. 

(5) Electroldnetic momentum of a circuit . . p 

(6) Electric current ....... C 

SECOND THREE PAIRS. 

Electrostatic Pair. 

(7) Electric displacement (measured by surface-density) . 3) 

(8) Electromotive force at a point . . . ( 

Magnetic Pair. 

(9) Magnetic induction * ..... 33 

(10) Magnetic force .; ..... $ 

Electrokinetic Pair. 

(11) Intensity of electric current at a point . . . ( 

(12) Vector potential of electric currents . . .51 

622.] The following relations exist between these quantities. 
In the first place, since the dimensions of energy are , and 

those of energy referred to unit of volume , we have the 

following equations of dimensions : 

(1) 

(2) 
Secondly, since e, p and 51 are the time-integrals of C, fi, and ( 



Thirdly, since E, 12, and p are the line-integrals of @, .>, and 91 
respectively, 



Finally, since e t C, and m are the surface-integrals of $), 6, and 
respectively, 



625.] THE TWO SYSTEMS OF UNITS. 241 

623.] These fifteen equations are not independent, and in order 
to deduce the dimensions of the twelve units involved, we require 
one additional equation. If, however, we take either e or m as an 
independent unit, we can deduce the dimensions of the rest in 
terms of either of these. 



(3) and (5) [j,] = M= 

(4) and (6) 



(10) 



624.] The relations of the first ten of these quantities may be 
exhibited by means of the following arrangement : 

e, 2), ), C and 12. E (, 33, m and p. 

The quantities in the first line are derived from e by the same 
operations as the corresponding quantities in the second line are 
derived from m. It will be seen that the order of the quantities 
in the first line is exactly the reverse of the order in the second 
line. The first four of each line have the first symbol in the 
numerator. The second four in each line have it in the deno 
minator. 

All the relations given above are true whatever system of units 
we adopt. 

625.] The only systems of any scientific value are the electro 
static and the electromagnetic system. The electrostatic system is 

VOL. II. ft 



242 DIMENSIONS OF UNITS. [626. 

founded on the definition of the unit of electricity, Arts. 41, 42, 
and may be deduced from the equation 



which expresses that the resultant force ( at any point, due to the 
action of a quantity of electricity e at a distance L, is found by 
dividing e by 7/ 2 . Substituting the equations of dimension (1) and 
(8), we find 



whence \e\ = \L* If* T^} , m = 
in the electrostatic system. 

The electromagnetic system is founded on a precisely similar 
definition of the unit of strength of a magnetic pole, Art. 374, 
leading to the equation ^ m 

* : = L* 

J/ 
whence 



e-] ri 
^J - \-^ J 



and [e] = 

in the electromagnetic system. From these results we find the 

dimensions of the other quantities. 

626.] Table of Dimensions. 

Dimensions in 

c, , , Electrostatic Electromagnetic 
Symbol Sygtem System 

Quantity of electricity .... e [Z* M * T~ l ] \L* M*\. 

Line-integral of electro- | ^ ^ M - T~^ \ti H* T~*\. 

motive force 3 

Quantity of magnetism -\ 

Electrokinetic momentum t . $ m I [tf M*\ \L* M* T~ 1 ]. 

of a circuit ) * 

Electric current C [L* M* T 



Magnetic potential ) {Q, 

Electric displacement | _ [T-^M^T~ l ~[ IT 

Surface-density 



Electromotive force at a point @ [^"M/^ 7 - 1 ] [Z*Jtf* I 7 " 2 ]. 

Magnetic induction 53 [IT^*] [i;-*^^- 1 ]. 

Magnetic force [L* M* T~*] [L~* M* I 1 ]. 

Strength of current at a point ( [Z~* If * T" 2 ] [^~^ If* T~ l ] . 

Vector potential 31 [Z-*!f*] 



628.] TABLE OF DIMENSIONS. 243 

627.] We have already considered the products of the pairs of 
these quantities in the order in which they stand. Their ratios are 
in certain cases of scientific importance. Thus 

Electrostatic Electromagnetic 
Symbol. System. System. 

e l~T 2 ~\ 

-=- = capacity of an accumulator . . q [Z] T~ I 

/coefficient of self-induction *\ 
-^- = j of a circuit, or electro- > L \~T~\ \f\* 

(. magnetic capacity J 

2) _ ( specific inductive capacity | ^ r _ 
=: ( of dielectric \ 

33 r^ 72 ! 

-- = magnetic inductive capacity . . ju y2 M- 

4P L^ J 

x? r- yr i p T 1 

- = resistance of a conductor .... R -=- "TT 

(S C specific resistance of a ) 
"T = : | substance } 

628.] If the units of length, mass, and time are the same in the 
two systems, the number of electrostatic units of electricity con 
tained in one electromagnetic unit is numerically equal to a certain 
velocity, the absolute value of which does not depend on the 
magnitude of the fundamental units employed. This velocity is 
an important physical quantity, which we shall denote by the 
symbol v. 

Number of Electrostatic Units in one Electromagnetic Unit. 
For*, C, 11, 5), , (, v. 

Form, ^ .0, 93, <, 21, - 

v 

For electrostatic capacity, dielectric inductive capacity, and con 
ductivity, v*. 

For electromagnetic capacity, magnetic inductive capacity, and 

resistance, 5- 

p 2 

Several methods of determining the velocity v will be given in 
Arts. 768-780. 

In the electrostatic system the specific dielectric inductive capa 
city of air is assumed equal to unity. This quantity is therefore 

represented by -^ in the electromagnetic system. 

R 2, 



244 DIMENSIONS OF UNITS. [629. 

In the electromagnetic system the specific magnetic inductive 
capacity of air is assumed equal to unity . This quantity is there 
fore represented by $ in the electrostatic system. 

Practical System of Electric Units. 

629.] Of the two systems of units, the electromagnetic is of the 
greater use to those practical electricians who are occupied with 
electromagnetic telegraphs. If, however, the units of length, time, 
and mass are those commonly used in other scientific work, such 
as the metre or the centimetre, the second, and the gramme, the 
units of resistance and of electromotive force will be so small that 
to express the quantities occurring in practice enormous numbers 
must be used, and the units of quantity and capacity will be so 
large that only exceedingly small fractions of them can ever occur 
in practice. Practical electricians have therefore adopted a set of 
electrical units deduced by the electromagnetic system from a large 
unit of length and a small unit of mass. 

The unit of length used for this purpose is ten million of metres, 
or approximately the length of a quadrant of a meridian of the 
earth. 

The unit of time is, as before, one second. 

The unit of mass is 10~~ n gramme, or one hundred millionth 
part of a milligramme. 

The electrical units derived from these fundamental units have 
been named after eminent electrical discoverers. Thus the practical 
unit of resistance is called the Ohm, and is represented by the 
resistance-coil issued by the British Association, and described in 
Art. 340. It is expressed in the electromagnetic system by a 
velocity of 10,000,000 metres per second. 

The practical unit of electromotive force is called the Volt, and 
is not very different from that of a DanielPs cell. Mr. Latimer 
Clark has recently invented a very constant cell, whose electro 
motive force is almost exactly 1.457 Volts. 

The practical unit of capacity is called the Farad. The quantity 
of electricity which flows through one Ohm under the electromotive 
force of one Volt during one second, is equal to the charge produced 
in a condenser whose capacity is one Farad by an electromotive 
force of one Volt. 

The use of these names is found to be more convenient in practice 
than the constant repetition of the words electromagnetic units, 



62 9 .] 



PEACTICAL UNITS. 



245 



with the additional statement of the particular fundamental units 
on which they are founded. 

When very large quantities are to be measured, a large unit 
is formed by multiplying the original unit by one million, and 
placing before its name the prefix mega. 

In like manner by prefixing micro a small unit is formed, one 
millionth of the original unit. 

The following table gives the values of these practical units in 
the different systems which have been at various times adopted. 



FUNDAMENTAL 
UNITS. 


PRACTICAL 
SYSTEM. 


B. A. REPORT, 
1863. 


THOMSON. 


WEBER. 


Length, 
Time, 

Mass. 


Earth s Quadrant, 
Second, 
10- 11 Gramme. 


Metre, 
Second, 
Gramme. 


Centimetre, 
Second, 
Gramme. 


Millimetre, 
Second, 
Milligramme. 


Resistance 


Ohm 


IO 7 


IO 9 


IO 1 


Electromotive force 


Volt 


IO 5 


IO 8 


10 U 


Capacity 
Quantity 


Farad 

Farad 
(charged to a Volt.) 


io- 7 
io- 2 


io- 9 
io- 1 


io- 10 

10 



CHAPTER XL 



ON ENERGY AND STRESS IN THE ELECTROMAGNETIC FIELD. 

Electrostatic Energy. 

630.] THE energy of the system may be divided into the Potential 
Energy and the Kinetic Energy. 

The potential energy due to electrification has been already con 
sidered in Art. 85. It may be written 

r=is(**), (i) 

where e is the charge of electricity at a place where the electric 
potential is ty, and the summation is to be extended to every place 
where there is electrification. 

If fj ffj Ji are the components of the electric displacement, the 
quantity of electricity in the element of volume dx dy dz is 



where the integration is to be extended throughout all space. 

631.] Integrating this expression by parts, and remembering 
that when the distance, r, from a given point of a finite electrified 
system becomes infinite, the potential ty becomes an infinitely small 
quantity of the order r* 1 , and that/, g, h become infinitely small 
quantities of the order r~ 2 , the expression is reduced to 



where the integration is to be extended throughout all space. 

If we now write P, Q, R for the components of the electromotive 

dty d^ city 

force, instead of -- , -- and -- =- , we find 
dx dy dz 

(5) 



633-] MAGNETIC ENERGY. 247 

Hence, the electrostatic energy of the whole field will be the same 
if we suppose that it resides in every part of the field where elec 
trical force and electrical displacement occur, instead of being 
confined to the places where free electricity is found. 

The energy in unit of volume is half the product of the electro 
motive force and the electric displacement, multiplied by the cosine 
of the angle which these vectors include. 

In Quaternion language it is 4/9(5 3). 

Magnetic Energy. 

632.] We may treat the energy due to magnetization in a similar 
way. If A, J5, C are the components of magnetization and a, /3, y 
the components of magnetic force, the potential energy of the 
system of magnets is, by Art. 389, 

Cy]dxdydz t (6) 

the integration being extended over the space occupied by mag 
netized matter. This part of the energy, however, will be included 
in the kinetic energy in the form in which we shall presently 
obtain it. 

633.] We may transform this expression when there are no elec 
tric currents by the following method. 

We know that da db do 

Hence, by Art. 97, if 

cm d& cm 

f. o .. ( R\ 

as is always the case in magnetic phenomena where there are no 
currents, 

=0, (9) 






the integral being extended throughout all space, or 

jjl{(a + lTtA)a + (P + lTtB)p + (y+ nC)y}dxdydz = 0. (10) 
Hence, the energy due to a magnetic system 



248 ENERGY AND STRESS. [634. 

Electrokinetic Energy. 

634.] We have already, in Art. 578, expressed the kinetic energy 
of a system of currents in the form 

T=\^(pi\ (12). 

where p is the electromagnetic momentum of a circuit, and % is 
the strength of the current flowing round it, and the summation 
extends to all the circuits. 

But we have proved, in Art. 590, that p may be expressed as 
a line-integral of the form 



where F, G, H are the components of the electromagnetic mo- 
mentum, C, at the point (xy z), and the integration is to be ex 
tended round the closed circuit s. We therefore find 



2 *" J \ ?<$ ds ds 

If ^, z;, w are the components of the density of the current at 
any point of the conducting circuit, and if S is the transverse 
section of the circuit, then we may write 

. dx .dy . dz 

i = uS, i^ = vS, 2- v = ^, (15) 

ds ds ds 

and we may also write the volume 

Sds = dxdydz, 
and we now find _ 

T = i / // (Fu + Gv + Hw) dxdydz, (16) 

where the integration is to be extended to every part of space 
where there are electric currents. 

635.] Let us now substitute for u, v, w their values as given by 
the equations of electric currents (E), Art. 607, in terms of the 
components a, /3, y of the magnetic force. We then have 

where the integration is extended over a portion of space including 
all the currents. 

If we integrate this by parts, and remember that, at a great 
distance r from the system, a, /3, and y are of the order of mag 
nitude r~ 3 , we find that when the integration is extended through 
out all space, the expression is reduced to 



/^ 7 dH \ f flG dF \] 7 



637.] ELECTROKINETIC ENERGY. 249 

By the equations (A), Art. 591, of magnetic induction, we may 
substitute for the quantities in small brackets the components of 
magnetic induction a, b, c, so that the kinetic energy may be 
written 1 /././. 

T= JJJ(aa + 6p + cy)da!dydz 9 (19) 

where the integration is to be extended throughout every part of 
space in which the magnetic force and magnetic induction have 
values differing from zero. 

The quantity within brackets in this expression is the product of 
the magnetic induction into the resolved part of the magnetic force 
in its own direction. 

In the language of quaternions this may be written more simply, 



where 33 is the magnetic induction, whose components are , b, c, 
and JQ is the magnetic force, whose components are a, (3, y. 

636.] The electrokinetic energy of the system may therefore be 
expressed either as an integral to be taken where there are electric 
currents, or as an integral to be taken over every part of the field 
in which magnetic force exists. The first integral, however, is the 
natural expression of the theory which supposes the currents to act 
upon each other directly at a distance, while the second is appro 
priate to the theory which endeavours to explain the action between 
the currents by means of some intermediate action in the space 
between them. As in this treatise we have adopted the latter 
method of investigation, we naturally adopt the second expression 
as giving the most significant form to the kinetic energy. 

According to our hypothesis, we assume the kinetic energy to 
exist wherever there is magnetic force, that is, in general, in every 
part of the field. The amount of this energy per unit of volume 

is -- S S3 $3, and this energy exists in the form of some kind 

o 77 

of motion of the matter in every portion of space. 

When we come to consider Faraday s discovery of the effect of 
magnetism on polarized light, we shall point out reasons for be 
lieving that wherever there are lines of magnetic force, there is 
a rotatory motion of matter round those lines. See Art. 821. 

Magnetic and Electrokinetic Energy compared. 
637.] We found in Art. 423 that the mutual potential energy 



250 ENERGY AND STRESS. [638. 

of two magnetic shells, of strengths $ and $ , and bounded by the 
closed curves s and / respectively, is 

cos e , 
as as , 

where e is the angle between the directions of ds and ds , and r 
is the distance between them. 

We also found in Art. 521 that the mutual energy of two circuits 
s and /, in which currents i and i flow, is 



-if 



cos e 7 .. f 
ds ds . 



If i, i are equal to (/>, </> respectively, the mechanical action 
between the magnetic shells is equal to that between the cor 
responding electric circuits, and in the same direction. In the case 
of the magnetic shells, the force tends to diminish their mutual 
potential energy, in the case of the circuits it tends to increase their 
mutual energy, because this energy is kinetic. 

It is impossible, by any arrangement of magnetized matter, to 
produce a system corresponding in all respects to an electric circuit, 
for the potential of the magnetic system is single valued at every 
point of space, whereas that of the electric system is many- valued. 

But it is always possible, by a proper arrangement of infinitely 
small electric circuits, to produce a system corresponding in all 
respects to any magnetic system, provided the line of integration 
which we follow in calculating the potential is prevented from 
passing through any of these small circuits. This will be more 
fully explained in Art. 833. 

The action of magnets at a distance is perfectly identical with 
that of electric currents. We therefore endeavour to trace both 
to the same cause, and since we cannot explain electric currents 
by means of magnets, we must adopt the other alternative, and 
explain magnets by means of molecular electric currents. 

638.J In our investigation of magnetic phenomena, in Part III 
of this treatise, we made no attempt to account for magnetic action 
at a distance, but treated this action as a fundamental fact of 
experience. We therefore assumed that the energy of a magnetic 
system is potential energy, and that this energy is diminished when 
the parts of the system yield to the magnetic forces which act 
on them. 

If, however, we regard magnets as deriving their properties from 
electric currents circulating within their molecules, their energy 



639-] AMPERE S THEORY OF MAGNETS. 251 

is kinetic, and the force between them is such that it tends to 
move them in a direction such that if the strengths of the currents 
were maintained constant the kinetic energy would increase. 

This mode of explaining magnetism requires us also to abandon 
the method followed in Part III, in which we regarded the magnet 
as a continuous and homogeneous body, the minutest part of which 
has magnetic properties of the same kind as the whole. 

We must now regard a magnet as containing a finite, though 
very great, number of electric circuits, so that it has essentially 
a molecular, as distinguished from a continuous structure. 

If we suppose our mathematical machinery to be so coarse that 
our line of integration cannot thread a molecular circuit, and that 
an immense number of magnetic molecules are contained in our 
element of volume, we shall still arrive at results similar to those 
of Part III, but if we suppose our machinery of a finer order, 
and capable of investigating all that goes on in the interior of the 
molecules, we must give up the old theory of magnetism, and adopt 
that of Ampere, which admits of no magnets except those which 
consist of electric currents. 

We must also regard both magnetic and electromagnetic energy 
as kinetic energy, and we must attribute to it the proper sign, 
as given in Art. 635. 

In what follows, though we may occasionally, as in Art. 639, &c., 
attempt to carry out the old theory of magnetism, we shall find 
that we obtain a perfectly consistent system only when we abandon 
that theory and adopt Ampere^s theory of molecular currents, as in 
Art. 644. 

The energy of the field therefore consists of two parts only, the 
electrostatic or potential energy 

W = \jjj(Pf + 

and the electromagnetic or kinetic energy 
T= ~ 



ON THE FORCES WHICH ACT ON AN ELEMENT OF A BODY PLACED 
IN THE ELECTROMAGNETIC FIELD. 

Forces acting on a Magnetic Element. 

639.] The potential energy of the element dx dy dz of a body 
magnetized with an intensity whose components are A, B, C, and 



252 ENERGY AND STRESS. [640. 

placed in a field of magnetic force whose components are a, /3, y, is 



Hence, if the force urging the element to move without rotation 
in the direction of a? is X 1 dxdydz, 



and if the moment of the couple tending to turn the element about 
the axis of x from y towards z is L dxdydz, 

L = By-C($. (2) 

The forces and the moments corresponding to the axes of y and 

z may be written down by making the proper substitutions. 

640. J If the magnetized body carries an electric current, of 

which the components are u 3 v, w, then, by equations C, Art. 60S, 

there will be an additional electromagnetic force whose components 

are X 2 , Y%, Z Z) of which X 2 is 

X 2 = VG wb. (3) 

Hence, the total force, X, arising from the magnetism of the 

molecule, as well as the current passing through it, is 



+vc-6. (4) 

dx dx 

The quantities a, 6, c are the components of magnetic induction, 
and are related to a, (3, y, the components of magnetic force, by 
the equations given in Art. 400, 

a = a -f 4 TT A, 

=/3 + 477., (5) 

C = 7+477(7. 

The components of the current, u, v, w, can be expressed in terms 
of a, /3, y by the equations of Art. 607, 



dy d(3 
4 TT u - j- 

dy dz 

da dy 

4;TTV = -= -~- 

dz dx 

dp da 

TT 4/7rw = -f- -T 

Hence dx dy 



(6) 



_ 
dx } dx n dx 

1 ( da -.da da 1 d 1 

= \a T +b+c~---- (a*+(3 2 +y 2 )}- (7) 

47T ( dx dy dz 2 dee. } 



641.] THEORY OF STRESS. 253 






Multiplying this equation, (8), by a, and dividing by 47i, we may 
add the result to (7), and we find 

(9) 



also, by (2), i = ((J-/3) y-(c-y)/3), (10) 

= ~(i v -eft), (11) 

where X is the force referred to unit of volume in the direction of 
#, and L is the moment of the forces about this axis. 

On the Explanation of these Forces by the Hypothesis of a Medium 
in a State of Stress. 

641 .] Let us denote a stress of any kind referred to unit of area 
by a symbol of the form P hk) where the first suffix, h , indicates that 
the normal to the surface on which the stress is supposed to act 
is parallel to the axis of h, and the second suffix, ft , indicates that 
the direction of the stress with which the part of the body on 
the positive side of the surface acts on the part on the negative 
side is parallel to the axis of k. 

The directions of h and k may be the same, in which case the 
stress is a normal stress. They may be oblique to each other, in 
which case the stress is an oblique stress, or they may be perpen 
dicular to each other, in which case the stress is a tangential 
stress. 

The condition that the stresses shall not produce any tendency 
to rotation in the elementary portions of the body is 

P - P 

^hk r Wi 

In the case of a magnetized body, however, there is such a 
tendency to rotation, and therefore this condition, which holds in 
the ordinary theory of stress, is not fulfilled. 

Let us consider the effect of the stresses on the six sides of 
the elementary portion of the body dx dy dz, taking the origin of 
coordinates at its centre of gravity. 

On the positive face dy dz, for which the value of % is \ dx, the 
forces are 



254 



ENERGY AND STRESS. 



[641. 



Parallel to x, 



dP. 



Parallel to y, (P xy + * -^f dx} dydz = Y +x , . 



(12) 



Parallel to 



(P+ 4 



The forces acting on the opposite side, X_ X9 Y_ x) and Z_ x , 
may be found from these by changing the sign of dx. We may 
express in the same way the systems of three forces acting on each 
of the other faces of the element, the direction of the force being 
indicated by the capital letter, and the face on which it acts by 
the suffix. 

If Xdxdydz is the whole force parallel to x acting on the element, 

Xdxdydz = X H 

,P. 



whence 



d 



dx dx 

^ P + ^ 

dy vx dz 



(13) 



If Ldxdydz is the moment of the forces about the axis of x 
tending to turn the element from y to 0, 
Ldxdydz = 



whence L = P yg P zy . (14) 

Comparing the values of X and L given by equations (9) and 
(11) with those given by (13) and (14), we find that, if we make 



= --_(aa-(<S 



1 

TTJ 
1 



p 

-*-%* A ~ 



~k 



= ~T- C ^ 






+r 



i 

4 77 
1 

= ^v a " / 

I 



(15) 



the force arising from a system of stress of which these are the 
components will be statically equivalent, in its effects on each 



642.] 



MAGNETIC STRESS. 



255 



element of the body, with the forces arising from the magnetization 
and electric currents. 

642.] The nature of the stress of which these are the components 
may be easily found, by making the axis of x bisect the angle 
between the directions of the magnetic force and the magnetic 
induction, and taking the axis of y in the plane of these directions, 
and measured towards the side of the magnetic force. 

If we put <) for the numerical value of the magnetic force, 33 for 
that of the magnetic induction, and 2 for the angle between their 
directions, 

a = *y cos e, /3 = ) sin e, y =. 0, 
a 33 cos e, b = 33 sin e, c 

1 - 2 i 2 

4 jf 



(17) 



p _ p p _ p _ o 

yz ~~ zx zy -* xz 

P xv = - 33 <> cos e sin e, 

P yx = - 33 4p cos e sin e. 
Hence, the state of stress may be considered as compounded of 

(1) A pressure equal in all directions = - & 2 . 

8 77 

(2) A tension along the line bisecting the angle between the 
directions of the magnetic force and the magnetic induction 

- 

(3) A pressure along the line bisecting the exterior angle between 
these directions = 33 sin 2 e. 

(4) A couple tending to turn every element of the substance in 
the plane of the two directions from the direction of magnetic 

induction to the direction of magnetic force - - 33 <) sin 2 e. 

When the magnetic induction is in the same direction as the 
magnetic force, as it always is in fluids and non-magnetized solids, 
then e = 0, and making the axis of x coincide with the direction of 
the magnetic force, 



256 



ENERGY AND STRESS. 



[643. 
(18) 



and the tangential stresses disappear. 

The stress in this case is therefore a hydrostatic pressure - - j 2 , 

combined with a longitudinal tension 33 <) along the lines of 

f 4 TT 

force. 

643.] When there is no magnetization, 33 = $3, and the stress is 
still further simplified, being a tension along the lines of force equal 

to - <) 2 , combined with a pressure in all directions at right angles 

. 1 

to the lines of force, numerically equal also to - 43 2 - The com 
ponents of stress in this important case are 

P xx = (a*-(3*-y 
P = ( 2 -a 2 -/3 

** 8 77 ^ 

yz zy ^^ 






(19) 



PX = Px = JL al3t 

4 7T 

The force arising from these stresses on an element of the medium 
referred to unit of volume is 
d d 

f -J-PVZ+ -rP> 

ay " dz 



Y _ d 

= 



1 C da d/3 dyl 1 ( d(3 dal 1 C dy da) 
^da d(3 dy\ 1 /da dy\ 



__ 



dy 



fa 



dy 



Now 



da d(3 dy 

-7- + ~r + -T 
dx dy dz 

da. dy 

-j- -y- 

dz dx 
dft da 

-j =- = 4 77 W- 

ax dy 
where m is the density of austral magnetic matter referred to unit 



645-] TENSION ALONG LINES OF FORCE. 257 

of volume, and v and w are the components of electric currents 
referred to unit of area perpendicular to y and z respectively. Hence, 
X = am+ vy wj3 



Similarly Y = fim + wa uy, 



(Equations of 
Electromagnetic (20) 

Force.) 
Zi = ym-i-vip va. 

644.] If we adopt the theories of Ampere and Weber as to the 
nature of magnetic and diamagnetic bodies, and assume that mag 
netic and diamagnetic polarity are due to molecular electric currents, 
we get rid of imaginary magnetic matter, and find that everywhere 

* = 0,and *? + *0 + ?y =0 , (21) 

dx dy dz 

so that the equations of electromagnetic force become, 
X = v y w /3, 

Ywa-uy } (22) 

Z = ujBva. 

These are the components of the mechanical force referred to unit 
of volume of the substance. The components of the magnetic force 
are a, /3, y, and those of the electric current are u, v, w. These 
equations are identical with those already established. (Equations 
(C), Art, 603.) 

645.] In explaining the electromagnetic force by means of a 
state of stress in a medium, we are only following out the con 
ception of Faraday"*, that the lines of magnetic force tend to 
shorten themselves, and that they repel each other when placed 
side by side. All that we have done is to express the value of 
the tension along the lines, and the pressure at right angles to 
them, in mathematical language, and to prove that the state of 
stress thus assumed to exist in the medium will actually produce 
the observed forces on the conductors which carry electric currents. 

We have asserted nothing as yet with respect to the mode 
in which this state of stress is originated and maintained in the 
medium. We have merely shewn that it is possible to conceive 
the mutual action of electric currents to depend on a particular 
kind of stress in the surrounding medium, instead of being a direct 
and immediate action at a distance. 

Any further explanation of the state of stress, by means of the 
motion of the medium or otherwise, must be regarded as a separate 
and independent part of the theory, which may stand or fall without 
affecting our present position. See Art. 832. 

* Esrp. Res., 3266, 3267, 3268. 
VOL. TT. S 



258 ENERGY AND STRESS. [646. 

In the first part of this treatise, Art. 108, we shewed that the 
observed electrostatic forces may be conceived as operating through 
the intervention of a state of stress in the surrounding medium. 
We have now done the same for the electromagnetic forces, and 
it remains to be seen whether the conception of a medium capable 
of supporting these states of stress is consistent with other known 
phenomena, or whether we shall have to put it aside as unfruitful. 

In a field in which electrostatic as well as electromagnetic action 
is taking place, we must suppose the electrostatic stress described 
in Part I to be superposed on the electromagnetic stress which we 
have been considering. 

646.] If we suppose the total terrestrial magnetic force to be 
10 British units (grain, foot, second), as it is nearly in Britain, then 
the tension perpendicular to the lines of force is 0.128 grains weight 
per square foot. The greatest magnetic tension produced by Joule * 
by means of electromagnets was about 140 pounds weight on the 
square inch. 

* Sturgeon s Annals of Electricity, vol. v. p. 187 (1840) ; or Philosophical Magazine, 
Dec., 1851. 



CHAPTER XII. 



CURRENT-SHEETS. 



647.] A CURRENT-SHEET is an infinitely thin stratum of con 
ducting matter, bounded on both sides by insulating 1 media, so that 
electric currents may flow in the sheet, but cannot escape from it 
except at certain points called Electrodes, where currents are made 
to enter or to leave the sheet. 

In order to conduct a finite electric current, a real sheet must 
have a finite thickness, and ought therefore to be considered a 
conductor of three dimensions. In many cases, however, it is 
practically convenient to deduce the electric properties of a real 
conducting sheet, or of a thin layer of coiled wire, from those of 
a current-sheet as defined above. 

We may therefore regard a surface of any form as a current-sheet. 
Having selected one side of this surface as the positive side, we 
shall always suppose any lines drawn on the surface to be looked 
at from the positive side of the surface. In the case of a closed 
surface we shall consider the outside as positive. See Art. 294, 
where, however, the direction of the current is defined as seen from 
the negative side of the sheet. 

The Current -function. 

648.] Let a fixed point A on the surface be chosen as origin, and 
let a line be drawn on the surface from A to another point P. Let 
the quantity of electricity which in unit of time crosses this line 
from left to right be $, then </> is called the Current-function at 
the point P. 

The current-function depends only on the position of the point P, 
and is the same for any two forms of the line AP, provided this 

s z 



260 CURRENT-SHEETS. [649. 

line can be transformed by continuous motion from one form to the 
other without passing through an electrode. For the two forms of 
the line will enclose an area within which there is no electrode, and 
therefore the same quantity of electricity which enters the area across 
one of the lines must issue across the other. 

If s denote the length of the line AP, the current across ds from 

left to right will be ds. 

If </> is constant for any curve, there is no current across it. Such 
a curve is called a Current-line or a Stream-line. 

649.] Let \}f be the electric potential at any point of the sheet, 
then the electromotive force along any element ds of a curve will be 

d^ , 
f-d*, 

ds 

provided no electromotive force exists except that which arises from 
differences of potential. 

If \^ is constant for any curve, the curve is called an Equi- 
potential Line. 

650.] We may now suppose that the position of a point on the 
sheet is defined by the values of </> and \[r at that point. Let ds l be 
the length of the element of the equipotential line ^ intercepted 
between the two current lines < and <j> + d<l>, and let ds 2 be the 
length of the element of the current line $ intercepted between the 
two equipotential lines ty and \fr + d\lf. We may consider ds } and ds z 
as the sides of the element dty d\^r of the sheet. The electromotive 
force d\l/ in the direction of ds 2 produces the current d<p across ds lf 

Let the resistance of a portion of the sheet whose length is ds 2t 
and whose breadth is ds l} be ds 2 

(T .- J 

0*1 

where <r is the specific resistance of the sheet referred to unit of 
area, then ds. 2 7 

*-*zf * 

, ds-, ds. 2 

whence jj- = <r yf - 

a<j) d\l/ 

651.] If the sheet is of a substance which conducts equally well 
in all directions, ds l is perpendicular to ds 2 . In the case of a sheet 
of uniform resistance or is constant, and if we make \jr = a\f/, we 
shall have ds : __ d(j> 

d9t~~ d+ * 

and the stream-lines and equipotential lines will cut the surface into 
little squares. 



652.] MAGNETIC POTENTIAL. 261 

It follows from this that if fa and i/r/ are conjugate functions 
(Art. 183) of cj) and \f/ t the curves fa may be stream-lines in the 
sheet for which the curves x/// are the corresponding equipotential 
lines. One case, of course, is that in which fa = \f/ and \j/i = <. 
In this case the equipotential lines become current-lines, and the 
current-lines equipotential lines *. 

If we have obtained the solution of the distribution of electric 
currents in a uniform sheet of any form for any particular case, we 
may deduce the distribution in any other case by a proper trans 
formation of the conjugate functions, according to the method given 
in Art. 190. 

652.] We have next to determine the magnetic action of a 
current-sheet in which the current is entirely confined to the sheet, 
there being no electrodes to convey the current to or from the 
sheet. 

In this case the current-function has a determinate value at 
every point, and the stream-lines are closed curves which do not 
intersect each other, though any one stream-line may intersect 
itself. 

Consider the annular portion of the sheet between the stream 
lines $ and <j)-{-b<p. This part of the sheet is a conducting circuit 
in which a current of strength 8 $ circulates in the positive direction 
round that part of the sheet for which c/> is greater than the given 
value. The magnetic effect of this circuit is the same as that of 
a magnetic shell of strength 8 $ at any point not included in the 
substance of the shell. Let us suppose that the shell coincides with 
that part of the current-sheet for which has a greater value than 
it has at the given stream-line. 

By drawing all the successive stream-lines, beginning with that 
for which $ has the greatest value, and ending with that for which 
its value is least, we shall divide the current-sheet into a series 
of circuits. Substituting for each circuit its corresponding mag 
netic shell, we find that the magnetic effect of the current-sheet 
at any point not included in the thickness of the sheet is the same 
as that of a complex magnetic shell, whose strength at any point 
is C-{-(f), where C is a constant. 

If the current-sheet is bounded, then we must make C 4- < = 
at the bounding curve. If the sheet forms a closed or an infinite 
surface, there is nothing to determine the value of the constant C. 

* See Thomson, Camb. and Dub. Math. Journ., vol. iii. p. 286. 



262 CURRENT -SHEETS. [653. 

653.] The magnetic potential at any point on either side of the 
current-sheet is given, as in Art. 415, by the expression 



= ^- 



where r is the distance of the given point from the element of 
surface dS, and Q is the angle between the direction of r, and that 
of the normal drawn from the positive side of dS. 

This expression gives the magnetic potential for all points not 
included in the thickness of the current-sheet, and we know that 
for points within a conductor carrying a current there is no such 
thing as a magnetic potential. 

The value of H is discontinuous at the current-sheet, for if &j_ 
is its value at a point just within the current-sheet, and Q, 2 its 
value at a point close to the first but just outside the current-sheet, 

& 2 = Hj + 4 TT $, 
where </> is the current-function at that point of the sheet. 

The value of the component of magnetic force normal to the 
sheet is continuous, being the same on both sides of the sheet. 
The component of the magnetic force parallel to the current-lines 
is also continuous, but the tangential component perpendicular to 
the current-lines is discontinuous at the sheet. If s is the length 
of a curve drawn on the sheet, the component of magnetic force 

T 



in the direction of ds is, for the negative side, T^J an d for the 



2 

positive side, =-^ ^ + 4 -n -f 
ds ds ds 

The component of the magnetic force on the positive side there 

fore exceeds that on the negative side by 4 TT -~ - At a given point 

ds 

this quantity will be a maximum when ds is perpendicular to the 
current-lines. 

On the Induction of Electric Currents in a Sheet of Infinite 

Conductivity. 
654.] It was shewn in Art. 579 that in any circuit 



where E is the impressed electromotive force, p the electrokinetic 
momentum of the circuit, R the resistance of the circuit, and i the 
current round it. If there is no impressed electromotive force and 

no resistance, then ~ = 0, or p is constant. 
tit 



656.] PLANE SHEET. 263 

Now 7;, the electrokinetic momentum of the circuit, was shewn 
in Art. 588 to be measured by the surface-integral of magnetic 
induction through the circuit. Hence, in the case of a current- 
sheet of no resistance, the surface-integral of magnetic induction 
through any closed curve drawn on the surface must be constant, 
and this implies that the normal component of magnetic induction 
remains constant at every point of the current-sheet. 

655.] If, therefore, by the motion of magnets or variations of 
currents in the neighbourhood, the magnetic field is in any way 
altered, electric currents will be set up in the current-sheet, such 
that their magnetic effect, combined with that of the magnets or 
currents in the field, will maintain the normal component of mag 
netic induction at every point of the sheet unchanged. If at first 
there is no magnetic action, and no currents in the sheet, then 
the normal component of magnetic induction will always be zero 
at every point of the sheet. 

The sheet may therefore be regarded as impervious to magnetic 
induction, and the lines of magnetic induction will be deflected by 
the sheet exactly in the same way as the lines of flow of an electric 
current in an infinite and uniform conducting mass would be 
deflected by the introduction of a sheet of the same form made 
of a substance of infinite resistance. 

If the sheet forms a closed or an infinite surface, no magnetic 
actions which may take place on one side of the sheet will produce 
any magnetic effect on the other side. 

Theory of a Plane Current-sJieet. 

656.] We have seen that the external magnetic action of a 
current-sheet is equivalent to that of a magnetic shell whose strength 
at any point is numerically equal to c/>, the current-function. When 
the sheet is a plane one, we may express all the quantities required 
for the determination of electromagnetic effects in terms of a single 
function, P, which is the potential due to a sheet of imaginary 
matter spread over the plane with a surface-density <. The value 

of P is of course r (*& 

< (1) 



where r is the distance from the point (x, y, z] for which P is cal 
culated, to the point x ", y , in the plane of the sheet, at which the 
element dx dif is taken. 

To find the magnetic potential, we may regard the magnetic 



264 CURRENT -SHEETS. [657. 



shell as consisting of two surfaces parallel to the plane of xy, the 

first, whose equation is z = J <?, having 1 the surface-density , and 

c 

the second, whose equation is z =\c, having the surface-density 



c 
The potentials due to these surfaces will be 

-P/ c \ and -- P/ cv- 
c (*-g) c (*+?) 

ft 

respectively, where the suffixes indicate that z -- is put for z 

s* 

in the first expression, and z 4- - for z in the second. Expanding 

2i 

these expressions by Taylor s Theorem, adding them, and then 
making c infinitely small, we obtain for the magnetic potential due 
to the sheet at any point external to it, 



657.] The quantity P is symmetrical with respect to the plane of 
the sheet, and is therefore the same when z is substituted for z. 
H, the magnetic potential, changes sign when z is put for z. 
At the positive surface of the sheet 

11 = - = 2770. (3) 

dz 
At the negative surface of the sheet 

a = - d f- = -2v< t> . (4) 

CIZ 

Within the sheet, if its magnetic effects arise from the magneti 
zation of its substance, the magnetic potential varies continu 
ously from 2ir<p at the positive surface to 2ir(p at the negative 
surface. 

If the sheet contains electric currents, the magnetic force 
within it does not satisfy the condition of having a potential. 
The magnetic force within the sheet is, however, perfectly deter 
minate. 

The normal component, 



is the same on both sides of the sheet and throughout its sub 
stance. 

If a and ft be the components of the magnetic force parallel to 



657.] VECTOR-POTENTIAL. 265 

x and to y at the positive surface, and a, j3 those on the negative 

surface dd> /<,% 

a = 27T-^ = a , (6) 



Within the sheet the components vary continuously from a and 
/3 to a and /3 . 



The equations -5 j 

dii dz 

i/ 



= _^, (8) 

dz dx dy 

,7 /~] 3 TJ! s7 (~\ 

(v \JT tt Jj Cu \L 

dx dy dz j 

which connect the components F, G, H of the vector-potential due 
to the current-sheet with the scalar potential 12, are satisfied if 
we make d P dP 

-j- > Cr = =- , JLL = 0. (9) 

dy dx 

We may also obtain these values by direct integration, thus for F, 



Since the integration is to he estimated over the infinite plane 
sheet, and since the first term vanishes at infinity, the expression is 
reduced to the second term ; and by substituting 

d I d \ 

- -- tor -j-? - , 
dy r ay r 

and remembering that (/> depends on x f and y f ^ and not on HP, y, z t 



If H is the magnetic potential due to any magnetic or electric 
system external to the sheet, we may write 

F=-J& dz, (10) 

and we shall then have 



for the components of the vector-potential due to this system. 



266 CURRENT- SHEETS. [658. 

658.] Let us now determine the electromotive force at any point 
of the sheet, supposing the sheet fixed. 

Let X and Zbe the components of the electromotive force parallel 
to x and to y respectively, then, by Art. 598, we have 



If the electric resistance of the sheet is uniform and equal to &, 
X = au, Y = (TV, (14) 

where u and v are the components of the current, and if < is the 

current-function, ^<f> ^ 

u = -f- t v = ~ (15) 

dy dx 

But, by equation (3), 



at the positive surface of the current-sheet. Hence, equations (12) 
and (13) may be written 

t (16) 



dy at 

* d+ , . 

c j ~ 



where the values of the expressions are those corresponding to the 
positive surface of the sheet. 

If we differentiate the first of these equations with respect to x, 
and the second with respect to ^, and add the results, we obtain 

The only value of \jf which satisfies this equation, and is finite 
and continuous at every point of the plane, and vanishes at an 
infinite distance, is ^ _ Q (19) 

Hence the induction of electric currents in an infinite plane sheet 
of uniform conductivity is not accompanied with differences of 
electric potential in different parts of the sheet. 

Substituting this value of ^, and integrating equations (16), 
(17), we obtain ^ dP dP c i P 

Since the values of the currents in the sheet are found by 



66O.] DECAY OF CURRENTS IN THE SHEET. 267 

differentiating 1 with respect to as or y, the arbitrary function of z 
and t will disappear. We shall therefore leave it out of account. 

If we also write for , the single symbol R, which represents 

277 

a certain velocity, the equation between P and P becomes 

4f-f+f- w 

659.] Let us first suppose that there is no external magnetic 
system acting on the current sheet. We may therefore suppose 
P / = 0. The case then becomes that of a system of electric currents 
in the sheet left to themselves, but acting on one another by their 
mutual induction, and at the same time losing their energy on 
account of the resistance of the sheet. The result is expressed 
by the equation dP dP 

-"3T = "77 

dz dt 
the solution of which is 

P=f(x,y,(z+Rty. (23) 

Hence, the value of P on any point on the positive side of the 
sheet whose coordinates are x, y> z, and at a time #, is equal to 
the value of P at the point #, y, (z + Rt] at the instant when tf=0. 

If therefore a system of currents is excited in a uniform plane 
sheet of infinite extent and then left to itself, its magnetic effect 
at any point on the positive side of the sheet will be the same 
as if the system of currents had been maintained constant in the 
sheet, and the sheet moved in the direction of a normal from its 
negative side with the constant velocity R. The diminution of 
the electromagnetic forces, which arises from a decay of the currents 
in the real case, is accurately represented by the diminution of the 
force on account of the increasing distance in the imaginary case. 
660.] Integrating equation (21) with respect to t, we obtain 



If we suppose that at first P and P are both zero, and that a 
magnet or electromagnet is suddenly magnetized or brought from 
an infinite distance, so as to change the value of P suddenly from 
zero to P , then, since the time-integral in the second member of 
(24) vanishes with the time, we must have at the first instant 

P = -P 
at the surface of the sheet. 

Hence, the system of currents excited in the sheet by the sudden 



268 CURRENT -SHEETS. [66 1. 

introduction of the system to which P f is due is such that at the 
surface of the sheet it exactly neutralizes the magnetic effect of 
this system. 

At the surface of the sheet, therefore, and consequently at all 
points on the negative side of it, the initial system of currents 
produces an effect exactly equal and opposite to that of the 
magnetic system on the positive side. We may express this by 
saying that the effect of the currents is equivalent to that of an 
image of the magnetic system, coinciding in position with that 
system, but opposite as regards the direction of its magnetization 
and of its electric currents. Such an image is called a negative 
image. 

The effect of the currents in the sheet on a point on the positive 
side of it is equivalent to that of a positive image of the magnetic 
system on the negative side of the sheet, the lines joining corre 
sponding points being bisected at right angles by the sheet. 

The action at a point on either side of the sheet, due to the 
currents in the sheet, may therefore be regarded as due to an 
image of the magnetic system on the side of the sheet opposite 
to the point, this image being a positive or a negative image 
according as the point is on the positive or the negative side of 
the sheet. 

661.] If the sheet is of infinite conductivity, R = 0, and the 
second term of (24) is zero, so that the image will represent the 
effect of the currents in the sheet at any time. 

In the case of a real sheet, the resistance R has some finite value. 
The image just described will therefore represent the effect of the 
currents only during the first instant after the sudden introduction 
of the magnetic system. The currents will immediately begin to 
decay, and the effect of this decay will be accurately represented if 
we suppose the two images to move from their original positions, in 
the direction of normals drawn from the sheet, with the constant 
velocity R. 

662.] We are now prepared to investigate the system of currents 
induced in the sheet by any system, M, of magnets or electro 
magnets on the positive side of the sheet, the position and strength 
of which vary in any manner. 

Let P , as before, be the function from which the direct action 
of this system is to be deduced by the equations (3), (9), &c., 

dp 
then j- b t will be the function corresponding to the system re- 



664.] MOVING TKAIL OP IMAGES. 269 

presented by -= 8 1. This quantity, which is the increment of M 

(it 

in the time bt, may be regardejl as itself representing a magnetic 
system. 

If we suppose that at the time t a positive image of the system 

- r b t is formed on the negative side of the sheet, the magnetic 
clt/ 

action at any point on the positive side of the sheet due to this 
image will be equivalent to that due to the currents in the sheet 
excited by the change in M during the first instant after the 
change, and the image will continue to be equivalent to the 
currents in the sheet, if, as soon as it is formed, it begins to move 
in the negative direction of z with the constant velocity E. 

If we suppose that in every successive element of the time an 
image of this kind is formed, and that as soon as it is formed 
it begins to move away from the sheet with velocity E, we shall 
obtain the conception of a trail of images, the last of which is 
in process of formation, while all the rest are moving like a rigid 
body away from the sheet with velocity E. 

663.] If P / denotes any function whatever arising from the 
action of the magnetic system, we may find P, the corresponding 
function arising from the currents in the sheet, by the following 
process, which is merely the symbolical expression for the theory 
of the trail of images. 

Let P T denote the value of P (the function arising from the 
currents in the sheet) at the point (x^y, z + Er], and at the time 
t T, and let P T denote the value of P (the function arising from 
the magnetic system) at the point (#, y, (z-\-E,r}) } and at the 

time*-T. Then dP r ^dP T dP T 

-= = JK-j --- T^J [251 

dr dz dt 

and equation (21) becomes 

dP, d^ 

lh = ^u (26) 

and we obtain by integrating with respect to T from r = to r = oo, 



as the value of the function P, whence we obtain all the properties 
of the current sheet by differentiation, as in equations (3), (9), &c. 

664.] As an example of the process here indicated, let us take 
the case of a single magnetic pole of strength unity, moving with 
uniform velocity in a straight line. 



270 CURRENT-SHEETS. [665. 

Let the coordinates of the pole at the time t be 



The coordinates of the image of the pole formed at the time 
t T are 

= U(*-T), 17 = 0, ^-(c + ttJtf-Tj + tfT), 
and if r is the distance of this image from the point (a?, y, z), 



To obtain the potential due to the trail of images we have to 
calculate d r dr 

7/7 7 7" 

If we write Q 2 = u 2 4- (R- U>) 2 , 

dr 1 



the value of r in this expression being found by making r = 0. 

Differentiating this expression with respect to t, and putting 
t = 0, we obtain the magnetic potential due to the trail of images, 



" ~Q 

By differentiating this expression with respect to x or 2, we 
obtain the components parallel to x or respectively of the mag 
netic force at any point, and by putting x = 0, z = c, and r 2c 
in these expressions, we obtain the following values of the com 
ponents of the force acting on the moving pole itself, 



665.] In these expressions we must remember that the motion 
is supposed to have been going on for an infinite time before the 
time considered. Hence we must not take n> a positive quantity, 
for in that case the pole must have passed through the sheet 
within a finite time. 

If we make u = 0, and ft) negative, X = 0, and 

z- -1: 

"^ * 9 



or the pole as it approaches the sheet is repelled from it. 
If we make n> = 0, we find Q 2 = u 



Y- 








668.] FOKCE ON MOVING POLE. 271 

The component X represents a retarding force acting on the pole 
in the direction opposite to that of its own motion. For a given 
value of R, X is a maximum when u == 1.2772. 

When the sheet is a non-conductor, R = oo and X = 0. 

When the sheet is a perfect conductor, R = and X = 0. 

The component Z represents a repulsion of the pole from the 
sheet. It increases as the velocity increases, and ultimately becomes 

- when the velocity is infinite. It has the same value when 

R is zero. 

666.] When the magnetic pole moves in a curve parallel to the 
sheet, the calculation becomes more complicated, but it is easy to 
see that the effect of the nearest portion of the trail of images 
is to produce a force acting on the pole in the direction opposite 
to that of its motion. The effect of the portion of the trail im 
mediately behind this is of the same kind as that of a magnet 
with its axis parallel to the direction of motion of the pole at 
some time before. Since the nearest pole of this magnet is of the 
same name with the moving pole, the force will consist partly of 
a repulsion, and partly of a force parallel to the former direction 
of motion, but backwards. This may be resolved into a retarding 
force, and a force towards the concave side of the path of the 
moving pole. 

667.] Our investigation does not enable us to solve the case 
in which the system of currents cannot be completely formed, 
on account of a discontinuity or boundary of the conducting 
sheet. 

It is easy to see, however, that if the pole is moving parallel 
to the edge of the sheet, the currents on the side next the edge 
will be enfeebled. Hence the forces due to these currents will 
be less, and there will not only be a smaller retarding force, but, 
since the repulsive force is least on the side next the edge, the pole 
will be attracted towards the edge. 

Theory of Arago^s Rotating Disk. 

668.] Arago discovered* that a magnet placed near a rotating 
metallic disk experiences a force tending to make it follow the 
motion of the disk, although when the disk is at rest there is 
no action between it and the magnet. 

This action of a rotating disk was attributed to a new kind 
* Annales de Chimie et de Physique, 1826. 



272 CURRENT -SHEETS. [668. 

of induced magnetization, till Faraday* explained it by means of 
the electric currents induced in the disk on account of its motion 
through the field of magnetic force. 

To determine the distribution of these induced currents, and 
their effect on the magnet, we might make use of the results already 
found for a conducting sheet at rest acted on by a moving magnet, 
availing ourselves of the method given in Art. 600 for treating the 
electromagnetic equations when referred to moving systems of axes. 
As this case, however, has a special importance, we shall treat it 
in a direct manner, beginning by assuming that the poles of the 
magnet are so far from the edge of the disk that the effect of the 
limitation of the conducting sheet may be neglected. 

Making use of the same notation as in the preceding articles 
(656-667), we find for the components of the electromotive force 
parallel to x and y respectively, 

dy d\js 

(1) 



a- u = y 

dt dx 



dx d\lf 

(TV = y-y; -- f"> j 

dt dy J 

where y is the resolved part of the magnetic force normal to the 
disk. 

If we now express u and v in terms of $, the current-function, 

,._**, (2) 

dx 

and if the disk is rotating about the axis of z with the angular 
velocity o>, dy d x 

1=.*, Jf. * (3) 

Substituting these values in equations (1), we find 

d<t> dty fA\ 

<T !- = ya># -y-, (4) 

dy dx 

d(f) d\jf ,.. 

o- -- = y a) y -- f- - (5) 

dx * J dy 

Multiplying (4) by x and (5) by y } and adding, we obtain 



Multiplying (4) by y and (5) by x, and adding, we obtain 

f d<b dfh^ d\ls d\b 

*(x-r- +y-r-} = -r- -V-r- 
V dx * dy dy J dx 

/ / 

* Exp. Res., 81. 



668.] ARAGO S DISK. 273 

If we now express these equations in terms of r and 0, where 

x r cos d } y = r sin 6, (8) 

they become a ~ = y o> r 2 r -^- > (9) 

du dr 



(10) 



Equation (10) is satisfied if we assume any arbitrary function 
of r and 0, and make d 



* = ar Tr- 
Substituting- these values in equation (9), it becomes 



Dividing by ar 2 , and restoring the coordinates SB and ^, this 
becomes d\ d*x _ /i 4 \ 

^ + d/ -<r y 

This is the fundamental equation of the theory, and expresses the 
relation between the function, x, and the component, y, of the mag 
netic force resolved normal to the disk. 

Let Q be the potential, at any point on the positive side of the 
disk, due to imaginary matter distributed over the disk with the 
surface-density x 

At the positive surface of the disk 






Hence the first member of equation ( 1 4) becomes 
dx 2 dy 2 ~ 2 77 dz 

*S iS 

But since Q satisfies Laplace s equation at all points external 
to the disk, d 2 0. d 2 0. d 2 , 17) 



dz* 



and equation (14) becomes 



j = coy. 

2 TT dz* 

Again, since Q is the potential due to the distribution x> the 

potential due to the distribution $, or -^ , will be . From this 

du clQ 

we obtain for the magnetic potential due to the currents in the disk, 



VOL. II. 



274 CURRENT- SHEETS. [669. 

and for the component of the magnetic force normal to the disk 
due to the currents, 

*._*.. (20) 

71 dz dedz* 

If f2 2 is the magnetic potential due to external magnets, and 

if we write r 

(21) 



the component of the magnetic force normal to the disk due to 
the magnets will be 



We may now write equation (18), remembering that 

y 
<r d*Q 



Integrating twice with respect to z, and writing R for - , 

2i TT 



(24) 



If the values of P and Q are expressed in terms of r, 6, and 
where 7? 

f=*--0, (25) 

0) 

equation (24) becomes, by integration with respect to (, 

(2G) 



669.] The form of this expression shews that the magnetic action 
of the currents in the disk is equivalent to that of a trail of images 
of the magnetic system in the form of a helix. 

If the magnetic system consists of a single magnetic pole of 
strength unity, the helix will lie on the cylinder whose axis is 
that of the disk, and which passes through the magnetic pole. 
The helix will begin at the position of the optical image of the 
pole in the disk. The distance, parallel to the axis between con- 

71 

secutive coils of the helix, will be 2 IT . The magnetic effect of 

CO 

the trail will be the same as if this helix had been magnetized 
everywhere in the direction of a tangent to the cylinder perpen 
dicular to its axis, with an intensity such that the magnetic moment 
of any small portion is numerically equal to the length of its pro 
jection on the disk. 






670.] SPHERICAL SHEET. 275 

The calculation of the effect on the magnetic pole would be 
complicated, but it is easy to see that it will consist of 

(1) A dragging force, parallel to the direction of motion of 
the disk. 

(2) A repulsive force acting from the disk. 

(3) A force towards the axis of the disk. 

When the pole is near the edge of the disk, the third of these 
forces may be overcome by the force towards the edge of the disk, 
indicated in Art. 667. 

All these forces were observed by Arago, and described by him in 
the Annales cle C/iimie for 1826. See also Felici, in Tortolinr s 
Annals, iv, p. 173 (1853), and v. p. 35 ; and E. Jochmann, in Crelle s 
Journal, Ixiii, pp. 158 and 329; and Pogg. Ann. cxxii, p. 214 
(1864). In the latter paper the equations necessary for deter 
mining the induction of the currents on themselves are given, but 
this part of the action is omitted in the subsequent calculation of 
results. The method of images given here was published in the 
Proceedings of the Eoyal Society for Feb. 15, 1872. 

Spherical Current- Sheet. 

670.] Let $ be the current-function at any point Q of a spherical 
current-sheet, and let P be the po 
tential at a given point, due to a 
sheet of imaginary matter distributed 
over the sphere with surface-density 
<p, it is required to find the magnetic 
potential and the vector-potential of 
the current-sheet in terms of P. 

Let a denote the radius of the 
sphere, r the distance of the given 
point from the centre, and p the 

reciprocal of the distance of the given point from the point Q on 
the sphere at which the current-function is (p. 

The action of the current-sheet at any point not in its substance 
is identical with that of a magnetic shell whose strength at any 
point is numerically equal to the current-function. 

The mutual potential of the magnetic shell and a unit pole placed 
at the point P is, by Art. 410, 



T 2 




276 CURllENT- SHEETS. 

Since p is a homogeneous function of the degree 1 mr and a, 

dp dp 
a-/- +r-f = p, 

da dr 



Since r and a are constant during the surface-integration, 



But if P is the potential due to a sheet of imaginary matter 
of surface-density $, 



and 12, the magnetic potential of the current-sheet, may be expressed 
in terms of P in the form 

a= _l-i(P,). 

a dr v 

671.] We may determine F, the ^-component of the vector- 
potential, from the expression given in Art. 416, 



where f , ry, f are the coordinates of the element dS, and I, m, n are 
the direction-cosines of the normal. 

Since the sheet is a sphere, the direction-cosines of the normal are 



dp . N o ^ 
and ^ = (y -,)y = -^, 

sothat _*=--- 



_z dp y dp m 
a dy a dz 

multiplying by (/> dS, and integrating over the surface of the sphere, 
we find z ( ].p y dp 

a dy a dz 



672.] FIELD OF UNIFORM FORCE. 277 

x (IP z dP 

Similarly G = - -= ---- 5 

a dz a ax 



--. 

a dx a dy 

The vector S(, wliose components are F, G, //, is evidently per 
pendicular to the radius vector r, and to the vector whose com- 

dP dP , dP TC 

ponents are -7- > =- . and -=- . It we determine the lines 01 inter- 
dx ay dz 

sections of the spherical surface whose radius is r, with the series of 
equipotential surfaces corresponding 1 to values of P in arithmetical 
progression, these lines will indicate by their direction the direction 
of [, and by their proximity the magnitude of this vector. 
In the language of Quaternions, 

21 = -7 P VP. 

a 

672.] If we assume as the value of P within the sphere 



where Y i is a spherical harmonic of degree i, then outside the sphere 

The current-function < is 

2i+l 1 
= AX A. 

47T tf 

The magnetic potential within the sphere is 



and outside & = i - A ( - ) Y, . 

a \r 

For example, let it be required to produce, by means of a wire 
coiled into the form of a spherical shell, a uniform magnetic force 
M within the shell. The magnetic potential within the shell is, in 
this case, a solid harmonic of the first degree of the form 

12, Mr cos 0, 
where M is the magnetic force. Hence A = ^ 2 J/, and 

d> = Ma cos 0. 

Sir 

The current-function is therefore proportional to the distance 
from the equatorial plane of the sphere, and therefore the number 
of windings of the wire between any two small circles must be 
proportional to the distance between the planes of these circles. 



278 CURRENT-SHEETS. [673. 

If N is the whole number of windings, and if y is the strength 
of the current in each winding, 

$ = \ Ny cos 0. 
Hence the magnetic force within the coil is 

47T Ny 

M = -- - 
3 a 

673.] Let us next find the method of coiling the wire in order 
to produce within the sphere a magnetic potential of the form of a 
solid zonal harmonic of the second degree, 



Here < = -A (f cos 2 



If the whole number of windings is N, the number between the 
pole and the polar distance is ^ j^sin 2 0. 

The windings are closest at latitude 45. At the equator the 
direction of winding changes, and in the other hemisphere the 
windings are in the contrary direction. 

Let y be the strength of the current in the wire, then within 

the shell 4 77 

fl = 

O 

Let us now consider a Conductor in the form of a plane closed 
curve placed anywhere within the shell with its plane perpendicular 
to the axis. To determine its coefficient of induction we have to 

find the surface-integral of -=- over the plane bounded by the 

clz 

curve, putting y = 1. 

Now ^ 



Ar 
and -=- = -= Nz. 

dz 5 a 2 

Hence, if S is the area of the closed curve, its coefficient of in 
duction is o 



If the current in this conductor is y, there will be, by Art. 583, 
a force Z } urging it in the direction of 0, where 
,dM 8 



and, since this is independent of x, y, z, the force is the same in 
whatever part of the shell the circuit is placed. 

674.] The method given by Poisson, and described in Art. 437, 



LINEAR CURRENT -FUNCTION. 279 

may be applied to current-sheets by substituting- for the body 
supposed to be uniformly magnetized in the direction of z with 
intensity 7, a current-sheet having the form of its surface, and for 
which the current-function is Xz. (1) 

The currents in the sheet will be in planes parallel to that of xy, 
and the strength of the current round a slice of thickness dz will be 
Idz. 

The magnetic potential due to this current-sheet at any point 
outside it will be T dV ( . 

~ ~dz 

At any point inside the sheet it will be 

rlV 

a=-4V/*-/^-. (3) 

dz 

The components of the vector-potential are 

F = -I cl ^, G = I~, 11=0. (4) 

dy dx 

These results can be applied to several cases occurring in practice. 

675.] (1) A plane electric circuit of any form. 

Let V be the potential due to a plane sheet of any form of which 
the surface-density is unity, then, if for this sheet we substitute 
either a magnetic shell of strength 7 or an electric current of 
strength I round its boundary, the values of H and of F, G, H will 
be those given above. 

(2) For a solid sphere of radius a, 

V= - when r is greater than a, (5) 

o T 

and 7= ~ (3a 2 r 2 ) when r is less than a. (6) 

o 

Hence, if such a sphere is magnetized parallel to z with intensity 
7, the magnetic potential will be 

H = I -3 z outside the sphere, (7) 

and II = I z inside the sphere. (8) 

3 

If, instead of being magnetized, the sphere is coiled with wire 
in equidistant circles, the total strength of current between two 
small circles whose planes are at unit distance being 7, then outside 
the sphere the value of H is as before, but within the sphere 



This is the case already discussed in Art. 672. 



280 CURRENT- SHEETS. [676. 

(3) The case of an ellipsoid uniformly magnetized parallel to 
a given line has been discussed in Art. 437. 

If the ellipsoid is coiled with wire in parallel and equidistant 
planes, the magnetic force within the ellipsoid will be uniform. 

(4) A Cylindric Magnet or Solenoid. 

676.] If the body is a cylinder having any form of section and 
bounded by planes perpendicular to its generating lines, and 
if F! is the potential at the point (a?, y, z) due to a plane area of 
surface-density unity coinciding with the positive end of the 
solenoid, and V z the potential at the same point due to a plane area 
of surface-density unity coinciding with the negative end, then, if 
the cylinder is uniformly and longitudinally magnetized with in 
tensity unity, the potential at the point (#,y, z) will be 

fi=r 1 -r 2 . (10) 

If the cylinder, instead of being a magnetized body, is uniformly 
lapped with wire, so that there are n windings of wire in unit 
of length, and if a current, y, is made to flow through this wire, 
the magnetic potential outside the solenoid is as before, 

but within the space bounded by the solenoid and its plane ends 

12 = ny(47rz + F! Fg). (12) 

The magnetic potential is discontinuous at the plane ends of the 
solenoid, but the magnetic force is continuous. 

If r lt r 2t the distances of the centres of inertia of the positive 
and negative plane end respectively from the point (a?, y, z), are 
very great compared with the transverse dimensions of the solenoid, 
we may write ^_ A v _ A 

where A is the area of either section. 

The magnetic force outside the solenoid is therefore very small, 
and the force inside the solenoid approximates to a force parallel to 
the axis in the positive direction and equal to 4 it n y. 

If the section of the solenoid is a circle of radius a, the values of 
F! and Fg may be expressed in the series of spherical harmonics 
given in Thomson and Tait s Natural Philosophy, Art. 546, Ex. II., 

V=2-n\ rQ l + a + ^ Q 2 ^<g 4 -f 1 1 3 ^Q 6 + & 

when r>a. (15) 



6;7-] SOLENOID. 281 

In these expressions r is the distance of the point (as, y, z) from 
the centre of one of the circular ends of the solenoid, and the zonal 
harmonics, Q l , Q 2 , &c., are those corresponding to the angle 6 which 
r makes with the axis of the cylinder. 

The first of these expressions is discontinuous when 6 = , but 

2 

we must remember that within the solenoid we must add to the 
magnetic force deduced from this expression a longitudinal force 
4 TT n y. 

677.] Let us now consider a solenoid so long that in the part 
of space which we consider, the terms depending on the distance 
from the ends may be neglected. 

The magnetic induction through any closed curve drawn within 
the solenoid is 4-nny A , where A is the area of the projection of 
the curve on a plane normal to the axis of the solenoid. 

If the closed curve is outside the solenoid, then, if it encloses the 
solenoid, the magnetic induction through it is 4 TT n y A, where A is 
the area of the section of the solenoid. If the closed curve does not 
surround the solenoid, the magnetic induction through it is zero. 

If a wire be wound ri times round the solenoid, the coefficient of 
induction between it and the solenoid is 

M 47rnn A. (16) 

By supposing these windings to coincide with n windings of the 
solenoid, we find that the coefficient of self-induction of unit of 
length of the solenoid, taken at a sufficient distance from its ex 
tremities, is L 4 Tin 2 A. (17) 

Near the ends of a solenoid we must take into account the terms 
depending on the imaginary distribution of magnetism on the plane 
ends of the solenoid. The effect of these terms is to make the co 
efficient of induction between the solenoid and a circuit which sur 
rounds it less than the value 4^nA } which it has when the circuit 
surrounds a very long solenoid at a great distance from either end. 

Let us take the case of two circular and coaxal solenoids of the 
same length L Let the radius of the outer solenoid be c 19 and let 
it be wound with wire so as to have % windings in unit of length. 
Let the radius of the inner solenoid be c 2) and let the number of 
windings in unit of length be n 2 , then the coefficient of induction 
between the solenoids, neglecting the effect of the ends, is 

M=G ff , (18) 

where G = 4 TTW, (19) 

and g = TT e ln z . (20) 



282 CURRENT-SHEETS. [678. 

678.] To determine the effect of the positive end of the solenoids 
we must calculate the coefficient of induction on the outer solenoid 
due to the circular disk which forms the end of the inner solenoid. 
For this purpose we take the second expression for V, as given 
in equation (15), and differentiate it with respect to r. This gives 
the magnetic force in the direction of the radius. We then multiply 
this expression by 2 TT r 2 dp, and integrate it with respect to ju, from 

pi = to jit = . _ - . This gives the coefficient of induction 

V ^ 2 + C l 2 

with respect to a single winding of the outer solenoid at a distance 
z from the positive end. We then multiply this by dz, and 
integrate with respect to z from z = I to z = 0. Finally, we 
multiply the result by % n. 2 , and so find the effect of one of the 
ends in diminishing the coefficient of induction. 

We thus find for the value of the coefficient of mutual induction 
between the two cylinders, 

M = 7i 2 n 1 n z c 2 2 (l2c 1 ci), (21) 



where r is put, for brevity, for \// 2 + c. 

It appears from this, that in calculating the mutual induction of 
two coaxal solenoids, we must use in the expression (20) instead of 
the true length I the corrected length I 2 c^ a, in which a portion 
equal to ac^ is supposed to be cut off at each end. When the 
solenoid is very long compared with its external radius, 



(23) 
i \ 

679.] When a solenoid consists of a number of layers of wire of 
such a diameter that there are n layers in unit of length, the 
number of layers in the thickness dr is n dr, and we have 







=4 Trfn*dr, and g = TT l\ n 2 r 2 dr. (24) 

If the thickness of the wire is constant, and if the induction take 
place between an external coil whose outer and inner radii are x and 
y respectively, and an inner coil whose outer and inner radii are 
y and z, then, neglecting the effect of the ends, 

Gg = $**ln*n*(x-y)(y*-z*). (25) 



68o.] INDUCTION COIL. 283 

That this may be a maximum, x and z being given, and y 
variable, z* , , 

* = *?-*;* ( 2G ) 

J 

This equation gives the best relation between the depths of the 
primary and secondary coil for an induction-machine without an 
iron core. 

If there is an iron core of radius z, then G remains as before, but 

g = TT ifn 2 (r 2 + 4 TT K z 2 ) dr, (27) 

-*)- (28) 



If y is given, the value of z which gives the maximum value of g is 

187TK ,, 

z = 4 v - I " J 

3y i87TK+l 

When, as in the case of iron, K is a large number, z = f y, nearly. 

If we now make x constant, and y and z variable, we obtain the 
maximum value of Gg when 

x \y\ z : : 4 : 3 : 2. (30) 

The coefficient of self-induction of a long solenoid whose outer 
and inner radii are x and y> and having a long iron core whose 
radius is z, is 

L = %7T 2 ln*(v-y) 2 (x 2 + 2xy + 3y 2 + 24;TTKZ 2 ). (31) 

680.] We have hitherto supposed the wire to be of uniform 
thickness. We shall now determine the law according to which 
the thickness must vary in the different layers in order that, for 
a given value of the resistance of the primary or the secondary coil, 
the value of the coefficient of mutual induction may be a maximum. 

Let the resistance of unit of length of a wire, such that n windings 
occupy unit of length of the solenoid, be p n 2 . 

The resistance of the whole solenoid is 

E = 2iilJ*rdr. (32) 

The condition that, with a given value of R, G may be a maximum 

. dG n dR . . 
is - T - =C~r- , where C is some constant. 
* _ dr l 

This gives n 2 proportional to - , or the diameter of the wire of 

the exterior coil must be proportional to the square root of the 
radius. 

In order that, for a given value of R, g may be a maximum 

*.0, + lS.. (33) 



284 CURRENT -SHEETS. [68 1. 

Hence, if there is no iron core, the diameter of the wire of the 
interior coil should be inversely as the square root of the radius, 
but if there is a core of iron having a high capacity for magneti 
zation, the diameter of the wire should be more nearly directly 
proportional to the square root of the radius of the layer. 

An Endless Solenoid. 

681.] If a solid be generated by the revolution of a plane area A 
about an axis in its own plane, not cutting it, it will have the form 
of a ring. If this ring be coiled with wire, so that the windings 
of the coil are in planes passing through the axis of the ring, then, 
if n is the whole number of windings, the current-function of the 

layer of wire is $ = n y 0, where 6 is the angle of azimuth about 

the axis of the ring. 

If 12, is the magnetic potential inside the ring and 12 that out 
side, then 12-12 = 47T( + <?= 2ny0 + C. 
Outside the ring 12 must satisfy Laplace s equation, and must 
vanish at an infinite distance. From the nature of the problem 
it must be a function of only. The only value of 12 which fulfils 
these conditions is zero. Hence 

12 = 0, 12 = 2ny8+C. 

The magnetic force at any point within the ring is perpendicular 

to the plane passing through the axis, and is equal to 2ny- 

where r is the distance from the axis. Outside the ring there is 
no magnetic force. 

If the form of a closed curve be given by the coordinates z, r, 
and of its tracing point as functions of s, its length from a fixed 
point, the magnetic induction through the closed curve is 

[ z dr 

2ny - -j- ds 
V r ds 

taken round the curve, provided the curve is wholly inside the ring. 
If the curve lies wholly without the ring, but embraces it, the 
magnetic induction through it is 

/" z dr _ , 
2 n y / -=-, ds = 2 n y a, 

J Q T (IS 

where the accented coordinates refer not to the closed curve, but to 
a single winding of the solenoid. 

The magnetic induction through any closed curve embracing the 



68 1.] ENDLESS SOLENOID. 285 

ring 1 is therefore the same, and equal to 2 n y a, where a is the linear 

/* z f dr 
-Tjds . If the closed curve does not embrace the 
/ ds 

ring, the magnetic induction through it is zero. 

Let a second wire be coiled in any manner round the ring, not 
necessarily in contact with it, so as to embrace it n f times. The 
induction through this wire is 2 n ri y a, and therefore M, the 
coefficient of induction of the one coil on the other, is M = 2 n ri a. 

Since this is quite independent of the particular form or position 
of the second wire, the wires, if traversed by electric currents, will 
experience no mechanical force acting between them. By making 
the second wire coincide with the first, we obtain for the coefficient 
of self-induction of the ring-coil 

L = 2 n 2 a. 



CHAPTER XIII. 



PARALLEL CURRENTS. 



Cylindrical Conductors. 

682.] IN a very important class of electrical arrangements the 
current is conducted through round wires of nearly uniform section, 
and either straight, or such that the radius of curvature of the axis 
of the wire is very great compared with the radius of the transverse 
section of the wire. In order to be prepared to deal mathematically 
with such arrangements, we shall begin with the case in which the 
circuit consists of two very long parallel conductors, with two pieces 
joining their ends, and we shall confine our attention to a part of 
the circuit which is so far from the ends of the conductors that the 
fact of their not being infinitely long does not introduce any 
sensible change in the distribution of force. 

We shall take the axis of z parallel to the direction of the con 
ductors, then, from the symmetry of the arrangements in the part 
of the field considered, everything will depend on //, the component 
of the vector-potential parallel to z. 

The components of magnetic induction become, by equations (A), 

m 



dH 



c 0. 

For the sake of generality we shall suppose the coefficient of 
magnetic induction to be p, so that a = /a a, b /u, /3, where a and (3 
are the components of the magnetic force. 

The equations (E) of electric currents, Art. GO 7, give 

u = 0, v = 0. 4 KW = -^ (3) 

dx dy 



683.] STRAIGHT WIRE. 287 

683.] If the current is a function of r, the distance from the axis 
of Zj and if we write 

os = r cos 0, and y = r sin 0, (4) 

and {3 for the magnetic force, in the direction in which 6 is measured 
perpendicular to the plane through the axis of z, we have 

4 =f + 10 = 1*08,). (5) 

dr r r dr ^ 

If C is the whole current flowing through a section bounded by 
a circle in the plane gey, whose centre is the origin and whose 

radius is r, /> 

<?= / 2trrwdr = %(3r. (6) 

JQ 

It appears, therefore, that the magnetic force at a given point 
due to a current arranged in cylindrical strata, whose common axis 
is the axis of z, depends only on the total strength of the current 
flowing through the strata which lie between the given point and 
the axis, and not on the distribution of the current among the 
different cylindrical strata. 

For instance, let the conductor be a uniform wire of radius a, 
and let the total current through it be C, then, if the current is 
uniformly distributed through all parts of the section, w will be 
constant, and C=7rwa 2 . (7) 

The current flowing through a circular section of radius r, r being 
less than a, is C = -nwr 2 . Hence at any point within the wire, 



C 

Outside the wire 8 = 2 . (9) 

f 

In the substance of the wire there is no magnetic potential, for 
within a conductor carrying an electric current the magnetic force 
does not fulfil the condition of having a potential. 

Outside the wire the magnetic potential is 

l = 2C0. (10) 

Let us suppose that instead of a wire the conductor is a metal 
tube whose external and internal radii are a-j, and a 2 , then, if (7 is 
the current through the tubular conductor, 

C = 7Tw(a l 2 -a. 2 2 ). (11) 

The magnetic force within the tube is zero. In the metal of the 
tube, where ; is between a-^ and a 2 , 

P= 2^-^-- 2 r-- 2 - 2 , (12) 



288 PARALLEL CURRENTS. [684. 

and outside the tube, c 

/3=2-, (13) 

the same as when the current flows through a solid wire. 

684.] The magnetic induction at any point is b = p (3, and since, 
by equation (2), fi - _ ^ (14) 

dr 



H^-jppdr. (15) 

The value of // outside the tube is 

A 2iJL Clogr, (16) 

where JU Q is the value of /x in the space outside the tube, and A is a 
constant, the value of which depends on the position of the return 
current. 

In the substance of the tube, 

a \ ~~ a -2 a i 

In the space within the tube H is constant, and 



#=^-2 Mo Clog 1 + M e(l + -lo gr ^). (18) 

U-^ U>2 i*^ 

685.] Let the circuit be completed by a return current, flowing 
in a tube or wire parallel to the first, the axes of the two currents 
being at a distance b. To determine the kinetic energy of the 
system we have to calculate the integral 

T = \ fjJHw dx cly dz. (19) 

If we confine our attention to that part of the system which lies 
between two planes perpendicular to the axes of the conductors, and 
distant I from each other, the expression becomes 



T= \l Hivdxdy. (20) 

If we distinguish by an accent the quantities belonging to the 
return current, we may write this 

^-!-=jJHw dx dy +jJH wdxcly + jJHwdxdy+jJll w dx dy . (21) 

Since the action of the current on any point outside the tube is 
the same as if the same current had been concentrated at the axis 
of the tube, the mean value of H for the section of the return 
current is A 2^C log I, and the mean value of H for the section 
of the positive current is A 2 /u G Y/ log b. 



687.] LONGITUDINAL TENSION. 289 

Hence, in the expression for T, the first two terms may be written 
AC -2n () CC log6 ) and A C-2 n CC logl>. 

Integrating the two latter terms in the ordinary way, and adding 
the results, remembering that C+ C = 0, we obtain the value of 
the kinetic energy T. Writing this \LC 2 , where L is the co 
efficient of self-induction of the system of two conductors, we find 
as the value of L for unit of length of the system 
L 



If the conductors are solid wires, a. 2 and a< are zero, and 

T /,2 

(23) 



a i a i 

It is only in the case of iron wires that we need take account of 
the magnetic induction in calculating their self-induction. In 
other cases we may make /x , /LI, and // all equal to unity. The 
smaller the radii of the wires, and the greater the distance between 
them, the greater is the self-induction. 

To find the Repulsion, X, between the Two Portions of Wire. 
686.] By Art. 580 we obtain for the force tending to increase b, 

*-<". 

= 2 MO |C">, (24) 

which agrees with Ampere s formula, when JU Q = 1, as in air. 

687.] If the length of the wires is great compared with the 
distance between them, we may use the coefficient of self-induction 
to determine the tension of the wires arising from the action of the 
current. 

If Z is this tension, 



In one of Ampere s experiments the parallel conductors consist 
of two troughs of mercury connected with each other by a floating 
bridge of wire. When a current is made to enter at the extremity 
of one of the troughs, to flow along it till it reaches one extremity 

VOL. II. U 



290 PAEALLEL CURRENTS. [688. 

of the floating wire, to pass into the other trough through the 
floating bridge, and so to return along the second trough, the 
floating bridge moves along the troughs so as to lengthen the part 
of the mercury traversed by the current. 

Professor Tait has simplified the electrical conditions of this 
experiment by substituting for the wire a floating siphon of glass 
filled with mercury, so that the current flows in mercury through 
out its course. 




Fig. 40. 

This experiment is sometimes adduced to prove that two elements 
of a current in the same straight line repel one another, and thus 
to shew that Ampere s formula, which indicates such a repulsion 
of collinear elements, is more correct than that of Grassmann, which 
gives no action between two elements in the same straight line ; 
Art. 526. 

But it is manifest that since the formulae both of Ampere and of 
Grassmann give the same results for closed circuits, and since we 
have in the experiment only a closed circuit, no result of the 
experiment can favour one more than the other of these theories. 

In fact, both formulae lead to the very same value of the 
repulsion as that already given, in which it appears that b, the 
distance between the parallel conductors is an important element. 

When the length of the conductors is not very great compared 
with their distance apart, the form of the value of L becomes 
somewhat more complicated. 

688.] As the distance between the conductors is diminished, the 
value of L diminishes. The limit to this diminution is when the 
wires are in contact, or when b = a l + a 2 . In this case 

fiV (26) 



689.] MINIMUM SELF-INDUCTION. 291 

This is a minimum when a^ = a 2t and then 
= 2 /(log 4 + 1), 
= 2^(1.8863), 

= 3.7726^. (27) 

This is the smallest value of the self-induction of a round wire 
doubled on itself, the whole length of the wire being 2 I. 

Since the two parts of the wire must be insulated from each 
other, the self-induction can never actually reach this limiting 
value. By using broad flat strips of metal instead of round wires 
the self-induction may be diminished indefinitely. 

On the Electromotive Force required to produce a Current of Varying 
Intensity along a Cylindrical Conductor. 

689.] When the current in a wire is of varying intensity, the 
electromotive force arising from the induction of the current on 
itself is different in different parts of the section of the wire, being 
in general a function of the distance from the axis of the wire 
as well as of the time. If we suppose the cylindrical conductor 
to consist of a bundle of wires all forming part of the same circuit, 
so that the current is compelled to be of uniform strength in every 
part of the section of the bundle, the method of calculation which 
we have hitherto used would be strictly applicable. If, however, 
we consider the cylindrical conductor as a solid mass in which 
electric currents are free to flow in obedience to electromotive force, 
the intensity of the current will not be the same at different 
distances from the axis of the cylinder, and the electromotive forces 
themselves will depend on the distribution of the current in the 
different cylindric strata of the wire. 

The vector-potential //, the density of the current w, and the 
electromotive force at any point, must be considered as functions of 
the time and of the distance from the axis of the wire. 

The total current, C, through the section of the wire, and the total 
electromotive force, JE, acting round the circuit, are to be regarded 
as the variables, the relation between which we have to find. 

Let us assume as the value of H, 

H= S+To + T^+bc. + T.r**, (1) 

where S, T , T lf &c. are functions of the time. 

Then, from the equation 

d 2 H , 1 dH f . 

-J-H- H -=- = 47TW, (2) 

dr 2 r dr 

we find -TIW = T l + &c + n*T n r Zn ~ 2 . (3) 

U 2 



292 PARALLEL CURRENTS. [690. 

If p denotes the specific resistance of the substance per unit of 
volume, the electromotive force at any point is p w, and this may be 
expressed in terms of the electric potential and the vector potential 
H by equations (B), Art. 598, 

dV dll , A . 

<> w = -^-w 

d3> dS dT Q clT^ dT n 

-? w = T* + Tt+-W + -W T +^ + ^? T (5) 

Comparing the coefficients of like powers of r in equations 

<s) nd(5) 






Hence we may write -=- = , (9) 

T _,dT _ 1 d T 

J 2 ^--pTt>- /B "?(iFaF 

690.] To find the total current (7, we must integrate w over the 
section of the wire whose radius is a, 

r a 

C=27T wrdr. (11) 

^o 

Substituting the value of itw from equation (3), we obtain 



(12) 

The value of H at any point outside the wire depends only on 
the total current C, and not on .the mode in which it is distributed 
within the wire. Hence we may assume that the value of H at the 
surface of the wire is A C, where A is a constant to be determined 
by calculation from the general form of the circuit. Putting H=AC 
when r = a, we obtain 

2n - (13) 



If we now write - = a, a is the value of the conductivity of 

P 
unit of length of the wire, and we have 



(15) 



690.] VARIABLE CURRENT. 293 

Eliminating T from these two equations, we find 
.dC dS, . dC 



. = o. (16) 

If I is the whole length of the circuit, R its resistance, and E the 
electromotive force due to other causes than the induction of the 
current on itself, dS E I 

Tl=-J a = K 
dC PcPC P fPC 



The first term, RC> of the right-hand member of this equation 
expresses the electromotive force required to overcome the resist 
ance according to Ohm s law. 

The second term, l(A + \)-;- , expresses the electromotive force 

dt 

which would be employed in increasing the electrokinetic momentum 
of the circuit, on the hypothesis that the current is of uniform 
strength at every point of the section of the wire. 

The remaining terms express the correction of this value, arising 
from the fact that the current is not of uniform strength at different 
distances from the axis of the wire. The actual system of currents 
has a greater degree of freedom than the hypothetical system, 
in which the current is constrained to be of uniform strength 
throughout the section. Hence the electromotive force required 
to produce a rapid change in the strength of the current is some 
what less than it would be on this hypothesis. 

The relation between the time-integral of the electromotive force 
and the time-integral of the current is 

(19) 

If the current before the beginning of the time has a constant 
value C 0) and if during the time it rises to the value C L , and re 
mains constant at that value, then the terms involving the differ 
ential coefficients of C vanish at both limits, and 

,\ (20) 

the same value of the electromotive impulse as if the current had 
been uniform throughout the wire. 



294 PARALLEL CURRENTS. [691. 

On the Geometrical Mean Distance of Two Figures in a Plane.* 

691.] In calculating the electromagnetic action of a current 
flowing in a straight conductor of any given section on the current 
in a parallel conductor whose section is also given, we have to find 
the integral 



where doc dy is an element of the area of the first section, dx dy an 
element of the second section, and r the distance between these 
elements, the integration being extended first over every element 
of the first section, and then over every element of the second. 
If we now determine a line R, such that this integral is equal to 



where A 1 and A 2 are the areas of the two sections, the length of R 
will be the same whatever unit of length we adopt, and whatever 
system of logarithms we use. If we suppose the sections divided 
into elements of equal size, then the logarithm of R, multiplied by 
the number of pairs of elements, will be equal to the sum of the 
logarithms of the distances of all the pairs of elements. Here R 
may be considered as the geometrical mean of all the distances 
between pairs of elements. It is evident that the value of R must 
be intermediate between the greatest and the least values of r. 

If R A and R B are the geometric mean distances of two figures, 
A and JB, from a third, C } and if RA+B is that of the sum of the two 
figures from C, then 

(A + B) log R A+B =A log R A + B log R B . 

By means of this relation we can determine R for a compound 
figure when we know R for the parts of the figure. 

692.] EXAMPLES. 

(1) Let R be the mean distance from the point to the line 
AB. Let OP be perpendicular to AB, then 

AB (log R + 1) = AP log OA + PB log OB+ OP AOB. 



i / 

Fig. 41. 
* Trans. R. S. Edin., 1871-2. 



692.] 



GEOMETRIC MEAN DISTANCE. 



295 



(2) For two lines (Fig. 42) of lengths a and b drawn perpendicu 
lar to the extremities of a line of length c and on the same side of it. 
(2 log 72 +3) = (c 2 - (a-b} 2 ) log+/c 2 + (a- &)* + c 2 log c 

4- (a 2 c 2 ) log \/a 2 + c 2 4- (b 2 c 2 ) log \/b 2 4- c 2 

/ z\ * i a ^ ~u - -b 

c(a o) tan" 1 



Fig. 42. 

(3) For two lines, PQ and RS (Fig. 43), whose directions inter 
sect at 0. 

PQ.RS(2logR+3) = logPR(20P.ORsin 2 0-PR 2 cosO) 
+ logQS(20Q.OSsin 2 0-QS 2 cosO) 
- log PS (2 OP. OS sin 2 - PS 2 cos 0) 



-sinO {OP 2 . SPR- OQ 2 . S QR+OR 2 . PltQ-OS 2 . PSQ}. 




Fig. 43. 

(4) For a point and a rectangle ABCD (Fig. 44). Let OP, 
OQ, OR, OS, be perpendiculars on the sides, then 
AB.AD (2 log 72+ 3) = 2.0P.OQ log OA + 2 .OQ. OR log OB 
+ 2. OR. OS log OC + 2.0S.OP logOD 



Fig. 44. 



296 PARALLEL CURRENTS. [693. 

(5) It is not necessary that the two figures should be different, for 
we may find the geometric mean of the distances between every pair 
of points in the same figure. Thus, for a straight line of length 0, 

log 72 = log af, 
or E = ae~%, 

R = 0.223130. 

(6) For a rectangle whose sides are a and d, 

} gR = logvV+^-iJiog /y/i + ^-^g V 1 + & 

+ ietan-i*+i-tan-i-. 

o a a b 

When the rectangle is a square, whose side is 0, 
log 5 = Iog0 + i log 2 + | -ff, 
R = 0.447050. 

(7) The geometric mean distance of a point from a circular line 
is equal to the greater of the two quantities, its distance from the 
centre of the circle, and the radius of the circle. 

(8) Hence the geometric mean distance of any figure from a 
ring bounded by two concentric circles is equal to its geometric 
mean distance from the centre if it is entirely outside the ring, but 
if it is entirely within the ring 



a l a 2 

where 0j and 2 are the outer and inner radii of the ring. R is 
in this case independent of the form of the figure within the ring. 

(9) The geometric mean distance of all pairs of points in the 
ring is found from the equation 

log R = ^0! 2 J^ log ^ 4- J *l ~\ . 

For a circular area of radius 0, this becomes 

log R = Iog0-i, 
or R = ae~*, 

R = 0.77880. 
For a circular line it becomes 

693.] In calculating the coefficient of self-induction of a coil of 
uniform section, the radius of curvature being great compared with 



693-] 



SELF-INDUCTION OF A COIL. 



297 



the dimensions of the transverse section, we first determine the 
geometric mean of the distances of every pair of points of the 
section by the method already described, and then we calculate the 
coefficient of mutual induction between two linear conductors of 
the given form, placed at this distance apart. 

This will be the coefficient of self-induction when the total cur 
rent in the coil is unity, and the current is uniform at all points of 
the section. 

But if there are n windings in the coil we must multiply the 
coefficient already obtained by n 2 , and thus we shall obtain the 
coefficient of self-induction on the supposition that the windings of 
the conducting wire fill the whole section of the coil. 

But the wire is cylindric, and is covered with insulating material, 
so that the current, instead of being uniformly distributed over the 
section, is concentrated in certain parts of it, and this increases the 
coefficient of self-induction. Besides this, the currents in the 
neighbouring wires have not the same action on the current in a 
given wire as a uniformly distributed current. 

The corrections arising from these considerations may be de 
termined by the method of the geometric mean distance. They 
are proportional to the length of the whole wire of the coil, and 
may be expressed as numerical quantities, by which we must 
multiply the length of the wire in order to obtain the correction 
of the coefficient of self-induction. 

Let the diameter of the wire be d. It is 
covered with insulating material, and wound 
into a coil. We shall suppose that the sections 
of the wires are in square order, as in Fig. 45, 
and that the distance between the axis of each 
wire and that of the next is D, whether in 
the direction of the breadth or the depth of 
the coil. D is evidently greater than d. 

We have first to determine th excess of 
self-induction of unit of length of a cylindric wire of diameter d 
over that of unit of length of a square wire of side D, or 
, R for the square 
Og * R for the circle 



o 


o 


o 


o 


o 


o 


o 


o 


o 



Fig. 45. 



D 



= 2 (log-T + 0.1380606) 



298 PARAkCEL CURRENTS. [693. 

The inductive action of the eight nearest round wires on the wire 
under consideration is less than that of the corresponding eight 
square wires on the square wire in the middle by 2x(. 01971). 

The corrections for the wires at a greater distance may be neg 
lected, and the total correction may be written 

2(log e -=- + 0.11835). 

The final value of the self-induction is therefore 
L n 2 M+ 2/(log e -j + 0.11835), 

where n is the number of windings, and I the length of the wire, 
M the mutual induction of two circuits of the form of the mean 
wire of the coil placed at a distance R from each other, where R is 
the mean geometric distance between pairs of points of the section. 
D is the distance between consecutive wires, and d the diameter 
of the wire. 



CHAPTER XIV. 



CIRCULAR CURRENTS. 



Magnetic Potential due to a Circular Current. 

694.] THE magnetic potential at a given point, due to a circuit 
carrying a unit current, is numerically equal to the solid angle sub 
tended by the circuit at that point ; see Arts. 409, 485. 

When the circuit is circular, the solid angle is that of a cone 
of the second degree, which, when the given point is on the axis 
of the circle, becomes a right cone. When the point is not on 
the axis, the cone is an elliptic cone, and its solid angle is 
numerically equal to the area of the spherical ellipse which it traces 
on a sphere whose radius is unity. 

This area can be expressed in finite terms by means of elliptic 
integrals of the third kind. We shall find it more convenient to 
expand it in the form of an infinite series of spherical harmonics, for 
the facility with which mathematical operations may be performed 
on the general term of such a series z 

more than counterbalances the trouble 
of calculating a number of terms suffi 
cient to ensure practical accuracy. 

For the sake of generality we shall 
assume the origin at any point on the 
axis of the circle, that is to say, on 
the line through the centre perpen 
dicular to the plane of the circle. 

Let (Fig. 46) be the centre of the 
circle, C the point on the axis which 
we assume as origin, H a point on the 
circle. 

Describe a sphere with C as centre, 
and CH as radius. The circle will lie 
on this sphere, and will form a small circle of the sphere of 
angular radius a. 




Fig. 46. 



300 CIRCULAR CURRENTS. [694. 

Let CH = c, 

OC = b c cos a, 
OH= a = c sin a. 

Let A be the pole of the sphere, and Z any point on the axis, and 
let CZ=z. 

Let R be any point in space, and let CR = r, and ACR = 6. 

Let P be the point when CR cuts the sphere. 

The magnetic potential due to the circular current is equal to 
that due to a magnetic shell of strength unity bounded by the 
current. As the form of the surface of the shell is indifferent, 
provided it is bounded by the circle, we may suppose it to coincide 
with the surface of the sphere. 

We have shewn in Art. 670 that if P is the potential due to a 
stratum of matter of surface-density unity, spread over the surface 
of the sphere within the small circle, the potential due to a mag 
netic shell of strength unity and bounded by the same circle is 

* = ii(rP). 

c dr ^ 

We have in the first place, therefore, to find P. 

Let the given point be on the axis of the circle at Z, then the 
part of the potential at Z due to an element dS of the spherical 
surface at P is $$ 

~ZP 

This may be expanded in one of the two series of spherical har 
monics, r], 



or ++&c. + < i + &c 



.j> 

the first series being convergent when z is less than c, and the 
second when z is greater than c. 

Writing dS = c 2 dp dfa 

and integrating with respect to < between the limits and 2?r, 
and with respect to //, between the limits cos a and 1, we find 



or P=2vQ dp + to>.+ r Q i dp. (O 

By the characteristic equation of Q i} 



695-] SOLID ANGLE SUBTENDED BY A CIRCLE. 301 



Hence ^ = . (2) 

J^ ^ ^(^ + l) dp 

This expression fails when i = 0, but since Q = 1, 



As the function -~ occurs in every part of this investigation we 
d //. 

shall denote it by the abbreviated symbol Q/. The values of Q/ 
corresponding to several values of i are given in Art. 698. 

We are now able to write down the value of P for any point R, 
whether on the axis or not, by substituting r for z, and multiplying 
each term by the zonal harmonic of 6 of the same order. For 
P must be capable of expansion in a series of zonal harmonics of 
with proper coefficients. When = each of the zonal harmonics 
becomes equal to unity, and the point E lies on the axis. Hence 
the coefficients are the terms of the expansion of P for a point on 
the axis. We thus obtain the two series 

(4) 



(4 ) 

695.] We may now find o>, the magnetic potential of the circuit, 
by the method of Art. 670, from the equation 



We thus obtain the two series 

(6) 



! C 2 i 

t? & ()& W + &c - + J+ 

The series (6) is convergent for all values of r less than c, and the 
series (6 r ) is convergent for all values of r greater than <?. At the 
surface of the sphere, where r c, the two series give the same 
value for <o when Q is greater than a, that is, for points not 
occupied by the magnetic shell, but when 6 is less than a, that is, 
at points on the magnetic shell, 

0/= CO+47T. (7) 

If we assume 0, the centre of the circle, as the origin of co 
ordinates, we must put a = - , and the series become 



302 



CIRCULAR CURRENTS. 

1 (n a _ 1 

- 



[696. 
. (8) 



where the orders of all the harmonics are odd *. 

0# the Potential Energy of two Circular Currents. 

696.] Let us begin by supposing the two magnetic shells which 
are equivalent to the currents to be portions of two concentric spheres, 
their radii being c^ and <? 2 , of which c^ is the greater (Fig. 47). 
Let us also suppose that the axes of the two shells coincide, and 

that QJ is the angle subtended by 
the radius of the first shell, and ez 2 
the angle subtended by the radius 
of the second shell at the centre C. 

Let o^ be the potential due to the 
first shell at any point within it, then 
the work required to carry the second 
shell to an infinite distance is the 
value of the surface-integral 

r/wco, 

JJ dr 
Hence 




Fig. 47. 
extended over the second shell. 






4** sin* a l( y< 



or, substituting the value of the integrals from equation (2), Art. 694, 



* The value of the solid angle subtended by a circle may be obtained in a more 
direct way as follows. 

The solid angle subtended by the circle at the point Z in the axis is easily shewn 



i-* (a)) + &c. 

C 



Expanding this expression in spherical harmonics, we find 

(cos a-l) + (Q, ^cosa-Qo (a))- +&c. + (<& (a) coso- 

C 



for the expansions of cw for points on the axis for which z is less than c or greater 
than c respectively. Remembering the equations (42) and (43) of Art. 132 (vol. i. 
p. 165), the coefficients in these equations are evidently the same as those we have 
now obtained in a more convenient form for computation. 



698.] POTENTIAL OF TWO CIRCLES. 303 

697.] Let us next suppose that the axis of one of the shells is 
turned about C as a centre,, so that it now makes an angle with 
the axis of the other shell (Fig. 48). We have only to introduce 
the zonal harmonics of into this expression for M, and we find for 
the more general value of M, 



This is the value of the potential energy due to the mutual 
action of two circular currents of unit strength, placed so that 
the normals through the centres of the circles meet in a point C 
in an angle 0, the distances of the circumferences of the circles from 
the point C being <? x and c 2 , of which c is the greater. 

If any displacement dx alters the value 
of M, then the force acting in the direc 
tion of the displacement is X = -= 

For instance, if the axis of one of the 
shells is free to turn about the point C, 
so as to cause to vary, then the moment 
of the force tending to increase & is 0, 
where _ dM 

Performing the differentiation, and remembering that 




dB 
where (j)/ has the same signification as in the former equations, 

= 4 7T 2 sin 2 a 1 sin 2 a 2 sin c 2 < J $/(%) /(a 2 ) Qi(Q) + &c. 

^ 1 



698.] As t 1 e values of Q{ occur frequently in these calculations 
the following table of values of the first six degrees may be useful. 
In this table /x stands for cos 0, and v for sin 6. 



304 CIRCULAR CURRENTS. [699. 

699.] It is sometimes convenient to express the series for M in 
terms of linear quantities as follows : 

Let a be the radius of the smaller circuit, I the distance of its 
plane from the origin, and c = \/a 2 -\-b 2 . 

Let A, B, and C be the corresponding- quantities for the larger 
circuit. 

The series for M may then be written, 
A 2 

M= 1.2.7T 2 ^0 2 COS0 

C 3 



4- 2.3.7T 2 -y=- a 2 b (cos 2 6- i sin 2 (9) 

+ 3.4.7T 2 A2 ( 2 -* A ^ a 2 (2_1 ^2)( COS 30_ 3 sin 2 fl cog tf) 
-f &C. 

If we make 0=0, the two circles become parallel and on the 
same axis. To determine the attraction between them we may 
differentiate M with respect to b. We thus find 

dM 

w=* 

700.] In calculating the effect of a coil of rectangular section 
we have to integrate the expressions already found with respect 
to A, the radius of the coil, and _Z?, the distance of its plane from 
the origin, and to extend the integration over the breadth and 
depth of the coil. 

In some cases direct integration is the most convenient, but 
there are others in which the following method of approximation 
leads to more useful results. 

Let P be any function of x and ^, and let it be required to find 
the value of P where 



T+i* r+4y 
Pxy = / / Pdxdy. 

J- J - 



In this expression P is the mean value of P within the limits of 
integration. 

Let P be the value of P when x = and y = 0, then, expanding 
P by Taylor s Theorem, 



Integrating this expression between the limits, and dividing the 
result by xy> we obtain as the value of P, 



7QI.J COIL OF EECT ANGULAR SECTION. 305 



In the case of the coil, let the outer and inner radii be A + \ , 
and A \^ respectively, and let the distance of the planes of the 
windings from the origin lie between JB + ^rj and B\TI, then the 
breadth of the coil is r\, and its depth these quantities being 
small compared with A or C. 

In order to calculate the magnetic effect of such a coil we may 
write the successive terms of the series as follows :-^- 






&C., &c. ; 

ft= 2 

= 277^ 



&c., &c. 

The quantities G , G 1 , G 2 , &c. belong to the large coil. The 
value of o> at points for which r is less than C is 

a, = _27T + 2G - ^ r Q l (0)- G^r* Q 2 ((9)-^-&c. 
The quantities g l9 g^ &c. belong to the small coil. The value of 
a/ at points for which r is greater than c is 



The potential of the one coil with respect to the other when the 
total current through the section of each coil is unity is 



To find M by Elliptic Integrals. 
701.] When the distance of the circumferences of the two circles 

VOL. II. * 



306 CIRCULAR CURRENTS. [701. 

is moderate as compared with the radii of the smaller, the series 
already given do not converge rapidly. In every case, however, 
we may find the value of M for two parallel circles by elliptic 
integrals. 

For let b be the length of the line joining the centres of the circles, 
and let this line be perpendicular to the planes of the two circles, 
and let A and a be the radii of the circles, then 

M 



/"/" 

= / / 



the integration being extended round both curves. 
In this case, 

r 2 = A 2 + a 2 + b 2 -2Aacos((j>-(l> ) 

e = $ , ds = 

27T 



M 



/ 
~J 



where c == 

and F and E are complete elliptic integrals to modulus c. 

From this we get, by differentiating with respect to b and re 
membering that c is a function of b, 



c 

If fj and / 2 denote the greatest and least values of r, 
rf =(A + of + V, r* =(A- a) 2 + b 2 , 

4* 

and if an angle y be taken such that cos y = , 



where F y and E y denote the complete elliptic integrals of the first 
and second kind whose modulus is sin y. 

If A a,j cot y = - , and 

i Cb 

-^-= 

The quantity -^- represents the attraction between two parallel 
circular currents, the current in each being unity. 



703.] LINES OF MAGNETIC FOKCE. 307 

Second Expression for M. 
An expression for M, which is sometimes more convenient, is got 

by making ^ = - - , in which case 
r i + r 2 

M = 4 



To draw the Lines of Magnetic Force for a Circular Current. 

702.] The lines of magnetic force are evidently in planes passing 
through the axis of the circle, and in each of these lines the value 
of M is constant. 

Calculate the value of K Q ,-= - = r^ from Legendre s 

(/sine A3in0) 

tables for a sufficient number of values of 0. 

Draw rectangular axes of so and z on the paper, and, with centre 
at the point x = \ a (sin + cosec d), draw a circle with radius 
\ a (cosec sin 0). For all points of this circle the value of e l will 
be sin 0. Hence, for all points of this circle, 



= ^ and A = 



Now A is the value of x for which the value of M was found. 
Hence, if we draw a line for which x = A, it will cut the circle 
in two points having the given value of M. 

Giving M a series of values in arithmetical progression, the 
values of A will be as a series of squares. Drawing therefore a 
series of lines parallel to z y for which x has the values found for A, 
the points where these lines cut the circle will be the points where 
the corresponding lines of force cut the circle. 

If we put m = 4 a a, and M = nm, then 

A x = n 2 K e a. 
We may call n the index of the line of force. 

The forms of these lines are given in Fig. XVIII at the end of 
this volume. They are copied from a drawing given by Sir W. 
Thomson in his paper on Vortex Motion*. 

703.] If the position of a circle having a given axis is regarded 
as defined by 6, the distance of its centre from a fixed point on 
the axis, and , the radius of the circle, then M, the coefficient 
of induction of the circle with respect to any system whatever 

* Trans. R. 8. t Edin., vol. xxv. p. 217 (1869). 
X 2 



308 CIRCULAR CURRENTS. [703. 

of magnets or currents, is subject to the following equation 

d 2 M d 2 M I dM fc x-v 

da 2 db 2 a da 

To prove this, let us consider the number of lines of magnetic 
force cut by the circle when a or b is made to vary. 

(1) Let a become a + ba, b remaining constant. During this 
variation the circle, in expanding, sweeps over an annular surface 
in its own plane whose breadth is 8 a. 

If V is the magnetic potential at any point, and if the axis of y 
be parallel to that of the circle, then the magnetic force perpen- 

dV 

dicular to the plane of the ring- is -7- 

dy 

To find the magnetic induction through the annular surface we 
have to integrate 



where 6 is the angular position of a point on the ring. 

But this quantity represents the variation of M due to the 

variation of #, or -= 8 a. Hence 
da 

dM ^ f 2n a dT d0 (2] 



(2) Let 6 become 6 + 85, a remaining constant. During this 
variation the circle sweeps over a cylindric surface of radius a and 
length 8. 

The magnetic force perpendicular to this surface at any point is 

-s- where r is the distance from the axis. Hence 
dr 

dM PIT dV JQ ... 

= / a-j-dB. (3) 

db JQ dr 

Differentiating equation (2) with respect to a, and (3) with 
respect to I, we get 

d*M Pd7 7 f* d z Y . 
- - = / -j-dO+l a- r de, (4) 

da 2 JQ dy J dr dy 

d*M r* d^v .... ( . 

- = a-f-j-dB, (5) 

oar J dr dy 



-j 

Hence ^ + - - = / -j-dO, (6) 

da* db 2 JQ dy 

\dM 

= a-da-^y^ 
Transposing the last term we obtain equation (1). 



704.] TWO PARALLEL CIRCLES. 309 

Coefficient of Induction of Two Parallel Circles when the Distance 
betiveen the Arcs is Small compared with the Hadlus of either 
Circle. 

704.] We might deduce the value of M in this case from the 
expansion of the elliptic integral already given when its modulus 
is nearly unity. The following method, however, is a more direct 
application of electrical principles. 

First Approximation. 

Let A and a be the radii of the circles, and b the distance between 
their planes, then the shortest distance 
between the arcs is 




We have to find M 19 the magnetic 
induction through the circle A, due to a 
unit current in a on the supposition that 
r is small compared with A or a. 

We shall begin by calculating the 
magnetic induction through a circle in 
the plane of a whose radius is a c, c being a quantity small com 
pared with a (Fig. 49). 

Consider a small element ds of the circle a. At a point in the 
plane of the circle, distant p from the middle of ds, measured in 
a direction making an angle 6 with the direction of ds, the magnetic 
force due to ds is perpendicular to the plane, and equal to 

s sin 6 ds. 
P 2 

If we now calculate the surface-integral of this force over the 
space which lies within the circle a, but outside of a circle whose 
centre is ds and whose radius is c, we find it 

*2asin0 j 

g sin 6 ds d0 dp = {log 8 a log c 2} ds. 

If c is small, the surface-integral for the part of the annular space 
outside the small circle c may be neglected. 

We then find for the induction through the circle whose radius 
is a c, by integrating with respect to ds, 

M ac = ^ -n a (logStf logc 2}, 
provided c is very small compared with a. 

Since the magnetic force at any point, the distance of which 
from a curved wire is small compared with the radius of curvature, 



/*JT /* 

/ / 

J J c 



310 CIRCULAR CURRENTS. [705. 

is nearly the same as if the wire had been straight, we can calculate 
the difference between the induction through the circle whose 
radius is a c, and the circle A by the formula 
M a AM ac = 4: 7t a {logo log r}. 

Hence we find the value of the induction between A and a to be 

M Aa = 4 77 a (log 8 a log r 2) 
approximately, provided r is small compared with a. 

705.] Since the mutual induction between two windings of the 
same coil is a very important quantity in the calculation of ex 
perimental results, I shall now describe a method by which the 
approximation to the value of M for this case can be carried to any 
required degree of accuracy. 

We shall assume that the value of M is of the form 



1 j . j vu j / r i j ** / A f w r f n 

where A = a -f ^i# + A 2 \- A 2 - \-A 3 -^-}-A 3 -~ + &c., 

a a, a* a* 

and B = 2a + B,uo+B 9 + B ^- + B^ + B^ +&c., 

2 # 2 & 3 2 3 2 

where and + o? are the radii of the circles, and y the distance 
between their planes. 

We ; have to determine the values of the coefficients A and B. 
It is manifest that only even powers of y can occur in these quan 
tities, because, if the sign of y is reversed, the value of M must 
remain the same. 

We get another set of conditions from the reciprocal property 
of the coefficient of induction, which remains the same whichever 
circle we take as the primary circuit. The value of M must there 
fore remain the same when we substitute a + % for a, and a? for a? 
in the above expression. 

We thus find the following conditions of reciprocity by equating 
the coefficients of similar combinations of x and y, 

A . A A 7?__JLJL 

^3 - """^2 ~~^3> -3 3 ~ 2 -< 



76.] 



COIL OF MAXIMUM SELF-INDUCTION. 



311 



From the general equation of M, Art. 703, 
d 2 M d*M 1 dM 



dx 2 dy* a + x dx 
we obtain another set of conditions, 

O // I O J .. A 

2 l" *^ 2 ~"~ ~^1 3 



2 + 2A 



= 2A 



2 



&c.; 



4 A 2 + A l = 



Solving these equations and substituting the values of the co 
efficients, the series for If becomes 



M 



log 



+ &C.J 



O ^^ 1. I 

^5 2 -p 



+ &c 



] 



To find the form of a coil for which the coefficient of self-in 
duction is a maximum, the total length and thickness of the 
wire being given. 

706.] Omitting the corrections of Art. 705, we find by Art. 673 

where n is the number of windings of the wire, a is the mean 
radius of the coil, and R is the geometrical mean distance of the 
transverse section of the coil from itself. See Art. 690. If this 
section is always similar to itself, R is proportional to its linear 
dimensions, and n varies as R z . 

Since the total length of the wire is 2 TT an, a varies inversely 
as n. Hence 

dn _ dR , da dR 

- = 2-^-, and = 2 -^- , 
n R a R 

and we find the condition that L may be a maximum 



312 CIRCULAR CURRENTS. [76- 

If the transverse section of the coil is circular, of radius <?, then, 

by Art. 6 9 2, R 

Iog 7 =-i, 

and log = ^, 

whence a = 3.22 c ; 

or, the mean radius of the coil should be 3.22 times the radius of 
the transverse section of the coil in order that such a coil may have 
the greatest coefficient of self-induction. This result was found by 
Gauss *. 

If the channel in which the coil is wound has a square transverse 
section, the mean diameter of the coil should be 3.7 times the side 
of the square section. 

* Werl-e, Gottingen edition, 1867, vol. v. p. 622. 



CHAPTER XV. 

ELECTROMAGNETIC INSTRUMENTS. 

Galvanometers. 

707.] A GALVANOMETER is an instrument by means of which an 
electric current is indicated or measured by its magnetic action. 

When the instrument is intended to indicate the existence of a 
feeble current, it is called a Sensitive Galvanometer. 

When it is intended to measure a current with the greatest 
accuracy in terms of standard units, it is called a Standard Galva 
nometer. 

All galvanometers are founded on the principle of Schweigger s 
Multiplier, in which the current is made to pass through a wire, 
which is coiled so as to pass many times round an open space, 
within which a magnet is suspended, so as to produce within this 
space an electromagnetic force, the intensity of which is indicated 
by the magnet. 

In sensitive galvanometers the coil is so arranged that its 
windings occupy the positions in which their influence on the 
magnet is greatest. They are therefore packed closely together 
in order to be near the magnet. 

Standard galvanometers are constructed so that the dimensions 
and relative positions of all their fixed parts may be accurately 
known, and that any small uncertainty about the position of the 
moveable parts may introduce the smallest possible error into the 
calculations. 

In constructing a sensitive galvanometer we aim at making the 
field of electromagnetic force in which the magnet is suspended as 
intense as possible. In designing a standard galvanometer we 
wish to make the field of electromagnetic force near the magnet 
as uniform as possible, and to know its exact intensity in terms 
of the strength of the current. 






314 ELECTROMAGNETIC INSTRUMENTS. [708. 

On Standard Galvanometers. 

708.] In a standard galvanometer the strength of the current 
has to be determined from the force which it exerts on the sus 
pended magnet. Now the distribution of the magnetism within 
the magnet, and the position of its centre when suspended, are not 
capable of being determined with any great degree of accuracy. 
Hence it is necessary that the coil should be arranged so as to 
produce a field of force which is very nearly uniform throughout 
the whole space occupied by the magnet during its possible motion. 
The dimensions of the coil must therefore in general be much larger 
than those of the magnet. 

By a proper arrangement of several coils the field of force within 
them may be made much more uniform than when one coil only 
is used, and the dimensions of the instrument may be thus reduced 
and its sensibility increased. The errors of the linear measurements, 
however, introduce greater uncertainties into the values of the 
electrical constants for small instruments than for large ones. It 
is therefore best to determine the electrical constants of small 
instruments, not by direct measurement of their dimensions, but 
by an electrical comparison with a large standard instrument, of 
which the dimensions are more accurately known ; see Art. 752. 

In all standard galvanometers the coils are circular. The channel 
in which the coil is to be wound is carefully turned. Its breadth 




Fig. 50. 

is made equal to some multiple, n, of the diameter of the covered 
wire. A hole is bored in the side of the channel where the wire is 



709.] MEASUKEMENT OF THE COIL. 315 

to enter, and one end of the covered wire is pushed out through 
this hole to form the inner connexion of the coil. The channel is 
placed on a lathe, and a wooden axis is fastened to it; see Fig. 50. 
The end of a long string is nailed to the wooden axis at the same 
part of the circumference as the entrance of the wire. The whole 
is then turned round, and the wire is smoothly and regularly laid 
on the bottom of the channel till it is completely covered by n 
windings. During this process the string has been wound n times 
round the wooden axis, and a nail is driven into the string at the 
^th turn. The windings of the string should be kept exposed 
so that they can easily be counted. The external circumference 
of the first layer of windings is then measured and a new layer 
is begun, and so on till the proper number of layers has been 
wound on. The use of the string is to count the number of 
windings. If for any reason we have to unwind part of the coil, 
the string is also unwound, so that we do not lose our reckoning 
of the actual number of windings of the coil. The nails serve 
to distinguish the number of windings in each layer. 

The measure of the circumference of each layer furnishes a test 
of the regularity of the winding, and enables us to calculate the 
electrical constants of the coil. For if we take the arithmetic mean 
of the circumferences of the channel and of the outer layer, and 
then add to this the circumferences of all the intermediate layers, 
and divide the sum by the number of layers, we shall obtain the 
mean circumference, and from this we can deduce the mean radius 
of the coil. The circumference of each layer may be measured by 
means of a steel tape, or better by means of a graduated wheel 
which rolls on the coil as the coil revolves in the process of 
winding. The value of the divisions of the tape or wheel must 
be ascertained by comparison with a straight scale. 

709.] The moment of the force with which a unit current in 
the coil acts upon the suspended apparatus may be expressed in 
the series ^ gin Q + ^ gin Q ^ ^ + &c ^ 

where the coefficients G refer to the coil, and the coefficients g to 
the suspended apparatus, being the angle between the axis of 
the coil and that of the suspended apparatus ; see Art. 700. 

When the suspended apparatus is a thin uniformly and longitud 
inally magnetized bar magnet of length 2 1 and strength unity, 
suspended by its middle, 

^i = 2^, #2 = 0, # 3 =2 3 , &c. 



316 ELECTROMAGNETIC INSTRUMENTS. [7 IQ - 

The values of the coefficients for a magnet of length 2 1 magnetized 
in any other way are smaller than when it is magnetized uni 
formly. 

710.] When the apparatus is used as a tangent galvanometer, 
the coil is fixed with its plane vertical and parallel to the direction 
of the earth s magnetic force. The equation of equilibrium of the 
magnet is in this case 

m^HcosO = my sin0 {6^+ G 2 $ 2 Q/^ + fec.}, 

where mg^ is the magnetic moment of the magnet, .7? the horizontal 
component of the terrestrial magnetic force, and y the strength 
of the current in the coil. When the length of the magnet is 
small compared with the radius of the coil the terms after the first 
in G and g may be neglected, and we find 

TT 

y = -= cot 0. 
G i 

The angle usually measured is the deflexion, b, of the magnet 
which is the complement of 0, so that cot = tan 8. 

The current is thus proportional to the tangent of the deviation, 
and the instrument is therefore called a Tangent Galvanometer. 

Another method is to make the whole apparatus moveable about 
a vertical axis, and to turn it till the magnet is in equilibrium with 
its axis parallel to the plane of the coil. If the angle between the 
plane of the coil and the magnetic meridian is 8, the equation of 
equilibrium is 

&c -l > 



whence y = -^ - 5 .sin 8. 

(G^-fec.) 

Since the current is measured by the sine of the deviation, the 
instrument when used in this way is called a Sine Galvanometer. 

The method of sines can be applied only when the current is 
so steady that we can regard it as constant during the time of 
adjusting the instrument and bringing the magnet to equi 
librium. 

711.] We have next to consider the arrangement of the coils 
of a standard galvanometer. 

The simplest form is that in which there is a single coil, and 
the magnet is suspended at its centre. 

Let A be the mean radius of the coil, its depth, rj its breadth, 
and n the number of windings, the values of the coefficients are 



712.] TANGENT GALVANOMETEE. 317 



4 = 0, &c. 
The principal correction is that arising 1 from G 3 . The series 



becomes G^ y t ( 1 | -p ^ (cos 2 J sin 2 0)) 

V 1 

The factor of correction will differ most from unity when the 
magnet is uniformly magnetized and when = 0. In this case it 

I 2 

becomes 1 2 ~^ It vanishes when tan = 2, or when the de- 
.4 

flexion is tan" 1 4, or 2634 . Some observers, therefore, arrange 
their experiments so as to make the observed deflexion as near 
this angle as possible. The best method, however, is to use a 
magnet so short compared with the radius of the coil that the 
correction may be altogether neglected. 

The suspended magnet is carefully adjusted so that its centre 
shall coincide as nearly as possible with the centre of the coil. If, 
however, this adjustment is not perfect, and if the coordinates of 
the centre of the magnet relative to the centre of the coil are os, y, z, 
z being measured parallel to the axis of the coil, the factor of 

correction is (l 4- 3 ) 

When the radius of the coil is large, and the adjustment of the 
magnet carefully made, we may assume that this correction is 
insensible. 

Gaugavn?* Arrangement. 

712.] In order to get rid of the correction depending on G 3 
Gaugain constructed a galvanometer in which this term was ren 
dered zero by suspending the magnet, not at the centre of the 
coil, but at a point on the axis at a distance from the centre equal 
to half the radius of the coil. The form of G is 



and, since in this arrangement B = \ A, G 3 = 0. 

This arrangement would be an improvement on the first form 
if we could be sure that the centre of the suspended magnet is 



318 ELECTROMAGNETIC INSTRUMENTS. [713. 

exactly at the point thus defined. The position of the centre of the 
magnet, however, is always uncertain, and this uncertainty intro 
duces a factor of correction of unknown amount depending on G 2 and 

of the form (l -r) , where z is the unknown excess of distance 

^4 

of the centre of the magnet from the plane of the coil. This 
correction depends on the first power of -j . Hence Gaugain s coil 

with eccentrically suspended magnet is subject to far greater un 
certainty than the old form. 

Helmholtz s Arrangement, 

713.] Helmholtz converted Gaugain s galvanometer into a trust 
worthy instrument by placing a second coil, equal to the first, at 
an equal distance on the other side of the magnet. 

By placing the coils symmetrically on both sides of the magnet 
we get rid at once of all terms of even order. 

Let A be the mean radius of either coil, the distance between 
their mean planes is made equal to A^ and the magnet is suspended 
at the middle point of their common axis. The coefficients are 

& = 



G 3 = 0.0512 (31 2 - 36rj 2 ), 



G B = -0.73728 

where n denotes the number of windings in both coils together. 

It appears from these results that if the section of the coils be 
rectangular, the depth being f and the breadth 17, the value of 
6r 3 , as corrected for the finite size of the section, will be small, and 
will vanish, if is to 77 as 36 to 31. 

It is therefore quite unnecessary to attempt to wind the coils 
upon a conical surface, as has been done by some instrument makers, 
for the conditions may be satisfied by coils of rectangular section, 
which can be constructed with far greater accuracy than coils 
wound upon an obtuse cone. 

The arrangement of the coils in Helmholtz s double galvanometer 
is represented in Fig. 54, Art. 725. 



715.] GALVANOMETER OF THREE COILS. 319 

The field of force due to the double coil is represented in section 
in Fig. XIX at the end of this volume. 

Galvanometer of Four Coils. 

714.] By combining four coils we may get rid of the coefficients 
G 2 , G 3 , G, G 5 , and G 6 . For by any symmetrical combinations 
we get rid of the coefficients of even orders Let the four coils 
be parallel circles belonging to the same sphere, corresponding 
to angles 6, (j>, TT <, and TT 0. 

Let the number of windings on the first and fourth coil be n y 
and the number on the second and third pn. Then the condition 
that G 3 = for the combination gives 

ft sin 2 q; (0) + ^ft sin 2 $ Q 9 (c/>) = 0, (1) 

and the condition that G 5 = gives 

ft sin 2 6 <2 5 (6) + pn sin 2 < Q/ (<#>) = 0, (2) 

Putting sin 2 = x and sin 2 $ = y^ (3) 

and expressing Q 3 and Q 5 (Art. 698) in terms of these quantities, 
the equations (1) and (2) become 

= 0, (4) 

= 0. (5) 

Taking twice (4) from (5), and dividing by 3, we get 

6# 2 -7# 3 + 6j?y 2 -7j^ 3 = 0. (6) 

Hence, from (4) and (6), 

_ x 5x 4_ x 2 7# 6 

P= y I=5j = /6=7^ 
and we obtain 

7 a? 6 32 7x 6 



= f 



4 

Both x and y are the squares of the sines of angles and must 
therefore lie between and 1 . Hence, either x is between and -f , 
in which case y is between -f- and 1, and p between co and ^%, 
or else x is between f and 1, in which case y is between and 
f, and p between and |f. 

Galvanometer of Three Colls. 

715.] The most convenient arrangement is that in which x = 1. 
Two of the coils then coincide and form a great circle of the sphere 
whose radius is C. The number of windings in this compound 
coil is 64. The other two coils form small circles of the sphere. 
The radius of each of them is \/ C. The distance of either of 



320 ELECTROMAGNETIC INSTRUMENTS. [715. 

them from the plane of the first is \/ i C. The number of windings 
on each of these coils is 49. 

1 20 

The value of G 1 is ~-^- < 
L> 

This arrangement of coils is represented in Fig. 51, 




Fig. 51. 

Since in this three-coiled galvanometer the first term after G 1 
which has a finite value is (r 7 , a large portion of the sphere on 
whose surface the coils lie forms a field of force sensibly uniform. 

If we could wind the wire over the whole of a spherical surface, 
as described in Art. 627, we should obtain a field of perfectly 
uniform force. It is practically impossible, however, to distribute 
the windings on a spherical surface with sufficient accuracy, even 
if such a coil were not liable to the objection that it forms a closed 
surface, so that its interior is inaccessible. 

By putting the middle coil out of the circuit, and making the 
current flow in opposite directions through the two side coils, we 
obtain a field of force which exerts a nearly uniform action in 
the direction of the axis on a magnet or coil suspended within it, 
with its axis coinciding with that of the coils; see Art. 673. For 
in this case all the coefficients of odd orders disappear, and since 



Hence the expression for the magnetic potential near the centre 
of the coil becomes 



^ Q G W + &C.J 



7 1 6.] THICKNESS OF THE WIRE. 321 

On the Proper Thickness of the Wire of a Galvanometer, the External 
Resistance being given. 

716.] Let the form of the channel in which the galvanometer 
coil is to be wound be given, and let it be required to determine 
whether it ought to be filled with a long thin wire or with a shorter 
thick wire. 

Let I be the length of the wire, y its radius, y + b the radius 
of the wire when covered, p its specific resistance, g the value of 
G for unit of length of the wire, and r the part of the resistance 
which is independent of the galvanometer. 

The resistance of the galvanometer wire is 

P i 

Jt= -- 5 

ity* 

The volume of the coil is 

7= 4l(y + b) 2 . 

The electromagnetic force is y G, where y is the strength of the 
current and G gl. 

If E is the electromotive force acting in the circuit whose 
resistance is R + r, E = y (R + r). 

The electromagnetic force due to this electromotive force is 

G 



which we have to make a maximum by the variation of y and I. 
Inverting the fraction, we find that 

_P _J_ r 
TT<? f gl 
is to be made a minimum. Hence 

pdy rdl 
& - o H 75 = 0. 

7T^ 3 I 2 

If the volume of the coil remains constant 

dl dy 

-y- + 2 -*- = 0. 

1 y + 6 

Eliminating dl and dy, we obtain 

p y + b _ r 



r 
or 



R y 

Hence the thickness of the wire of the galvanometer should be 
such that the external resistance is to the resistance of the gal 
vanometer coil as the diameter of the covered wire to the diameter 
of the wire itself. 

VOL. IT. Y 



322 ELECTROMAGNETIC INSTRUMENTS. [717. 

On Sensitive Galvanometers. 

717.] In the construction of a sensitive galvanometer the aim 
of every part of the arrangement is to produce the greatest possible 
deflexion of the magnet by means of a given small electromotive 
force acting between the electrodes of the coil. 

The current through the wire produces the greatest effect when 
it is placed as near as possible to the suspended magnet. The 
magnet, however, must be left free to oscillate, and therefore there 
is a certain space which must be left empty within the coil. This 
defines the internal boundary of the coil. 

Outside of this space each winding must be placed so as to have 
the greatest possible effect on the magnet. As the number of 
windings increases, the most advantageous positions become filled 
up, so that at last the increased resistance of a new winding 
diminishes the effect of the current in the former windings more 
than the new winding itself adds to it. By making the outer 
windings of thicker wire than the inner ones we obtain the greatest 
magnetic effect from a given electromotive force. 

718.] We shall suppose that the windings of the galvanometer 
are circles, the axis of the galvanometer passing through the centres 
of these circles at right angles to their planes. 

Let r sin Q be the radius of one of these circles, and r cos the 
distance of its centre from the centre of the galvanometer, then, 
if I is the length of a portion of wire coinciding with this circle, 

and y the current which flows in it, the 
magnetic force at the centre of the gal 
vanometer resolved in the direction of 
the axis is s i n Q 

y -p- 

If we write r 2 = x 2 sin 0, (1) 

this expression become^ y ^ 

x 

Hence, if a surface be constructed 
similar to those represented in section 
in Fig. 52, whose polar equation is 

r 2 = x* sin 0, (2) 

where a? x is any constant, a given length 
of wire bent into the form of a circular 
g arc will produce a greater magnetic 

effect when it lies within this surface than when it lies outside it. 




719.] SENSITIVE GALYANOMETEK. 323 

It follows from this that the outer surface of any layer of wire 
ought to have a constant value of x, for if x is greater at one place 
than another a portion of wire might be transferred from the first 
place to the second, so as to increase the force at the centre of the 
galvanometer. 

The whole force due to the coil is y G, where 



G 



n- 



the integration being extended over the whole length of the wire, 
x being considered as a function of I. 

719.] Let y be the radius of the wire, its transverse section will 
be 7r^ 2 . Let p be the specific resistance of the material of which 
the wire is made referred to unit of volume, then the resistance of a 

length I is ^ } and the whole resistance of the coil is 

*f 

/* 77 

(4) 

where y is considered a function of I. 

Let Y 2 be the area of the quadrilateral whose angles are the 
sections of the axes of four neighbouring wires of the coil by a 
plane through the axis, then Y 2 l is the volume occupied in the coil 
by a length I of wire together with its insulating covering, and 
including any vacant space necessarily left between the windings 
of the coil. Hence the whole volume of the coil is 






r=jYdl, 

where Y is considered a function of /. 

But since the coil is a figure of revolution 

V 2 TT jjr 2 sin dr dO, (6) 

or, expressing r in terms of x, by equation (2), 

V = 2 TT I j a? (sin 0)* dan dB. (7) 

Now 27T / (sill 0)$ dO is a numerical quantity, call it JV, then 
o 

where F" is the volume of the interior space left for the 
magnet. 

Let us now consider a layer of the coil contained between the 
surfaces x and x + das. 

Y 2, 



324 ELECTROMAGNETIC INSTRUMENTS. [7 J 9- 

The volume of this layer is 



x = Y 2 dl, (9) 

where dl is the length of wire in this layer. 

This gives us dl in terms of dx. Substituting this in equations 
(3) and (4), we find 



where f/(r and f/.S represent the portions of the values of G and of 
It due to this layer of the coil. 

Now if E be the given electromotive force, 



where r is the resistance of the external part of the circuit, in 
dependent of the galvanometer, and the force at the centre is 

G 



si 
We have therefore to make -= a maximum, by properly ad- 

JK -\- T 

justing the section of the wire in each layer. This also necessarily 
involves a variation of Y because Y depends on y. 

Let G and JR Q be the values of G and of R + r when the given 
layer is excluded from the calculation. We have then 



R +dR 

and to make this a maximum by the variation of the value of y for 
the given layer we must have 

,* 

( 13 > 



. 

ay 

C 1 
Since dx is very small and ultimately vanishes, ^- will be sensibly, 

**o 

and ultimately exactly, the same whichever layer is excluded, and 
we may therefore regard it as constant. We have therefore, by (10) 
and (11), X 2 Y dy. P R + r 

f + 7 3r) = 1-^- = constant - (14) 

If the method of covering the wire and of winding it is such 
that the proportion between the space occupied by the metal of 






720.] SENSITIVE GALVANOMETER. 325 

the wire bears the same proportion to the space between the wires 
whether the wire is thick or thin, then 



and we must make both y and Y proportional to x, that is to say, 
the diameter of the wire in any layer must be proportional to the 
linear dimension of that layer. 

If the thickness of the insulating covering is constant and equal 
to d, and if the wires are arranged in square order, 

Y=2(y + b\ (15) 

and the condition is 

= constant. (16) 



In this case the diameter of the wire increases with the diameter 
of the layer of which it forms part, but not in so high a ratio. 

If we adopt the first of these two hypotheses, which will be nearly 
true if the wire itself nearly fills up the whole space, then we may 
put y = ax, Y= $y, 

where a and ft are constant numerical quantities, and 



where a is a constant depending upon the size and form of the free 
space left inside the coil. 

Hence, if we make the thickness of the wire vary in the same 
ratio as as, we obtain very little advantage by increasing the 
external size of the coil after the external dimensions have become 
a large multiple of the internal dimensions. 

720.] If increase of resistance is not regarded as a defect, as 
when the external resistance is far greater than that of the gal 
vanometer, or when our only object is to produce a field of intense 
force, we may make y and Y constant. We have then 

N 

G = 71 (*-")> 

-p 1 Pf/*.3 n %\ 

~ 3 Yf> Jj * * 

where a is a constant depending on the vacant space inside the 
coil. In this case the value of G increases uniformly as the 
dimensions of the coil are increased, so that there is no limit to 
the value of G except the labour and expense of making the coil. 



326 



ELECTROMAGNETIC INSTRUMENTS, 



[721. 



On Suspended Coils. 

721.] In the ordinary galvanometer a suspended magnet is acted 
on by a fixed coil. But if the coil can be suspended with sufficient 
delicacy, we may determine the action of the magnet, or of another 
coil on the suspended coil, by its deflexion from the position of 
equilibrium. 

We cannot, however, introduce the electric current into the coil 
unless there is metallic connexion between the electrodes of the 
battery and those of the wire of the coil. This connexion may be 
made in two different ways, by the Bifilar Suspension, and by wires 
in opposite directions. 

The bifilar suspension has already been described in Art. 459 
as applied to magnets. The arrangement of the upper part of the 
suspension is shewn in Fig. 55. When applied to coils, the two 
fibres are no longer of silk but of metal, and since the torsion of 
a metal wire capable of supporting the coil and transmitting the 
current is much greater than that of a silk fibre, it must be taken 
specially into account. This suspension has been brought to great 
perfection in the instruments constructed by M. Weber. 

The other method of suspension is by means of a single wire 
which is connected to one extremity of the coil. The other ex 
tremity of the coil is connected to another wire which is made 
to hang down, in the same vertical straight line with the first wire, 
into a cup of mercury, as is shewn in Fig. 57, Art. 729. In certain 
cases it is convenient to fasten the extremities of the two wires to 
pieces by which they may be tightly stretched, care being taken 

that the line of these wires passes 
through the centre of gravity of the 
coil. The apparatus in this form 
may be used when the axis is not 
vertical ; see Fig. 53. 

722.] The suspended coil may be 
used as an exceedingly sensitive gal 
vanometer, for, by increasing the in 
tensity of the magnetic force in the 
field in which it hangs, the force due 
to a feeble current in the coil may 
be greatly increased without adding 
to the mass of the coil. The mag 
netic force for this purpose may be 




Fig. 53. 



produced by means of permanent magnets, or by electromagnets 



723-] SUSPENDED COIL. 327 

excited by an auxiliary current, and it may be powerfully concen 
trated on the suspended coil by means of soft iron armatures. Thus, 
in Sir W. Thomson s recording apparatus, Fig. 53, the coil is sus 
pended between the opposite poles of the electromagnets N and S, 
and in order to concentrate the lines of magnetic force on the ver 
tical sides of the coil, a piece of soft iron, 1), is fixed between the 
poles of the magnets. This iron becoming magnetized by induc 
tion, produces a very powerful field of force, in the intervals between 
it and the two magnets, through which the vertical sides of the 
coil are free to move, so that the coil, even when the current 
through it is very feeble, is acted on by a considerable force 
tending to turn it about its vertical axis. 

723.] Another application of the suspended coil is to determine, 
by comparison with a tangent galvanometer, the horizontal com 
ponent of terrestrial magnetism. 

The coil is suspended so that it is in stable equilibrium when 
its plane is parallel to the magnetic meridian. A current y is 
passed through the coil and causes it to be deflected into a new 
position of equilibrium, making an angle with the magnetic 
meridian. If the suspension is bifilar, the moment of the couple 
which produces this deflexion is I 1 sin 0, and this must be equal to 
HyffcosO, where His the horizontal component of terrestrial mag 
netism, y is the current in the coil, and g is the sum of the areas of 
all the windings of the coil. Hence 

F 

II y tan0. 

g 

If A is the moment of inertia of the coil about its axis of sus 
pension, and Tthe time of a single vibration, 
FT 2 = v*A, 

Ti^A 

and we obtain Hy = - tan 0. 



If the same current passes through the coil of a tangent galva 
nometer, and deflects the magnet through an angle 0, 

y 



where G is the principal constant of the tangent galvanometer, Art. 710, 
From these two equations we obtain 
7T tr /AGkaxid TT /A tan tan rf> 

: ~T A/Tte^T = T V -oT 

Tliis method wa^ given by F. Kohlrausch *. 

* r"ogg., Ann. cxxxviii, Feb. 18G9. 



328 ELECTROMAGNETIC INSTRUMENTS. [? 2 4- 

724.] Sir William Thomson has constructed a single instrument 
by means of which the observations required to determine H and y 
may be made simultaneously by the same observer. 

The coil is suspended so as to be in equilibrium with its plane 
in the magnetic meridian, and is deflected from this position 
when the current flows through it. A very small magnet is sus 
pended at the centre of the coil, and is deflected by the current in 
the direction opposite to that of the deflexion of the coil. Let the 
deflexion of the coil be 6, and that of the magnet 0, then the 
energy of the system is 

Hy g sm9 + my G sin (0 fy Hmcos Fcos 9. 

Differentiating with respect to and 0, we obtain the equa 
tions of equilibrium of the coil and of the magnet respectively, 

Hyg cos + my (7 cos (0 0) + F sin Q = 0, 
my G cos (6 0)-f Hm sin = 0. 

From these equations we find, by eliminating H or y } a quadratic 
equation from which y or // may be found. If m, the magnetic 
moment of the suspended mag-net, is very small, we obtain the 
following approximate values 

j _ IT / ^<?sin0cos(0 0) L mG cos (6 0) 

~T V g cos 6 sin 2 g cos0 

77 / ^4 sin sin ^m sin0 

" ~T V G g cos 6 cos (00) ~~ 2 7 cos ^ " 

In these expressions G and g are the principal electric constants 
of the coil, A its moment of inertia, T its time of vibration, m the 
magnetic moment of the magnet, H the intensity of the horizontal 
magnetic force, y the strength of the current, the deflexion of the 
coil, and that of the magnet. 

Since the deflexion of the coil is in the opposite direction to the 
deflexion of the magnet, these values of H and y will always be 
real. 

Weber s Electrody nanometer. 

725.] In this instrument a small coil is suspended by two wires 
within a larger coil which is fixed. When a current is made to 
flow through both coils, the suspended coil tends to place itself 
parallel to the fixed coil. This tendency is counteracted by the 
moment of the forces arising from the bifilar suspension, and it is 
also affected by the action of terrestrial magnetism on the sus 
pended coil. 



725.] ELECTRODYNAMOMETER. 329 

In the ordinary use of the instrument the planes of the two coils 
are nearly at right angles to each other, so that the mutual action 
of the currents in the coils may be as great as possible, and the 
plane of the suspended coil is nearly at right angles to the magnetic 
meridian, so that the action of terrestrial magnetism may be as 
small as possible. 

Let the magnetic azimuth of the plane of the fixed coil be a, 
and let the angle which the axis of the suspended coil makes with 
the plane of the fixed coil be Q + fi, where (3 is the value of this 
angle when the coil is in equilibrium and no current is flowing, 
and"* 6 is the deflexion due to the current. The equation of equi 
librium is 

Let us suppose that the instrument is adjusted so that a and j3 
are both very small, and that Hgy^ is small compared with F. 
We have in this case, approximately, 

(r^y 1 y 2 cos/3 Zfyy 2 sin(a-|-/3) HGg^y^y^ G 2 y 2 y l 2 y 2 2 smj3 

If the deflexions when the signs of y l and y 2 are changed are 
as follows : e when is , and , 



then we find 

F 

y l y 2 J (tan 0J + tan 2 tan 3 tan 4 ). 

If it is the same current which flows through both coils we may put 
y l y 2 = y 2 , and thus obtain the value of y. 

When the currents are not very constant it is best to adopt this 
method, which is called the Method of Tangents. 

If the currents are so constant that we can adjust /3, the angle 
of the torsion-head of the instrument, we may get rid of the 
correction for terrestrial magnetism at once by the method of sines. 
In this method /3 is adjusted till the deflexion is zero, so that 

0=_/3. 

If the signs of y 1 and y 2 are indicated by the suffixes of /3 as 
before, 

Fsin & = -Fsin P 3 = Gffy l y 2 + Hg y 2 sin a, 

F sin )3 2 = ^sin /3 4 = Gg y l y^ Rg y 2 sin a, 

F 

and Yl y 2 = - ^ (sin fa + sin fa - sin fa - sin fa). 



330 



ELECTROMAGNETIC INSTRUMENTS. 



[7 2 5< 




725.] ELECTRODYNAMOMETER. 331 

This is the method adopted by Mr. Latimer Clark in his use 
of the instrument constructed by the Electrical Committee of the 
British Association. We are indebted to Mr. Clark for the drawing 
of the electrodynamometer in Figure 54, in which Helmholtz s 
arrangement of two coils is adopted both for the fixed and for the 
suspended coil*. The torsion-head of the instrument, by which 
the bifilar suspension is adjusted, is represented in Fig. 55. The 




Fig. 55. 

equality of the tension of the suspension wires is ensured by their 
being attached to the extremities of a silk thread which passes over 
a wheel, and their distance is regulated by two guide-wheels, which 
can be set at the proper distance. The suspended coil can be moved 
vertically by means of a screw acting on the suspension-wheel, 
and horizontally in two directions by the sliding pieces shewn at 
the bottom of Fig. 55. It is adjusted in azimuth by means of the 
torsion-screw, which turns the torsion-head round a vertical axis 
(see Art. 459). The azimuth of the suspended coil is ascertained 
by observing the reflexion of a scale in the mirror, shewn just 
beneath the axis of the suspended coil. 

* In the actual instrument, the wires conveying the current to and from the coils 
are not spread out as displayed in the figure, but are kept as close together as pos 
sible, so as to neutralize each other s electromagnetic action. 



332 ELECTROMAGNETIC INSTRUMENTS. 



The instrument originally constructed by Weber is described in 
his Elektroctynamiscke Maasbeslimmungen. It was intended for the 
measurement of small currents, and therefore both the fixed and 
the suspended coils consisted of many windings, and the suspended 
coil occupied a larger part of the space within the fixed coil than in 
the instrument of the British Association, which was primarily in 
tended as a standard instrument, with which more sensitive instru 
ments might be compared. The experiments which he made with 
it furnish the most complete experimental proof of the accuracy of 
Ampere s formula as applied to closed currents, and form an im 
portant part of the researches by which Weber has raised the 
numerical determination of electrical quantities to a very high rank 
as regards precision. 

Weber s form of the electrodynarnometer, in which one coil is 
suspended within another, and is acted on by a couple tending 
to turn it about a vertical axis, is probably the best fitted for 
absolute measurements. A method of calculating the constants of 
such an arrangement is given in Art. 697. 

726.] If, however, we wish, by means of a feeble current, to 
produce a considerable electromagnetic force, it is better to place 
the suspended coil parallel to the fixed coil, and to make it capable 
of motion to or from it. 

The suspended coil in Dr. Joule s 
current- weigher, Fig. 56, is horizontal, 
and capable of vertical motion, and the 
force between it and the fixed coil is 
estimated by the weight which must 
be added to or removed from the coil 
in order to bring it to the same relative 
position with respect to the fixed coil 
that it has when no current passes. 

The suspended coil may also be 
fastened to the extremity of the hori- 
56< zontal arm of a torsion-balance, and 

may be placed between two fixed coils, one of which attracts it, 
while the other repels it, as in Fig. 57. 

By arranging the coils as described in Art. 729, the force acting 
on the suspended coil may be made nearly uniform within a small 
distance of the position of equilibrium. 

Another coil may be fixed to the other extremity of the arm 
of the torsion-balance and placed between two fixed coils. If the 




728.] 



CURRENT-WEIGHER. 



333 



two suspended coils are similar, but with the current flowing in 
opposite directions, the effect of terrestrial magnetism on the 




Fig. 57. 

position of the arm of the torsion-balance will be completely 
eliminated. 

727.] If the suspended coil is in the shape of a long solenoid, 
and is capable of moving parallel to its axis, so as to pass into 
the interior of a larger fixed solenoid having the same axis, then, 
if the current is in the same direction in both solenoids, the sus 
pended solenoid will be sucked into the fixed one by a force which 
will be nearly uniform as long as none of the extremities of the 
solenoids are near one another. 

728.] To produce a uniform longitudinal force on a small coil 
placed between two equal coils of much larger dimensions, we 
should make the ratio of the diameter of the large coils to the dis 
tance between their planes that of 2 to \/3. If we send the same 
current through these coils in opposite directions, then, in the ex 
pression for o>, the terms involving odd powers of r disappear, and 
since sin 2 a = -f and cos 2 a = f, the term involving /- 4 disappears 
also, and we have 



~ Q 2 (0) + V 



&c 



which indicates a nearly uniform force on a small suspended coil. 
The arrangement of the coils in this case is that of the two outer 
coils in the galvanometer with three coils, described at Art. 715. 
See Fig. 51. 



334 ELECTROMAGNETIC INSTRUMENTS. [? 2 9- 

729.] If we wish to suspend a coil between two coils placed 
so near it that the distance between the mutually acting wires is 
small compared with the radius of the coils, the most uniform force 
is obtained by making the radius of either of the outer coils exceed 

that of the middle one by - ^ of the distance between the planes 

v3 
of the middle and outer coils. 



CHAPTER XVI. 



ELECTROMAGNETIC OBSERVATIONS. 



730.] So many of the measurements of electrical quantities 
depend on observations of the motion of a vibrating body that we 
shall devote some attention to the nature of this motion, and the 
best methods of observing it. 

The small oscillations of a body about a position of stable equi 
librium are, in general, similar to those of a point acted on by 
a force varying directly as the distance from a fixed point. In 
the case of the vibrating bodies in our experiments there is also 
a resistance to the motion, depending on a variety of causes, such 
as the viscosity of the air, and that of the suspension fibre. In 
many electrical instruments there is another cause of resistance, 
namely, the reflex action of currents induced in conducting circuits 
placed near vibrating magnets. These currents are induced by the 
motion of the magnet, and their action on the magnet is, by the 
law of Lenz, invariably opposed to its motion. This is in many 
cases the principal part of the resistance. 

A metallic circuit, called a Damper, is sometimes placed near 
a magnet for the express purpose of damping or deadening its 
vibrations. We shall therefore speak of this kind of resistance 
as Damping. 

In the case of slow vibrations, such as can be easily observed, 
the whole resistance, from whatever causes it may arise, appears 
to be proportional to the velocity. It is only when the velocity 
is much greater than in the ordinary vibrations of electromagnetic 
instruments that we have evidence of a resistance proportional to 
the square of the velocity. 

We have therefore to investigate the motion of a body subject 
to an attraction varying as the distance, and to a resistance varying 
as the velocity. 



336 



ELECTROMAGNETIC OBSERVATIONS. 



731.] The following application, by Professor Tait*, of the 
principle of the Hodograph, enables us to investigate this kind 
of motion in a very simple manner by means of the equiangular 
spiral. 

Let it be required to find the acceleration of a particle which 
describes a logarithmic or equiangular spiral with uniform angular 
velocity o> about the pole. 

The property of this spiral is, that the tangent PT makes with 
the radius vector PS a constant angle a. 

If v is the velocity at the point P, then 

v . sin a = co . SP. 

Hence, if we draw SP parallel to PT and equal to SP, the velocity 
at P will be given both in magnitude and direction by 



v = 



sin a 



SP. 




Fig. 58. 

Hence P will be a point in the hodograph. But SP is SP turned 
through a constant angle TT a, so that the hodograph described 
by P is the same as the original spiral turned about its pole through 
an angle TT a. 

The acceleration of P is represented in magnitude and direction 

by the velocity of P multiplied by the same factor, -. 

Hence, if we perform on SP the same operation of turning it 
* Proc. R. S. Win., Dec. 16, 1867. 



732.] DAMPED VIBRATIONS. 337 

through an angle IT a into the position SP , the acceleration of P 
will be equal in magnitude and direction to 

- &, 



where SP is equal to SP turned through an angle 2 IT 2 a. 

If we draw PF equal and parallel to SP , the acceleration will be 

9 

PF, which we may resolve into 



sin 2 a 

J?LpS*n& -4-P*. 
sm*a sin^a 

The first of these components is a central force towards S pro 
portional to the distance. 

The second is in a direction opposite to the velocity, and since 

_, sin a cos a 
PK = 2 cos a PS = - 2 - v, 

0} 

this force may be written 

co cos a 

2. v. 

sin a 

The acceleration of the particle is therefore compounded of two 
parts, the first of which is an attractive force /ur, directed towards S, 
and proportional to the distance, and the second is 2 kv, a resist 
ance to the motion proportional to the velocity, where 

ft) 2 , 7 cos a 

a = . , and k = o> -. 

sin^ a sin a 

If in these expressions we make a = , the orbit becomes a circle, 

and we have JU G = o) 2 , and k = 0. 

Hence, if the law of attraction remains the same, ju = /ut , and 

co = o) sin a, 

or the angular velocity in different spirals with the same law of 
attraction is proportional to the sine of the angle of the spiral. 

732.] If we now consider the motion of a point which is the 
projection of the moving point P on the horizontal line XT, we 
shall find that its distance from S and its velocity are the horizontal 
components of those of P. Hence the acceleration of this point is 
also an attraction towards S, equal to /x, times its distance from S f 
together with a retardation equal to k times its velocity. 

We have therefore a complete construction for the rectilinear 
motion of a point, subject to an attraction proportional to the 
distance from a fixed point, and to a resistance proportional to 
the velocity. The motion of such a point is simply the horizontal 

VOL. II. Z 



338 ELECTROMAGNETIC OBSERVATIONS. [733. 

part of the motion of another point which moves with uniform 
angular velocity in a logarithmic spiral. 

733.] The equation of the spiral is 

r = Ce-$ CQia . 

To determine the horizontal motion, we put 
< = co ^, x = a-\-r sin </>, 
where a is the value of x for the point of equilibrium. 

If we draw BSD making an angle a with the vertical, then the 
tangents BX> DY, GZ, &c. will be vertical, and X, Y, Z, &c. will 
be the extremities of successive oscillations. 

734.] The observations which are made on vibrating bodies are 

(1) The scale-reading at the stationary points. These are called 

Elongations. 

(2) The time of passing a definite division of the scale in the 

positive or negative direction. 

(3) The scale-reading at certain definite times. Observations of 

this kind are not often made except in the case of vibrations 
of long period *. 
The quantities which we have to determine are 

(1) The scale-reading at the position of equilibrium. 

(2) The logarithmic decrement of the vibrations. 

(3) The time of vibration. 

To determine the Reading at the Position of Equilibrium from 
Three Consecutive Elongations, 

735.] Let #!, # 2 , # 3 be the observed scale-readings, corresponding 
to the elongations X, Y, Z, and let a be the reading at the position 
of equilibrium, S, and let r^ be the value of SB, 
# j a = /! sin a, 
$ 2 a = 1\ sin a e~* cot a , 
# 3 a = r l sina- 27rcota . 
From these values we find 

(*!-) 0*8 -) = 0* 2 -) 2 

, X-, 

whence a = 



vU\ "J~ 2/o " * . &O 

When a* 3 does not differ much from x^ we may use as an ap 
proximate formula 

a = }(a? 1 + 2a? a + a? 3 ). 

* See Gauss, Resultate des Magnetischen Vereins, 1836. II. 



LOGAKITHMIC DECREMENT. 339 

To determine the Logarithmic Decrement. 

736.] The logarithm of the ratio of the amplitude of a vibration 
to that of the next following is called the Logarithmic Decrement. 
If we write p for this ratio 



L is called the common logarithmic decrement, and A. the Napierian 
logarithmic decrement. It is manifest that 
A = L log e 10 = 77 cot a. 



Hence a = cot" 1 - 

77 



which determines the angle of the logarithmic spiral. 

In making a special determination of A we allow the body to 
perform a considerable number of vibrations. If c 1 is the amplitude 
of the first, and c n that of the n^ vibration, 



If we suppose the accuracy of observation to be the same for 
small vibrations as for large ones, then, to obtain the best value 
of A, we should allow the vibrations to subside till the ratio of c 1 to 
c n becomes most nearly equal to e, the base of the Napierian 

logarithms. This gives n the nearest whole number to - + 1 . 

A 

Since, however, in most cases time is valuable, it is best to take 
the second set of observations before the diminution of amplitude 
has proceeded so far. 

737.] In certain cases we may have to determine the position 
of equilibrium from two consecutive elongations, the logarithmic 
decrement being known from a special experiment. We have then 

_ #l + ^2 






Time of Vibration . 

738.] Having determined the scale-reading of the point of equi 
librium, a conspicuous mark is placed at that point of the scale, 
or as near it as possible, and the times of the passage of this mark 
are noted for several successive vibrations. 

Let us suppose that the mark is at an unknown but very small 
distance as on the positive side of the point of equilibrium, and that 

z 2 



340 ELECTROMAGNETIC OBSERVATIONS. [739. 

tfj is the observed time of the first transit of the mark in the positive 
direction, and 2 , ^ 3 , &c. the times of the following transits. 

If T be the time of vibration, and P 15 P 2 , P 3 , &c. the times of 
transit of the true point of equilibrium, 



where v lt v 29 &c. are the successive velocities of transit, which we 
may suppose uniform for the very small distance SB. 

If p is the ratio of the amplitude of a vibration to the next in 

succession, 1 , as x 

v 9 -- #T , and. = p 

P l /2 ^l 

If three transits are observed at times t i3 t 2 , 3 , we find 



The period of vibration is therefore 

2 P+1 
The time of the second passage of the true point of equilibrium is 

P 2 = i (^-f 2 ^ 2 + O ~i / " \z (*i 2 ^2 + ^)- 

Three transits are sufficient to determine these three quantities, 
but any greater number may be combined by the method of least 
squares. Thus, for five transits, 



The time of the third transit is, 



739.] The same method may be extended to a series of any 
number of vibrations. If the vibrations are so rapid that the time 
of every transit cannot be recorded, we may record the time of 
every third or every fifth transit, taking care that the directions 
of successive transits are opposite. If the vibrations continue 
regular for a long time, we need not observe during the whole 
time. We may begin by observing a sufficient number of transits 
to determine approximately the period of vibration, T, and the time 
of the middle transit, P, noting whether this transit is in the 
positive or the negative direction. We may then either go on 
counting the vibrations without recording the times of transit, 
or we may leave the apparatus un watched. We then observe a 



PERIODIC TIME OF VIBRATION. 341 

second series of transits,, and deduce the time of vibration T and 
the time of middle transit P , noting the direction of this transit. 

If T and T f , the periods of vibration as deduced from the two 
sets of observations, are nearly equal, we may proceed to a more 
accurate determination of the period by combining the two series 
of observations. 

Dividing P P by T, the quotient ought to be very nearly 
an integer, even or odd according as the transits P and P are 
in the same or in opposite directions. If this is not the case, the 
series of observations is worthless, but if the result is very nearly 
a whole number n, we divide P P by n, and thus find the mean 
value of T for the whole time of swinging. 

740.] The time of vibration T thus found is the actual mean 
time of vibration, and is subject to corrections if we wish to deduce 
from it the time of vibration in infinitely small arcs and without 
damping. 

To reduce the observed time to the time in infinitely small arcs, 
we observe that the time of a vibration of amplitude a is in general 
of the form T - T^(l + *c 2 ), 

where K is a coefficient, which, in the case of the ordinary pendulum, 
is -g^. Now the amplitudes of the successive vibrations are c, 
cp~ 1 f cp~ 2 , ... cp l ~ n , so that the whole time of n vibrations is 



where T is the time deduced from the observations. 

Hence, to find the time T^ in infinitely small arcs, we have 
approximately, 



n p-! 
To find the time T when there is no damping, we have 



sn a 







741.] The equation of the rectilinear motion of a body, attracted 
to a fixed point and resisted by a force varying as the velocity, is 

7 n j 

.^ + 2*^+*(*-)= s O, (1) 

where x is the coordinate of the body at the time t, and a is the 
coordinate of the point of equilibrium. 



342 ELECTROMAGNETIC OBSERVATIONS. [?4 2 - 

To solve this equation, let 

x-a = e-Vy; (2) 

then gl + ^.^^o; (3) 

the solution of which is 

y Ccos (\/oo 2 IP t-\-d), when k is less than <o ; (4) 

y = A + Bt, when k is equal to o> ; (5) 

and y C cos h ( Vk* o> 2 1 + a), when k is greater than o>. (6) 

The value of a? may be obtained from that of y by equation (2). 
When k is less than o>, the motion consists of an infinite series of 
oscillations, of constant periodic time, but of continually decreasing 
amplitude. As k increases, the periodic time becomes longer, and 
the diminution of amplitude becomes more rapid. 

When k (half the coefficient of resistance) becomes equal to or 
greater than o>, (the square root of the acceleration at unit distance 
from the point of equilibrium,) the motion ceases to be oscillatory, 
and during the whole motion the body can only once pass through 
the point of equilibrium, after which it reaches a position of greatest 
elongation, and then returns towards the point of equilibrium, con 
tinually approaching, but never reaching it. 

Galvanometers in which the resistance is so great that the motion 
is of this kind are called dead beat galvanometers. They are useful 
in many experiments, but especially in telegraphic signalling, in 
which the existence of free vibrations would quite disguise the 
movements which are meant to be observed. 

Whatever be the values of k and o>, the value of a, the scale- 
reading at the point of equilibrium, may be deduced from five scale- 
readings, p, q, r, s, t, taken at equal intervals of time, by the formula 






(p-2+r) (r- 2s + 1) - (q- 

On the Observation of the Galvanometer. 

742.] To measure a constant current with the tangent galvano 
meter, the instrument is adjusted with the plane of its coils parallel 
to the magnetic meridian, and the zero reading is taken. The 
current is then made to pass through the coils, and the deflexion 
of the magnet corresponding to its new position of equilibrium is 
observed. Let this be denoted by $. 

Then, if // is the horizontal magnetic force, G the coefficient of 
the galvanometer, and y the strength of the current, 

(I) 



744-] DEFLEXION OF THE GALVANOMETER. 343 

If the coefficient of torsion of the suspension fibre is r MH (see 
Art. 452), we must use the corrected formula 

JT 

y = -(tan$+r(j[>sec<). (2) 



Best Value of the Deflexion. 

743.] In some galvanometers the number of windings of the 
coil through which the current flows can be altered at pleasure. 
In others a known fraction of the current can be diverted from the 
galvanometer by a conductor called a Shunt. In either case the 
value of G, the effect of a unit-current on the magnet, is made 
to vary. 

Let us determine the value of , for which a given error in the 
observation of the deflexion corresponds to the smallest error of the 
deduced value of the strength of the current. 

Differentiating equation (1), we find 

dy H , . 

4 = ^ sec *- 

Eliminating G, -~ = sin 2 $. (4) 

This is a maximum for a given value of y when the deflexion is 
45. The value of G should therefore be adjusted till Gy is as 
nearly equal to H as is possible ; so that for strong currents it is 
better not to use too sensitive a galvanometer. 

On the Best Method of applying the Current. 

744.] When the observer is able, by means of a key, to make or 
break the connexions of the circuit at any instant, it is advisable to 
operate with the key in such a way as to make the magnet arrive 
at its position of equilibrium with the least possible velocity. The 
following method was devised by Gauss for this purpose. 

Suppose that the magnet is in its position of equilibrium, and that 
there is no current. The observer now makes contact for a short 
time, so that the magnet is set in motion towards its new position 
of equilibrium. He then breaks contact. The force is now towards 
the original position of equilibrium, and the motion is retarded. If 
this is so managed that the magnet comes to rest exactly at the 
new position of equilibrium,, and if the observer again makes con 
tact at that instant and maintains the contact, the magnet will 
remain at rest in its new position. 



344 ELECTROMAGNETIC OBSERVATIONS. [745. 

If we neglect the effect of the resistances and also the inequality 
of the total force acting in the new and the old positions, then, 
since we wish the new force to generate as much kinetic energy 
during the time of its first action as the original force destroys 
while the circuit is broken, we must prolong the first action of the 
current till the magnet has moved over half the distance from the 
first position to the second. Then if the original force acts while 
the magnet moves over the other half of its course, it will exactly 
stop it. Now the time required to pass from a point of greatest 
elongation to a point half way to the position of equilibrium is 
one-sixth of a complete period, or one-third of a single vibration. 

The operator, therefore, having previously ascertained the time 
of a single vibration, makes contact for one-third of that time, 
breaks contact for another third of the same time, and then makes 
contact again during the continuance of the experiment. The 
magnet is then either at rest, or its vibrations are so small that 
observations may be taken at once, without waiting for the motion 
to die away. For this purpose a metronome may be adjusted so as 
to beat three times for each single vibration of the magnet. 

The rule is somewhat more complicated when the resistance is of 
sufficient magnitude to be taken into account, but in this case the 
vibrations die away so fast that it is unnecessary to apply any 
corrections to the rule. 

When the magnet is to be restored to its original position, the 
circuit is broken for one-third of a vibration, made again for an 
equal time, and finally broken. This leaves the magnet at rest in 
its former position. 

If the reversed reading is to be taken immediately after the direct 
one, the circuit is broken for the time of a single vibration and 
then reversed. This brings the magnet to rest in the reversed 
position. 

Measurement l>y the First Swing. 

745.] When there is no time to make more than one observation, 
the current may be measured by the extreme elongation observed 
in the first swing of the magnet. If there is no resistance, the 
permanent deflexion $ is half the extreme elongation. If the re 
sistance is such that the ratio of one vibration to the next is p, and 
if is the zero reading, and d l the extreme elongation in the first 
swing, the deflexion, <, corresponding to the point of equilibrium is 



0Q+P0! 

9 1+p 



747-] SERIES OF OBSERVATION S. 345 

In this way the deflexion may be calculated without waiting for 
the magnet to come to rest in its position of equilibrium. 

To make a Series of Observations. 

746.] The best way of making a considerable number of mea 
sures of a constant current is by observing three elongations while 
the current is in the positive direction, then breaking contact for 
about the time of a single vibration, so as to let the magnet swing 
into the position of negative deflexion, then reversing the current 
and observing three successive elongations on the negative side, 
then breaking contact for the time of a single vibration and re 
peating the observations on the positive side, and so on till a suffi 
cient number of observations have been obtained. In this way the 
errors which may arise from a change in the direction of the earth s 
magnetic force during the time of observation are eliminated. The 
operator, by carefully timing the making and breaking of contact, 
can easily regulate the extent of the vibrations, so as to make them 
sufficiently small without being indistinct. The motion of the 
magnet is graphically represented in Fig. 59, where the abscissa 
represents the time, and the ordinate the deflexion of the magnet. 
If 1 . . . 6 be the observed elongations, the deflexion is given by the 
equation 8 = + 2 + 0_0_20 0. 




Fig. 59. 

Method of Multiplication. 

747.] In certain cases, in which the deflexion of the galvanometer 
magnet is very small, it may be advisable to increase the visible 
effect by reversing the current at proper intervals, so as to set 
up a swinging motion of the magnet. For this purpose, after 
ascertaining the time, T, of a single vibration of the magnet, the 
current is sent in the positive direction for a time T, then in the 
reversed direction for an equal time, and so on. When the motion 
of the magnet has become visible, we may make the reversal of the 
current at the observed times of greatest elongation. 

Let the magnet be at the positive elongation , and let the 
current be sent through the coil in the negative direction. The 



346 ELECTROMAGNETIC OBSERVATIONS. [748. 

point of equilibrium is then $, and the magnet will swing to a 
negative elongation 0, such that 



Similarly, if the current is now made positive while the magnet 
swings to 2 , P 2 = -0 1 + (p+ 1) 0, 

or P 2 2 = + (P+1) 2 4>; 
and if the current is reversed n times in succession, we find 



whence we may find < in the form 



***FTf=7*- 

If ^ is a number so great that p~ n may be neglected, the ex 
pression becomes n 1 



The application of this method to exact measurement requires an 
accurate knowledge of p, the ratio of one vibration of the magnet 
to the next under the influence of the resistances which it expe 
riences. The uncertainties arising from the difficulty of avoiding 
irregularities in the value of p generally outweigh the advantages 
of the large angular elongation. It is only where we wish to 
establish the existence of a very small current by causing it to 
produce a visible movement of the needle that this method is really 
valuable. 

On the Measurement of Transient Currents. 

748.] When a current lasts only during a very small fraction of 
the time of vibration of the galvanometer-magnet, the whole quan 
tity of electricity transmitted by the current may be measured by 
the angular velocity communicated to the magnet during the 
passage of the current, and this may be determined from the 
elongation of the first vibration of the magnet. 

If we neglect the resistance which damps the vibrations of the 
magnet, the investigation becomes very simple. 

Let y be the intensity of the current at any instant, and Q the 
quantity of electricity which it transmits, then 



= \ydt. (1) 



749-] TRANSIENT CURRENTS. 347 

Let M be the magnetic moment, and A the moment of inertia of 
the magnet and suspended apparatus, 

,72/9 

A "L^ + MHsm = MGy cos 0. (2) 

(It 

If the time of the passage of the current is very small, we may 
integrate with respect to t during this short time without regarding 
the change of 0, and we find 



=MG cos y dt + C = MGQ cos + C. (3) 

This shews that the passage of the quantity Q produces an angular 
momentum MGQ cos in the magnet, where is the value of 
at the instant of passage of the current. If the magnet is initially 
in equilibrium, we may make = 0. 

The magnet then swings freely and reaches an elongation 1 . If 
there is no resistance, the work done against the magnetic force 
during this swing is MR (I cosflj. 

The energy communicated to the magnet by the current is 



Equating these quantities, we find 

lf = 2 ^(l-cos<y, (4) 

s-IJ- a ^ * 

tf6- -^t 

dO /MH . 

whence -=- = 2 A / - sin J 0j 

^ \ A 

i\/rn 

Qby(3). (5) 



A 

But if T be the time of a single vibration of the magnet, 



T 



= " A/ 



(6) 



TT m 

and we find Q = - - 2 sin \ Q lt (7) 

where // is the horizontal magnetic force, Q- the coefficient of the 
galvanometer, T the time of a single vibration, and O l the first- 
elongation of the magnet. 

749.] In many actual experiments the elongation is a small 
angle, and it is then easy to take into account the effect of resist 
ance, for we may treat the equation of motion as a linear equation. 

Let the magnet be at rest at its position of equilibrium, let an 
angular velocity v be communicated to it instantaneously, and let 
its first elongation be O l . 



348 ELECTROMAGNETIC OBSERVATIONS. [750. 

The equation of motion is 



(8) 

= C^secpe-^t^Pcosfa t + p). (9) 

cl-t 

,j a 

When t = 0, 6 = 0, and = C(d l = v. 



dt 



When <*>!$ + p = -> 



Hence 0, = -- e v * cos/3. (11) 

ME 

JNow = or = o>i sec^/3, (12) 

^4 

x 

tan = -, wj^^, (13) 

7T jt-i 



Hence * 1 = , (l.) 

- 



which gives the first elongation in terms of the quantity of elec 
tricity in the transient current, and conversely, where T^ is the 
observed time of a single vibration as affected by the actual resist 
ance of damping. When A. is small we may use the approximate 
formula TT T 



Method of Recoil. 

750.] The method given above supposes the magnet to be at 
rest in its position of equilibrium when the transient current is 
passed through the coil. If we wish to repeat the experiment 
we must wait till the magnet is again at rest. In certain cases, 
however, in which we are able to produce transient currents of 
equal intensity, and to do so at any desired instant, the following 
method, described by Weber *, is the most convenient for making 
a continued series of observations. 

* Rcsullate des Magnetisckcn Vereins, 1838, p. 98. 



75O.] METHOD OF KECOIL. 349 

Suppose that we set the magnet swinging by means of a transient 
current whose value is Q Q . If, for brevity, we write 

G V^TT 2 -itan-i 

Jf~T~~ e n = jSr ( 18 ) 

then the first elongation 

^ = KQ, = ^ (say). (19) 

The velocity instantaneously communicated to the magnet at 
starting is jf Q 

v- ^rft- (20) 

When it returns through the point of equilibrium in a negative 
direction its velocity will be 

v 1 =ve~^. (21) 

The next negative elongation will be 

6 z = -6 1 e-* = b 1 . (22) 

When the magnet returns to the point of equilibrium, its velocity 
will be V2 = V() e- 2 \ (23) 

Now let an instantaneous current, whose total quantity is Q, 
be transmitted through the coil at the instant when the magnet is 
at the zero point. It will change the velocity v 2 into v 2 v, where 



If Q is greater than Q e~ 2 ^, the new velocity will be negative and 
equal to 



^^ VH5 "BO* 

The motion of the magnet will thus be reversed, and the next 
elongation will be negative, 

3 = K(Q Q 6~ 2A ) = c 1 = KQ + O^^. (25) 

The magnet is then allowed to come to its positive elongation 

and when it again reaches the point of equilibrium a positive 
current whose quantity is Q is transmitted. This throws the 
magnet back in the positive direction to the positive elongation 

or, calling this the first elongation of a second series of four, 

# 2 = KQ (1 <?~" 2A )-f a^e~^ K . (28) 

Proceeding in this way, by observing two elongations + and , 

then sending a positive current and observing two elongations 



350 



ELECTROMAGNETIC OBSERVATIONS. 



[75 1 - 



and -f , then sending a positive current, and so on, we obtain 
a series consisting of sets of four elongations, in each of which 



and 



(29) 



(30) 



If n series of elongations have been observed, then we find the 
logarithmic decrement from the equation 



and Q from the equation 







. (32) 



Fig, 60. 

The motion of the magnet in the method of recoil is graphically 
represented in Fig. 60, where the abscissa represents the time, and 
the ordinate the deflexion of the magnet at that time. See Art. 760. 

Method of Multiplication. 

751.] If we make the transient current pass every time that the 

magnet passes through the zero point, and always so as to increase 

the velocity of the magnet, then, if 1} 2 , &c. are the successive 

elongations, ^ = -KQ-e~* O lf (33) 

O s =-KQ-e-^e 2 . (34) 

The ultimate value to which the elongation tends after a great 

many vibrations is found by putting n = Q n -i > whence we find 

( 35 ) 



If A is small, the value of the ultimate elongation may be large, 
but since this involves a long continued experiment, and a careful 
determination of A, and since a small error in A introduces a large 
error in the determination of Q, this method is rarely useful for 



75I-] MISTIMING THE CURRENT. 351 

numerical determination, and should be reserved for obtaining- evi 
dence of the existence or non-existence of currents too small to be 
observed directly. 

In all experiments in which transient currents are made to act on 
the moving 1 magnet of the galvanometer, it is essential that the 
whole current should pass while the distance of the magnet from 
the zero point remains a small fraction of the total elongation. 
The time of vibration should therefore be large compared with the 
time required to produce the current, and the operator should have 
his eye on the motion of the magnet, so as to regulate the instant 
of passage of the current by the instant of passage of the magnet 
through its point of equilibrium. 

To estimate the error introduced by a failure of the operator to 
produce the current at the proper instant, we observe that the effect 
of a force in increasing the elongation varies as 



and that this is a maximum when = 0. Hence the error arising 
from a mistiming of the current will always lead to an under 
estimation of its value, and the amount of the error may be 
estimated by comparing the cosine of the phase of the vibration at 
the time of the passage of the current with unity. 






CHAPTER XVII. 



COMPARISON OF COILS. 

Experimental Determination of the Electrical Constants 
of a Coil. 

752.] WE have seen in Art. 717 that in a sensitive galvanometer 
the coils should he of small radius, and should contain many 
windings of the wire. It would he extremely difficult to determine 
the electrical constants of such a coil hy direct measurement of its 
form and dimensions, even if we could obtain access to every 
winding of the wire in order to measure it. But in fact the 
greater number of the windings are not only completely hidden 
by the outer windings, but we are uncertain whether the pressure 
of the outer windings may not have altered the form of the inner 
ones after the coiling of the wire. 

It is better therefore to determine the electrical constants of the 
coil by direct electrical comparison with a standard coil whose con 
stants are known. 

Since the dimensions of the standard coil must be determined by 
actual measurement, it must be made of considerable size, so that 
the unavoidable error of measurement of its diameter or circum 
ference may be as small as possible compared with the quantity 
measured. The channel in which the coil is wound should be of 
rectangular section, and the dimensions of the section should be 
small compared with the radius of the coil. This is necessary, not 
so much in order to diminish the correction for the size of the 
section, as to prevent any uncertainty about the position of those 
windings of the coil which are hidden by the external windings *. 

* Large tangent galvanometers are sometimes made with a single circular con 
ducting ring of considerable thickness, which is sufficiently stiff to maintain its form 
without any support. This is not a good plan for a standard instrument. The dis 
tribution of the current within the conductor depends on the relative conductivity 



753-] PRINCIPAL CONSTANTS OF A COIL. 353 

The principal constants which we wish to determine are 

(1) The magnetic force at the centre of the coil due to a unit- 
current. This is the quantity denoted by G 1 in Art. 700. 

(2) The magnetic moment of the coil due to a unit-current. 
This is the quantity ff 1 . 

753.] To determine G 1 . Since the coils of the working galva 
nometer are much smaller than the standard coil, we place the 
galvanometer within the standard coil, so that their centres coincide, 
the planes of both coils being vertical and parallel to the earth s 
magnetic force. We have thus obtained a differential galvanometer 
one of whose coils is the standard coil, for which the value of G 
is known, while that of the other coil is /, the value of which we 
have to determine. 

The magnet suspended in the centre of the galvanometer coil 
is acted on by the currents in both coils. If the strength of the 
current in the standard coil is y, and that in the galvanometer coil 
y , then, if these currents flowing in opposite directions produce a 
deflexion 6 of the magnet, 

#tan8= G^y -G l7 , (1) 

where H is the horizontal magnetic force of the earth. 

If the currents are so arranged as to produce no deflexion, we 
may find <?/ by the equation 

<?/= -, e,. ( 2 ) 

We may determine the ratio of y to y in several ways. Since the 
value of G l is in general greater for the galvanometer than for the 
standard coil, we may arrange the circuit so that the whole current 
y flows through the standard coil, and is then divided so that y 
flows through the galvanometer and resistance coils, the combined 
resistance of which is J? 13 while the remainder y y flows through 
another set of resistance coils whose combined resistance is E . 



of its various parts. Hence any concealed flaw in the continuity of the metal may 
cause the main stream of electricity to flow either close to the outside or close to the 
inside of the circular ring. Thus the true path of the current becomes uncertain. 
Besides this, when the current flows only once round the circle, especial care is 
necessary to avoid any action on the suspended magnet due to the current on its 
way to or from the circle, because the current in the electrodes is equal to that in 
the circle. In the construction of many instruments the action of this part of the 
current seems to have been altogether lost sight of. 

The most perfect method is to make one of the electrodes in the form of a metal 
tube, and the other a wire covered with insulating material, and placed inside the 
tube and concentric with it. The external action of the electrodes when thus arranged 
is zero, by Art. 683. 

VOL. II. A a 



354 COMPARISON OF COILS. [754- 

We have then, by Art. 276, 



or = . (4) 

V H -i 

and G ; = ^+^ Gl . (5) 

tf 2 

If there is any uncertainty about the actual resistance of the 
galvanometer coil (on account, say, of an uncertainty as to its tem 
perature) we may add resistance coils to it, so that the resistance of 
the galvanometer itself forms but a small part of H lt and thus 
introduces but little uncertainty into the final result. 

754.] To determine g lt the magnetic moment of a small coil due 
to a unit-current flowing through it, the magnet is still suspended 
at the centre of the standard coil, but the small coil is moved 
parallel to itself along the common axis of both coils, till the same 
current, flowing in opposite directions round the coils, no longer 
deflects the magnet. If the distance between the centres of the 
coils is r, we have now 

=2 4 + 3^+4^f +&c. ( 6 ) 

^.O ^.4 >O 

By repeating the experiment with the small coil on the opposite 
side of the standard coil, and measuring the distance between the 
positions of the small coil, we eliminate the uncertain error in the 
determination of the position of the centres of the magnet and 
of the small coil, and we get rid of the terms in g 2) g, &c. 

If the standard coil is so arranged that we can send the current 
through half the number of windings, so as to give a different value 
to G 19 we may determine a new value of r, and thus, as in Art. 454, 
we may eliminate the term involving g^ . 

It is often possible, however, to determine g z by direct measure 
ment of the small coil with sufficient accuracy to make it available 
in calculating the value of the correction to be applied to g^ in 
the equation i 



where # 3 = -ir0 a (6 2 -f 3f 2 2j 2 ), by Art. 700. 

o 



Comparison of Coefficients of Induction. 

755.] It is only in a small number of cases that the direct 
calculation of the coefficients of induction from the form and 



755-] 



MUTUAL INDUCTION OF TWO COILS. 



355 



position of the circuits can be easily performed. In order to attain 
a sufficient degree of accuracy, it is necessary that the distance 
between the circuits should be capable of exact measurement. 
But when the distance between the circuits is sufficient to prevent 
errors of measurement from introducing large errors into the result, 
the coefficient of induction itself is necessarily very much reduced 
in magnitude. Now for many experiments it is necessary to make 
the coefficient of induction large, and we can only do so by bringing 
the circuits close together, so that the method of direct measure 
ment becomes impossible, and, in order to determine the coefficient 
of induction, we must compare it with that of a pair of coils ar 
ranged so that their coefficient may be obtained by direct measure 
ment and calculation. 

This may be done as follows : 

Let A and a be the standard 
pair of coils, B and b the coils to 
be compared with them. Con 
nect A and B in one circuit, and 
place the electrodes of the gal 
vanometer, G, at P and Q, so 
that the resistance of PAQ is 
R, and that of QBP is S, K 
being the resistance of the gal 
vanometer. Connect a and b in 
one circuit with the battery. Fi g . 51. 

Let the current in A be , 

that in B, y> and that in the galvanometer, sc y, that in the battery 
circuit being y. 

Then, if M l is the coefficient of induction between A and , and 
M 2 that between B and b, the integral induction current through 
the galvanometer at breaking the battery circuit is 




x-y - y 



R" S 



1 + 



(8) 



. 

R "" 8 



By adjusting the resistances R and 8 till there is no current 
through the galvanometer at making or breaking the galvanometer 
circuit, the ratio of M 2 to M 1 may be determined by measuring that 
of S to R. 



A a 2 



356 COMPARISON OF COILS. [756. 

Comparison of a Coefficient of Self-induction with a Coefficient of 
Mu tual Induction . 

756.] In the branch AF of Wheatstone s Bridge let a coil be 

inserted, the coefficient of self-induc 
tion of which we wish to find. Let 
us call it L. 

In the connecting wire between A 
and the battery another coil is inserted. 
The coefficient of mutual induction be 
tween this coil and the coil in AF 
is M. It may be measured by the 
method described in Art. 755. 

If the current from A to F is #, and 

.p. 62 that from A to H is ^, that from Z 

to A, through B, will be oc+y. The 
external electromotive force from A to F is 




The external electromotive force along AH is 

A-H=Qy. (10) 

If the galvanometer placed between F and H indicates no current, 
either transient or permanent, then by (9) and (10), since I1 F=0, 



whence L = - (l + ~) M. (13) 

^o 

Since L is always positive, M must be negative, and therefore the 
current must flow in opposite directions through the coils placed 
in P and in B. In making the experiment we may either begin 
by adjusting the resistances so that 

PS=QR, (14) 

which is the condition that there may be no permanent current, 
and then adjust the distance between the coils till the galvanometer 
ceases to indicate a transient current on making and breaking the 
battery connexion ; or, if this distance is not capable of adjustment, 
we may get rid of the transient current by altering the resistances 
Q and S in such a way that the ratio of Q to S remains constant. 
If this double adjustment is found too troublesome, we may adopt 



757-] SELF-INDUCTION. 357 

a third method. Beginning with an arrangement in which the 
transient current due to self-induction is slightly in excess of that 
due to mutual induction, we may get rid of the inequality by in 
serting a conductor whose resistance is W between A and Z. The 
condition of no permanent current through the galvanometer is not 
affected by the introduction of W. We may therefore get rid of 
the transient current by adjusting the resistance of W alone. When 
this is done the value of L is 

. (15) 



. 

Comparison of the Coefficients of Self -induction of Two Coils. 

757.] Insert the coils in two adjacent branches of Wheatstone s 
Bridge. Let L and N be the coefficients of self-induction of the 
coils inserted in P and in R respectively, then the condition of no 
galvanometer current is 

(P* + l^)8y=Qy(X* + N%), (16) 

whence PS = QJR, for no permanent current, (17) 

and = , for no transient current. (18) 

JT J-l/ 

Hence, by a proper adjustment of the resistances, both the per 
manent and the transient current can be got rid of, and then 
the ratio of L to N can be determined by a comparison of the 
resistances. 



CHAPTER XVIIL 



ELECTROMAGNETIC UNIT OF RESISTANCE. 



On the Determination of the Resistance of a Coil in Electro 
nic Measure. 



758.] THE resistance of a conductor is defined as the ratio of the 
numerical value of the electromotive force to that of the current 
which it produces in the conductor. The determination of the 
value of the current in electromagnetic measure can be made by 
means of a standard galvanometer, when we know the value of the 
earth s magnetic force. The determination of the value of the 
electromotive force is more difficult, as the only case in which we 
can directly calculate its value is when it arises from the relative 
motion of the circuit with respect to a known magnetic system. 

759.] The first determination of the resistance of a wire in 
electromagnetic measure was made by Kirchhoff*. He employed 
two coils of known form, A 1 and A^ and calculated their coefficient 

of mutual induction from the geo 
metrical data of their form and 
position. These coils were placed 
in circuit with a galvanometer, 6r, 
and a battery, B, and two points 
of the circuit, P, between the coils, 
and Q, between the battery and 
galvanometer, were joined by the 
wire whose resistance, R, was to be measured. 

When the current is steady it is divided between the wire and 
the galvanometer circuit, and produces a certain permanent de 
flexion of the galvanometer. If the coil A 1 is now removed quickly 

* * Bestimmong Her Constanten von welcher die Intensitat inducirter elektrischer 
Strome abhangt. Pogg. Ann., Ixxvi (April 1849). 




759-] KIRCHHOFF S METHOD. 359 

from A 2 and placed in a position in which the coefficient of mutual 
induction between A l and A. 2 is zero (Art. 538), a current of induc 
tion is produced in both circuits, and the galvanometer needle 
receives an impulse which produces a certain transient deflexion. 

The resistance of the wire, R, is deduced from a comparison 
between the permanent deflexion, due to the steady current, and the 
transient deflexion, due to the current of induction. 

Let the resistance of QGA l P be K, of PA 2 Q, B, and of PQ, R. 

Let Lj M and N be the coefficients of induction of A l and A 2 . 

Let x be the current in (7, and y that in J3, then the current 
from P to Q is x y. 

Let E be the electromotive force of the battery, then 

)= o, (l) 



Rx + (B + R}y + -j- (Mx + Ny} = E. (2) 

When the currents are constant, and everything at rest, 

(K+R}x-Ry = 0. (3) 
If M now suddenly becomes zero on account of the separation of 
A 1 from A 2 , then, integrating with respect to t, 

J / "" \ / 

Mx = lEdt = 0. (5) 



whence x = M (B \. R] ^ ml ^2 ( 6 ) 

Substituting the value of y in terms of x from (3), we find 

6 = ~R (B + R)(K+R}-R? (7) 



When, as in Kirchhoff s experiment, both B and K are large 
compared with R, this equation is reduced to 

x _M 

~x~~R 

Of these quantities, x is found from the throw of the galvanometer 
due to the induction current. See Art. 768. The permanent cur 
rent, at, is found from the permanent deflexion due to the steady 
current; see Art. 746. M is found either by direct calculation 
from the geometrical data, or by a comparison with a pair of coils, 
for which this calculation has been made; see Art. 755. From 



360 UNIT OF RESISTANCE. [760. 

these three quantities R can be determined in electromagnetic mea 
sure. 

These methods involve the determination of the period of vibra 
tion of the galvanometer magnet, and of the logarithmic decrement 
of its oscillations. 

Weber s Method by Transient Currents*. 

760.] A coil of considerable size is mounted on an axle, so as to 
be capable of revolving about a vertical diameter. The wire of this 
coil is connected with that of a tangent galvanometer so as to form 
a single circuit. Let the resistance of this circuit be R. Let the 
large coil be placed with its positive face perpendicular to the 
magnetic meridian, and let it be quickly turned round half a revo 
lution. There will be an induced current due to the earth s mag 
netic force, and the total quantity of electricity in this current in 
electromagnetic measure will be 



where ff l is the magnetic moment of the coil for unit current, which 
in the case of a large coil may be determined directly, by mea 
suring the dimensions of the coil, and calculating the sum of the 
areas of its windings. If is the horizontal component of terrestrial 
magnetism, and R is the resistance of the circuit formed by the 
coil and galvanometer together. This current sets the magnet of 
the galvanometer in motion. 

If the magnet is originally at rest, and if the motion of the coil 
occupies but a small fraction of the time of a vibration of the 
magnet, then, if we neglect the resistance to the motion of the 
magnet, we have, by Art. 748, 

// T 

<2=^-2sinU (2) 

Cr 7T 

where G is the constant of the galvanometer, T is the time of 
vibration of the magnet, and 6 is the observed elongation. From 
these equations we obtain 

* = * 15& . (3) 

The value of H does not appear in this result, provided it is the 
same at the position of the coil and at that of the galvanometer. 
This should not be assumed to be the case, but should be tested by 
comparing the time of vibration of the same magnet, first at one of 
these places and then at the other. 

* ElcU. Moots*. ; or Pogg., Ann. Ixxxii, 337 (1851). 



762.] WEBER S METHOD. 361 

761.] To make a series of observations Weber began with the 
coil parallel to the magnetic meridian. He then turned it with its 
positive face north, and observed the first elongation due to the 
negative current. He then observed the second elongation of the 
freely swinging magnet, and on the return of the magnet through 
the point of equilibrium he turned the coil with its positive face 
south. This caused the magnet to recoil to the positive side. The 
series Was continued as in Art. 750, and the result corrected for 
resistance. In this way the value of the resistance of the combined 
circuit of the coil and galvanometer was ascertained. 

In all such experiments it is necessary, in order to obtain suffi 
ciently large deflexions, to make the wire of copper, a metal which, 
though it is the best conductor, has the disadvantage of altering 
considerably in resistance with alterations of temperature. It is 
also very difficult to ascertain the temperature of every part of the 
apparatus. Hence, in order to obtain a result of permanent value 
from such an experiment, the resistance of the experimental circuit 
should be compared with that of a carefully constructed resistance- 
coil, both before and after each experiment. 

Weber s Method by observing the Decrement of the Oscillations 
of a Magnet. 

762.] A magnet of considerable magnetic moment is suspended 
at the centre of a galvanometer coil. The period of vibration and 
the logarithmic decrement of the oscillations is observed, first with 
the circuit of the galvanometer open, and then with the circuit 
closed, and the conductivity of the galvanometer coil is deduced 
from the effect which the currents induced in it by the motion of 
the magnet have in resisting that motion. 

If T is the observed time of a single vibration, and A. the Na 
pierian logarithmic decrement for each single vibration, then, if we 

write ,, 

o> = ^> (1) 

and a = ~ , (2) 

the equation of motion of the magnet is of the form 

$ = Ce- at cos(o>t + (3}. (3) 

This expresses the nature of the motion as determined by observa 
tion. We must compare this with the dynamical equation of 
motion. 



362 UNIT OF RESISTANCE. [?62. 

Let M be the coefficient of induction between the galvanometer 
coil and the suspended magnet. It is of the form 

M = Giffi Qi TO + $222 $2 W + &c., (4) 

where G 1} G 2 , &c. are coefficients belonging to the coil, ff l3 g z , &c. 
to the magnet, and Q l (0), Q. 2 (Q), &c., are zonal harmonics. of the 
angle between the axes of the coil and the magnet. See Art. 700. 
By a proper arrangement of the coils of the galvanometer, and by 
building up the suspended magnet of several magnets placed side by 
side at proper distances, we may cause all the terms of M after the 
first to become insensible compared with the first. If we also put 

(f> = -- 0, we may write 

M = Gm sin$, (5) 

where G is the principal coefficient of the galvanometer, m is the 
magnetic moment of the magnet, and $ is the angle between the 
axis of the magnet and the plane of the coil, which, in this ex 
periment, is always a small angle. 

If I/ is the coefficient of self-induction of the coil, and R its 
resistance, and y the current in the coil, 

0, (6) 



or L~ -fj^y-f mcos( - = 0. (7) 

U/t Cit 

The moment of the force with which the current y acts on the 

magnet is y r , or Gmy cos $. The angle </> is in this experiment 

ct cp 

so small, that we may suppose cos < = 1 . 

Let us suppose that the equation of motion of the magnet when 
the circuit is broken is 



where A is the moment of inertia of the suspended apparatus, S~- 

Cvv 

expresses the resistance arising from the viscosity of the air and 
of the suspension fibre, &c., and C<$> expresses the moment of the 
force arising from the earth s magnetism, the torsion of the sus 
pension apparatus, &c., tending to bring the magnet to its position 
of equilibrium. 

The equation of motion, as affected by the current, will be 

A + sc 



762.] WEBER S METHOD. 363 

To determine the motion of the magnet, we have to combine this 
equation with (7) and eliminate y. The result is 



a linear differential equation of the third order. 

We have no occasion, however, to solve this equation, because 
the data of the problem are the observed elements of the motion 
of the magnet, and from these we have to determine the value 
of E. 

Let a and o) be the values of a and o> in equation (2) when the 
circuit is broken. In this case R is infinite, and the equation is 
reduced to the form (8). We thus find 

B=2Aa Q , C=A(a^ + ^). (11) 

Solving equation (10) for R, and writing 



we find 



o), where i=V I, (12) 



Since the value of co is in general much greater than that of a, 
the best value of R is found by equating the terms in i o>, 




2A(a a ) a-a 

We may also obtain a value of R by equating the terms not 
involving i, but as these terms are small, the equation is useful 
only as a means of testing the accuracy of the observations. From 
these equations we find the following testing equation, 



(co 2 -o> 2 ) 2 }. (15) 

Since LAv? is very small compared with G 2 m 2 , this equation 

a 2 -a 2 ; (16) 

and equation (14) may be written 

E= GV_ L 

2A(a-a ) r 

In this expression G may be determined either from the linear 
measurement of the galvanometer coil, or better, by comparison 
with a standard coil, according to the method of Art. 753. A is 
the moment of inertia of the magnet and its suspended apparatus, 
which is to be found by the proper dynamical method. o>, &> , a 
and a , are given by observation. 



364 UNIT OF RESISTANCE. [763. 

The determination of the value of m, the magnetic moment of 
the suspended magnet, is the most difficult part of the investigation, 
because it is affected by temperature, by the earth s magnetic force, 
and by mechanical violence, so that great care must be taken to 
measure this quantity when the magnet is in the very same circum 
stances as when it is vibrating. 

The second term of R, that which involves L, is of less import 
ance, as it is generally small compared with the first term. The 
value of L may be determined either by calculation from the known 
form of the coil, or by an experiment on the extra-current of in 
duction. See Art. 756. 

Thomson s Method by a Revolving Coil. 

763.] This method was suggested by Thomson to the Committee 
of the British Association on Electrical Standards, and the ex 
periment was made by M. M. Balfour Stewart, Fleeming Jenkin, 
and the author in 1863 *. 

A circular coil is made to revolve with uniform velocity about a 
vertical axis. A small magnet is suspended by a silk fibre at the 
centre of the coil. An electric current is induced in the coil by 
the earth s magnetism, and also by the suspended magnet. This 
current is periodic, flowing in opposite directions through the wire 
of the coil during different parts of each revolution, but the effect of 
the current on the suspended magnet is to produce a deflexion from 
the magnetic meridian in the direction of the rotation of the coil. 

764.] Let H be the horizontal component of the earth s mag 
netism. 

Let y be the strength of the current in the coil. 

g the total area inclosed by all the windings of the wire. 
G the magnetic force at the centre of the coil due to unit- 
current. 

L the coefficient of self-induction of the coil. 
M the magnetic moment of the suspended magnet. 
the angle between the plane of the coil and the magnetic 

meridian. 
</> the angle between the axis of the suspended magnet and 

the magnetic meridian 

A the moment of inertia of the suspended magnet. 
MHr the coefficient of torsion of the suspension fibre, 
a the azimuth of the magnet when there is no torsion. 
R the resistance of the coil. 

* See Report of (he British Association for 1863. 



765.] THOMSON S METHOD. 365 

The kinetic energy of the system is 

T=\Ly* -Hgy sm6-MGy sin (0 <f>) + MHcoaQ+b Atf>. (1 ) 

The first term, Jrj&y 2 , expresses the energy of the current as 
depending on the coil itself. The second term depends on the 
mutual action of the current and terrestrial magnetism, the third 
on that of the current and the magnetism of the suspended magnet, 
the fourth on that of the magnetism of the suspended magnet and 
terrestrial magnetism, and the last expresses the kinetic energy of 
the matter composing the magnet and the suspended apparatus 
which moves with it. 

The potential energy of the suspended apparatus arising from the 
torsion of the fibre is 

**-S*0. (2) 



The electromagnetic momentum of the current is 

clT 

(6-<t)), (3) 



dy 
and if R is the resistance of the coil, the equation of the current is 



or, since 6 = tot, (5) 

<p)cos(0(})). (6) 



765.] It is the result .alike of theory and observation that <, the 
azimuth of the magnet, is subject to two kinds of periodic variations. 
One of these is a free oscillation, whose periodic time depends on 
the intensity of terrestrial magnetism, and is, in the experiment, 
several seconds. The other is a forced vibration whose period is 
half that of the revolving coil, and whose amplitude is, as we shall 
see, insensible. Hence, in determining y, we may treat $ as 
sensibly constant. 

We thus find 

y = j/^tftf (Hcos6 + La> sin 0) (7) 

( (8) 



+ Ce * . (9) 

The last term of this expression soon dies away when the rota 
tion is continued uniform. 



366 UNIT OF RESISTANCE. [766. 

The equation of motion of the suspended magnet is 

d*T _dT_ f!F_ 
d<j> dt dfy dcf) 

whence A$ MGy cos (0 c/>)-f Jf # (sin c/> + r (c/> a)) = 0. (11) 

Substituting the value of y, and arranging the terms according 

to the functions of multiples of 6, then we know from observation 

that 

< r= c/> -f be~ lt cos nt + c cos 2 (0 /3), (12) 

where c/> is the mean value of c/>, and the second term expresses 
the free vibrations gradually decaying, and the third the forced 
vibrations arising from the variation of the deflecting current. 

TT~\T 

The value of n in equation (12) is j- secc/>. That of c, the am- 

A. 

n 2 

plitude of the forced vibrations, is J 3- sin c/>. Hence, when the 

co 

coil makes many revolutions during one free vibration of the magnet, 
the amplitude of the forced vibrations of the magnet is very small, 
and we may neglect the terms in (11) which involve c. 

Beginning with the terms in (11) which do not involve 0, we find 



MHGgu /z> J v 

5 CR cos cf> -f L co sin d> ) H ---- - - *-r- R 

2 ^ 



(cl> -a)). (13) 

Remembering that is small, and that L is generally small 
compared with Gg> we find as a sufficiently approximate value of R, 



766.] The resistance is thus determined in electromagnetic mea 
sure in terms of the velocity co and the deviation </>. It is not 
necessary to determine H, the horizontal terrestrial magnetic force, 
provided it remains constant during the experiment. 

M 

To determine we must make use of the suspended magnet to 

deflect the magnet of the magnetometer, as described in Art. 454. 
In this experiment M should be small, so that this correction be 
comes of secondary importance. 

For the other corrections required in this experiment see the 
Report of tli e British Association for 1863, p. 168. 



767.] JOULE S METHOD. 367 

Joule s Calorimetric Method. 

767.] The heat generated by a current y in passing through a 
conductor whose resistance is R is, by Joule s law, Art. 242. 

(1) 

where / is the equivalent in dynamical measure of the unit of heat 
employed. 

Hence, if R is constant during the experiment, its value is 

(2) 



This method of determining R involves the determination of ^, 
the heat generated by the current in a given time, and of y 2 , the 
square of the strength of the current. 

In Joule s experiments *, h was determined by the rise of tem 
perature of the water in a vessel in which the conducting wire was 
immersed. It was corrected for the effects of radiation, &c. by 
alternate experiments in which no current was passed through the 
wire. 

The strength of the current was measured by means of a tangent 
galvanometer. This method involves the determination of the 
intensity of terrestrial magnetism, which was done by the method 
described in* Art. 457. These measurements were also tested by the 
current weigher, described in Art. 726, which measures y 2 directly. 

The most direct method of measuring / y 2 dt y however, is to pass 

the current through a self-acting electrodynamometer (Art. 725) 
with a scale which gives readings proportional to y 2 , and to make 
the observations at equal intervals of time, which may be done 
approximately by taking the reading at the extremities of every 
vibration of the instrument during the whole course of the experi 
ment. 

* Report of the British Association for 1867. 



CHAPTER XIX. 

COMPARISON OF THE ELECTROSTATIC WITH THE ELECTRO 
MAGNETIC UNITS. 

Determination of the Number of Electrostatic Units of Electricity 
in one Electromagnetic Unit. 

768.] THE absolute magnitudes of the electrical units in both 
systems depend on the units of length, time, and mass which we 
adopt, and the mode in which they depend on these units is 
different in the two systems, so that the ratio of the electrical units 
will be expressed by a different number, according to the different 
units of length and time. 

It appears from the table of dimensions, Art. 628, that the 
number of electrostatic units of electricity in one electromagnetic 
unit varies inversely as the magnitude of the unit of length, and 
directly as the magnitude of the unit of time which we adopt. 

If, therefore, we determine a velocity which is represented nu 
merically by this number, then, even if we adopt new units of 
length and of time, the number representing this velocity will still 
be the number of electrostatic units of electricity in one electro 
magnetic unit, according to the new system of measurement. 

This velocity, therefore, which indicates the relation between 
electrostatic and electromagnetic phenomena, is a natural quantity 
of definite magnitude, and the measurement of this quantity is one 
of the most important researches in electricity. 

To shew that the quantity we are in search of is really a velocity, 
we may observe that in the case of two parallel currents the attrac 
tion experienced by a length a of one of them is, by Art. 686, 



F= 

o 

where (7, C are the numerical values of the currents in electromag- 



769.] 11ATIO EXPRESSED BY A VELOCITY. 369 

netic measure, and I the distance between them. If we make 
b = 2 a, then p _ CC\ 

Now the quantity of electricity transmitted by the current C in 
the time t is Ct in electromagnetic measure, or nCt in electrostatic 
measure, if n is the number of electrostatic units in one electro 
magnetic unit. 

Let two small conductors be charged with the quantities of 
electricity transmitted by the two currents in the time t, and 
placed at a distance r from each other. The repulsion between 
them will be CC n 2 t 2 

F = 72- 

Let the distance r be so chosen that this repulsion is equal to the 
attraction of the currents, then 



Hence r = nt-, 

or the distance r must increase with the time t at the rate n. 
Hence n is a velocity, the absolute magnitude of which is the 
same, whatever units we assume. 

769.] To obtain a physical conception of this velocity, let us ima 
gine a plane surface charged with electricity to the electrostatic sur 
face-density <r, and moving in its own plane with a velocity v. This 
moving electrified surface will be equivalent to an electric current- 
sheet, the strength of the current flowing through unit of breadth 

of the surface being- av in electrostatic measure, or - av in elec- 

n 

tromagnetic measure, if n is the number of electrostatic units in 
one electromagnetic unit. If another plane surface, parallel to the 
first, is electrified to the surface-density o- , and moves in the same 
direction with the velocity v , it will be equivalent to a second 
current-sheet. 

The electrostatic repulsion between the two electrified surfaces is, 
by Art. 124, 2 ir<r<r for every unit of area of the opposed surfaces. 

The electromagnetic attraction between the two current-sheets 
is, by Art. 653, 2 ituu for every unit of area, u and u being the 
surface-densities of the currents in electromagnetic measure. 

But u = - (TV. and u = - </v , so that the attraction is 
n n 

,vv 

27TO-0- jr. 

n 2 

VOL. II. B b 



370 COMPARISON OF UNITS. [770. 

The ratio of the attraction to the repulsion is equal to that of 
vv f to n 2 . Hence, since the attraction and the repulsion are quan 
tities of the same kind, n must be a quantity of the same kind as v, 
that is, a velocity. If we now suppose the velocity of each of the 
moving planes to be equal to %, the attraction will be equal to the 
repulsion, and there will be no mechanical action between them. 
Hence we may define the ratio of the electric units to be a velocity, 
such that two electrified surfaces, moving in the same direction 
with this velocity, have no mutual action. Since this velocity is 
about 288000 kilometres per second, it is impossible to make the 
experiment above described. 

770.] If the electric surface-density and the velocity can be made 
so great that the magnetic force is a measurable quantity, we may 
at least verify our supposition that a moving electrified body is 
equivalent to an electric current. 

It appears from Art. 57 that an electrified surface in air would 
begin to discharge itself by sparks when the electric force 2 TTO- 
reaches the value 130. The magnetic force due to the current-sheet 

v 

is 2 TTCT - The horizontal magnetic force in Britain is about 0.175. 
n 

Hence a surface electrified to the highest degree, and moving with 
a velocity of 100 metres per second, would act on a magnet with a 
force equal to about one-four-thousandth part of the earth s hori 
zontal force, a quantity which can be measured. The electrified 
surface may be that of a non-conducting disk revolving in the plane 
of the magnetic meridian, and the magnet may be placed close to 
the ascending or descending portion of the disk, and protected from 
its electrostatic action by a screen of metal. I am not aware that 
this experiment has been hitherto attempted. 

I. Comparison of Units of Electricity. 

771.] Since the ratio of the electromagnetic to the electrostatic 
unit of electricity is represented by a velocity, we shall in future 
denote it by the symbol v. The first numerical determination of 
this velocity was made by Weber and Kohlrausch *. 

Their method was founded on the measurement of the same 
quantity of electricity, first in electrostatic and then in electro 
magnetic measure. 

The quantity of electricity measured was the charge of a Leyden 
jar. It was measured in electrostatic measure as the product of the 

* Elektrodynamische Maasbestimmungen ; and Pogg. Ann. xcix, (Aug. 10, 1856.) 



77I-] METHOD OF WEBER AND KOHLRAUSCH. 371 

capacity of the jar into the difference of potential of its coatings. 
The capacity of the jar was determined by comparison with that of 
a sphere suspended in an open space at a distance from other 
bodies. The capacity of such a sphere is expressed in electrostatic 
measure by its radius. Thus the capacity of the jar may be found 
and expressed as a certain length. See Art. 227. 

The difference of the potentials of the coatings of the jar was mea 
sured by connecting the coatings with the electrodes of an electro 
meter, the constants of which were carefully determined, so that the 
difference of the potentials, U, became known in electrostatic measure. 

By multiplying this by c, the capacity of the jar, the charge of 
the jar was expressed in electrostatic measure. 

To determine the value of the charge in electromagnetic measure, 
the jar was discharged through the coil of a galvanometer. The 
effect of the transient current on the magnet of the galvanometer 
communicated to the magnet a certain angular velocity. The 
magnet then swung round to a certain deviation, at which its 
velocity was entirely destroyed by the opposing action of the 
earth s magnetism. 

By observing the extreme deviation of the magnet the quantity 
of electricity in the current may be determined in electromagnetic 
measure, as in Art. 748, by the formula 

// T 

Q = -^ - 2 sin i<9, 

where Q is the quantity of electricity in electromagnetic measure. 
We have therefore to determine the following quantities : 

U, the intensity of the horizontal component of terrestrial mag 
netism ; see Art. 456. 

G, the principal constant of the galvanometer; see Art. 700. 

T, the time of a single vibration of the magnet ; and 

6, the deviation due to the transient current. 

The value of v obtained by MM. Weber and Kohlrausch was 

v 310740000 metres per second. 

The property of solid dielectrics, to which the name of Electric 
Absorption has been given, renders it difficult to estimate correctly 
the capacity of a Ley den jar. The apparent capacity varies ac 
cording to the time which elapses between the charging or dis 
charging of the jar and the measurement of the potential, and the 
longer the time the greater is the value obtained for the capacity of 
the jar. 

B b 2 



372 COMPARISON OF UNITS. [772. 

Hence, since the time occupied in obtaining 1 a reading of the 
electrometer is large in comparison with the time during which the 
discharge through the galvanometer takes place, it is probable that 
the estimate of the discharge in electrostatic measure is too high, 
and the value of v, derived from it, is probably also too high. 

II. v expressed as a Resistance, 

772. J Two other methods for the determination of v lead to an 
expression of its value in terms of the resistance of a given con 
ductor, which, in the electromagnetic system, is also expressed as a 
velocity. 

In Sir William Thomson s form of the experiment, a constant 
current is made to flow through a wire of great resistance. The 
electromotive force which urges the current through the wire is mea 
sured electrostatically by connecting the extremities of the wire with 
the electrodes of an absolute electrometer, Arts. 217, 218. The 
strength of the current in the wire is measured in electromagnetic 
measure by the deflexion of the suspended coil of an electrodyna- 
mometer through which it passes, Art. 725. The resistance of the 
circuit is known in electromagnetic measure by comparison with a 
standard coil or Ohm. By multiplying the strength of the current 
by this resistance we obtain the electromotive force in electro 
magnetic measure, and from a comparison of this with the electro 
static measure the value of v is obtained. 

This method requires the simultaneous determination of two 
forces, by means of the electrometer and electrodynamometer re 
spectively, and it is only the ratio of these forces which appears in 
the result. 

773.] Another method, in which these forces, instead of being 
separately measured, are directly opposed to each other, was em 
ployed by the present writer. The ends of the great resistance coil 
are connected with two parallel disks, one of which is moveable. 
The same difference of potentials which sends the current through 
the great resistance, also causes an attraction between these disks. 
At the same time, an electric current which, in the actual experi 
ment, was distinct from the primary current, is sent through two 
coils, fastened, one to the back of the fixed disk, and the other to 
the back of the moveable disk. The current flows in opposite 
directions through these coils, so that they repel one another. By 
adjusting the distance of the two disks the attraction is exactly 
balanced by the repulsion, while at the same time another observer, 



774-] METHODS OF THOMSON AND MAXWELL. 373 

by means of a differential galvanometer with shunts, determines 
the ratio of the primary to the secondary current. 

In this experiment the only measurement which must he referred 
to a material standard is that of the great resistance, which must 
be determined in absolute measure by comparison with the Ohm. 
The other measurements are required only for the determination of 
ratios, and may therefore be determined in terms of any arbitrary 
unit. 

Thus the ratio of the two forces is a ratio of equality. 

The ratio of the two currents is found by a comparison of resist 
ances when there is no deflexion of the differential galvanometer. 

The attractive force depends on the square of the ratio of the 
diameter of the disks to their distance. 

The repulsive force depends on the ratio of the diameter of the 
coils to their distance. 

The value of v is therefore expressed directly in terms of the 
resistance of the great coil, which is itself compared with the Ohm. 

The value oft?, as found by Thomson s method, was 28.2 Ohms* ; 
by Maxwell s, 28.8 Ohmsf. 

III. Electrostatic Capacity in Electromagnetic Measure. 

774.] The capacity of a condenser may be ascertained in electro 
magnetic measure by a comparison of the electromotive force which 
produces the charge, and the quantity of electricity in the current 
of discharge. By means of a voltaic battery a current is maintained 
through a circuit containing a coil of great resistance. The con 
denser is charged by putting its electrodes in contact with those of 
che resistance coil. The current through the coil is measured by 
the deflexion which it produces in a galvanometer. Let $ be this 
deflexion, then the current is, by Art. 742, 

H 

TT = tan <f>, 

where H is the horizontal component of terrestrial magnetism, and 
G is the principal constant of the galvanometer. 

If R is the resistance of the coil through which this current is 
made to flow, the difference of the potentials at the ends of the 
coil is E= R-y, 



* Report of British Association, 1869, p. 434. 

t Phil. Trans., 1868, p. 643; and Report of British Association, 1869, p. 436. 



374 COMPAKISON OF UNITS. [775. 

and the charge of electricity produced in the condenser, whose 
capacity in electromagnetic measure is C, will he 



Now let the electrodes of the condenser, and then those of the 
galvanometer, be disconnected from the circuit,, and let the magnet 
of the galvanometer be brought to rest at its position of equili 
brium. Then let the electrodes of the condenser be connected with 
those of the galvanometer. A transient current will flow through 
the galvanometer, and will cause the magnet to swing to an ex 
treme deflexion 0. Then, by Art. 748, if the discharge is equal to 

the charge, jj f 

Q = 2sini0. 

(JT 7T 

We thus obtain as the value of the capacity of the condenser in 
electromagnetic measure 

C 2sin ^ 
TT It tan <p 

The capacity of the condenser is thus determined in terms of the 
following quantities : 

T t the time of vibration of the magnet of the galvanometer from 
rest to rest. 

R, the resistance of the coil. 

0, the extreme limit of the swing produced by the discharge. 

<, the constant deflexion due to the current through the coil ~R. 
This method was employed by Professor Fleeming Jenkin in deter 
mining the capacity of condensers in electromagnetic measure *. 

If c be the capacity of the same condenser in electrostatic mea 
sure, as determined by comparison with a condenser whose capacity 
can be calculated from its geometrical data, 

c = v*C. 

tan$ 



Hence v 2 

T 2 sm 

The quantity v may therefore be found in this way. It depends 
on the determination of R in electromagnetic measure, but as it 
involves only the square root of JR, an error in this determination 
will not affect the value of v so much as in the method of Arts. 
772, 773. 

Intermittent Current. 

775.] If the wire of a battery-circuit be broken at any point, and 

* Report of British Association, 1867. 



776.] WIPPE. 375 

the broken ends connected with the electrodes of a condenser, the 
current will flow into the condenser with a strength which dimin 
ishes as the difference of the potentials of the condenser increases, 
so that when the condenser has received the full charge corre 
sponding to the electromotive force acting on the wire the current 
ceases entirely. 

If the electrodes of the condenser are now disconnected from the 
ends of the wire, and then again connected with them in the 
reverse order, the condenser will discharge itself through the wire, 
and will then become recharged in the opposite way, so that a 
transient current will flow through the wire, the total quantity of 
which is equal to two charges of the condenser. 

By means of a piece of mechanism (commonly called a Commu 
tator, or wippe] the operation of reversing the connexions of the 
condenser can be repeated at regular intervals of time, each interval 
being equal to T. If this interval is sufficiently long to allow of 
the complete discharge of the condenser, the quantity of electricity 
transmitted by the wire in each interval will be 2 EC, where E is 
the electromotive force, and C is the capacity of the condenser. 

If the magnet of a galvanometer included in the circuit is loaded, 
so as to swing so slowly that a great many discharges of the con 
denser occur in the time of one free vibration of the magnet, the 
succession of discharges will act on the magnet like a steady current 
whose strength is 2 EC 

~~T~ 

If the condenser is now removed, and a resistance coil substituted 
for it, and adjusted till the steady current through the galvano 
meter produces the same deflexion as the succession of discharges, 
and if E is the resistance of the whole circuit when this is the case, 

E _2EC. 
~R- ~T~ 

R = TC- ( 2 ) 

We may thus compare the condenser with its commutator in 
motion to a wire of a certain electrical resistance, and we may make 
use of the different methods of measuring resistance described in 
Arts. 345 to 357 in order to determine this resistance. 

776.] For this purpose we may substitute for any one of the 
wires in the method of the Differential Galvanometer, Art. 346, or 
in that of Wheatstone s Bridge, Art. 347, a condenser with its com 
mutator. Let us suppose that in either case a zero deflexion of the 



376 COMPARISON OF UNITS. [777. 

galvanometer has been obtained, first with the condenser and com 
mutator, and then with a coil of resistance R L in its place, then 

T 

the quantity ^ will be measured by the resistance of the circuit of 
2 L> 

which the coil R l forms part, and which is completed by the re 
mainder of the conducting system including the battery. Hence 
the resistance, R, which we have to calculate, is equal to R 1 , that 
of the resistance coil, together with R 2 , the resistance of the re 
mainder of the system (including the battery), the extremities of 
the resistance coil being taken as the electrodes of the system. 

In the cases of the differential galvanometer and Wheatstone s 
Bridge it is not necessary to make a second experiment by substi 
tuting a resistance coil for the condenser. The value of the resist 
ance required for this purpose may be found by calculation from 
the other known resistances in the system. 

Using the notation of Art. 347, and supposing the condenser 
and commutator substituted for the conductor AC in Wheatstone s 
Bridge, and the galvanometer inserted in OA, and that the deflexion 
of the galvanometer is zero, then we know that the resistance of a 
coil, which placed in AC would give a zero deflexion, is 

* = J = *! (3) 

The other part of the resistance, R 2 , is that of the system of con 
ductors AO, OC, AB } BC and OB, the points A and C being con 
sidered as the electrodes. Hence 

R - ^( g 



In this expression a denotes the internal resistance of the battery 
and its connexions, the value of which cannot be determined with 
certainty ; but by making it small compared with the other resist 
ances, this uncertainty will only slightly affect the value of R 2 . 

The value of the capacity of the condenser in electromagnetic 
measure is ^ 

= 



777.] If the condenser has a large capacity, and the commutator 
is very rapid in its action, the condenser may not be fully discharged 
at each reversal. The equation of the electric current during the 
discharge is 



+SC = 0, (6) 

where Q is the charge, C the capacity of the condenser, R 2 the 



778.] CONDENSER COMPARED WITH COIL. 377 

resistance of the rest of the system between the electrodes of the 
condenser, and E the electromotive force due to the connexions 
with the battery. 

Hence Q = (Q Q + EC)e~W-EC, (7) 

where Q is the initial value of Q. 

If T is the time during which contact is maintained during each 
discharge, the quantity in each discharge is 




\+e 

By making c and y in equation (4) large compared with ft, a, or 
a, the time represented by R 2 C may be made so small compared 
with r, that in calculating the value of the exponential expression 
we may use the value of C in equation (5). We thus find 



- Ol (9) 

RJG" ~^T~ T 9 

where R is the resistance which must be substituted for the con 
denser to produce an equivalent effect. R 2 is the resistance of the 
rest of the system, T is the interval between the beginning of a 
discharge and the beginning of the next discharge, and r is the 
duration of contact for each discharge. We thus obtain for the 
corrected value of C in electromagnetic measure 



l+e * 2 T 
~ 



- 71 rri 

\e R z T 

IV. Comparison of the Electrostatic Capacity of a Condenser with 

the Electromagnetic Capacity of Self-induction of a Coil. 
778.] If two points of a conducting 
circuit, between which the resistance is 
R, are connected with the electrodes of 
a condenser whose capacity is (7, then, 
when an electromotive force acts on the 
circuit, part of the current, instead of 
passing through the resistance R, will 
be employed in charging the condenser. 
The current through R will therefore 
rise to its final value from zero in a 
gradual manner. It appears from the 
mathematical theory that the manner in which the current through 




378 COMPARISON OF UNITS. [77^. 

R rises from zero to its final value is expressed by a formula of 
exactly the same kind as that which expresses the value of a cur 
rent urged by a constant electromotive force through the coil of an 
electromagnet. Hence we may place a condenser and an electro 
magnet on two opposite members of Wheatstone s Bridge in such 
a way that the current through the galvanometer is always zero, 
even at the instant of making or breaking the battery circuit. 

In the figure, let P, Q, R, S be the resistances of the four mem 
bers of Wheatstone s Bridge respectively. Let a coil, whose coeffi 
cient of self-induction is It, be made part of the member AH, whose 
resistance is Q, and let the electrodes of a condenser, whose capacity 
is C, be connected by pieces of small resistance with the points F 
and Z. For the sake of simplicity, we shall assume that there is no 
current in the galvanometer G, the electrodes of which are con 
nected to F and //. We have therefore to determine the condition 
that the potential at F may be equal to that at H. It is only when 
we wish to estimate the degree of accuracy of the method that we 
require to calculate the current through the galvanometer when 
this condition is not fulfilled. 

Let x be the total quantity of electricity which has passed 
through the member AF, and z that which has passed through FZ 
at the time t, then x z will be the charge of the condenser. The 
electromotive force acting between the electrodes of the condenser 

is, by Ohm s law, R , so that if the capacity of the condenser 

. (i) 



Let y be the total quantity of electricity which has passed through 
the member AH, the electromotive force from A to H must be equal 
to that from A to F, or 



Since there is no current through the galvanometer, the quantity 
which has passed through HZ must be also y, and we find 

8% = X* (3) 

dt dt 

Substituting in (2) the value of x, derived from (1), and com 
paring with (3), we find as the condition of no current through the 
galvanometer 



779-] CONDENSER COMBINED WITH COIL. 379 

The condition of no final current is, as in the ordinary form of 
Wheatstone s Bridge, Qff _ $p (5) 

The condition of no current at making and breaking the battery 

connexion is r 

= RC. (6) 

Here -~ and RC are the time-constants of the members Q and R 

respectively, and if, by varying Q or R, we can adjust the members 
of Wheatstone s Bridge till the galvanometer indicates no current, 
either at making and breaking the circuit, or when the current is 
steady, then we know that the time-constant of the coil is equal to 
that of the condenser. 

The coefficient of self-induction, L> can be determined in electro 
magnetic measure from a comparison with the coefficient of mutual 
induction of two circuits, whose geometrical data are known 
(Art. 756). It is a quantity of the dimensions of a line. 

The capacity of the condenser can be determined in electrostatic 
measure by comparison with a condenser whose geometrical data 
are known (Art. 229). This quantity is also a length, c. The elec 
tromagnetic measure of the capacity is 



Substituting this value in equation (8), we obtain for the value 

of v 2 

v* = j QR, (8) 

where c is the capacity of the condenser in electrostatic measure, 
L the coefficient of self-induction of the coil in electromagnetic 
measure, and Q and R the resistances in electromagnetic measure. 
The value of v, as determined by this method, depends on the 
determination of the unit of resistance, as in the second method, 
Arts. 772, 773. 

V. Combination of the Electrostatic Capacity of a Condenser with 
the Electromagnetic Capacity of Self-induction of a Coil. 

779.] Let C be the capacity of the condenser, the surfaces of 
which are connected by a wire of resistance R. In this wire let the 
coils L and L be inserted, and let L denote the sum of their ca 
pacities of self-induction. The coil L is hung by a bifilar suspen 
sion, and consists of two coils in vertical planes, between which 



380 



COMPARISON OF UNITS. 



[779- 



passes a vertical axis which carries the magnet M, the axis of which 
revolves in a horizontal plane between the coils L L. The coil L 
has a large coefficient of self-induction, and is fixed. The sus 
pended coil IS is protected from the 
currents of air caused by the rota 
tion of the magnet by enclosing the 
rotating parts in a hollow case. 

The motion of the magnet causes 
currents of induction in the coil, and 
these are acted on by the magnet, 
so that the plane of the suspended 
coil is deflected in the direction of 
the rotation of the magnet. Let 
us determine the strength of the 
induced currents, and the magnitude 
of the deflexion of the suspended 
coil. 

Let x be the charge of electricity 
on the upper surface of the condenser C, then, if E is the electro 
motive force which produces this charge, we have, by the theory of 
the condenser, x CE. (1) 

We have also, by the theory of electric currents, 
d 




= 0, 



(2) 



where M is the electromagnetic momentum of the circuit L , when 
the axis of the magnet is normal to the plane of the coil,, and 6 is 
the angle between the axis of the magnet and this normal. 
The equation to determine x is therefore 



-n 

+CR-- +>== 
at 



- 

at 



(3) 



If the coil is in a position of equilibrium, and if the rotation of 
the magnet is uniform, the angular velocity being , 

6 = wt. (4) 

The expression for the current consists of two parts, one of which 
is independent of the term on the right-hand of the equation, 
and diminishes according to an exponential function of the time. 
The other, which may be called the forced current, depends entirely 
on the term in 0, and may be written 

x = A sin + cos 0. (5) 



779-] CONDENSER COMBINED WITH COIL. 381 

Finding the values of A and B by substitution, in the equation (3), 
we obtain RCn cos6-(l-CLn 2 )sm9 



The moment of the force with which the magnet acts on the coil 
L , in which the current x is flowing, is 

= x~(Mcos0) = Jfsin*- (7) 

dQ clt 

Integrating this expression with respect to t> and dividing by t, 
we find, for the mean value of 0, 
- 1 

~ * R* 

If the coil has a considerable moment of inertia, its forced vibra 
tions will be very small, and its mean deflexion will be proportional 
to 0. 

Let D 19 DD D 3 be the observed deflexions corresponding to an 
gular velocities n lt n 2 , n 3 of the magnet, then in general 



, (9) 

D \>n 

where P is a constant. 

Eliminating P and R from three equations of this form, we find 



/IJ 

If n 2 is such that CLn^ = 1, the value of -=- will be a minimum 

for this value of n. The other values of n should be taken, one 
greater, and the other less, than n 2 . 

The value of CL, determined from this equation, is of the dimen 
sions of the square of a time. Let us call it r 2 . 

If C 9 be the electrostatic measure of the capacity of the con 
denser, and L m the electromagnetic measure of the self-induction of 
the coil, both C 9 and L m are lines, and the product 

C 8 L m = v*C s L 8 = v*C m L m = vV ; (11) 

and f!-*^, (12) 

where r 2 is the value of C 2 Z 2 , determined by this experiment. The 
experiment here suggested as a method of determining v is of the 
same nature as one described by Sir W. R. Grove, PhU. Mag., 



382 COMPARISON OF UNITS. [780. 

March 1868, p. 184. See also remarks on that experiment, by the 
present writer, in the number for May 1868. 

VI. Electrostatic Measurement of Resistance. (See Art. 355.) 

780.] Let a condenser of capacity C be discharged through a 
conductor of resistance R, then, if x is the charge at any instant, 



_ 
Hence x = x Q e R. (2) 

If, by any method, we can make contact for a short time, which 
is accurately known, so as to allow the current to flow through the 
conductor for the time t, then, if E Q and J 1 are the readings of an 
electrometer put in connexion with the condenser before and after 
the operation, RC(log e E -log, E^ = t. (3) 

If C is known in electrostatic measure as a linear quantity, R 
may be found from this equation in electrostatic measure as the 
reciprocal of a velocity. 

If R s is the numerical value of the resistance as thus determined, 
and R m the numerical value of the resistance in electromagnetic 
measure, r> 

" 2 = Sr (4) 

Since it is necessary for this experiment that R should be very 
great, and since R must be small in the electromagnetic experi 
ments of Arts. 763, &c., the experiments must be made on separate 
conductors, and the resistance of these conductors compared by the 
ordinary methods. 



CHAPTER XX. 



ELECTROMAGNETIC THEORY OF LIGHT. 

781.] IN several parts of this treatise an attempt has been made 
to explain electromagnetic phenomena by means of mechanical 
action transmitted from one body to another by means of a medium 
occupying the space between them. The undulatory theory of light 
also assumes the existence of a medium. We have now to shew 
that the properties of the electromagnetic medium are identical with 
those of the luminiferous medium. 

To fill all space with a new medium whenever any new phe 
nomenon is to be explained is by no means philosophical, but if 
the study of two different branches of science has independently 
suggested the idea of a medium, and if the properties which must 
be attributed to the medium in order to account for electro 
magnetic phenomena are of the same kind as those which we 
attribute to the luminiferous medium in order to account for the 
phenomena of light, the evidence for the physical existence of the 
medium will be considerably strengthened. 

But the properties of bodies are capable of quantitative measure 
ment. We therefore obtain the numerical value of some property of 
the medium, such as the velocity with which a disturbance is pro 
pagated through it, which can be calculated from electromagnetic 
experiments, and also observed directly in the case of light. If it 
should be found that the velocity of propagation of electromagnetic 
disturbances is the same as the velocity of light, and this not only 
in air, but in other transparent media, we shall have strong reasons 
for believing that light is an electromagnetic phenomenon, and the 
combination of the optical with the electrical evidence will produce 
a conviction of the reality of the medium similar to that which we 
obtain, in the case of other kinds of matter, from the combined 
evidence of the senses. 



384 ELECTROMAGNETIC THEORY OF LIGHT. 

782.] When light is emitted, a certain amount of energy is 
expended by the luminous body, and if the light is absorbed by 
another body, this body becomes heated, shewing that it has re 
ceived energy from without. During the interval of time after the 
light left the first body and before it reached the second, it must 
have existed as energy in the intervening space. 

According to the theory of emission, the transmission of energy 
is effected by the actual transference of light-corpuscules from the 
luminous to the illuminated body,, carrying with them their kinetic 
energy, together with any other kind of energy of which they may 
be the receptacles. 

According to the theory of undulation, there is a material medium 
which fills the space between the two bodies, and it is by the action 
of contiguous parts of this medium that the energy is passed on, 
from one portion to the next, till it reaches the illuminated body. 

The luminiferous medium is therefore, during the passage of light 
through it, a receptacle of energy. In the undulatory theory, as 
developed by Huygens, Fresnel, Young, Green, &c., this energy 
is supposed to be partly potential and partly kinetic. The potential 
energy is supposed to be due to the distortion of the elementary 
portions of the medium. We must therefore regard the medium as 
elastic. The kinetic energy is supposed to be due to the vibratory 
motion of the medium. We must therefore regard the medium as 
having a finite density. 

In the theory of electricity and magnetism adopted in this 
treatise, two forms of energy are recognised, the electrostatic and 
the electrokinetic (see Arts. 630 and 636), and these are supposed 
to have their seat, not merely in the electrified or magnetized 
bodies, but in every part of the surrounding space, where electric 
or magnetic force is observed to act. Hence our theory agTees 
with the undulatory theory in assuming the existence of a medium 
which is capable of becoming a receptacle of two forms of energy *. 

783.] Let us next determine the conditions of the propagation 
of an electromagnetic disturbance through a uniform medium, which 
we shall suppose to be at rest, that is, to have no motion except that 
which may be involved in electromagnetic disturbances. 

* For my own part, considering the relation of a vacuum to the magnetic force, 
and the general character of magnetic phenomena external to the magnet, I am more 
inclined to the notion that in the transmission of the force there is such an action, 
external to the magnet, than that the effects are merely attraction and repulsion at a 
distance. Such an action may be a function of the aether; for it is not at all unlikely 
that, if there be an aether, it should have other uses than simply the conveyance of 
radiations. Faraday s Experimental Researches, 3075. 



783-] PROPAGATION OF ELECTROMAGNETIC DISTURBANCES. 385 

Let C be the specific conductivity of the medium, K its specific 
capacity for electrostatic induction, and //, its magnetic perme 
ability. 

To obtain the general equations of electromagnetic disturbance, 
we shall express the true current ( in terms of the vector potential 
$[ and the electric potential *. 

The true current ( is made up of the conduction current $ and 
the variation of the electric displacement 5), and since both of these 
depend on the electromotive force (, we find, as in Art. 611, 



But since there is no motion of the medium, we may express the 
electromotive force, as in Art. 599, 

@ = -Sl-V*. (2) 

Hence 6 =-(C + K*$ (f + V*). (3) 

But we may determine a relation between ( and 51 in a different 
way, as is shewn in Art. 616, the equations (4) of which may be 
written 47r M ( = V 2 2l + V/, (4) 



T dF dG dH , M 

where / = -=- + -y- -f -7- ( 5 ) 

das dy dz 

Combining equations (3) and (4), we obtain 

> (6) 

which we may express in the form of three equations as follows 



rf*x _ dJ 
dy> 






These are the general equations of electromagnetic disturbances. 

If we differentiate these equations with respect to #, y, and z 
respectively, and add, we obtain 



If the medium is a non-conductor, (7=0, and V 2 ^, which is 
proportional to the volume-density of free electricity, is independent 
of t. Hence / must be a linear function of ^, or a constant, or zero, 
and we may therefore leave / and ^ out of account in considering 
periodic disturbances. 

VOL. n. re 



386 ELECTROMAGNETIC THEORY OF LIGHT. 

Propagation of Undulations in a Non-conducting Medium. 
784.] In this case C~ 0. and the equations become 



The equations in this form are similar to those of the motion of 
an elastic solid, and when the initial conditions are given, the 
solution can be expressed in a form given by Poisson *, and applied 
by Stokes to the Theory of Diffraction f. 

Let us write V = == - (10) 



If the values of F, G, H, and of -=- > -j- > are given at every 

point of space at the epoch (t 0), then we can determine their 
values at any subsequent time, t, as follows. 

Let be the point for which we wish to determine the value 
of F at the time t. With as centre, and with radius Tt, describe 
a sphere. Find the initial value of J^at every point of the spherical 

surface, and take the mean, F, of all these values. Find also the 

j-pi 
initial values of -=- at every point of the spherical surface, and let 

dF 

the mean of these values be -j- 

dt 

Then the value of F at the point 0, at the time t, is 




Similarly G = ^(Gt)+ t-jr > \ (11) 



785.] It appears, therefore, that the condition of things at the 
point at any instant depends on the condition of things at a 
distance Vt and at an interval of time t previously, so that any 
disturbance is propagated through the medium with the velocity V. 

Let us suppose that when t is zero the quantities 1 and 21 are 

* Mem. de I A cad., torn, iii, p. 130. 

t Cambridge Transactions, vol. ix, p. 10 (1850). 



787.] VELOCITY OF LIGHT. 387 

zero except within a certain space S. Then their values at at 
the time t will be zero, unless the spherical surface described about 
as centre with radius Vt lies in whole or in part within the 
space S. If is outside the space S there will be no disturbance 
at until Vt becomes equal to the shortest distance from to the 
space S. The disturbance at will then begin, and will go on till 
Vt is equal to the greatest distance from to any part of S. The 
disturbance at will then cease for ever. 

786.] The quantity V, in Art. 793, which expresses the velocity 
of propagation of electromagnetic disturbances in a non-conducting 

medium is, by equation (9), equal to 



If the medium is air, and if we adopt the electrostatic system 
of measurement, K = I and jut = - T > so that V v, or the velocity 

of propagation is numerically equal to the number of electrostatic 
units of electricity in one electromagnetic unit. If we adopt the 

electromagnetic system. K = ^ and \L 1 , so that the equation 
V= v is still true. 

On the theory that light is an electromagnetic disturbance, pro 
pagated in the same medium through which other electromagnetic 
actions are transmitted, V must be the velocity of light, a quantity 
the value of which has been estimated by several methods. On the 
other hand, v is the number of electrostatic units of electricity in one 
electromagnetic unit, and the methods of determining this quantity 
have been described in the last chapter. They are quite inde 
pendent of the methods of finding the velocity of light. Hence the 
agreement or disagreement of the values of Fand of v furnishes a 
test of the electromagnetic theory of light. 

787.] In the following table, the principal results of direct 
observation of the velocity of light, either through the air or 
through the planetary spaces, are compared with the principal 
results of the comparison of the electric units : 



Velocity of Light (metres per second). 

Fizeau 314000000 

Aberration, &c., and) 



Sun s Parallax ) 



308000000 



Foucault .. .. 2983GOOOO 



Ratio of Electric Units. 
Weber 310740000 

Maxwell ... 288000000 
Thomson... 282000000. 



It is manifest that the velocity of light and the ratio of the units 
are quantities of the same order of magnitude. Neither of them 

c c 2 



388 ELECTROMAGNETIC THEORY OF LIGHT. 

can be said to be determined as yet with such a degree of accuracy 
as to enable us to assert that the one is greater or less than the 
other. It is to be hoped that, by further experiment, the relation be 
tween the magnitudes of the two quantities may be more accurately 
determined. 

In the meantime our theory, which asserts that these two quan 
tities are equal, and assigns a physical reason for this equality, is 
certainly not contradicted by the comparison of these results such 
as they are. 

788.] In other media than air, the velocity V is inversely pro 
portional to the square root of the product of the dielectric and the 
magnetic inductive capacities. According to the undulatory theory, 
the velocity of light in different media is inversely proportional to 
their indices of refraction. 

There are no transparent media for which the magnetic capacity 
differs from that of air more than by a very small fraction. Hence 
the principal part of the difference between these media must depend 
on their dielectric capacity. According to our theory, therefore, 
the dielectric capacity of a transparent medium should be equal to 
the square of its index of refraction. 

But the value of the index of refraction is different for light of 
different kinds, being greater for light of more rapid vibrations. 
We must therefore select the index of refraction which corresponds 
to waves of the longest periods, because these are the only waves 
whose motion can be compared with the slow processes by which 
we determine the capacity of the dielectric. 

789.] The only dielectric of which the capacity has been hitherto 
determined with sufficient accuracy is paraffin, for which in the solid 
form M.M. Gibson and Barclay found * 

K = 1.975. (12) 

Dr. Gladstone has found the following values of the index of 
refraction of melted paraffin, sp.g. 0.779, for the lines A, D and H : 



Temperature 
54C 



A 
1.4306 



57C 1.4294 



D 

1.4357 
1.4343 



H 

1.4499 
1.4493 



from which I find that the index of refraction for waves of infinite 

length would be about 1 422 

The square root of K is 1.405. 

The difference between these numbers is greater than can be ac- 

* Phil. Trans, 1871, p. 573. 



790.] PLANE WAVES. 389 

counted for by errors of observation, and shews that our theories of 
the structure of bodies must be much improved before we can 
deduce their optical from their electrical properties. At the same 
time, I think that the agreement of the numbers is such that if no 
greater discrepancy were found between the numbers derived from 
the optical and the electrical properties of a considerable number of 
substances, we should be warranted in concluding that the square 
root of 7T, though it may not be the complete expression for the 
index of refraction, is at least the most important term in it. 

Plane Waves. 

790.] Let us now confine our attention to plane waves, the front 
of which we shall suppose normal to the axis of z. All the quan 
tities, the variation of which constitutes such waves, are functions 
of z and t only, and are independent of x and y. Hence the equa 
tions of magnetic induction, (A), Art. 591, are reduced to 

dG dF 

a=-j-) b = -> c = 0, (13) 

dz dz 

or the magnetic disturbance is in the plane of the wave. This 
agrees with what we know of that disturbance which constitutes 
light. 

Putting pa, m/3 and /uty for a, b and c respectively, the equations 
of electric currents, Art. 607, become 



db d*F 

j- = -- Y~9 

dz dz 2 

da d*GL Y (14) 



4 71 U U = -- j- = 9 

dz dz 2 



4:7TfJiW = 0. 

Hence the electric disturbance is also in the plane of the wave, and 
if the magnetic disturbance is confined to one direction, say that of 
x, the electric disturbance is confined to the perpendicular direction, 
or that of y. 

But we may calculate the electric disturbance in another way, 
for iff, g, h are the components of electric displacement in a non 
conducting medium 

df dg dh 

u = 7t = ! " = 3r 

If P, Q, R are the components of the electromotive force 

-* - * (16) 



390 



ELECTROMAGNETIC THEORY OF LIGHT. 



[791. 



and since there is no motion of the medium, equations (B), Art. 598, 

Q = -*, R=- d -H. (17) 



become P = =- > 

at 



Hence u -^-=- , 



K 



K d 2 F 



, , 
(18) 



4 77 d 47T ^ 2 

Comparing 1 these values with those given in equation (14), we find 



> f 



(19) 



J 



The first and second of these equations are the equations of pro 
pagation of a plane wave, and their solution is of the well-known 
form F=A(z-Vt)+/ 2 (z+n),l 

o=A(*-rt)+M*+rf).\ (20) 

The solution of the third equation is 

KpH=A + t, (21) 

where A and B are functions of z. H is therefore either constant 
or varies directly with the time. In neither case can it take part 
in the propagation of waves. 

791.] It appears from this that the directions, both of the mag 
netic and the electric disturbances, lie in 
the plane of the wave. The mathematical 
form of the disturbance therefore, agrees 
with that of the disturbance which consti 
tutes light, in being transverse to the di 
rection of propagation. 

If we suppose G 0, the disturbance 
will correspond to a plane-polarized ray of 



light. 

The magnetic force is in this case paral- 

i ill? 
lei to the axis of y and equal to , , and 

the electromotive force is parallel to the 

dF 




axis of x and equal to 



dt 



The mag- 



Fig. 66. 



netic force is therefore in a plane perpen 
dicular to that which contains the electric force. 

The values of the magnetic force and of the electromotive force at 
a given instant at different points of the ray are represented in Fig. 66, 



793-] ENERGY AND STRESS OF RADIATION. 391 

for the case of a simple harmonic disturbance in one plane. This 
corresponds to a ray of plane-polarized light, but whether the plane 
of polarization corresponds ta the plane of the magnetic disturbance, 
or to the plane of the electric disturbance, remains to be seen. See 
Art. 797. 

Energy and Stress of Radiation. 

79.2.] The electrostatic energy per unit of volume at any point of 
the wave in a non-conducting medium is 



K, KdF 



i 



1 / p _ P2 _ 

2/ 877 8 77 dt 






(22) 
The electrokinetic energy at the same point is 

(23) 



8 77 877/X 

In virtue of equation (8) these two expressions are equal, so that at 
every point of the wave the intrinsic energy of the medium is half 
electrostatic and half electrokinetic. 

Let j9 be the value of either of these quantities, that is, either the 
electrostatic or the electrokinetic energy per unit of volume, then, 
in virtue of the electrostatic state of the medium, there is a tension 
whose magnitude is jo, in a direction parallel to #, combined with a 
pressure, also equal to^, parallel to y and z. See Art. 107. 

In virtue of the electrokinetic state of the medium there is a 
tension equal to p in a direction parallel to y, combined with a 
pressure equal to p in directions parallel to x and z. See Art. 643. 

Hence the combined effect of the electrostatic and the electro- 
kinetic stresses is a pressure equal to 2p in the direction of the 
propagation of the wave. Now 2/> also expresses the whole energy 
in unit of volume. 

Hence in a medium in which waves are propagated there is a 
pressure in the direction normal to the waves, and numerically 
equal to the energy in unit of volume. 

793.] Thus, if in strong sunlight the energy of the light which 
falls on one square foot is 83.4 foot pounds per second, the mean 
energy in one cubic foot of sunlight is about 0.0000000882 of a foot 
pound, and the mean pressure on a square foot is 0.0000000882 of a 
pound weight. A flat body exposed to sunlight would experience 
this pressure on its illuminated side only, and would therefore be 
repelled from the side on which the light falls. It is probable that 
a much greater energy of radiation might be obtained by means of 



392 ELECTROMAGNETIC THEORY OF LIGHT. [794. 

the concentrated rays of the electric lamp. Such rays falling- on a 
thin metallic disk, delicately suspended in a vacuum, might perhaps 
produce an observable mechanical effect. When a disturbance of 
any kind consists of terms involving sines or cosines of angles 
which vary with the time, the maximum energy is double of the 
mean energy. Hence, if P is the maximum electromotive force, 
and /3 the maximum magnetic force which are called into play 
during the propagation of light, 

JET 

P 2 = /3 2 = mean energy in unit of volume. (24) 

8 7T 8 77 

With Pouillet s data for the energy of sunlight, as quoted by 
Thomson, Trans. R.S.E., 1854, this gives in electromagnetic mea 
sure 

P = 60000000, or about 600 Darnell s cells per metre ; 

/3 = 0.193, or rather more than a tenth of the horizontal mag 
netic force in Britain. 



Propagation of a Plane Wave in a Crystallized Medium. 

794.] In calculating, from data furnished by ordinary electro 
magnetic experiments, the electrical phenomena which would result 
from periodic disturbances, millions of millions of which occur in a 
second, we have already put our theory to a very severe test, even 
when the medium is supposed to be air or vacuum. But if we 
attempt to extend our theory to the case of dense media, we become 
involved not only in all the ordinary difficulties of molecular theories, 
but in the deeper mystery of the relation of the molecules to the 
electromagnetic medium. 

To evade these difficulties, we shall assume that in certain media 
the specific capacity for electrostatic induction is different in dif 
ferent directions, or in other words, the electric displacement, in 
stead of being in the same direction as the electromotive force, and 
proportional to it, is related to it by a system of linear equations 
similar to those given in Art. 297. It may be shewn, as in 
Art. 436, that the system of coefficients must be symmetrical, so 
that, by a proper choice of axes, the equations become 

f=~K,P, ff = X,Q, * = K t R, (1) 

where K l , K 2 , and K 3 are the principal inductive capacities of the 
medium. The equations of propagation of disturbances are therefore 



796.] DOUBLE REFRACTION. 393 



^F__^G^ d*H ( d*F d 2 * 

~df^~dz*~ ~dx~dy dz~dx ~ 1/X \ dt 2 ~ dxdt 
d 2 F ,d 2 G d 2 * 



dz 2 dx z dy dz dxdy 2/ ^^ 2 dydt 
d 2 F d 2 G ,d 2 ff d 2 * 



dx 2 dy 2 dzdx dydz r \dt 2 dzdt } 

795.] If I, m, n are the direction-cosines of the normal to the 
wave-front, and V the velocity of the wave, and if 

Ix + my + nz~Pt = w, (3) 

and if we write F", G", H", V" for the second differential coeffi 
cients of F, G, //, ^ respectively with respect to w, and put 

1 1 1 






(4) 



where a, , c are the three principal velocities of propagation, the 
equations become 



n*-F"-lmG"-nlH"--rV = 0, 




-ImF" + (n 2 + 1*- ~G"-mnH"- VV = 0, (5) 

-nlF"- mn G" + (l 2 + m 2 - 
796.J If we write 

72 

we obtain from these equations 

rU(PF"-W) = 0,) 

(7) 



Hence, either V = 0, in which case the wave is not propagated at 
all ; or, U = 0, which leads to the equation for V given by Fresnel ; 
or the quantities within brackets vanish, in which case the vector 
whose components are F", G" , H" is normal to the wave-front and 
proportional to the electric volume-density. Since the medium is 
a non-conductor, the electric density at any given point is constant, 
and therefore the disturbance indicated by these equations is not 
periodic, and cannot constitute a wave. We may therefore consider 
*"= in the investigation of the wave. 



394 ELECTROMAGNETIC THEORY OF LIGHT. [797. 

797.] The velocity of the propagation of the wave is therefore 
completely determined from the equation U = 0, or 

I 2 m 2 n 2 . } 

7* -a 2 + T*^JP + F 2 -c 2 = 

There are therefore two, and only two, values of V 2 correspondiDg 
to a given direction of wave-front. 

If A, jot, v are the direction-cosines of the electric current whose 
components are u y v, w> 

A: M :,:::G":-", (9) 



then l\ + mn + nv=0; (10) 

or the current is in the plane of the wave-front, and its direction 
in the wave-front is determined by the equation 

l -(b 2 -c 2 } + ( c *-a*)+-(a*-6*) = 0. (11) 

A )U V 

These equations are identical with those given by Fresnel if we 
define the plane of polarization as a plane through the ray per 
pendicular to the plane of the electric disturbance. 

According to this electromagnetic theory of double refraction the 
wave of normal disturbance, which constitutes one of the chief 
difficulties of the ordinary theory, does not exist, and no new 
assumption is required in order to account for the fact that a ray 
polarized in a principal plane of the crystal is refracted in the 
ordinary manner *. 

Relation between Electric Conductivity and Opacity. 

798.] If the medium, instead of being a perfect insulator, is a 
conductor whose conductivity per unit of volume is C, the dis 
turbance will consist not only of electric displacements but of 
currents of conduction, in which electric energy is transformed into 
heat, so that the undulation is absorbed by the medium. 

If the disturbance is expressed by a circular function, we may 

write -t-qz), (1) 



for this will satisfy the equation 

, v 





provided q 2 -p z = ^Kn 2 , (3) 

and 2p = 1-ny.Cn. (4) 



* See Stokes Report on Double Refraction ; Brit. Assoc. Reports, 1862, p. 255. 



8oi.] CONDUCTIVITY AND OPACITY. 395 

The velocity of propagation is 

r=A (5) 

2 

and the coefficient of absorption is 

p = 27T/ACT. (6) 

Let R be the resistance, in electromagnetic measure, of a plate 
whose length is /, breadth #, and thickness z, 

*=-se- (7) 

The proportion of the incident light which will be transmitted by 

this plate will be 

i_v_ 

e -* p *=. e rMb B . (8) 

799.] Most transparent solid bodies are good insulators, and all 
good conductors are very opaque. There are, however, many ex 
ceptions to the law that the opacity of a body is the greater, the 
greater its conductivity. 

Electrolytes allow an electric current to pass, and yet many of 
them are transparent. We may suppose, however, that in the case 
of the rapidly alternating forces which come into play during the 
propagation of light, the electromotive force acts for so short a 
time in one direction that it is unable to effect a complete separation 
between the combined molecules. When, during the other half of 
the vibration, the electromotive force acts in the opposite direction 
it simply reverses what it did during the first half. There is thus 
no true conduction through the electrolyte, no loss of electric 
energy, and consequently no absorption of light. 

800.] Gold, silver, and platinum are good conductors, and yet, 
when formed into very thin plates, they allow light to pass through 
them. From experiments which I have made on a piece of gold 
leaf, the resistance of which was determined by Mr. Hockin, it 
appears that its transparency is very much greater than is con 
sistent with our theory, unless we suppose that there is less loss 
of energy when the electromotive forces are reversed for every semi- 
vibration of light than when they act for sensible times, as in our 
ordinary experiments. 

801.] Let us next consider the case of a medium in which the 
conductivity is large in proportion to the inductive capacity. 

In this case we may leave out the term involving K in the equa 
tions of Art. 783, and they then become 



396 ELECTROMAGNETIC THEORY OF LIGHT. [802. 



(1) 



Each of these equations is of the same form as the equation of the 
diffusion of heat given in Fourier s Traite de Chaleur. 

802.] Taking the first as an example, the component F of the 
vector-potential will vary according to time and position in the same 
way as the temperature of a homogeneous solid varies according 
to time and position, the initial and the surface-conditions being 
made to correspond in the two cases, and the quantity 47r/u,Cbeing 
numerically equal to the reciprocal of the thermometric conductivity 
of the substance, that is to say, the number of units of volume of 
the substance which would be heated one degree by the heat which passes 
through a unit cube of the substance, two opposite faces of which differ 
by one degree of temperature, while the other faces are impermeable to 
heat*. 

The different problems in thermal conduction, of which Fourier 
has given the solution, may be transformed into problems in the 
diffusion of electromagnetic quantities, remembering that F, G, H 
are the components of a vector, whereas the temperature, in Fourier s 
problem, is a scalar quantity. 

Let us take one of the cases of which Fourier has given a com 
plete solution t, that of an infinite medium, the initial state t)f 
which is given. 

The state of any point of the medium at the time t is found 
by taking the average of the state of every part of the medium, 
the weight assigned to each part in taking the average being 



where r is the distance of that part from the point considered. This 
average, in the case of vector-quantities, is most conveniently taken 
by considering each component of the vector separately. 

* See Maxwell s Theory of Heat, p. 235. 

t Traite de la Chalewr, Art. 384. The equation which determines the temperature, 
v, at a point (x, y, z) after a time t, in terms of /(a, 0, 7), the initial temperature at 
the point (0,0,7), is 



r C r do. d@ dy ( I 

v=/// r=- e * * M J (**& 

/// 2 3 \/^v 3 t 3 
/ j j 

where k is the thermometric conductivity. 



804.] ESTABLISHMENT OF THE DISTRIBUTION OF FORCE. 397 

803.] We have to remark in the first place, that in this problem 
the thermal conductivity of Fourier s medium is to be taken in 
versely proportional to the electric conductivity of our medium, 
so that the time required in order to reach an assigned stage in 
the process of diffusion is greater the higher the electric conduct 
ivity. This statement will not appear paradoxical if we remember 
the result of Art. 655, that a medium of infinite conductivity forms 
a complete barrier to the process of diffusion of magnetic force. 

In the next place, the time requisite for the production of an 
assigned stage in the process of diffusion is proportional to the square 
of the linear dimensions of the system. 

There is no determinate velocity which can be defined as the 
velocity of diffusion. If we attempt to measure this velocity by 
ascertaining the time requisite for the production of a given amount 
of disturbance at a given distance from the origin of disturbance, 
we find that the smaller the selected value of the disturbance the 
greater the velocity will appear to be, for however great the distance, 
and however small the time, the value of the disturbance will differ 
mathematically from zero. 

This peculiarity of diffusion distinguishes it from wave-propaga 
tion, which takes place with a definite velocity. No disturbance 
takes place at a given point till the wave reaches that point, and 
when the wave has passed, the disturbance ceases for ever. 

804.] Let us now investigate the process which takes place when 
an electric current begins and continues to flow through a linear 
circuit, the medium surrounding the circuit being of finite electric 
conductivity. (Compare with Art. 660). 

When the current begins, its first effect is to produce a current 
of induction in the parts of the medium close to the wire. The 
direction of this current is opposite to that of the original current, 
and in the first instant its total quantity is equal to that of the 
original current, so that the electromagnetic effect on more distant 
parts of the medium is initially zero, and only rises to its final 
value as the induction-current dies away on account of the electric 
resistance of the medium. 

But as the induction-current close to the wire dies away, a new 
induction -current is generated in the medium beyond, so that the 
space occupied by the induction-current is continually becoming 
wider, while its intensity is continually diminishing. 

This diffusion and decay of the induction-current is a pheno 
menon precisely analogous to the diffusion of heat from a part of 



398 ELECTROMAGNETIC THEORY OF LIGHT. [805. 

the medium initially hotter or colder than the rest. We must 
remember, however, that since the. current is a vector quantity., 
and since in a circuit the current is in opposite directions at op 
posite points of the circuit, we must, in calculating any given com 
ponent of the induction-current, compare the problem with one 
in which equal quantities of heat and of cold are diffused from 
neighbouring places, in which case the effect on distant points will 
be of a smaller order of magnitude. 

805.] If the current in the linear circuit is maintained constant, 
the induction currents, which depend on the initial change of state, 
will gradually be diffused and die away, leaving the medium in its 
permanent state, which is analogous to the permanent state of the 
flow of heat. In this state we have 

V 2 I< 7 = V 2 = y 2 #=0 (2) 

throughout the medium, except at the part occupied by the circuit, 
in which V 2 F= 4wM , 

V 2 =47r^,> (3) 

V 2 //=477^J 

These equations are sufficient to determine the values of F, G, R 
throughout the medium. They indicate that there are no currents 
except in the circuit, and that the magnetic forces are simply those 
due to the current in the circuit according to the ordinary theory. 
The rapidity with which this permanent state is established is so 
great that it could not be measured by our experimental methods, 
except perhaps in the case of a very large mass of a highly con 
ducting medium such as copper. 

NOTE. In a paper published in PoggendorfFs Annalen, June 1867, 
M. Lorenz has deduced from Kirchhoff s equations of electric cur 
rents (Pogg. Ann. cii. 1856), by the addition of certain terms which 
do not affect any experimental result, a new set of equations, indi 
cating that the distribution of force in the electromagnetic field 
may be conceived as arising from the mutual action of contiguous 
elements, and that waves, consisting of transverse electric currents, 
may be propagated, with a velocity comparable to that of light, in 
non-conducting media. He therefore regards the disturbance which 
constitutes light as identical with these electric currents, and he 
shews that conducting media must be opaque to such radiations. 

These conclusions are similar to those of this chapter, though 
obtained by an entirely different method. The theory given in 
this chapter was first published in the PUL Trans, for 1865. 






CHAPTER XXI. 



MAGNETIC ACTION ON LIGHT. 

806.] THE most important step in establishing a relation between 
electric and magnetic phenomena and those of light must be the 
discovery of some instance in which the one set of phenomena is 
aifected by the other. In the search for such phenomena we must 
be guided by any knowledge we may have already obtained with 
respect to the mathematical or geometrical form of the quantities 
which we wish to compare. Thus, if we endeavour, as Mrs. Somer- 
ville did, to magnetize a needle by means of light, we must re 
member that the distinction between magnetic north and south is 
a mere matter of direction, and would be at once reversed if we 
reverse certain conventions about the use of mathematical signs. 
There is nothing in magnetism analogous to those phenomena of 
electrolysis which enable us to distinguish positive from negative 
electricity, by observing that oxygen appears at one pole of a cell 
and hydrogen at the other. 

Hence we must not expect that if we make light fall on one end 
of a needle, that end will become a pole of a certain name, for the 
two poles do not differ as light does from darkness. 

We might expect a better result if we caused circularly polarized 
light to fall on the needle, right-handed light falling on one end 
and left-handed on the other, for in some respects these kinds of 
light may be said to be related to each other in the same way as 
the poles of a magnet. The analogy, however, is faulty even here, 
for the two rays when combined do not neutralize each other, but 
produce a plane polarized ray. 

Faraday, who was acquainted with the method of studying the 
strains produced in transparent solids by means of polarized light, 
made many experiments in hopes of detecting some action on polar 
ized light while passing through a medium in which electrolytic 
conduction or dielectric induction exists *. He was not, however, 
* Experimental Researches, 951-954 and 2216-2220. 



400 MAGNETIC ACTION ON LIGHT. [807. 

able to detect any action of this kind, though the experiments were 
arranged in the way best adapted to discover effects of tension, 
the electric force or current being at right angles to the direction 
of the ray, and at an angle of forty-five degrees to the plane of 
polarization. Faraday varied these experiments in many ways with 
out discovering any action on light due to electrolytic currents or 
to static electric induction. 

He succeeded, however, in establishing a relation between light 
and magnetism, and the experiments by which he did so are de 
scribed in the nineteenth series of his Experimental Researches. We 
shall take Faraday s discovery as our starting point for further 
investigation into the nature of magnetism, and we shall therefore 
describe the phenomenon which he observed. 

807.] A ray of plane-polarized light is transmitted through a 
transparent diamagnetic medium, and the plane of its polarization, 
when it emerges from the medium, is ascertained by observing the 
position of an analyser when it cuts off the ray. A magnetic force 
is then made to act so that the direction of the force within the 
transparent medium coincides with the direction of the ray. The 
light at once reappears, but if the analyser is turned round through 
a certain angle, the light is again cut off. This shews that the 
effect of the magnetic force is to turn the plane of polarization, 
round the direction of the ray as an axis, through a certain angle, 
measured by the angle through which the analyser must be turned 
in order to cut off the light. 

808.] The angle through which the plane of polarization is 
turned is proportional 

(1) To the distance which the ray travels within the medium. 
Hence the plane of polarization changes continuously from its posi 
tion at incidence to its position at emergence. 

(2) To the intensity of the resolved part of the magnetic force in 
the direction of the ray. 

(3) The amount of the rotation depends on the nature of the 
medium. No rotation has yet been observed when the medium is 
air or any other gas. 

These three statements are included in the more general one, 
that the angular rotation is numerically equal to the amount by 
which the magnetic potential increases, from the point at which 
the ray enters the medium to that at which it leaves it, multiplied 
by a coefficient, which, for diamagnetic media, is generally positive. 

809.] In diamagnetic substances, the direction in which the plane 



8io.] FARADAY S DISCOVERY. 401 

of polarization is made to rotate is the same as the direction in which 
a positive current must circulate round the ray in order to produce 
a magnetic force in the same direction as that which actually exists 
in the medium. 

Verdet, however, discovered that in certain ferromagnetic media, 
as, for instance, a strong solution of perchloride of iron in wood- 
spirit or ether, the rotation is in the opposite direction to the current 
which would produce the magnetic force. 

This shews that the difference between ferromagnetic and dia 
magnetic substances does not arise merely from the magnetic per 
meability being in the first case greater, and in the second less, 
than that of air, but that the properties of the two classes of bodies 
are really opposite. 

The power acquired by a substance under the action of magnetic 
force of rotating the plane of polarization of light is not exactly 
proportional to its diamagnetic or ferromagnetic magnetizability. 
Indeed there are exceptions to the rule that the rotation is positive for 
diamagnetic and negative for ferromagnetic substances, for neutral 
chromate of potash is diamagnetic, but produces a negative rotation. 

810.] There are other substances, which, independently of the 
application of magnetic force, cause the plane of polarization to 
turn to the right or to the left, as the ray travels through the sub 
stance. In some of these the property is related to an axis, as in 
the case of quartz. In others, the property is independent of the 
direction of the ray within the medium, as in turpentine, solution 
of sugar, &c. In all these substances, however, if the plane of 
polarization of any ray is twisted within the medium like a right- 
handed screw, it will still be twisted like a right-handed screw if 
the ray is transmitted through the medium in the opposite direction. 
The direction in which the observer has to turn his analyser in order 
to extinguish the ray after introducing the medium into its path, 
is the same with reference to the observer whether the ray comes 
to him from the north or from the south. The direction of the 
rotation in space is of course reversed when the direction of the ray is 
reversed. But when the rotation is produced by magnetic action, its 
direction in space is the same whether the ray be travelling north 
or south. The rotation is always in the same direction as that of 
the electric current which produces, or would produce, the actual 
magnetic state of the field, if the medium belongs to the positive 
class, or in the opposite direction if the medium belongs to the 
negative class. 

VOL. IT. D d 



402 MAGNETIC ACTION ON LIGHT. [8 1 I. 

It follows from this, that if the ray of light, after passing through 
the medium from north to south, is reflected by a mirror, so as to 
return through the medium from south to north,, the rotation will 
be doubled when it results from magnetic action. When the rota 
tion depends on the nature of the medium alone, as in turpentine, &c., 
the ray, when reflected back through the medium, emerges in the 
same plane as it entered, the rotation during the first passage 
through the medium having been exactly reversed during the 
second. 

811.] The physical explanation of the phenomenon presents con 
siderable difficulties, which can hardly be said to have been hitherto 
overcome, either for the magnetic rotation, or for that which 
certain media exhibit of themselves. We may, however, prepare 
the way for such an explanation by an analysis of the observed 
facts. 

It is a well-known theorem in kinematics that two uniform cir 
cular vibrations, of the same amplitude, having the same periodic 
time, and in the same plane, but revolving in opposite directions, 
are equivalent, when compounded together, to a rectilinear vibra 
tion. The periodic time of this vibration is equal to that of the 
circular vibrations, its amplitude is double, and its direction is in 
the line joining the points at which two particles, describing the 
circular vibrations in opposite directions round the same circle, 
would meet. Hence if one of the circular vibrations has its phase 
accelerated, the direction of the rectilinear vibration will be turned, 
in the same direction as that of the circular vibration, through an 
angle equal to half the acceleration of phase. 

It can also be proved by direct optical experiment that two rays 
of light, circularly-polarized in opposite directions, and of the same 
intensity, become, when united, a plane-polarized ray, and that if 
by any means the phase of one of the circularly-polarized rays is 
accelerated, the plane of polarization of the resultant ray is turned 
round half the angle of acceleration of the phase. 

812.] We may therefore express the phenomenon of the rotation 
of the plane of polarization in the following manner : A plane- 
polarized ray falls on the medium. This is equivalent to two cir 
cularly-polarized rays, one right-handed, the other left-handed (as 
regards the observer) . After passing through the medium the ray 
is still plane-polarized, but the plane of polarization is turned, say, 
to the right (as regards the observer) . Hence, of the two circularly- 
polarized rays, that which is right-handed must have had its phase 






8 14-] 



STATEMENT OF THE FACTS. 



403 



accelerated with respect to the other during its passage through the 
medium. 

In other words, the right-handed ray has performed a greater 
number of vibrations, and therefore has a smaller wave-length, 
within the medium, than the left-handed ray which has the same 
periodic time. 

This mode of stating what takes place is quite independent of 
any theory of light, for though we use such terms as wave-length, 
circular-polarization, &c., which may be associated in our minds 
with a particular form of the undulatory theory, the reasoning is 
independent of this association, and depends only on facts proved 
by experiment. 

813.] Let us next consider the configuration of one of these rays 
at a given instant. Any undulation, the motion of which at each 
point is circular, may be represented by a helix or screw. If the 
screw is made to revolve about its axis without any longitudinal 
motion, each particle will describe a circle, and at the same time the 
propagation of the undulation will be represented by the apparent 
longitudinal motion of the similarly situated parts of the thread of 
the screw. It is easy to see that if the screw is right-handed, and 
the observer is placed at that end towards which the undulation 
travels, the motion of the screw will appear to him left-handed, 
that is to say, in the opposite di 
rection to that of the hands of a 
watch. Hence such a ray has 
been called, originally by French 
writers, but now by the whole 
scientific world, a left-handed cir 
cularly-polarized ray. 

A right-handed circularly-polar 
ized ray is represented in like 
manner by a left-handed helix. 
In Fig. 67 the right-handed helix 
A, on the right-hand of the figure, 
represents a left-handed ray, and 
the left-handed helix B, on the left- 
hand, represents a right-handed 
ray. 

814.] Let us now consider two 
such rays which have the same 
wave-length within the medium. 




67< 



They are geometrically alike in 
B d i 



404 MAGNETIC ACTION OX LIGHT. [815. 

all respects, except that one is the perversion of the other, like its 
image in a looking-glass. One of them, however, say A, has a 
shorter period of rotation than the other. If the motion is entirely 
due to the forces called into play by the displacement, this shews 
that greater forces are called into play by the same displacement 
when the configuration is like A than when it is like B. Hence in 
this case the left-handed ray will be accelerated with respect to the 
right-handed ray, and this will be the case whether the rays are 
travelling from N to S or from S to N. 

This therefore is the explanation of the phenomenon as it is pro 
duced by turpentine, &c. In these media the displacement caused 
by a circularly-polarized ray calls into play greater forces of resti 
tution when the configuration is like A than when it is like B. 
The forces thus depend on the configuration alone, not on the direc 
tion of the motion. 

But in a diamagnetic medium acted on by magnetism in the 
direction SN 9 of the two screws A and B, that one always rotates 
with the greatest velocity whose motion, as seen by an eye looking 
from S to N, appears like that of a watch. Hence for rays from S 
to N the right-handed ray B will travel quickest, but for rays 
from N to 8 the left-handed ray A will travel quickest. 

815.] Confining our attention to one ray only, the helix B has 
exactly the same configuration, whether it represents a ray from S 
to N or one from N to S. But in the first instance the ray travels 
faster, and therefore the helix rotates more rapidly. Hence greater 
forces are called into play when the helix is going round one way 
than when it is going round the other way. The forces, therefore, 
do not depend solely on the configuration of the ray, but also on 
the direction of the motion of its individual parts. 

816.] The disturbance which constitutes light, whatever its 
physical nature may be, is of the nature of a vector, perpendicular 
to the direction of the ray. This is proved from the fact of the 
interference of two rays of light, which under certain conditions 
produces darkness, combined with the fact of the non-interference 
of two rays polarized in planes perpendicular to each other. For 
since the interference depends on the angular position of the planes 
of polarization, the disturbance must be a directed quantity or 
vector, and since the interference ceases when the planes of polar 
ization are at right angles, the vector representing the disturbance 
must be perpendicular to the line of intersection of these planes, 
that is, to the direction of the ray. 



817.] C1KCULARLY-POLAKIZED LIGHT. 405 

817.] The disturbance, being a vector, can be resolved into com 
ponents parallel to x and y, the axis of z being 4 parallel to the 
direction of the ray. Let f and 77 be these components, then, in the 
case of a ray of homogeneous circularly-polarized light, 

f = rcosO, rj = rsmO, (1) 

where = nt qz + a. (2) 

In these expressions, r denotes the magnitude of the vector, and 
the angle which it makes with the direction of the axis of x. 
The periodic time, r, of the disturbance is such that 

UT 27T. (3) 

The wave-length, A, of the disturbance is such that 

q\ = 27T. (4) 

The velocity of propagation is - 

The phase of the disturbance when t and z are both zero is a. 

The circularly-polarized light is right-handed or left-handed 
according as q is negative or positive. 

Its vibrations are in the positive or the negative direction of 
rotation in the plane of (no, y}^ according as n is positive or negative. 

The light is propagated in the positive or the negative direction 
of the axis of z, according as n and q are of the same or of opposite 
signs. 

In all media n varies when q varies, and -=- is always of the same 
sign with - 

Hence, if for a given numerical value of n the value of - is 

greater when n is positive than when n is negative, it follows that 
for a value of q, given both in magnitude and sign, the positive 
value of n will be greater than the negative value. 

Now this is what is observed in a diamagnetic medium, acted on 
by a magnetic force, y, in the direction of z. Of the two circularly- 
polarized rays of a given period, that is accelerated of which the 
direction of rotation in the plane of (#, y) is positive. Hence, of 
two circularly-polarized rays, both left-handed, whose wave-length 
within the medium is the same, that has the shortest period whose 
direction of rotation in the plane of xy is positive, that is, the ray 
which is propagated in the positive direction of z from south to 
north. We have therefore to account for the fact, that when in the 
equations of the system q and r are given, two values of n will 



406 MAGNETIC ACTION ON LIGHT. [8 1 8. 

satisfy the equations, one positive and the other negative, the 
positive value being numerically greater than the negative. 

818.] We may obtain the equations of motion from a considera 
tion of the potential and kinetic energies of the medium. The 
potential energy, F, of the system depends on its configuration, 
that is, on the relative position of its parts. In so far as it depends 
on the disturbance due to circularly-polarized light, it must be a 
function of r, the amplitude, and q, the coefficient of torsion, only. 
It may be different for positive and negative values of q of equal 
numerical value, and it probably is so in the case of media which 
of themselves rotate the plane of polarization. 

The kinetic energy, T, of the system is a homogeneous function 
of the second degree of the velocities of the system, the coefficients 
of the different terms being functions of the coordinates. 

819.] Let us consider the dynamical condition that the ray may 
be of constant intensity, that is, that r may be constant. 

Lagrange s equation for the force in r becomes 
d dT dT 



Since r is constant, the first term vanishes. We have therefore the 
equation dT dV . . 

Tr + ~dr = ( 

in which q is supposed to be given, and we are to determine the 
value of the angular velocity 0, which we may denote by its actual 
value, n. 

The kinetic energy, T, contains one term involving n 2 ; other 
terms may contain products of n with other velocities, and the 
rest of the terms are independent of n. The potential energy, T 7 , is 
entirely independent of n. The equation is therefore of the form 

An* + Bn+C = 0. (7) 

This being a quadratic equation, gives two values of n. It appears 
from experiment that both values are real, that one is positive and 
the other negative, and that the positive value is numerically the 
greater. Hence, if A is positive, both B and C are negative, for, 
if % and n 2 are the roots of the equation, 

^(% + O + -#=0. (8) 

The coefficient, _Z?, therefore, is not zero, at least when magnetic 
force acts on the medium. We have therefore to consider the ex 
pression Bn, which is the part of the kinetic energy involving the 
first power of n, the angular velocity of the disturbance. 



821.] MAGNETISM IMPLIES AN ANGULAR TELOCITY. 407 

820.] Every term of T is of two dimensions as regards velocity. 
Hence the terms involving- n must involve some other velocity. 
This velocity cannot be r or q, because, in the case we consider, 
r and q are constant. Hence it is a velocity which exists in the 
medium independently of that motion which constitutes light. It 
must also be a velocity related to n in such a way that when it is 
multiplied by n the result is a scalar quantity, for only scalar quan 
tities can occur as terms in the value of T, which is itself scalar. 
Hence this velocity must be in the same direction as n, or in the 
opposite direction, that is, it must be an angular velocity about the 
axis of z. 

Again, this velocity cannot be independent of the magnetic force, 
for if it were related to a direction fixed in the medium, the phe 
nomenon would be different if we turned the medium end for end, 
which is not the case. 

We are therefore led to the conclusion that this velocity is an 
invariable accompaniment of the magnetic force in those media 
which exhibit the magnetic rotation of the plane of polarization. 

8.21.] We have been hitherto obliged to use language which is 
perhaps too suggestive of the ordinary hypothesis of motion in the 
undulatory theory. It is easy, however, to state our result in a 
form free from this hypothesis. 

Whatever light is, at each point of space there is something 
going on, whether displacement, or rotation, or something not yet 
imagined, but which is certainly of the nature of a vector or di 
rected quantity, the direction of which is normal to the direction 
of the ray. This is completely proved by the phenomena of inter 
ference. 

In the case of circularly-polarized light, the magnitude of this 
vector remains always the same, but its direction rotates round the 
direction of the ray so as to complete a revolution in the periodic 
time of the wave. The uncertainty which exists as to whether this 
vector is in the plane of polarization or perpendicular to it, does not 
extend to our knowledge of the direction in which it rotates in right- 
handed and in left-handed circularly-polarized light respectively. 
The direction and the angular velocity of this vector are perfectly 
known, though the physical nature of the vector and its absolute 
direction at a given instant are uncertain. 

When a ray of circularly-polarized light falls on a medium under 
the action of magnetic force, its propagation within the medium 
is affected by the relation of the direction of rotation of the light to 



408 MAGNETIC ACTION ON LIGHT. [822. 

the direction of the magnetic force. From this we conclude, by the 
reasoning of Art. 821, that in the medium, when under the action 
of magnetic force, some rotatory motion is going on, the axis of ro 
tation being in the direction of the magnetic forces ; and that the 
rate of propagation of circularly-polarized light, when the direction 
of its vibratory rotation and the direction of the magnetic rotation 
of the medium are the same, is different from the rate of propaga 
tion when these directions are opposite. 

The only resemblance which we can trace between a medium 
through which circularly-polarized light is propagated, and a me 
dium through which lines of magnetic force pass, is that in both 
there is a motion of rotation about an axis. But here the resem 
blance stops, for the rotation in the optical phenomenon is that of 
the vector which represents^ the disturbance. This vector is always 
perpendicular to the direction of the ray, and rotates about it a 
known number of times in a second. In the magnetic phenomenon, 
that which rotates has no properties by which its sides can be dis 
tinguished, so that we cannot determine how many times it rotates 
in a second. 

There is nothing, therefore, in the magnetic phenomenon which 
corresponds to the wave-length and the wave-propagation in the op 
tical phenomenon. A medium in which a constant magnetic force 
is acting is not, in consequence of that force, filled with waves 
travelling in one direction, as when light is propagated through it. 
The only resemblance between the optical and the magnetic pheno 
menon is, that at each point of the medium something exists of 
the nature of an angular velocity about an axis in the direction of 
the magnetic force. 

On the Hypothesis of Molecular Vortices. 

822.] The consideration of the action of magnetism on polarized 
light leads, as we have seen, to the conclusion that in a medium 
under the action of magnetic force something belonging to the 
same mathematical class as an angular velocity, whose axis is in the 
direction of the magnetic force, forms a part of the phenomenon. 

This angular velocity cannot be that of any portion of the me 
dium of sensible dimensions rotating as a whole. We must there 
fore conceive the rotation to be that of very small portions of the 
medium, each rotating on its own axis. This is the hypothesis of 
molecular vortices. 

The motion of these vortices, though, as we have shewn (Art. 575), 



824.] MOLECULAR VOHTICES. 409 

it does not sensibly affect the visible motions of large bodies, may 
be such as to affect that vibratory motion on which the propagation 
of light, according to the undulatory theory, depends. The dis 
placements of the medium, during the propagation of light, will 
produce a disturbance of the vortices, and the vortices when so dis 
turbed may react on the medium so as to affect the mode of propa 
gation of the ray. 

823.] It is impossible, in our present state of ignorance as to the 
nature of the vortices, to assign the form of the law which connects 
the displacement of the medium with the variation of the vortices. 
We shall therefore assume that the variation of the vortices caused 
by the displacement of the medium is subject to the same conditions 
which Helmholtz, in his great memoir on Vortex-motion *, has 
shewn to regulate the variation of the vortices of a perfect liquid. 

Helmholtz s law may be stated as follows : Let P and Q be two 
neighbouring particles in the axis of a vortex, then, if in conse 
quence of the motion of the fluid these particles arrive at the 
points P Q , the line P Q will represent the new direction of the 
axis of the vortex, and its strength will be altered in the ratio of 
P Q to PQ. 

Hence if a, /3, y denote the components of the strength of a vor 
tex, and if f, 17, f denote the displacements of the medium, the value 
of a will become 

/ d d^ d ^ 

a = a + a -= f-p -= |-y -y- > 
ax ay dz 



We now assume that the same condition is satisfied during the 
small displacements of a medium in which a, (3, y represent, not 
the components of the strength of an ordinary vortex, but the 
components of magnetic force. 

824.] The components of the angular velocity of an element of 

the medium are Wl = \ (* - ^?) , ] 
dt V dy dz 

(2) 



* Crelle s Journal, vol. Iv. (1858). Translated by Tait, Phil. Mag., July, 1867. 



410 MAGNETIC ACTION ON LIGHT. [8 2 5- 

The next step in our hypothesis is the assumption that the 
kinetic energy of the medium contains a term of the form 

2<?(ao> 1 + /3a> 2 + y6> 3 ). (3) 

This is equivalent to supposing that the angular velocity acquired 
by the element of the medium during the propagation of light is a 
quantity which may enter into combination with that motion by 
which magnetic phenomena are explained. 

In order to form the equations of motion of the medium, we must 
express its kinetic energy in terms of the velocity of its parts, 
the components of which are f, 77, f We therefore integrate by 
parts, and find 

2 C 1 1 1 (acoj + /3a> 2 -f ya> 3 ) dx dy dz 

+ cff(aC- yfl dz dx + OJJ(ft- arj) dx dy 



The double integrals refer to the bounding surface, which may be 
supposed at an infinite distance. We may, therefore, while in 
vestigating what takes place in the interior of the medium, confine 
our attention to the triple integral. 

825.] The part of the kinetic energy in unit of volume, expressed 
by this triple integral, may be written 

**C(t+iiv + tw), (5) 

where u, v, w are the components of the electric current as given in 
equations (E), Art. 607. 

It appears from this that our hypothesis is equivalent to the 
assumption that the velocity of a particle of the medium whose 
components are f, r/, is a quantity which may enter into com 
bination with the electric current whose components are u, v, w. 

826.] Returning to the expression under the sign of triple inte 
gration in (4), substituting for the values of a, ft, y, those of 
a , /3 , /, as given by equations (1), and writing 
d d d d 



the expression under the sign of integration becomes 
dr d ,d d d sdr 



dk zdh Tz "" r/ dk dx dy 

In the case of waves in planes normal to the axis of z the displace- 



828.] DYNAMICAL THEOEY. 411 

ments are functions of z and t only, so that -77 = y -j- > and this 

dfi dz 
expression is reduced to 



^ 

The kinetic energy per unit of volume, so far as it depends on 
the velocities of displacement, may now be written 



where p is the density of the medium. 

827.] The components, X and Y 9 of the impressed force, referred 
to unit of volume, may be deduced from this by Lagrange s equa 
tions, Art. 564. 

(10) 



<> 

These forces arise from the action of the remainder of the medium 
on the element under consideration, and must in the case of an 
isotropic medium be of the form indicated by Cauchy, 



828.] If we now take the case of a circularly-polarized ray for 
which f = rcos(ntqz), r] = r sin (nt - qz\ (14) 

we find for the kinetic energy in unit of volume 

T \pr*n 2 Cyr 2 q*n , (15) 

and for the potential energy in unit of volume 



= r*Q, (16) 

where Q is a function of q 2 . 

The condition of free propagation of the ray given in Art. 820, 
equation (6), is dT _dV 

dr dr 

which gives P n 2 -2Cyq 2 n = Q, (18) 

whence the value of n may be found in terms of q. 

But in the case of a ray of given wave-period, acted on by 



412 MAGNETIC ACTION ON LIGHT. [829. 

magnetic force, what we want to determine is the value of -, when n 
is constant, in terms of ~ , when y is constant. Differentiating (1 8) 



(2pn 2Cyf)dn {-j^ + lCygnjdti ZCifndy = 0. (19) 

We thus find -f = - ^ ~f ( 2 ) 

ay pnCyq 2 an 

829.] If A is the wave-length in air, and i the corresponding 
index of refraction in the medium, 

q\ = 2ni, n\ = 2irv. (21) 

The change in the value of q, due to magnetic action, is in every 
case an exceedingly small fraction of its own value, so that we may 

^ %, (22) 



where qt is the value of q when the magnetic force is zero. The 
angle, 0, through which the plane of polarization is turned in 
passing through a thickness c of the medium, is half the sum of 
the positive and negative values of qc, the sign of the result being 
changed, because the sign of q is negative in equations (14). We 
thus obtain 

0=-cy^ (23) 

4 TT C i 2 . di x 1 



The second term of the denominator of this fraction is approx 
imately equal to the angle of rotation of the plane of polarization 
during its passage through a thickness of the medium equal to half 
a wave-length. It is therefore in all actual cases a quantity which 
we may neglect in comparison with unity. 

Writing ~ = m, (25) 

vp 

we may call m the coefficient of magnetic rotation for the medium, 
a quantity whose value must be determined by observation. It is 
found to be positive for most diamagnetic, and negative for some 
paramagnetic media. We have therefore as the final result of our 
theory *2 j; 

-x, (26) 



where 6 is the angular rotation of the plane of polarization, m a 



830.] FORMULA FOR THE ROTATION. 413 

constant determined by observation of the medium, y the intensity 
of the magnetic force resolved in the direction of the ray, c the 
length of the ray within the medium, X the wave-length of the 
light in air, and i its index of refraction in the medium. 

830.] The only test to which this theory has hitherto been sub 
jected, is that of comparing the values of for different kinds of 
light passing through the same medium and acted on by the same 
magnetic force. 

This has been done for a considerable number of media by M. 
Verdet "*, who has arrived at the following results : 

(1) The magnetic rotations of the planes of polarization of the 
rays of different colours follow approximately the law of the inverse 
square of the wave-length. 

(2) The exact law of the phenomena is always such that the pro 
duct of the rotation by the square of the wave-length increases from 
the least refrangible to the most refrangible end of the spectrum. 

(3) The substances for which this increase is most sensible are 
also those which have the greatest dispersive power. 

He also found that in the solution of tartaric acid, which of itself 
produces a rotation of the plane of polarization, the magnetic rotation 
is by no means proportional to the natural rotation. 

In an addition to the same memoir f Verdet has given the results 
of very careful experiments on bisulphide of carbon and on creosote, 
two substances in which the departure from the law of the inverse 
square of the wave-length was very apparent. He has also com 
pared these results with the numbers given by three different for 
mulae, f 2 Jj . 

(i) 0-. 

(II) e -. 
(ill) e-. 

w/v 

The first of these formulae, (I), is that which we have already ob 
tained in Art. 829, equation (26). The second, (II), is that which 
results from substituting in the equations of motion, Art. 826, equa- 

70 70 >> -70 

cL t\ cL * f] YI 

tions (10), (11), terms of the form -~ and -j^, instead of -=-5-3- 

cl/t dt dz dt 

* Recherches sur leg proprie te s optiques de veloppe es dans les corps transparents 
par Faction du magn^tisme, 4 me partie. Comptes JfawfttS, t. Ivi. p. 630 (6 April, 1863). 
t Comptes Rendw, Ivii. p. 670 (19 Oct., 1863). 



414 MAGNETIC ACTION ON LIGHT. [830. 



and -- j-A I am no ^ aware that this form of the equations has 
dz^dt 

been suggested by any physical theory. The third formula, (III), 
results from the physical theory of M. C. Neumann *, in which the 

equations of motion contain terms of the form ~ and -- t. 

dt dt 

It is evident that the values of 6 given by the formula (III) are 
not even approximately proportional to the inverse square of the 
wave-length. Those given by the formulae (I) and (II) satisfy this 
condition, and give values of 6 which agree tolerably well with the 
observed values for media of moderate dispersive power. For bisul 
phide of carbon and creosote, however, the values given by (II) differ 
very much from those observed. Those given by (I) agree better 
with observation, but, though the agreement is somewhat close for 
bisulphide of carbon, the numbers for creosote still differ by quan 
tities much greater than can be accounted for by any errors of 
observation. 

Magnetic Rotation of the Plane of Polarization (from Verdef). 

Bisulphide of Carbon at 24. 9 C. 

Lines of the spectrum C D E F G 

Observed rotation 592 768 1000 1234 1704 

Calculated by I. 589 760 1000 1234 1713 

II. 606 772 1000 1216 1640 

III. 943 967 1000 1034 1091 
Rotation of the ray E = 25. 28 . 

Creosote at 24. 3 C. 

Lines of the spectrum C D E F 

Observed rotation 573 758 1000 1241 1723 

Calculated by I. 617 780 1000 1210 1603 

II. 623 789 1000 1200 1565 

III. 976 993 1000 1017 1041 
Rotation of the ray E = 21. 58 . 

We are so little acquainted with the details of the molecular 

* Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires elec- 
tricas vel magneticas declinetur. Halis Saxonum, 1858. 

f* These three forms of the equations of motion were first suggested by Sir G. B. 
Airy (Phil. Mag., June 1846) as a means of analysing the phenomenon then recently 
discovered by Faraday. Mac Cullagh had previously suggested equations containing 

terms of the form in order to represent mathematically the phenomena of quartz. 

These equations were offered by Mac Cullagh and Airy, not as giving a mechanical 
explanation of the phenomena, but as shewing that the phenomena may be explained 
by equations, which equations appear to be such as might possibly be deduced from 
some plausible mechanical assumption, although no such assumption lias yet been 
made. 



831.] ARGUMENT OF THOMSON. 415 

constitution of bodies, that it is not probable that any satisfactory 
theory can be formed relating to a particular phenomenon, such as 
that of the magnetic action on light, until, by an induction founded 
on a number of different cases in which visible phenomena are found 
to depend upon actions in which the molecules are concerned, we 
learn something more definite about the properties which must be 
attributed to a molecule in order to satisfy the conditions of ob 
served facts. 

The theory proposed in the preceding pages is evidently of a 
provisional kind, resting as it does on unproved hypotheses relating 
to the nature of molecular vortices, and the mode in which they are 
affected by the displacement of the medium. We must therefore 
regard any coincidence with observed facts as of much less scientific 
value in the theory of the magnetic rotation of the plane of polari 
zation than in the electromagnetic theory of light, which, though it 
involves hypotheses about the electric properties of media, does not 
speculate as to the constitution of their molecules. 

831.] NOTE. The whole of this chapter may be regarded as an 
expansion of the exceedingly important remark of Sir William 
Thomson in the Proceedings of the Royal Society, June 1856 : The 
magnetic influence on light discovered by Faraday depends on the 
direction of motion of moving particles. For instance, in a medium 
possessing it, particles in a straight line parallel to the lines of 
magnetic force, displaced to a helix round this line as axis, and then 
projected tangentially with such velocities as to describe circles, 
will have different velocities according as their motions are round 
in one direction (the same as the nominal direction of the galvanic 
current in the magnetizing coil), or in the contrary direction. But 
the elastic reaction of the medium must be the same for the same 
displacements, whatever be the velocities and directions of the par 
ticles ; that is to say, the forces which are balanced by centrifugal 
force of the circular motions are equal, while the luminiferous 
motions are unequal. The absolute circular motions being there 
fore either equal or such as to transmit equal centrifugal forces to 
the particles initially considered, it follows that the luminiferous 
motions are only components of the whole motion ; and that a less 
luminiferous component in one direction, compounded with a mo 
tion existing in the medium when transmitting no light, skives an 
equal resultant to that of a greater luminiferous motion in the con 
trary direction compounded with the same non -luminous motion. 
I think it is not only impossible to conceive any other than this 



410 MAGNETIC ACTION ON LIGHT. 

dynamical explanation of the fact that circularly-polarized light 
transmitted through magnetized glass parallel to the lines of mag 
netizing force, with the same quality, right-handed always, or left- 
handed always, is propagated at different rates according as its 
course is in the direction or is contrary to the direction in which a 
north magnetic pole is drawn ; but I believe it can be demonstrated 
that no other explanation of that fact is possible. Hence it appears 
that Faraday s optical discovery affords a demonstration of the re 
ality of Ampere s explanation of the ultimate nature of magnetism ; 
and gives a definition of magnetization in the dynamical theory of 
heat. The introduction of the principle of moments of momenta 
(" the conservation of areas ") into the mechanical treatment of 
Mr. Rankine s hypothesis of " molecular vortices," appears to indi 
cate a line perpendicular to the plane of resultant rotatory mo 
mentum ("the invariable plane") of the thermal motions as the 
magnetic axis of a magnetized body, and suggests the resultant 
moment of momenta of these motions as the definite measure of 
the " magnetic moment." The explanation of all phenomena of 
electromagnetic attraction or repulsion, and of electromagnetic in 
duction, is to be looked for simply in the inertia and pressure of 
the matter of which the motions constitute heat. Whether this 
matter is or is not electricity, whether it is a continuous fluid inter- 
permeating the spaces between molecular nuclei, or is itself mole- 
cularly grouped ; or whether all matter is continuous, and molecular 
heterogeneousness consists in finite vortical or other relative mo 
tions of contiguous parts of a body ; it is impossible to decide, and 
perhaps in vain to speculate, in the present state of science. 

A theory of molecular vortices, which I worked out at consider 
able length, was published in the Phil. Mag. for March, April, and 
May, 1861, Jan. and Feb. 1862. 

I think we have good evidence for the opinion that some pheno 
menon of rotation is going on in the magnetic field, that this rota 
tion is performed by a great number of very small portions of 
matter, each rotating on its own axis, this axis being parallel to the 
direction of the magnetic force, and that the rotations of these dif 
ferent vortices are made to depend on one another by means of some 
kind of mechanism connecting them. 

The attempt which I then made to imagine a working model of 
this mechanism must be taken for no more than it really is, a de 
monstration that mechanism may be imagined capable of producing 
a connexion mechanically equivalent to the actual connexion of the 



831.] THEOBY OP MOLECULAK VORTICES. 417 

parts of the electromagnetic field. The problem of determining the 
mechanism required to establish a given species of connexion be 
tween the motions of the parts of a system always admits of an 
infinite number of solutions. Of these, some may be more clumsy 
or more complex than others, but all must satisfy the conditions of 
mechanism in general. 

The following results of the theory, however, are of higher 
value : 

(1) Magnetic force is the effect of the centrifugal force of the 
vortices. 

(2) Electromagnetic induction of currents is the effect of the 
forces called into play when the velocity of the vortices is changing. 

(3) Electromotive force arises from the stress on the connecting 
mechanism. 

(4) Electric displacement arises from the elastic yielding of the 
connecting mechanism. 



VOL. II. 



CHAPTER XXII 

FEBROMAQNETISM AND DIAMAGNETISM EXPLAINED BY 
MOLECULAR CURRENTS. 

On Electromagnetic Theories of Magnetism. 

832.] WE have seen (Art. 380) that the action of magnets on 
one another can be accurately represented by the attractions and 
repulsions of an imaginary substance called * magnetic matter. 
We have shewn the reasons why we must not suppose this magnetic 
matter to move from one part of a magnet to another through a 
sensible distance, as at first sight it appears to do when we 
magnetize a bar, and we were led to Poisson s hypothesis that the 
magnetic matter is strictly confined to single molecules oi" the mag 
netic substance, so that a magnetized molecule is one in which the 
opposite kinds of magnetic matter are more or less separated to 
wards opposite poles of the molecule, but so that no part of either 
can ever be actually separated from the molecule (Art. 430). 

These arguments completely establish the fact, that magnetiza 
tion is a phenomenon, not of large masses of iron, but of molecules, 
that is to say, of portions of the substance so small that we cannot 
by any mechanical method cut one of them in two, so as to obtain a 
north pole separate from a south pole. But the nature of a mag 
netic molecule is by no means determined without further investi 
gation. We have seen (Art. 442) that there are strong reasons for 
believing that the act of magnetizing iron or steel does not consist 
in imparting magnetization to the molecules of which it is com 
posed, but that these molecules are already magnetic, even in un- 
magnetized iron, but with their axes placed indifferently in all 
directions, and that the act of magnetization consists in turning 
the molecules so that their axes are either rendered all parallel to 
one direction, or at least. are deflected towards that direction. 



834-] AMPERE S THEORY. 419 

833.] Still, however, we have arrived at no explanation of the 
nature of a magnetic molecule, that is, we have not recognized its 
likeness to any other thing of which we know more. We have 
therefore to consider the hypothesis of Ampere, that the magnetism 
of the molecule is due to an electric current constantly circulating 
in some closed path within it. 

It is possible to produce an exact imitation of the action of any 
magnet on points external to it, by means of a sheet of electric 
currents properly distributed on its outer surface. But the action 
of the magnet on points in the interior is quite different from the 
action of the electric currents on corresponding points. Hence Am 
pere concluded that if magnetism is to be explained by means of 
electric currents, these currents must circulate within the molecules 
of the magnet, and must not flow from one molecule to another. 
As we cannot experimentally measure the magnetic action at a 
point in the interior of a molecule, this hypothesis cannot be dis 
proved in the same way that we can disprove the hypothesis of 
currents of sensible extent within the magnet. 

Besides this, we know that an electric current, in passing from 
one part of a conductor to another, meets with resistance and gene 
rates heat ; so that if there were currents of the ordinary kind round 
portions of the magnet of sensible size, there would be a constant 
expenditure of energy required to maintain them, and a magnet 
would be a perpetual source of heat. By confining the circuits to 
the molecules, within which nothing is known about resistance, we 
may assert, without fear of contradiction, that the current, in cir 
culating within the molecule, meets with no resistance. 

According to Ampere s theory, therefore, all the phenomena of 
magnetism are due to electric currents, and if we could make ob 
servations of the magnetic force in the interior of a magnetic mole 
cule, we should find that it obeyed exactly the same laws as the 
force in a region surrounded by any other electric circuit. 

834.] In treating of the force in the interior of magnets, we have 
supposed the measurements to be made in a small crevasse hollowed 
out of the substance of the magnet, Art. 395. We were thus led 
to consider two different quantities, the magnetic force and the 
magnetic induction, both of which are supposed to be observed in 
a space from which the magnetic matter is removed. We were 
not supposed to be able to penetrate into the interior of a mag 
netic molecule and to observe the force within it. 

If we adopt Ampere s theory, we consider a magnet, not as a 

E e 2 



420 ELECTE1C THEORY OF MAGNETISM. [835. 

continuous substance, the magnetization of which varies from point 
to point according to some easily conceived law, but as a multitude 
of molecules, within each of which circulates a system of electric 
currents, giving rise to a distribution of magnetic force of extreme 
complexity, the direction of the force in the interior of a molecule 
being generally the reverse of that of the average force in its neigh 
bourhood, and the magnetic potential, where it exists at all, being 
a function of as many degrees of multiplicity as there are molecules 
in the magnet. 

835.] But we shall find, that, in spite of this apparent complexity, 
which, however, arises merely from the coexistence of a multitude 
of simpler parts, the mathematical theory of magnetism is greatly 
simplified by the adoption of Ampere s theory, and by extending 
our mathematical vision into the interior of the molecules. 

In the first place, the two definitions of magnetic force are re 
duced to one, both becoming the same as that for the space outside 
the magnet. In the next place, the components of the magnetic 
force everywhere satisfy the condition to which those of induction 
are subject, namely, da dp, dy _ 
dx dy dz ~ 

In other words, the distribution of magnetic force is of the 
same nature as that of the velocity of an incompressible fluid, 
or, as we have expressed it in Art. 25, the magnetic force has no 
convergence. 

Finally, the three vector functions the electromagnetic momen 
tum, the magnetic force, and the electric current become more 
simply related to each other. They are all vector functions of no 
convergence, and they are derived one from the other in order, by 
the same process of taking the space-variation, which is denoted 
by Hamilton by the symbol V. 

836.] But we are now considering magnetism from a physical 
point of view, and we must enquire into the physical properties of 
the molecular currents. We assume that a current is circulating 
in a molecule, and that it meets with no resistance. If L is the 
coefficient of self-induction of the molecular circuit, and M the co 
efficient of mutual induction between this circuit and some other 
circuit, then if y is the current in the molecule, and y that in the 
other circuit, the equation of the current y is 

=-S r , (2) 



838.] CIRCUITS OF NO RESISTANCE. 421 

and since by the hypothesis there is no resistance, R = 0, and we 
get by integration 

Ly + My = constant, = Ly ot say. (3) 

Let us suppose that the area of the projection of the molecular 
circuit on a plane perpendicular to the axis of the molecule is A, 
this axis being defined as the normal to the plane on which the 
projection is greatest. If the action of other currents produces a 
magnetic force, X, in a direction whose inclination to the axis of 
the molecule is 0, the quantity My becomes XA cos0, and we have 
as the equation of the current 

Ly + XAco$e Ly , (4) 

where y is the value of y when X = 0. 

It appears, therefore, that the strength of the molecular current 
depends entirely on its primitive value y , and on the intensity of 
the magnetic force due to other currents. 

837.] If we suppose that there is no primitive current, but that 
the current is entirely due to induction, then 

* XA 

y = j cos 0. (o) 

Jj 

The negative sign shews that the direction of the induced cur 
rent is opposite to that of the inducing current, and its magnetic 
action is such that in the interior of the circuit it acts in the op 
posite direction to the magnetic force. In other words, the mole 
cular current acts like a small magnet whose poles are turned 
towards the poles of the same name of the inducing magnet. 

Now this is an action the reverse of that of the molecules of iron 
under magnetic action. The molecular currents in iron, therefore, 
are not excited by induction. But in diamagnetic substances an 
action of this kind is observed, and in fact this is the explanation of 
diamagnetic polarity which was first given by Weber. 

Weber s Theory of Diamagnetism. 

838.] According to Weber s theory, there exist in the molecules 
of diamagnetic substances certain channels round which an electric 
current can circulate without resistance. It is manifest that if we 
suppose these channels to traverse the molecule in every direction, 
this amounts to making the molecule a perfect conductor. 

Beginning with the assumption of a linear circuit within the mo 
lecule, we have the strength of the current given by equation (5). 



422 ELECTRIC THEORY OF MAGNETISM. [8 39. 

The magnetic moment of the current is the product of its strength 
by the area of the circuit, or yA, and the resolved part of this in the 
direction of the magnetizing force is yAcosO, or, by (5), 

Y //2 
-^-cos 2 0. (6) 



If there are n such molecules in unit of volume, and if their axes are 
distributed indifferently in all directions, then the average value of 
cos 2 will be J, and the intensity of magnetization of the substance 
will be ^nXA* , ? . 

L 
Neumann s coefficient of magnetization is therefore 



_ 

The magnetization of the substance is therefore in the opposite 
direction to the magnetizing force, or, in other words, the substance 
is diamagnetic. It is also exactly proportional to the magnetizing 
force, and does not tend to a finite limit, as in the case of ordinary 
magnetic induction. See Arts. 442, &c. 

839.] If the directions of the axes of the molecular channels are 
arranged, not indifferently in all directions, but with a preponder 
ating number in certain directions, then the sum 



Ju 

extended to all the molecules will have different values according 
to the direction of the line from which 6 is measured, and the dis 
tribution of these values in different directions will be similar to the 
distribution of the values of moments of inertia about axes in dif 
ferent directions through the same point. 

Such a distribution will explain the magnetic phenomena related 
to axes in the body, described by Pliicker, which Faraday has called 
Magne-crystallic phenomena. See Art. 435. 

840.] Let us now consider what would be the effect, if, instead 
of the electric current being confined to a certain channel within 
the molecule, the whole molecule were supposed a perfect conductor. 

Let us begin with the case of a body the form of which is acyclic, 
that is to say, which is not in the form of a ring or perforated 
body, and let us suppose that this body is everywhere surrounded 
by a thin shell of perfectly conducting matter. 

We have proved in Art. 654, that a closed sheet of perfectly 
conducting matter of any form, originally free from currents, be- 



842.] PERFECTLY CONDUCTING MOLECULES. 423 

comes, when exposed to external magnetic force, a current-sheet, the 
action of which on every point of the interior is such as to make 
the magnetic force zero. 

It may assist us in understanding this case if we observe that 
the distribution of magnetic force in the neighbourhood of such a 
body is similar to the distribution of velocity in an incompressible 
fluid in the neighbourhood of an impervious body of the same form. 

It is obvious that if other conducting shells are placed within 
the first, since they are not exposed to magnetic force, no currents 
will be excited in them. Hence, in a solid of perfectly conducting 
material, the effect of magnetic force is to generate a system of 
currents which are entirely confined to the surface of the body. 

841.] If the conducting body is in the form of a sphere of radius 
r, its magnetic moment is 

and if a number of such spheres are distributed in a medium, so 
that in unit of volume the volume of the conducting matter is Xf, 
then, by putting ^=1, and /x 2 = in equation (17), Art. 314, we find 
the coefficient of magnetic permeability, 

f\ n If 

(9) 



whence we obtain for Poisson s magnetic coefficient 

t=-\tf, (10) 

and for Neumann s coefficient of magnetization by induction 



Since the mathematical conception of perfectly conducting bodies 
leads to results exceedingly different from any phenomena which 
we can observe in ordinary conductors, let us pursue the subject 
somewhat further. 

842.] Returning to the case of the conducting channel in the 
form of a closed curve of area A, as in Art. 836, we have, for the 
moment of the electromagnetic force tending to increase the angle 0, 



n0 m (12) 

= ^-sin0cos0. (13) 

This force is positive or negative according as is less or greater 
than a right angle. Hence the effect of magnetic force on a per 
fectly conducting channel tends to turn it with its axis at right 



424 ELECTRIC THEORY OF MAGNETISM. [843. 

angles to the line of magnetic force, that is, so that the plane of the 
channel becomes parallel to the lines of force. 

An effect of a similar kind may be observed by placing a penny 
or a copper ring between the poles of an electromagnet. At the 
instant that the magnet is excited the ring turns its plane towards 
the axial direction, but this force vanishes as soon as the currents 
are deadened by the resistance of the copper *. 

843.] We have hitherto considered only the case in which the 
molecular currents are entirely excited by the external magnetic 
force. Let us next examine the bearing of Weber s theory of the 
magneto-electric induction of molecular currents on Ampere s theory 
of ordinary magnetism. According to Ampere and Weber, the 
molecular currents in magnetic substances are not excited by the 
external magnetic force, but are already there, and the molecule 
itself is acted on and deflected by the electromagnetic action of the 
magnetic force on the conducting circuit in which the current flows. 
When Ampere devised this hypothesis, the induction of electric cur 
rents was not known, and he made no hypothesis to account for the 
existence, or to determine the strength, of the molecular currents. 

We are now, however, bound to apply to these currents the same 
laws that Weber applied to his currents in diamagnetic molecules. 
We have only to suppose that the primitive value of the current y, 
when no magnetic force acts, is not zero but y . The strength of 
the current when a magnetic force, X, acts on a molecular current 
of area A, whose axis is inclined 6 to the line of magnetic force, is 



and the moment of the couple tending to turn the molecule so as 

to increase is X 2 A 2 

y XAsm0 + sin 26. (15) 

Hence, putting A 

Ay Q = m, /- = *, (16) 

^7o 

in the investigation in Art. 443, the equation of equilibrium becomes 
Xsin0 3X 2 sin0cos0 = Dsin(a-0). (17) 

The resolved part of the magnetic moment of the current in the 
direction of X is 

XA 2 

y A cosO = y Acos0 -- ^ cos 2 (9, (18) 

L 

= mcosO(l-3XcoaO). (19) 

* See Faraday, Exp. Res., 2310, &c. 



845-] MODIFIED THEORY OF INDUCED MAGNETISM. 425 

844.] These conditions differ from those in Weber s theory of 
magnetic induction by the terms involving the coefficient B. If 
BX is small compared with unity, the results will approximate to 
those of Weber s theory of magnetism. If BX is large compared 
with unity, the results will approximate to those of Weber s theory 
of diamagnetism. 

Now the greater y , the primitive value of the molecular current, 
the smaller will B become, and if L is also large, this will also 
diminish B. Now if the current flows in a ring channel, the value 

T> 

of L depends on log , where R is the radius of the mean line of 

the channel, and r that of its section. The smaller therefore the 
section of the channel compared with its area, the greater will be L, 
the coefficient of self-induction, and the more nearly will the phe 
nomena agree with Weber s original theory. There will be this 
difference, however, that as X, the magnetizing force, increases, the 
temporary magnetic moment will not only reach a maximum, but 
will afterwards diminish as X increases. 

If it should ever be experimentally proved that the temporary 
magnetization of any substance first increases, and then diminishes 
as the magnetizing force is continually increased, the evidence of 
the existence of these molecular currents would, I think, be raised 
almost to the rank of a demonstration. 

845.] If the molecular currents in diamagnetic substances are 
confined to definite channels, and if the molecules are capable of 
being deflected like those of magnetic substances, then, as the mag 
netizing force increases, the diamagnetic polarity will always increase, 
but, when the force is great, not quite so fast as the magnetizing 
force. The small absolute value of the diamagnetic coefficient shews, 
however, that the deflecting force on each molecule must be small 
compared with that exerted on a magnetic molecule, so that any 
result due to this deflexion is not likely to be perceptible. 

If, on the other hand, the molecular currents in diamagnetic 
bodies are free to flow through the whole substance of the molecules, 
the diamagnetic polarity will be strictly proportional to the mag 
netizing force, and its amount will lead to a determination of the 
whole space occupied by the perfectly conducting masses, and, if we 
know the number of the molecules, to the determination of the size 
of each, 



CHAPTER XXIII. 



THEORIES OF ACTION AT A DISTANCE. 



On the Explanation of Ampere s Formula given by Gauss and Weber. 

846.] The attraction between the elements ds and da of two 
circuits, carrying electric currents of intensity i and i t is, by 
Ampere s formula, 

ii ds ds dr dr\ ft \ 

3--; (1) 



zr _ . 

r 2 v ds ds ds ds 

the currents being estimated in electromagnetic units. See Art. 526. 
The quantities, whose meaning as they appear in these expres 
sions we have now to interpret, are 

dr dr . d 2 r 

cos e, -jr- -7-7 > and -= T> ; 
ds ds dsds 

and the most obvious phenomenon in which to seek for an inter 
pretation founded on a direct relation between the currents is the 
relative velocity of the electricity in the two elements. 

847.] Let us therefore consider the relative motion of two par 
ticles, moving with constant velocities v and v along the elements 
ds and ds respectively. The square of the relative velocity of these 
particles is U 2 = v z _ 2 vv cos e + v 2 -, (3) 

and if we denote by r the distance between the particles, 

dr dr ,dr ... 

v7 ~ v 7~+ v -r>> ( 4 ) 

^ ds ds 

.dr dr /9 /dr\ 2 /e v 

v 5 



848.] FECHNER S HYPOTHESIS. 427 

where the symbol <) indicates that, in the quantity differentiated, 
the coordinates of the particles are to be expressed in terms of the 
time. 

It appears, therefore, that the terms involving the product vv in 
the equations (3), (5), and (6) contain the quantities occurring in 
(1) and (2) which we have to interpret. We therefore endeavour to 

~~ 



and 2 But in order to 



express (1) and (2) in terms of ^ 2 , i 

do so we must get rid of the first and third terms of each of these 
expressions, for they involve quantities which do not appear in the 
formula of Ampere. Hence we cannot explain the electric current 
as a transfer of electricity in one direction only, but we must com 
bine two opposite streams in each current, so that the combined 
effect of the terms involving v 2 and v 2 may be zero. 

848.] Let us therefore suppose that in the first element, ds, we 
have one electric particle, , moving with velocity ?;, and another, e lt 
moving with velocity v l , and in the same way two particles, ef and 
e\, in ds t moving with velocities v and v L respectively. 

The term involving v 2 for the combined action of these particles 



Similarly 2 (t/W) = (v 2 e + v\ 2 e\) (e + ^) ; (8) 

and 2(vtfeS) = (ve + v^^v e + vYi). (9) 

In order that 2 (o 2 ee ) may be zero, we must have either 

/ + e\ = 0, or V 2 e + v 1 2 e 1 = 0. (10) 

According to Eechner s hypothesis, the electric current consists 
of a current of positive electricity in the positive direction, com 
bined with a current of negative electricity in the negative direc 
tion, the two currents being exactly equal in numerical magnitude, 
both as respects the quantity of electricity in motion and the velo 
city with which it is moving. Hence both the conditions of (10) 
are satisfied by Fechner s hypothesis. 

But it is sufficient for our purpose to assume, either 

That the quantity of positive electricity in each element is nu 
merically equal to the quantity of negative electricity ; or 

That the quantities of the two kinds of electricity are inversely 
as the squares of their velocities. 

Now we know that by charging the second conducting wire as a 
whole, we can make e -f e\ either positive or negative. Such a 
charged wire, even without a current, according to this formula, 
would act on the first wire carrying a current in which v 2 e -j- r 1 2 e l 



428 ACTION AT A DISTANCE. [849. 

has a value differing from zero. Such an action has never been 
observed. 

Therefore, since the quantity e + e\ may be shewn experimentally 
not to be always zero, and since the quantity v 2 e + v 2 1 e l is not 
capable of being experimentally tested, it is better for these specu 
lations to assume that it is the latter quantity which invariably 
vanishes. 

849.] Whatever hypothesis we adopt, there can be no doubt that 
the total transfer of electricity, reckoned algebraically, along the 
first circuit, is represented by 

ve-\-v 1 e i = dels; 

where c is the number of units of statical electricity which are 
transmitted by the unit electric current in the unit of time, so that 
we may write equation (9) 

2 (vv ee } = c 2 ii ds ds . (11) 

Hence the sums of the four values of (3), (5), and (6) become 

2 (ee n 2 ) = -2 c^ii ds ds cos e ; (12) 

^, (13) 

ds ds 



and we may write the two expressions (1) and (2) for the attraction 
between ds and ds 



850.] The ordinary expression, in the theory of statical electri- 

PP 

city, for the repulsion of two electrical particles e and e is - , and 



which gives the electrostatic repulsion between the two elements if 
they are charged as wholes. 

Hence, if we assume for the repulsion of the two particles either 
of the modified expressions 






we may deduce from them both the ordinary electrostatic forces, and 
the forces acting between currents as determined by Ampere. 



FORMULAE OF GAUSS AND WEBER, 429 

851.] The first of these expressions, (18), was discovered by 
Gauss * in July 1835, and interpreted by him as a fundamental law 
of electrical action, that Two elements of electricity in a state of 
relative motion attract or repel one another, but not in the same 
way as if they are in a state of relative rest. This discovery was 
not, so far as I know, published in the lifetime of Gauss, so that the 
second expression, which was discovered independently by W.Weber, 
and published in the first part of his celebrated Elektrodynamische 
Maasbe&timmungen^ , was the first result of the kind made known 
to the scientific world. 

852.] The two expressions lead to precisely the same result when 
they are applied to the determination of the mechanical force be 
tween two electric currents, and this result is identical with that 
of Ampere. But when they are considered as expressions of the 
physical law of the action between two electrical particles, we are 
led to enquire whether they are consistent with other known facts 
of nature. 

Both of these expressions involve the relative velocity of the 
particles. Now, in establishing- by mathematical reasoning the 
well-known principle of the conservation of energy, it is generally 
assumed that the force acting between two particles is a function of 
the distance only, and it is commonly stated that if it is a function 
of anything else, such as the time, or the velocity of the particles, 
the proof would not hold. 

Hence a law of electrical action, involving the velocity of the 
particles, has sometimes been supposed to be inconsistent with the 
principle of the conservation of energy. 

853.] The formula of Gauss is inconsistent with this principle, 
and must therefore be abandoned, as it leads to the conclusion that 
energy might be indefinitely generated in a finite system by physical 
means. This objection does not apply to the formula of Weber, for 
he has shewn J that if we assume as the potential energy of a system 
consisting of two electric particles, 



the repulsion between them, which is found by differentiating this 
quantity with respect to r, and changing the sign, is that given by 
the formula (19). 

* Werke (G-ottingen edition, 1867), \ol.v. p. 616. 
t Abh. Leibnizens Qes., Leipzig (1846). 
J Pogg. Ann., Ixxiii. p. 229 (1848). 



430 ACTION AT A DISTANCE. [8 54. 

Hence the work done on a moving particle by the repulsion of a 
fixed particle is ^o~"^i where \IT O and \//j are the values of \ff at the 
beginning and at the end of its path. Now \j/ depends only on the 
distance, r, and on the velocity resolved in the direction of r. If, 
therefore, the particle describes any closed path, so that its position, 
velocity, and direction of motion are the same at the end as at the 
beginning, ^ will be equal to \^ , and no work will be done on the 
whole during the cycle of operations. 

Hence an indefinite amount of work cannot be generated by a 
particle moving in a periodic manner under the action of the force 
assumed by Weber. 

854.] But Helmholtz, in his very powerful memoir on the Equa 
tions of Motion of Electricity in Conductors at Rest *, while he 
shews that Weber s formula is not inconsistent with the principle 
of the conservation of energy, as regards only the work done during 
a complete cyclical operation, points out that it leads to the conclu 
sion, that two electrified particles, which move according to Weber s 
law, may have at first finite velocities, and yet, while still at a finite 
distance from each other, they may acquire an infinite kinetic energy, 
and may perform an infinite amount of work. 

To this Weber f replies, that the initial relative velocity of the 
particles in Helmholtz s example, though finite, is greater than the 
velocity of light ; and that the distance at which the kinetic energy 
becomes infinite, though finite, is smaller than any magnitude which 
we can perceive, so that it may be physically impossible to bring two 
molecules so near together. The example, therefore, cannot be tested 
by any experimental method. 

Helmholtz J has therefore stated a case in which the distances are 
not too small, nor the velocities too great, for experimental verifica 
tion. A fixed non-conducting spherical surface, of radius &, is uni 
formly charged with electricity to the surface-density a. A particle, 
of mass m and carrying a charge e of electricity, moves within the 
sphere with velocity v. The electrodynamic potential calculated 
from the formula (20) is 

2 

l-, (21) 



and is independent of the position of the particle within the sphere. 
Adding to this V t the remainder of the potential energy arising 

* Crelle s Journal, 72 (1870). 

t Elektr. Maasl). inlmondere liber das Princip der Erhaltung der Energie. 

J Ikiiin Monatslericht, April 1872; Phil May., Dec. 1872, Supp. 



856.] POTENTIAL OF TWO CLOSED CURRENTS. 431 

from the action of other forces, and \mv 2 , the kinetic energy of the 
particle, we find as the equation of energy 

r* const. (22) 



Since the second term of the coefficient of v 3 may be increased in 
definitely by increasing a, the radius of the sphere, while the surface- 
density a remains constant, the coefficient of v 2 may be made negative. 
Acceleration of the motion of the particle would then correspond to 
diminution of its vis viva, and a body moving in a closed path and 
acted on by a force like friction, always opposite in direction to its 
motion, would continually increase in velocity, and that without 
limit. This impossible result is a necessary consequence of assuming 
any formula for the potential which introduces negative terms into 
the coefficient of v 2 . 

855.] But we have now to consider the application of Weber s 
theory to phenomena which can be realized. We have seen how it 
gives Ampere s expression for the force of attraction between two 
elements of electric currents. The potential of one of these ele 
ments on the other is found by taking the sum of the values of the 
potential \j/ for the four combinations of the positive and negative 
currents in the two elements. The result is, by equation (20), taking 

the sum of the four values of ,, 

di 



(23) 
r ds ds 

and the potential of one closed current on another is 

_ w /Yl d 4-~ds ds = ii M, (24) 

jj. r ds ds 

i I r*o^ p 
where M = 1 1 - dsds*, as in Arts. 423, 524. 

In the case of closed currents, this expression agrees with that 
which we have already (Art. 524) obtained"*. 

Weber s Theory of the Induction of Electric Currents. 

856.] After deducing from Ampere s formula for the action 
between the elements of currents, his own formula for the action 
between moving electric particles, Weber proceeded to apply his 
formula to the explanation of the production of electric currents by 

* In the whole of this investigation Weber adopts the electrodynamic system of 
units. Tn this treatise we always use the electromagnetic system. The electro-mag 
netic unit of current is to the electrodynamic unit in the ratio of A/2 to 1. Art. 526. 



432 ACTION AT A DISTANCE. [857. 

magneto-electric induction. In this he was eminently successful, 
and we shall indicate the method by which the laws of induced 
currents may be deduced from Weber s formula. But we must 
observe,, that the circumstance that a law deduced from the pheno 
mena discovered by Ampere is able also to account for the pheno 
mena afterwards discovered by Faraday does not give so much 
additional weight to the evidence for the physical truth of the law 
as we might at first suppose. 

For it has been shewn by Helmholtz and Thomson (see Art. 543), 
that if the phenomena of Ampere are true, and if the principle of 
the conservation of energy is admitted, then the phenomena of in 
duction discovered by Faraday follow of necessity. Now Weber s 
law, with the various assumptions about the nature of electric 
currents which it involves, leads by mathematical transformations 
to the formula of Ampere. Weber s law is also consistent with the 
principle of the conservation of energy in so far that a potential 
exists, and this is all that is required for the application of the 
principle by Helmholtz and Thomson. Hence we may assert, even 
before making any calculations on the subject, that Weber s law 
will explain the induction of electric currents. The fact,, therefore, 
that it is found by calculation to explain the induction of currents, 
leaves the evidence for the physical truth of the law exactly where 
it was. 

On the other hand, the formula of Gauss, though it explains the 
phenomena of the attraction of currents, is inconsistent with the 
principle of the conservation of energy, and therefore we cannot 
assert that it will explain all the phenomena of induction. In fact, 
it fails to do so, as we shall see in Art. 859. 

857.] We must now consider the electromotive force tending to 
produce a current in the element els , due to the current in ds, when 
ds is in motion, and when the current in it is variable. 

According to Weber, the action on the material of the conductor 
of which ds is an element, is the sum of all the actions on the 
electricity which it carries. The electromotive force, on the other 
hand, on the electricity in dts t is the difference of the electric forces 
acting on the positive and the negative electricity within it. Since 
all these forces act in the line joining the elements, the electro 
motive force on ds is also in this line, and in order to obtain the 
electromotive force in the direction of ds we must resolve the force 
in that direction. To apply Weber s formula, we must calculate 
the various terms which occur in it, on the supposition that the 



858.] WEBER S THEORY OF INDUCED CURRENTS. 433 

element ds is in motion relatively to els , and that the currents in 
both elements vary with the time. The expressions thus found 
will contain terms involving* v 2 , vv , v 2 , v, ?/, and terms not involv 
ing v or v , all of which are multiplied by ee . Examining, as we 
did before, the four values of each term, and considering first the 
mechanical force which arises from the sum of the four values, we 
find that the only term which we must take into account is that 
involving the product vv ee . 

If we then consider the force tending to produce a current in the 
second element, arising from the difference of the action of the first 
element on the positive and the negative electricity of the second 
element, we find that the only term which we have to examine is 
that which involves vee . We may write the four terms included in 
2 (veef), thus 

e (ve -f v l tfj) and e\ (ve + v l e^. 

Since e -\-e\ = 0, the mechanical force arising from these terms is 
zero, but the electromotive force acting on the positive electricity e 
is (ve + v- e^, and that acting on the negative electricity e\ is equal 
and opposite to this. 

858.] Let us now suppose that the first element ds is moving 

relatively to ds with velocity V in a certain direction, and let us 

A A 

denote by Yds and Yds , the angle between the direction of V and 

that of ds and of ds respectively, then the square of the relative 
velocity, u 9 of two electric particles is 

u 2 = v 2 +v 2 +7 2 -2vv cose+27vcosFds-27v cos7cti. (25) 

The term in vv is the same as in equation (3). That in v, on which 

the electromotive force depends, is 

A 
2 Fv cos Yds. 

We have also for the value of the time- variation of r in this case 

c) r dr f dr dr 

= v --- + >o + , (26) 

^t ds ds dt 

where ^- refers to the motion of the electric particles, and ^- to 

that of the material conductor. If we form the square of this quan 
tity, the term involving vif, on which the mechanical force depends, 
is the same as before, in equation (5), and that involving v, on which 
the electromotive force depends, is 

dr dr 
2v- r - rr > 
ds dt 

VOL. ii. r f 



434 ACTION AT A DISTANCE. [859. 

Differentiating (26) with respect to t, we find 



dv dr , dv dr d 2 r 
* v ~foTs + v ^di^di 2 
We find that the term involving vv is the same as before in (6). 

The term whose sign alters with that of v is -=7- -=- 

dt ds 

859.] If we now calculate by the formula of Gauss (equation (18)), 
the resultant electrical force in the direction of the second element 
ds y arising from the action of the first element ds, we obtain 
1 A A A A 

-y dsds i V (2 cos Yds 3 cos Vr cos r ds) coerdi. (28) 

As in this expression there is no term involving the rate of va 
riation of the current i, and since we know that the variation of 
the primary current produces an inductive action on the secondary 
circuit, we cannot accept the formula of Gauss as a true expression 
of the action between electric particles. 

860.] If, however, we employ the formula of Weber, (19), we 

obtain \ drdi .drdr.dr f . 

(29) 



., , 

r 2 S ds dt ds dt> ds 

dr dr d ,i\ 7 7 , , QA >. 

or -Y -j-, -j- (-) dsds . (30) 

ds ds dt\r 

If we integrate this expression with respect to s and /, we obtain 
for the electromotive force on the second circuit 

d . CCl dr dr , . 

s JJ ;***? 

Now, when the first circuit is closed, 
d 2 r 



ds ds 



= 0. 



/*! dr dr , f A dr dr d 2 r \ , /*cose T 

Hence / - T -^ ds = / (- + ~- 7 - 7 ) ds = - I - - ds. (32) 
J r ds ds J V ds ds dsds J r 

But fj^^dsds / = M, by Arts. 423, 524. (33) 

Hence we may write the electromotive force on the second circuit 

-*< *> (34) 

which agrees with what we have already established by experiment ; 
Art. 539. 



863.] KEYSTONE OF ELECTRODYNAMICS. 435 

On Weber s Formula^ considered as resulting from an Action transmitted 
from one Electric Particle to the other with a Constant Velocity. 

861.] In a very interesting letter of Gauss to W. Weber * he 
refers to the electrodynamic speculations with which he had been 
occupied long before, and which he would have published if he could 
then have established that which he considered the real keystone 
of electrodynamics, namely, the deduction of the force acting be 
tween electric particles in motion from the consideration of an action 
between them, not instantaneous, but propagated in time, in a 
similar manner to that of light. He had not succeeded in making 
this deduction when he gave up his electrodynamic researches, and 
he had a subjective conviction that it would be necessary in the 
first place to form a consistent representation of the manner in 
which the propagation takes place. 

Three eminent mathematicians have endeavoured to supply this 
keystone of electrodynamics. 

862. J In a memoir presented to the Royal Society of Gottingen 
in 1858, but afterwards withdrawn, and only published in Poggen- 
dorff s Annalen in 1867, after the death of the author, Bernhard 
Riemann deduces the phenomena of the induction of electric cur 
rents from a modified form of Poisson s equation 



where Fis the electrostatic potential, and a a velocity. 

This equation is of the same form as those which express the 
propagation of waves and other disturbances in elastic media. The 
author, however, seems to avoid making explicit mention of any 
medium through which the propagation takes place. 

The mathematical investigation given by Riemann has been ex 
amined by Clausiusf, who does not admit the soundness of the 
mathematical processes, and shews that the hypothesis that potential 
is propagated like light does not lead either to the formula of Weber, 
or to the known laws of electrodynamics. 

863.] Clausius has also examined a far more elaborate investiga 
tion by C. Neumann on the Principles of Electrodynamics J. Neu 
mann, however, lias pointed out that his theory of the transmission 
of potential from one electric particle to another is quite different 
from that proposed by Gauss, adopted by Riemann, and criticized 

* March 19, 1845, WerJse, bd. v. 629. Tubingen, 1868. 

t Pogg., bd. cxxxv. 612. Mathematische Annalen, i. 317. 



436 ACTION AT A DISTANCE. [864. 

by Clausius, in which the propagation is like that of light. There 
is, on the contrary, the greatest possible difference between the 
transmission of potential, according to Neumann, and the propaga 
tion of light. 

A luminous body sends forth light in all directions, the intensity 
of which depends on the luminous body alone, and not on the 
presence of the body which is enlightened by it. 

An electric particle, on the other hand, sends forth a potential, 

ed 

the value of which, , depends not only on <?, the emitting particle, 

but on e , the receiving particle, and on the distance r between the 
particles at the instant of emission. 

In the case of light the intensity diminishes as the light is pro 
pagated further from the luminous body ; the emitted potential 
flows to the body on which it acts without the slightest alteration 
of its original value. 

The light received by the illuminated body is in general only a 
fraction of that which falls on it ; the potential as received by the 
attracted body is identical with, or equal to, the potential which 
arrives at it. 

Besides this, the velocity of transmission of the potential is not, 
like that of light, constant relative to the aether or to space, but 
rather like that of a projectile, constant relative to the velocity of 
the emitting particle at the instant of emission. 

It appears, therefore, that in order to understand the theory of 
Neumann, we must form a very different representation of the pro 
cess of the transmission of potential from that to which we have 
been accustomed in considering the propagation of light. Whether 
it can ever be accepted as the construirbar Vorstellung of the 
process of transmission, which appeared necessary to Gauss, I cannot 
say, but I have not myself been able to construct a consistent 
mental representation of Neumann s theory. 

864.] Professor Betti*, of Pisa, has treated the subject in a 
different way. He supposes the closed circuits in which the electric 
currents flow to consist of elements each of which is polarized 
periodically, that is, at equidistant intervals of time. These polar 
ized elements act on one another as if they were little magnets 
whose axes are in the direction of the tangent to the circuits. The 
periodic time of this polarization is the same in all electric cir 
cuits. Betti supposes the action of one polarized element on an- 

* Nuovo Cimento, xxvii (1868). 



866.] A MEDIUM NECESSARY. 437 

other at a distance to take place, not instantaneously, but after a 
time proportional to the distance between the elements. In this 
way he obtains expressions for the action of one electric circuit on 
another, which coincide with those which are known to be true. 
Clausius, however, has, in this case also, criticized some parts of 
the mathematical calculations into which we shall not here enter. 

865.] There appears to be, in the minds of these eminent men, 
some prejudice, or a priori objection, against the hypothesis of a 
medium in which the phenomena of radiation of light and heat, 
and the electric actions at a distance take place. It is true that at 
one time those who speculated as to the causes of physical pheno 
mena, were in the habit of accounting for each kind of action at a 
distance by means of a special sethereal fluid, whose function and 
property it was to produce these actions. They filled all space 
three and four times over with aethers of different kinds, the pro 
perties of which were invented merely to save appearances, so that 
more rational enquirers were willing rather to accept not only New 
ton s definite law of attraction at a distance, but even the dogma of 
Cotes "*, that action at a distance is one of the primary properties of 
matter, and that no explanation can be more intelligible than this 
fact. Hence the undulatory theory of light has met with much 
opposition, directed not against its failure to explain the pheno 
mena, but against its assumption of the existence of a medium in 
which light is propagated. 

866.] We have seen that the mathematical expressions for electro- 
dynamic action led, in the mind of Gauss, to the conviction that a 
theory of the propagation of electric action in time would be found 
to be the very key-stone of electrodynamics. Now we are unable 
to conceive of propagation in time, except either as the flight of a 
material substance through space, or as the propagation of a con 
dition of motion or stress in a medium already existing in space. 
In the theory of Neumann, the mathematical conception called 
Potential, which we are unable to conceive as a material substance, 
is supposed to be projected from one particle to another, in a manner 
which is quite independent of a medium, and which, as Neumann 
has himself pointed out, is extremely different from that of the pro 
pagation of light. In the theories of Riemann and Betti it would 
appear that the action is supposed to be propagated in a manner 
somewhat more similar to that of light. 

But in all of these theories the question naturally occurs : If 
* Preface to Newton s Principia, 2nd edition. 



438 ACTION AT A DISTANCE. [866. 

something is transmitted from one particle to another at a distance, 
what is its condition after it has left the one particle and before 
it has reached the other ? If this something is the potential energy 
of the two particles, as in Neumann s theory, how are we to con 
ceive this energy as existing in a point of space, coinciding neither 
with the one particle nor with the other ? In fact, whenever energy 
is transmitted from one body to another in time, there must be 
a medium or substance in which the energy exists after it leaves 
one body and before it reaches the other, for energy, as Torricelli * 
remarked, is a quintessence of so subtile a nature that it cannot be 
contained in any vessel except the inmost substance of material 
things. Hence all these theories lead to the conception of a medium 
in which the propagation takes place, and if we admit this medium 
as an hypothesis, I think it ought to occupy a prominent place in 
our investigations, and that we ought to endeavour to construct a 
mental representation of all the details of its action, and this has 
been my constant aim in this treatise. 

* Lezioni Accademiche (Firenze, 1715), p. 25. 



INDEX. 



The References are to the Articles. 



ABERRATION of light, 78. 
Absorption, electric, 53, 227, 329. 

of light, 798. 

Accumulators or condensers, 50, 226-228. 
Action at a distance, 105, 641-646, 846- 

866. 

Acyclic region, 19, 113. 
^Ether, 782 n. 
Airy, Sir G. B., 454, 830. 
Ampere, Andr Marie, 482, 502-528, 

638, 687, 833, 846. 
Anion, 237. 
Anode, 237. 
Arago s disk, 668, 669. 
Astatic balance, 504. 
Atmospheric electricity, 221. 
Attraction, electric, 27, 38, 103. 

explained by stress in a medium, 105. 



Barclay and Gibson, 229, 789. 

Battery, voltaic, 232. 

Beetz, W., 255, 265, 442. 

Betti, E., 173, 864. 

Bifilar suspension, 459. 

Bismuth, 425. 

Borda, J. C., 3. 

Bowl, spherical, 176-181. 

Bridge, Wheatstone s*. 347, 756, 775, 778. 

electrostatic, 353. 

Bright, Sir C., and Clark, 354, 367. 

Brodie, Sir B. C., 359. 

Broun, John Allan, 462. 

Brush, 56. 

Buff, Heinrich. 271, 368. 



Capacity (electrostatic), 50, 226. 
of a condenser, 50, 87, 102, 196, 227- 
229, 771, 774-780. 



Capacity, calculation of, 102, 196. 

measurement of, 227-229. 

in electromagnetic measure, 774, 
775. 

Capacity (electromagnetic) of a coil, 706, 

756, 778, 779. 
Cathode, 237. 
Cation, 237. 
Cauchy, A. L., 827. 
Cavendish, Henry, 38. 
Cayley, A., 553. 
Centrobaric, 101. 
Circuits, electric, 578-584. 
Circular currents, 694-706. 

solid angle subtended by, 695. 
Charge, electric, 31. 

Clark, Latimer, 358, 629, 725. 
Classification of electrical quantities, 620- 

629. 

Clausius, R., 70, 256, 863. 
Clifford, W. K., 138. 
Coefficients of electrostatic capacity and 

induction, 87, 102. 

of potential, 87. 

of resistance and conductivity, 297, 
298. 

of induced magnetization, 426. 

of electromagnetic induction, 755. 

of self-induction, 756, 757. 
Coercive force, 424, 444. 
Coils, resistance, 335-344. 

electromagnetic, 694-706. 

measurement of, 708. 

comparison of, 752-757. 
Comparison of capacities, 229. 

of coils, 752-757. 

of electromotive forces, 358. 

of resistances, 345-358. 
Concentration, 26, 77. 
Condenser, 50, 226-228. 



* Sir Charles Wheatstone, in his paper on New Instruments and Processes, Phil. 
Trans., 1843, brought this arrangement into public notice, with due acknowledgment 
of the original inventor, Mr. S. Hunter Christie, who had described it in his paper on 
Induced Currents, Phil. Trans., 1833, under the name of a Differential Arrange 
ment. See the remarks of Mr. Latimer Clark in the Society of Telegraph Engineers, 
May 8, 1872. 



440 



I N D E X. 



Condenser, capacity of, 50, 87, 102, 196, 

227-229, 771, 774-780. 
Conduction, 29, 241-254. 
Conduction, linear, 273-284. 

superficial, 294. 

in solids, 285-334. 

electrolytic, 255-265. 

in dielectrics, 325-334. 
Conductivity, equations of, 298, 609. 

and opacity, 798. 
Conductor, 29, 80, 86. 

Conductors, systems of electrified, 84-94. 
Confocal quadric surfaces, 147-154, 192. 
Conjugate circuits, 538, 759. 

conductors, 282, 347. 

functions, 182-206. 

harmonics, 138. 

Constants, principal, of a coil, 700, 753, 

754. 

Conservation of energy, 92, 242, 262, 543, 
Contact force, 246. 
Continuity in time and space, 7. 

equation of, 36, 295. 
Convection, 55, 238, 248. 
Convergence, 25. 
Copper, 51, 360, 362, 761. 
Cotes, Roger, 865. 

Coulomb, C. A., 38, 74, 215, 223, 373. 
Coulomb s law, 79, 80. 
Crystal, conduction in, 297. 

magnetic properties of, 435, 436, 438. 

propagation of light in a, 794797. 
Gumming, James, 252. 

Curl, 25. 

Current, electric, 230. 

be.st method of applying, 744. 



induced, 582. 

steady, 232. 

thermoelectric, 249-254. 

transient, 232, 530, 536, 537, 582, 
748, 758, 760, 771, 776. 

Current- weigher, 726. 
Cyclic region, 18, 113, 481. 
Cylinder, electrification of, 189. 

magnetization of, 436, 438, 439. 

currents in, 682-690. 
Cylindric coils, 676-681. 



Damped vibrations, 732-742, 762. 
Damper, 730. 
Daniell s cell, .232, 272. 
Dead beat galvanometer, 741. 
Decrement, logarithmic, 736. 
Deflexion, 453, 743. 
Delambre, J. B. J., 3. 
Dellmann, F., 221. 
Density, electric, 64. 

of a current, 285. 

measurement of, 223. 
Diamagnetism, 429, 440, 838. 
Dielectric, 52, 109, 111, 229, 325-334, 

366-370, 784. 



[ Diffusion of magnetic force, 801. 
Dip, 461. 
Dipolar, 173, 381. 

Dimensions, 2, 42, 87, 278, 620-629. 
Directed quantities (or vectors), 10. 
Directrix, 517. 
Discharge, 55. 
Discontinuity, 8. 
Disk, 177. 

Arago s, 668, 669. 
Displacement, electric, 60, 75, 76, 111, 

328-334, 608, 783, 791. 
Dygogram, 441. 



Earnshaw, S., 116. 

Earth, magnetism of, 465-474. 

Electric brush, 56. 

charge, 31. 

conduction, 29. 

convection, 211, 238, 248, 255, 259. 

current, 230. 

discharge, 55-57. 

displacement, 60, 75, 76, 111, 328- 
324, 608, 783, 791. 

energy, 85. 
: glow, 55. 

induction, 28. 

machine, 207. 

potential, 70. 

spark, 57. 

tension, 48, 59, 107, 108, 111. 

wind, 55. 
Electrode, 237. 

Electrodynamic system of measurement, 

526. 

Electrodynamometer, 725. 
Electrolysis, 236, 255-272. 
Electrolyte, 237, 255. 
Electrolytic conduction, 255-272, 363, 

799. 

polarization, 257, 264-272. 
Electromagnetic force, 475, 580, 583. 

measurement, 495. 

momentum, 585. 

observations, 730-780. 

and electrostatic units compared, 768 
780. 

rotation, 491. 
Electromagnetism, dynamical theory of, 

568-577. 

Electrometers, 214-220. 
Electromotive force, 49, 69, 111, 241, 

246-254, 358, 569, 579. 
Electrophorus, 208. 
Electroscope, 33, 214. 
Electrostatic measurements, 214-229. 

polarization, 59, 111. 

attraction, 103-111. 

system of units, 620, &c. 
Electrotonic state, 540. 
Elongation, 734. 
Ellipsoid, 150, 302, 437, 439. 
Elliptic integrals, 149, 437, 701. 
Energy, 6, 85, 630-638, 782, 792. 



I N D E X. 



441 



Equations of conductivity, 298, 609. 

of continuity, 35. 

of electric currents, 607. 

of total currents, 610. 

of electromagnetic force, 603. 

of electromotive force, 598. 

of Laplace, 77. 

of magnetization, 400, 605. 

of magnetic induction, 591. 

of Poisson, 77. 

of resistance, 297. 
Equilibrium, points of, 112-117. 



False magnetic poles, 468. 
Faraday, M., his discoveries, 52, 55, 236, 
255, 530, 531, 534, 546, 668, 806. 

his experiments, 28, 429, 530, 668. 

his methods, 37, 82, 122, 493, 528, 
529, 541, 592, 594, 604. 

his speculations, 54, 60, 83, 107, 109, 
245, 429, 502, 540, 547, 569, 645, 782. 

Farad, 629. 

Fechner, G. T., 231, 274, 848. 
Felici, R., 536-539, 669. 
Ferromagnetic, 425, 429, 844. 
Field, electric, 44. 

electromagnetic, 585-619. 

of uniform force, 672. 
First swing, 745. 
Fizeau, H. L., 787. 
Fluid, electric, 36, 37. 

incompressible, 61, 111, 295, 329, 334. 

magnetic, 380. 
Flux, 12. 

Force, electromagnetic, 475, 580, 583. 

electromotive, 49, 69, 111, 233, 241, 
246-254, 358, 569, 579, 595, 598. 

mechanical, 92, 93, 103-111, 174, 580, 
602. 

measurement of, 6. 

acting at a distance, 105. 

lines of, 82, 117-123, 404. 
Foucault, L., 787. 

Fourier, J. B. J., 2w, 243, 332, 333, 801- 
805. 



Galvanometer, 240, 707. 

differential, 346. 

sensitive, 717. 

standard, 708. 

observation of, 742-751. 

Gases, electric discharge in, 55-77, 370. 

resistance of, 369. 
Gassiot, J. P., 57. 
Gaugain, J. M., 366, 712. 
Gauge electrometer, 218. 

Gauss, C. F., 18, 70, 131, 140, 144, 409, 
421, 454, 459, 470, 706, 733, 744, 851. 
Geometric mean distance, 691-693. 
Geometry of position, 421. 
Gibson and Barclay, 229, 789. 
Gladstone, Dr. J. H., 789. 
Glass, 51, 271, 368. 



Glow, electric, 55. 
Grassmann, H., 526, 687. 
Grating, electric effect of, 203. 
Green, George, 70, 89, 318, 439. 
Green s function, 88, 101. 
theorem, 100. 
Groove, electric effect of, 199. 
Grove, Sir W. R., 272, 779. 
Guard-ring, 201, 217, 228. 
Gutta-percha, 51, 367. 



Hamilton, Sir W. Rowan, 10, 561. 
Hard iron, 424, 444. 
Harris, Sir W. Snow, 38, 216. 
Heat, conduction of, 801. 

generated by the current, 242, 283, 
299. 

specific, of electricity, 253. 
Helix, 813. 

Helmholtz, H., 88, 100, 202, 421, 543, 

713, 823, 854. 

Heterostatic electrometers, 218. 
Hockin, Charles, 352, 360, 800. 
Holtz, W., electrical machine, 212. 
Hornstein, Karl, 471 n. 
Huygens, Christian, 782. 
Hydraulic ram, 550. 
Hyposine, 151. 



Idiostatic electrometers, 218. 
Images, electric, 119, 155-181, 189. 

magnetic, 318. 

moving, 662. 

Imaginary magnetic matter, 380. 
Induced currents, 528-552. 

in a plane sheet, 656-669. 

Weber s theory of, 856. 
Induced magnetization, 424-448. 
Induction, electrostatic, 28, 75, 76, 111. 

magnetic, 400. 
Inertia, electric, 550. 

moments and products of, 565. 
Insulators, 29. 

Inversion, electric, 162-181, 188, 316. 
Ion, 237, 255. 
Iron, 424. 

perchloride of, 809. 
Irreconcileable curves, 20, 421. 



Jacobi, M. H., 336. 

Jenkins, William, 546. See Phil Mag., 

1834, pt. ii, p. 351. 
Jenkin, Fleeming, 763, 774. 
Jochmann, E., 669. 
Joule, J. P., 242, 262, 448, 457, 463, 726, 

767. 



Keystone of electrodynamics, 861. 
Kinetics, 553-565. 

Kirchhoff, Gustav, 282, 316, 439, 758. 
Kohlrausch, Rudolph, 265, 365, 723, 771. 



442 



INDEX. 



Lagrange s (J. L.) dynamical equations, 

553-565. 

Lame", G., 17, 147. 
Lamellar magnet, 412. 
Laplace, P. S., 70. 
Laplace s coefficients, 128-146. 

equation, 26, 77, 144, 301. 

expansion, 140. 
Leibnitz, G. W., 18, 424. 
Lenz, E., 265, 530, 542. 

Light, electromagnetic theory of, 781-805. 

and magnetism, 806-831. 
Line-density, 64, 81. 

integral, 16-20. 

of electric force, 69, 622. 

of magnetic force, 401, 481, 498, 499, 
590, 606, 607, 622. 

Lines of equilibrium, 112. 

of flow, 22, 293. 

of electric induction, 82, 117-123. 

of magnetic induction, 404, 489, 529, 
541, 597, 702. 

Linnaeus, C., 23. 
Liouville, J., 173, 176. 
Listing, J. B., 18, 23, 421. 
Lorenz, L., 805 n. 
Loschmidt, J., 5. 



Magnecrystallic phenomena, 425, 435, 

839. 
Magnet, its properties, 371. 

direction of axis, 372-390. 

magnetic moment of, 384, 390. 

centre and principal axes, 392. 

potential energy of, 389. 
Magnetic action of light, 806. 

disturbances, 473. 

force, law of, 374. 

direction of, 372, 452. 

intensity of, 453. 

induction, 400. 
Magnetic matter, 380. 

measurements, 449-464. 

poles, 468. 

survey, 466. 

variations, 472. 
Magnetism of ships, 441. 

terrestrial, 465-474. 
Magnetization, components of, 384. 

induced, 424-430. 

Ampere s theory of, 638, 833. 

Poisson s theory of, 429. 

_ Weber s theory of, 442, 838. 
Magnus (G.) law, 251. 
Mance s, Henry > method, 357. 
Matthiessen, Aug., 352, 360. 
Measurement, theory of, 1. 

of result of electric force, 38. 

of electrostatic capacity, 226-229. 

of electromotive force or potential, 
216, 358. 

of resistance, 335-357. 

of constant currents, 746. 

of transient currents, 748. 



Measurement of coils, 70S, 752-757. 

magnetic, 449-464. 
Medium, electromagnetic, 866. 

lummiferous, 806. 
Mercury, resistance of, 361. 
Metals, resistance of, 363. 
Michell, John, 38. 
Miller, W. H., 23. 
Mirror method, 450. 

Molecular charge of electricity, 259. 

currents, 833. 

standards, 5. 

vortices, 822. 
Molecules, size of, 5. 

electric, 260. 

magnetic, 430, 832-845. 
Moment, magnetic, 384. 

of inertia, 565. 
Momentum, 6. 

electrokinetic, 578, 585. 
Mossotti, O. F., 62. 
Motion, equations of, 553-565. 
Moving axes, 600. 

conductors, 602. 

images, 662. 

Multiple conductors, 276, 344. 

functions. 9. 
Multiplication, method of, 747, 751. 



Neumann, F. E., coefficient of magnetiza 
tion, 430. 

magnetization of ellipsoid, 439. 

theory of induced currents, 542. 
Neumann, C. G., 190, 830, 863. 
Nicholson s Eevolving Doubler, 209. 
Nickel, 425. 

Null methods, 214, 346, 503. 



Orsted, H. C., 239, 475. 
Ohm, G. S., 241, 333. 
Ohm s Law, 241. 
Ohm, the, 338, 340, 629. 
Opacity, 798, 
Ovary ellipsoid, 152. 



Paalzow, A., 364. 
Paraboloids, confocal, 154. 
Paramagnetic (same as Ferromagnetic), 

425, 429, 844. 
Peltier, A., 249. 
Periodic functions, 9. 
Periphractic region, 22, 113. 
Permeability, magnetic, 428, 614. 
Phillips, S. E., 342. 
Plan of this Treatise, 59. 
Plane current-sheet, 656-669. 
Planetary ellipsoid, 151. 
Platymeter, electro-, 229. 
Plucker, Julius, 839. 
Points of equilibrium, 112. 
Poisson, S. D , 155, 431, 437, 674. 
Poisson s equation, 77, 148. 



INDEX. 



443 



Poisson s theory of magnetism, 427, 429, 
431, 441, 832. 

theory of wave-propagation, 784. 
Polar definition of magnetic force, 398. 
Polarity, 381. 

Polarization, electrostatic, 59, 111. 

electrolytic, 257, 264-272. 

magnetic, 381. 

of light, 381, 791. 

circular, 813, 
Poles of a magnet, 373. 

magnetic of the earth, 468. 

Positive and negative, conventions about, 

23, 27, 36, 37, 63, 68-81, 231, 374, 394, 

417, 489, 498. 
Potential, 16. 

electric, 45, 70, 220. 

magnetic, 383, 391. 

of magnetization, 412. 

of two circuits, 423. 

of two circles, 698. 

Potential, vector-, 405, 422, 590, 617, 

657. 

Principal axes, 299, 302. 
Problems, electrostatic, 155-205. 

electrokinematic, 306-333. 

magnetic, 431-441. 

electromagnetic, 647-706. 

Proof of the law of the Inverse Square, 

74. 
Proof plane, 223. 



Quadrant electrometer, 219. 
Quadric surfaces, 147-154. 
Quantity, expression for a physical, 1. 
Quantities, classification of electromag 
netic, 620-629. 

Quaternions, 11, 303, 490, 522, 618. 
Quinke, G., 316 n. 



Radiation, forces concerned in, 792. 
Rankine, W. J. M., 115, 831. 
Ray of electromagnetic disturbance, 791. 
Reciprocal properties, electrostatic, 88. 

electrokinematic, 281, 348. 

magnetic, 421, 423. 

electromagnetic, 536, 
_ kinetic, 565. 
Recoil, method of, 750. 
Residual charge, 327-334. 

magnetization, 444. 
Replenisher, 210. 

Resistance of conductors, 51, 275. 

tables of, 362-365. 

equations of, 297. 

unit of, 758-767. 

electrostatic measure of, 355, 780. 
Resultant electric force at a point, 68. 
Riemann, Bernhard, 421, 862. 

Right and left-handed systems of axes, 
23,498, 501. 

crcularly-polarized rays, 813. 
Ritchie, W., 542. 



Ritter s (J. W.) Secondary Pile, 271. 
Rotation of plane of polarization, 806. 
magnetism, a phenomenon of, 821. 
Riihlmann, R.., 370. 

Rule of electromagnetic direction, 477, 
494, 496. 



Scalar, 11. 

Scale for mirror observations, 450. 

Sectorial harmonic, 132, 138. 

Seebeck, T. J., 250. 

Selenium, 51, 362. 

Self-induction, 7. 

measurement of, 756, 778, 779. 

coil of maximum, 706. 
Sensitive galvanometer, 717. 
Series of observations, 746, 750. 

Shell, magnetic, 409, 484, 485, 606, 652, 

670, 694, 696. 
Siemens, C. W., 336, 361. 
Sines, method of, 455, 710. 
Singular points, 128. 
Slope, 17. 
Smee, A., 272. 
Smith, Archibald, 441. 
Smith, W. R., 123, 316. 
Soap bubble, 125. 
Solenoid, magnetic, 407. 

electromagnetic, 676-681, 727. 
Solenoidal distribution, 21, 82, 407. 
Solid angle, 409, 417-422, 485, 695. 
Space- variation, 17, 71, 835. 
Spark, 57, 370. 

Specific inductive capacity, 52, 83, 94, 
111, 229, 325, 334, 627, 788. 

conductivity, 278, 627. 

resistance, 277, 627. 

heat of electricity, 253. 
Sphere, 125. 

Spherical harmonics, 128-146, 391, 431. 
Spiral, logarithmic, 731. 
Standard electrometer, 217. 

galvanometer, 708. 
Stokes, G. G., 24, 115, 784. 
Stoney, G. J., 5. 
Stratified conductors, 319. 
Stress, electrostatic, 107, 111. 

electrokinetic, 641, 645, 646. 
Strutt, Hon. J. W., 102, 306. 
Surface-integral, 15, 21, 75, 402. 

density, 64, 78, 223. 

Surface, equipotential, 46. 

electrified, 78. 
Suspended coil, 721-729. 
Suspension, bifilar, 45S. 

Joule s, 463. 

Thomson s, 721. 

unifilar, 449. 



Tables of coefficients of a coil, 700. 

of dimensions, 621-629. 

of electromotive force, 358. 

of magnetic rotation, 830. 



444 



I N D E X. 



Tables for magnetization of a cylinder, 
439. 

of resistance, 363-365. 

of velocity of light and of electromag 
netic disturbance, 787- 

of temporary and residual magnetiza 
tion, 445. 

Tait, P. G., 25, 254, 387, 522, 687, 

731. 

Tangent galvanometer, 710. 
Tangents, method of, 454, 710. 
Telegraph cable, 332, 689. 
Temporary magneti/ation, 444. 
Tension, electrostatic, 48, 59, 107, 108. 

electromagnetic, 645, 646. 
Terrestrial magnetism, 465-474. 
Thalen, Tobias Robert, 430. 
Theorem, Green s, 100. 

Earnshaw s, 116. 

Coulomb s, 80. 

Thomson s, 98. 

Gauss , 144, 409. 
Theory of one fluid, 37. 

of two fluids, 36. 

of magnetic matter, 380. 

of magnetic molecules, 430, 832-845. 

of molecular currents, 833. 

of molecular vortices, 822. 

of action at a distance, 105, 641-646, 
846-866. 

Thermo-electric currents, 249-254. 
Thickness of galvanometer wire, 716, 

719. 
Thomson, Sir William, 

electric images, 43, 121, 155-181, 
173. 

experiments, 51, 57, 248, 369, 772. 

instruments, 127, 201, 210, 211, 216- 
222, 272, 722, 724. 

magnetism, 318, 398, 400, 407-416, 
428. 

resistance, 338, 351, 356, 763. 

thermo-electricity, 207, 242, 249, 252, 
253. 

theorems, 98, 138, 263, 299, 304, 
652. 

theory of electricity, 27, 37, 543, 831, 
856. 

vortex motion, 20, 100, 487, 702. 

Thomson and Tait s Natural Philoso 
phy, 132, 141, 144, 162, 303, 553, 
676. 

Time, periodic of vibration, 456, 738. 
Time-integral, 541, 558. 
Torricelli, Evangelista, 866. 
Torsion-balance, 38, 215, 373, 726. 



Transient currents, 232, 530, 536,* 537, 
582, 748, 758, 760, 771, 776. 



Units of physical quantities, 2. 

three fundamental, 3. 

derived, 6. 

electrostatic, 41, 625. 

magnetic, 374, 625. 

electrodynamic, 526. 

electromagnetic, 526, 620. 

classification of, 620-629. 

practical, 629. 

of resistance, 758-767. 

ratios of the two systems, 768-780. 



Variation of magnetic elements, 472. 

Varley, C. F., 210, 271, 332, 369. 

Vector, 1 0. 

Vector-potential, 405, 422, 590, 617, 657. 

Velocity represented by the unit of re 
sistance, 338, 628, 758. 

by the ratio of electric units, 768- 

780. 

of electromagnetic disturbance, 784. 

of light, 787. 

of the electric current, 569. 
Verdet, M. E., 809, 830. 
Vibration, time of, 456, 738. 
Volt, 629. 

Volta, A., 246. 

Voltameter, 237. 

Vortices, molecular, 822-831. 



Water, resistance of, 365. 
Wave-propagation, 784, 785. 
Weber, W., 231, 338, 346. 

electrodynamometer, 725. 

induced magnetism, 442-448, 838. 

unit of resistance, 760-762. 

ratio of electric units, 227, 771. 

electrodynamic formula, 846-861. 
Wertheim, W., 447. 
Wheatstone s Bridge, 347- 

electrostatic, 353, 756, 775, 778. 
Whewell, W., 237. 
Wiedemann, G., 236, 370, 446, 447. 
Wind, electric, 55. 

Wippe, 775. 
Work, 6. 



Zero reading, 735. 
Zonal harmonic, 132. 



FILA1TE g 



VOL. II . 



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UNIVERSITY OF CALIFORNIA, BERKELEY 
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