Application of Dynamic to Physics and Chemistry (1888)

7/30/2019 Application of Dynamic to Physics and Chemistry (1888)

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TO

PHYSICS AND CHEMIS'

BY

J. J. THOMSON, M.A., F.R.S.FELLOW OF TRINITY COLLEGE AND CAVENDISH PROF

OF EXPERIMENTAL PHYSICS, CAMBRIDGE.

MACMILLAN AND CO.AND NEW YORK.

1888


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PRINTED BY C. J. CLAY, M.A. & SONS,AT THE UNIVERSITY PRESS.


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PREFACE.THE following pages contain the substance of a

course of lectures delivered at the Cavendish Labora-tory in the Michaelmas Term of 1886.Some of the results have already been publishedin the Philosophical Transactions of the RoyalSociety for 1886 and 1887, but as they relate tophenomena which belong to the borderland betweentwo departments of Physics, and which are generallyeither entirely neglected or but briefly noticed intreatises upon either, I have thought that it mightperhaps be of service to students of Physics topublish them in a more complete form. I haveincluded in the book an account of some investiga-tions published after the delivery of the lectureswhich illustrate the methods described therein.There are two modes of establishing the connexionbetween two physical phenomena ; the most obviousas well as the most interesting of these is to startwith trustworthy theories of the phenomena in ques-tion and to trace every step of the connexion betweenthem. This however is only possible in an exceed-in lv limited number of cases and we are in eneral


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may be, they must be related to each other in sucha way that the existence of the one involves thatof the other.

It is the object of this book to develop methodsof applying general dynamical principles for thispurpose.The methods I have adopted (of which that usedin the first part of the book was suggested byMaxwell's paper on the Electromagnetic Field) makeeverything depend upon the properties of a singlefunction of quantities fixing the state of the system, aresult analogous to that enunciated by M. Massieuand Prof. Willard Gibbs for thermodynamic pheno-mena and applied by the latter in his celebrated paperon the "Equilibrium of Heterogeneous Substances"to the solution of a large number of problems inthermodynamics.

I wish in conclusion to thank my friend Mr L. R.Wilberforce, M.A., of Trinity College, for his kindnessin correcting the proofs and for the many valuablesuggestions he has made while the book was passingthrough the press.

J. J. THOMSON.TRINITY COLLEGE, CAMBRIDGE,

May ind, 1888.


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CONTENTS.CHAPTER I.

i

Preliminary Considerations

CHAPTER II.The Dynamical Methods to be employed .....

CHAPTER III.Application of these Principles to Physics

CHAPTER IV.Discussion of the terms in the Lagrangian Function

CHAPTER V.Reciprocal Relations between Physical Forces when the Systems

exerting them are in a Steady State .....CHAPTER VI.

Effect of Temperature upon the Properties of Bodies .

CHAPTER VII.Electromotive Forces due to Differences of Temperature


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Introductory to the Study of Reversible Scalar Phenomena . . 140

CHAPTER X.Calculation of the mean Lagrangian Function . . . -151

CHAPTER XI.Evaporation . -158

CHAPTER XII.Properties of Dilute Solutions . . . . . . -179

CHAPTER XIII.Dissociation 193

CHAPTER XIV.General Case of Chemical Equilibrium . . . . .215

CHAPTER XV.Effects produced by Alterations in the Physical Conditions on the

Coefficient of Chemical Combhiation. ..... 233CHAPTER XVI.

Change of State from Solid to Liquid ...... 243CHAPTER XVII.The Connexion between Electromotive Force and Chemical

Change 265


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APPLICATIONS OF DYNAMICS TOPHYSICS AND CHEMISTRY.

CHAPTER I.PRELIMINARY CONSIDERATIONS.

i. IF we consider the principal advances made in thePhysical Sciences during the last fifty years, such as theextension of the principle of the Conservation of .Energyfrom Mechanics to Physics, the development of the Kinetic,.Theory of Gases, the discovery of the Induction of ElectricCurrents, we shall find that one of their most conspicuouseffects has been to intensify the belief that all physicalphenomena can be explained by dynamical principles andto stimulate the search for such explanations.

This belief which is the axiom on which all ModernPhysics is founded has been held ever since men first beganto reason and speculate about natural phenomena, but, withthe remarkable exceptions of its successful application inthe Corpuscular and Undulatory Theories of Light, itremained unfruitful until the researches of Davy, Rumford,Joule, Mayer and others showed that the kinetic energypossessed by bodies in visible motion can be very readilyr-nriTr rtPrl into 1^ a^ Tnill rrmr =>nxri=v -nrnx?^ -! tl-iaf ttrl-ie>nenrov


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The ready conversion of kinetic energy into heat con-vinced these philosophers that heat itself is kineticenergy; and the invariable relation between the quantitiesof heat produced and of kinetic energy lost, showed thatthe principle of the Conservation of Energy, or of Vis-Vivaas it was then called, holds in the transformation of heat intokinetic energy and vice versa.

This discovery soon called attention to the fact thatother kinds of energy besides heat and kinetic energycan be very readily converted from one form into another,and this irresistibly suggested the conclusion that the variouskinds of energy with which we have to deal in Physics, suchfor example as heat and electric currents, are really forms ofkinetic energy though the moving bodies which are theseat of this energy must be indefinitely small in comparisonwith the moving pieces of any machine with which we areacquainted.

These conceptions were developed by several mathema-ticians but especially by v. Helraholtz, who, in his treatiseUeber die Erhaltung der Kraft, Berlin, 1847, applied thedynamical method of the Conservation of Energy to thevarious branches of physics and showed that by this prin-ciple many well-known phenomena are connected witheach other in such a way that the existence of the oneinvolves that of the other.

2. The case which from its practical importance at firstattracted the most attention was that of the transformationof heat into other forms of energy and vice versa.

In this case it was soon seen that the rinci le of the


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when certain changes, take place in the body, hut thatthese relations could be deduced by the help of anotherprinciple, the Second Law of Thermodynamics which statesthat if to a system where all the actions are perfectly rever-sible a quantity of heat dQ be communicated at the absolutetemperature 0, then

the integration being extended over any complete cycle ofoperations.

This statement is founded on various axioms by differentphysicists, thus for example Clausius bases it upon the"axiom" that heat cannot of itself pass from one body toanother at a higher temperature, and Sir William Thomsonon the "axiom" that it is impossible by means of inanimatematerial agency to derive mechanical effect from any por-tion of matter by cooling it below the temperature of thecoldest of the surrounding objects.

Thus the Second Law of Thermodynamics is derivedfrom experience and is not a purely dynamical principle.We might have expected a priori from dynamical con-siderations that the principle of the Conservation of Energywould not be sufficient by itself to enable us to deduce allthe relations which exist between the various properties ofbodies. For this principle is rather a dynamical result thana dynamical method and in general is not sufficient byitself to solve completely any dynamical problem.Thus we could not expect that for the dynamical treat-ment of Ph sics the rinci le of the Conservation of Ener


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been tried. Fortunately we possess other methods, such asHamilton's principle of Varying Action and the method ofLagrange's Equations, which hardly require a more detailedknowledge of the structure of the system to which theyare applied than the Conservation of Energy itself and yetare capable of completely determining the motion of thesystem.

3. The object of the following pages is to endeavourto see what results can be deduced by the aid of thesepurely dynamical principles without using the Second Lawof Thermodynamics.The advantages of this method in comparison with thatof the two laws of Thermodynamics are(1) that it is a dynamical method, and so of a much morefundamental character than that involving the use of theSecond Law;

(2) that one principle is sufficient instead of two;(3) that the method can be applied to questions in which

there are no transformations of other forms of energy fromor into heat (except the unavoidable ones due to friction),while for this case the other method degenerates into theprinciple of the Conservation of Energy, which is often notsufficient to solve the problem.The disadvantages of the method on the other hand arethat, since the method is a dynamical one, the results areexpressed

in terms of dynamical quantities, such as energy,momentum, or velocity, and so require further knowledgebefore we can translate them in terms of the physicaluantities we wish to measure such as intensit of a

- . rnybujtu. j_,a.uuia.i,uiy.


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For this reason there are some cases where the SecondLaw of Thermodynamics leads to more definite results thanthe dynamical methods of Hamilton or Lagrange. Evenhere I venture to thinlc the results of the application of thedynamical method will be found interesting, as they showwhat part of these problems can be solved by dynamics, andwhat has to be done by considerations which are the resultsof experience.

4. Many attempts have been made to show that theSecond Law of Thermodynamics is a consequence of theprinciple of Least Action ; none of these proofs seem quitesatisfactory ; but even if the connexion had been proved inan unexceptionable way it would still seem desirable toinvestigate the results of applying the principle of LeastAction, or the equivalent one of Lagrange's Equations,directly to various physical problems.

If these results agree with those obtained by the use ofthe Second Law of Thermodynamics, it will be a kind ofpractical proof of the connexion between this law and theprinciple of Least Action.

5. Considering our almost complete ignorance of thestructure of the bodies which form most of the dynamicalsystems with which we have to deal in physics, it mightseem a somewhat unpromising undertaking to attempt toapply dynamics to such systems. But we must rememberthat the object of this application is not to discover theproperties of such systems in an altogether a priori fashion,but rather to predict their behaviour under certain circum-d-anr c: nft r hflvinor nhs rv rl .it nnrl .r

behind the dial the various are connected a


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pointers byquantity of mechanism of the nature of which we areentirely ignorant. Then if we move one of the pointers, Asay, it may happen that we set another one, B, in motion.

If now we observe how the velocity and position of Bdepend on the velocity and position of A, we can by the aidof dynamics foretell the motion of A when the velocity andposition of B are assigned, and we can do this even thoughwe are ignorant of the nature of the mechanism connectingthe two pointers. Or again we may find that the motion ofB when A is assigned depends to some extent upon thevelocity and position of a third pointer C\ if in this case^veobserve the effect of the motion of C upon that of A and Bwe may deduce by dynamics the way in which the motionof C will be affected by the velocities and positions of thepointers A and B.

This illustrates the way in which dynamical considera-tions may enable us to connect phenomena in differentbranches of physics. For the observation of the motion ofB when that ofA is assigned may be taken to represent theexperimental investigation of some phenomenon in Physics,while the deduction by dynamics of the motion of A whenthat of B is assigned may represent the prediction by theuse of Hamilton's or Lagrange's principle of a new phenome-non which is a consequence of the one investigated experi-mentally.

Thus to take an illustration, suppose we investigateexperimentally the effect of a current of electricity bothsteady and variable upon the torsion of a longitudinally

The method is really equivalent to an extension and


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generalization of the principle of the equality of action andreaction, as when we have two bodies A and J3 acting uponeach other if we observe the motion of B which results whenA moves in a known way we can deduce by the aid of thisprinciple the motion of A when that of B is known. Themore general case which we have to consider in Physics iswhen instead of two bodies attracting each other we havetwo phenomena which mutually influence each other.


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CHAPTER II.THE DYNAMICAL METHODS TO BE EMPLOYED.6. As we do not know the nature of the mechanism of

the physical systems whose action we wish to investigate, allthat we can expect to get by the application of dynamicalprinciples will be relations between various properties ofbodies. And to get these we can only use dynamicalmethods which do not require an intimate knowledge of thesystem to which they are applied.

The methods introduced by Hamilton and Lagrangepossess this advantage and, as they each make the behaviourof the system depend upon the properties of a single func-tion, they reduce the subject to the determination of thisfunction. In general the way that we are able to connectvarious physical phenomena is by seeing from the behaviourof the system under certain circumstances that there must bea term of a definite kind in this function, the existenceof this term will then often by the application of Lagrangianor Hamiltonian methods point to other phenomena besidesthe one that led to its detection.

7. We shall now for convenience of reference collect thedynamical equations which we shall most frequently have touse.


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where T and V are respectively the kinetic and potentialenergies of the system, t the time, and q a coordinate ofany type- In this case f and /, are each supposed to beconstant.

In some cases it is convenient to use the equation in thisform but in others it is more convenient to use Lagrange'sEquations, which may be derived from equation (i) (Routh'sAdvanced Rigid Dynamics, p. 249) and which may bewritten in the form d dL dL , .

~dt~dt~~dq= V- ^2 ''

where L is written for T- V and is called the Lagrangia-nfunction and Q is the external force acting on the systemtending to increase q.

In the preceding equations the kinetic energy is sup-posed to be expressed in terms of the velocities of thecoordinates. In many cases however instead of workingwith the velocities corresponding to all the coordinates it ismore convenient to work with the velocities correspondingto some coordinates but with the momenta corresponding tothe others. This is especially convenient when some ofthe coordinates only enter the Lagrangian function throughtheir differential coefficients and do not themselves occurexplicitly in this function. In a paper " On some Applica-tions of Dynamical Principles to Physical Phenomena

"

(Phil. Trans. 1885, Part n.) I have called these "kinos-thenic " coordinates. In the following pages the term" s eed coordinates " will for the sake of brevit be used

For if be a since


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x speed coordinate,dL*r :we have by Lagrange's equation since no external forceacts on the system d dL , .#^ =0 (3)

as the momentum corresponding to x is dLjdx, this equa-tion shows that it is constant.

8. Routh (Stability of Motion i p. 61) has given a generalmethod which enables us to use the velocities of somecoordinates and the momenta corresponding to the remain-der, and which is applicable whether these latter coordinatesare speed coordinates or not.

The method is as follows : suppose that we wish to usethe velocities of the coordinates q^ ^a ...and the momentacorresponding to the coordinates 1} < s ...then Routh hasshown that if we use instead of Z the new function Z' givenby the equation.

, r . dT . dT .L = L~^m~^^r~^c 4)d


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part the kinetic energy corresponding to the coordi-nate ! then we see by (4) that Z' = the kinetic energy ofthe system minus its potential energy minus twice the kineticenergy corresponding to the coordinates whose velocities areeliminated.

9. If we do not know the structure of this system allthat we can determine by observing its behaviour will bethe Lagrangian function or its modified form, and smcethis function completely determines the motion of thesystem it is all we require for the investigation of itsproperties. We see however that when we calculate the'energy" corresponding to any physical condition the in-terpretation may be ambiguous if the energy is not entirelypotential. For what we really calculate is the Lagrangianfunction or its modified form and this is the kinetic energyminus the potential energy mimes twice the kinetic energycorresponding

to the coordinates whose velocities are elimi-nated. So that the term in the energy which we have cal-culated may be any one of these three things. Thus totake an example, it is said that the energy of a piece ofsoft iron of unit volume, throughout which the intensityof magnetization is uniform and equal to /, is - J2J2k,where k is the coefficient of magnetic induction of theiron, but all that this means is that the term I2j2k occursin the Lagrangian function (modified or otherwise) ofthe system whose motion or configuration produces thephenomenon of magnetisation. And without further con-siderations we do not know whether this represents ano mi-innf r f VinAfir> n rcn? T^lofs nr nnf ntial n rmr T^ltl'

o. amoiguHy ever occurthe out because


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system completely mapped by coordinates,in this case whenever we find a term in the Lagrangianfunction it must be expressed in terms of these coordinatesand their velocities, or it may be the momenta correspondingto them, and \ve can tell by inspection whether the termexpresses kinetic or potential energy. Two investigations inthe second volume of Maxwell's Electricity and Magnetismafford a good illustration of the way in which this ambiguityis cleared away by an increase in the precision of our ideasabout the configuration of the system. In the early part ofthe volume by considering the mechanical forces between twocircuits carrying electric currents, it is shown that two suchcircuits conveying currents i,j possess a quantity of potentialenergy Mij where,/!/ is a quantity depending on the shapeand size of the two circuits and their relative position. Lateron however when coordinates capable of fixing the electricalconfiguration of the system have been introduced it is shownthat the system instead of possessing - Mij units of potentialenergy really possesses + Mij units of kinetic.

n. The following considerations may be useful ashelping to show that this ambiguity is largely verbal and isprobably mainly due to our ignorance of what potentialenergy really is.

Suppose that we have a system fixed by n coordinates,qv qa,...qn of the ordinary kind, that is, coordinates whichoccur explicitly.in the expressions for the kinetic or potentialenergies, and which we shall call positional coordinates,and m kiuosthem'c or speed coordinates < i; < 2,...< OT. Letno fnrl-Vir cnnnns t-linf tVi r nr rin t rms in fli

for the q coordinates if instead of the ordinary Lagrangian


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function L which reduces in this case to the kinetic energywe use the modified function L' given by the equationL'=L-^........................ (6),aor, .dT ,dT . N................ (7),

and where < n < 2 are to be eliminated by the aid of theequations

Thus since the expression for L does not contain anyterms involving the product of the velocity of a q and a coordinate, L will be of the form

M??)~ -MM")'where T(ff) is the kinetic energy arising from the motion ofthe q or positional coordinates, T($$) that arising from themotion of the kinosthenic or speed coordinates.

By Routh's modification of the Lagrangian equationswe have

di do ( T( }, ,but

so that equation (8) reduces tod dT dT

equations of motion would have been ot the type,


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d dT _ dl^ = _ dV (10).dt d(} } dq x ay 1By comparing equations (9) and (10) we see that the

system fixed by the positional coordinates q will behaveexactly like a system whose kinetic energy is T(qq\ and whosepotential energy is T(^ .

Thus we may look on the potential energy of any systemas kinetic energy arising from the motion of systems con-nected with the original system the configurations of thesesystems being capable of being fixed by kinosthenic or speedcoordinates.

Thus from this point of view all energy is kinetic, and allterms in the Lagrangian function express kinetic energy, theonly thing doubtful being whether the kinetic energy is dueto the motion of ignored or positional coordinates- thiscan however be determined at once by inspection.

12. Some of the theorems in dynamics become verymuch simpler from this point of vieAv. Let us take forexample the principle of Least Action that for the uncon-strained motion of a system whose energy remains constant

Tdtis a minimum from one configuration to another and applyit to the system we have been considering in which all theenergy is kinetic but some of it is due to the motion of asystem whose configuration can be fixed by kinostheniccoordinates.

system will by its unguided motion go along the path whichwill take it from one to another in the least


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configurationpossible time. The material system must of course includethe kinosthenic systems whose motion produces the sameeffect as the potential energy of the original system; andtwo configurations are not supposed to coincide unless theconfiguration of these kinosthenic systems coincide also.

This view which regards all potential energy as reallykinetic has the advantage of keeping before us the ideathat it is one of the objects of Physical Science to explainnatural phenomena by means of the properties of matter inmotion. When we have clone this we have got a completephysical explanation of any phenomenon and any furtherexplanation must be rather metaphysical than physical. Itis not so however when we explain the phenomenon as dueto changes in the potential energy of the system ; for potentialenergy cannot be said, in the strict sense of the term, toexplain anything. It does little more than embody theresults of experiments in a form suitable for mathematicalinvestigations.The matter whose motion constitutes the kinetic energyof the kinosthenic systems, the "


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CHAPTER III.APPLICATION OF THESE PRINCIPLES TO PHYSICS.

13. IN our applications of Dynamics to Physics it willbe well to begin with the cases which are the most nearlyallied to those we consider in ordinary Rigid Dynamics.Now in this subject when there is no friction all the motionsare reversible and are chiefly relations between vector quan-tities. We shall therefore begin by considering reversiblevector effects and afterwards go on to reversible effectsinvolving scalar as well as vector relations ; those forexample in which a scalar quantity such as temperature isprominently involved : lastly we shall consider irreversibleeffects. Thus the order in which we shall consider thesubject will be

1. Reversible vector phenomena.2. Reversible scalar phenomena.3. Irreversible phenomena.

14. We shall begin by considering the relations betweenthe phenomena in elasticity, electricity, and magnetism andthe way in which these depend upon the motion andconfiguration of the bodies which exhibit the phenomena.

These phenomena differ from some we shall consider material and the saturation of magnets may impose). The

other on the other hand a multi-


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phenomena depend upontude of coordinates over whose individual motion we haveno control though we have some over their average motion.As the first kind of phenomena most closely resemble thosewe have to do with in ordinary dynamics we shall beginwith them.

15. The first thing we have to do when we wish to applydynamical methods to investigate the motion of a system isto choose coordinates which can fix its configuration.We shall find it necessary to give a more generalmeaning to the term "coordinate" than that which obtainsin ordinary Rigid Dynamics. There a coordinate is ageometrical quantity helping to fix the geometrical con-figuration of the system.

In the applications of Dynamics to Physics however,the configurations of the systems we consider have to be fixed,with respect to such things as distributions of electricity andmagnetism, for example, as well as geometrically, and to dothis we have in the present state of our knowledge to usequantities which are not geometrical.

Againthe coordinates which fix the configurations of the

systems in ordinary dynamics are sufficient to fix themcompletely, while we may feel pretty sure that the coordi-nates which we use to fix the configuration of the systemwith respect to many of its physical properties, though theymay fix it as far as we can observe it, are not sufficient tofix it in every detail ; that is they would not be sufficientto fix it if we had the power of observing differences

trations of this latter point For example, when a spheremoves through an incompressible fluid, we can express the


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kinetic energy of the system comprising both sphere andfluid in terms of the differential coefficients of the threecoordinates which fix the centre of the sphere, though itwould require a practically infinite number of coordinatesto fix the configuration of the fluid completely.Now Thomson and Tait (Natural Philosophy, vol. i.p. 320) have shown how we can "ignore" these coordinateswhen the kinetic energy can be expressed without them, andthat we may treat the system as if it were fully determinedby the coordinates in terms of whose differential coefficientsthe kinetic energy is expressed.

And again Larmor (Proceedings of London Mathemati-cal Society, xv. p. 173) has proved that if Z' be Routh'smodification of the Lagrangian function, that is q^ $..being the coordinates retained, < < 2,.--those ignored, Q1}Q2 ..., $1} $2 the momenta corresponding to these coordinatesrespectively, ifthen 8 fh L'dt=o (IT).Jto

If all the kinetic energy vanishes when the positionalcoordinates q^ ,,... are constant, as is the case when anumber of solids move through a perfect fluid in whichthere is no circulation, L' is the difference between thekinetic and potential energies of the system. If howeverthe kinetic energy does not vanish when the velocities ofthe positional coordinates all vanish, as for example when

and their first differential coefficients, then whatever thesequantities may be, we must have a series of equations of the


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type d dL' dL' _j, ~j~ ~~j O...., ......( 12 j.at aq aqThus we see that we may treat any variable quantities

as coordinates if the modified Lagrangian, function can beexpressed in terms of them and their first differential coeffi-cients. We shall take this as our definition of a coordinate.

1 6. When we introduce a symbol to fix a physicalquantity we may not at first sight be sure whether it is acoordinate or the differential coefficient of one with respectto the time.

For example, we might feel uncertain whether the symbolrepresenting the intensity of a current was a coordinate orthe differential coefficient of one. The simplest dynamicalconsiderations however will enable us to overcome thisdifficulty. Thus if when there is no dissipation of energyby irreversible processes, the quantity represented by thesymbol remains constant under the action of a constantforce tending to alter its value, the energy at the same timeremaining constant, then the symbol is a coordinate.

Again, if it remains constant and not zero when no forceacts upon it, there being no dissipation and the energyremaining constant, the symbol represents a velocity, that is,the differential coefficient of a coordinate with respect tothe time.Let us apply these considerations to the example men-tioned above ; as the intensity of a current flowing through

the rate of change of a coordinate an not by the coordinateitself.


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Specification of Coordinates.17. To fix the configuration of the system so far as the

phenomena we are considering are concerned we shall usethe following kinds of coordinates.

(1) Coordinates to fix the geometrical configuration ofthe system, i.e. to fix the position in space of any bodies offinite size which may be in the system. For this purposewe shall use the coordinates ordinarily used in Rigid Dy-namics and denote them by the letters xl} xz , ^3 ...; andwhen we want to denote a geometrical coordinate generallywithout reference to any one in particular we shall use theletter x.

(2) Coordinates to fix the configuration of the strainsin the system. We shall use for this purpose, as is ordinarilydone in treatises on elasticity, the components parallel tothe axes of x, y, z of the displacements of any small portionof the body, and denote them by the letters a, /3, y respec-tively. For the strains

da d]3 dydx' dy' dz

}

fdy dj!\ /da dy"\dy dz) ' \dz

we shall use the letters e, f, g, a, b, c respectively. It willbe convenient to have a letter typifying these quantitiesgenerally without reference to any one in particular, weshall use the letter w for this purpose.

an e ectr c displacement, ana in a con uc or ne r meintegral of a current flowing through some definite area.


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(4) Coordinates to fix the magnetic configuration. Wemight do this by specifying the intensity of magnetizationat each point, but it is clearer I think to regard the magneticconfiguration as depending, even in the simplest case, upontwo coordinates, one of which is a kinosthenic or speedcoordinate.

This way of looking at it brings it into harmony with thetwo most usual ways of representing the magnetization of abody, viz. Ampere's theory and the hypothesis of MolecularMagnets.

According to Ampere's theory the magnetization is dueto electric currents flowing through perfectly conductingcircuits in the molecules of the magnets. In this case thedifferential coefficient of the kinosthenic coordinate wouldfix the intensity of the current, and the other coordinate theorientation of the planes of the circuits.

According to the Molecular Magnet theory, any magnetof finite size is built up of a large number of small magnetsarranged in a polarized way. Here the momentum corre-sponding to the kinosthenic or speed coordinate may beregarded as fixing the magnetic moment of a little magnet,which it is well fitted to do by its constancy ; the other co-ordinate may be regarded as fixing the arrangement of thelittle magnets in space.

We shall denote the kinosthenic coordinate by andthe geometrical one by 77 and suppose that they are sochosen that the intensity of magnetization at any point is >?,nrViova ic f-Vii= momentum r>ni"ri=cr nnrliriar tr fViF> VinriefTi nir-

1 8. Having chosen me coordinates there are two waysin which we may proceed. We may either write down the


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most general expression for the Lagrangian function in termsof these coordinates and their differential coefficients, andthen investigate the physical consequences of each termin this expression. If these consequences are contradictedby experience we conclude that the term we are consideringdoes not exist in the expression for the Lagrangian function.

Or we may know as the result of experiment that theremust be a certain term in the expression for the Lagrangianfunction and proceed by the application of Lagrange's Equa-tions to develop the consequences of its existence. Thus, forexample, we know by considering the amount of workrequired to establish the electric field that there must be inthe Lagrangian function of unit volume of the dielectric aterm of the form

where K is the specific inductive capacity of the dielectricand D the resultant electric displacement. We can then byapplying Lagrange's equation to this term see what are theconsequences of the specific inductive capacity of thedielectric being altered by strain (see 39).We shall make use of both methods but commence withthe first as being perhaps the most instructive, and alsobecause we shall have a great many examples of the secondmethod later on since the scalar phenomena do not admitof being treated by the first method.

19. In using the first method the first thing we have to

and that we work with Routh's modification of the Lagran-


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gian function.The most general expression for the terms correspond-ing to the kinetic part of this function, which is the onlypart we can typify, is of the form

+ 2 yjt+ (se/o/) ob8 + (7717) if + () 2+ 2 (xy) xy + 2 (xw) xw + 2 (xrj) cti]+ 2 (x) x+ 2 (yw] yw 4- 2 (yrj) yrj+ 2

These terms may be divided into fifteen types.There are five sets which are quadratic functions of the

velocity or momentum corresponding to one kind of coordi-nate. Each of these five sets must exist in actual physicalsystems if there is anything analogous to inertia in thephenomena which the corresponding coordinates typify.

Again, there are ten sets of terms of the type(xy) xy or (xg) xg,

involving the product of two velocities or a velocity anda momentum of two coordinates of different kinds.

To determine whether any particular term of this typeexists or not we must determine what the physical conse-uences of it would be ; if these are found to be contrary

Lagrangian function of the type


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where/ and q may be any of the five kinds of coordinates weare considering.

Then we have by Routh's modification of Lagrange'sequations

d_d _d_ =p , Ndt dp dp

.................. v

where P is the external force of type/ acting on the system.Thus the effect of the term(pq)pq

is equivalent to the existence of a force of the type / equalto

{(#>*} -*{(*) t}.that is,

a force of type q equal to

KM *}-!{


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to some physical phenomenon and as it is clearer to take adefinite case to illustrate this, let us suppose that p is thegeometrical coordinate symbolized by x, and q the electricalcoordinate y.

Then if the term (xy) xy occurred in the expression forthe Lagrangian function, the mechanical force produced bya steady current would not be the same as that produced bya variable one momentarily of the same intensity. This isso because by the expression (14) there is the term

(xy) yin the expression for the force of type x, that is themechanical- force, and as y is zero if the current is steady,there would be a mechanical force depending on the rate ofvariation of the current if this term existed.

Again, we see from the term -=- (xy) y~ in the expressioncty(14), remembering that/ stands for x and q for y, that if(xy) were a function of y the current would produce amechanical force proportional to its square, so that the forcewould not be reversed if the direction of the current wasreversed.

Or again, if we consider the expression for the force oftype y or q, that is the electromotive force, we see that theexistence of this term implies the production of an electro-motive force by a body whose velocity is changing,depending upon the acceleration of the body ; this is shownb the existence of the term (x ) x in (15), the expression

one that would not be reversed when the direction of motioof the body was reversed.


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As none of these effects have been observed we conclucthat this term does not exist in the expression for tr.Lagrangian function of a physical system (see MaxweiElectricity and Magnetism, 574).

21. We shall now go through the various types of ternwhich involve the product of the velocities of two coorcnates of different kinds, or a velocity of one kind and tlmomentum of another, in order to see whether they exist o t-o rvm rrn afi r> fr rr*&C! rlot-voTM-It-nrr ii-rn-m ^-li**

y. Terms of the form


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Terms of this kind only involve the production of anelectromotive force in a varying magnetic field, the electro-motive force varying as the rate of change of the magneticfield. This is the well-known phenomenon of the produc-tion of an electromotive force round a circuit whenever thenumber of lines of magnetic force passing through it ischanged.As the term we are considering is the only one in theLagrangian function which could give rise to an effect ofthis kind without also giving rise to other effects which havenot been verified by experience, we conclude that this termdoes exist.

8. Terms of the form(wtf) wrj.

If we take any molecular theory of magnetism, such asAmpere's, where the magnetic field depends ' upon thearrangement of the molecules of the body, we should ratherexpect this term to exist. The consequences of its existencehave however not been detected by experiments.

If this term existed, then considering in the first placeits effect upon the magnetic configuration we see that avibrating body should produce magnetic effects dependingupon the vibrations. Secondly, considering the effects ofthis term on the strain configuration we see that thereshould be a distorting force depending upon the rate ofacceleration of the magnetic field. As neither of theseaffo M-o l-io rf> hoon i~J- ce>nrf>r1 tliova IP v>rs CLtrlAm^^^ nf t-l^n.

These would involve the existence of distorting forcesdepending upon the rate of change of the magnetic field,


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and we have no evidence of any such effect.10. Terms of the form

If we assume Ampere's hypothesis of molecule currents thisterm is of the same nature as the term (x) x which wediscussed before, so that unless the properties of thesemolecular circuits differ essentially from those of finite sizewith which we are acquainted this term cannot exist.

21. Summing up the results of the foregoing considera-tions, we arrive at the conclusion that the terms in theLagrangian function which represent the kinetic energydepending upon the five classes of coordinates we areconsidering must be of one or other of the following types :

(xx) x'

(wiv) i'.(17).

22. We might make a model with five degrees offreedom which would illustrate the connection betweenthese phenomena which are fixed by coordinates of fivetypes.And if we arrange the model so that its configuration

analogous to that possessed by the physical system, thenthe working of this model will illustrate the interaction of


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phenomena in electricity, magnetism, elasticity &c., and anyphenomenon exhibited by the model will have its counter-part in the phenomena exhibited in these subjects.When however we know the expression for the energyof such a model, there is no necessity to construct it inorder to see how it will work, as we can deduce all the rulesof working by the application of Lagrange's Equations.And from one point of view we may look upon the methodwe are using in this book as that of forming, not a model,but the expression for the Lagrangian function of a modelevery property of which must correspond to some actualphysical phenomenon.


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CHAPTER IV.DISCUSSION OF THE TERMS IN THE LAGRANGIANFUNCTION.

23. WE must now proceed to examine the terms in theexpression (17) more in detail, and find what coordinatesenter into the various coefficients (xx), (yy) When wehave proved that these coefficients involve some particularcoordinates we must go on to see what the physicalconsequences will be. In this way we shall be able toobtain many relations between the phenomena in electri-city, magnetism and elasticity.

24. The first term we have to consider is {xx} x2, Avhichcorresponds to the expression for the ordinary kinetic energyof a system of bodies. We know that {xx} may be a functionof the geometrical coordinate typified by x, but we need notstop to consider the consequences of this as they are fullydeveloped in treatises on the Dynamics of a System of RigidBodies.

Next {xx} may involve the electrical coordinate y, forin a paper "On the Effects produced by the Motion of

1 ,e a / o\-m+~ \v .................. (18),2 15 a J x "where


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permeability of the dielectric surrounding it.The existence in the kinetic energy of this term, which isdue to the "displacement currents" started in the surroundingdielectric by the motion of the electrification on the sphere,shows that electricity behaves in some respects very muchas. if it had mass. For we see by the expression (18) thatthe kinetic energy of an electrified sphere is the same asif the mass of the body had been increased by 4/x


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or about 4xio~ l /p,where p is the mean density of the substance enclosed by theelectrified surface.

Thus the alteration in mass, even if the mean densityinside the surface is as small as that of air at the atmosphericpressure and oC., is only about 5 x io~ 16 of the originalmass, and is much too small to be observed.

Let us now consider the electrical effects of this term.Let Q be the electromotive force acting on the sphere.The energy of the system, using the same notation as before,,

is/i 2 lie \ i(~m-\ zr + -\2 15 a J 25 a ) 2 Ka '

If v be increased by $v and e by Se, the increment in theenergy is

15 a ) 15 a Ka'and by the Conservation of Energy this must equal

QSe,so that

1 5 a ) 1 5 a Ka ^ ^ - Z>Since no mechanical force acts upon the system themomentum will be constant, so that

/ 4 * z\ 8 /j. CD-

So that the capacity of the sphere is increased in the


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ratio of r to i / i - i*.Kv2, or since according to theElectromagnetic Theory of Light ^K~ i / Vs where V is thevelocity of light through the dielectric, the capacity of the

/ 4 &2sphere is increased in the ratio of i to i / i - - -p. Thusthe capacity of a condenser in motion will not be the sameas that of the same condenser at rest, but as the differencedepends on the square of the ratio of the velocity of thecondenser to the velocity of light it will be exceedinglysmall.

If the earth does not carry the ether with it, a point onthe earth's surface will be moving relatively to the ether, andthe alterations in the velocity of such a point which occurduring the day will produce a small diurnal variation in thecapacities of condensers.

25. When we have two spheres of radius a and amoving with velocities v and if respectively the kineticenergy (see the paper on the " Effects produced by theMotion of Electrified Bodies," Phil. Mag., April, 1881),assuming Maxwell's theory, is

1 2 uA, g /I,2 u\ ,. ufl?'cose- m + - - }V* + - m + '-r-}l/i + --^2 15 a ) \2 15 a J $where R is the distance between the centres of the twospheres and e the angle between their directions of motion;

. .' J ntffi -i-rT- a -fi ra tr i-li n TIT nocoe nnrl r^no vnroo -^-f* +-1-* c*

equations in the way explained in 20, show that these termsrequire the existence of the following forces on the twospheres, i> and if being the accelerations of the spheres


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respectively. ,On the first sphere.a. An attraction

fj.ee' ,- ^ VV COS3^'along the line joining the centres of the spheres.

/fi. A force^eer .,'__ 713*

in the direction opposite to the acceleration of the secondsphere.

y. A forcei , , d f

in the direction opposite to the direction of motion of thesecond sphere.There are corresponding forces on the second sphere, andwe see that unless the two spheres move with equal and

uniform velocities in the same direction the forces on thetwo spheres are not equal and opposite. The sum of themomenta of the two spheres will not increase indefinitelyhowever, since the sum of the actions and reactions is notconstant but is a function of the accelerations.

We may easily prove that if x, j>, z are the coordinates ofthe centre of one sphere, d, y', z' those of the other, then(' 4 fjies p.ee' }dx ( , 4 /x/2 ju, ee' \ dx'

. un er e ec r ca . . .the conclusion that the coefficient {xx} is a function of theelectrification of the system.


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Thus according to Clausius' theory (Crelle, 82, p. 85)the forces between two small electrified bodies in motion arethe same as if, using the same notation as before, there wasthe term

, , cos e eeap ywj ----- _ --R KRin the expression for the Lagrangian function. The first ofthese is the same as the term we have just been considering.

The forces which according to Weber's theory (Abhand-lungen der Koniglich Sachsischen Gesellschaft der Wisscn-schaften, 1846,13.

211. Maxwell's Electricity and Magnetism,2nd Edit. vol. n. 853) exist between two electrified bodiesin motion may easily be shown to be the same as thosewhich would exist if in the Lagrangian function there wasthe term

where x, y, z, x', y', %'. are the coordinates of the centres ofthe electrified bodies and te, v, TV, u', v', w' the componentsof their velocities parallel to the axes of coordinates.

This term leads however to inadmissible results, as wecan see by taking the simple case when the bodies are movingin the same straight line which we may take as the axis ofx. In this case the term in the kinetic energy reduces to

zee IR since the coefficients of n and u are each increasedby half this amount. Hence if we take e and e of oppositesigns and suppose the electrifications are great enough to


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make zee'jR greater than the masses of one or both of thebodies, then one of the bodies at least will behave as if itsmass were negative. This is so contrary to experience thatwe conclude the theory cannot be right. This consequenceof Weber's theory was first pointed out by v. Helmholtz( Wissenschaftliche Abhandlungen, i. p. 647).

The forces which according to Riemann's theory, givenin his posthumous work Schwere^ Elektridtdt und Magnetis-mus, p. 326, exist between two moving electrified bodies mayeasily be shown to be the same as those which would exist ifthere were the termp&

-g {(U - U') tt +(V- 7/)' +(W- Wj} -KRin the expression for the Lagrangian function. We caneasily

see that this theory is open to the same objectionas Weber's, that is, it would make an electrified bodybehave in some cases as if its mass were negative.

27. If we regard the expression for the kinetic energyfrom the point of view of its bearing on electrical phenomenawe shall see that it shows that if we connect the terminalsof a battery to two spheres made of conducting material, thequantity of electricity on the spheres will depend upon theirvelocities.

We see from the expression (22) for the kinetic energy ofa moving conductor that if we have a number of conductorsmoving about in the electric field there will be a positive term

decrease in the potential energy produced by a gnelectrification, since an increase in the potential eneicorresponds to a decrease in the Lagrangian functii


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Thus the presence of the moving conductors is equivalito a diminution in the stiffness of the dielectric with respto alterations in its state of electrification. And therefithe speed with which electrical oscillations are propagalacross any medium will be diminished by the presencemolecules moving about in it; the diminution being pportional to the square of the ratio of the velocity of 1molecules to the velocity with which light is propagatacross the medium. Thus if the electromagnetic theorylight is true the result we have been discussing hasimportant bearing on the effect of the molecules of maton the rate of propagation of light.

28. We can see that {xx} may be a function of tstrain coordinates, for let us take the case when {xx} is tmoment of inertia of a bar about an axis through its centithen it is evident if the bar be compressed in the midcand pulled out at the ends that the moment of inertia vibe less than if the bar were unstrained, for the effectthe strain has

practicallybeen to bring the matter formi

the bar nearer to the axis. Thus the moment of inerand therefore {xx} may depend upon the strain coordinati

These coordinates will in general only enter {xx} throuthe expression for the alteration in the density of the strainbody, i.e. through

da. dfi dy . .*" (23)r

8 / (T- V] t=Qwe shall easily find that the presence of (23) in {xx}leads to the introduction of the so-called forces"


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"centrifugalinto the equation of elasticity for a rotating elastic solid.This however we shall leave as an exercise for the reader.

29. Let us now consider that part of the Lagrangianfunction which depends upon the velocities of the electricalcoordinates, i.e. the part denoted by

i!f \K.y y\) y? ~^~ 2 (y y) y\y ~*~ }Let us take the case of two conducting circuits whose

electrical configuration .is fixed by the coordinates j]5 j/2,where jp,, ya are the currents flowing through the circuitsrespectively.

This part of the Lagrangian function may in this case beconveniently written

Now we can fix the geometrical configuration of the twocircuits if we have coordinates which can fix the positionof the centre of gravity and the shape and situation of thefirst circuit, the shape of the second circuit and its positionrelatively to the first.Let us denote by x2 - x l any coordinate which helps tofix the position of one circuit relatively to the other, and by

2 coordinates helping to fix the shape of the first andsecond circuits respectively.

It is evident that the kinetic energy must be expressiblein terms of these coordinates, for the only coordinates neces-sarv tn fix fli svstem whir.h we have omitted are those fixine-

these coordinates.If we write for a moment x instead of x2 --xl (a coordi-

nate helping to fix the position of one circuit relatively to


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the other) then by Lagrange's Equations we see that theseterms in the kinetic energy correspond to the existence of aforce tending to increase x equal to

i dL . dM . . i dN .We see from this expression that dLjdx, and dN/dxmust vanish, otherwise there would be a force between the

two circuits even though the current in one of themvanished. The quantities L and N are by definition thecoefficients of self-induction of the two circuits, and hencewe see that the coefficient of the self-induction of a circuit isindependent of the position of other circuits in its neighbour-hood and is therefore the same as if these circuits were re-moved 1 .

By (16) the force tending to increase x isdM .that is there is a force between the two circuits proportionalto the product of the currents flowing through them, and alsoto the differential coefficient with respect to the coordinatealong which the force is reckoned of a function which doesnot involve the electrical coordinates. This correspondsexactly to the mechanical forces which are actually observed

1 This is quite consistent with the apparent diminution in the self-induction caused bv a neirrlihnrmno- rirr.nif -when an alternatin current-

. . _ . . . -ical forces which two circuits carrying currents are known toexert upon each other proves that the term Mj> }j>2 exists inthe expression for the Lagrangian function.


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Let us now go on to consider the effect of these termson the electrical configuration of the two circuits.

By Lagrange's Equation for the coordinate y } we haved dL dU T , '_. __ _ _ _ T/dt dj> l dy } ~ Y ^

where y, is the external electromotive force tending toincrease y^ . Now as we shall prove directly dL'jdy l = o, sothat the effects on the electrical configuration of the firstcircuit, arising from the term

are the same as would be produced by an external electro-motive force tending to increase y l equal to

(26).

Thus if any of the four quantities L, M, y^ yz vary invalue there is an electromotive force acting round thecircuit through which the current y^ flows. And theexpression (26) gives the E. M. F. produced either by themotion of neighbouring circuits conveying currents or byalterations in the magnitudes of the currents flowing throughthe circuits.

This example is given in Maxwell's Electricity andMagnetism, vol. n. part iv. chapter vi., and it is one whichillustrates the Dower of the dvnamical method verv well.

law of the induction of currents follows at once by theapplication of Lagrange's Equations.The problem we have just been considering is dynamic-


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ally equivalent to finding the equations of motion of aparticle with two degrees of freedom when under the actionof any forces. We know that these cannot be deduced bythe aid of the principle of the Conservation of Energy alone,for to take the simplest case of all, that in which no forcesact upon the particle, the principle of the Conservation ofEnergy is satisfied if the velocity is constant whether theparticle moves in a straight line or not. From this analogywe see that when we have two circuits the principle of theConservation of Energy is not sufficient to deduce theequations of motion, and that some other principle must beassumed implicitly in those proofs which profess to deducethese equations by means of the Conservation of Energyalone.

30. There is no experimental evidence to show that\yy] is a function of the electrical coodinates y, and itcertainly is not when the electrical systems consist of aseries of conducting circuits, for if it were the coefficients ofself and mutual induction would depend upon the length oftime the currents had been flowing through the circuits.And in any case it would require the existence of electro-motive forces which would not be reversed if the directionof all the electric displacements in the field were re-versed.

31. Similar reasoning will show that \yy\ cannot be afunction of the ma netic coordinates, for if it were there

of the strain coordinates or not. If it is then the coefficientsof self and mutual induction of a number of circuits mustdepend upon the state of strain of the wires forming the


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circuits. This result though not impossible has never beendetected, and it is contrary to Ampere's hypothesis that theforce exerted by a current depends only upon its strengthand position and not upon the nature or state of thematerial through which it flows.Then again, if we consider what the effect on the elasticproperties of the substance would be if {yy} were a functionof the strain coordinates, we see at once that it wouldindicate that the elastic properties of a wire would bealtered while an electric current was passing through it.

The evidence of various experimenters on this point issomewhat conflicting. Both Wertheim {Ann. de Chim. etde Phys. [3] 12, p. 610, Wiedemann's Elektricitdt, u. p. 403)and Tomlinson have observed that the elasticity of a wire isdiminished when a current passes through it and that thisdiminution is not due to the heat generated by the current.Streintz ( Wien. Ber. [2] 67, p. 323, Wiedemann's Elektricitdt,n. p. 404) on the other hand was unable to detect any sucheffect.

But even if this effect were indisputably established itwould not prove rigorously that {yy} is a function of thestrain coordinates, for as we shall endeavour to show whenwe consider electrical resistance this effect might have beendue to another cause.

To sum up we see that {yy} is a function of theeometrical coordinates but not of the electric or ma netic

electrical or strain coordinates. Thus the terms we aabout to consider in the Lagrangian function of urvolume of a substance are those we have denoted by


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we may have in addition to these terms' arising from tlpotential energy.

In order to begin with as simple a case as possible 1us suppose that all the magnetic changes take place indenitely slowly ; in this case we may neglect the term

and confine our attention to the terms

or as it is more convenient to write them

Let us take first the case when the magnetizationparallel to one of the axes, x for example, and let us denothe magnetic force parallel to this direction by H and tlintensity of magnetization by /, where by definition

f=rf........... - ............ (28).The investigation in 389 of Maxwell's Electricity at

Magnetism, shows that if we suppose that all the energy .the magnetic field is resident in the magnets, there is in tlLagrangian function for unit volume of a magnet the term

HI.

but we have seen in 9 that the question whether energydetermined in this manner is kinetic or potential is reallyleft unsettled : what is actually proved is that a certain term


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exists in the Lagrangian function.If we suppose that the energy is distributed throughout

the whole of the magnetic field, including unmagnetizedsubstances as well as magnets, then the investigation in 635of the Electricity and Magnetism shows that the Lagrangianfunction of unit volume anywhere in the magnetic field con-tains the term

where B is the magnetic induction.These two ways of regarding the energy in the magnetic

field lead to identical results; and as we shall for thepresent confine our attention to the magnetized substanceswe shall find it more convenient to adopt the first methodof looking at the question.We have seen that the Lagrangian function for unitvolume of a magnet contains the term

HI,or in our notation

and this is the term we previously denoted byitf}*

Since the magnetic changes are supposed to take placeindefinitely slowly, Lagrange's equation for the t] coordinatereduces to

and since is supposed to remain constant and therefore%d-r\ = dl,

this be written


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may

So that if k be the coefficient of magnetic induction anddefined by the equationwe have by (31)

I=-3>and therefore

(33).If we know the way in which / varies with H we could

by this equation express A as a function of 7. The relationbetween /and H'v$> however in general so complicated thatthere seems but little advantage to be gained by taking someempirical formula which connects the two and determiningA by its help.For small values of ff, Lord Rayleigh (Phil. Mag. 23,p. 225, 1887) has shown that IIH is constant, so byequation (33) A in this case is also constant.

34. The mechanical force parallel to the axis of xacting on unit volume of the magnet is

' dL'

or

1

dx.dH


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and this is the mechanical force parallel to the axis of xacting on unit volume of the magnet. This expression mayalso be written

k^-2 dxwith similar expressions for the components parallel to theaxes ofy and z.

These are the same expressions for this force as thosegiven in Maxwell's Electricity and Magnetism, vol. IT. p. 70,the consequences of which are as is well known in harmonywith Faraday's investigations on the way in which para-magnetic and diamagnetic bodies move when placed in avariable magnetic field.

35. We have just investigated the mechanical forcesproduced by a magnetic field ; we shall now proceed toinvestigate some of the stresses produced by it.

Let us take the case of a cylindrical bar of soft ironwhose axis coincides with the axis of x, and suppose that itis magnetized along its axis. Let e, f, g be the dilatationsof the bar parallel to the axes of x, y, z respectively. Weshall at present assume that there is no torsion in the bar.We shall suppose that the changes in the strains take placeso slowly that we may neglect the kinetic energy arisingfrom them.

The potential energy due to these strains is

neglecting those depending on the rate of variation 01 thequantities which rate we shall assume to be indefinitely smaThe experiments of Villari and Sir William Thomsi(Wiedemann's Elektricttat, in. p. 70 j) have shown that


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depends upon the strain in the magnet, hence by equatii(32) A will be a function of the strains. We shall piceed to investigate the stresses which arise in consequenof this. Using the Hamiltonian principle

t,

and substituting da/dx, dftjdy, dy/dz for e, f, g respective!we get the following equations by equating to zero tlvariation caused by changing a into a + Sa

d_L __ d d_L _~dx dx de ~ ;inside the bar, dLat the boundary.

By equating to zero the variation caused by changinginto /? + 8/3 we get dL _d_d ==0 .

dy dy df' ^i)

inside the bar,dL-af

= (38)at the boundary.

dL d dL . .~dk- -dzTs = * .................. (39)inside the bar ;


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dL^r ^

at the boundary.The first and second terms in the equations (35), (37)

and (39) may conveniently be considered separately. SinceH is the only quantity in the expression for L which caninvolve the coordinates x, y, or z explicitly the terms

dL dL dLdx' ~d~y' ~ds

reduce to.dH ,dff .dHdH dH JfT dHor kH-j-. kH r- , kff -j-dx ay -az

respectively.

These are the expressions for the components of themechanical force acting on the body, and it is shown inMaxwell's Electricity and Magnetism^ 642, that this dis-tribution of force would strain the body in the same way as"a hydrostatic pressure ff2/Sir combined with a tensionBPll^Tr along the lines of force," B being the magneticinduction. Thus we may suppose that the strains arisingf -nrr> 4-Uar-o + -I- T om rn^ m Tf a f rf o vo f 1- o o frO i V. c- Ano.

DYNAMICS.


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de#dA_' 'df '

J

Solving these equations and putting rj -I we get2.111 d , , ^ m n(d, , r^ d

dg

>- (^)ldg )dIf the magnet is symmetrical about its axis we have

/=


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d

mde __dl~ 3111- 11 ~k 3

m - nyn - n1H + 11

dfdk m + 11 I dkde yn n I? ~df

So that these equations become\ dk_He

df in 11 i2H dl* ~ yn-n lcNow if the coefficient of magnetization depends upon

the strains, the intensity of magnetization of the barwhen under the action of a constant magnetizing forcewill be altered by strain, and in order to compare theformulae with the results of experiments \ve shall find itmore convenient to express de/dl2, dfldP in terms of thechanges which take place in the intensity of magnetizationwhen the bar is stretched rather than in terms of dk\dcand dk\df.We have I = kff,so that when H is supposed to be constant

dl ^dk,- H -,- +de de dl de 'and equations (46) may be written

de / dk\ / m i dl mdl ~ \ dl)\yn n kl dedf f , r dk\f m-n i dl

(47),

11 i dl\ 1

f - r dk\f m ~ n i dT, = T--H-JJ. ( = >-?->? r iu 11 1> T f

yn-nkl df) I ,> (40).mn i al\ x

Kirchhoff (Wied. Ann. xxiv. p. 52, xxv. p. 601) hasinvestigated the effect of this on a small soft iron sphereplaced in a uniform magnetic field and has shown that itwould an elongation of the sphere along the lines


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produceof force and a contraction at right angles to them. We maytherefore assume that in general this distribution of stresscauses an expansion of the magnet in the direction of thelines of force and a contraction in all directions perpen-dicular to this.

The expressions for the strains in a magnetizablesubstance placed in the magnetic field have also been in-vestigated by v. Helmholtz (Wied. Ann. xui. p. 385). Theobject of the investigations of v. Helmholtz and Kirchhoffwas rather different from that of Maxwell. Maxwell's objectwas to show that his distribution of stress would producethe same forces between magnetized bodies as those whichare observed in the magnetic field, while v. Helmholtz andKirchhoff' s object was to show that it follows from the prin-ciple of the Conservation of Energy that, whatever theoryof electricity and magnetism we assume, the bodies in theelectric or magnetic field must be strained as if they wereacted upon by a certain distribution of stress which in thesimplest case is the same as that given by Maxwell.We have in addition to the strain produced by thesestresses, the strains depending upon the alteration of theintensity of magnetization with stress along and perpen-dicular to the lines of force.

The effect of stress along the lines of force on theof iron has been Villari

the intensity of magnetization is diminished by stretching.Sir William Thomson also investigated the effect of

stress at right angles to the lines of magnetic force onthe intensity of magnetization and found that this was in


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general opposite to that of tension along the lines of force,so that for small values of the magnetizing force extensionat right angles to the lines of force diminishes the mag-netization, while for larger values of this force it increasesit. The critical value of the force in this case however ishigher than that for tension along the lines of force.

Thus, except when the magnetizing force is between thecritical values, dllde and dlldf have opposite signs, hencewe see by equation (48) that except in this case, sinceProf. Swing's measurements show that Hdkldl is alwaysless than unity,

de dldF anCi dehave the same sign, and

df dl, ,a ano. ,dl z de

have opposite signs.Now dllde is positive or negative according as themagnetizing force is less or greater than the critical value,so that when the magnetizing force is less than the criticalvalue the extension we are investigating will increase withthe magnetic force, but when the magnetizing force isgreater than this value the extension will diminish as theforce increases.

As we mentioned before the strain produced b Maxwell's

less than the critical value this strain and the strain we havejust investigated act in the same way, but when the force isgreater they act in opposite directions.

Joule's investigations (Phil Mag. 30, pp. 76, 225, 1847)


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prove that the length of an iron bar increases when itis magnetized and as far as the experiments went theincrease in the length was proportional to -the square of themagnetizing force. Mr Shelford Bidwell (Proc. Roy. Soc.XL. p. 109) however has lately shown that when the mag-netizing force is very large the magnet shortens as themagnetizing force increases.

Comparing these experimental results with our theoreticalconclusions we see that they are in accordance when themagnetizing force is small, and that when the magnetizingforce is large they indicate that the strains due to the samecause as that which causes the intensity of magnetization toalter with strains are more powerful than those arising fromMaxwell's distribution of stress. Prof. Swing's experiments6n the effect of strain on magnetization (" ExperimentalResearches in Magnetism," Phil. Trans. 1885, part n. p.585) would seem to show that this must be the case. ForKirchhoff (Wiedemann's Annalen, xxv. p. 60 1) has shownthat the greatest increase in length which Maxwell's stressescan produce in a soft iron sphere whose radius is J?, placedin a uniform magnetic field where the force at an infinitedistance from the sphere is .H, is

i 7 67r E 'where E is one of the constants of elasticit for soft iron

Now according to Prof. Ewing's experiments the intensityof magnetization of a soft iron wire which was represented by


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181 when there was no load was increased to 237 when thewire was loaded with a kilogramme, so that in this case8/i- = -... nearly ............ (50).

The diameter of the wire was such that the load, of akilogramme corresponded to a stress of about 2 x 1.0" persquare centimetre in C.G.S. units, so that if q be Young'smodulus for the wire and 8e the extension produced by theload qc = 2 x io 8 :for wrought iron q\n is about 2-5, so that

7zS


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from strain has been very much underestimated. IndeProf. Ewing (loc. at.) expresses his opinion that " ifdeal only with very small stresses it is doubtful whetlany reversal of the positive effect of stress wouldreached even at the highest obtainable value of the magnczation." By the positive effect of stress Prof. Ewing meaan increase of magnetization with an increase of stress, tmagnetizing force remaining constant.

Bidwell's discovery that defdl3 is negative when tmagnetization exceeds a certain value, in conjunction wthe theoretical results we have been investigating in tparagraph, shows that when the magnetization reaches tvalue the positive effects of stress must be reversed. Tmagnet in this case however is not free from stresses as ilacted on by those called into play by the magnetization.

36. If the dilatation in volume e+ zf be denoted bythen the part of 8 due to the same cause as that whimakes the intensity of magnetization depend upon strainby (48) given by the equation

_^ _ i i_ fdj_ dl\f _rrdk_dl* ~ yn- it kl \de + 2 df) V M ~dlJoule's experiments show that the dilatation in t

volume if it exists at all must be very small compared wfliA *lrmnoftr*n oo TIP* nm c nr^f n Klo. fr rlof^r*f if f IT -MI rrli

dl dlde + 2 ctf

must be small, so that dllde and dl\df must have oppositesigns except when they are very small. This agrees with


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the results of Sir William Thomson's experiments on theeffects of traction along and perpendicular to the lines offorce on the intensity of magnetization ; as, except in theneighbourhood of the critical magnetic forces when dlldcand dlldf are both small, traction along and perpendicularto the lines of force produced opposite results.

If we assume that Joule's experiments prove that thereis no change in volume then by equation (52)

dl dland equation (48) reduces to

de i dl dk* , .(53) "37. The critical value of the intensity of magnetization,

i.e. the intensity when the magnetization is neither increasednor decreased by a small strain, will, since by (32) and (47)

dk\ dl d d . ... , ............. (54),e d

be given by the equation

* In my paper on " Some Applications of Dynamics to PhysicalPhenomena," Part I. Phil. Trans. Part n. 1885 this equation has thewrong sign, which was carried down from equation (51) in the same

itAjde must be a function of


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Now if the term ye' exists, the coefficient of es in theLagrangian function will contain the term \yl" and so Avillinvolve the state of magnetization of the body. Thecoefficients of elasticity however are linear functions of thecoefficient of e 3 in the Lagrangian function so that if thislatter quantity depends upon the state of magnetization, thecoefficients of elasticity will do the same. We concludetherefore that the elasticity of an iron bar must be altered bymagnetization. This effect does not seem to have beenobserved.

If in the expression for A we neglect powers above thesecond we have

~-flde

'

and therefored dde d'P

The right hand side of this equation changes sign whene passes through the value

Now the effects of strain on the intensity of magnetizationand of magnetization upon strain depend by (32) and (47) expect magnetization on

strain of an iron rod to depend upon the strain previouslyexisting in the rod, in such a way that when the strain wasless than a critical value magnetization would increase thelength of the rod, and when it was greater than this value


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magnetization would have the contrary effect and tend toshorten the rod. This agrees with the result of Joule'sexperiments as he found that when the soft iron wires werestretched beyond a certain limit they became shorter insteadof longer when they were magnetized.38. So far we have only considered the effect of expan-sion and contraction upon the intensity of magnetizationand vice-versa. We can however in a similar way discussthe effects of torsion upon the magnetic properties of ironwire.

Let us now suppose that twist is the only strain in an ironwire which is longitudinally magnetized and has a twist cabout its axis, then, using the same notation as before, theterms in the Lagrangian function depending upon strainand magnetization are

By a similar method to that employed in the case ofdilatations we can prove that the twist c due to the samecause as that which makes the intensity of magnetizationalter with the torsion is given by the equation

dr~n dNow when a twisted bar is magnetized it untwists toa certain extent if the ma netization is intense but the twist

This shows that if A be expanded in powers of c thefirst power must be absent, otherwise by equation (58) anuntwisted bar would twist when it was magnetized. HenceA must contain a term in c~ and therefore the coefficient ofr in the Lagrangian function must contain a term


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proportionalto 7s. Now the coefficient of r\n the Lagrangian functionis proportional to the coefficient of rigidity and hence we seethat the rigidity of iron wire will be altered by magnetization.

Since the twist diminishes with strong magnetization wesee by equation (58) that the coefficient of c'\n

must be negative when / is large and hence that the co-efficient of c* in A must be negative. Let us call thiscoefficient y, the coefficient of f in the Lagrangianfunction is

-iy'/ 3 - ;l,but the apparent coefficient of rigidity is twice the coefficientof - r in the Lagrangian function so that in this case theapparent coefficient of rigidity is

n ' J 'Thus in this case the effect of strong magnetization is toincrease the rigidity, so that the same couple will not twistthe wire as much when it is strongly magnetized as when itis unmagnetized.

When the intensity of magnetization is small the oppositewill be the case, as in this case the twist in a wire increaseswhen it is longitudinally magnetized.

d d_dP dcwe have


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or approximately since d z(AP}lndc" is very smalldc _ i dl ( dk\~;~f~n T~T ~ 7 1 " " 7~ ) ( 00 /dP nkl dc \ dc) v 'We see by this equation that when the magnetization is

so strong that magnetizing the wire diminishes the twist init, then twisting the wire will diminish the intensity ofmagnetization. On the other hand when the intensity ofmagnetization is so small that magnetizing the wire increasesthe twist in it then twisting the wire will increase theintensity of magnetization.The reciprocal relations between torsion and magnetiza-tion have been experimentally investigated by Wiedemann(Lehre von der Elektricitat, in. p. 692) and he arrangesthe corresponding results in parallel columns. These arealso quoted in Prof. Chrystal's article on Magnetism in theEncyclopedia Britannica. The following is one set of thecorresponding statements.

" 5. If a wire under the influence of a twisting strain ismagnetized, the twist increases with weak but diminisheswith strong magnetization."

"V. If a bar under the influence of a longitudinalmagnetizing force is twisted the magnetization increaseswith small twists but decreases with large ones."

Comparing these statements with the results we have

part of V is true if the magnetization -is weak, but the oppositeis true if the magnetization is strong. Further since by Vthe influence of twist on magnetization depends upon thesize of the twist, it follows by equation (60) that the influence


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of magnetization upon twist must depend upon the size ofthe twist so that 5 is only true when the twist is on oneside of a critical value, when it is on the other side thecontrary is true.The existence of a critical twist as well as a criticalmagnetization makes the verbal enunciation of the relationsbetween torsion and magnetization cumbrous; they are allhowever expressed by equation (60).

39. Strains in a dielectric produced by theelectric field. The strains produced in a dielectric bythe electric field can be found by a method so similar tothat used in the last two paragraphs that we shall considerthem here though they have no connexion with the terms inthe Lagrangian function which we have been considering.Let/, y, r be the electric displacements parallel to theaxes ofx,y, z respectively, then if the body is isotropic, theterms in the Lagrangian function of unit volume of thedielectric which depend upon the coordinates fixing thestrains and electric configuration if the dielectric is freefrom torsion are,

%n (t +/2 +/ - tef- 2eg~ 2/)...(6i)where e f are the dilatations arallel to the axes of x v z

spectivelyva ue equations oecome re-

*2 -h q* + r2 } ~ ~ H- ;;/ (e +/+ ) + (,? ~f~S] = o,2 " >,


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+ f- + r^ir (62).

Now _/) -So that ifwe get from equations (62)

i' ' 477 '

je2 =x2 + Y- + z\47T

dK . (dK dK\\n - -(m-ti) (-.- + -, )h.de v ' \df dg)}with symmetrical expressions for g and h,The expansion in volume

e+f + gIs given by the equation

, R 3 i (dK dK dK\*+/+"=-o \ -r +-jf + -r t (64).STT $m n^ae df dg ) ^Just as in the analogous case of magnetism these are

not the only strains produced in the dielectric by theelectric field. The term (Xp -f Yq + Zr) which occurs inthe Lagrangian function can be shown to involve the same

exist in the electric field, viz. a tension JZR^lBir along thelines of force and a pressure of the same intensity at rightangles to them. The effect of this distribution of stresswill be of the same character for all dielectrics, and its

the distribution of


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nature depends more upon forcethroughout the electric field than upon the nature of thedielectric. The experiments of Quincke (Phil. Mag. x. p.30, 1880) and others show that the behaviour of differentdielectrics when placed in the same electric field is verydifferent. Thus, for example, though most dielectricsexpand when placed in an electric field, the fatty oils onthe contrary contract. This difference of behaviour showsthat in many cases at any rate, the strains due to the samecause as that which makes the specific inductive capacitydepend upon the strain are greater than those produced byMaxwell's distribution of stress.

Quincke has shown that the coefficients of elasticity of adielectric are altered when an electric displacement is pro-duced in it, this shows that i/AT when expanded in powersof e must contain a term in


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of the dielectric than the coefficient of magnetic inductionis of the molecular state of soft iron'. Thus there is a com-paratively small difference between the specific inductivecapacities of various substances, while the coefficient ofmagnetic induction of iron is enormously greater than thatof any other substance. Again, the coefficient of magneticinduction is known to be much affected by changes intemperature; while some recent experiments made by MrCassie in the Cavendish Laboratory have shown that theeffect of changes of . temperature on the specific inductivecapacities of ebonite, mica and glass is small, amount-ing in the case of glass, for which it is largest, to i part in400 for each degree centigrade of temperature. No experi-ments seem to have been made on the effect of torsion onelectrification or of electrification upon torsion.

40. Influence of inertia on magnetic pheno-mena. In the preceding investigations we have supposedthe magnetic changes to take place so slowly that theeffects of inertia may be neglected. If however a change inthe magnetization involves, as it does according to allmolecular theories of magnetism, motion of the moleculesof the magnet, then magnetism must behave as if it possessedinertia.

In soft iron and steel the conditions are made so com- cations. Ine ettect, n it exists, would probably be detected

most easily in the case of crystals, as only one of these,quartz, has ever been suspected of showing residual mag-netism (see Tumlirz, Wied. Ann. xxvir. p. 133, 1886). Theeffect of inertia would be to introduce into the equations


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of magnetization a term

where / is the intensity of magnetization. The equationsof magnetization would therefore be of the form

where Jfis the external magnetic force.If #"is periodic and varies as


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ay )

dxdydzdx'dy'dz'...(t>*i\

where p is the reciprocal of the distance between theelements dxdydz and dx'dy'dz'.Now we represent the intensity of magnetization byt\% where is the momentum corresponding to a kinosthenicor speed coordinate and 77 is a vector quantity.

Since T? is a vector quantity it may be resolved into com-ponents parallel to the axes of x, y, z. Let us denote thesecomponents by X, /A, v respectively, then we may putA = \, jB = rf, C=v.Making this substitution we haveT f / dp dp\ ( dp dpL=\ 21 ( /A -f-, - V /-, ) +V (v --, - \ -frL V dz dy J \ dx dz

w -, - .dy r dxSo that these terms are of the form

Considering the Lagrangian Equation for the electricalcoordinate, we see that there is an electromotive force

with corresponding expressions for the electromotive forces-to the axes of and z.


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parallel yThese are the usual expressions for the electromotiveforces due to the variations of the magnetic field.

The magnetic force parallel to x acting on the elementdx'dy'dz is by 33 equal toidLd\so that the magnetic force parallel to x per unit volumeis equal to

,*with similar expressions for the magnetic forces parallel tothe axes of y and z. These expressions agree with thosegiven by Ampere for the magnetic force produced by asystem of currents.

Again there is a mechanical force acting on the elementdxdydz whose component parallel to the axis of x isdL

dx 'If we call

f-j 'j >j / ( T? dp ndp\&x dy dz or [B -^ - C-^,J

f .dp * dp'or I A -4-. - B - -. MECHANICAL FORCE DUE TO A CURRENT. 69

quantities denoted by the same symbols in Maxwell'sElectricity and Magnetism.

Since the force on the element dxdydz is


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dLdx'we see that the force on unit volume may be writtendF dG dHU -- +V . \-W ----- ,dx dx dxor(dG dF\ (dF .dm dF dFv \'j ~j~ \~ w }~j j~ ( + u -r+v-r + -=-.{dx ay) \dz dx J dx dy dzThis differs from Maxwell's expression for the same force

by the term dF dF dF 'U -=- +V -3-- +W -=-.ax ay dz. du dv dwSince -7- + + -.- = odx dy dz

it follows thatdF dF dF\.,.-^+v-^+wf-} dxdydz odx dy dz J

if all the circuits are closed. So that as long as the circuitsare closed the effect of the translator}' forces is the same asif they were given by Maxwell's expressions.

In the above investigation we have assumed that wecould move the element without altering the current ; if wesuppose the current to move

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Application of Dynamic to Physics and Chemistry (1888)