Electromagnetic Theory, Vol. 1 (1893) — 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. General 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 J0 and g0, not upon e0 and h0. 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 electromagnetic waves.

Let there be any impressed forcive e0 in a stationary medium. Its activity is e0J per unit volume, where J is the electric cur- rent, and the total activity is 2e0J, where the 2 indicates sum- mation through all space, or at any rate so far as to include every place where e0 exists. Its equivalent is Q0, the total rate of waste of energy, and the rate of increase of the total stored energies, say U0 and T0. Thus,

T0 (34).

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. Ill

Now, the value of the summation is zero if e0 has no curl, or is irrotational — that is, an irrotational forcive does no work upon a circuital 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 e0 has no curl ; hence 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 2e0J necessarily vanishes under the circum- stance stated of e0 being irrotational.

Now return to equation (34), and suppose that the initial state of things is absence of E and H everywhere, so that Q0, U0 and T0 are all zero. Next start any irrotational e0, and see what will happen. The left side of (34) being zero, the right side must also be and continue zero. But Q0, U0 and T0 when not zero are essentially positive. Now if (U0 + T0) becomes positive, Qo should become negative, in order to keep the right number of (34) at zero. This negativity of Q0 being impos- sible, these quantities Q0, U0 and T0 must all remain zero. Con- sequently E and H must remain zero. That is, an irrotational forcive can produce 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 space, provided it be restricted to be of the irrotational type, so 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 e0, and similarly G with h0), and the positivity of energy ; and, more strikingly, the independence

112

ELECTROMAGNETIC THEORY.

CH. II.

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 bounding circuit, where there is a circuital distribution of g0 the curl of e0, of strength V.

Now, our present proposition asserts that the disturbances due to V over the surface S are in every respect the same as 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

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 113

the other passes through a conductor ; or that one surface is 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 St 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. S2 is another surface of impressed force, also reaching into the conductor. Now all these forcives (each 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 expressed in terms of E and H. The distribution of energy, for example, and the stresses, 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. This being

W = V(E-e0)(H-h0),

where we deduct the intrinsic forces to obtain the forces effec- tive in transferring energy, we see that every change made in the distribution of the intrinsic sources affects the flux of energy, in spite of the independence of E 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 itself.

Production of Steady State due to Impressed Forcive by crossing of Electromagnetic Waves. Example of a Cir- 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 e0 and h0 altogether, and regard the circuital vectors g0 and j0 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

i

114

ELECTROMAGNETIC THEORY.

CII. II.

infinitely rapidly there might be some difficulty in recognising the property, because 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 c and /x- in the medium ; and conduction does not alter this property, although by its attenuating and distorting effects it may profoundly alter the nature of the resultant phenomena. 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 from the real sources of disturbance. Thus, let the source be

FIG. 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 vt 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- creases to JAB overlapping commences, 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 — tha* «e, within the shaded region the magnetic

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 115

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 superimposition 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 difference between sources of energy and sources of disturbance. The source of energy only works when there is electric current at the spot, and this comes to it from the vortex lines (the circuital sources of dis- turbance). Thus, in the last figure, at time vt, when the ring of disturbance is of thickness 2vt, 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 Aac and B6c?, but is inoperative in cd, having done its work. Similarly when the sheet of impressed force has any other shape. The impressed force only works when it is allowed to work by the electromagnetic wave reaching it.

To emphasize the matter, take another case. Let the impressed force in any telegraph circuit be confined to a single plane 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 the 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 circuit, provided only that it is bounded by the original vortex line, in touch with the conductor.

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 ELECTROMAGNETIC THEORY. CH. II.

All impressed voltages and gaussages are more or less difficult to understand. But there need be no doubt as to the general truth of the property if the circuitality of the current can be trusted.

The Eruption of "47r"s.

§ 90. It may have been observed that the equations employed by me in the preceding differ from those in use in all mathe- matical treatises on the subject in a certain respect (amongst others), inasmuch as the constant 47r, which is usually so obtrusively prominent, has been conspicuous by its absence. This constant 4?r 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-Maxwellian 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

(1)

and, since this was proved by mathematics, it seemed that the- 4?r 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 4:r was purely conventional ; it was- 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 4?r 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 cosine* of the normal to an element of the surface. It was funny — very funny. How ever the 4?r managed to find its way in was- the puzzle in these and similar results ; for instance, in the well-known

, ...... (2)

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

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 117

mystery was carefully covered up by the mathematics. It was as hard to understand as the monarch found it to explain the presence of the apple in the dumpling — how did the goodwife manage to get it in ? Nor was the matter rendered plainer by Maxwell's great treatise. Maxwell thought his theory of electric displacement explained the meaning of the 47r (in the corresponding electric theorem), as if it were a matter of physics, instead, as is the fact, of irrationally chosen units.

As the present chapter has been mainly devoted to a general outline of electromagnetic theory expressed in formulae involv- ing rational units, it will be fitting, in bringing it to a conclu- sion, to explain here the relation these rational units bear to the ordinary irrational units.

The Origin and Spread of the Eruption.

§ 91. The origin of the 4?r absurdity lay in the wisdom of our ancestors, — literally. The inverse square law being recog- nised, say that two charges ql and q2 repelled one another with a force varying inversely as the square of the distance between them; thus,

...... (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 ? Our ancestors could not see into the future, — that is to say, beyond their noses, — 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 : —

118 ELECTROMAGNETIC THEORY. CII. II.

" The strength of the pole 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 mv must be proportional to the product mmr The force f 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, unless an absurd and useless coefficient be introduced,

f=mmJD*.» ..... ..... (4)

Observe the words which I have italicised. When it is con- sidered what Maxwell had then done in the way of framing a broad theory of electromagnetism, it is marvellous that he should have written in that way. By mere force of habit one might, indeed, not consider there to be anything anomalous about the 4?r in Coulomb's and Gauss's theorems. But did not Maxwell's electrostatic energy K^2/8?r and his magnetic energy pffi/Sir 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 framing a permanent system of practical units, which was what Maxwell and his colleagues were about ? 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 his theory of electric displacement, we are told that " the theory completely accounts for the theorem of Art. 77, that the total induction through a closed surface is equal to the total quantity of electricity within the surface multiplied by 4?r. For what we have called the induction through the surface is simply the electric displacement multiplied by 47r, and the total displace- ment outward is necessarily equal to the total charge within the surface." That is, his theory of electric displacement accounts for the 47r. So it seems to do; and yet the 4?r 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 all theories of electricity. It depends upon something much more fundamental.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 119

The Cure of the Disease by Proper Measure of the Strength of Sources.

§92. When looked into carefully, the question is simply 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 absorb 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

...... (5)

if C be the heat flux (per unit area) at distance r from the source of strength S. If we knock out the 4?r 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 regarded as a source, we should naturally measure its strength in a similar manner, viz., by the current in the pipe, or by 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- matically analogous cases. We find, for instance, that electro- static 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 surface 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.

120 ELECTROMAGNETIC THEORY. CH. II.

Using, temporarily, the language of lines of force, or of tubes of force (which, however, 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 circulation of the magnetic force is proportional to the current through the circuit ; and the natural pleasure of the strength of the current is the circuitation itself, without, as usual, dividing by 4n-. Now this division by 4?r 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 4r 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 = ?/47rr2, B = m/47rr2, . . . (6)

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

. . (7)

if q/c and m/p 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/27rr ...... (8)

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

Obnoxious Effects of the Eruption.

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

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 121

changing from irrational to rational units is to introduce 4?r. 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 4:r in the formulae 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 formulae where 4?r should be are principally scholastic formulae and little used; the many formulae where it is forced out are, on the contrary, useful formulae of actual practice, and of the practice of theory. A practical theorist would knock them out merely from the trouble they give, let alone the desire to see things in their right places. Furthermore, it should be remarked that the irrationality of the formulae is a great impediment in the way of a clear understanding of electromagnetic theory. The interpretation of equation (2) above, for instance, or of the similar well-known equation

$ = £) + 47r|ji,

presents some difficulty even to a student of ability, unless he be given beforehand a hint or two to assist him. For if K is a rational physical quantity, then u. cannot be ; or if /A is right, then K must be wrong. Or ^, ^, and ^L must be incongruous. The 4?r is also particularly inconvenient in descriptive matter relating to tubes of force or flux, and in everything connected with them.

This difficulty in the way of understanding the inner mean- ing of theory is still further increased by the 4n- not entering into the magnetic formulae in the same way as into the electric. Thus, it is jp and ^/4w which are analogues. Again, in many of the irrational formulae the irrationality appears to disappear. For instance, in J^/o, in J31C/, in J($5» n ^®j an(i in some others. This is because both the factors are irrational, and the two irrationalities cancel. It looks as if J) were the flux belonging to ($, but it is not. In reality, we have, if x = (47r),

therefore, ED =

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

122 ELECTROMAGNETIC THEORY. CH. II.

A Plea for the Removal of the Eruption by the Radical Cure.

§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 they are to be enforced, it is to b& 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, acre, pole, horse-power, etc., etc., has been repeatedly lamented by would-be reformers, who would introduce the common-sense decimal 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 measures with its absurd 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 ! 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, etc., are now legalised, so that, as I am informed, it is too late to alter them. This is a non seq., 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 physical laboratory determinations will be needed. The value of TT has been calculated to- hundreds of places of decimals ; so that rational ohms, volts, etc., can be now fixed with the same degree of accuracy as the irrational ones, by any calculator.

When I first brought up this matter in The Electrician ia 1882-3, explaining the origin of the 4?r absurdity and its cure, I did not go further than to use rational units in explaining the

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 123

theory of potentials, scalar and vector, and similar matters; then returning to irrational units in order to preserve harmony with the formulae 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 ignorance and want of interest in the subject. But much has happened since then. The spread of electrical know- ledge has been immense, concurrent 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 preceding, wholly avoided the irrational units ab initio ; and shall continue 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 formulae, 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 converting 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 formula) are used, then, in addition, you must first insert the constant 4?r 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 " 4?r 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 formulae in the two systems. Let us denote quantities in

124 ELECTROMAGNETIC THEORY. CH. II.

rational units as in the preceding (except that when vector rela- tions are not in question we need not employ special type), and the corresponding quantities in irrational units by the same symbols with the suffix f; thus, q and &. Also denote (47r)J by x. Let qi be the irrational charge which repels an equal irrational charge at distance r with the same force F as a rational charge q repels an equal clyirge at the same distance.1 Then,

.... (9)

Therefore

2 = *% ....... (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 formulae. By (10) we shall have

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

Since D = cE, and D^cEJiTr, .... (12) whilst D = #Di by (11), we have also

_E< A, V, et P, x13x

-fi-A-V-7-F

where A is the time-integral of E, V the line-integral of E, or voltage, e impressed electric force, P electric potential or poten- tial difference.

The permittivity c is the same in both systems. So is the ratio c/c0 of the permittivity to that of ether, or the specific inductive capacity (electric). The rational permittance of a unit cube condenser is c, and the irrational is cjkir. If S is the permittance of any condenser

S-^S* ...... (14)

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 125

and its energy is

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

JED = JEiDi = JcE2 = icEt2/47r, . . . (16)

(17) (18)

the first and second sets relating to electric energy, the third to magnetic energy. By Ohm's law

V = RC, and V« = RC,; . . (19)

therefore by (11) and (13)

*2=l=-^> ..... (20) where r is resistivity, so that

. . (21)

Rational v. Irrational 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 imu- tional poles mi and mi5 we have

(22)

where /x. is the inductivity of the medium, common to the two systems, as is likewise /A//AO the ratio of p. to that of the ether, or the permeability. So

m = xmi ....... (23)

Here some discrepancies come in. For

B = /*H, and 3, = ^; . . (24)

, Tfl Hi Bv l2v lit /ner\

so we have *=_ = _! = _!= £- ^ . . . (25)

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

and l^phjtf, . . . (26)

126 ELECTROMAGNETIC THEORY. CH. II.

from which, and by (25), we have

I = *Ii , - - (27)

The equation B = /*H, in the sense used by me, expands to B = /x(h + F), (28)

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

^ = ^0(1+K) = ^(1+4™,), . . . (29)

.€0 we have, by (28) and (26),

B = I + fi0F + /v<F, (30)

where /*0KF is the induced magnetisation.

If L is inductance, ^LC^L^2, whence »=£ = J^ if

L M

M is mutual inductance. In the common equation,

Bi = Fi + 47rIi (31),

the intrinsic and induced magnetisations are lumped together for one thing, and it is assumed that fi0 = l. The quantity K is thus essentially a numeric, whilst /* is only a numeric by assumption. But whilst p = i*>i, we have K = x*Ki. 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 Bf. In (27) also, I may be either intrinsic or induced, so far as the ratio x goes. The common equation, div "Di = p{, becomes

divD=/>, ...... (32)

and the characteristic equation of Poisson becomes

V2P=_/)/c, (33)

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

by P = 2/>/47iTC (34)

The definition of current density,

curl Hf = 471-0^, becomes, curlH = C (35)

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 127

Other changes readily follow. But now that I have explicitly stated the relation between my rational formulae and the ordinary, I leave the irrationals — for good, I hope — and return to the rational and simplified formulae, which are so much superior.

APPENDIX.

THE ROTATIONAL ETHER IN ITS APPLICATION TO ELECTROMAGNETISM.