Electromagnetic Theory, Vol. 2 (1899) — part 9 of 31

It is natural to ask what part do the stresses play in the propagation of disturbances ? The stresses and accompanying strains in an elastic body are materially concerned in the trans- mission of motion through them, and it might be thought that it would be the same here. But it does not appear to be so from the electromagnetic equations and their dynamical con- sequences— that is to say, we represent the propagation of dis- turbances by particular relations between the space- and the time- variations of £ and H ; and the electromagnetic stress and possible bodily motions seem to be accompaniments rather than the main theme.

Dependence of the Fluxes due to an Impressed Forcive upon its Curl only. GenepJ. Demonstration of this Property.

§ 87. In § 83 it was remarked that the flux induction due to an intrinsic magnetic forcive depends not upon itself directly, but upon its curl, and in § 84 a similar property was pointed out connecting the flux displacement and intrinsic electric force. That is to say, the fluxes depend upon the vectors and not upon eg and h^. This remarkable and, at first sight, strange property, which is general, admits of being •demonstrated in a manner which shall make its truth evident in a wider sense, and lead to a connected property of consider^ able importance in the theory of electromaj^netic waves.

Let there be any impressed forcive e^ in a stationary medium. Its activity is OqJ per unit volume, where J is the electric cur- rent, and the total activity is J, where the 2 indicates sum- mation through all space, or at any rate so far as to include every place where exists. Its equivalent is the total rate of waste of energy, and the rate of increase of the total stored energies, say and Tq. Thus,

2M"=Qo + Uo + Tp (34).

OUTLINE OF BLBCTROUAGNETIC OOimECTIONS. Ill

Now, the value of the aummation is zero if has no ourl, or is irrotational — that is, an irrotationa) forcive does no work upon a cirouital flux. This proposition may be rendered evident by employing a particular method of effecting the space sum- mation, viz., instead of the Cartesian method of cubic sub- •division of space, or any method employing co-ordinates, divide space into the elementary circuital tubes belonging to the flux. ^Here it is J.) Fixing the attention upon any one of these circuits, in which the flux is a constant quantity, we see at •once that the part of the summation belonging to it is the product of the impressed voltage in the circuit and the flux therein. But there is no impressed voltage, because Og has no curl ; heuce the circuit contributes nothing to the summation. Further, since this is true for every one of the elementary cir- •cuits, and inclusion of them all includes all space, we see that the summation SoqJ necessarily vanishes under the circum- stance stated of Cq being irrotational.

Now return to equation (34), and suppose that the initial state of things is absence of E and H everywhere, so that Qq, Uq .jind Tq are all zero. Next start any irrotational Cq, and see what will happen. The left side of (34) being zero, the right side must also be and continue zero. But Q,,, Uq and Tq when not zero are essentially positive. Now if (U^ + Tq) becomes positive, Qq should become negative, in order to keep the right number of (34) at zero. This negativity of Q^ being impos- Jiible, these quantities Qq, Uq and Tq must all remain zero. Con- ^sequently E and H must remain zero. That is, an irrotational forcive can pioduce no fluxes at all, if the flux corresponding to the force is restricted to be circuital.

It should be observed that this proposition applies not merely to the steady distribution appropriate to the given forcive, but to all intermediate stages, involving both electric and magnetic force, and flux of energy. Nothing happens, in fact, when any •distribution of impressed force is made to vary in time and in q>ace, provided it be restricted to be of the irrotational type, «o that the voltage (or the gaussage) in every circuit is nil.

Notice further the dependence of the property upon cir- 'Cuitality of the factor with which the impressed force is associated (thus J with e^, and similarly Gr with h^), and the positivity of energy ; and, more strikingly, the independence

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ELECTROMAGNETIC THEORY.

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of such details as are not concerned in equation (34), such as the distribution of conductivity, permittivity, &c., or of the forces being linear functions of the fluxes.

Identity of the Disturbances due to Impressed Forcives having the same CurL Example: — A Single Circuital Source of Disturbance.

§ 88. Returning to (34) again, we see that any two impressed forcives produce the same results in every particular, as regards the varying states of £ and H gone through, if their curl is the same. For the difference of the two forcives is an irrota- tional forcive, and is inoperative. Here it is desirable to take-

Fig. 6.

an explicit example for illustration of the meaning and effect. Describe a linear circuit in space, and a surface bounded by the circuit (Fig 6). Over this surface, which call S, let an impressed force V act normally, V being the same all over the surface. This system of force is irrotational everywhere except at the bomiding circuit, where there is a circuital distribution of g^y the curl of Cq, of strength V.

Now, our present proposition asserts that the disturbances due to V over the surface S are in every respect the same a& those due to V (the same normal impressed force), spread over any other surface bounded by the same circuit. The comprehensiveness of this property may be illustrated by sup- posing that one surface is wholly in a non-conductor, whilst

OUTUNE OF £L£CTRCMAGNBTIO OONNECTIONS.

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the other passes through a conductor ; or that one surface 18 wholly within a conductor, whilst the other passes out of it and through another conductor insulated from the first. Thus, in the diagram, let S be the first surface (in section), and Sj the second, passing through a conductor represented by the shaded region. The points A, B are where the common boundary of the two surfaces cuts the paper, whilst the arrows serve to show the direction of action of the impressed force. S, is another surface of impressed force, also reaching into the conductor. Now all these forcives (eaoh by itself) will produce the same final state of displacement in the dielectric and electrification of the conductor, and will do so in the same manner; that is, the electromagnetic disturbances generated will be the same when expreased in terms of E and H. The distribution of energy, for example, and the ttreaaet, will be the same.

But there must be some difference made by thus shifting the source of energy. Obviously the nature of the flux of energy is changed, inus being

W-V(B-eo)(H-ho),

where we deduct the intrinsic forces to obtain the forces effec- tive in transferring energy, we see that every change made in tiie distribution of the intrinsic sources affects the fiux of energy, in spite of the independence of £ and H of their distri- butions (subject to the limitation mentioned). In our example, however, the only change is in the sheet of impressed force itselt

Ftednction of Steady State due to Impressed Forcive by erossiiig of Etoctromagnetic Wavea. Ezample of a Oir- cular Source. Distinction between Source of Energy and of Disturbance.

§ 89. It may be readily suspected from the preceding, that, as far as the production of electromagnetic disturbances goes, we may ignore e© and ho altogether, and regard the circuital vectors and as the real sources. This is, in fact, the case when ultimately analysed. In the example just taken the cir- cuit ABA is the source of the disturbances. That is, they emanate from this line. If disturbances were propagated

114

KJDOTBOMACDmiO THBOBT.

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infinitely rapidly there might be some difficulty in recognising the property, beoaose the steady state appropriate to the instantaneous state of the impressed force would exist (if the conductor were away) ; but, as we shall see later, the speed of propagation is always finite, depending upon the values of e and /i in the medium ; and conduction does not alter this property, although by its attenuating and distorting e€RBct8 it may profoundly alter the nature of the resultant pheuomena. With, then, a finite speed of propagation, we have merely to cause the impressed force to vary or fluctuate sufficiently rapidly to obtain distinct evidence of the emanation of waves the real sources of disturbance. Thus, let the source be

FiQ, 7.

circular in a plane perpendicular to the paper (Fig. 7), and A, B the points where it cuts the paper. When the source is suddenly started, the circle ABA is the first line of magnetic force. At any time t later, such that is less than the distance JAB, the electromagnetic disturbance will be confined wholly.within a ring-shaped region having the circular source for core and of radius vt round A or round B. ' But when the distance vt in- eresfles to ^AB overlapping oonunences, and a little later there is (as in the shaded part of the figure) a region occupied by two coincident waves crossing one another. Now, the union of these waves produces the steady state of displacement without magnetic force— that is, within the shaded region the magnetic

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OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

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force vanishes, and the displacement is that which belongs to the final steady state. As time goes on, of course the shaded region enlarges itself indefinitely, although outside it is still a region occupied by electromagnetic disturbances going out to infinity. This supposes that there is no conductor in the way. Should there be one, then, as soon as it is struck by the initial electromagnetic wave, it becomes and continues to be a secondary source of disturbance, and the final steady state, •different from the former, now arises from the supeiimposition of the primary waves and the secondary.

It is, of course, impossible to go into detail at present^ the •object being merely to point out the difierence between sources of energy and sources of disturbance. The source of energy only works when there is electric current at the spot, and this <x>me8 to it from the vortex lines (the circuital sources of dis- turbance). Thus, in the last figure, at time when the ring of disturbance is of thickness 2vtf if the impressed force be in a circular disc whose trace is the straight line AB, the force is working in Aa and in B6, but inoperative elsewhere. Again, after overlapping has begun it is still working in Aae and Bbd, but is inoperative in cdt having done its work. Similarly when the sheet of impressed force has any other shape. The impressed force only works when tt is allowed to work by the electromagnetic wave reaching it.

To emphasize the matter, take another case. Let tiie impressed force in any telegraph circuit be confined to a single pUme section across the conductor, so that the vortex line is a line on its surface, going round it. If this be at Valencia, we may shift the " seat of the E.M.F.'' to Newfound- land, provided we preserve continuity of connexion with the -vortex line, as before explained; for instance, by extending ^lie sheet over the whole surface of the conductor between the two places. The sheet at Newfoundland and the surface sheet will together produce the same effects as the original sheet at Valencia. Or, we may have the sheet entirely outside the <;ircuit, provided only that it is bounded by the original vortex line, in touch with the conductor.

j What the practical interpretation of this extraordinary property is in connection with the "seat of E.M.F.'' of galvanic batteries and in electrolysis generally still remains obscure.

i2

116

BLBOXBOMAONBnO THBOBT.

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All impraned Toltages and gaiunageB are more or less difficult to undeiBtand. But there need be no doubt as to the general troth of the property if the circoitali^ of the ourient can be trusted.

The Sruption of "4n-*'8.

§ 90. It may have been obeerred that the equations employed by me in the preceding differ from those in use in all madie- matioal treatises on the subject in a certain respect (amongst others), inasmuch as the constant 49r, which is usually so obtrusively prominent, has been conspicuous by its absence. This constant 4ir was formerly supposed to be an essential part of all electric and magnetic theories. One of the earliest results to which a student of the mathematics of electricity was introduced in pre-Mazwellian days was Coulomb's law of the relation between the density of the electric layer on a- conductor and the intensity of electric force just outside it, say

E«4ir«r; (1)

and, since this was proved by mathematics, it seemed that the 47r was an essential ratio between two physical quantities, viz.,. electricity and the force it exerted on other electricity. Never a hint was given that this 47r was purely conventional ; it waa not, indeed, even recognised to be conventional, and is not at the present day in some quarters. Then, again, at the begin- ning of magnetism, was Gauss's celebrated theorem proving mathematically that the total flux of magnetic force outward through a closed surface equalled precisely iir times the total amount of magnetism enclosed within the surface ; and, for all that might appear to the contrary, this remarkable result flowed out necessarily from the properties of the potential function and its derivatives, and of the three direction cosinea of the normal to an element of the surface. It was funny — very funny. How ever the 4r managed to find its way in wa» the puzzle in these and similar results ; for instance, in the- well-known

/*=l + 4jrK, (2)

where is the permeability and k another physical property,, the susceptibility to magnetisation of a substance. The dark.

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OUTLINS OF SLBOI^MAONBIIG CONNECTIONS.

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mystery was carefully covered up by the mathematics. It was as hard to understand as the monarch found it to explain the preMDoe of the apple in the dumpling — how did the goodwife manage to get it in ? Nor was the matter rendered plainer by Maxwell's greac treatise. Maxwell thought his theory of eleotric displacement explained the meaning of the 4r (in the corresponding electric theorem), as if it were a matter of physics, instead, as is the fact, of irrationally chosen nnits.

As the present chapter has been mainly devoted to a general ontline of electromagnetic theory expressed in formulse involv- ing rational units, it will be fittings in bringing it to a oondu- flion, to explain here the relation these rational units bear to the orduiaiy irrational imits.

The Origin and Spread of the Eruption,

§ 91. The origin of the 4r absurdity lay in the wisdom of our ancestors, — ^literally. The inverse square law beuag recog- nised, say that two charges and repelled one another with a force varying inversely as the square of the distance between them; thus,

F^aq.qjr^ (3)

where a is a constant ; what was more natural than to make the expression of the law as simple as possible by giving the constant a the value unity, if, indeed, it were thought of at all f Our ancestors could not see into the future, — that is to say, beyond their noseSy^and perceive that this system would work out absurdly. They were sufficiently wise in their generation, and were not to blame.

But, after learning that certain physical quantities bear to one another invariable relations, we should, in forming a syste- matic representation of the same, endeavour to avoid the intro- duction of arbitrary and unnecessary constants. This valuable principle was recognised to a small extent by our ancestors, as above ; it was emphasised by Maxwell and Jenkin in their little treatise on units in one of the Reports of the B. A. Committee on Electrical Standards (1863, Appendix C ; p. 59 'of Spon's Reprint). Thus, referring to the magnetic law of inverse squares, we have the following : —

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EL£0TK0MAGN£T1C THEOBT.

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" The strength of the polo is necessarily defined as propor- tional to the force it is capable of exerting on any other pole. Hence the force / exerted between two poles of the strengths m and wij, must be proportional to the product mniy The force / is also found to be inversely proportional to the square of the distance, D, separating the poles, and to depend on no other quantity ; hence, we have, wUessan c^surd and useless coejicient be introdttced,

f'-mm^l^/* (4)

Observe the words which I have italioised. When it is con- sidered what Maxwell had then done m the way of framing a* broad theory of electromagnetism, it is marvellous that he should have written in that way. By mere foroe of habit on» might, indeed, not consider there to be anything anomaloi^ about the \ir in Coulomb's and Gauss's theorems. But did not Maxwell's eleotrostatio energy KC^^/Stt and his magnetic energy /A$^/87r per unit volume loudly proclaim that there was some- thing radically wrong in the system to lead to such a mode of expression, which fault should be attended to and set right ab- initio, especially in frammg a permanent system of practical units, which was what Maxwell and his colleagues were about f It would seem that the proclamation fell upon deaf ears, for not only were the units irrationally constructed, but in his Treatise, which followed some years later, we find the following statement (p. 155, second edition). After an account of hi»> theory of electric displacement, we are told that "the theory completely accounts for the theorem of Art. 77, that the totaX induction through a closed surface is equal to the total quantity* of electricity within the surface multiplied by iv. For what' we have called the induction through the surface is simply the electric displacement multiplied by 4ir, and the total displaoe>> ment outward is necessarily equal to the total charge within the surface." That is, his liieory of electric displacement- accounts for the 4ir. So it seems to do; and yet the has no essential connection with his theory, or with any one else's. Though by no means evident until it is pointed out, it is entirely a question of the proper choice of units, and is independent of nil theories of electricity. It depends upon something much> more fundamental.

OUTLUfS OF ELECTBOMAQNETIO CX>NNBCIIONS. 119

ThB Oure of the Disease by Proper Measure of the Strength

of Sources.

§92. When looked into oarefullj, the question is shnply this : What is the natural measure of the strength of a source? Suppose, for instance, we have a source of heat in a medium which does not ahsorb heat, how should we measure the intensity of the source ? Plainly by the amount of heat emitted per second, passing out through any surface enclosing the source. If the flux of heat be isotropically regular, its density will vary inversely as the square of the distance from a point source, giving

C = S/47rr2, (5)

if C be the heat flux (per unit area) at distance r from the source of strength S. If we knock out the 4ir we shall obviously have an unnatural measure of the strength of the source.

Similarly, if we send water along a pipe and let it flow from its end, which may therefore be regBodeA as a source, we should naturally measure its strength in a similar manner, viss., by the current in the pipe, or bj the total outflow.

Now in an electric field, or in a magnetic field, or in the field of any vector magnitude, we have everywhere mathe- matical^ analogous cases. We find, for instance^ that electro- statio force is distributed like velocity in an incompressible liquid, except at certain places, where it is, by analogy, generated, or has its source. If, then, we observe that the flux of force through a closed surfoee is not zero» there must be sources within the region enclosed, and the natural measure of the total strength of the sources is the total flux of force itself. I have put this in terms of electric force rather than of electric displacement, merely to exemplify that the matter has no particular connexion with electric displacement. In the former case it is a source of " electric force " that is considered ; in the latter, it would be of displacement ; and the principle concerned in a rational reckoning of the strength of a source is the same in either case. It is a part of the fitness of things, and holds good in the abstract theory of the space-variation of vector magnitudes, apart from all physical application.

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BLBOTBOKAGNlSnO THBOBT.

CH. n.

Using, temporarily, the language of linee of force, or of tabes of force (which, howerer, as here, sometimes works out rather nonsensically), we may say that a unit pole sends out one line of force, or one tube, when rationally estimated.

Next there is the proper measure of the strength of cir- cuital fluxes to be considered ; electric current, for instance. The universal property here is that the circuitation of the magnetic force is proportional to the current through the circuit ; and the natural measure of the strength of the current is the circuitation itself, without, as usual, dividing by 4r. Now this division by 47r arises out of the irrational reckoning of the strength of point sources. We may therefore expect that when the point sources are measured rationally, the 47r will disappear from the reckoning of electric current, making the circuitation of magnetic force the proper reckoning. This is so, as may be easily seen by substituting for a linear electric current an equivalent magnetic shell, and so bringing in point sources distributed over its two faces.

Thus, in rational units, if we have a point source q of dis- placement, and a point source m of induction, we have

D = 9/47rT«, B«m/4irr8, ... (6)

to express the displacement and induction at distance r, when the fluxes emanate isotropically ; and

E«(j/c)/4irr«, H - (m/,i)/4rr». . • (7)

if qfc and m/fx are the measures of the sources of electric and magnetic force respectively. In the magnetic case m may repre- sent the strength of a pole, on the understanding that (since the induction is really circuital) we ignore the flux coming to the pole (as along a filamentary magnet), and consider only the diverging induction. Also,

H-C/2jrr (8)

expresses the intensity of magnetic force at distance r from a long solitary straight current of strength C.

Obnozioiis Effects of the Emption.

§ 93. Considering merely the formuln belonging to point eources with uniform divergence, we see that the effect of

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OUTLINE OF ELBCTROMAQNETIO CONNECTIONS. 121

changing from irrational to rational units is to introduce iir. If this were all, we might overlook the fundamental irration- ality and use irrational units for practical convenience. But, ■as a matter of fact, it works out quite differently. For the unnatural suppression of the 47r in the formulte of central force, where it has a right to be, drives it into the blood, there to multiply itself, and afterwards break out all over the body of electromagnetic theory. The few formulas where iir should be are principally scholastic f ormolo and little used ; the many formulse where it is forced out are, on the contrary, useful lormulsD of actual practice, and of the practice of theory. A fvactical theorist would knock them out merely from the trouble they give, let alone the desire to eee thiuf^s in their right places. Furthermore, it should be remarked that the imtionalitj of the formulsB is a great impediment in the way •of a clear understandhig of electromagnetic theory. The interpretation of equation (2) above, for instance, or of the eimilar well-known equation

(ireaents some difficulty even to a student of ability, unless he be given beforehand a hint or two to assist hiuL For if k is ■a rational physical quantity, then m cannot be ; or if is right, then K must be wrong. Or and $ must be moongruous. The 4ir is also particularly inconvenient in descriptive matter relating to tubes of force or flux, and in everytliing connected with them.

This difficulty in the way of understanding the inner mean- ing of theory is still further increased by the 4ir not entering into the magnetic formulsB in the same way as into the electric Thus, it is ^ and $/4ir which are analogues. Again, in many of the irrational formulae the irrationality appears to disappear. For instance, in J^/o, in ^S^CH^, in ^<^P, in ii<!P, and in some others. This is because both the &ctors are irrational, and the two irrationalities oanoeL It looks as if p were the flux belonging to but it is not. In reality, we have, if 2 = (^tt)^,

therefore, EDb<I9>

whilst c is the same in both irrational and rational units.

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ELECTEOMAQNETIC THEOBT.

Cli. IL

A Plea for tlie Bemoval of the EmptioiL by the Sadical Omu

§94. The question now is, what is to be done? Are we modern pigmies, who by looking over the shoulders of the giants can see somewhat further than they did, to go on perpe- trating and perpetuating their errors for ever and ever, and even legalising them ? If tliey are to be enforced, it is to be hoped that it will not be made a penal offence not to use the legal and imperial units.

The " brain-wasting perversity " of the British nation in sub- mitting year after year to be ruled by such a heterogeneous and incongruous collection of units as the yard, foot, inch, mile, knot, pound, ounce, pint, quart, gallon, acrey pole, horse-poweiv- eto., etc., has been repeatedly lamented by would-be refoniiein» who would introduce the oommon-sense deouual system ; and amongst them have been prominent electricians who hoped tO' insert the thin end of the wedge by means of the decimal sub- division of the electrical units, and their connection with the metre and gramme, and thus lead to the abolition of the present British system of weights and mearores with its absord and useless arbitrary connecting constants. But what a satire it is upon their labours that they should have fallen into the -very- pit they were professedly avoiding I The perverse British nation — ^practically the British engineers — ^have surely a right to expect that the electricians will first set their own house in* order.

The ohm and the volt, eto., are now legalised, so that, as I am informed, it is too late to alter them. This is a non teq., however, for the yard and the gallon are legalised ; and if it is not too late to alter them, it cannot be too late to put the new- fangled ohm and its companions right. It is never too late to- mend. No new physiMl laboratory determinations will be needed. The value of w has been calculated to hundreds of places of decimals ; so that rational ohms, volts, etc, can be now fixed with the same degree of aoouiaqy as the irrational ones,, by any calculator.

When I first brought up this matter in The Electrician in 1882-3, explaining the origin of the 47r absurdity and its cure, I did not go further than to use rational units in explaining the*

OUTLINB OF SLEGTBOMAGNSTIO OONNECTIONS. 12^

theofy of potentials, scalar and vector, and siniilar matters; then returning to irrational units in order to preBerve harmony with tha formula} in Maxwell's treatise, and I did not think a change was practicable, on account of the B. A. Committee's work, and the general ignonnce and want of interest in the subject. But much has happened since then. The spread of electrical know- ledge has been immense, ooncurrent with the development of electrical industries, to say nothing of theoretical and experi> mental developments, I therefore now think the change is perfectly practicable. At any rate, some one must set the example, if the change is to occur. I have, therefore, in the preoeding, wholly avoided the irrational units ah initio ; and shall oontinne to use rational units in the remainder of this work.

So far as theoretical papers and treatises are concerned, there is no difficulty. Every treatise on Electricity should be done in rational formula), their connexion with the irrational

(so long as they exist) being explained separately (in a chapter at the end of the book, for instance), along with the method of oonverting into volts, amperes, etc., the present practical units. At present you have to first settle whether to use the electro- static or the electromagnetic units, and then introduce the appropriate powers of 10. If rational formulas are used, then, in addition, you must first insert the constant 4ir in certain places, so long as the irrational units last.

When, however, the real advantages of the rational system become widely recognized and thoroughly assimilated, then will come a demand for the rationalisation of the practical units. Even at present the poor practician is complaining that he cannot even pass from magnetomotive force to am- pere-turns without a "stupid" 47r coefficient getting in the way. That the practical electrical units should be reformed as a preliminary to the general reform of the British units requires no argument to maintain. That this general reform is coming I have not the least doubt. Even the perversity of the British engineer has its limits.

Rational v. Irrational Electric Poles.

95. We may now briefly compare some of the more impor- tant formulsd in the two systems. Let us denote quantities in

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vatioiial units as in the preoeding (except that when vector rela- tions aie not in question we need not employ special type), and the corresponding quantities in irrational units by the same symbols with the suffix thus, q and g^. Also denote (4s-)' by c Let ^{ be the irrational charge which repels an equal irrational charge at distance r with the mmt force F as a rational change 2 repels an equal charge at the same distance. Then,

^-w--m- ••■•'=>

Therefore

ff-*?!- (10)

As regards the ratio of the units, it is sufficient merely to observe that the magnitude of the number expressing a quantity varies inversely as the size of the unit. This applies throughout, so that we need not bring in units at all, but keep to the concretes appearing in formulas. By (10) we shall have

a;«^ = r«_ = _« . • . • (11) Pi o-< 1>< C,

if p is volume-density of electrification, o- surface density, D displacement, C electric current density (or else the total displacement and current).

Since D = cE, and D< = cEi/47r, .... (12) whilst D » xDi by (11), we have also

where A is the time-integral of Y the line-integral of £| or voltage, e impressed eleotrio force, P eleotric potential or poten- tial difference.

. The permittivi^ e is the same in both systems. So is the ratio o/c^ of the permittivity to that of ether, or the specific inductive capacity (electric). The rational permittance of a unit cube condenser is e, and the irrational is cjiv. If S is

the permittance of any condenser

S««2S<, (14)

OUTLINI OV BLBOTROMAONBnO OONNXCTIONB. 12&

and its energy is

JVQ-iV^-JSV»-iS.V/, . . . (16)

if Q is its charge and V its voltage. We also have

iED-iE<D|-}eES-ieE«V4'r,. . . (16)

iPp^iPiP. (17)

}AC-iA<q|, (18)

the first and second sets relfiting to electric energy, the third

to magnetic energy. By Ohm's law

V«RC, and V<-RC,; . . (19) therefore by (11) and (13)

where r is resistivity, so that

EC-E,C4-RC«-R«C<« . . . (21)

Bational ▼. Irzational Magnetic Poles.

§ 96. If the repulsion between two magnetic poles m and m is F at distance r, and this is also the repulsion between irra- tional poles and m^, we have

where /x is the inductivity of the medium, common to the twa systems, as is likewise the ratio of /a to that of the ether, or the permeability. So

m = xmt, •••••• (23)

Here some disorepanoies oome in. For

B = /xH, and B< = fiH<; . . (24)

«o we have .-^-Hi-|l-|-^ . . . (25)

where H is magnetic potential or gaussage. If I be intrinsic magnetisation (intensity of), then

Imfih, and I<-fiV«^ • • • (26)

126 ELBOTROXAGNVnO THBOBT. CH. IL

from which, and by (25), we have

l^xl^. (27)

The equation B ^-fiH, in the sense used by me, expands to

B^li(h + T), (28)

where li is intrinsic, and F is the magnetic force of the field. Also

fi-^(l + K)-^(l + 4ir#c,), . . . (29)

so we have, by (28) and (26),

B-I+f*oP+f«o»cP, (30)

where (/.qkY is the induced magnetisation. U L is inductance, ihC^^mOi^, whence a:3»^<»^if

M is mutual inductance. In the conunon equation,

B|»F<+4irI| (31),

the intrinsic and induced magnetisations are lumped together for one thing, and it is assumed that /Aq = 1. The quantity hc is thus essentially a numeric, whilst /x is only a numeric by assumption. But whilst /x = /^i, we have k^x^k^. Although magnetisation, whether intrinsic or induced, are of the same kind as induction, yet the reckoning is discrepant. Compare (27) with (25) for B and B^. In (27) also, I may be either

intrinsic or induced, so far as the ratio x goes. The common equation, div|Dj = 4;r/}^ becomes

div|D-p, (32)

and the characteristic equation of Poissou becomes

V«P--/^1^, (33)

the 4^ going out by the rationalisation. But P itself is given

by P«2/o/4imj. (34)

The dehnition of current density,

curl H<«4irO^

becomes . curl H = 0.

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OUTUNS OF XLSCIROIUOKBTIO CONNBCTIONS. 127

•Other ohangee readily follow. Bat now that I have ezplidtly stated the relation between my rational formulse and tb^ "(ffdinary, I leave the irrationalB — ^for good, I hope — and retnm to the rational and amplified fonmulse, which ace so much «nperior.

APPENDIX.

THE ROTATIONAL ETHER IN ITS APPLICATION TO

ELEGTEOMAGNETISM.

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