Or there is still another most interesting way of defining velocity, in which the analysis into space and time is not made at all, but velocity is directly measured by building up the given velocity by physical addition of a unit velocity selected arbitrarily. This matter is discussed at some length in my book "Dimensional Analysis",[16] but is of sufficient pertinence here to describe briefly. We may in the first place construct a concrete standard for velocity, as, for example, by stretching a string between two pegs on a board with a fixed weight. If we strike the string, a disturbance travels along the string which we can follow with the eye, and we define unit velocity as the velocity of this disturbance. An object has greater than unit velocity if it outruns the disturbance, and less if it lags behind. We may now duplicate our standard, making another board with pegs and stretched string, and check the equality of the two velocities by observing that the two disturbances run together. We now define two units of velocity as the velocity of anything which runs with the disturbance of the string of the second board, when the second board is made to move bodily with such a velocity that it runs with the disturbance of the first string. The process may be extended indefinitely, and any velocity measured.

If either of these two alternative definitions of velocity were adopted, it would be found that the velocity of light is infinite. Further, there would be no limit to the velocity which can be imparted to material bodies on giving them unlimited energy, which is what we are prepared to regard from ordinary experience is natural and simple. The infinite velocity for light, on the other hand, is most unnatural, particularly if we favor a medium point of view. We are here faced with a dilemma—all sorts of phenomena cannot at the same time be treated simply. If we attach the most fundamental significance to the behavior of material bodies, we shall do well to adopt one of the alternative definitions of velocity. If, on the other hand, we regard the phenomena of light as the most fundamental, we shall endeavor to form our definition so that the properties of light are simple. This was precisely the point of view of Einstein; it is characteristic of his entire scheme of restricted relativity that light is the fundamental thing, and this influenced him in adopting the first definition of velocity. Now one can have no quarrel with this desire to make light fundamental (the wisdom of doing this is to be justified by the results), and if the properties of light are to be treated mathematically, one can easily see the desirability of getting rid of infinite attributes, and so admit the desirability of making the velocity of light finite. But all this involves another very insidious assumption which we ourselves have tacitly used in all our preceding discussion, namely, that the notion of velocity properly pertains to light at all. Einstein has very definitely adopted this point of view, and so determined the character of the entire structure of relativity. I believe, on the contrary, that it is very gravely to be questioned whether the identification of light with a thing travelling, which is involved in applying the velocity concept, should be made. This discussion must be postponed, however, until we deal with the properties of light. The important points for us to notice at present are that the definition of velocity actually used involves the concept of extended time, and that it would be possible to define velocity in different ways, which would give quite a different complexion to phenomena at high velocities, but which would leave untouched our ordinary experience.

The velocities at which the precise form of definition becomes important are higher than can be reached in ordinary mechanical experiments. Such velocities can be attained in terrestrial laboratories only with electrified particles, as in experiments in high vacua or with radioactive disintegrations. It is interesting to notice that we very seldom attempt a direct measurement of velocity in such experiments by following a discrete particle in its flight and finding the time required to pass over a measured distance, but the velocities are measured indirectly, by calculation from the equations of electrodynamics and in terms of such immediately observed things as curvature of path. It is true that one or two experiments have attempted a more direct measure of velocity, but it seems there is room for more work here.

THE CONCEPTS OF FORCE AND MASS

Another concept of great importance is that of force. Since the usual analysis finds a connection between force and acceleration, and acceleration involves velocity, this is a natural place for the discussion of force. This concept has been subjected to much analysis by various writers. In origin the concept doubtless arises from the muscular sensations of resistance experienced from external bodies. This crude concept may at once be put on a quantitative basis by substituting a spring balance for our muscles, or instead of the spring balance we may use any elastic body, and measure the force exerted by it in terms of its deformation. Of course, the various precautions which must be taken in carrying out this idea physically are complex; the matter of precautions against temperature changes, for example, is one of the most easily understood. The concept of force so defined is limited to static systems; it is the task of statics to find the relation between the forces in systems at rest. We next extend the force concept to systems not in equilibrium, in which there are accelerations, and we must conceive that at first all our experiments are made in an isolated laboratory far out in empty space, where there is no gravitational field. We here encounter a new concept, that of mass, which as it is originally met is entangled with the force concept, but may later be disentangled by a process of successive approximations. The details of the various steps in the process of approximation are very instructive as typical of all methods in physics, but need not be elaborated here. Suffice it to say that we are eventually able to give to each rigid material body a numerical tag characteristic of the body, such that the product of this number and the acceleration it receives under the action of any given force applied to it by a spring balance is numerically equal to the force, the force being defined, except for a correction, in terms of the deformation of the balance, exactly as it was in the static case. In particularly, the relation found between mass, force, and acceleration applies to the spring balance itself by which the force is applied, so that a correction has to be applied for a diminution of the force exerted by the balance arising from its own acceleration.

We now extend the scope of our measurements by bringing our laboratory into the gravitational field of the earth, and immediately our experience is extended, in that we continually see bodies accelerated with no spring balance (that is, no force) acting on them. We extend the concept of force, and say that any body accelerated is acted on by a force, and the magnitude of this force is defined as that which would have been necessary to produce in the same body the same acceleration with a spring balance in empty space. There is physical justification for this extension in that we find we can remove the acceleration which a body acquires in a gravitational field by exerting on it with a spring balance a force of exactly the specified amount in the opposite direction. This extended idea of force may also be applied to systems in which there are electrical actions.

We thus see that in extending the notion of force from bodies in rest to bodies in motion, the character of the concept has changed, because the operations by which force is measured change—the force acting on a body is now measured in terms of its acceleration. But in determining the force from the acceleration, we have to know the mass. This mass has to be independently measured with the original concept of force; otherwise we have no basis for such simple statements as that the force of gravity on a body is proportional to its mass. All this applies to the ordinary range of experiments with low velocities. If now we extend the range of measurements, we find phenomena which we had not expected; for example, there seem to be difficulties in the way of indefinitely increasing the velocity of a material body, as of a charged atom. We begin to ask searching questions: is the force of gravity independent of velocity at high velocities, or is the mass independent of velocity under the same conditions or independent of the gravitational field, etc.?

In attempting to answer these new questions, we find difficulty with the concepts in terms of which they are formulated. There are no operations by which we can find whether force is independent of velocity unless we first know the mass, or any operations by which a mass can be measured unless we know a force. The purely mechanical systems with the highest velocities of which we have any experimental knowledge are the heavenly bodies. The motion of these is, with the important exception of Mercury, that predicted by the ordinary laws of mechanics, so that at first it might appear that we have here confirmation of the laws of mechanics for bodies with comparatively high velocities. But it must be remembered that all we can observe of the heavenly bodies is their positions, and that we cannot perform on these bodies all the operations by which we can check the laws of mechanics for terrestrial phenomena. If, for example, mass and the force with which gravity acts on mass were both equally affected by velocity, the motion of the heavenly bodies would be exactly the same as that observed now. Hence as we increase the range of velocity, the concepts of force and mass simultaneously lose their definiteness, and become partially fused. This is typical of what we have now come always to expect near the limit of the experimentally attainable; experience becomes less rich, the choice of physical operations more restricted, concepts change and become fewer in number. If we are to retain the same formal number of concepts, we must introduce arbitrary conventions or definitions. These definitions are to be determined largely by convenience. In the case of mechanical systems, this motive of convenience is supplied by considerations from outside the domain of mechanical phenomena. The highest velocities of practice are not reached in mechanical, but in electrical systems, in experiments with vacuum tubes, etc. Considerations of convenience are therefore dictated from the electrical point of view. These considerations will be gone into in much more detail later; the conclusion is all that we need here, which is that it is convenient to assume for the charge of the electron a constant number, independent of the velocity, and this involves making its mass variable in a definite way with velocity. Now if the principle of relativity is accepted, the mass of mechanical objects must vary with velocity in the same way as the mass of electrical charges. Since the variability of this latter is fixed, mechanical mass becomes a definite function of velocity, and the force is therefore also fixed in any specific physical case.

The fundamental definition of force given above is highly academic, involving as it does hypothetical experiments in laboratories situated far out in empty space. Some sort of procedure like this seems to correspond to more or less explicit statements to be found in the literature of mechanics. The meaning in terms of actual operations to be given to such definitions involves complicated inferential reasoning. We would make much closer connection with the conditions of actual experiment if in the definition we substituted for the hypothetical operations in empty space more or less approximately realizable operations on bodies sliding on level table tops without friction. I suppose our instinctive feeling for the laws of mechanics is such that we are convinced that definitions in terms of an interstellar space laboratory or a level table top are actually the same. But in principle we must recognize that when the operations are different, the concepts are different, and if we adopt something equivalent to the table top definition, as it seems we are physically forced to do, we must leave open in our thinking the possibility of finding in the present penumbra, when our accuracy is sufficiently increased, such phenomena perhaps as directional attributes of mass in a gravitational field.

We have just considered the sort of problem that we encounter on ordinary scales of magnitude on going from low to high velocities; what becomes of the concepts of force and mass when we go to a very small scale? Down to the atomic scale we may at least slur over the new physical difficulties, for although we cannot of course experiment with actual atoms, we can nevertheless make measurements of the Brownian[17] movement of suspensions in liquids settling in a gravitational field, for example, and the extrapolation to the atom is not a very great one. The mass of each individual atom is obtained by what is equivalent to a process of counting, assuming the law of conservation of mass on an atomic scale. This is justified by all chemical experience. The mass of the component parts of the atoms, the electrons, may also perhaps be given a unique significance after we have decided on the laws of the electrical field, by experiments on acceleration in electrical fields. The question which interests in principle here is what meaning, if any, shall be attached to the mass of the elements of the electron.

It is evident that we here go beyond any possible experience, at least for the present, and that experience has again become poorer and our concepts fewer in number. All that we can now demand is that certain combinations of numbers, some of which represent mechanical mass and others electrical charge, have proper relations to each other when integrated throughout the entire body of the electron. Similar questions confront us when we ask what are the forces which the parts of the electron exert on each other. We return to this question in considering the nature of the electrical concepts. In any event, the concepts of both force and mass are entirely altered in this domain.

It is interesting to note, in passing, that present electrical theory gives no meaning to the mass of the elements of the electron, since the total electromagnetic mass of the electron is built up from the mutual terms in the action of the elements—the total mass is not a linear resultant of the action of the elements.

THE CONCEPT OF ENERGY

In examining the concept of energy, we start with purely mechanical energy. In isolated mechanical systems, in which there are only conservative forces, the sum of kinetic and potential energy is constant. The kinetic energy may be defined as ∑ ½ mv², formed for all parts of the body. The potential energy is determined by the position of the parts of the system, and has physical significance only with reference to a datum position, that is, only changes of potential energy have meaning in terms of operations. The total energy ascribed to the system has therefore an element of arbitrariness in that the datum position may be chosen at random, and energy acquires meaning only on tracing the history back to the epoch of the datum position.

The concept of energy may be extended from mechanical systems to all systems with which we are acquainted; the operations by which meaning is given to the extended energy concept involve the generalized conservation principle, or the first law of thermodynamics. The extension to thermal systems is immediate; the inclusion of optical and electrical systems in the scheme was a most important physical step, which of course required careful experimental justification. Because of its wide range of application, the energy concept has now come to be regarded as one of the most important in physics; this idea was held by Ostwald[18] twenty and more years ago, and is now much to the front because of the connection between mass and energy indicated by the theory of relativity, and the important rôle assigned to energy levels in spectrum analysis.

What now is the precise nature and significance of the general energy concept? In the first place the conservation property of energy is one of the simplest and most obvious of the properties of matter, so that in this property of energy is seen a reason for ascribing to it certain of the properties of matter, in particular and most important, that of localization in space. We must recognize, however, that this idea of a location in space is injected into the situation entirely by ourselves, and corresponds to nothing directly given by the operations of experiment. The idea has had a most important effect, however. Witness, for instance, the importance ascribed to the discovery by Kelvin of a function by which the total energy of an electric field can be represented as distributed through space;[19] this was one of the most important props of the medium point of view.

A more critical examination is likely to diminish considerably our satisfaction with this naive analogy drawn between matter and energy. With regard to matter, we may still be tolerably satisfied with our ascription to matter of location in space, but it is quite different with regard to conservation of matter. In just what sense is matter conserved? Certainly not in terms of mass, as we at one time thought. Nevertheless we undeniably have a feeling that there is some sort of conservation property here, and are driven to formulate it badly in terms of a hypothetically constant number of protons and electrons. I have long thought that Newton was groping after some very similar idea when he so far forgot himself as to define mass as quantity of matter, a definition perfectly meaningless to a rigorous and unsympathetic interpretation. On the other hand, whatever meaning may reside in our idea of conservation of matter, it certainly is not, in at least one important respect, like the conservation of energy. For the energy of an isolated mechanical system is a function of the frame of reference in which it is described; merely by giving velocity to the reference frame and altering in no way the mechanical system we may change its kinetic, and so its total, energy by any direct amount. This does not even remotely resemble ordinary matter. I cannot see that the operations which are equivalent to the energy concept justify us in saying more than that energy is a property of a material system; the operations do not seem to give any unique meaning to a location associated with energy.

We now ask what significance is to be ascribed to the sort of conservation that energy does have. We restrict ourselves first to mechanical systems. The motions of a mechanical system satisfy certain differential equations of the second order, and the actual motion is to be found by an integration of the equations. In the integral of a differential equation certain constants appear which are determined by the initial conditions, and are therefore the same during all the future motion of the system; obviously these constants of motion correspond to conservative properties. This reasoning can of course be at once extended. Any system, mechanical or not, whose motion is determined by differential equations, will have certain conservative properties. For the systems of mechanics energy is one of the conservative functions; others are momentum and moment of momentum. Energy is particularly simple, in that it is connected with measurable properties of the system by a simple formula (∑ ½ mv²), and is furthermore scalar, which is also a property of quantity of matter. But to go further and ascribe to energy other properties of matter, such as localization in space, is entirely overlooking the essential difference in the character of the operation by which matter and quantity of energy are measured, that is, overlooking the essential difference in their physical character.

The possible extension of the energy concept from mechanics to thermodynamics receives a sufficient physical explanation in terms of our views of the essentially mechanical character of thermal phenomena. That the idea can be extended also to simple electrical or magnetic systems, in which the effect of velocity of propagation is neglected, is a consequence of the fact that in these systems the equations of motion remain of the same general mechanical type, it having been shown by Maxwell that the equations of such systems may be written in the generalized Lagrangean form. When, however, we extend our formulas to systems in which the velocity of propagation is important (that is, when we consider the field equations in their general form) we find that the Lagrangean equations no longer apply to matter taken by itself, and energy is no longer conserved in the original sense. A new function appears, however, which behaves mathematically in the same way that the energy did before. The equations of motion of the system remain Lagrangean in form if the mechanical parts of the system are supplemented by the electric and magnetic fields in space. In this extended form we have, therefore, a conservative function as before, and the energy concept may be retained in this enlarged aspect. The physical operations by which energy is determined are entirely altered, however, and the physical character of the concept is changed. No more than before is there justification for localizing energy in space, or ascribing to it other properties of matter. Yet the materialization of the energy concept, and the consequent desire that energy be localized in space, is one of the strongest arguments in many minds for the existence of a medium.

As far as I can see, therefore, the existence of conservative functions is involved in the possibility of describing natural phenomena with differential equations. That further there is a conservative function of the precise form found in mechanics is a consequence of the particular form of the equations and the nature of the forces. The question of the significance of the fact that the forces of nature appear to be conservative, with respect to this particular function of mechanics, is of much interest, but it is not our immediate concern now. We are interested rather to ask under what general conditions we shall have conservative functions. Quantum theory strongly suggests that when we pass to phenomena on a small enough scale, we may no longer be able to employ differential equations in our descriptions, and hence the previous reason for the existence of constants connected with the motion disappears. Now there is one obvious remark to be made about this more general situation. Whenever the future history of a system is so connected with its present condition that we can retrace our way to the present from any future configuration, we shall always have conservative functions. For any future configuration contains certain fixed (or conservative) features, in that we can reconstruct the unique present from any future state. There is no reason to expect that the operations by which we find the fixed features will always be simple, as in the mechanical case. Now the determination of the future by the present, and conversely the possibility of reconstructing the present from the future (or the past from the present), is, we are convinced, a property which is at least approximately true of phenomena down to a smaller scale of magnitude than we have yet reached, and so we expect to find these conservative functions in systems whose ultimate laws of motion are much more general than any with which we are yet familiar. The particular form of the conservative function depends on the character of the system. That there is a scalar conservative function for ordinary systems depends of course on particular properties of the system, but we are at least prepared to find that a scalar conservative function does not necessarily mean a differential equation of the second order.

The potential energy of a system has a particular significance with respect to this point of view. In an ordinary mechanical system, the potential energy simply measures the work done by the applied forces in being displaced from the initial to the final positions; that is, the potential energy is a measure of the deviation from the initial position, and so measures a certain feature of the history of the system. In the more general system, in which we may not have differential equations, we may look for something analogous to the potential energy which shall measure the displacement of the system from its initial configuration. Such a measure is always possible as long as the past can be reconstructed from the present (or the present from the future). We recall a remark of Poincaré's[20] to the effect that we of necessity must always have conservation, because if we have a system in which conservation apparently fails, we merely have to invent a new form of potential energy. This remark is obviously not of entire generality, but applies only to such systems as those considered here, in which the past may be reconstructed from the present.

Of late there has been much discussion of the advisability, on the basis of certain quantum phenomena, of giving up conservation as a principle applied to the details of the emission and absorption of light, retaining it only in a statistical sense. It seems to me that the question here in the minds of physicists was always merely one of convenience, and that few, if any, doubted the ultimate applicability of the principle of Poincaré, or thought that we were here concerned with a system of such great generality that the past could not be reconstructed from the present. The question was merely whether those variables in terms of which the potential energy is defined make close enough connection with other things of immediate experimental significance, or whether on the whole the retention of a potential energy is not more trouble than is justified by its convenience, making a treatment from the statistical point of view preferable. However, this is all now a matter of more or less past history, because with the recent extension of the experiments of Compton,[21] we seem to have experimental evidence for the validity of the conservation law in detail for elementary quantum processes, with a corresponding simple potential energy.

Going still deeper, however, there are quantum phenomena which still may have to be treated by statistical methods, and this may mean giving up conservation in detail. We have no experimental evidence, for example, of what the electron is doing while jumping from one quantum orbit to another. A situation like this merely means that those details which determine the future in terms of the past may lie so deep in the structure that at present we have no immediate experimental knowledge of them, and we may for the present be compelled to give a treatment from a statistical point of view based on considerations of probability. But I suppose that no one, except perhaps Norman Campbell,[22] will maintain that such a situation is more than temporary, or will cease to search for consequences of these details of structure which may be open to experimental verification.

Similarly, we cannot permanently be satisfied with a picture of radioactive phenomena which represents radioactive disintegration as a matter of chance.

The general conclusion to which all this discussion leads is that energy is probably not entitled to the fundamental position that physical thought is inclined to give it, but that it is a more or less incidental consequence of more deep-seated properties, and that the character of these deep-seated properties is subject to only the most general restrictions, so that very little about the nature of the details can be inferred from the existence of any energy function.

THE CONCEPTS OF THERMODYNAMICS

We shall not be concerned here with the many technical questions which are the proper subject of treatises on thermodynamics, but shall attempt an examination only of some fundamental concepts.

The most fundamental of these, which sets thermodynamics off apart from the simpler subjects, is probably that of temperature. In origin this concept was without question physiological, in much the same way as the mechanical concept of force was physiological. But just as the force concept was made more precise, so the temperature concept may be more or less divorced from its crude significance in terms of immediate sensation and be given a more precise meaning. This precision may be obtained through the notion of equilibrium states. We have in the first place the fundamental experimental fact that when a small body is placed inside a large system, which we recognize by crude means as comparatively invariable in temperature as time goes on, the small body very soon acquires a steady condition, that is, it comes to equilibrium with its surroundings. We now have the further experimental fact that if the small body A is in equilibrium with its environment, and body B is also in equilibrium with the same environment, there will be no change of condition of A and B when they are brought into contact with each other—that is, A and B are each in equilibrium with the other and also with the environment and therefore, by definition, at the same temperature as the environment. The temperature of the environment is now measured in terms of some of the properties of A and B which crude experience has shown change with the physiological temperature of A and B. The physiological notion of temperature is thus made more precise by being connected with the physical phenomenon of equilibrium.

Now it is at once evident that stated in this way without qualification we have said things that are not true. It is not true in general that, when A is in equilibrium with an environment and B is in equilibrium with the same environment, A will be in equilibrium with B. Suppose, for example, that the environment is a stream of water and A is a tiny water wheel moving freely in its bearings, and that B is a similar wheel with much friction. Then we know that B will become warm, and will not be in equilibrium with A when brought into contact with it. Or we may choose for A a mercury thermometer with bulb covered with putty, and for B a similar thermometer with bulb sheathed in platinum, and we know that the two thermometers will not register the same temperature in the water stream. Or still more simply, we may try to read the temperature of the air in our garden on any bright day with a silvered and with a blackened bulb thermometer; we know that the two thermometers will read different temperatures. It is evident, therefore, that we shall have to specify much more carefully the conditions under which equilibrium holds if we are to give precise significance to the temperature concept.

It seems fairly evident in the first place that we shall have to rule out systems in which there is large scale mechanical motion; the simple notion of temperature does not apply to a system moving with respect to us. Only when the two thermometers A and B move with the same velocity as the stream do we have three-fold equilibrium between the stream, A and B. We may state this in another way by saying that the temperature of a moving body must be measured on a thermometer stationary with respect to the body, but this is only a matter of words, and properly speaking the temperature concept applies only to a certain aspect of the relation between two bodies mutually at rest. We here entirely neglect relativity questions such, for example, as the proper way of correcting for the change of dimensions of moving thermometers.

If now the body whose temperature we are measuring does not move with the same velocity in all its parts, we may still give a meaning to local temperature by dividing the body into parts so small that the velocity of each part is essentially uniform, and measuring the temperature of each part with a thermometer stationary with respect to it. We are now confronted with the question of how far to carry the process of subdivision. Suppose we have a fluid whose motion is completely turbulent when measured with instruments of the ordinary scale of magnitude. For such a fluid the fundamental equilibrium proportions hold between two measuring bodies A and B and the fluid, provided that the bodies A and B are so large that the motion is completely turbulent on their scale of magnitude. We may then define the temperature of the turbulent fluid from the standpoint of these large scale bodies. But we may also define the temperature from the small scale point of view as the average of the temperatures recorded by sufficiently small thermometers, each moving with the velocity of a local bit of the fluid. These two temperatures will in general be different, and we must more or less arbitrarily select one which we define as the true temperature. It would seem that the small scale temperature is the better one to choose, because there is a certain degree of arbitrariness in specifying the scale from which the motion shall be judged completely turbulent But on the other hand, there are difficulties in the small scale definition, because the turbulence may become more and more fine grained, until we end with the motion of the molecules themselves, when the operations certainly fail which give meaning to the temperature concept. In this case of molecular turbulence, we are driven back to the large scale definition, which obviously corresponds to ordinary physical practice.

It appears then that the temperature concept is not a clean cut thing, which can be made to apply to all experience, but that it is more or less arbitrary, involving the scale of our measuring instruments. In any special case, the meaning of the temperature concept must be set by special convention. In practice this does not often make difficulty, because in the majority of cases there is no large scale motion with respect to the thermometers.

Consider now the other aspect of the equilibrium problem suggested by the thermometers with blackened and silvered bulbs in the sunshine. Our common experience tells us how to deal with this situation effectively enough for ordinary purposes. We recognize that the possibility of temperature equilibrium is disturbed by the radiation, and we protect the bulbs of the thermometers from the sun's radiation by appropriate shields. But this only minimizes the difficulty. For the shield is warmed by the sun, and in turn warms to a less degree by its radiation the bulb of the thermometer within. We must recognize that every body, no matter what its temperature, is always emitting radiation, so that the bulb of our thermometer is always in a radiation field. At first this puts us in a serious quandary as to the whole question of equilibrium and the meaning of temperature. The situation is saved by the experimental observation that there is a particular radiation field which affects all thermometers equally, namely, the field inside an infinite body all at the same temperature. Logically this looks like the vicious circle again, for we have not yet defined what we mean by the same temperature. But actually we avoid the circle here, as in so many other physical cases, by a process of asymptotic approximation. The procedure is perhaps something like this: we find that if we experiment with larger and larger bodies, isolated and at great distances from other bodies at approximately the same temperature as judged by crude physiological sensations, two thermometers, identical except that the bulb of one is blackened and that of the other is silvered, record more nearly the same temperature as time goes on and as the thermometers are sunk to greater depths in the body. In actual practice, of course, the radiational opacity of most bodies is so high that these precautions against the effects of external radiation can usually be entirely ignored. At high temperatures, on the other hand, radiation has to be explicitly dealt with.

The conclusion for us from these considerations is that operationally the concept of temperature is tied up with that of radiation—the equilibrium concept of temperature is strictly never exactly applicable; it is only a limiting sort of concept applicable when the radiation field is of a special sort, namely, that of a black body.

In spite of the explicit recognition which we have to give radiation in defining temperature, we usually entirely lose sight of it in thinking about the mechanism of ordinary physical processes, as for instance when we picture the temperature of a gas as determined by the kinetic energy of its molecules. Now I have no doubt that negligence of this sort can be justified, but the necessary logical analysis is apparently complicated, and involves a great many different sorts of experiment by methods of asymptotic approximation, by which we establish the existence of various sorts of physical constants, such as constants of emission and absorption and reflection and scattering and fluorescence and thermal conductivity. We do not need to make the analysis here, but I believe that some time it would be worth while to attempt it. Such an analysis will justify the principle so often used: that if a body is in thermal equilibrium the various processes involved, such as radiation or thermal conductivity, must when taken separately also be in equilibrium. Doubtless, if our experience had been confined to higher temperatures, like that of the sun, this notion of different mechanisms acting independently would have been more difficult to acquire.

We next consider another fundamental concept of thermodynamics, that of quantity of heat. We are at first perhaps inclined to think of this as a comparatively straightforward concept, given immediately in terms of experience, but an analysis of the operations by which we measure quantity of heat will show that the situation is really most complicated. Consider, for example, Joule's experiment in which the mechanical equivalent of heat was measured by determining the rise of temperature of the water in a container when stirred by paddles driven by a falling weight. We do not question that the rise of temperature of the water has its origin in the mechanical work done on it by the paddles. But what about the rise of temperature of the container? We shall doubtless say that part of this rise comes from heat communicated to it by the warmer water in contact with it, and part from mechanical work done on it by turbulent impact of the water. But by what operations shall we measure what part of the energy communicated to the container is heat and what part mechanical work? We try to give an idealized answer to this question in terms of Maxwell demons stationed at all parts of the boundary of the containing vessel with small scale measuring instruments. To measure the heat entering at any point I can see nothing else for the Lilliputian observers to do but to determine the temperature gradient at every point of the boundary from temperature observations at two different levels, and calculate the heat inflow from the gradient and the thermal conductivity of the material of the walls—there seems no way of measuring a flow of heat as such. The inflow of mechanical energy must be calculated from a detailed knowledge of the elastic waves and other large scale deformations of the walls. Here again there is an arbitrary element in our procedure; if our mechanical measuring instruments are on too gross a scale, we may miss mechanical energy which we would catch with finer instruments.

This situation which we have just submitted to detailed analysis is, I believe, typical of the general situation; it is not possible in the general case to find anything which we can call heat as such. Without further explicit examination, we can unambiguously speak of a body losing or gaining heat only when there has been no energy interchange of any other sort with other bodies. In such a case the heat is measured in terms of the temperature change of the body. The heat concept is in the general case a sort of wastebasket concept, defined negatively in terms of the energy left over when all other forms of energy have been allowed for.

The essential fact that a quantity of heat can by itself be defined only in terms of a drop of temperature is somewhat obscured by the usual method of thermodynamic analysis. In describing a Carnot engine, for example, it is specified that the engine shall work between a source and a sink so large that their temperature is not changed by the heat given out or absorbed by them, so that the impression is natural that heat may in some way be measured apart from temperature changes. This of course is not the case; we merely require that the source and sink be so large that their temperature changes are of a different order of magnitude from those in the working substance itself, so that with respect to the working substance, source and sink may be considered to be at constant temperature.

Assuming now that we are able to measure quantity of heat in those cases in which the concept has meaning, let us examine the first law of thermodynamics, which we write in the form:

dQ + dW = dE

Here dQ is the heat imparted to a given body by other bodies, dW is the work of all kinds done on it from outside, and dE is the increase of internal energy. Now if this equation says what appears at a naïve first glance, it should say that we find experimentally that the relation written always holds between the measured quantities dQ, dW, and dE. We have seen that in the general case it is not possible to assign a unique operational significance to dQ and dW, and presumably not to their sum. We ignore for the present difficulties of this kind and confine attention on dE; how shall we measure it? I believe it does not take much examination to convince us that there are no physical operations for measuring dE as such, and that therefore the equation expressing the first law must have a different significance from that which appears on the surface. This is often recognized in the statement that the essence of the first law is that dE is an exact differential determined only by the variables which fix the internal condition of the body, and not a function of the path by which the body is carried from one condition to another. But what shall we mean by internal condition, and how shall we be sure that we have found all the variables required to specify it completely? Internal condition may be a most complicated thing and require many variables, as shown by a piece of iron with a complicated magnetic history or by a piece of aluminum about to undergo recrystallization after overstrain. Here again I believe there is no physical procedure by which general meaning can be given to this concept of internal condition. In specific cases we can state what the variables are which determine internal condition, and the criterion that we have found the correct internal variables is that dE shall be a complete differential in terms of them. The first law of thermodynamics properly understood is not at all a statement that energy is conserved, for the energy concept without conservation is meaningless. The essence of the first law is contained in the statement that the energy concept exists (or has meaning in terms of operations).

The first law is often thought to be one of the most general of physics, but in a paradoxical sense it is the most special of all laws, because no general meaning can be given to the energy concept, but only specific meaning in special cases. The first law owes its complete generality to the fact that no specific case has yet been found of so broad a character that it cannot be included under one or another special case.

Examination will at once justify this view. Thus we find a great many systems which are adequately described in terms of two variables, pressure and temperature, in that a function of p and t can be found such that its differential equals dQ + dW. There are other systems in which the six components of stress and t completely fix the internal condition in the sense that they determine a dE. In other systems the specification of a magnetic field may be necessary, or an electric, or a gravitational field. No case is known which cannot be handled in terms of the action of external forces of the proper kind, but there is no general procedure, and the first law owes its generality to the exhaustive cataloging of special cases.

We may now return to the question left in abeyance above of the ambiguity in dQ + dW. In all the cases in which the specific variables can be found which define dE, dQ and dW also have meaning. Consider, for example, a gas, the internal condition of which may be characterized in terms of t and p. The mere fact that the internal condition can be specified in terms of two variables, one a mechanical variable, shows that the substance is mechanically homogeneous. Being mechanically homogeneous, we do not have the possibility of ambiguous values of dW varying with the scale of the measuring instruments, and in fact we know that dW = p dv. Similarly the gas being homogeneous and at rest as a whole allows unique values for dQ. Of course this cannot obscure the physical fact that even in such a gas, when we go to a small enough scale, we find inhomogeneities arising from the Brownian movement, etc. Practically our statement means that the inhomogeneities are so fine grained that over a very wide range of scale of the measuring instruments we find the same definite results. The same sort of considerations apply to more complicated systems. If dE is a complete differential in terms of t and six stress components, this means again that the body is homogeneous, its condition is determined by temperature and stress, which are the same throughout the body, and again there is no possible ambiguity from the scale of the instruments which measure dW and dQ. It seems in general, then, that if the body allows operations by which dE acquires meaning, at the same time dQ and dW are provided for. In working out this idea in full detail, some care must be given to the question of order of differentials. dQ, for unit time and unit volume, is strictly equal to k∇²t, where k is thermal conductivity, so that in determining dQ the second derivatives of temperature are involved.

If the body is obviously not homogeneous, it is still a matter of experience that it can be divided into small pieces, each of which are by themselves sufficiently homogeneous, and the first law in its usual form may be applied to each of the pieces.

Finally, we emphasize a fact already implicitly mentioned, namely, that no physical significance can be directly given to flow of heat, and there are no operations for measuring it. All we can measure are temperature distributions and rates of rise of temperature. As at present defined, a heat current is a pure invention, without physical reality, for any determined heat flow may always be modified by the addition of a solenoidal vector, with change in no measurable quantity. If someone states that throughout all space there is a uniform heat current of 10⁶ cal./cm.² sec. in the direction of Polaris, no disproof can be given, for such a stream is solenoidal, and as much heat flows out of every closed surface in unit time as flows in. Such a solenoidal flow is meaningless in terms of operations; we could give meaning to such a flow only in terms of some slight modification of the solenoidal condition introduced by the measuring instrument. In all ordinary conditions the flow of heat given by the simple relation q = k Grad t corresponds exactly to what our atomic pictures lead to expect in those cases where the details of the picture can be worked out. But there may be cases where it is advantageous to supplement the ordinary heat flow (= k Grad t) by the pure fiction of a solenoidal flow, because in this way it may be possible to account for new phenomena which appear when the solenoidal conditions are slightly departed from. Thus if in a conductor at uniform temperature carrying a steady electric current we say that a heat current is also flowing proportional to the electric current and therefore solenoidal, we may provide the possibility for a simple correlation of phenomena found under those more complicated conditions when an electric current flows in a conductor of non-uniform temperature in a magnetic field. If it should turn out that the heat current is uniquely determined by considerations of this character, then we have taken the first step away from the pure formalism which this sort of thing otherwise is in the direction of giving physical reality to the invention of "heat current."

There are other interesting questions of a fundamental thermodynamic character, such for example, as whether the entropy concept has any general significance apart from the scale of our measuring instruments, and what is the operational significance of applying thermodynamic concepts to radiation, but we shall not consider these questions here.

ELECTRICAL CONCEPTS

We now set ourselves the problem of finding the meaning of the various concepts in terms of which we describe the behavior of electrical systems, assuming that we understand what we mean by "electrical." We start with the simplest electrical systems, namely, those in which we deal with static phenomena on a large scale. In such systems there are independent physical operations by which we may find the magnitude of any charge, provided that it is effectively concentrated in a geometrical point. The measurements involved in these operations are measurements of ordinary mechanical forces; we assume that our knowledge of mechanics has already taught us how to make such measurements. An electrically charged body experiences forces, which may be measured by tying a string to it and pulling on the string with a spring balance hard enough to keep the body in equilibrium. Three charges are numerically equal if when each is placed at unit distance from another, in the absence of the third (or other charge), the forces are always the same. If furthermore the forces are of unit magnitude, the charges are defined as unit charge. Having obtained unit charge, we define the magnitude of any other charge as equal to the force which it experiences when placed at unit distance from unit charge. This of course is all very trite; the important thing for us is merely that magnitude of charge, or quantity of electricity, is an independent physical concept, and that unique operations exist for determining it. These operations presuppose the ability to perform certain operations of mechanics. Having now learned how to measure electrical quantities, we discover experimentally the inverse square law of force, and later arrive at the concept of the electric field. As we have seen, the field is an invention; here we shall use this concept only for the purpose for which the invention was made, and shall not involve ourselves in any of the implications of ascribing physical reality to the field. Notice that as long as we deal only with point charges we do not have to define field strength in terms of the limiting procedure of making the exploring charge smaller, for the limiting small charge is necessary only to avoid the reaction of the exploring charge on the positions of the charges which generate the field. All this again is trite; the important point is that the operations by which the inverse square law and the concept of the field are established presuppose that the charge is given as an independent concept, since the operations involve a knowledge of charges. The operations also involve the measurement of forces by the ordinary static procedure of mechanics with spring balances. With the means now at our command we establish one very important property of electric charges, namely that the total amount of charge on an isolated body of finite size is conserved, no matter how the charge is forced to rearrange itself by the motion of charges on adjacent bodies.

By procedures exactly like those outlined above, we may treat all the corresponding magnetic quantities; there is formal parallelism between the two sets of phenomena, but there is the physical difference that we have to realize a single magnetic pole by the device of using a very long slender magnet.

We now give our electrical system more freedom, in that we allow the charges to be in motion with respect to each other. Perhaps the most immediate question which we now have to ask is whether charge continues to be conserved when set in motion, or whether the total charge on an isolated body is a function of its velocity? To answer this question we must generalize the procedure by which we assigned a numerical value to a stationary charge. Perhaps the simplest way is to allow two unit charges each to move with constant velocity, remaining at unit distance apart, and measure with a spring balance the force required to keep them at constant distance apart. Now we immediately find that the force is altered under these conditions, so that our first impulse is to say that the charge is a function of the velocity. But as we experiment further, we find that the state of affairs is very complicated; the force between the two charges at any moment of their motion depends not only on the charges, their distance apart, and their velocities, but also on the angle between the line joining them and the direction of motion in the lines. Further experiment of other kinds yields other information; it requires a force to maintain a charge in uniform motion in a magnetic field, or to maintain a magnetic pole in motion in an electric field. A moving electric charge exerts a force on a stationary magnetic pole, so that by definition the moving charge is surrounded by a magnetic field, and similarly a moving magnetic pole is surrounded by an electric field. Returning to our two moving electric charges, we are impelled to ask whether, if all these complications are possible, the numerical constant (unity for static charges) in the inverse square law of force is a function of velocity as well as the magnitude of the charges themselves? If we broaden the question in this way, as we apparently must, our problem becomes indeterminate, for we are trying to answer two different questions with a single kind of measurement, namely of the force between moving charges. I have had no better luck on trying other methods of measurement. Apparently the operations do not exist by which unique meaning can be given to the question of whether the magnitude of a charge is a function of its velocity. On realizing this situation, we are at first embarrassed to know how to proceed, but we reflect that the embarrassment is not of our own making, but corresponds to a physical fact. The concept of charge as a unique and independent thing essentially pertains only to static systems. We may extend the concept to moving systems if we wish, as a matter of convenience to ourselves, but must recognize that such an extension is an invention of ours and not a reality of nature. Now we do make such an extension, and we make it in the simplest possible way, that is, we define the charge on an isolated body in motion as that which we should find on it if we reduced it to rest and made measurements according to the regular static procedure. That this is a convenient thing to do depends on the experimental result that the charge so found is independent of the way in which velocity is imparted to or removed from the body; in other words, whenever the body is reduced to rest, the same charge is always found on it.

Although this is pure definition on our part, it turns out to have a most simple and convenient connection with experimental facts which were discovered after the decision to treat a moving charge in this way was made; the discovery is of the atomic structure of electricity. If then we agree to call each elementary charge a constant independent of the velocity, the total charge on a body becomes merely proportional to the count of the total number of atomic charges on the body, which is certainly highly convenient and suggestive.

Having now fixed what we mean by the magnitude of a moving charge, we are ready to turn to the general problem of the behavior of any system of charged bodies in motion. For the present we consider only phenomena of the scale of everyday experience. The most general problem that has meaning here is to determine all measurable properties of the system in terms of those data which experiment shows can be arbitrarily specified. Now we have already emphasized that the electromagnetic field itself is an invention, and is never subject to direct observation. What we observe are material bodies, with or without charges (including eventually in this category electrons), their positions, motions, and the forces to which they are subject. The forces are to be measured according to definition in mechanical terms, either by the strains in members of a framework if the system is in equilibrium, or in terms of accelerations and masses if it is not in equilibrium. The electromagnetic field as such is not the final object of our calculations, but the calculation of it is only an intermediate auxiliary step, convenient to make because our mathematical formulation gives so simple a connection between electromagnetic field, charges, and mechanical action that the latter can be calculated at once in terms of the former. In fact the connection is so simple that in many cases we have come to regard our problem as solved if we can compute the electromagnetic field, overlooking the fact that the field has no immediate meaning in terms of experience.

Electromagnetic theory now presents us with a solution of the general problem; this solution is contained in the four-field equations of Maxwell, the constitutive equations, and those additional equations (quite often lost sight of) which give the forces exerted by the field on electric charges, or currents, or dielectrics. Let us inquire how we may set about testing the physical correctness of these equations. We may begin with one of the simplest possible tests, and inquire whether the equations are correct in stating that the force acting on a charge moving in an electric field is simply the product of the charge and the field strength. This, on the face of it, is a surprising statement. The field itself is affected by the motion of the charges which generate it, and it is natural to expect a converse effect. If, furthermore, we have sympathy with the medium point of view, it is easy to think that whatever it is in the medium that gets hold of a charge and exerts a force on it will find it harder to take hold when the charge is in motion.

In attempting to check our statement experimentally, the only additional complication, as compared with the static case which we have already checked, is afforded by the motion of the charge, for we have defined the magnitude of a charge in motion, so there is no difficulty here, and we may furthermore suppose that the field is generated by stationary charges, so that we need not trouble to inquire whether the procedure by which the field was originally defined is here applicable. The task of checking the equation then reduces to the simple physical task of measuring the force on the moving charge. How shall we do this? If the velocity is low, we may tie a string to the charge and measure the force with a spring balance (or its equivalent). But now an examination of the equations shows that in more complicated phenomena perceptible deviations from the static behavior are to be expected only at much higher velocities than can be attained by towing charges with a string and a spring balance, so that it is evidently necessary to check the simple equation for the force on a moving charge also at high velocity. Since at high velocity the spring balance method for measuring forces fails, we are driven to the only procedure that we have, namely a measurement in terms of the resultant acceleration, calculating the force by Newton's first law of mechanics. But this involves a knowledge of the mass of the moving body, which we recognize in general may be a function of the velocity. Now we have already seen, in discussing the concepts of mechanics, that the operations by which mechanical mass is defined cannot be carried out at high velocities, so that either the concept of mechanical mass becomes meaningless at high velocities, or we must adopt another definition. In attempting to give this new definition of mass at high velocities, we are driven to a result of special relativity theory, namely that all mass, mechanical or electrical, must be the same function of velocity. If now electrical mass can be found in terms of velocity, our immediate problem is solved and we shall be in a position to complete the experimental check of the equation. But as a matter of fact, in order to determine electrical mass, we have to use that equation which we are now engaged in trying to establish. Logically we have again the vicious circle, the physical significance of which is that independent operations do not exist for giving unique meaning to the concept of force on a charge at high velocity.

We seemed so close to our goal a minute ago; that we may allow ourselves to jump the logical chasm, and assume that the equation is correct. Electrical mass now becomes a definite function of velocity, mechanical mass the same function, and we are in a position to compare the actual acceleration received by a charge in a field with that calculated by the equation. Our conviction, on the basis of all experience up to the present, is that the two accelerations will be found to agree.

The equation then does somehow make correct connection with experience in that a consequence of the equation can be verified experimentally, in spite of the fact that as the equation stands it is meaningless, because the operations do not exist by which meaning can be given to the individual terms. At low velocities the equation really says what it seems to say, because the individual terms have meaning in terms of operations; and, what is more, what the equation says agrees with experiment. At high velocities the equation does not mean at all what appears on the surface; by itself it has no meaning; it has meaning only when considered as a member of a system of equations, and only in so far as the system of equations makes by implication statements about nature that have meaning in terms of operations that can be carried out physically. The individual terms of the equation of the system do not have meaning at high velocities, and in fact there are more terms than there are independent physical operations.

An exact analysis from the operational point of view of the significance of the equations at high velocities has perhaps never been made, and is not necessary for our immediate purpose. The discussion has brought out, however, that the number of physically independent concepts has been cut down by two at least, in that we have made purely formal definitions of the meaning of quantity of electricity, and of the force exerted by a field on a charge at high velocity. There is no reason to think that there is anything unique about this analysis, or that formal definitions might not have been given to other concepts than charge and force. We can only state that as far as physical content goes the equations have at least two degrees of freedom. It should then be possible to find quite different sorts of equations which agree equally well with experience. In particular, since we have seen that the force on a moving charge has no meaning in terms of independent operations, it should be possible by arbitrary definition to make this force any function of velocity that we please (of course reducing to the proper value at low velocities), and then to determine the other equations so that the entire group of equations is consistent with experiment. So far as I know, no one has tried to give such a modified set of equations, and indeed there is no particular reason why anyone should bother to do this, because the present equations are simple enough, and the modified equations, although perhaps differing greatly in appearance from the present ones, would have no advantage in any greater or different physical content.

But there is no reason to think that the present state of affairs will always continue. We have seen that the decrease in the number of concepts corresponds to our inability to measure as many sorts of physical things at high velocities as at low. Now it is the task of the future experimenter so to refine the possibilities of measurement at high velocities as to restore these two degrees of freedom. In particular, mass should be made measurable in mechanical terms at high velocities. When this restoration has been made, and all the quantities in our equations receive independent physical meaning, the significance of the equations in terms of operations will be quite altered, although the formal appearance will be unchanged. We must then be prepared to find, as always when we change the range of phenomena, that the equations in their present form do not correspond to the facts at all, and that one of the alternative forms allowed by our present two degrees of freedom is the correct form. But until the new experimental facts have been obtained, it seems hardly worth while to attempt to specify the doubly infinite variety of forms which the equations might have consistently with present experiment.[23]