This equation was applied by Thomson (article Elasticity) to find the relation between what he called the kinetic modulus of elasticity and the static modulus, that is, between the modulus for adiabatic strain and the modulus for isothermal strain.

The augmentation of the strain produced by raising the temperature 1 is e, and therefore edT, that is, -TedS?K?, is the increase of strain due to the sudden rise of temperature dT. This added to the isothermal strain produced by dS will give the whole adiabatic strain. Thus if M be the static or isothermal modulus, the adiabatic strain is dS?M - TedS?K?. If M' denote the kinetic or adiabatic modulus its value is dS divided by the whole adiabatic strain, that is, M' = M?(1 - MTe?K?) and the ratio M'?M = 1?(1 - MTe?K?).

It is well known and easy to prove, without the use of any theorem which can be properly called thermodynamic, that this ratio of moduli is equal to the ratio of the specific heat K of the substance, under the condition of constant stress, to the specific heat N under the condition of constant strain of the corresponding type. This, indeed, is self-evident if two changes of stress, one isothermal the other adiabatic, _which produce the same steps of displacement ds_, be considered, and it be remembered that the step ?T of temperature which accompanies the adiabatic change may be regarded as made up of a step -dT of temperature, accompanying a displacement ds effected at constant stress, and then two successive steps dT and ?T effected, at constant strain, along with the steps of stress dS. The ratio M'?M is easily seen to have the value (?T + dT)?dt, and since -KdT + N(?T + dT) = 0, by the adiabatic condition, the theorem is proved.

Laplace's celebrated result for air, according to which the adiabatic bulk-modulus is equal to the static bulk-modulus multiplied by the ratio of the specific heat of air pressure constant to the specific heat of air volume constant, is a particular example of this theory.

Thomson showed in the Elasticity article how, by the value of M'?M, derived as above from thermodynamic theory, the value of K?N could be obtained for different substances and for different types of stress, and gave very interesting tables of results for solids, liquids, and gases subjected to pressure-stress (bulk-modulus) and for solids subjected to longitudinal stress (Young's modulus).

The discussion as to the relation of the adiabatic and isothermal moduli of elasticity is part of a very important paper on "Thermoelastic, Thermomagnetic, and Thermoelectric Properties of Matter," which he published in the _Philosophical Magazine_ for January 1878. This was in the main a reprint of an article ent.i.tled, "On the Thermoelastic and Thermomagnetic Properties of Matter, Part I," which appeared in April 1855 in the first number of the _Quarterly Journal of Mathematics_. Only thermoelasticity was considered in this article; the thermomagnetic results had, however, been indicated in an article on "Thermomagnetism"

in the second edition of the _Cyclopaedia of Physical Science_, edited and in great part written by Professor J. P. Nichol, and published in 1860. For the same Cyclopaedia Thomson also wrote an article ent.i.tled, "Thermo-electric, Division I.--Pyro-Electricity, or Thermo-Electricity of Non-conducting Crystals," and the enlarged _Phil. Mag._ article also contained the application of thermodynamics to this kind of thermoelectric action.

This great paper cannot be described without a good deal of mathematical a.n.a.lysis; but the student who has read the earlier thermodynamical papers of Thomson will have little difficulty in mastering it. It must suffice to say here that it may be regarded as giving the keynote of much of the general thermodynamic treatment of physical phenomena, which forms so large a part of the physical mathematics of the present day, and which we owe to Willard Gibbs Duhem, and other contemporary writers.

Thomson had, however, previous to the publication of this paper, applied thermodynamic theory to thermoelectric phenomena. A long series of papers containing experimental investigations, and ent.i.tled, "Electrodynamic Qualities of Metals," are placed in the second volume of his _Mathematical and Physical Papers_. This series begins with the Bakerian Lecture (published in the _Transactions of the Royal Society_ for 1856) which includes an account of the remarkable experimental work accomplished during the preceding four or five years by the volunteer laboratory corps in the newly-established physical laboratory in the old College. The subjects dealt with are the Electric Convection of Heat, Thermoelectric Inversions, the Effects of Mechanical Strain and of Magnetisation on the Thermoelectric Qualities of Metals, and the Effects of Tension and Magnetisation on the Electric Conductivity of Metals. It is only possible to give here a very short indication of the thermodynamic treatment, and of the nature of Thomson's remarkable discovery of the electric convection of heat.

It was found by Seebeck in 1822 that when a circuit is formed of two different metals (without any cell or battery) a current flows round the circuit if the two junctions are not at the same temperature. For example, if the two metals be rods of antimony and bis.m.u.th, joined at their extremities so as to form a complete circuit, and one junction be warmed while the other is kept at the ordinary temperature, a current flows across the hot junction in the direction from bis.m.u.th to antimony.

Similarly, if a circuit be made of a copper wire and an iron wire, a current pa.s.ses across the warmer junction from copper to iron. The current strength--other things being the same--depends on the metals used; for example, bis.m.u.th and antimony are more effective than other metals.

It was found by Peltier that when a current, say from a battery, is sent round such a circuit, that junction is cooled and that junction is heated by the pa.s.sage of the current, which, being respectively heated and cooled, would without the cell have caused a current to flow in the same direction. Thus the current produced by the difference of temperature of the junctions causes an absorption of heat from the warmer junction, and an evolution of heat at the colder junction.

This naturally suggested to Thomson the consideration of a circuit of two metals, with the junctions at different temperatures, as a heat engine, of which the hot junction was the source and the cold junction the refrigerator, while the heat generated in the circuit by the current and other work performed, if there was any, was the equivalent of the difference between the heat absorbed and the heat evolved. Of course in such an arrangement there is always irreversible loss of heat by conduction; but when such losses are properly allowed for the circuit is capable of being correctly regarded as a reversible engine.

Shortly after Seebeck's discovery it was found by c.u.mming that when the hot junction was increased in temperature the electromotive force increased more and more slowly, at a certain temperature of the hot junction took its maximum value, and then as the temperature of the hot junction was further increased began to diminish, and ultimately, at a sufficiently high temperature, in most instances changed sign. The temperature of maximum electromotive force was found to be independent of the temperature of the colder junction. It is called the temperature of the neutral point, from the fact that if the two junctions of a thermoelectric circuit be kept at a constant small difference of temperature, and be both raised in temperature until one is at a higher temperature than the neutral point, and the other is at a lower, the electromotive force will fall off, until finally, when this point is reached, it has become zero.

Thus it was found that for every pair of metals there was at least one such temperature of the hot junction, and it was a.s.sumed, with consequences in agreement with experimental results, that when the temperature was the neutral temperature there was neither absorption nor evolution of heat at the junction. But then the source provided by the thermodynamic view just stated had ceased to exist. The current still flowed, there was evolution of heat at the cold junction, and likewise Joulean evolution of heat in the wires of the circuit in consequence of their resistance. Hence it was clear that energy must be obtained elsewhere than at the junctions. Thomson solved the problem by showing that (besides the Joulean evolution of heat) there is absorption (or evolution) of heat when a current flows in a conductor along which there is a gradient of temperature. For example, when an electric current flows along an unequally heated copper wire, heat is evolved where the current flows from the hot parts to the cold, and heat is absorbed where the flow is from cold to hot. When the hot junction is at the temperature of zero absorption or evolution of heat--the so-called neutral temperature--the heat absorbed in the flow of the circuit along the unequally heated conductors is greater than that evolved on the whole, by an amount which is the equivalent of the energy electrically expended in the circuit in the same time.

It was found, moreover, that the amount of heat absorbed by a given current in ascending or descending through a given difference of temperature is different in different metals. When the current was unit current and the temperature difference also unity, Thomson called the heat absorbed or evolved in a metal the specific heat of electricity in the metal, a name which is convenient in some ways, but misleading in others. The term rather conveys the notion that electricity has a material existence. A substance such as copper, lead, water, or mercury has a specific heat in a perfectly understood sense; electricity is not a substance, hence there cannot be in the same proper sense a specific heat of electricity.

However, this absorption and evolution of heat was investigated experimentally and mathematically by Thomson, and is generally now referred to in thermoelectric discussions as the "Thomson effect."

Part VI (_Trans. R.S._, 1875) of the investigations of the electrodynamic qualities of metals dealt with the effects of stretching and compressing force, and of torsion, on the magnetisation of iron and steel and of nickel and cobalt.

One of the princ.i.p.al results was the discovery that the effect of longitudinal pull is to increase the inductive magnetisation of soft iron, and of transverse thrust to diminish it, so long as the magnetising field does not exceed a certain value. When this value, which depends on the specimen, is exceeded, the effect of stress is reversed. The field-intensity at which the effect is reversed is called the Villari critical intensity, from the fact, afterwards ascertained, that the result had previously been established by Villari in Italy. No such critical value of the field was found to exist for steel, or nickel, or cobalt.

In some of the experiments the specimen was put through a cycle of magnetic changes, and the results recorded by curves. These proved that in going from one state to another and returning the material lagged in its return path behind the corresponding states in the outward path.

This is the phenomenon called later "hysteresis," and studied in minute detail by Ewing and others. Thomson's magnetic work was thus the starting point of many more recent researches.

CHAPTER IX

HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM ENERGY--VORTEX MOTION

Thomson devoted great attention from time to time to the science of hydrodynamics. This is perhaps the most abstruse subject in the domain of applied mathematics, and when viscosity (the frictional resistance to the relative motion of particles of the fluid) is taken into account, pa.s.ses beyond the resources of mathematical science in its present state of development. But leaving viscosity entirely aside, and dealing only with so-called perfect fluids, the difficulties are often overwhelming.

For a long time the only kind of fluid motion considered was, with the exception of a few simple cases, that which is called irrotational motion. This motion is characterised by the a.n.a.lytical peculiarity, that the velocity of an element of the fluid in any direction is the rate of variation per unit distance in that direction of a function of the coordinates (the distances which specify the position) of the particle.

This condition very much simplifies the a.n.a.lysis; but when it does not hold we have much more serious difficulties to overcome. Then the elements of the fluid have what is generally, but quite improperly, called molecular rotation. For we know little of the molecules of a fluid; even when we deal with infinitesimal elements, in the a.n.a.lysis of fluid motion, we are considering the fluid in ma.s.s. But what is meant is elemental rotation, a rotation of the infinitesimal elements as they move. We have an example of such motion in the air when a ring of smoke escapes from the funnel of a locomotive or the lips of a tobacco-smoker, in the motion of part of the liquid when a cup of tea is stirred by drawing the spoon from one side to the other, or when the blade of an oar is moving through the water. In these last two cases the depressions seen in the surface are the ends of a vortex which extends between them and terminates on the surface. In all these examples what have been called vortices are formed, and hence the name vortex motion has been given to all those cases in which the condition of irrotationality is not satisfied.

The first great paper on vortex motion was published by von Helmholtz in 1858, and ten years later a memoir on the same subject by Thomson was published in the _Transactions of the Royal Society of Edinburgh_. In that memoir are given very much simpler proofs of von Helmholtz's main theorems, and, moreover, some new theorems of wide application to the motion of fluids. One of these is so comprehensive that it may be said with truth to contain the whole of the dynamics of a perfect fluid. We go on to indicate the contents of the princ.i.p.al papers, as far as that can be done without the introduction of a.n.a.lysis of a difficult description.

In Chapter VI reference has been made to the "Notes on Hydrodynamics"

published by Thomson in the _Cambridge and Dublin Mathematical Journal_ for 1848 and 1849. These Notes were not intended to be entirely original, but were composed for the use of students, like Airy's Tracts of fifteen years before.

The first Note dealt with the equation of continuity, that is to say, the mathematical expression of the obvious fact that if any region of s.p.a.ce in a moving fluid be considered, the excess of rate of flow into the s.p.a.ce across the bounding surface, above the rate of flow out, is equal to the rate of growth of the quant.i.ty of fluid within the s.p.a.ce.

The proof given is that now usually repeated in text-books of hydrodynamics.

The second Note discussed the condition fulfilled at the bounding surface of a moving fluid. The chief mathematical result is the equation which expresses the fact, also obvious without a.n.a.lysis, that there is no flow of the fluid across the surface. In other words, the component of the motion of a fluid particle in the immediate neighbourhood of the surface at any instant, taken in the direction perpendicular to the surface, must be equal to the motion of the surface in that direction at the same instant.

The third Note, published a year later (February 1849), is of considerable scientific importance. It is ent.i.tled, "On the Vis Viva of a Liquid in Motion." What used to be called the "vis viva" of a body is double what is now called the energy of motion, or kinetic energy, of the body. The term liquid is merely a brief expression for a fluid, the ma.s.s of which per unit volume is the same throughout, and suffers no variation. The fluid, moreover, is supposed devoid of friction, that is, the relative motions of its parts are unresisted by tangential force between them. The chief theorem proved and discussed may be described as follows.

The liquid is supposed to fill the s.p.a.ce within a closed envelope, which fulfils the condition of being "simply continuous." The condition will be understood by imagining any two points A, B, within the s.p.a.ce, to be joined by two lines ACB, ADB both lying within the s.p.a.ce. These two lines will form a circuit ACBDA. If now this circuit, however it may be drawn, can be contracted down to a point, without any part of the circuit pa.s.sing out of the s.p.a.ce, the condition is fulfilled. Clearly the s.p.a.ce within the surface of an anchor-ring, or a curtain-ring, would not fulfil this condition, for one part of the circuit might pa.s.s from A to B round the ring one way, and the other from A to B the other way.

The circuit could not then be contracted towards a point without pa.s.sing out of the ring.

Now let the liquid given at rest in such a s.p.a.ce be set in motion by any arbitrarily specified variation of position of the envelope. The liquid within will be set in motion in a manner depending entirely on the motion of the envelope. It is possible to conceive of other motions of the liquid than that taken, which all agree in having the specified motion of the surface. Thomson's theorem a.s.serts that the motion actually taken has less kinetic energy than that of any of the other motions which have the same motion of the bounding surface.

The motion produced has the property described by the word "irrotational," that is, the elements of the fluid have no spinning motion--they move without rotation. A small portion of a fluid may describe any path--may go round in a circle, for example--and yet have no rotation. The reader may imagine a ball carried round in a circle, but in such a way that no line in the body ever changes its direction.

The body has translation, but no spin.

Irrotationality of a fluid is secured, as stated above, when the velocity of each element in any direction is the rate of variation per unit distance in that direction of a certain function of the coordinates, the distances, taken parallel to three lines perpendicular to one another and drawn from a point, which specify the position of the particle. In fact, what is called a velocity-potential exists, similar to the potential described in Chapter IV above, for an electric field.

This condition, together with the specified motion of the surface, suffices to determine the motion of the fluid.

Two important particular consequences were pointed out by Thomson: (1) that the motion of the fluid at any instant depends solely on the form and motion of the bounding surface, and is therefore independent of the previous motion; and (2) that if the bounding surface be instantaneously brought to rest, the liquid throughout the vessel will also be instantly brought to rest.

This theorem was afterwards generalised by Thomson (_Proc. R.S.E._, 1863), and applied to any material system of connected particles set into motion by specified velocities simultaneously and suddenly imposed at selected points of the system. It was already known that the kinetic energy of a system of bodies connected in any manner, and set in motion by impulses applied at specified points, was either a maximum or a minimum, as compared with that for any other motion compatible with these impulses, and with the connections of the system. This was proved by Lagrange in the _Mecanique a.n.a.lytique_ as a generalisation of a theorem given by Euler for a rigid body set into rotation by an impulse.

Bertrand proved in 1842 that when the impulses applied are given in amount, and are applied at specified points, the system starts off with kinetic energy greater than that of any other motion which is consistent with the given impulses and the connections of the system. This other motion must be such as could be produced in the system by the given impulses, together with any other set of impulses capable of doing no work on the whole.

Thomson's theorem is curiously complementary to Bertrand's. Let the system be acted on by impulses applied at certain specified points, and by no other impulses of any kind; and let the impulses be such as to start those selected points with any prescribed velocities. The system will start off with kinetic energy which is less than that of any other motion which the system could have consistently with the prescribed velocities, and which it could be constrained to take by impulses which do no work on the whole. In each case the difference of energies is the energy of the motion which must be compounded with one motion to give the other which is compared with it.

A simple example, such as might be taken of the particular case considered by Euler, may help to make these theorems clear. Imagine a straight uniform rod to lie on a horizontal table, between which and the rod there is no friction. Let the rod be struck a blow at one end in a horizontal direction at right angles to the length of the rod. If no other impulse acts, the end of the rod will move off with a certain definite velocity, and the other parts of the rod (which is supposed perfectly unbending) will be started by the connections of the system.

It is obvious that any number of other motions of the rod can be imagined, all of which give the same motion of the extremity struck. But the actual motion taken is one of turning about that point of the rod which is two-thirds of the length from the end struck. If the reader will consider the kinetic energy for any other horizontal turning motion consistent with the same motion of the end, he will find that the kinetic energy is greater than that of the motion just specified. This motion could be produced by applying at the point about which the rod turns the impulse required to keep that point at rest. The impulse so applied would do no work. The actual value is 1?8mv, where m denotes the ma.s.s of the rod and v the velocity of the end. If the motion taken were one of rotation about a point of the rod at distance x from the end struck, the kinetic energy would be m(4l - 6lx + 3x)v?6x, where 2l is the length of the rod, and this has its least value 1?8mv for x = 4l?3. For example, x = 2l gives 1?6mv, which is greater than the value just found.

Bertrand's theorem applied to this case of motion is not quite so easy, perhaps, to understand. The motion which is said to have maximum energy is one given by a specified impulse at the end struck, and this, in the absence of any other impulses, would be a motion of minimum energy. But let the alternative motion, which is to be compared with that actually taken, be one constrained by additional impulses such as can together effect no work, and the existence of the maximum is accounted for.

The kinetic energy produced is one-half the product of the impulse into the velocity of the point struck, that is Iv, and it has just been seen that this is the product of (1?6)mv by the factor (4l - 6lx + 3x)?x. This factor is 3I?mv, and is a minimum when x = 4l?3. Thus for a given I, v will have its maximum value when the factor referred to is least, and Iv will then be a maximum.

The bar can be constrained to turn about another point by a fixed pivot there situated. An impulse will be applied to the rod by the pivot, simultaneously with the blow; and it is obvious that this impulse does no work, since there is no displacement of the point to which it is applied.

The two theorems are consequences of one principle. The constraint in each case increases what may be called the effective inertia, which may be taken as I?v. Thus when v is given, I is increased by any constraint compelling the rod to rotate about a particular axis, and so Iv, or the kinetic energy, is increased. On the other hand, when I is given the same constraint diminishes v, and so Iv is diminished.

A short paper published in the B. A. Report for 1852 points out that the lines of force near a small magnet, placed with its axis along the lines of force in a uniform magnetic field, as it would rest under the action of the field, are at corresponding points similar to those of the field of an insulated spherical conductor, under the inductive influence of a distant electric change. Further, the fact is noted that, if the magnet be oppositely directed to the field, the lines of force are curved outwards, just as the lines of flow of a uniform stream would be by a spherical obstacle, at the surface of which no eddies were caused. This is one of those instructive a.n.a.logies between the theory of fluid motion and other theories involving perfectly a.n.a.logous fundamental ideas, which Thomson was fond of pointing out, and which helped him in his repeated attempts to imagine mechanical representations of physical phenomena of different kinds.

With these may be placed another, which in lectures he frequently dwelt on--a simple doublet, as it is called, consisting of a point-source of fluid and an equal and closely adjacent point-sink. A short tube in an infinite ma.s.s of liquid, which is continually flowing in at one end and out at the other, may serve as a realisation of this arrangement. The lines of flow outside the tube are exactly a.n.a.logous to the lines of force of a small magnet; and if at the same time there exist a uniform flow of the liquid in the direction of the length of the tube, the field of flow will be an exact picture of the field of force of the small magnet, when it is placed with its length along the lines of a previously existing uniform field. The flow in the doublet will be with or against the general flow according as the magnet is directed with or against the field.

The paper on vortex-motion has been referred to above, and an indication given of the nature of the fluid-motion described by this t.i.tle. There are, however, two cases of fluid-motion which are referred to as vortices, though the fundamental criterion of vortex-motion--the non-existence of a velocity-potential--is satisfied in only one of them.

The exhibition of one of these was a favourite experiment in Thomson's ordinary lectures, as his old students will remember. If water in a large bowl is stirred rapidly with a teaspoon carried round and round in a circle about the axis of the bowl, the surface will become concave, and the form of the central part will be a paraboloid of revolution about the vertical through the lowest point, that is to say, any section of that part of the surface made by a vertical plane containing the axis will be a parabola symmetrical about the axis. The motion can be better produced by mounting the vessel on a whirling-table, and rotating it about the vertical axis coinciding with its axis of figure; but the phenomenon can be quite well seen without this machinery. In this case the velocity of each particle of the water is proportional to its distance from the axis, and the whole ma.s.s, when relative equilibrium is set up, turns, as if it were rigid, about the axis of the vessel. Each element of the fluid in this "forced vortex," as it is called, is in rotation, and, like the moon, makes one turn in one revolution about the centre of its path. This is, therefore, a true, though very simple, case of vortex-motion.

On the other hand, what may be called a "free vortex" may exist, and is approximated to sometimes when water in a vessel is allowed to run off through an escape pipe at the bottom. The velocity of an element in this "vortex" is inversely proportional to its distance from the centre, and the form of the free surface is quite different from that in the other case. The name "free vortex" is often given to this case of motion, but there is no vortex-motion about it whatever.