371. 1Properties of a magnet when acted on by the earth.
372. Definition of the axis of the magnet and of the direction of magnetic force
373. Action of magnets on one another. Law of magnetic force
374. Definition of magnetic units and their dimensions
375. Nature of the evidence for the law of magnetic force.
376. Magnetism as a mathematical quantity
377. The quantities of the opposite kinds of magnetism in a magnet are always exactly equal
378. Effects of breaking a magnet
379. A magnet is built up of particles each of which is a magnet
380. Theory of magnetic ‘matter’
381. Magnetization is of the nature of a vector
382. Meaning of the term ‘Magnetic Polarization’
383. Properties of a magnetic particle
384. Definitions of Magnetic Moment, Intensity of Magnetization, and Components of Magnetization
385. Potential of a magnetized element of volume
386. Potential of a magnet of finite size. Two expressions for this potential, corresponding respectively to the theory of polarization, and to that of magnetic ‘matter’
387. Investigation of the action of one magnetic particle on another
388. Particular cases
389. Potential energy of a magnet in any field of force
390. On the magnetic moment and axis of a magnet
391. Expansion of the potential of a magnet in spherical harmonics
392. The centre of a magnet and the primary and secondary axes through the centre
393. The north end of a magnet in this treatise is that which points north, and the
394. The direction of magnetic force is that in which austral magnetism tends to move, that is, from south to north, and this is the positive direction of magnetic lines of force. A magnet is said to be magnetized from its south end towards its north end
CHAPTER II.magnetic force and magnetic induction
395. Magnetic force defined with reference to the magnetic potential
396. Magnetic force in a cylindric cavity in a magnet uniformly magnetized parallel to the axis of the cylinder
397. Application to any magnet
398. An elongated cylinder.—Magnetic force
399. A thin disk.—Magnetic induction
400. Relation between magnetic force, magnetic induction, and magnetization
401. Line-integral of magnetic force, or magnetic potential
402. Surface-integral of magnetic induction
403. Solenoidal distribution of magnetic induction
404. Surfaces and tubes of magnetic induction
405. Vector-potential of magnetic induction
406. Relations between the scalar and the vector-potential
CHAPTER III.magnetic solenoids and shells
407. Definition of a magnetic solenoid
408. Definition of a complex solenoid and expression for its potential at any point
409. The potential of a magnetic shell at any point is the product of its strength multiplied by the solid angle its boundary subtends at the point
410. Another method of proof
411. The potential at a point on the positive side of a shell of strength Φ exceeds that on the nearest point on the negative side by 4πΦ
412. Lamellar distribution of magnetism
413. Complex lamellar distribution
414. Potential of a solenoidal magnet
415. Potential of a lamellar magnet
416. Vector-potential of a lamellar magnet
417. On the solid angle subtended at a given point by a closed curve
418. The solid angle expressed by the length of a curve on the sphere
419. Solid angle found by two line-integrations
420. Π expressed as a determinant
421. The solid angle is a cyclic function
422. Theory of the vector-potential of a closed curve
423. Potential energy of a magnetic shell placed in a magnetic field
CHAPTER IV.induced magnetization
424. When a body under the action of magnetic force becomes itself magnetized the phenomenon is called magnetic induction
425. Magnetic induction in different substances
426. Definition of the coefficient of induced magnetization
427. Mathematical theory of magnetic induction. Poisson’s method
428. Faraday’s method.
429. Case of a body surrounded by a magnetic medium
430. Poisson’s physical theory of the cause of induced magnetism
CHAPTER V.particular problems in magnetic induction
431. Theory of a hollow spherical shell
432. Case when κ is large.
433. When i = 1.
434. Corresponding case in two dimensions. Fig. XV.
435. Case of a solid sphere, the coefficients of magnetization being different in different directions
436. The nine coefficients reduced to six. Fig. XVI
437. Theory of an ellipsoid acted on by a uniform magnetic force.
438. Cases of very flat and of very long ellipsoids
439. Statement of problems solved by Neumann, Kirchhoff, and Green
440. Method of approximation to a solution of the general problem when κ is very small. Magnetic bodies tend towards places of most intense magnetic force, and diamagnetic bodies tend to places of weakest force
441. On ship’s magnetism
CHAPTER VI.weber’s theory of induced magnetism
442. Experiments indicating a maximum of magnetization
443. Weber’s mathematical theory of temporary magnetization
444. Modification of the theory to account for residual magnetization
445. Explanation of phenomena by the modified theory
446. Magnetization, demagnetization, and remagnetization
447. Effects of magnetization on the dimensions of the magnet
448. Experiments of Joule
CHAPTER VII.magnetic measurements
449. Suspension of the magnet
450. Methods of observation by mirror and scale. Photographic method
451. Principle of collimation employed in the Kew magnetometer
452. Determination of the axis of a magnet and of the direction of the horizontal component of the magnetic force.
453. Measurement of the moment of a magnet and of the intensity of the horizontal component of magnetic force.
454. Observations of deflexion
455. Method of tangents and method of sines
456. Observation of vibrations
457. Elimination of the effects of magnetic induction
458. Statical method of measuring the horizontal force
459. Bifilar suspension
460. System of observations in an observatory
461. Observation of the clip-circle.
462. J. A. Broun’s method of correction
463. Joule’s suspension
464. Balance vertical force magnetometer
CHAPTER VIII.on terrestrial magnetism
465. Elements of the magnetic force
466. Combination of the results of the magnetic survey of a country
467. Deduction of the expansion of the magnetic potential of the earth in spherical harmonics
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.
469. Gauss’ calculation of the 24 coefficients of the first four harmonics
470. Separation of external from internal causes of magnetic force
471. The solar and lunar variations
472. The periodic variations
473. The disturbances and their period of 11 years
474. Reflexions on magnetic investigations
PART IV.ELECTROMAGNETISM
CHAPTER I.electromagnetic force
475. ¨ Orsted’s discovery of the action of an electric current on a magnet
476. The space near an electric current is a magnetic field
477. Action of a vertical current on a magnet
478. Proof that the force due to a straight current of indefinitely great length varies inversely as the distance
479. Electromagnetic measure of the current
480. Potential function due to a straight current. It is a function of many values
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
482. A small circuit acts at a great distance like a magnet
483. Deduction from this of the action of a closed circuit of any form and size on any point not in the current itself
484. Comparison between the circuit and a magnetic shell
485. Magnetic potential of a closed circuit
486. Conditions of continuous rotation of a magnet about a current
487. Form of the magnetic equipotential surfaces due to a closed circuit. Fig. XVIII
488. Mutual action between any system of magnets and a closed current
489. Reaction on the circuit
490. Force acting on a wire carrying a current and placed in the magnetic field
491. Theory of electromagnetic rotations
492. Action of one electric circuit on the whole or any portion of another
493. Our method of investigation is that of Faraday.
494. Illustration of the method applied to parallel currents
495. Dimensions of the unit of current
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.
497. Action of an infinite straight current on any current in its plane
498. Statement of the laws of electromagnetic force. Magnetic force due to a current
499. Generality of these laws
500. Force acting on a circuit placed in the magnetic field
501. Electromagnetic force is a mechanical force acting on the conductor, not on the electric current itself
CHAPTER II.amp`ere’s investigation of the mutual action of electric currents
502. Amp`ere’s investigation of the law of force between the elements of electric currents
503. His method of experimenting
504. Amp`ere’s balance
505. Amp`ere’s first experiment. Equal and opposite currents neutralize each other
506. Second experiment. A crooked conductor is equivalent to a straight one carrying the same current
507. Third experiment. The action of a closed current as an element of another current is perpendicular to that element
508. Fourth experiment. Equal currents in systems geometrically similar produce equal forces
509. In all of these experiments the acting current is a closed one
510. Both circuits may, however, for mathematical purposes be conceived as consisting of elementary portions, and the action of the circuits as the resultant of the action of these elements
511. Necessary form of the relations between two elementary portions of lines
512. The geometrical quantities which determine their relative position
513. Form of the components of their mutual action
514. Resolution of these in three directions, parallel, respectively, to the line joining them and to the elements themselves
515. General expression for the action of a finite current on the element of another
516. Condition furnished by Amp`ere’s third case of equilibrium
517. Theory of the directrix and the determinants of electrodynamic action
518. Expression of the determinants in terms of the components of the vector-potential of the current.
519. The part of the force which is indeterminate can be expressed as the spacevariation of a potential
520. Complete expression for the action between two finite currents
521. Mutual potential of two closed currents
522. Appropriateness of quaternions in this investigation
523. Determination of the form of the functions by Amp`ere’s fourth case of equilibrium
524. The electrodynamic and electromagnetic units of currents
525. Final expressions for electromagnetic force between two elements
526. Four different admissible forms of the theory
527. Of these Amp`ere’s is to be preferred
CHAPTER III.on the induction of electric currents
528. Faraday’s discovery. Nature of his methods
529. The method of this treatise founded on that of Faraday
530. Phenomena of magneto-electric induction
531. General law of induction of currents
532. Illustrations of the direction of induced currents
533. Induction by the motion of the earth
534. The electromotive force due to induction does not depend on the material of the conductor
535. It has no tendency to move the conductor
536. Felici’s experiments on the laws of induction
537. Use of the galvanometer to determine the time-integral of the electromotive force
538. Conjugate positions of two coils.
539. Mathematical expression for the total current of induction
540. Faraday’s conception of an electrotonic state
541. His method of stating the laws of induction with reference to the lines of magnetic force
542. The law of Lenz, and Neumann’s theory of induction
543. Helmholtz’s deduction of induction from the mechanical action of currents by the principle of conservation of energy
544. Thomson’s application of the same principle
545. Weber’s contributions to electrical science
CHAPTER IV.on the induction of a current on itself
546. Shock given by an electromagnet
547. Apparent momentum of electricity
548. Difference between this case and that of a tube containing a current of water
549. If there is momentum it is not that of the moving electricity
550. Nevertheless the phenomena are exactly analogous to those of momentum.
551. An electric current has energy, which may be called electrokinetic energy
552. This leads us to form a dynamical theory of electric currents
CHAPTER V.on the equations of motion of a connected system
553. Lagrange’s method furnishes appropriate ideas for the study of the higher dynamical sciences
554. These ideas must be translated from mathematical into dynamical language
555. Degrees of freedom of a connected system
556. Generalized meaning of velocity.
557. Generalized meaning of force
558. Generalized meaning of momentum and impulse
559. Work done by a small impulse
560. Kinetic energy in terms of momenta, (Tp)
561. Hamilton’s equations of motion
562. Kinetic energy in terms of the velocities and momenta, (Tpq˙)
563. Kinetic energy in terms of velocities, (Tq˙)
564. Relations between Tp and Tq˙ , p and q˙
565. Moments and products of inertia and mobility
566. Necessary conditions which these coefficients must satisfy
567. Relation between mathematical, dynamical, and electrical ideas
CHAPTER VI.dynamical theory of electromagnetism
568. The electric current possesses energy
569. The current is a kinetic phenomenon
570. Work done by electromotive force
571. The most general expression for the kinetic energy of a system including electric currents
572. The electrical variables do not appear in this expression.
573. Mechanical force acting on a conductor
574. The part depending on products of ordinary velocities and strengths of currents does not exist
575. Another experimental test
576. Discussion of the electromotive force
577. If terms involving products of velocities and currents existed they would introduce electromotive forces, which are not observed
CHAPTER VII.theory of electric circuits
578. The electrokinetic energy of a system of linear circuits
579. Electromotive force in each circuit
580. Electromagnetic force
581. Case of two circuits
582. Theory of induced currents
583. Mechanical action between the circuits
584. All the phenomena of themutual action of two circuits depend on a single quantity, the potential of the two circuits.
CHAPTER VIII.exploration of the field by means of the secondary circuit
585. The electrokinetic momentum of the secondary circuit
586. Expressed as a line-integral
587. Any system of contiguous circuits is equivalent to the circuit formed by their exterior boundary
588. Electrokinetic momentum expressed as a surface-integral
589. A crooked portion of a circuit equivalent to a straight portion
590. Electrokinetic momentum at a point expressed as a vector, A
591. Its relation to the magnetic induction, B. Equations (A)
592. Justification of these names
593. Conventions with respect to the signs of translations and rotations
594. Theory of a sliding piece
595. Electromotive force due to the motion of a conductor
596. Electromagnetic force on the sliding piece
597. Four definitions of a line of magnetic induction
598. General equations of electromotive force, (B)
599. Analysis of the electromotive force
600. The general equations referred to moving axes
601. The motion of the axes changes nothing but the apparent value of the electric potential
602. Electromagnetic force on a conductor
603. Electromagnetic force on an element of a conducting body. Equations (C).
CHAPTER IX. general equations of the electromagnetic field
604. Recapitulation
605. Equations of magnetization, (D)
606. Relation between magnetic force and electric currents
607. Equations of electric currents, (E)
608. Equations of electric displacement, (F)
609. Equations of electric conductivity, (G)
610. Equations of total currents, (H)
611. Currents in terms of electromotive force, (I)
612. Volume-density of free electricity, (J)
613. Surface-density of free electricity, (K)
614. Equations of magnetic permeability, (L)
615. Amp`ere’s theory of magnets
616. Electric currents in terms of electrokinetic momentum
617. Vector-potential of electric currents
618. Quaternion expressions for electromagnetic quantities
619. Quaternion equations of the electromagnetic field
CHAPTER X.dimensions of electric units
620. Two systems of units
621. The twelve primary quantities
622. Fifteen relations among these quantities
623. Dimensions in terms of [e] and [m]
624. Reciprocal properties of the two systems
625. The electrostatic and the electromagnetic systems
626. Dimensions of the 12 quantities in the two systems
627. The six derived units
628. The ratio of the corresponding units in the two systems
629. Practical system of electric units. Table of practical units
CHAPTER XI.on energy and stress in the electromagnetic field
630. The electrostatic energy expressed in terms of the free electricity and the potential
631. The electrostatic energy expressed in terms of the electromotive force and the electric displacement
632. Magnetic energy in terms of magnetization and magnetic force
633. Magnetic energy in terms of the square of the magnetic force
634. Electrokinetic energy in terms of electric momentum and electric current
635. Electrokinetic energy in terms of magnetic induction and magnetic force
636. Method of this treatise
637. Magnetic energy and electrokinetic energy compared
638. Magnetic energy reduced to electrokinetic energy
639. The force acting on a particle of a substance due to its magnetization
640. Electromagnetic force due to an electric current passing through it
641. Explanation of these forces by the hypothesis of stress in a medium
642. General character of the stress required to produce the phenomena
644. Force acting on a conductor carrying a current
645. Theory of stress in a medium as stated by Faraday
646. Numerical value of magnetic tension
CHAPTER XII.current-sheets
647. Definition of a current-sheet
648. Current-function.
649. Electric potential
650. Theory of steady currents.
651. Case of uniform conductivity
652. Magnetic action of a current-sheet with closed currents
653. Magnetic potential due to a current-sheet
654. Induction of currents in a sheet of infinite conductivity
655. Such a sheet is impervious to magnetic action
656. Theory of a plane current-sheet
657. The magnetic functions expressed as derivatives of a single function
658. Action of a variable magnetic system on the sheet
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
660. The currents, excited by the instantaneous introduction of a magnetic system, produce an effect equivalent to an image of that system
661. This image moves away from its original position with velocity R
662. Trail of images formed by a magnetic system in continuous motion
663. Mathematical expression for the effect of the induced currents
664. Case of the uniform motion of a magnetic pole
665. Value of the force acting on the magnetic pole
666. Case of curvilinear motion
667. Case of motion near the edge of the sheet
668. Theory of Arago’s rotating disk
669. Trail of images in the form of a helix
670. Spherical current-sheets
671. The vector- potential
672. To produce a field of constant magnetic force within a spherical shell
673. To produce a constant force on a suspended coil
674. Currents parallel to a plane
675. A plane electric circuit. A spherical shell. An ellipsoidal shell
676. A solenoid
677. A long solenoid
678. Force near the ends
679. A pair of induction coils
680. Proper thickness of wire
681. An endless solenoid
CHAPTER XIII.parallel currents
682. Cylindrical conductors
683. The external magnetic action of a cylindric wire depends only on the whole current through it
684. The vector-potential
685. Kinetic energy of the current
686. Repulsion between the direct and the return current
687. Tension of the wires. Amp`ere’s experiment
688. Self-induction of a wire doubled on itself
689. Currents of varying intensity in a cylindric wire
690. Relation between the electromotive force and the total current
691. Geometrical mean distance of two figures in a plane
692. Particular cases
693. Application of the method to a coil of insulated wires
CHAPTER XIV.circular currents
694. Potential due to a spherical bowl
695. Solid angle subtended by a circle at any point
696. Potential energy of two circular currents
697. Moment of the couple acting between two coils
698. Values of P i
699. Attraction between two parallel circular currents
700. Calculation of the coefficients for a coil of finite section
701. Potential of two parallel circles expressed by elliptic integrals
702. Lines of force round a circular current. Fig. XVIII
703. Differential equation of the potential of two circles
704. Approximation when the circles are very near one another
705. Further approximation.
706. Coil of maximum self-induction
CHAPTER XV.electromagnetic instruments
707. Standard galvanometers and sensitive galvanometers
708. Construction of a standard coil
709. Mathematical theory of the galvanometer
710. Principle of the tangent galvanometer and the sine galvanometer
711. Galvanometer with a single coil
712. Gaugain’s eccentric suspension
713. Helmholtz’s double coil. Fig. XIX
714. Galvanometer with four coils
715. Galvanometer with three coils
716. Proper thickness of the wire of a galvanometer
717. Sensitive galvanometers
718. Theory of the galvanometer of greatest sensibility
719. Law of thickness of the wire
720. Galvanometer with wire of uniform thickness
721. Suspended coils. Mode of suspension
722. Thomson’s sensitive coil
723. Determination of magnetic force by means of suspended coil and tangent galvanometer
724. Thomson’s suspended coil and galvanometer combined
725. Weber’s electrodynamometer
726. Joule’s current-weigher
727. Suction of solenoids.
728. Uniform force normal to suspended coil
729. Electrodynamometer with torsion-arm
CHAPTER XVI.electromagnetic observations
730. Observation of vibrations
731. Motion in a logarithmic spiral
732. Rectilinear oscillations in a resisting medium
733. Values of successive elongations
734. Data and quæsita
735. Position of equilibrium determined from three successive elongations
736. Determination of the logarithmic decrement
737. When to stop the experiment
738. Determination of the time of vibration from three transits
739. Two series of observations
740. Correction for amplitude and for damping
741. Dead beat galvanometer
742. To measure a constant current with the galvanometer
743. Best angle of deflexion of a tangent galvanometer
744. Best method of introducing the current
745. Measurement of a current by the first elongation
746. To make a series of observations on a constant current
747. Method of multiplication for feeble currents
748. Measurement of a transient current by first elongation
749. Correction for damping
750. Series of observations Zur¨uckwerfungs methode
751. Method of multiplication
CHAPTER XVII.comparison of coils
752. Electrical measurement sometimes more accurate than direct measurement
753. Determination of G1
754. Determination of g1
755. Determination of the mutual induction of two coils
756. Determination of the self-induction of a coil
757. Comparison of the self-induction of two coils
CHAPTER XVIII.electromagnetic unit of resistance
758. Definition of resistance
759. Kirchhoff’s method
760. Weber’s method by transient currents.
761. His method of observation
762. Weber’s method by damping.
763. Thomson’s method by a revolving coil
764. Mathematical theory of the revolving coil
765. Calculation of the resistance
766. Corrections
767. Joule’s calorimetric method
CHAPTER XIX.comparison of the electrostatic with the electromagnetic units
768. Nature and importance of the investigation
769. The ratio of the units is a velocity
770. Current by convection
771. Weber and Kohlrausch’s method
772. Thomson’s method by separate electrometer and electrodynamometer
773. Maxwell’s method by combined electrometer and electrodynamometer
774. Electromagnetic measurement of the capacity of a condenser.Jenkin’s method
775. Method by an intermittent current.
776. Condenser and Wippe as an arm of Wheatstone’s bridge
777. Correction when the action is too rapid
778. Capacity of a condenser compared with the self-induction of a coil
779. Coil and condenser combined
780. Electrostatic measure of resistance compared with its electro magnetic measure
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
782. Energy of light during its propagation
783. Equation of propagation of an electromagnetic disturbance.
784. Solution when the medium is a non-conductor
785. Characteristics of wave-propagation
786. Velocity of propagation of electromagnetic disturbances
787. Comparison of this velocity with that of light
788. The specific inductive capacity of a dielectric is the square of its index of refraction
789. Comparison of these quantities in the case of paraffin
790. Theory of plane waves
791. The electric displacement and the magnetic disturbance are in the plane of the wave-front, and perpendicular to each other.
792. Energy and stress during radiation.
793. Pressure exerted by light
794. Equations of motion in a crystallized medium
795. Propagation of plane waves
796. Only two waves are propagated
797. The theory agrees with that of Fresnel
798. Relation between electric conductivity and opacity
799. Comparison with facts
800. Transparent metals
801. Solution of the equations when the medium is a conductor
802. Case of an infinite medium, the initial state being given.
803. Characteristics of diffusion
804. Disturbance of the electromagnetic field when a current begins to flow
805. Rapid approximation to an ultimate state
CHAPTER XXI.magnetic action on light
806. Possible forms of the relation between magnetism and light
807. The rotation of the plane of polarization by magnetic action
808. The laws of the phenomena
809. Verdet’s discovery of negative rotation in ferromagnetic media
810. Rotation produced by quartz, turpentine, &c., independently of magnetism
811. Kinematical analysis of the phenomena
812. The velocity of a circularly-polarized ray is different according to its direction of rotation
813. Right and left-handed rays
814. In media which of themselves have the rotatory property the velocity is different for right and left-handed configurations
815. In media acted on by magnetism the velocity is different for opposite directions of rotation.
816. The luminiferous disturbance, mathematically considered, is a vector
817. Kinematic equations of circularly-polarized light
818. Kinetic and potential energy of the medium.
819. Condition of wave-propagation
820. The action of magnetism must depend on a real rotation about the direction of the magnetic force as an axis
821. Statement of the results of the analysis of the phenomenon.
822. Hypothesis of molecular vortices
823. Variation of the vortices according to Helmholtz’s law
824. Variation of the kinetic energy in the disturbed medium
825. Expression in terms of the current and the velocity
826. The kinetic energy in the case of plane waves
827. The equations of motion
828. Velocity of a circularly-polarized ray
829. The magnetic rotation
830. Researches of Verdet
831. Note on a mechanical theory of molecular vortices
CHAPTER XXII.ferromagnetism and diamagnetism explained by molecular currents
832. Magnetism is a phenomenon of molecules
833. The phenomena of magnetic molecules may be imitated by electric currents
834. Difference between the elementary theory of continuous magnets and the theory of molecular currents
835. Simplicity of the electric theory
836. Theory of a current in a perfectly conducting circuit
837. Case in which the current is entirely due to induction
838. Weber’s theory of diamagnetism
839. Magnecrystallic induction.
840. Theory of a perfect conductor
841. A medium containing perfectly conducting spherical molecules
842. Mechanical action of magnetic force on the current which it excites
843. Theory of a molecule with a primitive current
844. Modifications of Weber’s theory
845. Consequences of the theory
CHAPTER XXIII.theories of action at a distance
846. Quantities which enter into Amp`ere’s formula
847. Relative motion of two electric particles
848. Relative motion of four electric particles. Fechner’s theory
849. Two new forms of Amp`ere’s formula
850. Two different expressions for the force between two electric particles in motion
851. These are due to Gauss and to Weber respectively
852. All forces must be consistent with the principle of the conservation of energy
853. Weber’s formula is consistent with this principle but that of Gauss is not
854. Helmholtz’s deductions from Weber’s formula
855. Potential of two currents
856. Weber’s theory of the induction of electric currents
857. Segregating force in a conductor
858. Case of moving conductors
859. The formula of Gauss leads to an erroneous result
