A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 26 of 27

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 Vt 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 \mv2, the kinetic energy of the particle, we find as the equation of energy

r* const. (22)

Since the second term of the coefficient of v3 may be increased in definitely by increasing a, the radius of the sphere, while the surface- density a remains constant, the coefficient of v2 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 v2.

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 d4-~ds ds' = ii' M, (24)

jj. r ds ds

i I ro^ 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* v2, 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 vl tfj) and e\ (ve + vl 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, u9 of two electric particles is

u2 = v2+v'2+72-2vv'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 fdr 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 d2r *v~foTs + v ^di^di2' 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)

., — ,

r2 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, d2r

ds ds'

= 0.

/*! dr dr , f A dr dr d2r \ , /*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-

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-

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,

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, 794—797. 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,

Electrodynamometer, 725. Electrolysis, 236, 255-272. Electrolyte, 237, 255. Electrolytic conduction, 255-272, 363,

— 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,

'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,

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,

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,

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,

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.