Electromagnetic Theory, Vol. 2 (1899) — part 6 of 31

§ 61. The nature of a spherical eleofcromagnetio eheet expanding or contracting at speed v, when it is charged in an arbitrary manner, may also be readily seen by the foregoing. So far, when there has been electrification on the sheet, it has Ateea a solitary point-charge or else a pair, at opposite poles. Now, the general case of an arbitrary distribution of electrifi- cation can be followed up from the case of a pair of charges not at opposite ends of a diameter, and each of these may be taken by itself by means of an opposite charge at the centre or externally, so that integration does the rest of the work. When there is a pair of equal charges of opposite sign we do not need any external or internal complementary electrification, but we may make use of them argumentatively ; or we may let one charge leak outward, the other inward, and have the external and internal elec- trification in reality. The leakage should be of the isotropic character always. But the internal electrification need not be at the central point. It may be uniformly distributed upon a concentric sphere. This, again, may be stationary, or it may itself be in motion, expanding or contracting at the speed v. The external electrification, too^ may be on a concentric spherical surface, which may be in similar motion. The sheets, too, may be of finite depth, and arbitrarily electrified, so that we have any yolume-distribution of electrificati<m moving in space in radial lines to or from a centre, aooom* panied by electromagnetto disturbances arranged in spherical eheets.

There is thos a great wie^ of ways of making up problems ^tids character, the nature of whose solutions can be readily irictured mentally.

Two charges, and for example, on a spherical sheet One way is to put - q^ and - q^ at the centre, and superpose the two solutions. Or the changes - (qi and may be put extern nally, with isotropic leakage. Or pare may be inside, and part outside. Or we may haye no complementary electr^cation at ' all, but lead the diq>]acement away into plane waves touching the sphere.

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Only when the total charge on the spherical sheet is zero oaQ we dispense with these external aids (which to use depending upon the conditions of the problem) ; then the displacement has sources and sinks on the sheet which balance one another. The corresponding indi;ction ia always perpendicular to the resultant tangential displacement, and is given by the above formula, or Bo-yxYTDoi where Dq is the tangential displace- ment in the sheet (volume density x depth).

One case we may notice. If the density of electrification be uniform over the surfiice, there is no induction at alL That is, it is not an electromagnetio sheet, but only a sheet of electrifi- cation,, without tangential displacement, and therefore without induction.

So^ with a condenser consisting of a pair of concentric shells uniformly electrified, either or both may expand or contract without magnetic force. This is, however, not peculiar to the case of motion at speed «. Any speed will do. But in general, if the speed be not exactly v, there result diffused disturb- ances. The electromagnetio waves are no longer of the same pure type.

General Bemarks of. tbe CHreiiltal Laws. Ampere's Bule for deriving the Magnetic Force from the Ourrent. Bational Onrrent-element.

§62. The two laws of circuitation did not start into full activity all at once. On the contrary, although they express the fundamental electromagnetic principles concerned in the most concise and clear manner, it was comparatively late in the history of electromagnetism that they became clearly re- cognised and explicitly formularised. We have not here, how- ever, to do the work of the electrical Todhunter, but only to notice a few points of interest.

The first law had its first beginnings in the discovery of Oersted that the electric conflict acted in a revolving manner, and in the almost simultaneous remarkable investigations of Ampere. It did not, however, receive the above used form of expression. In fact, in the long series of investigations in electro-dynamics to which Oersted's discovery, and the work of Ampere, Henry, and Faraday, gave rise, it was customary to consider an element of a conduction current as generating a

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certain field of magnetic force. Natural as this course may have seemed, it was an unfortunate one, for it left the question of the closure of the current open ; and it is quite easy to see now that this alone constitated a great hindrance to progress. But so far as closed currents are concerned, in a medium of uniform inductivity, this way of regarding the relation between current and magnetic force gives equivalent results to those obtained from the first law of circuitation in the limited form suitable to the circumstances stated.

If 0 is the density of conduction current at any place, the corresponding field of magnetic force is given by

at distance r from the current-element 0, if r^ be a unit vector along r from the element to the point where H is reckoned. The intensity of H thus follows the law of the inverse square of the distance along any radius vector proceeding from the current-element; but, in passing from one radius vector to another, we have to consider its inclination to the aids of the current by means of the factor sin $ (where $ is the angle between r and the axis), involved in the vector product. Also, H is perpendicular to the plane containing r and the axis of the oorrent-element^ or the lines of H are circles about this axis.

But from the Maxwellian point of view this field of H is that corresponding to a certain circuital distribution of electric current, of which the current-element mentioned is only » part ; this complete current being related to the current- element in the same way as the induction of an elementary magnet is to the intensity of magnetisation of the latter. Calling the complete system of electric current a rational current-element, it may be easily seen that in a circuital distri- bution of rational current-elements the external portion of the ' current disappears by mutual cancelling, and there is left only the circuital current made up of the elements in the older sense. We may, therefore, employ the formula (1) to calculate without ambiguity the magnetic force of any circuital distri- bution of current. This a[)plies not merely to conduction current (which was all that the older electricians reckoned), but to electric current in the wider sense introduced by

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Maxwell. But the result will not be the real magnenc force unless the distribution of inductivity is uniform. When /* varies, we may regard the magnetic force of the current thus obtained as an impressed magnetic force, and then calculate what induction it sets up in the field of varying iuduotivitj. This may be regarded as an independent problem.

In passing, we may remark that we can mount from magnetic current to electric force by a formula precisely similar to (1), but subject to similar reservations.

The Cardinal Feature of Maxwell's System. Advice to

anti-Mazwellians.

§63. Bat this method of mounting from current to magnetic foroe (or equivalent methods employinpf potentials) is quite unsuitable to the treatment of electromagnetic waves, and is then usually of a quite unpractical nature. Besides that, the function electric current" is then often a quite subsidiary and unimportant quantity. It is the two fluxes, induction and displacement (or equivalently the two forces to correspond), that are important and significant; and if we wish to know the electric current (which may be quite a useless piece of informa- tion) we may derive it readily from the magnetic foroe by difierentiation ; the simplicity of the process being in striking contrast to that of the integrations by which we may mount from current to magnetic force.

To exemplify, consider the illustrations of plane and spherical eleotromagnetio waves of the simplest type given in §53 to 61, and observe that whilst the results are rationally and simply describable in terms of the fluxes or forces, yet to describe in terms of electric current (and derive the rest from it) would introduce such complications and obscurities as would tend to anything but intelligibility.

Now, Maxwell made the first law of circuitation (not, how- ever, in its complete form) practically the definition of electric current. This involves very important and far-reaching con- sequences. That it makes the electric current always circuital at once does away with a host of indeterminate and highly speculative problems relating to supposititious unclosed cur- Tents. It also necessitates the existence of electric current in })erfect non-conductors or insulators. This has always been a

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€tumbling-block to practicians who think themselves ]iractical. But Maxwell's innovation was really the most practical im- provement in electrical theory conceivable. The electric current in a nonconductor was the very thing wanted to co- ordinate electrostatics and electrokinetics, and consistently harmonise the equations of electromagnetism. It is the cardinal feature of Maxwell's system, and, when properly followed up, makes the insulating medium the true medium in the transmission of disturbances, and explains a multitude of phenomena that are inconceivable in any other theory (unless it be one of the same type). But let the theoretical recom- mendations (apart from modern experiments), which can only be appreciated after a pretty close study of theoiy, Maxwellian and otherwise, stand aside, and only let the tree be judged by its fruit. Is it not singular that there should be found people, the authors of works on Electricity, who are so intensely pre- judiced against the Maxwellian view, which it would be quite natural not to appreciate from the theoretical standpoint, as to be apparently quite unable to recognise that the fruit has any good flavour or savour, but think it no better than Dead Sea fruit? The subject is quite sufficiently difficult to render understandable popularly, without the unnecessary obstruc- tion evidenced by a carping and unreceptive spirit. The labours of many may be required before a satisfactory elemen- tary presentation of the theory can be given. So much the more need, therefore, is there for the popular writer to recognise the profound significance of the remarkable experimental work of late years, a significance he appears to have so sadly missed!^ When that is done, then will be the time for an understanding of Maxwell's views. Let him have patience, and believe that it is not all speculative metaphysics because it is not to his present taste. Never mind the ether disturbances playing pranks with the planets. They can take care of themselves.

OluuigeB in the Fonn of the First Oircnital Law.

§ 64. Two or three changes I have made* in Maxwell's form of the first circuital law. One is of a formal character, the introduction of the h term to express the intrinsic force of

• " Electromagnetic Induction and its Propagation," The Ekctrician^

• 1886, January 3, and later : or reprint.

F 2

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ma^etisation. This somewhat simplifies the mathematioSi and places the essential relations more clearly before the eye. Connected with this is a different reckoniog of the energy of an intrinsic magnet, in order to get oonsietent results.

The second change, which is not merely one of a formal character, but is an extension of an obligatory character, is the introduction of the h term to represent the motional magnetio force. In general problems relating to electromagnetic waves it is equally important with the motional electric force.

The third change is the introduction (first done, I think, by Prof. Fitzgerald) of the term to explicitly represent the oon- Tection-current or electrification in motion as a part of the true current. Although Maxwell did not himself explicitly repre* sent this, which was a remarkable oversight, he was strongly insistent upon the circuital nature of electric current, and would doubtless have seen the oversight the moment it waa suggested to him. Now, there are spots on the sun, and I see no good reason why the many faults in Maxwell's treatise should be ignored. It is most objectionable to stereotype the work of a great man, apparently merely because it was so great an advance, and because of the great respect thereby induced* The remark applies generally ; to the science of Quaternions, for instance, which, if I understand rightly, Prof. Tait would preserve in the form given to it by Hamilton. In application to Maxwell's theorv, I am sure that it is in a measure to the recognition of the faults in his treatise that a clearer view of the theory in its broader sense is due. ;

Introdnction of the Second Circuital Law.

§65. The second circuital law, like the first, had an experi- mental origin, of course, and, like the first, was long in approxi- mating to its present form — much longer, in fact, though in a different manner. Tiie experimental foundation was Faraday's recognition that the voltage induced in a conducting circuit was conditioned by the variation of the number of lines of force through it.

But, rather remarkably, mathematicians did not put this straight into symbols for an elementary circuit, but went to work in a more roinidabout way, and expressed it through the medium of an integration extended along a concrete circuit

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or else, of an equation of eleotromotiye force oontainiug a fancUon called the vector potential of the current, and another potential, the electrostatic, working together not altogether in the most harmoniously intelligible manner — in plain English, muddling one another. It is, I believe, a fact which has been veoognised that not even Maxwell himself quite understood how they operated in his ** general equations of propagation." We need not wonder, then, that Maxwell's followers have not found ft a vety easy task to understand what his theory really meant, and how to work it out. I had occasion to remark, some years eince, that it was very much Maxwell's own fault that his views obtained such slow acceptance; and, in now repeating the remark, do not abate one jot of my appreciation of his work, which increases daily. For he devoted the greater part of his treatise to tlie working out and presentation of results which <;ould be equally well done in terms of other theories, and gave only a very cursory and incomplete exposition of what were peculiarly his own views and their consequences, which are of the utmost importance. At the same time, it is easily to be recognised that he was himself fully aware of their importance, by the tone of quiet confidence in which he wrote concerning them.

Finding these equations of propagation containing the two -potentials unmanageable, and also not sufficiently comprehen- sive, I was obliged to dispense with them ; and, going back to first fjrinciples, introduced'"' what I term the second circuital law as a fundamental equation, the natural companion to the first. The change is, I believe, a practical one, and enables us to con- siderably simplify and clarify the treatment of geiieral ques- tions, whilst bringing to light interesting relations which were formerly hidden from view by the intervention of the vector |)otential A, and its parasites J and "4^.

Another rather curious point relates to the old German •electro-dynamic investigations and their extensions to endeavour to include, supersede, or generalise Maxwell by anti-Max wellian oethods. It would be slaying the slain to attack them ; but one point about them deserves notice. The main causes of the variety of formulaB, and the great complexity of the investiga- tions, were — first, the indefiniteness produced by the want of

• See footnote, p. 67>

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cirooitality in the current, and next the potential methods employed. [J. J. Thomaon's " Report on Electrical Theories contains an account of many. As a very full example, that most astoundingly complex investigation of Clausius, in the* second volume of his " Mechanische Warmetheorie," may be referred to.] But if the critical reader will look through these- investigations! and eliminate the potentials, he will find, as a- useful residuum, the second circuital law; and, bringing it into full view, will see that many of these investigations axe purely artificial elaborations, devoid of physical significance > gropings after mares* nests, so to speak. Now, using this law in the investigations, it will be seen to involve, merely as a matter of the mathematical fitness of things, the use of another, viz., the first circuital law, and so to justify Maxwell in his doctrine of the circuitalit^ of the current. The useful moral to be deduced is, I think, that in the choice of vaiiables to ex- press physical phenomena, one should keep as close as possible to those with which we are experimentally acquainted, and which are of dynamical significance, and be on one's guard against being led away from the straight and narrow path in the pursuit of the WilW-the-wisp.

As regards the terms in the expression of the second law which stand for unknown properties, they may be regarded merely as mathematical extensions which, by symmetrizing the ei] nut ions, render the correct electric and maji^netic analogies plainer, and sometimes assist working out. lint some other extensions of meaning, yet to be considered, have a more sub- stantial foundation.

Meaning of Tme Onrrent. Oriterion.

§66. One of these extensions refers to the meaning to be attached to the terra current, electric or magnetic respectively. As we have seen, electric current, which was originally conduc- tion current only, had a second part, the displacement current, added to it by Maxwell, to produce a circuital flux; and further, when there is electrification in motion, we must, working to the same end, add a third term, the convectiou current, to preserve circuital ity.

A fourth term may now be added to make up the " true " current, when the medium supporting the fluxes is in motion*

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Thus, let us separate the motional electric and magnetic forces from all other intrinsic forces that go with them in the two circuital laws (voltaic, thermo-electric, &c., § 39) ; denoting the former by e and h, and the latter by Bq and TIq. The two equations of circuitation 1(12), (13) § 38, and (6), (7) § 52)] now. read

curl(H-lio-li) = J=0 + i) + u/), . . . (1) -curl(E-e^-e) »OBK + B«fw<r. . . (2)

Now transfer the e and h terms to the ri^rht side, producing curl(H-Jlio)«Jo«0 + i} + iip+j, . . (3) -ourl(E-6o) -OoaK+B+wcr-l-g, . . (4) where j and g are the auxiliaries before used, given by

J «curl li, g** - curl e.

It is the new yectors and Qq ^^ich should be regarded as the true carrente when the medium moves.

As the auxiliaries J and g are themselyes oirouital yectorSi there may not at first sight appear to be any reason for further complicating the meaning of true current when separated into component parts, for the current J is circuital, and so is Jg, which is of course the same as J when the medium is station- ary. The extension would appear to be an unnecessary one, of a merely formal character.

On the other hand, it is to be observed that from the Maxwellian method of regarding the current as a function of the magnetic force, the extended meaning of true current is not a farther complication, but is a sunplification. For whereas in the equation (2) we deduct from the force E of the flux not only the intrinsic force 00 but also the motional force • to obtain the effective force whose curl measures the current ; on the other hand, in (4) we deduct only the intrinsic force. Away, therefore, from the sources of energy which are inde- pendent of the motion of the medium, the force whose curl is taken in (4) is the force of the flux, which specifies the elec- tric state of the medium, whether it be stationary or moving.

To sliow the effect of the change, consider the example of § 56, in which the medium moves past the charge, and there is continuously changing displacement outside a certain sphere. . If we consider J the true current, we should say there is elec-

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trie current, although there is no magnetic force. But accord- in*,' to the other way there is no true current, and the absence of magnetic force implies the absence of true electric current.

But still this question remains, so far, somewhat of a con- ventional one. Is there any test to be applied which shall effectually discriminate between J and Jq as the true current ? There would appear to be one and only one ; viz., that being an intrinsic foroe, its activity, whatever it be, say e^x, expresses the rate of communication of energy to the electromagnetic system from a source not included therein, and not connected with the motional foroe 6. When the medium is stationary x is J, and the question is, is x to be J or Jq (or anything else) when the medium moves f

Now this question can only be answered by making it a part of a much larger and more important one ; that is to say, by a comprehensive examination of all the fluxes of energy concerned in tlie equations (1), (2), or (3), (4), and their mutual harmo- nisation. This being done, the result is that OgJo is the activi^ of e^, so that it is that is the measure of the true current, with the simpler relation to the foroe of the flux; and it is naturally suggested that in case of further possible extensions, we should follow in the same track, and consider the true cur- rent to be always the curl of (H - ho), independently of the make up of its component parts.

This is not the place for a full investigation of this complex question, but the main steps can be given ; not for the exhibit tion of the mathematical working, which may either be taken for granted, or filled in by those who can do it, but especially with a view to the uftderstanding in a broad manner of the course of the argument. Up to the present I have used the notation of vectors for the condse and plain presentation of principles and results, but not for working purposes. This course will be continued now. How to work vectors may form the subject of a future chapter. It is not so hard when you know how to do it.

Tlie Pendstenoe of Energy. Oontiniiity in Time and Space

and Flux of Energy.

§ 67. The principle of the conservation or persistence of energy is certainly as old as Newton, when viewed from the

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standpoint of theoretical dynamics. But only (roughly speak- ing) about the middle of the present century did it become recognised by scientific men as a universal truth, extending to all the phenomena of Nature. This arose from the experi- mental demonstrations given (principally those of Joule), that in various cases in which energy disappeared from one form, it was not lost, but made its appearance iu other forms, and to the same amount.

The principle is now-a-days a sure article of scientific faith. Its ultimate basis is probably the conviction (a thoroughly reasonable one) that the laws of motion hold good in the in- visible world as well as in the visible, confirmed by the repeated experimental verification. But the principle is now believed in •quite generally ; by the ordinary unscientific man, for example, who oould not tell you correctly what the principle meant, or energy either, to save his life, although he might tell you that he oould not conceive the possibility of energy being destroyed. He goes by faith, having taken it in when young.

So it will be with the modem view of the ether as the medium through which energy is being sent when a wire sup- {>ort8 a current. Only train up the young to believe this, and they will afterwards look upon the notion of its going through the conductor as perfectly absurd, and will wonder how anyone ever could have believed it. But before this inevitable state of things comes to pass, an intermediate period must be passed through, in which scientific men themselves are learning to believe the modern view thoroughly from the evidence of its truth, and to teach it to others. This period is certainly not yet come to its end ; for almost weekly we may read about " the velocity of electricity in wires," such language emanating irom. scientific men who are engaged in repeating and extend- ing Hertz's experiments.

The principle of the continuity of energy is a special form of that of its conservation. In the ordinary understanding of the oonservation principle it is the integral amount of energy that 18 conserved, and nothing is said about its distribution or its motion. This involves continuity of existence in time, but not aeoessarily in space also.

But if we can localise energy definitely in space, then we are Iwund to ask how energy gets from place to place. If it

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poBseBsed continuity in time only, it might go out of existence at one place and come into existeuce Himultaneously at another. This is sufficient for its conservation. This view, howerer, does not recommend itself. The alternative is to assert continuity of existence iu space also, and to enunciate the principle thus : —

When energy goes from place to place, it traverses the intermediate space.

This is 80 intelligible and practical a form of the principle^ that we should do our utmost to carry it out.

The idea that energy has position, therefore, naturally involves the idea of a flux of energy. Let A be the vector flux of energy, or the amount transferred per unit of time per unit of area normal to the direction of its transfer ; and let T be the density of the energy. We may then write

_ »

convA — T, (5)

or divA=-T (6)

That is, the convergence of the flux of energy is accounted for by increased energy in the unit volume where it converges ; or, the divergence of the flux of energy, or the rate at which it leaves the unit volume, is accompanied by simultaneous and corresponding decrease in the density of the energy.

Examples. Convection of Energy and Flux of Energy due to an active Stress. Gravitational di&culty.

§ 68. ^owj there are numerous cases in which the flux of energy is perfectly plain, and only needs to be pointed out to be recognised. The simplest of all is the mere convection of . energy by motion of the matter with which it is associated. A body in translational motion, for instance, carries its energy with it; not merely the kinetic energy of its translational. motion, but also its rotational energy if it rotates, and like- wise the energy of any internal vibratory or rotatory motions it may possess, and other energy not Imown to be kinetic^ and therefore included under "potential energy," which we- may have good reason to localise in the body. Evea if part of the energy be outside the body, yet, if it be- associated with the body, it will travel with it, on the whole..

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•nd ao (at least lometimes) may be considered to be rigidly attached to it. This case may be regarded as the oonTection of energy, if we do not go too closely into detail.

When energy is thns conveyed, the energy flux is qT, where q is the velocity and T the density of the energy conveyed (of any kind). If, then, all energy were conveyed, equation (5) would become

conv<iT = t (7)

Now this equation, if we regard T as the density of matter, is the well known equation of continuity of matter used in hydro- dynamics and elsewhere.

This brings us to Prof. Lodge's theory of the identity of energy. {Phil. Mag., 1885.) Has energy personal identity, like matter! I cannot see it, for one ; and think it is pushing the principle of continuity of energy, which Prof. Lodge was writing about, too far. It is difficult to endow energy with objectivity, or thinginess, or personal identity, like matter. The rdiativi^ of motion seems to be entirely against the idea. Not are we able to write the equation (7) in all cases. Energy may be transferred in other ways than by convection of associated matter. If we atomise the energy we can then imagine the q above to be the velocity of the energy, not of the matter. But as the science of dynamics is at present understood, we cannot make use of this idea profitably, I think.

The chief reason whv the founders of the modem science of energy did not explicitly make use of the idea of the continuity of energy was probably the very obscure nature of gravitational energy. Where was it before it became localised as the kinetic energy of a mass 1 It is of no use to call it potential energy if that is to explain anything, which it does not. It would seem that the energy must have been in the ether, somewhere, and was transferred into the body, somehow. This makes the etlier the great store-house of all gravitational energy. But we are entirely ignorant of its distribution in the ether, and of its mode of transference.

Observe here that ether must be regarded as a form of matter, because it is the recipient of energy, and that is the oliaracteristic of ordinary matter. There is an unceasing enormous flux of energy through the ether from the Sun, for

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inBtance, and we know that it takes seTeral minutes to come ; 4c comes through the ether, without bringing the ether with it; it is not a convection of energy, therefore. Of course, to avoid confusion, it is well to distinguish ether from-ordinaiy matter

by its separate nanie ; but it is important to note that it haa some of the characteristics of ordinary matter. It need not be gravitating matter ; it is, perhaps, more likely to be the medium of gravitational action than to gravitate itself.

In default of ability to represent the flux of gravitational energy its entry into a body must be otherwise represented than by the convergence of a flux. Turn (7) into

£Q + oonvA«T. (8)

Here fq is the activity of the force f, and indicates energy com- municated to the unit volume in an unstated manner, generated within the unit volume, so to speak. The force f is thus an impressed force. By this device we can allow for unknown fluxes of energy. We should also explicitly represent the convective flux. Thus,

fa+conv((iT + A) = t, .... (9)

where A is the flux of energy other than convective, not asso- ciated with fl

When a stress works, for example, we shall have a flux of energy, A, which can hardly be considered to be convective, in the sense explained, a flux of energy through matter or ether, as in wave propagation through an elastic solid, or radiation through the ether. We can, however, sometimes reduce it to the form (iTj, and then it would appear to be convection of energy.

Tliis occurs in a moving frictionless fluid, for example, when the stress is an isotropic pressure jo. The Ilux of energy other than convective is ^q. But when we pass to a less ideal case, as an elastic solid, in which the stress is of a more general character, the energy flux expressive of the activity of the stress takes the form

A=-P,?, (10)

where is the vector expressing the pull per unit area on the plane perpendicular to the velocity q. In this equation q is the tensor of q.

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QpedaUsed fann of ezpresBion of the Oontinnity of Energy*

§ 69. Passing to more practical and specialised forms of expression of the continuity of energy, it is to be observed first that we are not usually concerned with a resultant flux of energy from all causes, but only with the particular forms^ relating to the dynamical question that may be under con- sideration. Secondly, that it is convenient to divide the energy above denoted by T into different i)arts, denoting potential energy, and kinetic energy, and wasted energy* Thus we have the following as a practical form,

f<l + conv[A + (i(U + T)l = Q+U + T, . (11)

where the terms in the square brackets indicate the energy flux, partly convective, with the factor q, U being the density of potential energy, and T that of kinetic energy, and partly non-convective, viz., A, which may be due to a working stress. Also fq is the activity of impressed force (here merely a trans- lational force). Thus on the left side of equation (11) we have a statement of the supply of energy to the unit volume fixed in space. On the right side we accotmt for it by the rates of increase of the stored potential and kinetic energy, and by Q, which means the rate of waste of energy in the unit volume. The wasted energy is also stored, at least temporarily ; but not recoverably, so that we may ignore energy altogether after it is once wasted.

In place of the term fq, in which the impressed force is translational, we may have other terms possessing a similar meaning, indicating a supply of energy from certain sources. These sources, too, may be not external, but internal or intrin- sic, as for instance when there is a thermo-electric or voltaic source of energy within the unit volume considered.

Also, the terms A, U, T, and Q may have to be split up* into different parts, according to the nature of the dynamical connections. The important thing to be grasped is, that when- ever we definitely localise energy we can obtain an equation showing its continuity in space and time, and that when we- can only partially localise it, we can still, by proper devices,, allow for the absence of definiteness. The equation is simply the equation of activity of the dynamical system, suitably

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arranged to show the flux of energy, and can always be obtained when the equations of motion are known and also the nature of the stored and wasted energies.