Heaviside Electromagnetic Induction And Its Propagation Sec XXI 2 XXIV

SECTION XXI A NETWORK OF LINEAR DIELECTRIC CONDUCTORS, OR OF SHUNTED CONDENSERS.

Let any number of points be connected by linear conductors, thus forming a network of any degree of complexity. They will be referred to as Branches. Let each branch consist of any number of conductors in sequence, to be called the Shunts. Let every shunt have its ends joined to the poles of a condenser by wires whose resistances we do not count. This makes the combination complete. We have a linear combination of inductive branches exactly similar to the conductive; they are side by side, as it were, and in connection at certain points. We may regard the conductors as shunts to the condensers, or the other way, as we please, but the former plan is perhaps the best.

Instead of thus shunting the condensers by external conductors, we may do away with the shunts, giving instead equal conductances to the condensers themselves, thus making a combination of unshunted leaky condensers.

Or, we may abolish the external condensers, and give equal dielectric capacity (uniformly distributed in the wires) to the former shunts themselves, thus making a network of dielectric conductors. The theory is in all cases the same, with certain exceptions as regards the effects of impressed force and of magnetic induction. In many respects the theory is most simply expressed by having the two qualities, conductivity and dielectric capacity, coincident, as in the dielectric conductors, instead of side by side, as in the case of the shunted condensers.

If the shunted condensers were all disconnected from one another so as to form independent circuits, partly conductive and partly dielectric, any charges they might have would discharge through their shunts, each charge at its proper rate depending upon the time-constant of the condenser concerned, which is rp, if r be the resistance of the shunt andp the capacity of the condenser. (Inertia is ignored.) When they are in connection, as above described, the time-constants of the normal systems will all lie between the greatest and the least of the time- constants of the separate shunted condensers. If these be all equal, there is but one time-constant, viz., the common value of rp. In this case, if the condensers be charged in any manner and then be left in connection without impressed force anywhere in the system, the charges will at once readjust themselves to a new distribution, to be found by the two considerations that the E.M.F. in any circuit in the new state is zero, and that the charges that disappear form a system of circuital displacement in the combination. This new state will then subside uniformly everywhere, each condenser discharging through its own shunt. If an impressed force be introduced at the junction of two shunted condensers, say in an infinitely short wire joining one shunt to the next, it sets up the appropriate state of conduction current in the branches and of charge in the condensers instantly; these charges are equal in all the condensers in one branch, and in different branches are simply proportional to the currents in the branches. The same will be true if the impressed force be in a shunt, if there be an equal and similarly directed impressed force in the corresponding condenser. (In the case of the dielectric wires there is no need for this reservation.) But if the impressed force be in a shunt only, the charge of its condenser will be opposite to that of the others in the same branch, and there will be retardation according to the common time-constant. Thus, if the condensers be charged in any manner which could be produced by impressed forces in any of or all the branches (equally in shunts and condensers), and be left to themselves, the subsidence is instantaneous. There is only retardation in connection with those parts of the system of charges which could not be produced in the described manner. And, considering electromagnetic induction, if it operate equally on conductor and dielectric, as in the case of the dielectric wires, it will oidy affect the discharge of the parts that before subsided instantly, the subsidence being 110 longer immediate; whilst the other parts will subside just as before, independently of inertia; for as the conductive and dielectric currents are equal and opposite, there is no true current and no magnetic induction in connection therewith.

The above are conclusions from the general theory in the last section. As regards the proof of the limits between which the time-constants must lie, let E be the E.M.F. in any condenser, p its capacity, and k the conductance of its shunt. Then, in any normal system, if Q be the dissipativity and U the potential energy,

2 (k +pn)E2 = Q+2nU= 0,

the 2 to include all the shunted condensers. If n do not lie between the greatest and least values of - kjp, the summation cannot vanish, as it must; therefore every n does lie between these limits.

The differential equation of the combination, and the determinantal equation of the rates of subsidence, are most directly found by the method used in the paper on Induction in Cores, when treating of combinations of coils [p. 415]. The sum of the steps of potential in any circuit must be zero; get, then, the expression for the step of potential between any two points in terms of the currents, and we have one equation for every circuit. Eliminate the currents by their conditions of continuity, and the result is the differential equation, or the determinantal equation, according as we treat d/dt as the differentiating operator or as a constant.

In the present case, if T be the sum of the currents in a shunt and in its condenser, reckoned the same way in both (or the true current in the dielectric conductor), we have

T = (k +pn)E,

if ii stand for djdt. Every condenser has an equation of this form. Here E is the fall of potential through the shunt and through the condenser. Since T is the same along the whole of any one branch, the fall of potential between its ends is

r2(+^)-i,

the 2 to include all the condensers in the branch. Hence, if the combination consist of only one closed circuit,

2 (k +pn)~1 = 0,

when cleared of fractions, is the differential, or the determinantal equation, according as n is d/dt or algebraical. That is, we equate the (generalised) resistance of the circuit to zero. Thus, if there be three condensers, and y stand for k +pn, the equation is

(?/i)_1 + (?A>)_1 + (>h)~l = 0 ; or, yxy2 + y,y3 + y.x = 0.

The determinantal equation of m condensers in one circuit is of the (m - l)th degree ; one freedom is lost. The missing root is the negative infinity root of the instantaneous subsidence. To bring it to finiteness, put in the circuit a conductor without condenser. Then its y is its real conductance, say k0, and the equation is = 0.

This has the full number m of roots.

Let qv q.2, be the charges, and Ev E2, ..., be the E.M.F.S of the condensers (of capacity pv p2, whose shunts have conductances kv k2, ...,) going round the circuit in the + direction of F. A charge is + when the displacement is in the + direction. Then we have

Qi ~Pi^'i> (Z2 etc.,

or, Qi^P^Kh +Pxn), Q<2^P2Tl(h+P2n\ etc.

Therefore, in a normal system, the ratios of the charges are

q1; q2 ; ... = (k1jp1 + %) 1 : (k2/p2 + n) 1 : =pjyi :pjl/'i >

and thus we have m sets of ratios, by giving to n its m values in succession. To determine the absolute size of a particular normal system, use U12 = 0, taking pjyv ... as the normal functions. If

A be their common multiplier, we get

A = pDy-') + {?py-*)t

if l)2, etc., be the initial charges.

When, by shortening the wire kQ, we send the root depending upon its presence to - oo , the above ratios become 1 : I : 1 : ... in the normal system of this root. Taking, then, 1, 1, 1, etc., as the normal functions,

A = (2 Dp~l) + 2j9'1

gives the common charge of all the condensers that instantly disappears. It is the charge due to the initial E.M.F. in the circuit. Its disappearance makes the electric force polar. If all the time-constants pjk of the separate condensers are equal we have y = 0 repeated (m 1) times. The charges at time t are therefore

i = (-^i -^)coj = (^2"" ^)o> etc.,

where e0 = and A is given by the previous equation.

So far relating to a single closed circuit, the next simplest case is that of any number of branches uniting two points. Here the sum of the currents leaving either point is zero. If Tj, T2, ..., be the currents in the branches, all reckoned parallel, ^12, ..., the capacities in the first branch, p2V p22, ..., in the second, with a similar notation for the other quantities, we have

Tj = (kn/pn + n)qu = (^u/Pu w)^i2 =

■^2 = 2l/P21 **■ ~ (^22/^22 n^l22 = etc., and the sum of the Ps is zero. These give the determinantal equation

(2y1-i)-i + (Sya-i)-i + (2y8-i)-i+...=0I

each summation to include all the ys in one branch only. That is, the sum of the generalised conductances of the branches in parallel is zero.

The number of missing roots is one less than the number of branches. The full number, equal to the number of condensers, may be got by inserting condenserless conductors in all the branches except one; if in that one also, it makes no difference in their number, though altering their magnitude.

If there be no inserted condenserless conductors, it will be necessary to determine what part of each charge instantly disappears. We have to make the electric force polar, and therefore equalise the E.M.F.s in the different branches reckoned the same way between the two points, and do it by making equal changes in the initial charges in any one branch. The charges q at time t, after they were given Di are, if kj-p is the same for every condenser, given by

5ii = (^11 *^i)o ?2i ~ (^21 -^2)o> etc->

?I2 = (-^12 ^l)c0 ?22 = (-^22 ■^2)e0> etc.,

etc. etc.

where e0 is, as before, the time-function of the repeated root, and there is one A to be found in each branch. It will not be necessary to take up space by describing how they are got in this special case, or in writing them out, as the following method, applicable to any combination, will apply.

In any network of linear conductors there is a certain number of degrees of freedom, i.cthe number of branches in which the currents must be given in order that they may be known in all the rest. Thus, in the common Bridge, the currents in three branches being given, those in the rest follow.

(If in points be joined by hm(m - 1) conductive branches, the number of current-freedoms is \(m ~ l)(m - 2). This is (m-1) less than the number of branches.) This number of current-freedoms is just the number of the missing ( - oo ) roots in the determinantal equation when the branches have condensers connected along them as described at the beginning. As for the equation itself, if the characteristic function of the conductive combination be known, it may be got by turning every k into k +pn in it, and equating the result to zero.

(The characteristic function is of the degree (in - 1) in terms of the conductances (one less than the number of points); hence, when for k we put (k+pn), the determinantal equation is of the degree (m - 1) in n, so that the roots are fewer in number than the branches by the number of current-freedoms in the conductive network, if there be but one condenser in each branch, and fewer in number than the condensers by the number of current-freedoms in the conductive network if there be many condensers in each branch. If, on the other hand, there are no condensers, but we take account of the self-induction of every branch, we get the determinantal equation by turning k into (k~1 + iiii)~1, if s be the inductance of a branch. There will now be (m- 1) fewer roots than the number of branches.)

Now, suppose k/p is the same for every condenser, and we want to know how the initial charges subside. Let us number the branches 1, 2, 3, etc., and choose (arbitrarily) a certain direction in each for the positive direction in which the current, E.M.F., and charges (displacements) are reckoned. Let every capacity have two suffixes, the first to denote which branch is referred to, the second to show its position in the branch; ancl do the same with the charges and the conductances. The currents T only want one suffix, to show which branch is referred to. We have, then, in the case of the infinity roots, if Au, etc., are the charges that disappear,

Axx = AX2 = A1S = .. - = Av say,

^ o| // 22 ^ ^ ^ 9, SSL} j

etc.; that is, the same portion of the charge of every condenser in one branch disappears instantly. Besides that, Av A0, ..., in the different branches, are connected together by the same conditions of continuity as the currents in the different branches. That is, Av A2, ..., form a system of circuital displacement.

The solution is therefore of the same form as in the previous equations, and we have only to find one A for each branch to complete the solution. Initially, we have

Ai = ^\ + (Ai ^l) Ai=-^2"M-^2i -^2)* etc->

-^12 ~ ^ 1 (-^12 ^l) ■^22 = ^2 ("^22 -^2) etc.,

■^13 = A\ -t* (^13 A-y)j D^ = A2 + (-^23 A^), etc., etc. etc.

The system of the As is circuital. That of the (D-A)s is such that its electric force is polar. The mutual energy of the latter and any circuital displacement is therefore zero. The mutual energy of the Ds and any circuital displacement is therefore equal to that of the As and the same. Let this any circuital displacement be the charges set up in the system by a unit impressed force in any branch (equally in shunt and condenser). For instance, let <Zn, dvp dV6, etc., be the charges of the condensers in branches 1, 2, 3, etc., due to unit impressed force in branch 1. Then the mutual energy of the If s and ds equal that of the As and ds. But the latter equals twice the product of the impressed force of the ds into the displacement of the As; or, since the impressed force is unity and is in the branch 1 only, it equals 2Al itself. Hence Ax is one half the mutual energy of the Ds and ds. Or,

Ai = DJPY + dl2^ D2fp2 + D,Jps + ...,

where the first 2 relates to branch 1, the second to branch 2, and so on.

Similarly, if diV <Z22, d.ni ..., are the charges in 1, 2, 3, ..., due to unit impressed force in 2, we have

A2 = d21S D1/pl + d22~ D2/p2 + d23Z D2s/p3 + ....

Thus the As are known in terms of initial charges and of the ds. The latter may be found in precisely the same manner as the current in the branches due to the unit impressed forces. In fact, instead of mutual energy, we may employ the idea of mutual dissipativity and activity. Let yn, y19, y13, etc., be the currents in 1, 2, 3, ... due to unit impressed force in 1. Then

^ i ~ Tii*-' 'f' 7i2^ ^2/^2 yis- * *

is an alternative form of Av In the reasoning we .should now imagine

the Ds to be currents (not closed), and the As closed currents, and speak of mutual dissipativity or of activity. The matter is therefore reduced to the problem of finding how current due to impressed force in any branch divides through the conductive system.

(When the time-constants of the condensers (shunted) are not equal, the charges that are left after the first readjustment require to be decomposed into their proper normal systems, to be done by the lT12 = 0 property. This does not present anything unusual.)

The following is the tridimensional representative of the above method of finding the As. Referring now everything to the unit volume, let D0 be the initial displacement, and p the capacity. Divide D0 into two parts, of which one is circuital, whilst the electric force of the other is polar. That is, let A be the circuital displacement, so that

divA = 0, curl (D0 - A)Jp = 0 ;

find A. The mutual energy of D0 and any circuital displacement equals that of A and the same, because the force (D0- A)/p is polar. Let the any circuital displacement be that due to unit e at any point, and call it d.

Then 2 dJ)^ = 2 ^ = 2 (e + f if f is the polar force of e,

= 2 eA = tensor of A at the place of e, if e be parallel to A.

Thus we know the distribution of A as soon as we know the displacement due to impressed force.

SECTION. XXII. THE MECHANICAL FORCES AND STRESSES. PRELIMINARY. THE

SIMPLE MAXWELLIAN STRESS.

As this is not a treatise upon the theory of Elasticity, it will be only necessary to say so much on the subject of stresses in general as will serve to introduce us to the principal formulae connecting stresses with the corresponding mechanical forces, which we may find useful hereafter. This can be done very briefly.

A simple stress is either a tension or a pressure acting in a certain line. It implies the existence of mutual force between contiguous parts of the substance in which it resides, and of a corresponding state of strain, with storage of energy in the potential form, i.e., depending upon configuration, though perhaps ultimately resolvable into kinetic energy. Thus if we fasten a cord to a beam, and hang a weight to its free end, the cord is slightly stretched, the work done by gravity during the stretching is somehow stored in the altered configuration, and the cord is put into a state of tension. At its lower end the tension in the cord is equal to the weight attached, at its upper end to the same plus the weight of the cord. The state of strain of course extends to the beam, and to the beams attachments, and so round to the earth, to which we ascribe the gravitational force, which is somehow stressed across the air to the weight, and from the weight to the earth.

If the stretched cord be in motion in its own line, as when a horse tugs a barge along a canal, there is, besides the transfer of energy through space by the onward motion of the horse, rope, barge, and dragged water, carrying their kinetic energy with them, a transfer of energy through the rope from the horse to the barge, and through the strained barge to the water, where it is wasted in friction. The rate of transfer per second equals the product of the tension of the rope into its speed, and the direction of transfer is against the direction of motion. If motion be transmitted from one machine to another by means of a horizontal endless band, the transfer of energy is through the stretched half of the band, and is again proportional to its speed (and against its motion), and to the difference of tensions of the two sections; a uniform tension meaning continuously stored potential energy.

A pressure is a negative tension. If the tension or pressure in a cord or rod be not uniform in amount across every section, we see at once that any small piece of the cord is pulled in opposite directions by forces of different amounts. Their difference is the mechanical force on the small piece considered, and measures its rate of acceleration of momentum. Thus, if P be the tension across a section at distance x from one end, dPjdx is the mechanical force at that place, per unit length of cord, acting in the direction of x positive. Unless otherwise balanced, it increases the momentum of the cord. Thus - dP/dx is the force that must be applied to keep the stress-difference from working. Examples:(1). A vertically hanging cord; unequal tension; applied force gravity. (2). In the endless horizontal band moving with uniform speed, there is no force resulting from its tension except where it changes in intensity, for example where it passes over pulleys. At a pulley where the band gives out energy, the force and velocity product is positive, and where it receives energy, negative. These forces of the stress-variation are the negatives of the forces the pulleys exert on the band.

The most general stress considered in the common theory of elasticity consists of three simple stresses (pressures or tensions) acting in three lines at right angles to one another in a substance. When, as is necessary in general, the axes of reference are not the lines of action of the mutually perpendicular simple stresses, the following notation is the most convenient. Although a tension or a pressure is not a vector in the usual sense, since it, although acting in a certain line, acts both ways, yet we may consider only one side of a stress at a time, and so represent the stress on any plane by a vector. On this understanding, let P1, P2, P3, be the vector stresses per unit area on planes whose normals are x, y, z respectively. These are the forces exerted by the matter on the positive side on that on the negative side of the three planes, and, being forces, are vectors. Let the scalar components of PL be Pn, P12, P13, etc.; and i, j, k be unit vectors parallel to x, y, z. Then

(1*0

(The first of a double suffix fixes the plane, and the second the direction of the force.) Here there are nine components in a general stress. But examination of the force on a unit cube arising from this stress shows at-once that the transverse stresses must be equal in pairs, P12 = P21, etc., if the force is to be purely translational, thus reducing the number to six. Then, the translational force due to the stress is

i div PT + j div P2 + k div P3; (2a)

i.e., the ^-component is div P15 etc.

Should, however, we admit the possibility of nine coefficients (as we may do, at least on paper, in some kinds of magnetic and electric stresses), the re-component of the translational force is not the divergence of Pj, but of its conjugate ; thus

^-component is not = dPn/dx + dPl2/dy + dP13Jdz,

but is = dPu/dx + dP2l/dy + dPsl/dz ; (3a)

a distinction which disappears when P12 = P2V etc. Besides this, there is rotational force arising from the stress, whose vector moment per unit volume is [the torque per unit volume]

i(^23 - P3,) + j(^3, - *) + k(A* - **). )

which also vanishes when P21 = P12, etc. Should this be the case, the negative of (2a) is the applied force required for equilibrium. If not, then the negative of (3a) is the ^-component of the applied force required to balance the translational force, and the negative of (4a) is required to balance the torque.

There are three applications of this theory of stress. The first is in the dynamical theory of elastic bodies; the second is, after Faraday and Maxwell, in the explanation of forces of unknown origin by means of stress in a medium; and the third application consists in the use of the stresses, not for explanation, but for purposes of investigation.

Thus, as from a given state of stress we derive the corresponding mechanical forces by differentiations, so we may obtain a state of stress that will produce a given distribution of force of any origin by integrations. The former is an exact process; the latter is to a certain extent indefinite; for we may clearly add to the state of stress that gives rise to certain forces any state of stress that gives rise to no forces. We should naturally choose the simplest forms that present themselves, unless there should be reasons against this. We have a choice of formulae for yet another reason, viz., when it is not the exact distribution of force that is known, but only its resultant effect on a solid body, of which examples will occur later. We need not bind ourselves to the hypothesis that a certain state of stress really exists in a certain case, but merely use the stress-vectors as auxiliary functions to assist the reasoning, if the investigations should be assisted thereby.

The gravitational application made by Maxwell requires a pressure along a line of gravitational force combined with an equal tension in all directions perpendicular to it. But the intensity of the stress is something stupendous, being at the earths surface

1 / 614 x 1023 \2 imo ,

= 8ir\3928.(637 x 10)v =5 * 10 dy"eS Per ** Cra

or 3770 tons per square inch. (Maxwell made it ten times as much, so

I give the above figures, in which I do not see any error; the unit mass is that of 3928 grams, 614 x 1026 that of the earth, and 637 x 106 cm. its radius.) In the earth, on the supposition of uniform density, it would be proportional to the square of the distance from the centre. But a very severe mental tension is caused by an endeavour to imagine this stress to really exist. Yet the action of gravitation must be transmitted somehow.

First Electromagnetic Application.

No objection on the score of enormously great stresses being required applies when the electric and magnetic mechanical forces are in question. In fact the method seems peculiarly fitted for their explanation. The cause of this would seem to be twofold. Gravitational matter is all attractive, and is collected in great lumps. The electric and magnetic matters, on the other hand, are comparatively superficial affairs, and are always in equal amounts of opposite kinds. If, as some suppose, the earth is full of electricity, it might as well not be there, for all the good it does.

As a first simple application, let us confine ourselves to a portion of space in which there are no impressed electric or magnetic forces, and the dielectric capacity, the conductivity, and the magnetic permeability are all constants, i.e., a homogeneous isotropic medium. There are three mechanical forces to be accounted for by a state of stress. (1) . The mechanical force on electrification. This is, per unit volume,

E/j = E div D = //E div E, (5a)

if E and D are the electric force and displacement, and p^cjiir the condenser capacity per unit volume. It acts parallel to E, which is the force per unit density. (2) . The mechanical force called by Maxwell the Electromagnetic force. This is,

VrB = Y(lE+pE)^H, (6)

k being the conductivity, // the permeability, H and B the magnetic force and induction, and T the true current, the sum of the conduction current and that of elastic displacement. It is perpendicular to both the current and the induction, and is in strength equal to the product of their tensors into the sine of the angle between their directions. Its existence in a dielectric is speculative, but it is difficult to do without it.

(3) . A mechanical force that we may call the Magnetoelectric Force. It is

4irVDG = VDB = jt;/xYEH, (7a)

H.F..P.VOL. I. 2M

where G is the magnetic current, or the time-variation of the magnetic induction -f Air. Its existence anywhere is speculative, but it is absolutely needed as a companion to the last. It is perpendicular to the electric displacement and to the magnetic current. If v be the velocity, the activity of this force is

4ttvVDG = 47rGVvD ; (8a)

hence 47rVvD is the magnetic force of induction, due to the motion, the second form of (8a) expressing its activity. The existence of this magnetic force due to motion in an electric field was concluded before by general reasoning. [See p. 446. Notice that the impressed force required to balance the magnetoelectric force is the negative of (7a); so that the motional magnetic force, regarded as impressed, is the negative of (8a). Similarly as regards the electromagnetic force and the motional electric force, p. 448. Some worked out examples will follow.]

The magnetoelectric force can only exist in transient states. The electromagnetic force exists in steady states as well, but then there must be dissipation of energy going on. The force on electrification is independent of whether the state is steady or transient.

The forces (1) and (3) are explained by a simple Maxwellian stress, electric ; whilst (2) is explained by a similar magnetic stress. A simple Maxwellian stress consists of a tension along a certain line combined with an equal lateral pressure. Let Ul and 1\ be the electric and magnetic energies per unit volume, or A/>E- and i/jtH-/47r. Then is the intensity of the electric stress, and Tx that of the magnetic stress. The tension is parallel to the electric force in the one case, and to the magnetic force in the other, the pressures being perpendicular to their directions.

The electric stress on any plane, defined by its unit vector normal N, is

(EN)D - Z/jN, (9a)

that is, a force parallel to D of intensity EN x tensor of D, combined with a normal pressure of intensity Uv Similarly the magnetic stress on the plane is

(HNJB^tt-T-jN, (10a)

i.e., a force parallel to B of intensity HN x tensor of B/4tt, combined with a normal pressure of intensity 1\. By taking N = i, j, k, in succession, we may obtain the corresponding three stress vectors on their planes. But the simple Maxwellian stress is fully defined by the single expression (9a) or (10a), according as it is electric or magnetic, N being in any direction we please.

To prove that these stresses give the required forces, it is sufficient to differentiate them. The divergence of the N-plane stress-vector is the N-component of the mechanical force due to the stress. Thus the divergence of (9a) gives the N-component of the forces (1) and (3), whilst that of (10<r) gives the force (2). As the transformations will occur later in a more general manner, space will not be occupied by them here. Whether these stresses be realities or not (physically), there can be no doubt as to their appro]triateness.

The medium is in equilibrium in all places where there is 110 electrification, or electric or magnetic current. Thus, in the region outside a wire supporting a steady conduction current, the strcss-vectors have no divergence, and there is no mechanical force arising from the stresses. On the plane containing both the electric and magnetic forces, the stress is a normal pressure, of intensity Ux + Tx, acting in the line of transfer of energy. If, further, the electric and magnetic forces are perpendicular, as when the circuit lies in one plane, we have a tension - 'J\ in the line of the electric, and a tension 7\ - Ux in the line of the magnetic force. Lastly, if also Ux and 2\ are equal, we have left only the previously mentioned simple pressure.

Generally, let the normal N to any plane make an angle. 0 with E and with H; then the force 011 the N-plane is compounded of a normal tension

Ux cos 20 + 7\ cos 2(f),

and two tangential forces

sin 20, and Tx sin 2</>,

the first being in the plane of E and N, the second in that of H and N.

There is another important case (not a steady state) in which the stress reduces to a pressure in one line, viz., in the propagation of a plane wave through a homogenous isotropic nonconducting medium. Let z be measured in the direction of propagation, x and y at right angles to z; then if E is parallel to x, H is parallel to y. If we look along z in the + direction, and E be + upward, H will be + to the right. They keep time together in all their variations of intensity at any place, and are of such relative magnitude that Ux and 'J\ are equal. Thus,

E = fxrYKN, H = c<-VNE, ^ = (Ua)

if v be the speed of the wave and N a unit vector parallel to z. Or if Et) and I[n are the tensors (magnitudes, apart from direction) of E and H,

E0 = MIW //,, = crE{).

Here, E and H being perpendicular and such that Ux = Tv the stress is a simple pressure P=2UX = 2 Tx in the line of z. The only mechanical force arising therefrom is one parallel to z, due to the variation of 1 along z. This force is the sum of the electromagnetic and magneto- electric forces, which are equal, and parallel to z, each represented by

- l<lP/d.~, per unit volume.

Since the medium is not in equilibrium under the stress I\ there is translatory motion in the line of z. This requires the medium to be compressible. Thus a wave of compression travels with the electromagnetic wave. The compression is, however, only an effect of, not the electromagnetic disturbance itself. Thus, in the case of a simple harmonic wave, there is a translatory to-and-fro motion ot the parts of the medium in the line of propagation, accompanying the wave ; having double its frequency, as there are two maxima of pressure in a wavelength. Assuming a light-ray to be an electromagnetic wave of this kind, and taking the amplitude of H to be 02 c.g.s. in strong sunlight, requiring the amplitude of E to be 6 x 10s, or 6 volts per cm., with an electric current-density of 240 c.g.s., the maximum translational force,

- dPjch parallel to z, is about 5 dynes per cubic cm. It is here supposed that the disturbance is simple-harmonic, and that vkj-n- = 4 x 105, if k is the wave-length. The translational momentum

parallel to z is, in general, 2IIJV+ a constant independent of the time.

This motion of the medium parallel to z, not to be confounded with the internal motions of the disturbance, must react upon the electromagnetic wave. For, if v1 be the z-velocity (vector), the electric force induced by the motion is /xYHv1, and the magnetic force induced by the motion is cVVjE; to a first approximation E and H are altered to these extents. If we compare these expressions with (1 \a) above, we see that rN = vl makes the electric and magnetic forces induced by the motion equal to the original electric and magnetic forces. This result of a uniform speed of motion of the medium in the direction of propagation does not, however, mean more than the expression of the fact that if we travel with a wave, and at the same speed, the wave will appear stationary.

The size of vx depends upon the density of the medium, varying inversely with it. But vl is not likely to be anything but a very minute fraction of the velocity of propagation, and therefore negligible, unless we artificially increase the electromagnetic or magnetoelectric forces by passing a ray of light through a strong magnetic or electric field. Noticing that H = -02 is quite small, we can greatly multiply the electromagnetic force by sending a ray across the lines of force of a strong magnetic field, whilst keeping its direction the same (along the ray). If, on the other hand, we send a ray parallel to the lines of force of the field, there is transverse electromagnetic force, and transverse motion produced, far exceeding the original in amount. Under such circumstances we might expect the effect of vx to be not negligible. These remarks, it will be noticed, rest upon the existence of the supposed stress.

SECTION XXIII. THE MECHANICAL ACTION BETWEEN TWO REGIONS.

Summary of some results of Vector Analysis.

In order to keep the present section within limits, it will be desirable to first give a short summary of certain general relations, discussed at length in previous articles. Let there be a distribution throughout space of a vector H, to be mentally realised by drawing lines following its direction, packing them closely where H is strong, loosely where it is weak. H may be the intensity of electric or magnetic force, the mechanical force on the unit of the corresponding matter. The field of H is decomposable into two fields of very different natures, say, H = F + K, such that F is a polar force, its line-integral round any circuit being zero, and K has no divergence, or is circuital. From this property it follows that 2FK through all space is zero, making The divergence of H is that of F only; the curl of H is that of K only. Let

div H = 47rp, curl H = 47rr ■ (12a)

then p is the density of the matter55 of F, and T is the density of the current of K. (Say magnetic matter and electric current, or electric matter and negative magnetic current; but, so far as the present section is concerned, the matter and current are simply defined by (12a). Let P and A be the potentials at any point, Q, of the matter and the current, according to

P= 2/>//■, A = 21>f (13ft)

r being the distance from p or T to the point Q where P or A is reckoned. Then

F=-VP, K = curlA, (14a)

show the derivation of F and K from the corresponding potentials. Also

F = - 2 pf, K = - 2 yrf, (15ft)

if f be the vector force at the place of p or T, due to unit matter at Q, where F or K is reckoned. If we call the quantity 2 H-/^ the energy of H, the total energies of F and K are

2 = 2 hPp, 2 hK-jiTT = 2 |Ar (16ft)

The mechanical force per unit volume is

pK + YTK (17ft)

Limitation to a bounded region.

The above referring to all space, in order to apply the results to a bounded region we must suppose H = 0 outside it, for our temporary purpose, but without altering H within the region. This makes p and r zero in the outer region, keeps them unaltered in the inner region, and, owing to the sudden cessation of H at the boundary, introduces new matter and current there. Let o-j and y1 be the surface representatives of p and r. They are given by

N1H = 4ircr1, VN1H = 4iry1, (18ft)

corresponding to (12), being the unit normal from the boundary to the inner region. If, now, we include this surface matter and current in the former p and T, the results are applicable to the inner region outside which H has been abolished, or to all space if we like to keep the outer H zero. P must be the potential of p and o-j, and A the potential of T and yv

As H in the inner region is the same as before H in the outer region was abolished, it follows that the force in the inner region due to the surface matter and current is identically the same as that due to the abolished p and F in the outer region. And, as there is now no force in the outer region, it follows that the force in the outer region due to the surface matter and current is the negative of that due to the matter and current in the inner region. If, for example, there was originally no p and no T in the outer region, the force in the inner region due to the surface matter is the negative of that due to the surface current, so that together they produce no force in the inner region, whilst in the outer region their joint effect is the negative of that of the p and T in the inner region.

Similarly we may treat the outer region as self-contained, by making H = 0 in the inner region, and introducing surface matter and current given by

N2H = 4 wo-.,, YN,H = 4:ry2,

No being the unit normal from the boundary to the outer region. Since N., = - Nl} cr2 and y2 are the negatives of the former cr1 and yr The force in the inner region due to <r2 and y2 is the negative of that due to the p and T in the outer region, whilst in the outer region it is the same as that due to the abolished p and T in the inner region. It will be convenient to have a fixed way of reckoning the normal and the surface matter and current. Let the normal be always + from the boundary to the outer region, and be called N, the same as the former N2. Similarly call the surface matter and current cr and y, corresponding to N, being the former o\2 and y2, so that o-j = - cr, and yl ~ - y.

Internal and External Energies.

Let pv Fj, and p0, I1., be the matter and current densities in the inner and the outer regions, er and y the surface matter and current (as just defined), 1\, Av J\2, A2, and i'0, A0, the potentials of the inner, the outer, and the boundary matter and current, after (13a). Then the energies in the internal and the external regions are

^ i(At - A{))(l\ -y) + ? l{l\ - A)(/>i - -M (VJa)

ami 2(-A-j + A0)(F2 + 7) + a + o')(P-2 + (J)> I

respectively, by (lGfl). But also, by disregarding the boundary altogether, the total energy in both regions is

2 i(Ai + A2)(r1 + r2) + ^ },(I\ + zy(/>1+,g. (2Q)

So the sum of the two expressions in (1 Ua) equated to that in (^0</) gives us the necessary relation

+ P1P2) = - {y(A--> - Aj) + o-(ZJ2 - 1\) + A0y + l\p} ;

both members being expressions for the mutual energy of the matter and current in the two regions. If there is 110 p or T in the outer region, this is equivalent to

0 = - (Ay + ZV - A0y - Z, and the external and the internal energies are

^ },(Al)y+ /, and 1' },(AV+Pp) - ^ \(Aity + Z,

respectively. The first of these we may also write as -IPcr, if H = -VP, as it can be expressed; or else as y, if curfA = H, which is possible when 2cr^0.

Mechanical Force between the Regions.

There being matter and current in either or both regions, the resultant force 011 the inner region is

2 (PlK + "VTjH), (2 la)

the summation extending throughout the region. It is zero 011 any region in which there is no matter or current. Similarly

2(p2H + Vr,H) (22 a)

is the resultant force 011 the outer region, the summation extending through it. Since the summations in (21a) and (22a) together include all space, the one sum is the negative of the other. Or, as (21 a) is the resultant force of the outer on the inner region, and (22a) is that of the inner on the outer region, the one is the negative of the other. That is to say, action and reaction are equal and opposite; or, stress is mutual; or, a complete dynamical system cannot set itself moving, when taken as a whole. Both (21a) and (22a) are expressed by

2 i(erH + YyH), (23<0

taken over the boundary, using the + sign to express the force of the outer 011 the inner region, and the sign to express that of the inner region on the outer. Comparing (23rt) with (22a) and (2b/), we see that boumlaiy matter and current take the place of the matter and current in the inner or the outer region as the case may be. We may verify the equivalence of (23a) to the others by differentiation, applying the perennially useful and labour-saving Theorem of Convergence to either region with the common boundary; but the reason of (23ft) and its necessity may be more simply seen thus. When we abolish the external field, and put cr and - y on the boundary, we make the inner region, with the boundary matter and current, a complete system, on which there is 110 resultant force. The resultant force 011 the surface matter and current - a- and - y is therefore the negative of that on the internal p and V. But the surface matter and current are, so far as the mechanical force between the regions is concerned, equivalent to the external matter and current. Hence the resultant force on a- and y is the same as that of the p and T in the outer region 011 the p and I in the inner. Hence, in (2lit) we may put o- and y for pl and Tj. But then, since by this we turn the volume-integral into a surface-integral, we must take the mean value of H through the infinitely thin layer of the surface matter and current. This is iH, since H = 0 outside when the surface matter and current are taken instead of the external p2 and I",. Hence the presence of tlie J in (23a).

The vector in that expression, viz.,

i(<rH + VyH), (24//)

is the vector stress, according to the last section, regarding one side of it only, the force of the outer on the inner region per unit area of the boundary. Putting <r = NH/47T, y = YNH/47T, it takes the form

H(HN/4TT) - N(H78TT) (25a)

Hence it is a simple Maxwellian stress of intensity H2/87r.

It is not necessary, in reckoning the resultant mutual force between the regions, to take H in the formulae, i.e., to take the intensity of the force due to the matter and current in both regions. Thus, H1 being the force-intensity due to Pl and F15 and H2 that due to p.2 and T2, the resultant forces on the inner and outer regions are

SfoHs + Vr^) and 2 (p.Hj + Vrj^) (26a)

respectively. Comparing with (21 a) and (22a) we see that this is equivalent to saying that

0 = 2^ + VrjHj), 0 = 2 (p2H2 + Vr,H.,) (27fl)

Remember that in the inner region H0 is the same as the force- intensity due to a- and - y; whilst in the outer region Hx is the same as that due to +<x and + y. The (26a) expressions are equivalent to the boundary summations

2(o-H2 + VyH2) and - 2 (crH1 + YyHj); or, in terms of the surface Hj and H2 only, to

2 {H^NH*) + H^NHj) - N(H1H2)}/4tr (28a)

H and P in either region due to p and T in the other.

For distinctness, let there be no p or T in the outer region. H is then given by

H = - 2 (pi + YTf) = - X"Zpp + curl 2pT, (29a)

H being reckoned at a fixed point Q, and p and f being the potential and force-intensity at the place of p and F due to unit matter at Q. That is,

p = \/r, i=X.Jr\

if /■ be the distance from Q, to the place of P or F, and r2 a unit vector along r from Q. For P and F we may substitute the surface matter and current, when the point Q is in the outer region; thus,

H = - 2 (<rf + Yyf) = V2 per + curl 2jjy (30a)

Then, since <r and y depend only upon the boundary H, if H be given only over a closed surface, we know H through the whole external space, so far as it depends upon p and T within the surface, and therefore definitely throughout the external space when there is P and F only within the surface (or, in the extreme, upon it). As we change the form of the boundary, the distributions of the surface cr and y change. There may be matter only, viz., when H is normal everywhere, or the boundary is an equipotential surface (say due to an electrically charged conductor). There would be current only if H could be everywhere tangential, but this is not possible if H be magnetic force, at least without having current in the outer region. In general, we have both matter and current. Although the external field is definitely fixed by the surface H, we can get no information from it as to the internal field, except that ~cr = 2p, or the matter on the surface is the same in amount as that within it. If this be not zero, there is matter of amount - ~o- or - p on the surface at infinity, if we stop the extension of space. A closed current is equivalent to equal amounts of + and - matter; so is a magnet. In the second form of (30ft), the force H is derived from the scalar potential of o- and the vector potential of y. We can, however, derive it from a scalar potential only, thus:Since the surface H, say H0, is given, the surface potential can be found, except as regards a constant, by a line-integration on the surface from a fixed point, whose potential is taken as zero. Thus, P0 being the surface potential, (31)

H0ds being the scalar product of H0 and ds, the vector element of the line of integration. Then, if P is the potential at the external point Q, we shall have

P = ^(r-!'(fN)P0/47r,

and H = - Vi. The first ^ gives that part of the external potential due to the surface matter, <t = H0N/4 77. The second part is the scalar potential of the current; that is, if it be electric current, it is its magnetic potential. Or, it is the external magnetic potential of a closed magnetic shell, normally magnetised to strength Pc,4ir. For, if I be the vector magnetic moment of a small magnet, its potential at Q is

- If; in the present case I = NP0/47r, and Pc/4-n- is the moment per unit area. It is the same as Maxwells current-function of a current- sheet. Test (32a) by the Convergence Theorem. The indeterminateness of P0 as per (31) as regards a constant does not affect (32(/), the external potential of a closed magnetic shell of uniform strength being zero. The external P in terms of the Surf are P0.

Let it be the surface P0 that is given, not H0 the surface force. A part of P is known, viz., the second term on the right of (32a). But the first term being in terms of the force, through o-, must be got rid of. Let x denote the operator that finds the external potential of the magnetic shell; that is, = - - (fN)/4?r operating upon P0, finds xP0 the external potential of the shell of strength P0 47r; this potential rP0 is not the same as P. Let x^P0 denote the surface value of xP0, not the same as P0. Then we may denote by x0~1 the operator which, acting upon P0, finds xQ~lP0, the strength of shell whose potential is P0 at the surface. Then, at an external point,

Then we have

I\) + (1 _■'0)^0 + (1 _;>o)2Pn + !> (33r/)

giving the external P in terms of the surface P0 by direct operations, when for x we put its rational equivalent - 2(fN)/47r.

The first approximation is xPn. The next is x(P0 - X^PQ), and so un. It is a process of exhaustion. But it( only works when 2^ = 0. (33*/) therefore solves the problem of finding the potential throughout a region bounded by a closed surface (taking N as the inward normal) in terms of the surface potential, the p and T being on the other side. And, when p and T are inside, it finds the external potential if = 0.

Annihilation of the Surface Current.

To get rid of y and substitute matter giving the same external potential. Thus, given H0. The first of (30a) gives H in terms of H() through tr and y. The first distribution of matter is o- = NH0/47t, and the force due to it is 2f(NH0)/4-. Call this ^Hu, and let its surface value be //0H0. Then

H = U!k ~1H0 = !J{ HQ + (1 _ //o)H0 + (1 - ?/o)JH0 +-..}> (34 a)

in direct operations.

Or thus, H0 gives a first cr and y. Find the field due to the first y. It has a normal and a tangential component, and therefore gives a second cr and y. Find the field due to the second y, which gives a third cr and y; and so on. As we proceed, the y left gets smaller and smaller, and is finalty annihilated, leaving a distribution of matter, the sum of all the crs, say cr(>. Then H = - 2 ftr0, where cr0 is given by

<r0 = N (1 --1 H() 47r, w here z = 2 fVN / 47r,

01, tr0 = N(H0 + :H0 + -2H0 + *..) 4?r (35j)

Annihilation nf the Surface Matter, when jiossihle.

Starting with F0, it gives a first cr and y, and the H due to the latter is

2 Vfy = 2 VfVWHy/4- = ir H0 say.

Let ruF0 be the surface value. Then

H = M-0-1H0 = -{H + (1 - "V)H0 + (1 - '0)-H0+ ... | (3lw)

Or thus; in the manner (35</) was got, but annihilating <r instead of y, we shall have H = 2 Vfy0, where y0 is the fiually-arrived-at current given by

y0 = VN(1 - >j)~ JH0,47r; where // = - - f.(N/47r).

Or y0 = VfVN(H0+ //H0 + //-Ho+ ...) Air (37^)

J lere y is the same as in (34a).

V in Terms of the Sin'far/ H0.

When the p and F are in the inner region, the fir^t approximation to the value of V at the external point Q is 47r, say ?fH : and the force due to this is - The complete external potential is

P = u( - V%0)" = u{ 1 + (1 + VmJ + ... }H0, (38fl).

But if the point Q be inside, and the p and T outside, this formula does not give the internal P (with N as the inward normal), but leaves it indeterminate as regards a constant. A in, Ternm of the Surface II0.

The first approximation to A (such that H = curl A) being 2^VNH0/47r, say yH0, and the force due to this being curl yH0, the complete external A at (j, when p and T are inside, and ^ p = 0, is

A = g(curl.<fo)-1H = (/{1 +(1 - curl .q0)+ (3tV)

But if ^ p or ^ a* is not zero, we shall have

H = curl A - Vi'p

where A is given by (39a), and l\ is the potential due to ~/> distributed equipotentially. The external energy will now be expressed by

2iA y + U\^p

where y = VNH0/47r; the mutual energy of the fields of A and 1\ being zero.

Putmarks on these for make.

The process indicated by the right member of (33</) consists in the substitution of magnetic shells for matter, and finding their potentials. Let the final result be Pv and PV) be its surface value. Then P 1(l = 7*0 if 2p = 0. Otherwise, their difference must be a constant, say P0 -J\o = P%r I 01' ■-'-Z>20 lllu-t be zero, therefore P.M is constant. It is the potential of the surface when the quantity of matter is distributed over it equipotentially. ►Similarly, in the (36<), (37a) annihilation of the surface matter, mi far as is possible, we arrive at the force which differs from the real force by that due to the matter ~o- which is left, and which is distributed equipotentially.

It will be observed that whilst the first <r and y together produce no field on one side of the surface, the annihilation of either completely alters this. The complete <x, for example, produces a field on both sides, although it is the same on one side as that due to cr and y, i.e., on the side where H was under investigation.

There is always a distinction between the external and the internal regions as regards the determination of P from the surface force. It fixes the external force, when p and T are inside, and it also fixes the potential, so as to vanish at an infinite distance. It also fixes the internal force when the p and F are outside, but cannot then fix the potential. (P ■!/■■, no surface force, yet a constant internal potential, depending upon external matter.) This puts a difficulty in the way of the estimation of the external V in teims of the surface J0 when the p and F are inside, and IIp is not zero, e\en when we apply Green's method We do not arrive at the proper Greenian distribution ot matter, but at another, giving a surface potential differing from what is wanted by a known constant, so that we have to find another distribution, to give this constant potential.

SECTION XXIV. ACTION BETWEEN A MAGNET AND A MAGNET, OR BETWEEN A

MAGNET AND A CONDUCTOR SUPPORTING AN ELECTRIC CURRENT. THE CLOSURE OF THE ELECTRIC CURRENT. ITS NECESSITY.

The section before the last being preliminary to the subject of the stresses, and the last section being of a perfectly abstract nature, the one to follow this will be on the magnetic stress in general, as modified by differences of permeability and other causes. The present section is of an intermediate nature. Though dealing with the magnetic stress outside magnets, its principal object is to direct attention to the vexed question of the closure of the electric current; which I endeavour, as far as I can, to bring down to a question of definition.

Let us forget, if possible, for a time, all knowledge of the electric currentor rather, let us make no use of it. Suppose that we are fully acquainted with the mechanical actions of rigid magnets upon one another. That there is probably no such thing as a perfectly rigid magnet (that is, in the larger theory, an unmagnetisable magnet, whose permeability is unity, or the same as that of the enveloping medium) is immaterial. We suppose there is. Except that the argument would be more complex, it would not alter our general conclusions to take a magnetisable magnet.

We may put any collection of little rigid magnets together to form a complex magnet, having any distribution of magnetisation. Its external field of force is to be got by observing the mechanical force it exerts upon one pole of an exceedingly weak and slender magnetised filament, uniformly and longitudinally magnetised, so as to localise its poles strictly at its ends. Upon the basis of the definition of a unit pole, that it repels a similar unit pole at unit distance with unit force, we can map out the external field of force. Let F be the magnetic force intensity, that is, the force on a unit pole placed in the field. F is subject to the conditions

div F = 0, curl F = 0,

outside the magnet. The lines of F start from the surface of the magnet, proceed in curved paths through the air, and end upon its surface again at other places.

If we describe a closed surface in the air, completely enclosing the magnets, of whose position within the surface we might be ignorant, w'e should be entirely unable from our examination of the field outside the surface, to determine the interior distribution of magnetisation, or even to determine the situation of its poles, that is, the distribution of the imaginary magnetic matter, to which, if we ascribe self-repulsive force according to the inverse-square lawT, we may attribute the external force. We might ascribe F to a distribution of matter tr over the surface itself, of total quantity zero; or to a distribution of a vector quantity y over the surface in closed lines, producing external F according to another law, viz., Vfy = force at a point Q due to the element y; f being the force at y due to unit matter at Q; or to combinations of both cr and y, one of which is unique, inasmuch as it produces no internal field. Contract the surface until it reaches the magnet itself. Then we may ascribe the external field to <r, or to y, or to combinations, definitely, on the surface of the magnet, or in its interior in various way, but not definitely. The distribution of magnetisation is, a fortiori, still more undiscoverable, for there may be distributions corresponding to no magnetic matter, giving no external force.

But, making use of our knowledge of the effect of building up a magnet from smaller ones, we may suppose that the magnetisation is known. Let the vector magnetisation be I, and its convergence be p, the density of the magnetic matter. Then

div F = 47rpy curl F = 0

fully determine F. It is of considerable practical utility in theoretical reasoning not to treat the surface- and the volume-densities separately, but to include them both in p, the volume-density. Thus, if a magnet be quite uniformly magnetised, there is, strictly speaking, no p. The convergence of I is on the surface. But, by supposing I not to cease abruptly on reaching the surface, but gradually, however rapidly, through a thin surface-layer, we make the matter have a space distribution, of the same total amount ner unit of surface as the surface distribution it represents. We can always derive the surface expressions from those of the volume with great ease, when we want them ; whilst our work is much simpler without them when we do not want them.

The resultant force between any two magnets may now be easily represented. ^F^ is the resultant force on a magnet whose matter density is Pv due to another magnet whose polar force is F2. And 2F,p2 is the force of the first magnet 011 the second. (We need not trouble about the forces of rotation at present, which are nearly as simply represented.) These are equal and opposite. Also SF^^O and 2 F= 0 ; that is, a magnet cannot translate itself (01* rotate itself either). So if F = Fj + F2, making F the actual force of the field, the forces on the magnets are 2 Fpl and 2 Fp2 respectively. Describe any closed surface separating one magnet from the other; let the first magnet be inside. The force 011 it may, as in the last section, be represented by the surface-integral

2-J(<7F + VyF),

if O- = NF/4TT, y = VNF/47r, N being a unit normal from the inner to the outer region. Or, which is the same thing, by

2{F(NF) - N(iF)}/47r,

in terms of the Maxwellian stress.

The following is the process of showing that this stress gives rise to the required mechanical forces. The quantity summed up is the vector stress on the plane whose normal is N. If we fix the direction of N, its divergence is the N-component of the force per unit volume, by the principles of varying stress.

Now div (F(NF)} = (NF) div F + FV(NF),

and div {N(.,.F2)} = A(}rF2) = F~,

aii\~ J (In

if n be length measured along N.

Also, F-['(F - V(NF) j = m F curl F,

[dn. j

so, S being the vector stress 011 the N plane,

4TT div S = (NF)div F NVF curl F,

= N{F div F - VFcurl F|,

div S ■-= N{Fp + VFF},

if F = curlF/47r. This being the N-component of the force, the force itself is + yrF per unit volume; or, since F = 0, the magnetic force of the magnets being polar, simply Fp per unit volume. The length of this process depends upon our wishing to develop the term V I F. If it were not for that, we would see at once that

V(NF) ='(F

when F is polar, N being in any direction, and so get the force Fp immediately. The question now asks itself (remembering that we are ignorant of the electric ciumit), what is this F, whose vanishing cuts the work short at the beginning, in our case of F being polar. Can it really represent any physical magnitude ?

It is defined by curlF = 47tF, and is necessarily zero in the case of magnets. It indicates closed lines of F, which are impossible with a strictly polar force in all space. F is a vector which is necessarily circuital. This is a mathematical consequence of its definition.

Furthermore, whilst F, as a polar force, with F = 0, is a special kind of distribution of a vector, if we allow V to be not zero F becomes of the most general type possible, any distribution of force, or any field of force, without the polar limitation. Given the divergence, and the curl of a vector, the vector itself is fixed, if it is to vanish at infinity.

Supposing, then, we allow that V can exist, we can predict what the mutual force between a magnet and it will be. Let there be I1 in a certain region, outside of which there is only p, therefore only magnets. The resultant force 011 this region is equivalent to YFF per unit volume. We cannot localise the forcewe can only know its total amount. It is necessarily - VFF. The reason we cannot say that VFF is the force on unit volume is that F is necessarily circuital, and so we cannot work down to a unit volume without having current in the external region as well, whic h is against our previous knowledge.

So far, r has a merely speculative existence. It is got by making the assumption that there can be circuital magnetic force. Admit that there can be, the laws of T follow. F is necessarily circuital. The magnetic force it produces at Q is 1' fr, if f be the force at V due to unit matter at Q. The force between one T and another, and between F and n follow, viz., that the resultant force on any region containing n and closed I1 is H being the actual intensity of the field due to all the p and T.

Now, we do know what T is, under certain circumstances. By the researches of Ampere, the father of electrodynamics, we know that F measures the density of current in a conductor when it is steady. His researches were conducted in a very different manner, and are indeed very difficult to follow, like most novel researches, but the results are exactly these, without his hypotheses as to the action between different elements of a current. We virtually measure the strength of current in a conductor, when we use a galvanometer, by the line-integral of H round a current, and that is the amount of the quantity r = curlH/47r, passing through the line of integration.

Also, steady currents are closed. So far, then, we identify V with the conduction current.

But our r is closed under any circumstances. We know also that conduction currents are not always closed; for instance, when we charge a condenser. We still measure the conduction current in its transient state by I1. We do so by the continuously changing iustantaneous magnetic force if the charge or discharge be slow enough; otherwise, by the ballistic method, which is virtually the same. I' being then unclosed in the conductor, has necessarily its exact complement, to close it, outside the conductor, i.e., our F has, though it may be only called current when in the conductor.

But our r, being identified with current when in a conductor, both in steady states when the current is closed in the conductor, and in transient states when it is closed through the dielectric, and this T in the dielectric being related to the magnetic force in the same way as if it were conduction current, why should we not call it electric current also ? As it demonstrably exists, we see that the closure of the current is reduced to a question of a name. It would be positively illogical not to call it electric current.

To sum up :

1. From magnetic knowledge only, there should be no circuital magnetic force in the space outside magnets.

2. If we admit the existence of circuital magnetic force, or, say generally, if we admit that the line-integral of the force in a circuit in air can be finite, we arrive at a vector quantity I' having also the property of being circuital, and we can find the mechanical force between it and magnets or other 1, and can thus definitely measure 1.

3. This T we know to be conduction current, when steady.

4. But in transient states, conduction currents are not always circuital.

5. But a part of T still measures the conduction current.

6. The other part, the complement of the conduction current, is outside the conductor, continuous vvifii the conduction current, and closing it.

7. Then why not call it electric current?

We see that it is not a question for experiment, for no amount of experimenting could alter this reasoning, but of definition, an agreement to call a certain function of the magnetic force always by one name, viz., the electric current, which, if in a conductor, heats it and wastes energy, whilst in a nonconductor does not, energy being stored potentially. It is, of course, needless to add that this current in a nonconductor is Maxwells current of displacement, D, the rate of increase of the displacement, whilst ED is the activity of the electric force E to match, and iED the stored potential energy of displacement.

Note on equation (32a) [p. 553].For mnemonical purposes, the following is a concise form of this equation. The potential being given = P0 over a closed surface, due to matter or current within or on it, the potential P at any external point Q is along the normal outward.

[The second half, Sections 25 to 47, of this Article, is in vol. 2].