A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 12 of 27

If the change of the relative position of the coils is made by moving the primary circuit instead of the secondary, the result is found to be the same.

539.] It follows from these experiments that the total induction current in B during the simultaneous motion of A from Al to A2J and of B from Bl to B.2, while the current in A changes from ^ to y2, depends only on the initial state AI} Bl, yl5 and the final state A2, B2, y2, and not at all on the nature of the intermediate states through which the system may pass.

Hence the value of the total induction current must be of the form F(A2, B2, y2) - F(Alf £19 7l),

where F is a function of A, B, and y.

With respect to the form of this function, we know, by Art. 536, that when there is no motion, and therefore Al = A2 and Bl = B2,

540.] ELECTROTONIC STATE. 173

the induction current is proportional to the primary current. Hence y enters simply as a factor, the other factor being a func tion of the form and position of the circuits A and J9.

We also know that the value of this function depends on the relative and not on the absolute positions of A and B, so that it must be capable of being1 expressed as a function of the distances of the different elements of which the circuits are composed, and of the angles which these elements make with each other.

Let M be this function, then the total induction current may be written C {Ml7l-M2y.2},

where C is the conductivity of the secondary circuit, and M^ y1 are the original, and M2, y2 the final values of M and y.

These experiments, therefore, shew that the total current of induction depends on the change which takes place in a certain quantity, My, and that this change may arise either from variation of the primary current y, or from any motion of the primary or secondary circuit which alters M.

540.] The conception of such a quantity, on the changes of which, and not on its absolute magnitude, the induction current depends, occurred to Faraday at an early stage of his researches*. He observed that the secondary circuit, when at rest in an electro magnetic field which remains of constant intensity, does not shew any electrical effect, whereas, if the same state of the field had been suddenly produced, there would have been a current. Again, if the primary circuit is removed from the field, or the magnetic forces abolished, there is a current of the opposite kind. He therefore recognised in the secondary circuit, when in the electromagnetic field, a ' peculiar electrical condition of matter,' to which he gave the name of the Electrotonic State. He afterwards found that he could dispense with this idea by means of considerations founded on the lines of magnetic force f, but even in his latest researches J, he says, ( Again and again the idea of an electrotonic state § has been forced upon my mind.'

The whole history of this idea in the mind of Faraday, as shewn in his published researches, is well worthy of study. By a course of experiments, guided by intense application of thought, but without the aid of mathematical calculations, he was led to recog nise the existence of something which we now know to be a mathe matical quantity, and which may even be called the fundamental

• Exp. Res., series i. 60. % Ib., 3269.

t Ib., series ii. (242). § Ib., 60, 1114, 1661, 1729, 1733.

174 MAGNETO-ELECTRIC INDUCTION. [541*

quantity in the theory of electromagnetism. But as he was led up to this conception by a purely experimental path, he ascribed to it a physical existence, and supposed it to be a peculiar con dition of matter, though he was ready to abandon this theory as soon as he could explain the phenomena by any more familiar forms of thought.

Other investigators were long afterwards led up to the same idea by a purely mathematical path, but, so far as I know, none of them recognised, in the refined mathematical idea of the potential of two circuits, Faraday's bold hypothesis of an electrotonic state. Those, therefore, who have approached this subject in the way pointed out by those eminent investigators who first reduced its laws to a mathematical form, have sometimes found it difficult to appreciate the scientific accuracy of the statements of laws which Faraday, in the first two series of his Researches, has given with such wonderful completeness.

The scientific value of Faraday's conception of an electrotonic state consists in its directing the mind to lay hold of a certain quantity, on the changes of which the actual phenomena depend. Without a much greater degree of development than Faraday gave it, this conception does not easily lend itself to the explanation of the phenomena. We shall return to this subject again in Art. 584.

541.] A method which, in Faraday's hands, was far more powerful is that in which he makes use of those lines of magnetic force which were always in his mind's eye when contemplating his magnets or electric currents, and the delineation of which by means of iron filings he rightly regarded * as a most valuable aid to the experimentalist.

Faraday looked on these lines as expressing, not only by their direction that of the magnetic force, but by their number and concentration the intensity of that force, and in his later re searches f he shews how to conceive of unit lines of force. I have explained in various parts of this treatise the relation between the properties which Faraday recognised in the lines of force and the mathematical conditions of electric and magnetic forces, and how Faraday's notion of unit lines and of the number of lines within certain limits may be made mathematically precise. See Arts. 82, 404, 490.

In the first series of his Researches J he shews clearly how the direction of the current in a conducting circuit, part of which is

• Exp. lies., 3234. t Ib., 3122. $ Ib., 114.

LINES OF MAGNETIC INDUCTION. 175

moveable, depends on the mode in which the moving1 part cuts through the lines of magnetic force.

In the second series* he shews how the phenomena produced by variation of the strength of a current or a magnet may be explained, by supposing the system of lines of force to expand from or contract towards the wire or magnet as its power rises or falls.

I am not certain with what degree of clearness he then held the doctrine afterwards so distinctly laid down by him f, that the moving conductor, as it cuts the lines of force, sums up the action due to an area or section of the lines of force. This, however, appears no new view of the case after the investigations of the second series J have been taken into account.

The conception which Faraday had of the continuity of the lines of force precludes the possibility of their suddenly starting into existence in a place where there were none before. If, therefore, the number of lines which pass through a conducting circuit is made to vary, it can only be by the circuit moving across the lines of force, or else by the lines of force moving across the circuit. In either case a current is generated in the circuit.

The number of the lines of force which at any instant pass through the circuit is mathematically equivalent to Faraday's earlier con ception of the electrotonic state of that circuit, and it is represented by the quantity My.

It is only since the definitions of electromotive force, Arts. 69, 274, and its measurement have been made more precise, that we can enunciate completely the true law of magneto -electric induction in the following terms : —

The total electromotive force acting round a circuit at any instant is measured by the rate of decrease of the number of lines of magnetic force which pass through it.

When integrated with respect to the time this statement be comes : —

The time-integral of the total electromotive force acting round any circuit, together with the number of lines of magnetic force which pass through the circuit, is a constant quantity.

Instead of speaking of the number of lines of magnetic force, we may speak of the magnetic induction through the circuit, or the surface-integral of magnetic induction extended over any surface bounded by the circuit.

• Exp. Res., 238. t Ib., 3082, 3087, 3113.

£ Ib., 217, &c.

176 MAGNETO-ELECTRIC INDUCTION. [542.

We shall return again to this method of Faraday. In the mean time we must enumerate the theories of induction which are founded on other considerations.

Lenz's Law.

542.] In 1834, Lenz* enunciated the following' remarkable relation between the phenomena of the mechanical action of electric currents, as defined by Ampere's formula, and the induction of electric currents by the relative motion of conductors. An earlier attempt at a statement of such a relation was given by Ritchie in the Philosophical Magazine for January of the same year, but the direction of the induced current was in every case stated wrongly. Lenz's law is as follows. —

If a constant current flows in the primary circuit A, and if, by the motion of A, or of the secondary circuit B, a current is induced in B, the direction of this induced current wilt be such that, by its electromagnetic action on A, it tends to oppose the relative motion of the circuits.

On this law J. Neumann f founded his mathematical theory of induction, in which he established the mathematical laws of the induced currents due to the motion of the primary or secondary conductor. He shewed that the quantity M, which we have called the potential of the one circuit on the other, is the same as the electromagnetic potential of the one circuit on the other, which we have already investigated in connexion with Ampere's formula. We may regard J. Neumann, therefore, as having completed for the induction of currents the mathematical treatment which Ampere had applied to their mechanical action.

543.] A step of still greater scientific importance was soon after made by Helmholtz in his Essay on the Conservation of Force J, and by Sir W. Thomson §, working somewhat later, but independently of Helmholtz. They shewed that the induction of electric currents discovered by Faraday could be mathematically deduced from the electromagnetic actions discovered by Orsted and Ampere by the application of the principle of the Conservation of Energy.

Helmholtz takes the case of a conducting circuit of resistance R, in which an electromotive force A, arising from a voltaic or thermo-

• Pogg., Ann. xxxi. 483 (1834).

t Berlin Acad., 1845 and 1847.

£ Kead before the Physical Society of Berlin, July 23, 1847. Translated in Taylor's 'Scientific Memoirs,' part ii. p. 114.

§ Trans. Brit. Ass., 1848, and Phil. Mag., Dec. 1851. See also his paper on 'Transient Electric Currents,' Phil. Mag., 1853. .

543-1 HELMHOLTZ AND THOMSON. 177

electric arrangement, acts. The current in the circuit at any instant is /. He supposes that a magnet is in motion in the neighbourhood of the circuit, and that its potential with respect to the conductor is F, so that, during any small interval of time dt, the energy communicated to the magnet by the electromagnetic action

is

The work done in generating heat in the circuit is, by Joule's law, Art. 242, I2 Belt, and the work spent by the electromotive force A, in maintaining the current / during the time dt, is A Idt. Hence, since the total work done must be equal to the work spent,

at whence we find the intensity of the current

Now the value of A may be what we please. Let, therefore, A = 0, and then 1

or, there will be a current due to the motion of the magnet, equal

dV

to that due to an electromotive force =- •

dt

The whole induced current during the motion of the magnet from a place where its potential is V^ to a place where its potential is Fo, is

and therefore the total current is independent of the velocity or the path of the magnet, and depends only on its initial and final positions.

In Helmholtz's original investigation he adopted a system of units founded on the measurement of the heat generated in the conductor by the current. Considering the unit of current as arbitrary, the unit of resistance is that of a conductor in which this unit current generates unit of heat in unit of time. The unit of electromotive force in this system is that required to produce the unit of current in the conductor of unit resistance. The adoption of this system of units necessitates the introduction into the equa tions of a quantity «, which is the mechanical equivalent of the unit of heat. As we invariably adopt either the electrostatic or

VOL. II. N

178 MAGNETO-ELECTRIC INDUCTION. [544.

the electromagnetic system of units, this factor does not occur in the equations here given.

544.] Helmholtz also deduces the current of induction when a conducting circuit and a circuit carrying a constant current are made to move relatively to one another.

Let Rlt R2 be the resistances, I19 I2 the currents, Alt A2 the external electromotive forces, and V the potential of the one circuit on the other due to unit current in each, then we have, as before,

4 /! + A, I2 = I^R, + L?R.> + /, 7, ~ •

If we suppose 7X to be the primary current, and 72 so much less

than /u that it does not by its induction produce any sensible

^ alteration in 715 so that we may put 7X = -— , then

a result which may be interpreted exactly as in the case of the magnet.

If we suppose J2 to be the primary current, and I± to be very much smaller than /2, we get for Ilt

A-I^ T AI L* dt

This shews that for equal currents the electromotive force of the first circuit on the second is equal to that of the second on the first, whatever be the forms of the circuits.

Helmholtz does not in this memoir discuss the case of induction due to the strengthening or weakening of the primary current, or the induction of a current on itself. Thomson * applied the same principle to the determination of the mechanical value of a current, and pointed out that when work is done by the mutual action of two constant currents, their mechanical value is increased by the same amount, so that the battery has to supply double that amount of work, in addition to that required to maintain the currents against the resistance of the circuits f.

545.] The introduction, by W. Weber, of a system of absolute

• Mechanical Theory of Electrolysis, Phil. Mag., Dec., 1851.

t Nichol's Cyclopaedia of Physical Science, ed. 1860, Article 'Magnetism, Dy namical Relations of,' and Reprint, § 571.

545-1 WEBER. 179

units for the measurement of electrical quantities is one of the most important steps in the progress of the science. Having already, in conjunction with Gauss, placed the measurement of magnetic quan tities in the first rank of methods of precision, Weher proceeded in his Electrodynamic Measurements not only to lay down sound principles for fixing the units to be employed, but to make de terminations of particular electrical quantities in terms of these units, with a degree of accuracy previously unattempted. Both the electromagnetic and the electrostatic systems of units owe their development and practical application to these researches.

Weber has also formed a general theory of electric action from which he deduces both electrostatic and electromagnetic force, and also the induction of electric currents. We shall consider this theory, with some of its more recent developments, in a separate chapter. See Art. 846.

N 2

CHAPTER IV.

ON THE INDUCTION OF A CURRENT ON ITSELF.

546.] FARADAY has devoted the ninth series of his Researches to the investigation of a class of phenomena exhibited by the current in a wire which forms the coil of an electromagnet.

Mr. Jenkin had observed that, although it is impossible to pro duce a sensible shock by the direct action of a voltaic system consisting of only one pair of plates, yet, if the current is made to pass through the coil of an electromagnet, and if contact is then broken between the extremities of two wires held one in each hand, a smart shock will be felt. No such shock is felt on making contact.

Faraday shewed that this and other phenomena, which he de scribes, are due to the same inductive action which he had already observed the current to exert on neighbouring conductors. In this case, however, the inductive action is exerted on the same conductor which carries the current, and it is so much the more powerful as the wire itself is nearer to the different elements of the current than any other wire can be.

547.] He observes, however *, that ' the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire.' Indeed, when we consider one particular wire only, the phenomena are exactly analogous to those of a pipe full of water flowing in a continued stream. If while the stream is flowing we suddenly close the end of the tube, the momentum of the water produces a sudden pressure, which is much greater than that due to the head of water, and may be sufficient to burst the pipe.

If the water has the means of escaping through a narrow jet

• Exp. Res., 1077-

55O.] ELECTRIC INERTIA. 181

when the principal aperture is closed, it will be projected with a velocity much greater than that due to the head of water, and if it can escape through a valve into a chamber, it will do so, even when the pressure in the chamber is greater than that due to the head of water.

It is on this principle that the hydraulic ram is constructed, by which a small quantity of water may be raised to a great height by means of a large quantity flowing down from a much lower level.

548.] These effects of the inertia of the fluid in the tube depend solely on the quantity of fluid running through the tube, on its length, and on its section in different parts of its length. They do not depend on anything outside the tube, nor on the form into which the tube may be bent, provided its length remains the same.

In the case of the wire conveying a current this is not the case, for if a long wire is doubled on itself the effect is very small, if the two parts are separated from each other it is greater, if it is coiled up into a helix it is still greater, and greatest of all if, when so coiled, a piece of soft iron is placed inside the coil.

Again, if a second wire is coiled up with the first, but insulated from it, then, if the second wire does not form a closed circuit, the phenomena are as before, but if the second wire forms a closed circuit, an induction current is formed in the second wire, and the effects of self-induction in the first wire are retarded.

549.] These results shew clearly that, if the phenomena are due to momentum, the momentum is certainly not that of the electricity in the wire, because the same wire, conveying the same current, exhibits effects which differ according to its form ; and even when its form remains the same, the presence of other bodies, such as a piece of iron or a closed metallic circuit, affects the result.

550.] It is difficult, however, for the mind which has once recognised the analogy between the phenomena of self-induction and those of the motion of material bodies, to abandon altogether the help of this analogy, or to admit that it is entirely superficial and misleading. The fundamental dynamical idea of matter, as capable by its motion of becoming the recipient of momentum and of energy, is so interwoven with our forms of thought that, when ever we catch a glimpse of it in any part of nature, we feel that a path is before us leading, sooner or later, to the complete under standing of the subject.

182 SELF-INDUCTION. [551-

551.] In the case of the electric current, we find that, when the electromotive force begins to act, it does not at once produce the full current, but that the current rises gradually. What is the electromotive force doing during the time that the opposing re sistance is not able to balance it ? It is increasing the electric current.

Now an ordinary force, acting on a body in the direction of its motion, increases its momentum, and communicates to it kinetic energy, or the power of doing work on account of its motion.

In like manner the unresisted part of the electromotive force has been employed in increasing the electric current. Has the electric current, when thus produced, either momentum or kinetic energy ?

We have already shewn that it has something very like mo mentum, that it resists being suddenly stopped, and that it can exert, for a short time, a great electromotive force.

But a conducting circuit in which a current has been set up has the power of doing work in virtue of this current, and this power cannot be said to be something very like energy, for it is really and truly energy.

Thus, if the current be left to itself, it will continue to circulate till it is stopped by the resistance of the circuit. Before it is stopped, however, it will have generated a certain quantity of heat, and the amount of this heat in dynamical measure is equal to the energy originally existing in the current.

Again, when the current is left to itself, it may be made to do mechanical work by moving magnets, and the inductive effect of these motions will, by Lenz's law, stop the current sooner than the resistance of the circuit alone would have stopped it. In this way part of the energy of the current may be transformed into mechanical work instead of heat.

552.] It appears, therefore, that a system containing an electric current is a seat of energy of some kind ; and since we can form no conception of an electric current except as a kinetic pheno menon *, its energy must be kinetic energy, that is to say, the energy which a moving body has in virtue of its motion.

We have already shewn that the electricity in the wire cannot be considered as the moving body in which we are to find this energy, for the energy of a moving body does not depend on anything external to itself, whereas the presence of other bodies near the current alters its energy.

• Faraday, Eocp. Res. (283.)

552.] ELECTROKINETIC ENEKGY. 183

We are therefore led to enquire whether there may not be some motion going1 on in the space outside the wire, which is not occupied by the electric current, but in which the electromagnetic effects of the current are manifested.

I shall not at present enter on the reasons for looking in one place rather than another for such motions, or for regarding these motions as of one kind rather than another.

What I propose now to do is to examine the consequences of the assumption that the phenomena of the electric current are those of a moving system, the motion being communicated from one part of the system to another by forces, the nature and laws of which we do not yet even attempt to define, because we can eliminate these forces from the equations of motion by the method given by Lagrange for any connected system.

In the next five chapters of this treatise I propose to deduce the main structure of the theory of electricity from a dynamical hypothesis of this kind, instead of following the path which has led Weber and other investigators to many remarkable discoveries and experiments, and to conceptions, some of which are as beautiful as they are bold. I have chosen this method because I wish to shew that there are other ways of viewing the phenomena which appear to me more satisfactory, and at the same time are more consistent with the methods followed in the preceding parts of this book than those which proceed on the hypothesis of direct action at a distance.

CHAPTER V.

ON THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM.

553.] IN the fourth section of the second part of his Mecanique Analytique, Lagrange has given a method of reducing the ordinary dynamical equations of the motion of the parts of a connected system to a number equal to that of the degrees of freedom of the system.

The equations of motion of a connected system have been given in a different form by Hamilton, and have led to a great extension of the higher part of pure dynamics *.

As we shall find it necessary, in our endeavours to bring electrical phenomena within the province of dynamics, to have our dynamical ideas in a state fit for direct application to physical questions, we shall devote this chapter to an exposition of these dynamical ideas from a physical point of view.

554.] The aim of Lagrange was to bring dynamics under the power of the calculus. He began by expressing the elementary dynamical relations in terms of the corresponding relations of pure algebraical quantities, and from the equations thus obtained he deduced his final equations by a purely algebraical process. Certain quantities (expressing the reactions between the parts of the system called into play by its physical connexions) appear in the equations of motion of the component parts of the system, and Lagrange's investigation, as seen from a mathematical point of view, is a method of eliminating these quantities from the final equations.

In following the steps of this elimination the mind is exercised in calculation, and should therefore be kept free from the intrusion of dynamical ideas. Our aim, on the other hand, is to cultivate

• See Professor Cayley's ' Report on Theoretical Dynamics,' British Association, 3 857 ; and Thomson and Tait's Natural Philosophy.

555-] GENERALIZED COORDINATES. 185

our dynamical ideas. We therefore avail ourselves of the labours of the mathematicians, and retranslate their results from the lan guage of the calculus into the language of dynamics, so that our words may call up the mental image, not of some algebraical process, but of some property of moving bodies.

The language of dynamics has been considerably extended by those who have expounded in popular terms the doctrine of the Conservation of Energy, and it will be seen that much of the following statement is suggested by the investigation in Thomson and Tait^s Natural Philosophy, especially the method of beginning with the theory of impulsive forces.

I have applied this method so as to avoid the explicit con sideration of the motion of any part of the system except the coordinates or variables, on which the motion of the whole depends. It is doubtless important that the student should be able to trace the connexion of the motion of each part of the system with that of the variables, but it is by no means necessary to do this in the process of obtaining the final equations, which are independent of the particular form of these connexions.

The Variables.

555.] The number of degrees of freedom of a system is the number of data which must be given in order completely to determine its position. Different forms may be given to these data, but their number depends on the nature of the system itself, and cannot be altered.

To fix our ideas we may conceive the system connected by means of suitable mechanism with a number of moveable pieces, each capable of motion along a straight line, and of no other kind of motion. The imaginary mechanism which connects each of these pieces with the system must be conceived to be free from friction, destitute of inertia, and incapable of being strained by the action of the applied forces. The use of this mechanism is merely to assist the imagination in ascribing position, velocity, and momentum to what appear, in Lagrange's investigation, as pure algebraical quantities.

Let q denote the position of one of the moveable pieces as defined by its distance from a fixed point in its line of motion. We shall distinguish the values of q corresponding to the different pieces by the suffixes u 2, &c. When we are dealing with a set of quantities belonging to one piece only we may omit the suffix.

186 KINETICS. [556.

When the values of all the variables (q) are given, the position of each of the moveable pieces is known, and, in virtue of the imaginary mechanism, the configuration of the entire system is determined.

The Velocities.

556.] During the motion of the system the configuration changes in some definite manner, and since the configuration at each instant is fully defined by the values of the variables (q), the velocity of every part of the system, as well as its configuration, will be com pletely defined if we know the values of the variables (q), together

with their velocities (-— , or, according to Newton's notation, q) •

The Forces.

557.] By a proper regulation of the motion of the variables, any motion of the system, consistent with the nature of the connexions, may be produced. In order to produce this motion by moving the variable pieces, forces must be applied to these pieces.

We shall denote the force which must be applied to any variable qr by Fr. The system of forces (F) is mechanically equivalent (in virtue of the connexions of the system) to the system of forces, whatever it may be, which really produces the motion.

The Momenta.

558.] When a body moves in such a way that its configuration, with respect to the force which acts on it, remains always the same, (as, for instance, in the case of a force acting on a single particle in the line of its motion,) the moving force is measured by the rate of increase of .the momentum. If F is the moving force, and p the momentum,

whence p = / Fdt.

The time-integral of a force is called the Impulse of the force ; so that we may assert that the momentum is the impulse of the force which would bring the body from a state of rest into the given state of motion.

In the case of a connected system in motion, the configuration is continually changing at a rate depending on the velocities (q\ so

559-] IMPULSE AND MOMENTUM. 187

that we can no longer assume that the momentum is the time- intesral of the force which acts on it.

o

But the increment bq of any variable cannot be greater than qbt, where 8^ is the time during which the increment takes place, and q is the greatest value of the velocity during that time. In the case of a system moving from rest under the action of forces always in the same direction, this is evidently the final velocity.

If the final velocity and configuration of the system are given, we may conceive the velocity to be communicated to the system in a very small time §t, the original configuration differing from the final configuration by quantities bqlt §£2, &c., which are less than q^btj ^25^, &c., respectively.

The smaller we suppose the increment of time 8£, the greater must be the impressed forces, but the time-integral, or impulse, of each force will remain finite. The limiting value of the impulse, when the time is diminished and ultimately vanishes, is defined as the instantaneous impulse, and the momentum p, corresponding to any variable q, is defined as the impulse corresponding to that variable, when the system is brought instantaneously from a state of rest into the given state of motion.

This conception, that the momenta are capable of being produced by instantaneous impulses on the system at rest, is introduced only as a method of defining the magnitude of the momenta, for the momenta of the system depend only on the instantaneous state of motion of the system, and not on the process by which that state was produced.

In a connected system the momentum corresponding to any variable is in general a linear function of the velocities of all the variables, instead of being, as in the dynamics of a particle, simply proportional to the velocity.

The impulses required to change the velocities of the system suddenly from yl9 q.2, &c. to £/, q2', &c, are evidently equal to Pi — p, Pz — J°2> ^ne cbaBgcs of momentum of the several variables.

Work done by a Small Impulse.

559.] The work done by the force Fl during the impulse is the space-integral of the force, or

W

=j

188 KINETICS. [560.

If fa is the greatest and q" the least value of tlie velocity q-^ during the action of the force, W must be less than

2i< Fdt

or

and greater than q"\Fdt or q(p\—p)>

If we now suppose the impulse / Fdt to be diminished without

limit, the values of q{ and q" will approach and ultimately coincide with that of qlt and we may write p{—p^ = §pi, so that the work done is ultimately 7ir

or, the work done by a very small impulse is ultimately the product of the impulse and the velocity.

Increment of the Kinetic Energy.

560.] When work is done in setting a conservative system in motion, energy is communicated to it, and the system becomes capable of doing an equal amount of work against resistances before it is reduced to rest.

The energy which a system possesses in virtue of its motion is called its Kinetic Energy, and is communicated to it in the form of the work done by the forces which set it in motion.

If T be the kinetic energy of the system, and if it becomes T 4- 8 T} on account of the action of an infinitesimal impulse whose components are 8^15 5j02, &c., the increment 8 T must be the sum of the quantities of work done by the components of the impulse, or in symbols, IT = &*& + js 8A + &c.,

= 2&8j»). (1)

The instantaneous state of the system is completely defined if the variables and the momenta are given. Hence the kinetic energy, which depends on the instantaneous state of the system, can be expressed in terms of the variables (q), and the momenta (/>). This is the mode of expressing T introduced by Hamilton. When T is expressed in this way we shall distinguish it by the suffix p) thus, Tp.

The complete variation of Tp is

^=2^+Ss?. (2)

561.] HAMILTON'S EQUATIONS. 189

The last term may be written

which diminishes with 8£, and ultimately vanishes with it when the impulse becomes instantaneous.

Hence, equating- the coefficients of bp in equations (1) and (2), we obtain . = ^ (s)

or, the velocity corresponding to the variable q is the differential coefficient of Tp with respect to the corresponding momentum p.