A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 13 of 27

We have arrived at this result by the consideration of impulsive forces. By this method we have avoided the consideration of the change of configuration during the action of the forces. But the instantaneous state of the system is in all respects the same, whether the system was brought from a state of rest to the given state of motion by the transient application of impulsive forces, or whether it arrived at that state in any manner, however gradual.

In other words, the variables, and the corresponding velocities and momenta, depend on the actual state of motion of the system at the given instant, and not on its previous history.

Hence, the equation (3) is equally valid, whether the state of motion of the system is supposed due to impulsive forces, or to forces acting in any manner whatever.

We may now therefore dismiss the consideration of impulsive forces, together with the limitations imposed on their time of action, and on the changes of configuration during their action.

Hamilton's Equations of Motion. 561.] We have already shewn that

dT

(4)

Let the system move in any arbitrary way, subject to the con ditions imposed by its connexions, then the variations of p and q are

(5)

190 KINETICS. [562.

and the complete variation of Tp is

But the increment of the kinetic energy arises from the work done by the impressed forces, or

IT, = 2 (Fig). (8)

In these two expressions the variations bq are all independent of each other, so that we are entitled to equate the coefficients of each of them in the two expressions (7) and (8). We thus obtain

where the momentum^ and the force Fr belong to the variable qr. There are as many equations of this form as there are variables. These equations were given by Hamilton They shew that the force corresponding to any variable is the sum of two parts. The first part is the rate of increase of the momentum of that variable with respect to the time. The second part is the rate of increase of the kinetic energy per unit of increment of the variable, the other variables and all the momenta being constant.

The Kinetic Energy expressed in Terms of the Momenta and

Velocities.

562.] Let pl9 p2, &c. be the momenta, and ql} q2, &c. the velocities at a given instant, and let px, p2, &c., qx, q2, &c. be another system of momenta and velocities, such that

Pi = *Pi> 4i = »0n &c- (10)

It is manifest that the systems p, q will be consistent with each other if the systems p, q are so.

Now let n vary by bn. The work done by the force Fl is

Fi*h = 4i8Pi = Jiftntn. (11)

Let n increase from 0 to 1, then the system is brought from a state of rest into the state of motion (qp), and the whole work expended in producing this motion is

-)/

But

ri

/ ndn = ,

Jn

564.] LAGRANGE'S EQUATIONS. 191

and the work spent in producing the motion is equivalent to the kinetic energy. Hence

TP*= iC^ift + ^fc + fcC'). (13)

where Tp$ denotes the kinetic energy expressed in terms of the momenta and velocities. The variables ql, q% , &c. do not enter into this expression.

The kinetic energy is therefore half the sum of the products of the momenta into their corresponding velocities.

When the kinetic energy is expressed in this way we shall denote it by the symbol Tp^ . It is a function of the momenta and velo cities only, and does not involve the variables themselves.

563.] There is a third method of expressing the kinetic energy, which is generally, indeed, regarded as the fundamental one. By solving the equations (3) we may express the momenta in terms of the velocities, and then, introducing these values in (13), we shall have an expression for T involving only the velocities and the variables. When T is expressed in .this form we shall indicate it by the symbol T^ . This is the form in which the kinetic energy is expressed in the equations of Lagrange.

564.] It is manifest that, since Tp, T$9 and Tp^ are three different expressions for the same thing,

Tp+Tt-2Tp(l = 0, or Tp + Tt-Piii-toto-ke. = °- (14)

Hence, if all the quantities jo, q, and q vary,

The variations 8jt? are not independent of the variations bq and bq, so that we cannot at once assert that the coefficient of each variation in this equation is zero. But we know, from equations

(3)'that g-ft = o,fa, do)

so that the terms involving the variations bp vanish of themselves. The remaining variations bq and bq are now all independent, so that we find, by equating to zero the coefficients of bqlt &c ,

192 KINETICS. [565.

or, the components of momentum are the differential coefficients of T^ with respect to the corresponding velocities.

Again, by equating to zero the coefficients of 8^15 &c.,

^+^ = 0; (.8)

dch d^

or, the differential coefficient of the kinetic energy with respect to any variable ql is equal in magnitude but opposite in sign when T is expressed as a function of the velocities instead of as a function of the momenta.

In virtue of equation (18) we may write the equation of motion (9),

p djiw,

dt dql

p i'VtW (20)

at dql dql

which is the form in which the equations of motion were given by Lagrange.

565.] In the preceding investigation we have avoided the con sideration of the form of the function which expresses the kinetic energy in terms either of the velocities or of the momenta. The only explicit form which we have assigned to it is

TP* = 4 (PiJi + J»2? + &c.), (21)

in which it is expressed as half the sum of the products of the momenta each into its corresponding velocity.

We may express the velocities in terms of the differential co efficients of Tp with respect to the momenta, as in equation (3),

This shews that Tp is a homogeneous function of the second degree of the momenta pl} p2, &c.

We may also express the momenta in terms of T$ , and we find

«-&§+*§ + *"•) <23>

which shews that T$ is a homogeneous function of the second degree with respect to the velocities <?15 q2, &c. If we write

Pn for ^, P12 for ^-±« &c.

and Qn for - ? , Q12 for -^ — /- , &c. ;

567.] MOMENTS AND PRODUCTS OF INERTIA. 193

then, since both T(j and Tp are functions of the second degree of q and of p respectively, both the P's and the Q's will be functions of the variables q only, and independent of the velocities and the momenta. We thus obtain the expressions for I\

2 TI = Pn tf + 2P12 q, q2 + &c., (24)

2Tp= QuPi2 + 2 Qi2PiP2 + &c- (25)

The momenta are expressed in terms of the velocities by the

linear equations ^ = pn ^ + P12 ^ + &c., (26)

and the velocities are expressed in terms of the momenta by the linear equations ^ = Qn p± + Q12p2 + &c. (27)

In treatises on the dynamics of a rigid body, the coefficients corresponding to Pn, in which the suffixes are the same, are called Moments of Inertia, and those corresponding to P12, in which the suffixes are different, are called Products of Inertia. We may extend these names to the more general problem which is now before us, in which these quantities are not, as in the case of a rigid body, absolute constants, but are functions of the variables

In like manner we may call the coefficients of the form Qn Moments of Mobility, and those of the form Q12, Products of Mobility. It is not often, however, that we shall have occasion to speak of the coefficients of mobility.

566.] The kinetic energy of the system is a quantity essentially positive or zero. Hence, whether it be expressed in terms of the velocities, or in terms of the momenta, the coefficients must be such that no real values of the variables can make T negative.

We thus obtain a set of necessary conditions which the values of the coefficients P must satisfy.

The quantities Pn, P22, &c., and all determinants of the sym metrical form

P P P

12 22 '

p p p

•• 13 •• 23 •* q

which can be formed from the system of coefficients must be positive or zero. The number of such conditions for n variables is 2n— 1.

The coefficients Q are subject to conditions of the same kind.

567.] In this outline of the fundamental principles of the dy namics of a connected system, we have kept out of view the mechanism by which the parts of the system are connected. We

VOL. n. o

194 KINETICS. [567.

have not even written down a set of equations to indicate how the motion of any part of the system depends on the variation of the variables. We have confined our attention to the variables, their velocities and momenta, and the forces which act on the pieces representing- the variables. Our only assumptions are, that the connexions of the system are such that the time is not explicitly contained in the equations of condition, and that the principle of the conservation of energy is applicable to the system.

Such a description of the methods of pure dynamics is not un necessary, because Lag-range and most of his followers, to whom we are indebted for these methods, have in general confined them selves to a demonstration of them, and, in order to devote their attention to the symbols before them, they have endeavoured to banish all ideas except those of pure quantity, so as not only to dispense with diagrams, but even to get rid of the ideas of velocity, momentum, and energy, after they have been once for all sup planted by symbols in the original equations. In order to be able to refer to the results of this analysis in ordinary dynamical lan guage, we have endeavoured to retranslate the principal equations of the method into language which may be intelligible without the use of symbols.

As the development of the ideas and methods of pure mathe matics has rendered it possible, by forming a mathematical theory of dynamics, to bring to light many truths which could not have been discovered without mathematical training, so, if we are to form dynamical theories of other sciences, we must have our minds imbued with these dynamical truths as well as with mathematical methods.

In forming the ideas and words relating to any science, which, like electricity, deals with forces and their effects, we must keep constantly in mind the ideas appropriate to the fundamental science of dynamics, so that we may, during the first development of the science, avoid inconsistency with what is already established, and also that when our views become clearer, the language we have adopted may be a help to us and not a hindrance.

CHAPTER VI.

DYNAMICAL THEORY OF ELECTROMAGNETISM.

568.] WE have shewn, in Art. 552, that, when an electric current exists in a conducting circuit, it has a capacity for doing a certain amount of mechanical work, and this independently of any external electromotive force maintaining the current. Now capacity for performing work is nothing else than energy, in whatever way it arises, and all energy is the same in kind, however it may differ in form. The energy of an electric current is either of that form which consists in the actual motion of matter, or of that which consists in the capacity for being set in motion, arising from forces acting between bodies placed in certain positions relative to each other.

The first kind of energy, that of motion, is called Kinetic energy, and when once understood it appears so fundamental a fact of nature that we can hardly conceive the possibility of resolving it into anything else. The second kind of energy, that depending on position, is called Potential energy, and is due to the action of what we call forces, that is to say, tendencies towards change of relative position. With respect to these forces, though we may accept their existence as a demonstrated fact, yet we always feel that every explanation of the mechanism by which bodies are set in motion forms a real addition to our knowledge.

569.] The electric current cannot be conceived except as a kinetic phenomenon. Even Faraday, who constantly endeavoured to emancipate his mind from the influence of those suggestions which the words ' electric current' and ' electric fluid' are too apt to carry with them, speaks of the electric current as ' something progressive, and not a mere arrangement ' *.

• Exp. Res., 283.

O 2

196 ELECTROKINETICS. _S7°-

The effects of the current, such as electrolysis, and the transfer of electrification from one body to another, are all progressive actions which require time for their accomplishment, and are there fore of the nature of motions.

As to the velocity of the current, we have shewn that we know nothing about it, it may be the tenth of an inch in an hour, or a hundred thousand miles in a second *. So far are we from knowing its absolute value in any case, that we do not even know whether what we call the positive direction is the actual direction of the motion or the reverse.

But all that we assume here is that the electric current involves motion of some kind. That which is the cause of electric currents has been called Electromotive Force. This name has long been used with great advantage, and has never led to any inconsistency in the language of science. Electromotive force is always to be understood to act on electricity only, not on the bodies in which the electricity resides. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity in them. If we ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the relation between electromotive force and ordinary force.

570.] When ordinary force acts on a body, and when the body yields to the force, the work done by the force is measured by the product of the force into the amount by which the body yields. Thus, in the case of water forced through a pipe, the work done at any section is measured by the fluid pressure at the section multiplied into the quantity of water which crosses the section.

In the same way the work done by an electromotive force is measured by the product of the electromotive force into the quantity of electricity which crosses a section of the conductor under the action of the electromotive force.

The work done by an electromotive force is of exactly the same kind as the work done by an ordinary force, and both are measured by the same standards or units.

Part of the work done by an electromotive force acting on a conducting circuit is spent in overcoming the resistance of the circuit, and this part of the work is thereby converted into heat. Another part of the work is spent in producing the electromag netic phenomena observed by Ampere, in which conductors are made to move by electromagnetic forces. The rest of the work

• Exp. Res., 1648.

KINETIC ENEEGY. 197

is spent in increasing the kinetic energy of the current, and the effects of this part of the action are shewn in the phenomena of the induction of currents observed by Faraday.

We therefore know enough about electric currents to recognise, in a system of material conductors carrying currents, a dynamical system which is the seat of energy, part of which may be kinetic and part potential.

The nature of the connexions of the parts of this system is unknown to us, but as we have dynamical methods of investigation which do not require a knowledge of the mechanism of the system, we shall apply them to this case.

We shall first examine the consequences of assuming the most general form for the function which expresses the kinetic energy of the system.

571.] Let the system consist of a number of conducting circuits, the form and position of which are determined by the values of a system of variables #15 x9) &c., the number of which is equal to the number of degrees of freedom of the system.

If the whole kinetic energy of the system were that due to the motion of these conductors, it would be expressed in the form

T = i (#! ffj a?!2 -f &c. + (^ a?2) ^ x2 -f &c.,

where the symbols (^15 a:lf &c.) denote the quantities which we have called moments of inertia, and (#1} sc29 &c.) denote the products of inertia.

If X' is the impressed force, tending to increase the coordinate x, which is required to produce the actual motion, then, by Lagrange's equation, d dT dT _

dt dx dx ~

When T denotes the energy due to the visible motion only, we shall indicate it by the suffix TO, thus, Tm.

But in a system of conductors carrying electric currents, part of the kinetic energy is due to the existence of these currents. Let the motion of the electricity, and of anything whose motion is governed by that of the electricity, be determined by another set of coordinates y^ y2, &c., then T will be a homogeneous function of squares and products of all the velocities of the two sets of coordinates. We may therefore divide T into three portions, in the first of which, Tm, the velocities of the coordinates x only occur, while in the second, Te, the velocities of the coordinates y only occur, and in the third, Tme, each term contains the product of the velocities of two coordinates of which one is as and the other y.

198 ELECTROKINETICS.

We have therefore T — T _L T -4- T

•* — -Lm-T--Le^r Lme)

where Tm = | (^ ^) ^2 -f &c. + (^ #2) ^ #2 + &c->

572.] In the general dynamical theory, the coefficients of every term may be functions of all the coordinates, both x and y. In the case of electric currents, however, it is easy to see that the coordinates of the class y do not enter into the coefficients.

For, if all the electric currents are maintained constant, and the conductors at rest, the whole state of the field will remain constant. But in this case the coordinates y are variable, though the velocities y are constant. Hence the coordinates y cannot enter into the expression for T, or into any other expression of what actually takes place.

Besides this, in virtue of the equation of continuity, if the con ductors are of the nature of linear circuits, only one variable is required to express the strength of the current in each conductor. Let the velocities y^yz, &c. represent the strengths of the currents in the several conductors.

All this would be true, if, instead of electric currents, we had currents of an incompressible fluid running in flexible tubes. In this case the velocities of these currents would enter into the expression for T, but the coefficients would depend only on the variables x, which determine the form and position of the tubes.

In the case of the fluid, the motion of the fluid in one tube does not directly affect that of any other tube, or of the fluid in it. Hence, in the value of T6, only the squares of the velocities y, and not their products, occur, and in T^ any velocity y is associated only with those velocities of the form x which belong to its own tube.

In the case of electrical currents we know that this restriction does not hold, for the currents in different circuits act on each other. Hence we must admit the existence of terms involving products of the form y±y^ and this involves the existence of something in motion, whose motion depends on the strength of both electric currents y^ and y2. This moving matter, whatever it is, is not confined to the interior of the conductors carrying the two currents, but probably extends throughout the whole space surrounding them. 573.] Let us next consider the form which Lagrange's equations of motion assume in this case. Let X' be the impressed force

573-] ELECTROMAGNETIC FORCE. 199

corresponding- to the coordinate a?, one of those which determine the form and position of the conducting- circuits. This is a force in the ordinary sense, a tendency towards change of position. It is given by the equation

x/_ cl_dT^dT dt dx dx

We may consider this force as the sum of three parts, corre sponding to the three parts into which we divided the kinetic energy of the system, and we may distinguish them by the same suffixes. Thus -%•' _ T

The part X'm is that which depends on ordinary dynamical con siderations, and we need not attend to it.

Since T0 does not contain x, the first term of the expression for X'e is zero, and its value is reduced to

J' dT*

«~ ~ dx '

This is the expression for the mechanical force which must be applied to a conductor to balance the electromagnetic force, and it asserts that it is measured by the rate of diminution of the purely electrokinetic energy due to the variation of the coordinate x. The electromagnetic force, Xe, which brings this external mechanical force into play, is equal and opposite to it, and is therefore measured by the rate of increase of the electrokinetic energy corresponding to an increase of the coordinate x. The value of Xe, since it depends on squares and products of the currents, remains the same if we reverse the directions of all the currents.

The third part of X' is

d dTme dT^

_ me~ dt dx dx

The quantity Tme contains only products of the form xy, so that

dT

me is a linear function of the strengths of the currents i/. The

first term, therefore, depends on the rate of variation of the strengths of the currents, and indicates a mechanical force on the conductor, which is zero when the currents are constant, and which is positive or negative according as the currents are in creasing or decreasing in strength.

The second term depends, not on the variation of the currents, but on their actual strength. As it is a linear function with respect to these currents, it changes sign when the currents change

200

ELECTROKINETICS.

[574.

sign. Since every term involves a velocity x, it is zero when the conductors are at rest.

We may therefore investigate these terms separately. If the conductors are at rest, we have only the first term to deal with. If the currents are constant, we have only the second.

574.] As it is of great importance to determine whether any part of the kinetic energy is of the form Tme, consisting of products of ordinary velocities and strengths of electric currents, it is de sirable that experiments should be made on this subject with great care.

The determination of the forces acting on bodies in rapid motion is difficult. Let us therefore attend to the first term, which depends on the variation of the strength of the current.

If any part of the kinetic energy depends on the product of an ordinary velocity and the strength of a current, it will probably be most easily ob served when the velocity and the current are in the same or in opposite directions. We therefore take a circular coil of a great many windings, and suspend it by a fine vertical wire, so that its windings are horizontal, and the coil is capable of rotating about a vertical axis, either in the same direction as the current in the coil, or in the opposite direction.

We shall suppose the current to be conveyed into the coil by means of the suspending wire, and, after passing round the windings, to com plete its circuit by passing downwards through a wire in the same line with the suspending wire and dipping into a cup of mercury.

Since the action of the horizontal component pj 33 of terrestrial magnetism would tend to turn

this coil round a horizontal axis when the current flows through it, we shall suppose that the horizontal com ponent of terrestrial magnetism is exactly neutralized by means of fixed magnets, or that the experiment is made at the magnetic pole. A vertical mirror is attached to the coil to detect any motion in azimuth.

Now let a current be made to pass through the coil in the direction N.E.S.W. If electricity were a fluid like water, flowing along the wire, then, at the moment of starting the current, and as

574-1 HAS AN" ELECTRIC CURRENT TRUE MOMENTUM.7? 201

long as its velocity is increasing, a force would require to be supplied to produce the angular momentum of the fluid in passing round the coil, and as this must be supplied by the elasticity of the suspending wire, the coil would at first rotate in the opposite direction or W.S.E.N., and this would be detected by means of the mirror. On stopping the current there would be another movement of the mirror, this time in the same direction as that of the current.

No phenomenon of this kind has yet been observed. Such an action, if it existed, might be easily distinguished from the already known actions of the current by the following peculiarities.

(1) It would occur only when the strength of the current varies, as when contact is made or broken, and not when the current is constant.

All the known mechanical actions of the current depend on the strength of the currents, and not on the rate of variation. The electromotive action in the case of induced currents cannot be confounded with this electromagnetic action.

(2) The direction of this action would be reversed when that of all the currents in the field is reversed.

All the known mechanical actions of the current remain the same when all the currents are reversed, since they depend on squares and products of these currents.

If any action of this kind were discovered, we should be able to regard one of the so-called kinds of electricity, either the positive or the negative kind, as a real substance, and we should be able to describe the electric current as a true motion of this substance in a particular direction. In fact, if electrical motions were in any way comparable with the motions of ordinary matter, terms of the form Tme would exist, and their existence would be manifested by the mechanical force Xm, .

According to Fechner's hypothesis, that an electric current con sists of two equal currents of positive and negative electricity, flowing in opposite directions through the same conductor, the terms of the second class Tme would vanish, each term belonging to the positive current being accompanied by an equal term of opposite sign belonging to the negative current, and the phe nomena depending on these terms would have no existence.

It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of a material fluid, we must carefully avoid

202

ELECTROKINETICS.

[575-

making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as mea sured in feet per second.

A knowledge of these things would amount to at least the begin nings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phe nomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the objects of study.

575.] The experimental investigation of the second term of Xme,

dT

namely -- r — , is more difficult, as it involves the observation of ax

the effect of forces on a body in rapid motion.

Fig. 34.

The apparatus shewn in Fig. 34, which I had constructed in 1861, is intended to test the existence of a force of this kind.

575-] EXPERIMENT OF ROTATION. 203

The electromagnet A is capable of rotating' about the horizontal axis BB', within a ring which itself revolves about a vertical axis.

Let A, J5, C be the moments of inertia of the electromagnet about the axis of the coil, the horizontal axis BB' , and a third axis CC' respectively.

Let 6 be the angle which CG' makes with the vertical, </> the azimuth of the axis BB', and \f/ a variable on which the motion of electricity in the coil depends.

Then the kinetic energy of the electromagnet may be written

2 T = A & sin2 0 + B 62 + <7<j>2 cos2 0 + E (<£ sin 6 + ^)2,

where E is a quantity which may be called the moment of inertia of the electricity in the coil.

If 0 is the moment of the impressed force tending to increase 0, we have, by the equations of dynamics,

d2Q • . . .

0 = B -r^— {(A— C)02sm0cos0 + ^(£cos0((/>sm<9 + //)}. (It

By making % the impressed force tending to increase \j/t equal to zero, we obtain

<£ sin 0 -f x//- = y,

a constant, which we may consider as representing the strength of the current in the coil.

If C is somewhat greater than A, 0 will be zero, and the equi librium about the axis BB' will be stable when

Ey sin 0 = - — — r •

This value of 0 depends on that of y, the electric current, and is positive or negative according to the direction of the current.

The current is passed through the coil by its bearings at B and B', which are connected with the battery by means of springs rubbing on metal rings placed on the vertical axis.

To determine the value of 0, a disk of paper is placed at C, divided by a diameter parallel to BB' into two parts, one of which is painted red and the other green.

When the instrument is in motion a red circle is seen at C when 0 is positive, the radius of which indicates roughly the value of 0. When 0 is negative, a green circle is seen at C.

By means of nuts working on screws attached to the electro magnet, the axis CC' is adjusted to be a principal axis having its moment of inertia just exceeding that round the axis A, so as

204 ELECTROKINETICS. [5?6.

to make the instrument very sensible to the action of the force if it exists.

The chief difficulty in the experiments arose from the disturbing action of the earth's magnetic force, which caused the electro magnet to act like a dip-needle. The results obtained were on this account very rough, but no evidence of any change in 6 could be obtained even when an iron core was inserted in the coil, so as to make it a powerful electromagnet.

If, therefore, a magnet contains matter in rapid rotation, the ang'ular momentum of this rotation must be very small compared with any quantities which we can measure, and we have as yet no evidence of the existence of the terms Tme derived from their me chanical action.

576.] Let us next consider the forces acting on the currents of electricity, that is, the electromotive forces.

Let Y be the effective electromotive force due to induction, the electromotive force which must act on the circuit from without to balance it is Y'= — Yt and, by Lagrange's equation,

Y= -r= — — —.

dt dy dy

Since there are no terms in T involving the coordinate ^, the second term is zero, and Y is reduced to its first term. Hence, electromotive force cannot exist in a system at rest, and with con stant currents.

Again, if we divide Y into three parts, Ym, Ye, and Yme, cor responding to the three parts of T, we find that, since Tm does not contain^, Ym = 0.

•W -C A V d dTe

We also find F, = — - , : -=-* •

dt dy

dT

Here -^-? is a linear function of the currents, and this part of dy

the electromotive force is equal to the rate of change of this function. This is the electromotive force of induction discovered by Faraday. We shall consider it more at length afterwards. 577.] From the part of T, depending on velocities multiplied by

currents, we find Ymc = — ^- •

dt du

dT

Now — -j^ is a linear function of the velocities of the conductors. dy

If, therefore, any terms of Tme have an actual existence, it would be possible to produce an electromotive force independently of all existing currents by simply altering the velocities of the conductors.

577-] ELECTROMOTIVE FORCE. 205

For instance, in the case of the suspended coil at Art. 559, if, when the coil is at rest, we suddenly set it in rotation about the vertical axis, an electromotive force would be called into action proportional to the acceleration of this motion. It would vanish when the motion became uniform, and be reversed when the motion was retarded.

Now few scientific observations can be made with greater pre cision than that which determines the existence or non-existence of a current by means of a galvanometer. The delicacy of this method far exceeds that of most of the arrangements for measuring the mechanical force acting on a body. If, therefore, any currents could be produced in this way they would be detected, even if they were very feeble. They would be distinguished from ordinary currents of induction by the following characteristics.

(1) They would depend entirely on the motions of the conductors, and in no degree on the strength of currents or magnetic forces already in the field.

(2) They would depend not on the absolute velocities of the con ductors, but on their accelerations, and on squares and products of velocities, and they would change sign when the acceleration be comes a retardation, though the absolute velocity is the same.

Now in all the cases actually observed, the induced currents depend altogether on the strength and the variation of currents in the field, and cannot be excited in a field devoid of magnetic force and of currents. In so far as they depend on the motion of con ductors, they depend on the absolute velocity, and not on the change of velocity of these motions.

We have thus three methods of detecting the existence of the terms of the form Ttne, none of which have hitherto led to any positive result. I have pointed them out with the greater care because it appears to me important that we should attain the greatest amount of certitude within our reach on a point bearing so strongly on the true theory of electricity.

Since, however, no evidence has yet been obtained of such terms, I shall now proceed on the assumption that they do not exist, or at least that they produce no sensible effect, an assumption which will considerably simplify our dynamical theory. We shall have occasion, however, in discussing the relation of magnetism to light, to shew that the motion which constitutes light may enter as a factor into terms involving the motion which constitutes mag netism.

CHAPTER VII.