The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 39 of 39

640 The Electrical Structure of Matter [ch. xxi

Equation (799) now becomes

T

2V(nr2) (V^-Vrj)2' and each side is at once seen to be equal to

T + T

rx + r2 '

The elimination of 1\ and r2 from this equation and the two equations

(798) gives

2ir2me2E2

w-r^iT'tt (800)-

(T + T fIV

We obtain all possible values for W by giving integral values to r and r. It is at once seen that no new values are introduced beyond those already discovered in equation (791).

By a well-known formula the eccentricity e of the orbit is given by

^'tSS-p^ji (801)-

and, since r and t' are necessarily integrals, it is clear that only definite values are permissible for the eccentricity. The semi-axes a, b of the orbit are related by

-- = 1-

a2

e"

so that equation (801) shews that bja will be commensurable for all orbits which can be described.

By a well-known theorem, the energy of an elliptic orbit is equal to that of a circular orbit whose radius is equal to the semi-major-axis of the ellipse. It follows from equation (790) that the semi- major-axis a of an elliptic orbit is given by

n

h°-

a = 4^> <802>'

where n is written for t + t, while from equation (801) the eccentricity is given by

e2 = l--, (803).

n-

The orbits n = 1 (t = 1), n = 2 (t = 1 or 2), n = 3 (t = 1, 2 or 3) and n = 4

(t= 1, 2, 3 or 4) are shewn in fig. 140*. The distance of closest approach to the nucleus is a (1 — e) which is given by

^2

a (-1 ~ 6) = A 2 J? n(n~ W - T2).

4nr2eEm

• Reproduced by permission from a paper by Bohr {Nature, July 7, 1923).

718-720]

Buhrs Theory

041

The expression on the right has its minimum value when t = 1 and n = oo , namely

Thus the electron never approaches the nucleus to within a distance less than half of the radius of the smallest circular orbit.

Fig. 140.

719. Since the extension to elliptical orbits has introduced no new values of W, it follows that the spectrum will consist of those lines which were pre- dicted by the simple theory of circular orbits and no others. Nevertheless the possibility of elliptical orbits has introduced an essential difference into the spectrum. Since an orbit of any permissible energy Wj or W2 can now be described in more than one way, it follows that a fall from energy W2 to energy Wx can occur in more than one way, so that the spectral line which is produced by a fall from energy W2 to energy Wx may appropriately be thought of as the superposition of a number of lines, all of which, although having precisely the same frequency, are produced by different events.

720. It is possible to separate out these coincident lines in a variety of ways. Perhaps the simplest is by placing the radiating atoms in a magnetic field. Each electronic orbit is affected by the field, and different orbits, even though of the same energy before the field was put on, will be affected in different ways. It follows that the spectral lines which were originally coincident will be displaced to different extents, as is observed in the Zeeman effect. It can be shewn that the explanation of the normal Zeeman effect which has already been given in § 635 holds valid even when the quantum- restrictions are applied to the electronic orbits. The anomalous Zeeman

.t 41

642 The Electrical Structure of Matter [ch. xxi

effect presents a more complicated problem which can hardly yet be said to have been satisfactorily solved.

The placing of the radiating matter in a powerful electrostatic field also results in a separation of the originally coincident lines, this being known as the Stark effect. The dynamical theory just explained provides a calculation of the separations to be expected in this case, and the predicted separations are found to agree very closely with those actually observed.

721. But perhaps the most interesting feature of all is that there is a slight separation even when external magnetic and electric fields are entirely absent. In the analysis of § 713, we treated m the mass of the electron as an absolute constant, although we already knew (§ 660) that the mass varies with the velocity of motion of the electron. For circular orbits this does not matter much; the mass of course remains strictly constant throughout the description of any single circular orbit, although varying slightly from one orbit to another. But in an elliptic orbit the mass varies from one part to another of the same orbit. When allowance is made for this, the orbit is no longer strictly ellip- tical, and formula (800) only provides a first approximation to its energy. When the necessary additional terms are included, it is found that the value of W no longer depends solely on t + t', but on t and r' separately. It follows that each of the lines which our simple theory treated as a superposition of coincident lines must in actual fact shew a "fine-structure" of adjacent slightly separated lines. Such "fine-structures" are easily observed in a powerful spectroscope. The theoretical separations to be expected have been calculated by Sommerfeld and others, and although the observed separations are so small as to make exact measurement exceedingly difficult, there seems no room for doubt that they agree with those predicted by theory.

The dynamical theory of these phenomena is not given in the present book. The reader who wishes to study it is referred to the original papers of Bohr, Sommerfeld and others, or to the author's " Dynamical Theory of Gases."

722. The new dynamics, as has now been seen, allots a definite size to the atom and so provides a mechanism by which atoms have a permanent existence, instead of radiating away all their energy and collapsing. For the hydrogen atom the minimum energy is found by taking r + r=l in equation (800), and since t cannot be zero, this requires that r = 1 and t = 0. Thus the orbit of minimum energy is the circular orbit of radius (cf. equation (802))

A2 a = -=-£5- (804).

The electrons in hydrogen atoms can describe circular orbits of radii 4, 9, 16, ... times this and a variety of non-circular orbits as well, but this equation defines the hydrogen atom in its normal state of minimum energy.

720-723] Bohr's Theory 643

On inserting the numerical values already given, we find a = 0'53 x 10-8 cms., which is in good agreement with the radius of the hydrogen atom as found in other ways.

Dewar found the density of solid hydrogen at 132° absolute to be 00763. Thus a cubic centimetre of hydrogen at this temperature has mass 0-0763 grammes and consists of atoms of hydrogen each of which is known to have a mass of 1662 x 1024 grammes. It follows that the number of hydrogen atoms in a cubic centimetre of solid hydrogen is 4-59 x 1022, so that the space occupied by each is 2-18 x 1023 cubic centimetres. This is the space that would be occupied if the atoms were spheres arranged in cubical packing, each being of radius 140 x 108 cms. In § 150 we found that the dielectric constant of hydrogen is the same as if the molecules were spheres of radius 0'916 x 10-8 cms. A similar calculation would have suggested that the hydrogen atom might be regarded as having a radius equal to l//2 times this or 0*723 x 108 cms. Neither of these calculations can lay claim to great accuracy; a far more accurate determination of atomic dimensions is obtained from the Kinetic Theory of Gases. If the molecules of hydrogen are regarded as spheres, the three phenomena of viscosity, conduction of heat and diffusion agree in assigning to these spheres a radius of 0 68 x 108 cms., while obser- vations on the deviations from Boyle's law suggest the slightly lower value of 0*64 x 10-8 cms. Again the hydrogen atom may be supposed to have a radius equal to l//2 times that of the molecule, so that the two values of the atomic radius are respectively 054 x 10-8 and 051 x 108 cms., in close agreement with the value 0"53 x 10-8 required by the electrical structure of the atom.

It must, however, be noticed that the Kinetic Theory requires the atom to occupy a three-dimensional volume, whereas on Bohr's theory the hydrogen atom is at most a disc. If we imagine this disc, the orbit of the negative electron, to be continually changing its orientation in space we pass naturally to the conception of the hydrogen atom reserving for itself, or perhaps clearing for itself, a spherical space equal in radius to the orbit of the electron. This conception is in accordance with the known facts of crystal structure.

723. The theory of structures of more than two constituent parts is far less advanced, no satisfactory mechanism having yet been devised for either the helium atom or the hydrogen molecule. A large amount of consistent evidence suggests that the electrons of complex atoms are arranged in shells or rings corresponding to different quantum numbers, but the method of arrangement has not yet been brought within the scope of mathematical treatment.

Questions of electrical conductivity and of the optical and dispersive pro- perties of substances are clearly subjects for treatment by the new dynamics, but only meagre progress has so far been made. The same applies to the problem of the nature of radiation. There is at present a divergence of

41—2

T/2 T2 vt TVVry T3>

and equations (805) and (806) now shew that the possible frequencies of radiation are given by

v = sn (807).

If the motion of the electron, describing its orbit with frequency n, had been analysed by Fourier's theorem, and the resulting radiation calculated by the classical electrodynamics, we should have found radiations of frequencies

n, 2n, Sn, ...,

so that according to the classical electrodynamics also, the frequencies of the emitted radiation would be given by equation (807).

It accordingly appears that in the limiting case in which t and r' are both large, the old classical electrodynamics and the new quantum dynamics agree in predicting the same frequencies for the spectrum of emitted radiation. In this limiting case, the radius of the electron orbit is infinite, successive radii

644 The Electrical Structure of Matter [ch. xxi

opinion as to whether radiation is propagated in accordance with Maxwell's equations or in the form of "atomic" packets of energy which travel through space without spreading out in the manner demanded by the classical electro- magnetic theory.

The Correspondence Principle of Bohr.

724. In conclusion we may refer to a procedure which holds out some hope of bridging the gulf between the classical electrodynamics and the new electrodynamics of quanta.

When an electron describes a circular orbit about a nucleus, the number

of revolutions per second in this orbit, n, is equal to B\1it, whence, from

equations (786) and (790),

47r2e2#*m n = T3/,3 (805>

The period of an elliptic orbit is known to be the same as that of a circular orbit of the same energy, so that the same equation will give the frequency of revolution in an elliptic orbit of total quantum number t.

The frequencies of the radiation which can be emitted on the electron dropping from this orbit to one of lower quantum number r' are, from equation (792),

V = ^~{^-^ (806)-

If the integers t and r are both large, and differ only by a small number s,

which must of course also be integral, the approximate value of (-^ -)

will be

723, 724] The Correspondence Principle 645

only differ by an infinitesimal fraction of each, and orbits of all eccentricities are possible. Thus the electron is just on the verge of becoming a free electron. This limiting case provides a bridge between the old mechanics and the new; on one side of the bridge the classical electrodynamics holds undisputed sway, but as we cross the bridge and advance into the territory on the other side, the additional restrictions imposed by the quantum dynamics become ever more important until finally they may be considered to govern the whole situation. The exploration of the territory on the far side of the bridge will provide work for a new generation of mathematical physicists; the present work attempts only to bring the reader as far as the bridge, and to make clear to him that if he crosses it he must expect to find different conditions prevailing on the other side.

INDEX

The numbers refer to the pages, [pp. 1—299, Electrostatic Problems, pp. 300— end, Current and Magnetic.']

Aberration, 593, 609 Abraham, 525, 584, 589 Absorption of light, 545, 554, 555

,, bands, 554

Accelerated electron, 577, 592 Action at a distance, 140

,, mechanical, see Mechanical action, Mechanical force

,, principle of least, 488, 517, 579 Adams, E. P., 515

Alternating currents, 456, 465, 477, 501, 528 Amber, electrification of, 1 Ampere, 3, 507

(unit of current), 305, 530, 531 Ampere's law (field of a current), 439, 514 Angle of conductor, lines of force near, 61 Anion, 308

Anisotropic media, 134, 152, 535 Anode, 308 Argand diagram, 262 Argument of a complex quantity, 262 Arons, 362

Atom, structure of, 22, 630, 634 ; see also Molecule Atomic heat, 556

,, nature of electricity, 21, 309, 631, 633

,, numbers, 630 Attracted-disc electrometer, 105

Ballistic galvanometer, 437 Balmer's Series, 636 Barnes, E. W., 199 Batteries, work done by, 104, 506 Biaxal harmonics, 241 Bohr, 634 ff ., 644 Boscovitch, 141 Bound-charge, 126, 361, 552 Boundary-conditions, in dielectrics (electro- static), 121, 178

) > ))

j

conductors, 346

!> >5

i>

magnetic media, 413

                                 I) 

)>

propagation of light, 539

Boussinesq, 505

Bowl, electrified spherical,

250

Brace, 595

Bridge, Wheatstone's,

31o:

,316

Bucherer, 20, 590

Cable, submarine, 79, 319, 332, 351, 505 Capacity, coefficients of, 93, 96, 97

,, inductive, see Inductive capacity ,, of a conductor, 67, 94 ,, ,, a condenser, 115 ,, ,, a circular disc, 249 ,, ,, an ellipsoid, 248 ,, ,, an elliptic disc, 249 ,, ,, a Leyden Jar, 77, 277 ,, ,, a parallel plate condenser, 77, 274 ,, ,, a spherical bowl, 250 ,, ,, ,, condenser, 71

,, ,, a spheroid, 248 ,, ,, a submarine cable, 351, 504 ,, ,, a telegraph wire, 195, 505 Cascade, condensers in, 76 Cathode, 308 Cation, 308

Cavendish. 13, 37, 74, 115, 250 Cavendish's proof of law of force, 13, 37 Chapman, S., 403

Charge, electric, see Electric charge and Elec- trification Charge, moving electric, magnetic field of, 513 Chree, 402 Circular current, 431

,, cylinders, 73, 267 „ disc, 249 ,, ring, 225 Coefficients of Potential, Capacity and Induc- tion (electrostatics), 93, 96, 97 ,, of self and mutual induction (cir-

cuits), 443 Collinear charges, 57 Complex quantities, 262 Condenser, 71-78, 99 ; see also Capacity

,, discharge of a, 88, 331, 361, 458,

498 Conditions at boundary, see Boundary-conditions Conduction in solids, 300, 306, 557, 643 ,, ,, liquids, 307

„ ,, gases, 311

,, see also Electron

Conductors and insulators, 5 ,, systems of, 88

,, see also Capacity

Confocal coordinates, 244, 257

648

Index

Conformal representation, 264, 280 Conjugate functions, 261-279, 286

,, conductors, 328 Contact difference of potential, 303

,, conductors in, 101, 303, 317 Continuity, equation of, 344, 476, 559 Contracted coordinates, 583 Contractile electron, 589, 595 Contraction hypothesis (Lorentz-Fitzgerald),

594, 597, 606 Correspondence principle, 614 Coulomb's torsion balance, 11, 365 law (R = 4t«t), 45, 121 ,, (unit of charge), 530

Crystalline media, 134, 152, 535 Current- sheets, 480 Currents of electricity, 22, 300, 306

,, in linear conductors, 300, 452, 496,

499, 502 „ ,, continuous media, 311, 473, 502,

526, 544, 555, 557 ,, ,, dielectrics, 358, 510, 550

induction of, 452, 473, 496, 562 magnetic field of, 425, 438, 513, 514, 573 ,, measurement of, 305, 314 ,, slowly-varying, 331 Curvilinear coordinates, 242 Cylindrical conductors and condensers, 67, 73, 187, 195, 257-279

D'Arsonval galvanometer, 436

Debye's theory of specific heats, 556

Declination, magnetic, 401

Deformable electron, 589, 596, 611

Diamaguetism, 410, 505

Dielectrics, 74, 115

,, boundary of, 121, 178

,, currents in, 358, 510, 550

,, images in, 200

,, inductive capacity of, 74, 115, 532

,, molecular action in, 126, 551

,, stresses and mechanical action in,

172-181, 201, 579, 619 ,, time of relaxation of, 359

Dip, magnetic, 401

Disc, circular or elliptic, 248, 249

Discharge of condenser, 88, 331, 361, 458, 498

Dispersion of light, 532, 553, 643

Displacement (electrostatic), 117, 153, 552 -currents, 155, 510, 514, 528 ,, -theory of Maxwell, 153, 510, 514

Dolazalek electrometer, 110

Doppler-effeet, 609

Doublet, electric, 50, 168, 193, 215, 232, 551

Drude's theory of conduction, 557

Dynamical theory of currents, 485

»

Dynamo, action of, 458, 465

Earnshaw's theorem, 167 Eddington, 623, 624 Eichenwald, 605

Einstein, 597, 598, 599, 600, 602, 621 Electric charges, force between, 11, 12, 13, 37 ,, ,, equilibrium of, 23, 167

„ currents, see Currents „ intensity, 24, 31, 117, 121, 571, 575 „ potential, 26, 31, 121, 569, 570, 575 ,, screening, 62, 97, 548 Electricity, measurement of quantity of, 8, 77, 109, 437 ,, positive and negative, 8

,, theories of, 19, 20

Electrification, 5

,, at surfaces and boundaries, 18,

21, 45, 61, 194, 347 by friction, 1, 9 by induction, 16, 125, 186 ,, line of zero, 88, 194

Electrokinetic momentum, 498 Electrolytic conduction, 307 Electromagnetic field, general equations of, 568,

602 ,, Weyl's theory of the, 623 mass, 585, 596, 611, 613 momentum, 583, 615, 617, 620 theory of light, 3, 526, 532 ff. units, 427, 528 ,, waves, 524, 525

Electrometers, 105, 107 Electromotive force, 303, 453 Electron, charge and mass of, 20, 590 ,, internal mechanics of, 590 ,, motion of, in conduction, 306, 307, 320, 343, 496, 549, 557, 562, 563 ,, ,, ,, in free space, 559 ff.

,, size of, 586, 590, 612 ,, structure of, 589, 590, 596 ,, theory of conduction, 306. 549,555,557 ,, ,, ,, dispersion, 553, 554

Electrophorus, 17 Electropositive, electronegative, 10 Electroscope, gold-leaf, 7, 17 Electrostriction, 181 Ellipsoidal analysis, 230, 244, 251 ,, conductors, 246, 253

,, harmonics, 251

Elliptic cylinders, 270

„ disc, 248 Energy, conservation of, 28, 32 „ flow of, 519, 617

„ localisation of, 151, 399, 415, 443, 494, 516, 576, 617, 620

»

»> »>

» »>

Index

649

Energy, mass of, 613

momentum of, 615

of conductors and condensers, 83, 106 ,, light-waves, 537, 617 „ magnetic field, 396, 399, 415, 507 ,, magnetised bodies, 377, 380, 381 ,, systems of currents, 443 Equilibrium, points of, 59, 167 Equipotential surfaces, 29, 47-62, 370 Equivalent stratum (Green's), 182, 361, 375 Ewing, 422 Expansions in harmonics, 211

,, ,, Legendre's coefficients, 223

,, ,, sines and cosines, 259

Farad (unit of capacity), 77, 530

Faraday, 3, 74, 115, 116, 126, 140, 155, 308,

402, 514, 618 Finite current sheets, 481 Fitzgerald, 594

Fizeau's water-tube experiment, 593, 607 Flame, conducting power of, 6, 125 Flux of energy, 519, 617 Force, lines of, 25, 29, 43, 47-58, 62, 370

,, magnetic, 381

,, mechanical, see Mechanical force

„ tubes of, 44, 47-58, 117, 371 Fourier's theorem, 259 Franklin, 19 Fresnel, 536

Galvanometer, 433 Gases, conduction in, 311

,, inductive capacity of, 132, 532 ,, velocity of light in, 533 Gauss' theorem, 33, 118, 161, 162, 370, 386 Generalised coordinates, 489 ,, forces, 493

„ momenta, 493

„ relativity, 598, 621

Generation of electricity, 9

,, heat, 320, 348 Gravitation, 620, 621, 629 Green, analytical theorem of, 156

equivalent stratum of, 182, 361, 375 ,, reciprocation theorem of, 92, 163 Guard-ring, 78, 106

Hagen and Kubens, 548 Hall effect, 563 Hamilton's principle, 487 Harmonic potential, 224 Harmonics, biaxal, 241

,, ellipsoidal, 251

spherical, 206-223, 233-242, 243

tesseral, 237

zonal, 233

Harmonics, tables of—

harmonics of integral degrees, 258 Legendre's coefficients, 219 tesseral harmonics, 240

Heat, generation of, 320, 348

Heaviside, 505

Helmholtz, stresses in dielectrics, 177

Hertzian vibrator, 578

Holtz influence machine, 18

Hurmuzescu, 525

Hydrogen atom, 631, 632, 636, 641

Hyperbolic cylinders, 267, 270

Hysteresis, magnetic, 412

Images in electrostatics, 185-201, 258, 281, ! Impulsive forces, 493

Induction, coefficients of (electrostatics), 93,96,

97 ,, ,, ,, (circuits), 443

„ electrification by, 16, 125, 186

,, magnetic, 384

,, of currents, 452, 562

Inductive capacity of dielectric, 74, 115, 134,

532 ,, ,, ,, crystals, 135

,, „ ,, gases, 132, 533

,, „ ,, liquids, 75, 360

,, ,, in terms of molecular struc-

ture, 130, 134, 553 Infinite conductors, resistance in, 350 Infinity, field at, 56 Insulators and conductors, 5, 545 Intensity (electric), 24, 32, 33, 117, 121, 571, 575 ,, of magnetisation, 368 Intersecting planes, 188. 206

,, spheres, 206

Inverse square, law of, 13, 31, 37, 168, 365 Inversion, 202, 258, 286 Ion, 308

,, velocity of, 310 Ionisation, 311

Jamin, 544

Joule effect in conductors, 320

Kamerlingh Onnes, 558

Kaufmann, 590

Kelvin (Lord), 193, 199, 249, 250, 335, 365,

469 Ketteler-Helmholtz formula, 553 Kirchhoff, 198, 287

,, 's Laws, 311

,, solution of wave-equation, 522

Lagrange's equations, 489, 492, 493 Lame's functions, 252

650

Index

Laplace's equation, 40, 42, 120, 243, 245

,, ,, solution in spherical har-

monics, 206 ,, ,, solution in ellipsoidal har-

monics, 251 ,, ,, solution in spheroidal har-

monics, 206 Larmor, 553, 565, 577 Law of force, 13, 31, 37, 168, 365

,, ,, between current elements, 441

Least action, 488, 517, 579 Lebedew, 538

Legendre's coefficients, 217, 225, 231 Lenz's law of induction of currents, 453 Leyden jar, 77, 277 Lienard, 575

Light, electromagnetic theory of, 3, 526, 532 „ velocity of, 526, 532 ,, dispersion of, 532, 553, 643 Lightning conductor, 61, 479 Lindemann, 556

Lines of force (electrostatic), 25, 29, 43, 47, 62 i, ), ,, (magnetic), 370 „ „ flow, 341 ,, ,, induction, 386 Liouville, solution of wave-equation, 521 Lorentz (H. A.), 553, 575, 589, 592, 594, 600, 602 Lorenz (L.), 553

Maclaurin, 553

Maclean, 525

Magnetic field, 369

,, ,, produced by currents, 425

„ energy of, 396, 415, 494, 507 ,, ,, ofmovingelectrons,514,561,573

,, matter, Poisson's imaginary, 375 ,, particle, 366 ,, ,, potential of, 372

,, ,, potential energy of, 377

,, ,, resolution of, 372

,, ,, vector-potential of, 393

shell, 376, 426 ,, ,, potential of, 376

,, ,, potential energy of, 380

,, ,, vector-potential of, 395

Magnetised body, 367

,, ,, potential of, 372

,, ,, potential energy of, 381

,, „ measurement of force inside

a, 381

Magnetism, physical facts of, 364, 408, 425 ,, terrestrial, 400

theories of, 3, 418, 508

Magnetostriction, 417

Magri, 553

Majorana, 608

Mass, electromagnetic, 585, 611, 613

»> »i

o

»)

J » J)

Matter, structure of, 20, 130, 134; see also Electron and Molecule ,, imaginary magnetic, 375 Maxwell, 2, 3 et passim

,, displacement theory, 153, 510 ,, theory of induced magnetism, 421 theory of light, 3, 526, 532 Measurements :

charge of electricity, 8, 77, 109, 437 current of electricity, 314, 433 inductive capacity, 74, 360 potential difference, 106, 107 resistance, 314 Mechanical action in the ether, 3, 140, 579, 618 ,, ,, dielectrics, 172 ,, ,, magnetic media, 415 force on a circuit, 439, 505 conductor, 102 ,, dielectric, 124, 172 ,, moving electron, 561 ff., 579, 581, 611 , , ,, ,, surface, 79, 178

Medium between conductors, 140, 151 Metallic media, reflection and refraction of light in, 546, 555 ,, ,, absorption in, 545

Michelson, 526

and Morley, 593, 618, 620 Millikan, 20

Mirror galvanometer, 437 Molecular theory of dielectric action, 126, 133, 361 „ „ ,, magnetism, 3, 366, 409,

418, 421, 508 ,, ,, ,, light propagation, 551

Molecule and Atom, structure of, 133, 168, 232, 550, 567, 630, 643 ,, radiation of light from, 577, 632 Moment of a magnet, 366 Momentum, electrokinetic, 498

„ electromagnetic, 583, 615, 617, 620

,, generalised, 493

Mossotti's theory of dielectric action, 127, 168 Moving charge, field of a, 513, 573

,, ,, force on a, 560 Multiple-valued potentials, 279, 429 Muraoka, 362

Nernst and Lindemann, 556 Network of conductors, steady currents in, 311,

316, 322 ,, ,, ,, oscillations in, 499

Neumann's law of current induction, 453 Nichols and Hull, 538 Nicholson, 556

Oersted, 425

Index

651

Ohm (unit of resistance), 305, 530 Ohm's law, 301, 307, 309, 343 Oscillations in a network of conductors, 499 Oscillatory discharge of a condenser, 460

Parabolic cylinders, 267, 269

Parallel plate condenser, 77, 115, 272, 274

Paramagnetism, 410, 413

Particle, magnetic, .366, 372, 377, 393

Pender, 515

Permeability, magnetic, 410

Perot and Fabry, 525

Perrotin, 526

Physical dimensions of electric quantities, 14,

531 Picard, 505 Planck, 634, 636

Plane conductors and condensers, 69, 185, 194, 272 ,, current sheets, 480, 482 ,, semi-infinite (electrified), 266, 273, 282 ,, waves of light, 534 Poincare\ 505, 591 Poisson's equation, 40, 121

„ imaginary magnetic matter, 375, 418 ,, theory of induced magnetism, 127, 418 Polarisation (electrostatic), 117, 118, 126, 155, 232, 536 of light, 535, 543, 565 Polarising angle of light, 543 Polarity of molecules, 126 Potential (electrostatic), 26, 31, 121, 345

,, ,, maxima and minima,

43, 167 „ (electric), 570, 625

(magnetic), 370, 413, 429 (vector), 393, 625 ,, coefficients of, 93, 96, 97 Poyntings theorem, 518 Practical units, 530 Pressure of radiation, 538, 615 Principal coordinates, 550 Pulse of electric action, 578

Quadrant electrometer, 107

Quadric, stress-, 147

Quantity of electricity, 7, 8, 77, 109, 437

Quantum theory, 558

Quincke, 181, 416, 417

Radiant Energy, mass of, 613, 614, 616

„ ,, momentum of, 615, 616

,, ,, nature of, 644

Radiation of Energy, 577, 592, 617

,, pressure of, 538, 615

„ from electrons, 576, 592

Radius of atom, 642, 643

„ molecule, 132, 613 Rapidly alternating currents, 477, 501 Rayleigh (Lord), 358, 595 Recalescence, 412

Reciprocation theorem of Green, 92, 163 Reflection of light, 540, 541, 542, 546, 548

,, coefficients of metals, 548 Rafraction of light, 539, 541, 543

,, ,, lines of force, 123

„ „ „ flow, 346 Refractive index, 532, 553 Relativity, theory of, 593 ff. Relaxation, time of (for a dielectric), 359 Residual discharge, 361 Resistance of a conductor, 301, 314, 355, 557

,, measurement of, 314

, , specific, 342

-box, 314 Resolution of a magnetic particle, 372 Retentiveness (magnetic), 412, 422 Riemann, 527

,, 's surface, 280 Rontgen, 514

,, rays, 311, 578 Rosa and Dorsey, 525 Rowland, 514, 515

,, and Nichols, 361 Russell, A., 199 Rutherford, 630

Saturation (magnetic), 411

Saunders, 525

Schott, 575

Schuster, 403, 555

Schwarz's transformation, 271

Screening, electric, 62, 97, 548

Searle, 584

Self-induction, 456

Sellmeyer's dispersion formula, 553

Shell, magnetic, see Magnetic shell

Signals, transmission of, 332, 502

Sine-galvanometer, 435

Soap-bubble, electrification of, 81

Solenoid, magnetic, 432

Solenoidal vector, 158

Sommerfeld, 283

Specific heats, 556

Specific inductive capacity, see Inductive capacity

Spherical conductors and condensers, 66, 71, 99, 100, 189, 192, 196, 226, 228, 231, 264 „ bowl, 250

„ harmonics (theory), 206, 233, 243 ,, ,, (applications), 224, 401

Spheroidal conductor, 248

652

Index

Spheroidal harmonics, 254, 257

Stark effect, 642

Stewart, Balfour, 402

Stokes, 578

Stokes' theorem, 388

Stresses, general theory of, 142

,, electrostatic, 146, 169

,, in dielectrics, 175

,, „ electromagnetic field, 582, 618, 619

,, ,, magnetic media, 415 Submarine cable, 79, 319, 332, 351, 505 Superposition of fields, 90, 191 Surface-electrification in conductors, 18, 21, 37,

45, 61, 121, 194

„ „ ,, dielectrics, 125

,, harmonics, 208 Susceptibility, magnetic, 410

Tangent galvanometer, 434 Telegraph wire, capacity of, 195

,, ,, transmission of signals along,

317, 332, 502 Telegraphic equation, 504 Terrestrial magnetism, 400 Tesseral harmonics, 237 Thomson, J. J., 286 Time of relaxation, 359 Torsion balance, 11, 365 Transformer, theory of, 465 Trouton and Rankine, 595 Trowbridge and Duane, 525 Tubes of force (electrostatic), 44, 46, 47, 117

Tubes of force (magnetic), 371 ,, ,, flow, 341 „ ,, induction, 386

Unicuraal curves, 269 Uniformly magnetised body, 373 Uniqueness of solution, 89. 163 Units, 14, 77, 305, 365, 427, 522, 528 „ ratio of electrical, 525, 528, 530

Vector-potential, 393, 438, 474, 625

Velocity of electromagnetic waves, 505, 524, 525

„ light, 526, 532 Volt (unit of potential), 305, 530 Volta's law, 303 Voltaic cell, 302 Voltmeter, 314

Water-tube experiment, 593, 607

Wave-propagation, equation of, 520, 525, 534, 571 in dielectrics, 524 ,, metals, 527, 545 ,, crystalline media, 536 „ „ velocity of, 524, 525

Weber's theory of magnetism, 3, 418, 508

Weyl, 623

Wheatstone's bridge, 315, 316

Wiechert, 575

Wilson, H. A., 605

Zeeman effect, 564, 641 Zonal harmonics, 233

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