The aim of a Mechanical theory of a special class of natural phenomena has been to represent the changes of which the phenomena consist in terms of mass and motion. Other concepts those of force work and energy are also employed in most such theories; and each one of these may be taken to be an independent concept or else as derivative according to the special form in which an abstract theory is stated and also according to the special class of phenomena with which the theory deals. In the case of the changes of position and motion of molar bodies the Mechanical theory employed is founded upon the Classical Dynamics of Galileo and Newton; and as I have previously shown this theory may be stated in a form in which force work and kinetic energy are derivative conceptions defined in terms of the fundamental concepts of mass space and time. In the more developed forms of this theory certain general principles have been deduced as consequences of the fundamental assumptions of the theory and one or other of these principles has been tentatively assumed as the basis of more extended mechanical theories independently of the original assumption that all the forces of a system of bodies are central forces. I propose in the present lecture to give some account of more general mechanical theories which have been set up in this manner. Mechanical theories have been devised to describe other physical phenomena such as the elastic deformations of solid bodies the phenomena of sound and vibrations optical thermal and electromagnetic phenomena. For some purposes not ponderable matter but an imponderable ether of some special type has been assumed as the field of masses in motion. In such cases the character of the relation between ordinary matter and the assumed ether forces itself upon the attention as urgently requiring elucidation if ether and matter are to be considered together in one mechanical scheme. But as we shall see it is possible to set up an abstract mechanical theory without making a complete set of detailed assumptions as to the precise nature of all the connections of the system. The relations between ether and matter have been speculatively conceived in a variety of ways. For example in Kelvin's vortex atom theory the basis of ordinary matter consists of vortex rings in an ether conceived as having the properties of a perfect fluid. A view which was propounded by Riemann makes the smallest element of matter a singular point in the ether at which a continuous annihilation of ether takes place. In Larmor's rotational ether an electron is a point of the ether at which a special kind of singularity exists. In some theories in which ether and matter are combined the Newtonian law of the equality of action and reaction is not satisfied when the matter alone is taken into account so that the interaction of matter and ether forms an essential part of the dynamical scheme.

# X; Mechanical Theories and Thermodynamics

X

Mechanical Theories and Thermodynamics

In all mechanical or quasi-mechanical theories the masses whether of ponderable matter or of imponderable ethers have usually been conceived in accordance with the realism of common sense not as pure concepts but as independently existent entities. As I have before urged however the validity and the usefulness of these theories are in no way invalidated if we refrain from all realistic assumptions and regard the elements employed in the theories as concepts only.. A strong underlying motive which has usually dominated investigators who have built up these theories has been the desire to give explanations which should make the stages of physical processes accessible to the sensuous imagination. Any elements in a theory which fail to satisfy this requisite have been accepted with reluctance. I have already referred to a striking instance of this in the unwillingness which has been shown to regard the notion of action at a distance as anything more than a provisional makeshift to be reduced if possible to the apparently more plausible notion of action by contact by means of some articulated mechanism involving the propagation of action through some medium connecting the bodies whose interaction is to be explained. That action by contact is itself in need of elucidation has however gradually forced itself upon the minds of men of Science though apparently not in all cases upon the minds of scholars for whom Greek Philosophy contains the quintessence of all wisdom. The failure to attain to anything except an indefinite regress of attempts to reduce contact action to a form which would satisfy the craving for efficient causation has been one of the factors which have led to the removal of the category of efficient causation from Natural Science. A great advantage of the modern view that a scientific theory is a purely conceptual scheme is that such a theory is emancipated from the somewhat narrow limitations imposed by the necessity that its form should be such that a sensuous representation of phenomena is provided. Of this freedom some modern theories especially those of Geometry and the theory known as the Einstein theory of relativity have availed themselves in a strikingly large measure. The latter theory is not a mechanical theory in the sense in which I here employ the term and I accordingly postpone any discussion of it. In the Newtonian Dynamics as originally conceived mass and force were taken as fundamental and independent concepts; force being regarded as a cause which produced as its effect change in the amount of motion defined as product of mass into velocity. I have however in an earlier lecture pointed out that this causal relation must be removed from its position in the theory in order that the theory may be stated as a conceptual scheme which becomes a deductive one when the requisite definitions and postulations have been fixed in precise form. The notion of force is then not independent of the concepts of mass time and space since force is used as a synonym for the product of mass into acceleration if indeed the term continues to be employed. In fact if the scheme originally stated by Newton in his laws of motion his definitions and deductions be taken in a revised form as the basis of a kinetic theory of the motions of molar bodies the existence of forces and the existence of accelerations are not two facts of observation but one only. That the earth and the sun move towards one another with accelerations in a definite ratio is the one fact as regards their relation to one another which is relevant to the dynamical theory. The supposed cause of this their so-called mutual attraction is not an independent fact but merely an assumption made in accordance with the supposed necessity of accounting for the first fact as due to causation. Thus in Newtonian Dynamics as a theory of the motion of such bodies as those of the solar system force is not an independent concept; the theory operates with the three independent concepts of time space and mass; motion being regarded as a transformation involving elements of space and time.

There exists however the department of Statics which is much more ancient than the Dynamics of Galileo and Newton. In accordance with the Newtonian Dynamics the weight of a body is the product of its mass into the acceleration with which it falls to the ground. But if the body be supported by a spring balance or in other manner in accordance with the principles of Statics the weight is regarded as a force still in existence but balanced by another force due to the support. If we refrain from formulating any theory as to the state of the support such as a kinetic theory of its corpuscles the notion of force as a stress or pressure is requisite as an independent concept. Moreover when the small-scale phenomena which occur in the smallest parts of bodies are taken into consideration the notion of force as an independent concept is indispensable in default of a complete kinetic theory of the corpuscles or smallest parts of such bodies. Especially in the theory of the elastic deformations of solid bodies the bodies are frequently treated as continuous distributions of mass; and the conception of force in the form of systems of stresses between adjacent parts of a body is employed as a necessary element in the construction of a theory of the strains or elastic displacements within the body.

In order to comprehend the character of the mechanical schemes that have been tentatively applied to the small phenomena of Physics it is necessary to consider the later developments of Newtonian Dynamics and the mode in which some of these have been extended by generalization into schemes of Dynamics that are free from some of the special restrictions involved in the original treatment of the subject by Newton and Galileo. In view of what I have said the notions of force and mass as independent conceptions or else the equivalent conceptions of work and mass are in general required in the basis of mechanical schemes. A conception which is usually regarded as requisite in the more general formulation of Dynamics as for example in the system developed by Hertz is that of inexorable constraints in a system of particles or of gross bodies; these constraints have the effect of diminishing the number of possible independent modes of motion of which a system is capable. They are usually represented on the perceptual side by rigid connections inextensible cords or rods connecting different parts of a system. A method of formulating Dynamics so that it may be applicable to systems which include such inexorable constraints was introduced by d'Alembert. He regarded the reversed mass-accelerations of all the elementary parts of a system as in equilibrium with the forces acting on the system whether from without or between parts of the system. All the conditions which determine the motions of the system are then included in one formulation of a statical character; the condition namely that no work is done by the equilibrating forces in any displacements consistent with the preservation of the postulated set of connections of the parts of the system. The deduction of d'Alembert's principle from Newton's theory in its conceptual form is subject to considerable logical difficulties. A method has been given by Boltzmann in his lectures on Dynamics by which these difficulties may be overcome. He takes as the basis of his treatment a finite set of masses concentrated at points; between each pair of such points he considers forces to act of equal magnitude and in opposite directions along the line joining them so that each of the points has an acceleration towards the other inversely proportional to the assigned masses of the points in accordance with Newton's third law of motion. Boltzmann then builds up the equations of motion of a system of bodies in which there may be inexorable constraints by treating each body as consisting of a very large but finite number of mass-points. The constraints are represented in the same manner by sets of mass-points between which forces act that are functions of the distances between pairs of these mass-points of such a character that these functions have very great magnitude whenever the standard distance between a pair of the points is changed either by excess or by defect. External forces acting on a system he takes as also due to external mass-points. In this manner Boltzmann deduces d'Alembert's principle and in fact the whole dynamical scheme for a system of bodies in a manner which avoids the logical difficulties of the way in which that extension of Newtonian Dynamics has usually been made. But the greatest advance in the direction of setting up a unified scheme of dynamics in a form so general that it is applicable to a system in which many of the details concerning the connections of the system may be unknown was made by Lagrange and published by him in 1788 in his great work the

*Mécanique Analytique*.In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in connection with physical theories. A striking example of this is to be found in Lagrange's abstract formulation of Dynamics as given in the

*Mécanique Analytique.*In order to characterize the spirit in which this great work is conceived I cannot do better than quote the words of Lagrange himself from the Preface. He writes:We have already various treatises on Mechanics but the plan of this one is entirely new. I intend to reduce this Science and the art of solving problems relating to it to general formulae the simple development of which provides all the equations necessary for the solution of each problem. I hope that the manner in which I have tried to attain this object will leave nothing to be desired. No diagrams will be found in this work. The methods that I explain require neither geometrical nor mechanical constructions or reasoning but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it and will be grateful to me for having thus extended its domain.

Lagrange's procedure was to express d'Alembert's variational equation in a form in which a certain number of variables of the most general kind are employed. The number of the variables is the number of independent motions which are allowed to the system by the constraints or connections contained in it. This number is what we now call the number of degrees of freedom of the system and the independent variables which specify the configuration of the system are called the generalized coordinates; the generalized force-component which corresponds to each of the generalized coordinates is defined as the coefficient of the variation of that generalized coordinate in the final expression for the virtual work of the forces which act on the system or between its parts. In the case in which the forces of the system form what is known as a conservative system that is when they are the gradients of that single function of the generalized coordinates which we call the potential energy Lagrange's equations of motion of the system are such that only a knowledge of the forms of two functions is required to make a determination of the positions of the system possible when its configuration and motion at one specified time are given. These two functions are the kinetic energy of the system expressed as a quadratic function of the generalized velocities that is of the gradients of the generalized coordinates with respect to the time and the potential energy which is a function of the generalized coordinates only. It will be observed that a conservative system includes as a particular case that in which all the forces are functions of the distances only between pairs of particles between which they act as in the original Newtonian scheme.

The Lagrangian equations of motion are equivalent to the statement that the possible paths of a conservative system are the extremals of the time-integral of the single function which is expressed as the difference between the kinetic energy and the potential energy of the system. An important modification of the Lagrangian dynamical scheme was made by Helmholtz by E. J. Routh and by Thomson and Tait in order to make a formulation suitable to the case in which the system contains parts that are independently in rotation. By elimination of the coordinates and velocities corresponding to these freely rotating parts of the system their effect is taken account of by a modification of the Lagrangian function. In the modified form the velocities of the rotating parts of the system give a contribution to the potential energy of the system. This so-called method of “

*Pignoration of coordinates*”—a technical term which has given rise to some misunderstanding—by indicating that part of the potential energy of a system may be regarded as really dependent upon the kinetic energy of motions that are concealed within the system has been sometimes regarded as a step in the direction of reducing all potential energy to kinetic energy. To those who attach paramount importance to the direct correlation of all the conceptual elements of a scientific scheme with sensuous intuition the ultimate reduction of potential energy to kinetic energy has usually been regarded as an ideal to be striven after. This aim is however of subordinate importance for those who are willing to accept as valid and satisfactory a scientific conceptual scheme in which some of the concepts employed do not correspond directly with anything that is accessible to sensuous intuition.The analytical Mechanics founded by Lagrange was extended and generalized in the two fundamentally important memoirs on the subject published by Sir William Hamilton. In accordance with the Hamiltonian scheme the whole of Dynamics is subsumed under what are known as the principles of least and of varying action. In two alternative forms a single function either the action or the characteristic function according to the alternative adopted has the property that the whole of the possible motions of a system are disclosed from a complete knowledge of the form of the function by means of the variation of an integral in which the function is the integrand. Like most such general principles the principle of least action has a previous history: it was originally formulated for simple cases of motion by Maupertuis without any adequate foundation. On Maupertuis' discovery Whewell

^{1}writes:Maupertuis conceived that he could establish

*a priori*by theological arguments that all mechanical changes must take place in the world so as to occasion the least possible quantity of*action.*In asserting this it was proposed to measure the action by the product of velocity and space; and this measure being adopted the mathematicians though they did not generally assent to Maupertuis' reasonings found that his principle expressed a remarkable and useful truth which might be established on known mechanical grounds.The Hamiltonian principle can only be deduced from the principles of Dynamics as formulated by Newton and Galileo by the employment of certain restrictions on the nature of the forces and the connections in the system. Although these restrictions are of a very general character they imply various restrictions upon the nature of the motions which can be deduced from the Hamiltonian principle. When the principles of Mechanics as formulated by Lagrange Hamilton and Jacobi are taken as the basic principles of the Science it is unnecessary to assume

*a priori*that their applications are restricted in the manner to which I have referred. The principles of energy and of least and varying action may be accepted hypothetically for the purposes of conceptual description of actual motions; the test of the descriptive value of the principles as in all such cases can only be that of experience. The fundamental conceptions with which the scheme operates are those of space time energy and mass; the last of these appears on the abstract side only in the form of coefficients in the energy function. In this scheme the concept of force does not appear as an independent notion but only as a derivative conception that of a gradient of potential energy. Although this has the advantage of being free from the various difficulties connected with the conception of force in the Dynamics of Newton and Galileo the identification and formal representation of the various forms of energy that are required in connection with various physical phenomena constitute the main difficulty in the employment of the general dynamical scheme in which the conception of energy is fundamental. The Hamiltonian principle in either of its equivalent forms gives a complete account of the transformations of the energy of a system between its various forms by means of the employment of a single principle whenever we are in possession of the formal expressions for the kinetic and potential energies of a particular dynamical system. One great advantage of this general dynamical scheme is that it affords the means of discovering the main features of the various motions that occur in a system without requiring the possession of a complete knowledge of the details of the mechanism of the system. In fact an indefinite variety of actual mechanistic systems can always be imagined for all of which the forms of the kinetic and potential energy functions are identical; and the consequences of the fundamental principle are applicable to the description of the changes in all such systems.Until about the middle of the last century most of the theories which were set up for the description of the various physical phenomena consisted of attempts to reduce them to cases of forces acting at a distance between material atoms. In fact the Newtonian system of gravitating forces between the bodies of the solar system formed the model and the inspiration of such attempts. In the second half of the nineteenth century the concentration of the attention of men of Science upon the principle of the Conservation of Energy owing to the brilliant verification by Joule of that principle in the case of transformation of mechanical work into heat led to the adoption of the transformations of energy as fundamental in the newer physical theories. In these newer theories especially in Thermodynamics and in the hands of Maxwell in Electromagnetics the generalized scheme of Dynamics associated with the names of Lagrange and Hamilton found a wide field of application. As regards the Hamiltonian principle in either of its forms considered as an hypothetical scheme for the description of physical processes the chief question which arises is as to its scope; that is whether it is capable of representing all the motions which take place in an isolated physical system. There is one important restriction of the principle which appears to limit its applicability even in the case of the motion of gross bodies. The connections of a system are expressed by means of equations connecting the coordinates which represent the positions of the bodies of the systems; but in some cases these equations essentially involve the gradients of these coordinates with respect to the time that is the velocities. In this latter case the Hamiltonian principle at least in its original form is not applicable and if it be employed it may lead to results which are not in accordance with the actual motions which take place in such a system. Attention has been drawn by Hertz in the introduction to his attempt to formulate anew the principles of Dynamics to a comparatively simple case in which the Hamiltonian principle is in default. This is the case of a spherical body rolling freely on a horizontal plane sufficiently rough to prevent all sliding motion. If the initial and final positions of the body are arbitrarily assigned there is always one mode of motion such that the Hamiltonian integral is a minimum. But in point of fact there are initial and final positions such that even with initial velocities at our free choice the body will not move into its final position unless external forces are applied to it to compel it to do so. Even if the initial and final positions are so chosen that a natural motion from one into the other is possible this motion is not the one for which the time of the motion is a minimum as it should be in accordance with the Hamiltonian principle. We may attempt to explain this discrepancy by denying the possibility of a motion of rolling absolutely without sliding but there certainly exist natural motions in which this condition is very approximately satisfied and we should consequently expect that the Hamiltonian condition would give in such cases an approximation to the actual motion which appears not to be the case. The general result is that in some natural motions the connections of the system are of such a character that the Hamiltonian principle is not applicable to the description of these motions.

There is however another restriction on the scope of the Hamiltonian Dynamics at least in its original form which must be taken into account in attempting to form an estimate of its range of applicability. The kinetic energy of a system expressed in terms of the generalized velocities is a homogeneous quadratic function of those velocities which accordingly is unchanged in value if the signs of all the velocities be reversed without changing their magnitudes. It follows that the motions in the system are all capable of being described in the reverse order without being otherwise changed; in other words the motions described are all reversible motions. There is however evidence of overwhelming strength which emerged originally in connection with the theory of Heat that some small-scale motions which occur in nature must be regarded as irreversible at all events that we are unable by any means at our disposal to realize such motions in the reverse direction. This difficulty Helmholtz endeavoured to combat in his investigations on cyclical systems. He showed that by elimination of the coordinates which represent certain concealed stationary motions in a system the kinetic energy is expressed in a form which involves not only quadratic but also linear terms in the remaining velocities of the system; and in that case the Lagrangian equation would lead to motions which are irreversible in default of means for reversing the concealed motions of small parts of the system. In this connection Helmholtz has pointed out the necessity of considering more general forms than the original one of the expression for what he terms the kinetic potential of a system; by which is denoted in the original scheme the difference between the potential and the kinetic energy. The attempts made by Helmholtz and others to construct a mechanics based on the conception of energy and on the hypotheses of the conservation of energy and the principle of least action only without having recourse to atomic assumptions which should be applicable to thermal electrical and chemical phenomena have been only partially successful especially as the difficulties connected with the interpretation of the irreversibility of many physical processes have not been overcome.

The most completely developed physical theory dependent upon the principle of the Conservation of Energy supplemented by another principle not immediately obtainable from Classical Mechanics is the theory of Heat known as Thermodynamics. Before the time of Rumford and Davy heat was regarded for the most part as an indestructible substance caloric. It was supposed that when caloric entered a body the effect of combination was in general an expansion of the body; even when contraction was the result of the combination the analogy of certain chemical combinations such as that of potassium and oxygen could be appealed to. The phenomenon of conduction of heat as transference of caloric presented no difficulty. The difference of specific heat of various substances was explained by assuming that they required different amounts of caloric to be mixed with them in order to produce equal changes of temperature. Black's theory of the latent heat of water assumed that water differs from ice at the same temperature in containing an admixture of a definite equivalent of caloric which was represented by a molecular state of the body which does not exhibit itself in the form of a rise of temperature. Thus the theory of caloric provided a plausible explanation of the most prominent thermal phenomena with however the important exception of the production of heat by friction or concussion frequently recognized by the adherents of the theory as not capable of satisfactory explanation.

The theory that heat is representable as motion instead of as a substance was established by the experiments on friction of which Count Rumford published an account in the

*Phil. Trans*for 1798. He pointed out that friction led to an inexhaustible supply of heat and that this is inconsistent with the theory of heat as a substance but consistent with the idea that heat consists of motion. The theory of heat as motion was still more lucidly developed by Davy in a tract published in 1799 in which he gives an account of his experiment in which two pieces of ice were rubbed together until both were almost entirely melted. The general law of the communication of heat was laid down by Davy in his*Chemical Philosophy*published in 1812 in the proposition that “The immediate cause of the phenomenon of heat then is motion and the laws of its communication are precisely the same as the laws of the communication of motion.”The foundations of modern Thermodynamics were first laid down by Sadi Carnot in his essay

*Réflexions sur la puissance motrice du feu*published in 1824. He recognized the important fact that in order to produce Work by heat it is necessary to have two bodies at different temperatures and he pointed out the analogy of work done when there is a fall of temperature with the case of work done by a fall of water from a higher to a lower level. He introduced the notion of a cycle of operations in which the initial and final states of a body are identical as regards temperature density and molecular condition. Carnot's theory of the heat engine is injuriously affected by the fact that he still held the theory of heat as caloric but he stated the important result that to obtain the maximum of work in a cycle that cycle must be reversible; which means that heat must only pass from a body to another body at very nearly the same temperature. He showed that the ratio of the work done by a reversible engine to the heat taken from the source is a function of the temperatures of the source and condenser only; when this difference is very small the ratio is that known as Carnot's function which depends only on the temperature of the source. In 1848 this conception was made by Lord Kelvin the basis of his absolute thermometric scale independent of the properties of any particular substance.The development of Thermodynamics based on the rejection of the theory of caloric was carried out by Rankine Clausius and Kelvin almost simultaneously. Rankine based his investigations on the hypothesis that the motion which represents the heat in a body consists of molecular vortices or circulating streams. He supposes that the whirling matter is diffused in the form of atmospheres round nuclei and that radiation whether of light or heat consists in the transmission of a vibratory motion of the nuclei by means of forces which they exert on one another. By means of this special hypothesis of molecular vortices he deduced from general dynamical principles what is termed the general equation of the mechanical action of heat. In later works he introduced the function known as the Thermodynamic function and applied his principles of Thermodynamics to various practical questions relating to the steam engine and other heat engines.

The theories of Clausius and Kelvin have the advantage of being independent of any special theory of the character of the motion which exhibits itself as heat but instead of being deducible from the general principles of Dynamics they make use of a fact of observation known as the second law of Thermodynamics or the principle of Clausius. The first law of Thermodynamics is taken to be the principle of Conservation of Energy as applied to the equivalence of heat and mechanical work the amount of which was determined by Joule and later experimenters. The principle of Clausius asserts that in a series of transformations in which the final is identical with the initial state it is impossible for heat to pass from a colder to a warmer body unless some other accessory phenomenon occurs at the same time. A more precise statement of the principle called by Clausius “the law of the equivalence of transformations” is to the effect that:

in all cases in which a quantity of heat is transformed into work and the bodies by means of which that transformation is effected return at the end of the operation to their original condition; another quantity of heat must at the same time pass from a hotter to a colder body; and the proportion which this latter quantity of heat bears to the former depends solely upon the temperatures of the bodies between which it passes and not upon the nature of the intervening bodies.

A function called “entropy” was introduced by Clausius whose value is found by dividing the quantity of heat expended in producing a given change in a given substance by the absolute temperature as measured by a perfect gas thermometer. The conception of entropy is a case of a fundamental Concept essential to the scheme of Thermodynamics which does not directly represent anything accessible to sensuous perception. The second law of Thermodynamics involves the postulation that the entropy of a thermally isolated system always tends to increase. The change in entropy is quite distinct from change in temperature and from the change which consists in loss or gain of heat. For example in chemical reactions the entropy increases without any heat being supplied to the substances. When a perfect gas expands in a vacuum its entropy increases and yet the temperature does not change and the gas has neither yielded nor received heat. A difficulty in the conception of entropy is that it is not possible to define the equality of entropy of two bodies chemically different although it is possible to compare the variations of entropy to which the bodies are separately subject.

The principle that entropy tends continually to increase has been stated by Perrin in the form that “an isolated system never passes twice through the same state.” It involves the postulation that the course of physical phenomena is so to speak in a definite direction which is never reversed. This principle has been generalized by Clausius and Kelvin in a form in which it universe. It is said that the entropy of the universe is continually increasing. Thus although the whole energy of the universe is regarded as remaining constant through all transformations it becomes increasingly unavailable since the energy is transformed gradually into heat uniformly distributed at an everywhere identical temperature. The final state of the universe would then be one in which nothing would happen because no energy would anywhere be available for the purposes of the chemical and thermal transformations by which all change is conditioned.

This theory of the dissipation of energy is open to the very serious criticism applicable to all statements made about the physical universe as a whole. The extension of a principle asserted in the first instance to apply to a finite isolated system is made hypothetically to one which we are not warranted in regarding as finite. If we consider ever larger portions of the universe it may be the case that the energy and the entropy of a portion increase indefinitely as the portion is continually increased and in that case the assertion ceases to have a definite meaning. Besides the range of validity of the principle even as applied to an isolated finite system has not really been ascertained. Kelvin himself expressly excluded living organisms in his statement that “it is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” Moreover the view is held by those who have considered the matter from the point of view of molecular or atomic theories that the principle of increase of entropy is only a statistical principle based upon the laws of probability. The tendency of these statistical methods developed by Willard Gibbs and by Boltzmann is to regard the principle as pointing out that a given system tends towards the configuration presented by the maximum probability; and the entropy of the system is expressible in terms of the numerical value of this probability. This maximum probability increases with the number of molecules concerned but does not reach absolute certainty.

Thermodynamics in the hands of Willard Gibbs and later investigators has attained to very great success in its application to the ascertainment of laws regulating changes of chemical constitution or of physical state. After Gibbs Helmholtz introduced into the domain of Chemistry the conception that energy can be divided into two parts; the first free energy capable of undergoing all transformations and of producing external action; the second bound energy only manifesting itself by giving out heat. It is the variation of the free energy not that of the total energy which is efficient in determining chemical reactions. The utilization of the theory of Gibbs expressed chiefly in what is known as the Phase Law was due to Van der Waals Van t'Hoff and Roozeboom in the discussion of complicated chemical reactions.

On the whole it may be asserted that the principles of Thermodynamics have proved a most valuable tool for the coordination of a great number of physical and chemical properties of matter; but the laws must still be considered as hypotheses the precise range of applicability of which is not known and has not yet been definitely delimited.

Both in those theories which depend upon corpuscular hypotheses and in those in which no molecular or atomic hypothesis is made the conception of Energy as a measurable quantity capable of continuous transformation has been employed as a fundamental element. In any theory based upon the Classical Dynamics even when extended in accordance with the conceptions of Hamilton and others the transference of energy from one form to another is regarded as essentially continuous in amount. This is also the case when the fact of the irreversibility or apparent irreversibility of certain motions is taken into account and formulated abstractly in a theory of entropy. Certain facts have however recently emerged which throw very serious doubts upon the adequacy of any theory in which continuous transformations of energy are admitted for the purpose of representing certain classes of phenomena. A theory of Quanta in which portions of energy are transformed by jumps that is discontinuously has arisen as the result of an attempt to represent these facts. This theory associated with the name of Planck arose in the first instance from an investigation of the spectrum of black-body radiation; and it also has bearings upon the theories of the line-spectra of the elements and of the specific heats of solid bodies. The result up to the present time of the discussion of this matter has been to show that there are exceedingly strong grounds for the assertion that certain kinds of phenomena involving the transformation of energy are incapable of being described by any conceptual scheme of the kind which we call Newtonian Dynamics even in a generalized form; but the subject is still in the region of controversy. The theory of radiation starts from the assumption of the (conceptual) existence of an ether which must be regarded as continuous or at least as very much more finely grained than ordinary matter. On this assumption it can be proved that in accordance with the dynamical theory a state of thermal equilibrium between say a piece of iron and the surrounding ether can only be attained when all or nearly all the energy of the motion of the parts of the iron has been drawn from the body into the surrounding ether. There is in fact a tendency for the whole energy of moving systems immersed in a medium of any type like that which must be assigned to the ether to be transferred to the medium and ultimately to be found in the shortest vibrations which that medium is capable of executing. It is now a matter for observation to determine whether this is what actually happens; and the observed facts relating to thermal equilibrium between a black-body and the surrounding medium appear to give decisive evidence that what should happen in accordance with the dynamical theory does not actually occur. A law of thermal equilibrium between a black-body and the surrounding medium was obtained by Planck from thermodynamical considerations; and this law is inconsistent with the total or almost total absorption of the energy by the ether. It has been shown that Planck's law is in close agreement with the results of observation. A demonstration has been given by Poincaré which has been widely accepted as valid that Planck's law of partition of energy between the black-body and the surrounding medium is not consistent with any scheme of continuous transference of energy but necessarily involves the assumption that the energy is transferred discontinuously by jumps. This would also be the case if Planck's law be taken to express only an approximation to the actual law of partition. It would thus appear that the motion of the medium must be governed by laws which involve the quantum-theory; and this negatives the possibility of describing such motion in accordance with the Dynamics of Newton and Galileo or with any extension of it in which the treatment of the transference of energy by continuous amounts is fundamental.

The general result of these recent investigations is to indicate that limits of the applicability of Newtonian and Post-Newtonian Dynamics exist. The dynamical scheme has been found sufficient with certain reservations indicated by the Einstein theory of relativity for the description of the large-scale phenomena of Physics; but there never existed any cogent reason for assuming that the scheme need necessarily prove adequate for the description of all the small-scale phenomena.

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