Electromagnetic Theory, Vol. 2 (1899) — part 3 of 31

Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in soine respects a very limited, in others a very impor- tant part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity. Facts are of QOt much use, considered as facts. They bewilder by their number and their apparent inooherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter. Theory is the essence of facts. Without theoiy scientifio knowledge would be only worthy of the mad- house.

In some branches of knowledge, the facts have been so far refined into theory that mathematical reasoning becomes ap- plicable on a most extensive scale. One of these branches is Eleotromagnetism, that most extensive science which presents such a remarkable two-sidedness, showing the electric and the magnetic aspects either separately or together, in stationary conditions, and a third condition when the electric and mag* netio forces act suitably in dynamical combination, with equal development of the electric and magnetic energies, the state of electromagnetic waves.

It goes without saying that there are numerous phenomena connected with electricity and magnetism which are very imperfectly understood, and which have not been formularised, except perhaps in an empirical manner. Such is particularly the case where the sciences of Electricity and Chemistry meet. Chemistry is, so far, eminently un mathematical (and therefore a suitable study for men of large capacity, who may be nearly destitute of mathematical talent — but this by the way), and it appears to communicate a part of its complexity and vagueness to electrical science whenever electrical phenomena which we can study are accompanied by chemical changes, lint generally speaking, excepting electrolytic phenomena and other compli-

INTRODUCTION.

1^

cations {e.g., the transport of matter in rarefied media when electrical discharges occur), the phenomena of electromagnetism are, in the main, remarkably well known, and amenable to- mathematical treatment.

§ 13. Ohm (a distinguished mathematician, be it noted)- brought into order a host of puzzling facts connecting electro- motive force and electric current in conductors, which all pre- vious electricians had only succeeded in loosely binding together qualitatively under some rather vague statements. Even as- late as 20 years ago, "quantity" and " tension " were much used by men who did not fully appreciate Ohm's law. (Is it not rather remarkable that some of Germany's best men of genius> should have been, perhaps, unfairly treated % Ohm ; Mayer Reis ; even von Helmholts has mentioned the difficulty he had in getting recognised. But perhaps it is the same all the- world over.) Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a- schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed to theory. Another large part became virtually annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's- investigations. Poisson attacked the difficult problem of in- duced magnetisation, and his results, though differently exjMressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena- into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of" experimentalists, on electrostatics and electrodynamics and the- induction of currents. These were rather long in being brought from the crude experimental state to a compact system, ex- pressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. II can scarcely be doubted that had he been one, he would have been greatly assisted in his researches, have saved himself much useless speculation, and would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected

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ELECTROMAGNBTIC THEORY.

CH. X.

work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

Passing over the other developments which were made in the theory of electricity and magnetism, without striking new departures, we come to about 1860. There was then a collec- tion of detached theories, but loosely connected, and embedded in a heap of unnecessary hypotheses, scientifically valueless, and entirely opposed to the spirit of Faraday's ways of thinking, and, in fact, to the spirit of the time. AU the useless hypo- theses had to be discarded, for one thing ; a complete and harmonious theory had to be made up out of the useful re- mainder, for another; and, in particular, the physics of the subject required to be rationalised, the supposed mutual attrac- tions or repulsions of electricity, or of magnetism, or of elements of electric currents upon one another, abolished, and electro- magnetic effects accounted for by continuous actions through a medium, propagated in time. All this, and much more, was done. The crowning achievement was reserved for the heaven- sent Maxwell, a man whose fame, gsent as it is now, has, com- paratively speaking, yeb to oome.

§ 14. It will have been observed that I have said next to nothing upon the study of pure mathematics ; this is a matter with which we are not concerned. But that I have somewhat dilated (and I do not think needlessly) upon the advantages attending the use of mathematical methods by the materialist to assist him in his study of the laws governing the material universe, by the proper co-ordination of known and the dis- covery of unknown (but not unknowable) phenomena.

It was discovered by mathematical reasoning that when an electric current is started in a wire, it begins entirely upon its skin, in fact upon the outside of its skin ; and that, in conse- quence, sufficiently rapidly impressed fluctuations of the current keep to the skin of the wire, and do not sensibly penetrate to its interior.

Now very few (if any) unmatheiuatical electricians can understand this fact ; many of them neither understand it nor believe it. Even many who do believe it do so, I believe, simply , because they are told so, and not because they can in the least

INTKUUUCTION.

15

feel positive about its truth of their own knowledge. As an eminent ])ractician reuiarked, after prolonged scepticism, " When Sir W. Thomson says so, who can doubt it ? " What a world of worldly wisdom lay in that remark !

Now I do admire this characteristically stubborn English way of being determined not to be imposed upon by any absurd theory that goes against all one's most cherislied convictions, and which cannot be properly understood without mathe- matics. For without the mathematics, and with only the sure knowledge of Ohm's law and the old-fashioned notions concerning the function of a conducting wire to guide one, no one would think of such a theory. It is quite preposterous from this point of view. Nevertheless, it is true ; and the view was not put forward as a hypothesis, but as a plain matter of fact.

The case in question is one in which we can be very sure of all the fundamental data of any importance, and the laws con- •cemed. We can, for instance, by straightforward experiment, especially with properly constructed induction balances ad- mitting of exact interpretation of results, readily satisfy our- selves that a high degree of accuracy must obtain not merely for Ohm's law, but also for Faraday's law of E.M.F. in circuits, and even in iron for Poisson's law of induced magnetisation, within certain limits. We have, therefore, all the conditions wanting for the successful application of mathematical reasoning of a precise character, and justification for the confidence that mathematicians can feel in the results theoretically deduced m a legitimate manner, however difficult it may be to give an easily intelligible account of their meaning to the unmathe- matical.

This, however, I will say for the sceptic who has the courage of his convictions, and writes openly against what Is, to him, pure nonsense. He is doing, in his way, good service in the •cause of truth and the advancement of scientific knowledge, by stimulating interest in the subject and causing people to inquire and read and think about these things, and form their own judgment if possible, and modify their old views if they should be found wanting. Nothing is more useful than open and free ■criticism, and the truly earnest and disinterested student of science always welcomes it.

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ELECTBOMAONETIC TUEOUT.

CH. I.

§ 15. The following may assist the unmathematical reader to an understanding of the subject. It is not demonstrative, of course, but is merely descriptive. If, however, it be translated into mathematical language and properly worked out, it will be found to be demonstrative, and to lead. to a complete theory of the functions of wires in general.

Start with a very long solenoid of fine wire in circuit with a source of electrical energy. Let the material inside the solenoid be merely air, that is to say, ether and air. If we examine the nature of the fluctuations of current in the coil in relation to the fluctuations of impressed force on it, we find that the cur- rent in the coil behaves as if it were a material fluid possessing inertia and moving against resistance. The fanatics of Ohm's law do not usually take into aeoonnt the inertia. It is as if the current in the coil could not move without simultaneously setting into rotation a rigid material core filling the solenoid, and free to rotate on its axis.

If we take the air out of the solenoid and substitute any other non conducting material for a core the same thing happens; only the inertia varies with the material, according to its mag- netic indactiTity.

But if wo use a conducting core we get new phenomena^ for wo find that there is no longer a definite resistance and a definite inertia. There is now frictional resistance in the oore, and this increases the effective resistance of the coil. At the same time the inertia is reduced.

On examining the theory of the matter (on the basis of Ohm's law and Faraday's law applied to the conducting core) we find that we can now account for things (in our analogy) by supposing that the rigid solid core first used is replaced by a viscous fluid core, like treacle. On starting a current in the coil it cannot now turn the core round bodily at once, but only its external portion. In time, however, if the source of current be steadily operating, the motion will penetrate fliroughout ttie viscous core, which will finally move as the former rigid core did. If, however, the current in the coil fluctuate in strength very rapidly, the corresponding fluctuations of motion in the core will be practically confined to its skin. The effective inertia is reduced because the core does not move as a rigid body ; the eflcctive resistance is increased by the viscosity generating heat.

INTRODUCTION,

17

Now, returning to the solenoid, wo have perfect symmetry with respect to its axis, since the core is supposed to be exceed- ingly long^ and uniformly lapped with wire. The situation of the source of energy, as regards the core itself, is plainly on its boondaiy, where the coil is placed ; and it is therefore a matter of oommon-Bense that in the communication of energy to the core either when the coirent is steady or when it varies, the transfer of energy takes place transversely, that is, from the boundary to the axis, in planes perpendicular to the axis, and therefore perpendicular to the current in the core itself. This is oonfirmed by the electromagnetic equations.

But the electromagnetic equations go further than this, and assert that the transference of eneigy in any isotropic electrical conductor always takes place across the lines of conduction cur- rent, and not merely in the case of a core uniformly lapped with wire, where it is nearly self-evident that it must be so. This is a my important result^ being the post-finger pointing to a dear understanding of electromagnetism. Passing to the case of a ▼eiy long straight round wire supporting an electric cunrent, we are bound to conclude that the transference of energy takes place tnnsversely, not longitudinally ; that is, across the wire instead of along it. The source of eneigy must, therefore, first supply the dielectric surrounding the wire before the substance of the wire itsell can be influenced ; that is, the dielectric must be the real primary agent in the electromagnetic phenomena con- nected with the electric current in the wire.

Beyond this transverse transference of energy, there does not, however, at first sight, appear to be much analogy between the case of the solenoid with a core and the straight wire in a dielectric. The source of energy in one case is virtually brought right up to the siurface of the core in a uniform manner. But in the other case the source of energy — the battery, for instance — may be miles away at one end of the wire, and there is no im'^iediately obvious uniform application of the source to the skin of the wire. But observe that in the former case the magnetic force is axial, and the electric current circular, whilst in the latter case the electric current (in the straight wire) is axial and the magnetic force circular. Now an examination of the electromagnetic equations shows that the conditions of propagation of axial magnetic force and circular current are the

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■LBCTBOJfAaNXnO THBORT.

GH. I.

same as those for axial current and circular magnctio force. We therefore furbher ooDclude that (with the exchanges made) the phenomena concerned in the core of the solenoid and in the long straight wire are of the same chaiacter.

Furthermore, if we go into detail, and consider the influence of the surroundings of the wire (which go to determine the ▼alue of the inductance of the .cuvuit) we shall find that not only is the character of the phenomena the same, but that they may be made similar in detail (so as to be represented by dollar curves, for example).

The source of energy, therefore, is yirtually transferred instantly from its real place to the whole skin of the wire, over which it is uniformly spread, just as in the case of the con- ducting core within a solenoid.

§ 16. So far we can go by Ohm's law and Faraday's law of E.M.F., and, if need be^ Poisson's law of induced magnetisa- tion (or its modem equivalent practically). But it is quite impossible to stop here. Even if we had no knowledge of electrostatics and of the properties of condensers, we should, by the above course of inquiry, be irresistibly led to a theory of transmission of electrical disturbances through a medium surrounding the wire, instead of through the wire. Maxwell's theory of dielectric displacement furnishes what is wanted to explain results which are in some respects rather unintelligible when deduced in the above manner without reference to elec- trostatic phenomena.

We learn from it that the battery or other source of energy acts upon the dielectric primarily, producing electric displace- ment and magnetic induction ; that disturbances are propa- gated through the dielectric at the speed of light ; that the manner of propagation is similar to that of displacements and motions in an incompressible elastic solid ; that electrical conductors act, as regards the internal propagation, not as conductors but rather as obstructors, though they act as con- ductors in another sense, by guiding the electromagnetic waves along definite paths in space, instead of allowing them to be immediately spread away to nothing by spherical en- largement at the speed of light; that when we deal with steady states, or only slowly varying states, involving immensely

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INTRODUOnON.

19

grest wave length in the dieleotrio, the resulting magnetic phenomena are jnst such as would arise were the speed of propagation infinitely great instead of being finite; that if we make our oscillations ft«ter, we shall begin to get signs of propagation in the manner of waves along wires, with, however, great distortion and attenuation by the resistance of the wires ; that if we make them much faster we shall obtain a comparatively undistorted transmission of waves (as in long- distance telephony over copper wires of low resistance) ; and that if we make our oscillations very fast indeed, we shall have practically mere skin conduction of the waves along the wires «t the speed of light (as in some of Lodge's lightning-conductor experiments, and more perfectly with Hertzian waves).

Now all these things have been worked out theoretically, and, AS is now well known, most of them have been proved experi- mentally ; and yet I hear someone say that Hertz's experiments don't prove anything in particular !

Lastly, from millions of vibrations per second, proceed to billions, and we come to light (and heat) radiation, which are, in Maxwell's theory, identified with electromagnetic disturb- ances. The great gap between Hertzian waves and waves of light has not yet been bridged, but I do not doubt that it will be done by the discovery of improved methods of generating and obseiving very short waves.

CHAPTER 11

OUTLINE OF THE ELECTROMAGNETIO CONNECTIONS.

Electric and Magnetic Force; Displacement and Induction; BlastiTitj and P«nnltttvitj, IndoctlvitF and SeloctlYitr.

§ 20. Our primaiy knowledge of electricity, in its qnantita- tbe aspect, is foonded upon the observation of the meohanioal foroes experienced by an electrieallj ohaiged body, by a mag- netised body, and by a body supporting electric oorrent. In the study of these mechanical forces we are led to the more abstract ideas of electric force and magnetic foroe, apart hom electrification, or magnetisation, or electrio current, to work upon and produce visible effects. The oonoeption of fidds of force naturally follows, with the mapping out of space* by means of lines or tubes of foroe definitely distributed. A ftirther and very important step is the recognition that the two- vectors, electric force and magnetic foroe, represent, or are- capable of measuring, the actual physical state of the medium concerned, fh>m the electromagnetic point of view, when taken in conjunction with other quantities experimentally recognis- able as properties of matter, showing that different sub8tance& are affected to different extents by the same intensity of electrio or magnetic force. Electric force is then to be conceived a» producing or being invariably associated with a flux, the electric displacement ; and similarly magnetic force as producing a^ second flux, the magnetic induction.

If £ be the electric force at any point and D the displaoe* ment, we have

D = cE; ....... (1)

OUTLINE OF ELE0TR03iAGNETIC CONNECTIONS.

21

and similarly, if H be the magnetic force and B the induction, then

Here the ratios e and fi represent physioal properties of the medium. The one (/x), which indicates capacity for supporting magnetio indaction, is its inductivity ; whilst the other, indi- cating the capacity for pnmitting eleotrio displacement! is its permittiTity (or pennittanoy). Otherwise, we may write

and now the ratio e"^ is the elastivity and fiT^ is the rdnotivity (or relootancy). Sometimes one way is preferable^ sometimes the other*

§ 21. All space must be conceived to be filled with a medium which can support displacement and induction. In the former aspect only it is a dielectric. It is, however, equally necessary to consider the magnetic side of the matter, and we may, with- out coining a new word, generally understand by a dielectric a medium which supports both the fluxes mentioned.

Away from matter (in the ordinary sense) the medinm con- cerned is the ether, and fi and e are absolute constants. The presence of matter, to a first approximation, merely altera the Talue of these constants. The permittivity is always increased, ■o &r as is known. On the other hand, the inductivity may be either increased or reduced, thero being a very small increase or decrease in most substances, but a very large increase in a few, the so-called magnetic metals* The range within which the proportionality of flux to force obtains is then a limited one, which, however, contains some important practical applications.

When the fluxes vary, their rates of increase B and D are the velocities corresponding to the forces £ and H, provided no

other effects are produced. The activity of £ is £D, and that of H is HB. The work spent in producing the fluxes (not counting what may be done simultaneously in other ways) is^ thaiefore —

B-fiH.

(2)

E = c-iD, .

(3) (4)

Electric and Magnetic Energy.

22

ELECTROMAGNETIC THEORY.

CH. n.

where U is the electric and T the magnetic energy per unit volume.

When /i and c are constants, these give

U - iED - ^£>, (6)

T ^ iHB = J/xH2, (7)

to express the energy stored in the medium, electrio and mag- netic respectively.

When /X and c are not constants, the previous expressions (5) will give definite values to the energy provided there be a de- finite relation between a force and a flux. If, however, there be no definite relation (which means that other circumstances liian the value of the force control the value of the flux), the energy stored will not be strictly expressible in terms simply of the force and the flux, and there will be usually a waste of energy in a cyclical process, as in the case of iron, so closely studied by Ewing. This does not come within the scope of a precise mathematical theory, which must of necessity be a sort of skeleton framework, with which complex details have to be separately adjusted in the most feasible manner that presents itself.

Eolotropic BelatioiiB.

§ 22. But a precise theory nevertheless admits of consider- able extension from the above with /a and e regarded as scalar constants. All bodies are strained more or less, and are thereby usually made eolotropic, even if they be not naturally eolo- tropic. The force and the flux are not then usually concurrent, or identical in durection. But at any point in an eolotropic substance there are always (if force and flux be proi)ortional) three mutually perpendicular axes of concurrence — the prin- cipal axes — when we have (refemug to displacement)

Di-CjEp D,-c,E^ Dj-CsEs,

if the c*s' are the principal permittivities, the E's the corre- sponding effective components of the electric force, and the D's those of the displacement If E be parallel to a principal axis, so is ]>. In general, by compounding the force compo- nents, we obtain E the actual force, and by compounding the flux components obtain D the displacement to correspond, which can only concur with £ in the above-mentioned special

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 23

cases if the principal permittivities be all different. But should a pair be equal, then E and D concur in the plane containing the equal permittivities, for the permittivity is the same for any axis in this plane.

The energy stored is still half the scalar product of the force and the flux, or |ED, understanding that the scalar product of two vectoi*s, which is the product of their tensors (or magnitudes) when they concur, is the same multiplied by the cosine of their included angle in the general case.

Vector-analysis is, I think, most profitably studied in the concrete application to physical questions, for which, indeed, it is specially adapted. Nevertheless, it will be convenient, a little later, to give a short account of the very elements pf the subject, in order not to have to too frequently interrupt our electromagneUc arguments by mathematical explanations. In the meantime, oonaider the dielectric medium further.

Distinction between Absolute and Belative Permittivity or

Inductivity.

§ 23. The two quantities c and /i are to be regarded as known data, given over all space, usually absolute constants ; but when the simpler properties of the ether are complicated by the presence of matter, then vaxying in value from place to place in isotropic but heterogenous substances ; or, m case of eolotropy, the three principal values must be given, as well as the direction of their axes, for every point considered. Keeping to the case of isotropy, the ratios c/co and [x/fM^ of the permittivity and inductivity of a body to that of the standard ether are the specific inductive capacities, electric and magnetic respectively ; and are mere numerics, of course. They do not express physical pi;operties themselves except in the limited sense of telling us how many times as great something is in one case than in another. This is an important point. It is like the difference between density and specific gravity. It is possible to so choose the electric and magnetic units that 1, c=l in ether; then /x and c in all bodies are mere numerics. But although this system (used by Hertz) has some evident recommendations, I do not think its adoption is desirable, at least at present. 1 do not see how it is possible for any medium to have less than two physical properties effective in

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EUMTBOXAONBTIO THBOBT*

CE. II.

the propagation of mfM. If this be admitted, I think it may ako be admitted to be desirable to explicitly admit their exist- ence and symbolise them (not as mere nnmerics, but as physical

magnitudes in a wider sense), although their precise interpre- tation may long remain unknown.

If, for example, H be imagined to be the velocity of a sub- stance, then J/xH^ is its kinetic energy, and /i its density. And if E be a torque, then c"^ (the elastivity) is the corre- sponding coefficient of elasticity, tiie rigidity, or gMasi-rigidity, as the case may be ; whilst c is the coefficient of compliance, or the compliancy ; and ^c£^ is the stored energy of the strain.

Dissipation of Energy. The Conduction-current ; Ckmdne- tivity and Resistivity. The Electric Onrrent.

§ 24. Besides influencing the values of the ether constants as above deecribed, we have also to admit that in certain kinds of matter, when under the influence of electric forces energy is dissipated coiUinwmdy, besides being stored. These are adled electrical oonduotors. When the conduction is of the simplest (metallic) Qrpe, the waste of energy takes place at a rate pro- portional to the square of the electric force. Thus, if Q| be the Joulean wastes

This new flux 0 is the conduction current, and k is the conduc- tivity (electric). Its reciprocal is the resistivity.

The termination -ivity is used in connection with specific pcoperties. It does not always sound well at firsts but that wears ofi". Sometimes the termination -ancy does as welL

The conductivity k is constant (at one temperature), or is a linear operator, as in the previous cases with respect to /i and c. The dissipation of energy does not imply its destruction, but simply its rejection or waste, so far as the special electro- magnetic aflairs we are concerned with. The conductor is heated, and the heat is radiated or conducted away. This is also (most probably) an electromagnetic process, but of a different order. Only in so far as the effect of the heat alters the conductivi^, &c., or, by differences of temperature^ causes

Qi = Z;E2 = E0,

(8) (9)

if

OUTLINB OF XLIOIBOXAONSnO OOKNECnONS. 25

thormo-eleotrio foroe, are we ooncemed with energy wasted according to Joule's law.

The activity of the electric force^ when there is waste, as well . as storage, is

E(0 + D) = Qi+U (10)

The sam 0+D is the electric enxxent, when tiie medium is at rest. When it is in motion, a further term has sometimes to be added, viz., the conyection current.

PIctitioiis Magnetic Oondnction-cnrrent and Beal Magnetic

Onirentb

§25. If a substance were found which could not support magnetic force without a continuous dissipation of energy, such a substance would (by analogy) be a magnetic conductor. Let, for instance,

K«5'H, (11)

then K is the density of the magnetic conduction current, and the rate of waste of energy is

Q2-HK-i,rH«. (12)

The activity of the magnetic foroe is now

H(K + B) = Qa + T (13)

Cknnpare this equation with (10). The magnetic current is K+B.

As there is (I believe) no evidence that the property sym* bolised by g has any existence, it is needless to invent a special oaine for it or its reciprocal, but to simply call g the magnetic conductivity. The idea of a magnetic current is a very useful

one, nevertheless. The magnetic current B is of course real ; it is the part K that is speculative. It plays an important part in the theory of the transmission of waves in conductors.

Forces and Fluxes.

§ 26. So far we have considered the two forces, electric and magnetic^ producing four fluxes, two involving storage and two waste of energy, and we have defined the terminology when the state of things at a point is concerned. We reckon forces per

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ELECTROMAGNETIC THEOBT.

CH. II.

Qttit length, fluxes per unit area, and energies or wastes per unit volume. It is thus a unit cube tliat is referred to, whose edge, side, and volume are utilised. Bat a unit cube does not mean a cube whose edge is 1 centim. or any other concrete length ; it may indeed be of any size if the quantities concerned are uniformly distributed throughout it, but as they usually vary from place to place, the unit cube of reference should be imagined to be infinitely small. The next step is to display the equivalent relations, and develop the equivalent terminology, when any finite volume is concerned, in those cases that admit of the same simple representation in the form of linear equa- tions.

Line-Integral of a Foree. Voltage and Oanssage.

§ 27. The line-integral of the electric force from one point to another along a stated path is the electromotive force along that path ; this was abbreviated hy Fleeming Jenkin to E.M.F. He was a practical man, as well as a practician. When ex- pressed in terms of a certain unit called the volt, electromotive force may bo, and often is, called the voltage. This is much better than " the volts." I think, however, that it may often be conveniently termed the voltage irrespective of any par- ticular unit. We might put it in this way. Volta was a. distinguished man who made important researches connected with electromotive force, which is, therefore, called voltage, whilst a certain unit of voltage is called a volt. At any rate, we may try it and see how it works.

The line-integral of the magnetic force from one point to* another along a stated path is sometimes called the magneto* motive force. The only xecommendation of this cumbrous term is that it is correctly correlated with the equally cum- brous electromotive force. Magnetomotive force may be called the gaussage [pr. gowsage], after Gauss, who distinguished him- self in magnetic researches; and a certain unit of gaussage may be called a gauss [pr. gowoe]. I believe this last has already been done, though it has not been formally sanctioned. Gaussage may also be experimented with.

The voltage or the gaussage along a line is the sum of the- effective electric or magnetio forces along the line ; the effec- tive force being merely the tangential component of the reaL

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OUTLINE OF ELECTIIOMACNKTIC CONNECTIONS.

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force. Thus, electric force is the voltage per unit length, and magnetic force the gaussage per unit length along lines of force.

Snrfoce-integral of a Flux. Density and Intensity.

§ 28. Next as regards the fluxes, when considered with reference to anv area. The flux through an area is the sum of tlio effective fluxes through its elementary units of nrea ; the effective flux beiugj the normal component of the flux or the component perpendicular to the area. We do not, I think, need a number of new words to distinguish flaxes through a surface from fluxes per unit surface. Thus, we may speak of the induction through a surface (or through a cirouit bounding it) ; or of the current through a surface (as across the section of a wire); or of the displacement through a surface (as in a con- denser), without any indeflniteness, meaning in all cases the surface integral of the flux in question.

In contradistinction to this, it may be sometimes convenient to speak of the density of the current, or of the induction, or of the displacement, that is, the amount per unit area. Similarly, we may sometimes speak of the intensity of the electric or magnetic force, using "density" for a flux and ** intennty " for a force.

It may be observed by a thoughtful reader that there is a good deal of the conventional in thus associating one set of vectors with a line, and another set with a surface, and other quantities with a volume. It is, however, of considerable prac- tical utility to carry out these distinctions, at least in a mathe- matical treatment. But it should never be forgotten that electric force, equally with displacement^ is distributed through- out volwnesy and not merely along lines or over areas.

Oonductance and Besistance.

§ 29. Conductivity gives rise to conductance, and resistivity to resistance. For explicitness, let a conducting mass of any shape be perfectly insulated, except at two places, A and B, to be conductively connected with a source of voltage. Let the voltage established between A and B through the conductor be V, and let it be the same by any path. This will be the case when the current is steady. Also let C be this steady current, in at A and out at B. We shall have

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SLEOTBOMAGNSTIO THBORT.

CH. II.

V-RC, C-KV, (1)