Fig. 13.Fig. 13.
Fig. 13.
In one end of a packing-box, about 2ft. long by 18in. wide and 18in. deep, a circular hole is cut, and the edges of the hole are thinned down to a blunt edge. This can be closed at pleasure by a piece of board. The opposite end is removed, and a sheet ofcanvas stretched tightly in its place, and tacked to the ends of the sides. Through two holes bored in one of the sides the mouths of two flasks with bent necks protrude into the box. One of these flasks contains ammonia, the other hydrochloric acid. When the hole at one end is closed up by a slip of tinplate, and the liquids are heated with a spirit-lamp, the vapours form a cloud of sal-ammoniac within the box, which is retained during its formation. The hole is then opened, and the canvas struck smartly with the palm of the open hand. Immediately a beautiful ring of smoke emerges, clear-cut and definite as a solid, and moves across the room. (See Fig.13.) Of course, it is a ring of air, made visible by the smoke carried with it. By varying the shape of the aperture—for example, by using instead of the hole cut in the wood, a slide of tinplate with an elliptic hole cut in it—the vortex-rings can be set in vibration as they are created, and the vibrations studied as the vortex moves.
Still more beautiful vortices can be formed in water by using a long tank of clear water to replace the air in which the vortex moves, and a compartment at one end filled with water coloured with aniline, instead of the smoke-box. A hole in the dividing partition enables the vortex to be formed, and a piston arrangement fitted to the opposite side enables the impulse to the water to be given from without.
From the account of the nature of vortex-motion given above, it will be clear that vortices in a perfect fluid once existent must be ever existent. To create a vortex within a mass of irrotationally moving perfectfluid is physically impossible. It occurred to Thomson, therefore, that ordinary matter might be portions of a perfect fluid, filling all space, differentiated from the surrounding fluid by the rotation which they possess. Such matter would fulfil the law of conservation, as it could neither be created nor destroyed by any physical act.
The results of such experiments led Thomson to frame his famous vortex-atom theory of matter, a theory, however, which he felt ultimately was beset with so many difficulties as to be unworkable.
The paper on vortex-motion also deals with the modification of Green's celebrated theorem of analysis, which, it was pointed out by Helmholtz, was necessary to adapt it to a space which is multiply continuous. The theorem connects a certain volume-integral taken throughout a closed space with an integral taken over the bounding surface of the space. This arises from the fact noticed above that in multiply continuous space (for example, the space within an endless tube) the functions which are the subject of integration may not be single valued. Such a function would be the velocity-potential for fluid circulating round the tube—cyclic motion, as it was called by Thomson. If a closed path of any form be drawn in such a tube, starting from a pointP, and doubling back so as to return toPwithout making the circuit of the tube, the velocity-potential will vary along the tube, but will finally return to its original value when the starting point is reached. And the circulation round this circuit will be zero. But if the closed path make the circuit of the tube, the velocity-potential will continuously vary along the path, until finally, whenPisreached again, the value of the function is greater (or less) than the value assumed for the starting point, by a certain definite amount which is the same for every circuit of the space. If the path be carried twice round in the same direction, the change of the function will be twice this amount, and so on. The space within a single endless tube such as an anchor-ring is doubly continuous; but much more complicated cases can be imagined. For example, an anchor-ring with a cross-connecting tube from one side to the other would be triply continuous.
Thomson showed that the proper modification of the theorem is obtained by imagining diaphragms placed across the space, which are not to be crossed by any closed path drawn within the space, and the two surfaces of each of which are to be reckoned as part of the bounding surface of the space. One such diaphragm is sufficient to convert a hollow anchor-ring into a singly continuous space, two would be required for the hollow anchor-ring with cross-connection, and so on. The number of diaphragms required is always one less than the degree of multiplicity of the continuity.
The paper also deals with the motion of solids in the fluid and the analogous motions of vortex-rings and their attraction by ordinary matter. These can be studied with vortex-rings in air produced by the apparatus described above. Such a ring made to pass the re-entrant corner of a wall—the edge of a window recess, for example—will appear to be attracted. A large sphere such as a large terrestrial globe serves also very well as an attracting body.
Two vortex-rings projected one after the other alsoact on one another in a very curious manner. Their planes are perpendicular to the direction of motion, and the fluid is moving round the circular core of the ring. There is irrotational cyclic motion of the fluid through the ring in one direction and back outside, as shown in Fig.13, which can be detected by placing a candle flame in the path of the centre. The first ring, in consequence of the existence of that which follows it, moves more slowly, and opens out more widely, the following ring hastens its motion and diminishes in diameter, until finally it overtakes the former and penetrates it. As soon as it has passed through it moves ahead more and more slowly, until the one which has been left behind begins to catch it up, and the changes which took place before are repeated. The one penetrating becomes in its turn the penetrated, and so on in alternation. Great care and skill are, however, necessary to make this interesting experiment succeed.
We have not space to deal here with other hydrodynamical investigations, such as the contributions which Thomson made to the discussion of the many difficult problems of the motion of solids through a liquid, or to his very numerous and important contributions to the theory of waves. The number and importance of his hydrodynamical papers may be judged from the fact that there are no less than fifty-two references to his papers, and thirty-five to Thomson and Tait'sNatural Philosophyin the latest edition of Lamb'sHydrodynamics, and that many of these are concerned with general theorems and results of great value.
InDecember 1851 Thomson communicated an important paper to thePhilosophical Magazineon "The Mechanical Theory of Electrolysis," and "Applications of Mechanical Effect to the Measurement of Electromotive Forces, and of Galvanic Resistances, in Absolute Units."
In the first of these he supposed a machine of the kind imagined by Faraday, consisting of a metal disk, rotating uniformly with its plane at right angles to the lines of force of a uniform magnetic field, and touched at its centre and its circumference by fixed wires, to send a current through an electrochemical apparatus, to which the wires are connected. A certain amount of workWwas supposed to be spent in a given time, during which a quantity of heatHwas evolved in the circuit, and a certain amount of workMspent in the chemical apparatus in effecting chemical change. IfHbe taken in dynamical units,W=H+M.
The work done in driving the disk, if the intensity of the field isI, the current producedc, the radius of the discr, and the angular velocity of turningw, is ½Ir2cw.
Thomson assumed that the work done in the electrochemical apparatus was equal to the heat of chemicalcombination of the substance or substances which underwent the chemical action, taken with the proper sign according to the change, if more compound substances than one were acted on. HenceMrepresented this resultant heat of combination.
The electrochemical apparatus was a voltameter containing a definite compound to be electrolysed, or a voltaic cell or battery. And by Faraday's experiments on electrolysis it was known that the amount of chemical action was proportional to the whole quantity of electricity passed through the cell in a given time, so that the rate at which energy was being spent in the cell was at any instant proportional to the current at that instant.
The chemical change could be measured by considering only one of the elements set free, or made to combine, by the passage of the current, and considering the quantity of heatθ, say, for the whole chemical change in the cell corresponding to the action on unit mass of that element. Thus ifEdenote the whole quantity of that element operated on the heat of combination in the vessel wasθE. IfEbe taken for unit of time, and ε denote the quantity set free by the passage of unit quantity of electricity, thenE=εc, since a current conveyscunits of electricity in one second. The numberεis a definite quantity of the element, and is called its electrochemical equivalent. Again, from Joule's experiments,H=Rc2, ifRdenote the resistance of the current, and so
and
The quantity ½Ir2wis the electromotive force due to the disk.
Thuscwas positive or negative according as ½Ir2wwas greater or less thanθε, and was zero when ½Ir2w=θε. Thus the electromotive force of the disk was opposed by a back electromotive forceθεdue to the chemical action in the voltameter or battery, to which the wires from the disk were connected.
The conclusion arrived at therefore was that the electromotive force (or, as it was then termed, the intensity) of the electrochemical action was equal to the dynamical value of the whole chemical change effected by a current of unit strength in unit of time.
From this result Thomson proceeded to calculate the electromotive forces required to effect chemical changes of different kinds, and those of various types of voltaic cell. Supposing a unit of electricity to be carried by the current through the cell, he considered the chemical changes which accompanied its passage, and from the known values of heats of combination calculated their energy values. In some parts the change was one of chemical combination, in others one of decomposition of the materials, and regard had to be paid to the sign of the heat-equivalent. By properly summing up the whole heat-equivalents a net total was obtained which, according to Thomson, was the energy consumed in the passage of unit current, and was therefore the electromotive force. The theory was incomplete, and required to be supplemented by thermodynamic theory, which shows that besides the electromotive force there must be included in the quantity set against the sum of heats a term represented by the product of the absolute temperaturemultiplied by the rate of variation of electromotive force with alteration of temperature. Thus the theory is only applicable when the electromotive force is not affected by variation of temperature. The necessary addition here indicated was made by Helmholtz.
In the next paper, which appeared in the same number (December 1851) of thePhilosophical Magazine, the principle of work is applied to the measurement of electromotive forces and resistances in absolute units. The advantages of such units are obvious. Nearly the whole of the quantitative work of the older experimenters was useless except for those who had actually made the observations: it was hardly possible for one man to advance his researches by employing data obtained by others. For the results were expressed by reference to apparatus and materials in the possession of the observers, and to these others could obtain access only with great difficulty and at great expense—to say nothing of the uncertainty of comparisons made to enable the results of one man to be linked on to those made elsewhere, and with other apparatus, by another. It was imperative, therefore, to obtain absolute units—units independent of accidents of place and apparatus—for the expression of currents, electromotive forces, and resistances, so as to enable the results of the work of experiments all over the world to be made available to every one who read the published record. (See Chap.XIII.)
The magneto-electric machine imagined in the former paper gave a means of estimating the electromotive force of a cell or battery in absolute units. The same kind of machine is used here, in the simpler form of a sliding conductor connecting a pair of insulatedrails laid with their plane perpendicular to the lines of force of a uniform magnetic field. If the rails be connected by a wire, and the slider be moved so as to cut across the lines of force, a current will be produced in the circuit. The current can be measured in terms of the already known unit of current, that current which flowing in a circle of radius unity produces a magnetic field at the centre of 2π units. This current,c, say, in strength, flowing in the circuit, renders a dynamical forcecIlnecessary to move the slider of lengthlacross the lines of force of the field of intensityI, and if the speed of the slider required for the currentcbev, the rate at which work is done in moving the slider iscIlv. This must be the rate at which work is done in the circuit by the current, and if the only work done be in the heating of the conductor, we havecIlv=Rc2, orIlv=Rc, so thatIlvis the electromotive force. Any electromotive force otherwise produced, which gave rise to the same current, must obviously be equal toIlv, so that the unit of electromotive force can thus be properly defined.
Thomson used a foot-grain-second system of units; but from this arrangement are now obtained the C.G.S. units of electromotive force and resistance. IfIis one C.G.S. unit,lone centimetre, andvone centimetre per second, we have unit electromotive force in the C.G.S. system. Also in one C.G.S. unit of resistance ifcbe unity as well asIlv.
The idea of the determination of a resistance in absolute units on correct principles was due to W. Weber, who also gave methods of carrying out the measurement; and the first determination was made by Kirchhoff in 1849. Thomson appears, however,to have been the first to discuss the subject of units from the point of view of energy. This mode of regarding the matter is important, as the absolute units are so chosen as to enable work done by electric and magnetic forces to be reckoned in the ordinary dynamical units. A vast amount of experimental resource and skill has been spent since that time on the determination of resistance, though not more than the importance of the subject warranted. We shall have to return to the subject in dealing with the work of the British Association on Electrical Standards, of which Thomson was for long an active member.
In his famous tract on the conservation of energy, published in 1847, von Helmholtz discussed some puzzling results obtained by Riess in the magnetisation of iron wires by the current of a Leyden jar discharge flowing in a coil surrounding them, and by the fact, observed by Wollaston, that when water was decomposed by Leyden jar discharges a mixture of oxygen and hydrogen appeared at each electrode, and suggested that possibly the discharge was oscillatory in character.
In 1853 the subject was discussed mathematically by Thomson, in a paper which was to prove fruitful in our own time in a manner then little anticipated. The jar is given, let us say, with the interior coating charged positively, and the exterior coating charged negatively. A coil or helix of wire has its ends connected to the two coatings, and a current immediately begins in the wire, and gradually (not slowly) increases in strength. Accompanying the creation of the current is the production of a magnetic field, thatis, the surrounding space is made the seat of magnetic action. The magnetic field, as we shall see from another investigation of Thomson's, almost certainly involves motion in or of a medium—the ether—filling the space where the magnetic action is found to exist. The charge of the jar consists of a state of intense and peculiar strain in the glass plate between the coatings. When the plates are connected by the coil, this state of strain breaks down and motion in the medium ensues, not merely between the plates, but also in the surrounding space—in fact, in the whole field. This motion—which is not to be confused with bodily displacement of finite parts of the medium—is opposed by something akin to inertia of the medium (the property that confers energy on matter when in motion), so that when the motion is started it persists, until it is finally wiped out by resistance of the nature of friction. The inertia here referred to depends on the mode in which the coil is wound, or whether it contains or not an iron core.
If the work done in charging a Leyden jar or electric condenser, by bringing the charge to the condenser in successive small portions, is considered, it is at once clear that it must be proportional to the square of the whole quantity of electricity brought up. For whatever the charge may be, let it be brought up from a great distance in a large numberNof equal instalments. The larger the whole amount the larger must each instalment be, and therefore the greater the amount accumulated on the condenser when any given number of instalments have been deposited. But the greater any charge that is being brought up, and also the greater the charge that has already arrived,the greater is the repulsion that must be overcome in bringing up that instalment, in simple proportion in each case, and therefore the greater the work done. Thus the whole work done in bringing up the charge must be proportional toQ2. We suppose it to be ½Q2⁄C, whereCis a constant depending on the condenser and called its capacity.
The idea of the charge as a quantity of some kind of matter, brought up and placed on the insulated plate of the condenser, has only a correspondence to the fact, which is that the medium between the plates is the seat, when the condenser is charged, of a store of energy, which can only be made available by connecting the plates of the condenser by a wire or other conductor. The charge is only a surface aspect of the state of the medium, apparently a state of strain, to which the energy belongs.
When a wire is used to connect the plates the state of strain disappears; the energy comes out from the medium between the plates by motion sideways of the tubes of strain (so that the insulating medium is under longitudinal tension and lateral pressure) which, according to Faraday's conception of lines of electric force connecting the charge on a body with the opposite charges on other bodies, run from plate to plate, when the condenser is in equilibrium in the changed state. These tubes move out with their ends on the wire, carrying the energy with them, and the ends run towards one another along the wire; the tube shortens in the process, and energy is lost in the wire. The ends of a tube thus moving represent portions of the charges which were on the plates, and the oppositely-directed motions of the opposite charges represent acurrent along the wire from one conductor to the other. The motion of the tubes is accompanied by the development of a magnetic field, the lines of force of which are endless, and the direction of which at every point is perpendicular at once to the length of the tube and to the direction in which it is there moving. In certain circumstances the tube, by the time its ends have met, will have wholly disappeared in the wire, and the whole energy will have gone to heat the wire: in other circumstances the ends will meet before the tube has disappeared, the ends will cross, and the tube will be carried back to the condenser and reinserted in the opposite direction. At a certain time this will have happened to all the tubes, though they will have lost some of their energy in the process; and the condenser will again be charged, though in the opposite way to that in which it was at first. Then the tubes will move out again, and the same process will be repeated: once more the condenser will be charged, but in the same direction as at first, and once more with a certain loss of energy. Again the process of discharge and charge will take place, and so on, again and again, until the whole energy has disappeared. This process represents, according to the modern theory of the flow of energy in the electromagnetic field, with more or less accuracy, what takes place in the oscillatory discharge of a condenser.
The motion of the tubes with their ends on the wire represents a certain amount of energy, commonly regarded as kinetic, and styled electrokinetic energy. Ifcdenote the current, that is, the rate, −dQ/dt, at which the charge of the condenser is being changed, andLa quantity called self-inductance, dependingmainly on the arrangement of the connecting wire—whether it is wound in a coil or helix, with or without an iron core, or not—the electrokinetic energy will be ½Lc2. This is analogous to the kinetic energy ½mv2of a body (say a pendulum bob) of massmand velocityv, so thatLrepresents a quantity for the conducting arrangement analogous to inertia, andcis the analogue of the velocity of the body. The whole energy at any instant is thus
½Q2⁄C+ ½Lc2, or ½Q2⁄C+ ½L(dQ⁄dt)2.
The loss of energy due to heating of the conducting connection is not completely understood, though its quantitative laws have been quite fully ascertained and expressed in terms of magnitudes that are capable of measurement. It was found by Joule to be proportional to the second power, or square, of the current, and to a quantityRdepending on the conductor, and called its resistance. The generation of heat in the conductor seems to be due to some kind of frictional action of particles of the conductor set up by the penetration of the Faraday tubes into it. A conductor is unable to bear any tangential action exerted upon it by Faraday tubes, which, however, when they exist, begin and end at material particles, except when they are endless, as they may be in the radiation of energy. When the Faraday tubes are moving with any ordinary speed they are not at their ends perpendicular to the conducting surface from which they start or at which they terminate, but are there more or less inclined to the surface, and consequently there is tangential action which appears to displace the particles (not merely at the surface, unless the alternation is very rapid)relatively to one another and so cause frictional generation of heat.
The time rate of generation of heat is thusRc2, orR(dQ⁄dt)2, when the units in whichRandcare expressed are such as to make this quantity a rate of doing work in the true dynamical sense. This is the rate at which the sum of energy already found is being diminished, and so the equation
holds, or leaving out the common factordQ⁄dt, the equation
This last equation was established by Thomson, and is precisely that which would be obtained for a pendulum bob of massL, pulled back towards the position of equilibrium with a forceQ⁄C, whereQis the displacement from the middle position, and having its motion damped out by resisting force of amountRper unit of the velocity.
It is more instructive perhaps to take the oscillatory motion of a spiral spring hung vertically with a weight on its lower end, as that which has a differential equation equivalent to the equation just found. When the stretch is of a certain amount, there is equilibrium—the action of the spring just balances the weight,—and if the spring be stretched further there will be a balance of pull developed tending to bring the system back towards the equilibrium position. If left to itself the system gets into motion, which, if the resistance isnot too great, is added to until the equilibrium position is reached; and the motion, which is continued by the inertia of the mass, only begins to fall off as that position is passed, and the pull of the spring becomes insufficient to balance the weight. Thus the mass oscillates about the position of equilibrium, and the oscillations are successively smaller and smaller in extent, and die out as their energy is expended finally in doing work against friction.
If the resisting force for finite motion is very great, as for example when the vibrating mass of the pendulum or spring is immersed in a very viscous fluid, like treacle, oscillation will not take place at all. After displacement the mass will move at first fairly quickly, then more and more slowly back to the position of equilibrium, which it will, strictly speaking, only exactly reach after an infinite time. The resisting force is here indefinitely small for an indefinitely small speed, but it becomes so great when any motion ensues, that as the restoring force falls off with the displacement, no work is finally done by it, except to move the body through the resisting medium.
The differential equation is applicable to the spring ifQis again taken as displacement from the equilibrium position,Las the inertia of the vibrating body, 1 ⁄Cas the pull exerted by the spring per unit of its extension (that is, the stiffness of the spring), andRhas the same meaning as before.
In this case of motion, as well as in that of the pendulum, energy is carried off by the production of waves in the medium in which the vibrator is immersed. These are propagated out from the vibrator as their source, but no account of them is taken in the differentialequation, which in that respect is imperfect. There is no difficulty, only the addition of a little complication, in supplying the omission.
The formation of such waves by the spiral spring vibrator can be well shown by immersing the vibrating body in a trough of water, and the much greater rate of damping out of the motion in that case can then be compared with the rate of damping in air.
It has been indicated that the differential equation does not represent oscillatory motion if the value ofRis too great. The exact condition depends on the roots of the quadratic equationLx2+Rx+ 1 ⁄C= 0, obtained by writing 1 forQ, andxford⁄dt, and then treatingxas a quantity. These roots are −R⁄ 2L± √(R2⁄ 4L2− 1 ⁄CL), and are therefore real or imaginary according as 4L⁄Cis less or greater thanR2. If the roots are real, that is, ifR2be greater than 4L⁄C, the discharge will not be oscillatory; the Faraday tubes referred to above will be absorbed in the wire without any return to the condenser. The corresponding result happens with the vibrator whenRis sufficiently great, orL⁄Csufficiently small (a weak spring and a small mass, or both), to enable the condition to be fulfilled.
If, however, the roots of the quadratic are imaginary, that is, if 4L⁄Cbe greater thanR2(a condition which will be fulfilled in the spring analogue, by making the spring sufficiently stiff and the mass large enough to prevent the friction from controlling the motion) the motion is one in whichQdisappears by oscillations about zero, of continually diminishing amplitude. A complete discussion gives for the period of oscillation4πL⁄ √(4L⁄C−R2), or ifRbe comparatively small, 2π√(LC). The chargeQfalls off by the fractione−RT⁄2L(whereeis the number 2.71828...) in each periodT, and so gradually disappears.
Thus electric oscillations are produced, that is to say, the charged state of the condenser subsides by oscillations, in which the charged state undergoes successive reversals, with dissipation of energy in the wire; and both the period and the rate of dissipation can be calculated ifL,C, andRare known, or can be found, for the system. These quantities can be calculated and adjusted in certain definite cases, and as the electric oscillations can be experimentally observed, the theory can be verified. This has been done by various experimenters.
Returning to the pendulum illustration, it will be seen that the pendulum held deflected is analogous to the charged jar, letting the pendulum go corresponds to connecting the discharging coil to the coatings, the motion of the pendulum is the analogue of that motion of the medium in which consists the magnetic field, the friction of the air answers to the resistance of the wire which finally damps out the current. The inertia or mass of the bob is the analogue of what Thomson called the electromagnetic inertia of the coil and connections; what is now generally called the self-inductance of the conducting system. The component of gravity along the path towards the lowest point, answers to the reciprocal, 1 ⁄C, of the capacity of the condenser.
It appears from the analogy that just as the oscillations of a pendulum can be prevented by immersing the bob in a more resisting medium, such as treacle or oil, so that when released the pendulum slips down tothe vertical without passing it, so by properly proportioning the resistance in the circuit to the electromagnetic inertia of the coil, oscillatory discharge of the Leyden jar may also be rendered impossible.
All this was worked out in an exceedingly instructive manner in Thomson's paper; the account of the matter by the motion of Faraday tubes is more recent, and is valuable as suggesting how the inertia effect of the coil arises. The analogy of the pendulum is a true one, and enables the facts to be described; but it is to be remembered that it becomes evident only as a consequence of the mathematical treatment of the electrical problem. The paper was of great importance for the investigation of the electric waves used in wireless telegraphy in our own time. It enabled the period of oscillation of different systems to be calculated, and so the rates of exciters and receivers of electric waves to be found. For such vibrators are really Leyden jars, or condensers, caused to discharge in an oscillatory manner.
This application was not foreseen by Thomson, and, indeed, could hardly be, as the idea of electric waves in an insulating medium came a good deal later in the work of Maxwell. Yet the analogy of the pendulum, if it had then been examined, might have suggested such waves. As the bob oscillates backwards and forwards the air in which it is immersed is periodically disturbed, and waves radiate outwards from it through the surrounding atmosphere. The energy of these waves is exceedingly small, otherwise, as pointed out above, a term would have to be included in the theory of the resisted motion of the pendulum to account for this energy of radiation. So likewise when the electricvibrations proceed, and the insulating medium is the seat of a periodically varying magnetic field, electromagnetic waves are propagated outwards through the surrounding medium—the ether—and the energy carried away by the waves is derived from the initial energy of the charged condenser. In strictness also Thomson's theory of electric oscillations requires an addition to account for the energy lost by radiation. This is wanting, and the whole decay of the amount of energy present at the oscillator is put down to the action of resistance—that is, to something of the nature of frictional retardation. Notwithstanding this defect of the theory, which is after all not so serious as certain difficulties of exact calculation of the self-inductance of the discharging conductor, the periods of vibrators can be very accurately found. When these are known it is only necessary to measure the length of an electrical wave to find its velocity of propagation. When electromagnetic waves were discovered experimentally in 1888 by Heinrich Hertz, it was thus that he was able to demonstrate that they travelled with the velocity of light.
Thomson suggested that double, triple and quadruple flashes of lightning might be successive flashes of an oscillatory discharge. He also pointed out that if a spark-gap were included in a properly arranged condenser and discharging wire, it might be possible, by means of Wheatstone's revolving mirror, to see the sparks produced in the successive oscillations, as "points or short lines of light separated by dark intervals, instead of a single point of light, or of an unbroken line of light, as it would be if the discharge were instantaneous, or were continuous, or of appreciable duration."
This anticipation was verified by experiments made by Feddersen, and published in 1859 (Pogg. Ann., 108, 1859). The subject was also investigated in Helmholtz's laboratory at Berlin, by N. Schiller, who, determining the period for condensers with different substances between the plates, was able to deduce the inductive capacities of these substances (Pogg. Ann., 152, 1874). [The specific inductive capacity of an insulator is the ratio of the capacity of a condenser with the substance between the plates to the capacity of an exactly similar condenser with air between the plates.]
The particular case of non-oscillatory discharge obtained by supposingCandQboth infinitely great and to have a finite ratioV(which will be the potential, p.34, of the charged plate), is considered in the paper. The discharging conductor is thus subjected to a difference of potential suddenly applied and maintained at one end, while the other end is kept at potential zero. The solution of the differential equation for this case will show how the current rises from zero in the wire to its final steady value. Ifcbe put as before for the current −dQ⁄dt, and the constant valueVforQ⁄C, the equation is
which gives, sincec= 0 whent= 0,
Thus, when an infinite time has elapsed the current has becomeV⁄R, the steady value.
Thomson concludes by showing how, by measuring the non-oscillatory discharge of a condenser (the capacity of which can be calculated) by means of an electrodynamometer and an ordinary galvanometer arranged in series, what W. Weber called the duration of the discharging current may be determined. From this Thomson deduced a value for the ratio of the electromagnetic unit of electricity to the electrostatic unit, and indicated methods of determining this ratio experimentally. This ratio is of fundamental importance in electromagnetic theory, and is essentially of the nature of a speed. According to Maxwell it is the speed of propagation of electromagnetic waves in an insulating medium for which the units are defined. It was first determined in the Glasgow laboratory by Mr. Dugald McKichan, and has been determined many times since. It is practically identical with the speed of light as ascertained by the best experiments.
Professor Taitwas appointed to the Chair of Natural Philosophy in the University of Edinburgh in 1860, and came almost immediately into frequent contact with Thomson. Both were Peterhouse men, trained by the same private tutor—William Hopkins—both were enthusiastic investigators in mathematical as well as in experimental physics, they taught in the sister universities of Edinburgh and Glasgow, and had much the same kind of classes to deal with and the same educational problems to solve. Tait was an Edinburgh man—an old school-fellow of Clerk Maxwell at the Edinburgh Academy—and had therefore been exposed to that contact, in play and in work, with compeers of like age and capabilities, which is one of the best preparations for the larger school and more serious struggles of life. Thomson's early education, under his father's anxious care, had no doubt certain advantages, and his early entrance into college classes gave him to a great extent that intercourse with others for which such advantages are never complete compensation. The two men had muchcommunity of thought and experience, and the literary partnership into which they entered was hailed as one likely to do much for the progress of science.
In some ways, however, Thomson and Tait were very different personalities. Thomson troubled himself little with metaphysical subtleties, his conceptions were like those of Newton, absolutely clear so far as they went; he never, in his teaching at least, showed any disposition to discuss the "foundations of dynamics," or the conception of motion in a straight line. These were taken for granted like the fundamental ideas in a book on geometry; and the student was left to do what every true dynamical student must do for himself sooner or later—to compare the abstractions of dynamics with the products of his experience in the world of matter and force. Perhaps a little guidance now and then in the difficulties about conceptions, which beset every beginner, might not have been amiss: but Thomson was so intent on the concrete example in hand—pendulum or gyrostat, or what not—that he left each man to form or correct his own ideas by the lessons which such examples afford to every one who carefully examines them.
Tait, on the other hand, though he continually denounced metaphysical discussion, was in reality much more metaphysical than Thomson, and seemed to take pleasure in the somewhat transcendental arguments with regard to matters of analysis which were put forward, especially in theElements of Quaternions, by Sir William Rowan Hamilton, of Dublin, a master whom he much revered. But there is metaphysics and metaphysics! and the pronouncements of professed metaphysicians were often characterisedas non-scientific and fruitless, which no doubt they were from the physical point of view.
Then Tait was strongly convinced of the importance for physics of the quaternion analysis: Thomson was not, to say the least; and this was probably the main reason why the vectorial treatment of displacement, velocities, and other directed quantities, has no place in the joint writings of the two Scottish professors. In controversy Tait was a formidable antagonist: when war was declared he gave no quarter and asked for none, though he never fought an unchivalric battle. He admired foreign investigators—and especially von Helmholtz—but he was always ready to put on his armour and place lance in rest for the cause of British science. Thomson was much less of a combatant, though he also could bravely splinter a spear with an opponent on occasion, as in the memorable discussion with Huxley on the Age of the Earth.
Tait's professorial lectures were always models of clear and logical arrangement. Every statement bore on the business in hand; the experimental illustrations, always carefully prepared beforehand, were called for at the proper time and were invariably successful. With Thomson it was otherwise: his digressions, though sometimes inspired and inspiring, were fatal to the success of the utmost efforts of his assistants to make his lectures successful systematic expositions of the facts and principles of elementary physics.
As has been stated in Chapter IV, two books were announced in 1863 as in course of preparation for the ensuing session of College. These were not published until 1867 and 1873; the first issued was the famousTreatise on Natural Philosophy, the second was entitledElements of Natural Philosophy, and consisted in the main of part of the non-mathematical or large type portions of theTreatise. The scheme of the latter was that of an articulated skeleton of statements of principles and results, printed in ordinary type, with the mathematical deductions and proofs in smaller type. As was to be expected, theElements, to a student whose mathematical reading was wide enough to tackle theTreatise, was the more difficult book of the two to completely master. But the continued large print narrative, as it may be called, is extremely valuable. It is a memorial of a habit of mind which was characteristic of both authors. They kept before them always the idea or thing rather than its symbol; and thus the edifice which they built up seemed never obscured by the scaffolding and machinery used in its erection. And as far as possible in processes of deduction the ideas are emphasised throughout; there is no mere putting in at one end and taking out at the other; the result is examined and described at every stage. As in all else of Thomson's work, physical interpretation is kept in view at every step, and made available for correction and avoidance of errors, and the suggestion of new inquiries.
The book as it stands consists of "Division I, Preliminary" and part of "Division II, Abstract Dynamics." Division I includes the chapter on Kinematics already referred to, a chapter on Dynamical Laws and Principles, chapters on Experience and Measures and Instruments. Division II is represented only by Chapter V, Introductory; Chapter VI, Statics of a Particle and Attractions; and Chapter VII, Statics of Solids and Fluids. Thus Abstract Dynamics iswithout the more complete treatment of Kinetics to which, as well as to Statics, the discussion of Dynamical Laws and Principles was intended to be an introduction. But to a considerable extent, as we shall see, Kinetics is treated in this introductory chapter: indeed, the discussion of the general theorems of dynamics and their applications to kinetics is remarkably complete.
In Volume II it was intended to include chapters on the kinetics of a particle and of solid and fluid bodies, on the vibrations of solid bodies, and on wave-motion in general. It was expected also to contain a chapter much referred to in Volume I, on "Properties of Matter." That the work was not completed is a matter of keen regret to all physicists, regret, however, now tempered by the fact that many of the subjects of the unfulfilled programme are represented by such works as Lord Rayleigh'sTheory of Sound, Lamb'sHydrodynamics, and Routh'sDynamics of a System of Rigid Bodies. But all deeply lament the loss of the "Properties of Matter." No one can ever write it as Thomson would have written it. His students obtained in his lectures glimpses of the things it might have contained, and it was most eagerly looked for. If that chapter only had been given, the loss caused by the discontinuance of the book would not have been so irreparable.
The first edition of the book was published by the Clarendon Press, Oxford. It was printed by Messrs. Constable, of Edinburgh, and is a beautiful specimen of mathematical typography. In some ways the first edition is exceedingly interesting, for it is not too much to say that its issue had an influence on dynamical science, and its exposition in this country, only secondto that due to Newton'sPrincipia. Three other works, perhaps, have had the same degree and kind of influence on mathematical thought—Laplace'sMécanique Céleste, Lagrange'sMécanique Analytique, and Fourier'sThéorie Analytique de la Chaleur.
The second edition was issued by the Cambridge University Press as Parts I and II in 1878 and 1883. Various younger mathematicians now of eminence—Professor Chrystal, of Edinburgh, and Professor Burnside, of Greenwich, may be mentioned—read the proofs, and it is on the whole remarkably free from typographical and other errors. With the issue of Part II, the continuation was definitely abandoned.
In the second edition many topics are more fully discussed, and the contents include a very valuable account of cycloidal motion (or oscillatory motion, as it is more usually called), and of a revised version of the chapter on Statics which forms the concluding portion of the book, and which discusses some of the great problems of terrestrial and cosmical physics.
Various speculations have been indulged in, from time to time, as to the respective parts contributed to the work by the two authors, but these are generally very wide of the mark. The mode of composition of the sections on cycloidal (oscillatory) motion gives some idea of Thomson's method of working. His proofs (of "T and T-dash" as the authors called the book) were carried with him by rail and steamer, and he worked incessantly (without, however, altogether withdrawing his attention from what was going on around him!) at corrections and additions. He corrected heavily on the proofs, and then overflowed into additional manuscript. Thus, when he came to theshort original § 343, he greatly extended that in the first instance, and proceeded from section to section until additions numbered from § 343ato § 343p, amounting in all to some ten pages of small print, had been interpolated. Similarly § 345 was extended by the addition of §§ 345 (i) to 345 (xxviii), mainly on gyrostatic domination. The method had the disadvantage of interrupting the printers and keeping type long standing, but the matter was often all the more inspiring through having been produced under pressure from the printing office. Indeed, much was no doubt written in this way which, to the great loss of dynamical science, would otherwise never have been written at all.
The kinematical discussion begins with the consideration of motion along a continuous line, curved or straight. This naturally suggests the ideas of curvature and tortuosity, which are fully dealt with mathematically, before the notion of velocity is introduced. When that is done, the directional quality of velocity is not so much insisted on as is now the case: for example, a point is spoken of as moving in a curve with a uniform velocity; and of course in the language of the present time, which has been rendered more precise by vector ideas, if not by vector-analysis, the velocity of a point which is continually changing the direction of its motion, cannot be uniform. The same remark may be made regarding the treatment of acceleration: in both cases the reference of the quantity to three Cartesian axes is immediate, and the changes of the components, thus fixed in direction, are alone considered.
There can be no doubt that greater clearness is obtained by the process afterwards insisted on by Tait,of considering by a hodographic diagram the changes of velocity in successive intervals of time, and from these discovering the direction and magnitude of the rate of change at each instant. This method is indeed indicated at § 37, but no diagram is given, and the properties of the hodograph are investigated by means of Cartesians. The subject is, however, treated in theElementsby the method here indicated.
Remarkable features of this chapter are the very complete discussion of simple harmonic or vibratory motion, the sections on rotation, and the geometry of rolling and precessional motion, and on the curvature of surfaces as investigated by kinematical methods. A remark made in § 96 should be borne in mind by all who essay to solve gyrostatic problems. It is that just as acceleration, which is always at right angles to the motion of a point, produces a change in the direction of the motion but none in thespeedof the point (it does influence thevelocity), so an action, tending always to produce rotation about an axis at right angles to that about which a rigid body is already rotating, will change the direction of the axis about which the body revolves, but will produce no change in the rate of turning.20
A very full and clear account of the analysis of strains is given in this chapter, in preparation for the treatment of elasticity which comes later in the book; and a long appendix is added on Spherical Harmonics, which are defined as homogeneous functions of the coordinates which satisfy the differential equation of the distribution of temperature in a medium in which there is steady flow of heat, or of distribution of potential in an electrical field. This appendix is within its scope one of the most masterly discussions of this subject ever written, though, from the point of view of rigidity of proof, required by modern function-theory, it may be open to objection.
In the next chapter, which is entitled "Dynamical Laws and Principles," the authors at the outset declare their intention of following thePrincipiaclosely in the discussion of the general foundations of the subject. Accordingly, after some definitions the laws of motion are stated, and the opportunity is taken to adopt and enforce the Gaussian system of absolute units for dynamical quantities. As has been indicated above, the various difficulties more or less metaphysical which must occur to every thoughtful student in considering Newton's laws of motion are not discussed, and probably such a discussion was beyond the scheme which the authors had in view. But metaphysics is not altogether excluded. It is stated that "matter has an innate power of resisting external influences, so that every body, as far as it can, remains at rest, or moves uniformly in a straight line," and it is stated that this property—inertia—is proportional to the quantity of matter in the body. This statement is criticised by Maxwell in his review of theNatural PhilosophyinNaturein 1879 (one of the last papers that Maxwell wrote). He asks, "Is it a fact that 'matter has any power, either innate or acquired, of resisting external influences'? Does not every force which acts on a body always produce that change in the motion of the body by which its value, as a force, is reckoned? Is a cup of tea to be accused of resisting the sweetening influence of sugar, because it persistently refuses to turn sweet unless the sugar is put into it?"
This innate power of resisting is merely themateriæ vis insitaof Newton's "Definitio III," given in thePrincipia, and the statement to which Maxwell objects is only a free translation of that definition. Moreover, when a body is drawn or pushed by other bodies, it reacts on those bodies with an equal force, and this reaction is just as real as the action: its existence is due to the inertia of the body. The definition, from one point of view, is only a statement of the fact that the acceleration produced in a body in certain circumstances depends upon the body itself, as well as on the other bodies concerned, but from another it may be regarded as accounting for the reaction. The mass or inertia of the body is only such a number that, for different bodies in the same circumstances as to the action of other bodies in giving them acceleration, the product of the mass and the acceleration is the same for all. It is, however, a very important property of the body, for it is one factor of the quantum of kinetic energy which the body contributes to the energy of the system, in consequence of its motion relatively to the chosen axes of reference, which are taken as at rest.
The relativity of motion is not emphasised so greatlyin theNatural Philosophyas in some more modern treatises, but it is not overlooked; and whatever may be the view taken as to the importance of dwelling on such considerations in a treatise on dynamics, there can be no doubt that the return to Newton was on the whole a salutary change of the manner of teaching the subject.
The treatment of force in the first and second laws of motion is frankly causal. Force is there thecauseof rate of change of momentum; and this view Professor Tait in his own writings has always combated, it must be admitted, in a very cogent manner. According to him, force is merely rate of change of momentum. Hence the forces in equations of motion are only expressions, the values of which as rates of change of momentum, are to be made explicit by the solution of such equations in terms of known quantities. And there does not seem to be any logical escape from this conclusion, though, except as a way of speaking, the reference to cause disappears.
The discussion of the third law of motion is particularly valuable, for, as is well known, attention was therein called to the fact that in the last sentences of theScholiumwhich Newton appended to his remarks on the third law, the rates of working of the acting and reacting forces between the bodies are equal and opposite. Thus the whole work done in any time by the parts of a system on one another is zero, and the doctrine of conservation of energy is virtually contained in Newton's statement. The only point in which the theory was not complete so far as ordinary dynamical actions are concerned, was in regard to work done against friction, for which, when heat wasleft out of account, there was no visible equivalent. Newton's statement of the equality of what Thomson and Tait called "activity" and "counter-activity" is, however, perfectly absolute. In the completion of the theory of energy on the side of the conversion of heat into work, Thomson, as we have seen, took a very prominent part.
After the introduction of the dynamical laws the most interesting part of this chapter is the elaborate discussion which it contains of the Lagrangian equations of motion, of the principle of Least Action, with the large number of extremely important applications of these theories. The originality and suggestiveness of this part of the book, taken alone, would entitle it to rank with the great classics—theMécanique Céleste, theMécanique Analytique, and the memoirs of Jacobi and Hamilton—all of which were an outcome of thePrincipia, and from which, with thePrincipia, the authors of theNatural Philosophydrew their inspiration.
It is perhaps the case, as Professor Tait himself suggested, that no one has yet arisen who can bend to the fullest extent the bow which Hamilton fashioned; but when this Ulysses appears it will be found that his strength and skill have been nurtured by the study of theNatural Philosophy. Lagrange's equations are now, thanks to the physical reality which the expositions and examples of Thomson and Tait have given to generalised forces, coordinates, and velocities, applied to all kinds of systems which formerly seemed to be outside the range of dynamical treatment. As Maxwell put it, "The credit of breaking up the monopoly of the great masters of the spell, and making all their charms familiar in our ears as household words, belongsin great measure to Thomson and Tait. The two northern wizards were the first who, without compunction or dread, uttered in their mother tongue the true and proper names of those dynamical concepts, which the magicians of old were wont to invoke only by the aid of muttered symbols and inarticulate equations. And now the feeblest among us can repeat the words of power, and take part in dynamical discussions which a few years ago we should have left to our betters."
A very remarkable feature in this discussion is the use made of the idea of "ignoration of coordinates." The variables made use of in the Lagrangian equations must be such as to enable the positions of the parts of the system which determine the motion to be expressed for any instant of time. These parts, by their displacements, control those of the other parts, through the connections of the system. They are called the independent coordinates, and sometimes the "degrees of freedom," of the system. Into the expressions of the kinetic and potential energies, from which by a formal process the equations of motion, as many in number as there are degrees of freedom, are derived, the value of these variables and of the corresponding velocities enter in the general case. But in certain cases some of the variables are represented by the corresponding velocities only, and the variables themselves do not appear in the equations of motion. For example, when fly-wheels form part of the system, and are connected with the rest of the system only by their bearings, the angle through which the wheel has turned from any epoch of time is of no consequence, the only thing which affects the energy of the systemis the angular velocity or angular momentum of the wheel. The system is said by Thomson and Tait in such a case to be under gyrostatic domination. (See "Gyrostatic Action," p.214below.)
Moreover, since the force which is the rate of growth of the momentum corresponding to any coordinate is numerically the rate of variation with that coordinate of the difference of the kinetic and potential energies, every force is zero for which the coordinate does not appear; and therefore the corresponding momentum is constant. But that momentum is expressed by means of the values of other coordinates which do appear and their velocities, with the velocities for the absent coordinates; and as many equations are furnished by the constant values of such momenta as there are coordinates absent. The corresponding velocities can be determined from these equations in terms of the constant momenta and the coordinates which appear and their velocities. The values so found, substituted in the expressions for the kinetic and potential energies, remove from these expressions every reference to the absent coordinates. Then from the new expression for the kinetic energy (in which a function of the constant momenta now appears, and is taken as an addition to the potential energy) the equations of motion are formed for the coordinates actually present, and these are sufficient to determine the motion. The other coordinates are thus in a certain sense ignored, and the method is called that of "ignoration of coordinates."
Theorems of action of great importance for a general theory of optics conclude this chapter; but of these it is impossible to give here any account, withouta discussion of technicalities beyond the reading of ordinary students of dynamics.
In an Appendix to Part I an account is given of Continuous Calculating Machines. Ordinary calculating machines, such as the "arithmometer" of Thomas of Colmar, carry out calculations and exhibit the result as a row of figures. But the machines here described are of a different character: they exhibit their results by values of a continuously varying quantity. The first is one for predicting the height of the tides for future time, at any port for which data have been already obtained regarding tidal heights, by means of a self-registering tide-gauge. Two of these were made according to the ideas set forth in this Appendix; one is in the South Kensington Museum, the other is at the National Physical Laboratory at Bushy House, where it is used mainly for drawing on paper curves of future tidal heights, for ports in the Indian Ocean. From these curves tide-tables are compiled, and issued for the use of mariners and others.
Another machine described in this Appendix was designed for the mechanical solution of simultaneous linear equations. It is impossible to explain here the interesting arrangement of six frames, carrying as many pulleys, adjustable along slides (for the solution of equations involving six unknown quantities), which Thomson constructed, and which is now in the Natural Philosophy Department at Glasgow. The idea of arranging the first practical machine for this number of variables, was that it might be used for the calculation of the corrections on values already found for the six elements of a comet or asteroid. The machine was made, but some mechanical difficultiesarose in applying it, and the experiments with it were not at the time persevered with. Very possibly, however, it may yet be brought into use.