Duhem, Pierre. Treaty of energetics or general thermodynamics Volume 1 Gauthier-Villars Paris 1911 ------------------------------------------------------------------------ 4% leu li ------------------------------------------------------------------------ ENERGY OF GENERAL THERMODYNAMICS. TREATY or ------------------------------------------------------------------------ PARIS. IMPRIMERIE GAUTUIER-VILLARS, 46(108 Quai des Grands-Augustins, 05. ------------------------------------------------------------------------ OF ENERGETIC 01'. OF GENERAL THERMODYNAMICS PAK Professor of Theoretical Physics at the University of Bordeaux. RATIONAL MECHANICS. -GENERAL STATIC DISPLACEMENT OF THE EQUILIBRIUM. r.AUTHIER-VILLARS, IMPRIMEUR-LIBRAIRE 1)11 BURHAli DKS LONUITUDKS, 1)15 I.'ÉCOLK I' O I. Y T KO H M Q l Quai des Grands-Augustins. 55. TREATY Pierre DUHEM, Correspondent of the Institut de France, TOME I. CONSERVATION OF ENKROIK. PARIS, 191 1 ------------------------------------------------------------------------ All translation, reproduction and adaptation rights reserved for all countries. ------------------------------------------------------------------------ TREATY OF ENERGY. INTRODUCTION. 1. Thermodynamics or Energetics. Theoretical physics represents by means of quantities the properties of the bodies it studies. The methods of measurement allow to make correspond, with a more or less great approximation, each intensity of a property to a particular determination of the quantity which represents this property. By the methods of measurement, each physical phenomenon corresponds to a group of numbers, each physical law to one or more algebraic relations between various quantities, each set of concrete bodies to a system of quantities, to an abstract and mathematical scheme. Theoretical Physics has, unceasingly, to solve the following problem From given physical laws draw new physical laws; either it proposes to show that these last laws, already known directly, are only consequences of the first ones or it proposes to announce laws that the experimenter has not yet noticed. To deal with this problem, Theoretical Physics combines the given laws, which are particular to certain physical properties and to certain bodies, with rules derived from general principles, which it assumes to be true for all physical properties and for all bodies For example, she wants to show that. if we know the law of saturated vapor pressure of a liquid, the laws of eompressibility ------------------------------------------------------------------------ The law according to which the heat of vaporization varies can be determined; for this purpose, it combines the first times according to the rules of the principle of the conservation of energy and of Carnnt's principle, principles which it supposes to be applicable to all bodies and to all their properties. G is the system of these general principles that we propose to expose. For a long time, physicists assumed that all properties of bodies were reduced, in the last analysis, to combinations of figures and local motions; the general principles to which all physical properties must be subjected were, then, none other than the principles that govern local motion, principles that compose Rational Mechanics. Rational Mechanics was the code of the general principles of Physics. The reduction of all physical properties to combinations of figures and motions, or, according to the name in use, the mechanical explanation of the universe, seems to be condemned today. It is not condemned by a priori, metaphysical or mathematical reasons. It is condemned because it has been until now only a project, only a dream, and not a reality. In spite of immense efforts, physicists have never succeeded in conceiving an arrangement of geometrical figures and local motions which, treated according to the rules of rational mechanics, gives a satisfactory representation of a somewhat extended set of physical laws. Is the attempt to reduce all of Physics to rational Mechanics, an attempt that has always been in vain in the past, destined to succeed one day? Only a prophet could answer this question affirmatively or negatively. Without prejudging the meaning of this answer, it seems wiser to give up, at least temporarily, these efforts, so far fruitless, towards the mechanical explanation of the Universe. We shall therefore attempt to formulate the body of general laws to which all physical properties must obey, without assuming a priori that these properties are all reducible to the geometrical figure and to local motion. The body of these general laws will then no longer be reduced to rational mechanics. ------------------------------------------------------------------------ In truth, the geometrical figure and the local motion remain physical properties; they are even those properties which are the most immediately accessible to us. Our body of general laws will have to apply these properties and, applied to these properties, it will have to give us back the rules which dominate the local motion, the rules of rational Mechanics. Rational Mechanics must therefore result from the body of general laws that we propose to constitute; it must be what we obtain when we apply these general laws to particular systems where we only take into account the figure of the bodies and their local motion. The code of the general laws of Physics is known today under two names : Thermodynamics and Energetics. The name Thermodynamics is closely connected with the history of this science; its two most essential principles, Carnot's principle and the principle of the conservation of energy, were discovered while studying the motive power of fire engines. This name is further justified by the fact that the two notions of work and quantity of heat are constantly at play in the reasoning by which this doctrine is developed. The name Energetics is due to Hankine (' ); the idea of energy being the first one that this doctrine has to define, the one to which most of the other notions it uses are connected, this name seems no less well chosen than the name Thermodynamics. Without deciding whether one of these two terms should be considered preferable to the other, we will use them both as equivalent. 2. On the logical meaning of the principles of Energetics. We must not forget the logical character of the principles that we are going to formulate and group ( i. (') J. Macu>oiin Kankink, Outlines of tlw Science of Energclics {Glasgow Philosophical Society l'roceedings. Vol. III, n" (5, a May iS5f>). - J. Macquokx RANKtNF, Misceltaneous scientific Pa/xtrs, p. -o.). (J) We limit ourselves here to give a very concise summary of what we ------------------------------------------------------------------------ have, ueveioppe in 1 following work: La théorie physique, soit objet et sa structure, Paris, 1906. This work can be seen as a kind of logical introduction to the present treatise. These principles are pure postulates; we can state them as we please, provided that the statement of none of them is contradictory in itself and that the statements of the various principles do not contradict each other. The character of a good physical theory is the following: by applying this set of principles to formulas which represent exact experimental laws, one derives new formulas which, in their turn, represent other exact experimental laws. The experimental control of all the principles of Energetics is thus the only criterion of truth of this doctrine. This control can, moreover, only be carried out on the whole of the principles of Energetics taken in its integrity or, at least, on very extended parts of this whole. It would be impossible to submit to the control of the experiment one of these principles, taken in isolation, or even a small number of these principles. Any experiment, however simple, invokes, in its interpretation, very numerous and very diverse principles. We will have many occasions to recognize this in the course of this presentation. The experimental control can therefore only concern the set of ultimate consequences of the theory; it assesses whether or not this set of consequences gives a satisfactory representation of the experimental data; but as long as the theory has not produced the set of its final consequences, one should not call upon this control, because this call would be premature; In the course of its exposition, a physical theory is free to choose the way it likes, provided it avoids any logical contradiction; in particular, it does not have to take into account the facts of experience; it is only when it has reached the end of its development that its ultimate consequences can and must be compared with the experimental laws. To say that the principles of Energetics are pure postulates ------------------------------------------------------------------------ and that no logical constraint limits our right to choose them arbitrarily, this does not mean that we will formulate them at random. On the contrary, we shall be very closely guided in the choice of these statements, knowing full well that it would be enough to modify anything in them for the experimental verification of the consequences to become faulty in some place. This guidance is assured by the knowledge we have of the past of Science. Principles have been formulated which have been found to be in gross contradiction with experience; other principles have been substituted for them, which have obtained a partial, but still perfect, confirmation; they have then been modified, corrected, ensuring by each change a more exact agreement of their corollaries with the facts. We are assured that the garment whose forms we cut will fit exactly the body it is to cover because the pattern has been tried on and retouched many times. Each of the principles that we will state does not therefore include any logical demonstration; but it would include a historical justification; we could, before stating it, enumerate the principles of different form that had been tried before it, that could not be modelled exactly on reality, that had to be rejected or retouched until the whole system of Energetics was adapted in a satisfactory way to the whole of the physical laws. The fear of an excessive length will forbid us to expose this historical justification. ------------------------------------------------------------------------ CHAPTER I. PRELIMINARY DEFINITIONS. I. Of absolute time and absolute motion. We will assume Geometry and Kinematics; we will borrow from these sciences all the information we need. Kinematics uses the two words time and motion; we will also use these two words when we deal with Energetics; they will not have exactly the same value in these two doctrines; it is important to specify here in which sense we will use them. Although the consciousness provides us with a certain notion of equal times, this notion is not precise enough to serve for the construction of a science; we have to refer to the measure of time given by a certain clock, either natural or artificial; the choice of this clock is limited by only one condition it is that the times that it marks as equal appear also equal, or more or less, to our conscience but the information provided by our conscience and by the conscience of our fellow men are, in this respect, so little precise and so little concordant that they leave the choice of this clock an extreme latitude. As for movement, all that the senses and memory allow us to observe is that with time, the relalive position of the various bodies that surround us experiences certain changes or, in other words, that these bodies are in movement with respect to one of them taken as a term of comparison. We can choose a certain reference trihedron invariably linked to the body which serves as term of comparison : experience, completed by abstraction, will allow us to determine, at ------------------------------------------------------------------------ At each moment, the coordinates of any material point we want to consider, these coordinates being related to this reference trihedron. Experience, even with the help of abstraction, only provides us, as far as time is concerned, with the notion of time relalij to a certain clock, arbitrarily chosen, and, as far as motion is concerned, with the notion (motion, relalij to a certain reference trihedron, arbitrarily chosen. These two notions are sufficient to construct Kinematics. Let us take any theorem of Kinematics, for example the following theorem: When a point describes a circle of radius R, with a uniform motion of which u> is the angular velocity, the acceleration is, at each instant, directed from the point to the center of the circle and has the value Rio2. It is clear that we will not be able to judge that a point describes a circle, unless we relate its motion to a certain reference trihedron (for example, to a trihedron that is invariant to the Sun), and that we will not be able to declare that the motion of this point is uniform, unless we follow this motion by reading the time on a certain clock (for example, on a clock that is set to the motion of the stars and that gives the sidereal time). The velocity and acceleration referred to in the statement of the theorem will be the velocity and acceleration relative to this reference trihedron and to this clock. Let us now change our reference trihedron and our clock; let us take, for example, a trihedron invariably linked to the Earth and read the time on a sundial. Let us consider a point which, relative to our first reference trihedron and our first clock, was moving with a circular and uniform motion and was thus in the conditions required for the previous theorem to be applicable to it; it will move, relative to our new reference trihedron and our new clock, according to a completely different law; its motion will no longer be n-i circular nor uniform, so that the previous theorem can no longer be applied to it. But this theorem will still be true. If we take ------------------------------------------------------------------------ a point whose motion, relative to our new reference trihedron and to our new clock, is a circular, uniform motion of radius 11 and angular velocity ̃ w, the acceleration, relative to this reference trihedron and to this clock, will still be directed from the point to the center of the circle and will have the magnitude Roj What we have just said about this theorem can be repeated for all the theorems of Kinematics, which allows us to state the following proposition The statement of a proposition of Kinematics would have no sense if one did not suppose that before stating it one had made a choice of a certain clock which serves to measure time and of a reference trihedron to which the movements are related; but the exactness of this proposition is independent of the way in which this double choice was made; it is not altered if one replaces this choice by another. This proposal can be stated more briefly as follows The notions of time relative to a certain clock and of motion relative to a certain reference trihedron are sufficient to constitute Kinematics. The same is not true of the science whose principles we are going to expose. Let us take, for example, the following theorem, consequence of the laws to which we will be led The center of gravity of an absolutely isolated body moves in a rectilinear and uniform motion. Obviously, to judge that the trajectory of a point is rectilinear, we must relate the position of this point to a certain reference trihedron, and to judge that this trajectory is described by a uniform motion, we must suppose that time is given by a certain clock. Let us imagine that we have taken a reference trihedron invariably linked to the Sun, a clock giving the sidereal time, and that, with this choice, the preceding law is verified for a certain absolutely isolated body. ------------------------------------------------------------------------ Let's change the reference trihedron and change the clock. Let us take, for example, a trihedron invariably linked to the Earth and a sundial. The point which, relatively to the first trihedron and to the first clock, moved with a rectilinear and uniform motion, describes now, relatively to the new trihedron and to the new clock, a complicated trajectory following a complicated law. But, on the other hand, the change of reference trihedron, and of clock does not prevent the considered point from remaining the center of gravity of the studied body and does not prevent this body from remaining an absolutely, isolated body. This change therefore leaves the point in question in the conditions where it must be for the stated theorem to be applicable to it. So, speaking about change of trihedron: of reference and of clock, the stated theorem, which was true, has become false. By generalizing this remark, we arrive at the following consequence The accuracy of the laws of Energetics is not independent of the choice of the clock to which the time is related and of the reference trihedron to which the levels are related. If we suppose that all these laws are exact with a certain choice of the clock and the trihedron, many of them will become false, in general, when we make a choice of a new clock that is not set on the first one and of a new trihedron that is not invariably linked to the first one. Of the clock to which we suppose the time related when we affirm the exactitude of the laws of the Energetics, we say that it measures the absolute time or that it is an absolute clock! he lrihedron of reference to which we suppose the related movements is said trihedron absolutely, fixed; any movement, related to this lrihedron and to this clock is said absolute movement: a body whose position with regard to this lrihedron is independent of the time is said in absolute rest. We can therefore say that qne,/w definition, absolute clock and reference trihedron absolutely fixed its/the clock and reference trihedron with respect to which the principles of Energetics are assumed true. ------------------------------------------------------------------------ This definition would suffice if Energetics were only to be a logically arranged algebraic and geometrical construction. But this is not the purpose of Energetics; Energetics must be a physical theory. In other words, the propositions which compose this mathematical construction must be equivalent in an approximate way to the experimental laws, provided that one establishes in a suitable way an approximate correspondence between the elements of the mathematical construction and the physical properties accessible to the experiment. Therefore, we have to answer the following two questions What is the concrete system whose deformations measure in an approximate way the absolute time, which realizes approximately the absolute clock? What) is the concrete body which can be regarded as being more or less at rest, to which the absolutely fixed trihedron of the abstract theory can be regarded as almost invariably bound? According to the principles that we recalled a moment ago, the answer to these two questions should be directed by the following rule The concrete variable system that will be made to correspond to the abstract absolute clock of the theory, the concrete body that will be made to correspond to the absolutely fixed abstract body of the theory will have to be chosen in such a way that the propositions of r Energetics provide a sufficiently approximate representation of the experimental laws. This principle shows immediately that the choice of the absolutely fixed trihedron and the absolute clock is subordinated to three conditions `: i" The mathematical constitution of the energetic system is considered to be fixed, not only in its general principles, but also in its ultimate consequences, which are the only ones comparable to experimental laws -V It is also assumed that the measurement methods are fixed, which alone allow a physical property to be matched, ------------------------------------------------------------------------ accessible to observation, to each of the mathematical elements that are used to build the theory; 3" In other words, the degree of approximation with which the abstract propositions of Energetics are required to represent the experimental laws is assumed to be fixed. For example, the ancients considered the Earth as the absolutely fixed body to which absolute movements must be related; this assumption could be maintained, but on the condition of adopting an energetic system quite different from the one we are going to expose and much less simple than this one. So far, the branch of Energetics which deals with the movements of celestial bodies (Celestial Mechanics) has agreed satisfactorily with the laws of observation by taking sidereal time as absolute time; however, very precise observations point to some disagreements. There are two ways to make these disagreements disappear: either change the absolute clock and look at the sidereal day as slightly variable; or change one of the hypotheses on which the branch of Energetics that deals with celestial motions is based. In short, the choice of the concrete clock which must correspond to the abstract absolute clock, and of the concrete trihedron which must correspond to the abstract absolutely fixed Irièdro, is a part of the overall operation which consists in comparing the whole system of physical theory with the whole system of experimental laws. As we have often stated, no part of this operation can, logically, be separated from the others; they must be supposed to be done all at once, all at once; only the necessities of exposition and of I teaching compel us to break up the comparison of the theory with the experiment cl to present separately the various fragments of it, to the detriment of logical ̃rigueur. It is by virtue of this method, and by anticipation, that we will state the following proposals In most applications of V Kncrin''li, "', "', of their velocities. These are all independent numbers; their definitions allow to choose arbitrarily their values. But between the values of the components of the velocities and the derivatives with respect to time of the coordinates, there are relations, since we have by definition dx dy y dz dx' u - - - v - -7- > w = -7- i u - -y- > dt dt dt dt 2" The system is a polarized dielectric. Its physical properties at the moment are known when we give ourselves the components A, H, C, of the polarization at each point (x, y, z) and the components u, v, n> of the displacement flux at the same point. These numbers A, H, C, u, v, ir are independent numbers; they can be assigned any value without contradicting their definitions. But these same definitions lead to relations between the components of the displacement flux and the derivatives with respect to time of the polarization, because they teach us that we have at OH dC ¡Jl\. ()B ac 11. = - i> = - "< = --- dt iif. Ot 3" The system is a good electrically conductive body. To represent its physical properties at the instant t, we have to consider Ja superficial electric density E at each point of the surface which limits the conductor, the solid electric density e at each point (x. y, :) of the volume which it occupies, finally the components u, c, .) a m -t- {i p -j- y iv 4 - - = o. u it V (V 7 t = 0. When independent numbers are such that there is a relation between the values taken by some of them, at a certain instant, and the values taken, at the same instant, by the derivatives of some order of the others with respect to time, we will say that there is a second order dependence between these numbers. t is well understood that this second-order dependence must be, like the first-order dependence, a purely logical dependence, deriving, as in the examples we have cited, from the very definition of the numbers by which the properties of the system are represented; it must not result from a relationship intended to represent a physical law. Let us take, for example, a homogeneous mixture of oxygen, hydrogen and water vapor; fi is the temperature and p the density; x is the ratio of the mass of water vapor contained in the mixture to the mass of water vapor that it would contain if the combination were pushed as far as possible. iNot only are the numbers fi, o, x independent numbers, but there is no second-order dependence between them; without contradicting their definitions, one can not only give them arbitrary values, but also give values Y- ̃ dx do d() arbitrary a -r-y ~r-, -r- dt cil dt However, we will be led to formulate this law: The speed of combination is determined when we know the temperature, the density and the degree of combination of the system. This law will be expressed by a relation of the form (lx fy x ). ~=/(~~). ------------------------------------------------------------------------ But this relation will not be a dependence of second order, because it does not result from the simple delinquency of the three numbers fï, p, -r- it has for goal to represent the results of the experiment. When numbers representing the physical properties of a system have no dependence between them, neither of the first order nor of the second order, we say that they are absolutely independent. Once these definitions have been made, we can indicate which conditions must be fulfilled by the numbers so that we can say that these numbers define Y state of a given system at a given time l. These conditions are as follows T i" The numbers considered represent physical properties of the given system at time t; time t is not considered as such a property: these numbers vary in a continuous way with their first derivatives with respect to l exist and are always finite; These numbers are, by their definition, absolutely independent of each other cl absolutely independent of t\ If other nonibr. also represent properties (physical of the same system inManl. t, either these new ̃numbers have some relation of the first order to the previous ones; or these new numbers are related to some derivative .with respect to t of the first ones or vice versa, which constitutes a relation of the second order. Let us take, for example, a solid conductor, immobile and electrified; we can say that its calibration at time t. is determined by In solid electric density e at each point (.r, r, :̃) of the volume it occupies and by the surface density E at each point of the surface which limits it. Indeed i" The numbers e. E. represent properties of the conductor at the instalment a" their definition, these numbers are absolutely independent between them and independent* of time t -We are led to consider another physical property of the same body at the moment, namely the electric Jhu' in each ------------------------------------------------------------------------ point; but the components u, e, "v of this llux are related to e and E by the second order relations (i) and (a). Let us also take as an example the homogeneous mixture of oxygen, hydrogen and water vapour which we spoke about a moment ago; we could not say that the state of this mixture is determined by its density o and its degree of combination x; we have indeed to consider a third physical property of this mixture, the temperature, represented by the number (), and this number is absolutely independent of the numbers p and x; we can say, on the contrary, that the state of this mixture is determined by the three numbers j, 0, x. Apart from the restrictions we have just mentioned, General Energetics gives us no indication as to how to choose the properties that define the state of a system. In each of the Chapters of Physics, one fixes, by means of hypotheses particular to that Chapter, the quantities which will serve to define the state of the system which one proposes to study; once this state is thus determined, one applies to it the rules traced by general Energetics. In order to determine which are the physical properties that must be used to define the state of a system, we have no other guiding principle than the knowledge of the object that the physical theory proposes to build, by means of mathematical notions, a kind of scheme that represents, with a given approximation, the laws to which a certain set of concrete bodies is subjected. As we have already explained in the preceding paragraph, we do not propose, in general, to construct a mathematical scheme which represents, at once, all the known physical laws to which the set of concrete bodies under consideration is subject; such a scheme would, in general, be frighteningly complicated. We simply propose to represent with a certain approximation some of these laws, while ignoring the others. This makes it possible not to include in the definition of the state of the system the representation of all the physical properties which could appear there. We only consider some of these properties, ignoring the others. We obtain from ------------------------------------------------------------------------ In this way, a simplified scheme is created which represents with a certain approximation a certain number of laws among those which govern our concrete set. If, later on, one proposes either to represent the same laws with a greater approximation, or to represent, in addition, some of the laws from which one had, at first, made abstraction, one is naturally led to make use of a new mathematical scbeine more complicated than the first one; one takes back then, to form the new definition of the state of the system, some of the physical properties which one had neglected in the first definition. Let us suppose, for example, that we want to represent the laws of electrical distribution on a conductor made of an alloy of copper and tin. To represent the state of such a conductor at a given moment, we could give us not only the solid electric density at any point of the volume occupied by the conductor and the superficial electric density at each point of the surface, but also, at any point of the conductor, the density of the material, its chemical composition and its temperature. Nevertheless, physicists have recognized that many of the laws of electrical distribution can be represented with a fair degree of approximation without including the last three quantities in the definition of the state of the system, and by simply determining this state by knowing the two electrical densities. If, however, we want the approximation with which these laws are represented to exceed a certain term, we are obliged to give up this definition, simplified to the extreme, of the state of the system. Let us imagine, however, that we stick to this definition. Then, between the corollaries of the theory and the results of the experiment, which has been made very precise, a host of disagreements appear - differences in potential on contact, thermo-electric effects, etc. To make these disagreements disappear, to establish a more exact concordance between the experimental laws and their theoretical representation, one appeals to a more complicated mathematical scheme; one takes back, in the definition of the stall, of the system, the physical properties which one had initially abandoned : the density of the matter, the chemical composition, the temperature. ------------------------------------------------------------------------ There is, we guess, no a priori principle t j 1 that decides whether, in the definition of the standard of a system, one must include this physical property or whether it is permissible to disregard it; the success of the mode of definition adopted, the satisfactory agreement between the consequences of the theory and the teachings of experience, is the only character to which we can recognize that we have been right in adopting such a definition in preference to another. It can also happen that a certain physical property is considered, in one branch of theoretical physics, as one of the essential characteristics of the state of the studied systems, and that in another branch of theoretical physics, this same property is completely ignored. Thus, the branch of theoretical physics called Rational Mechanics, when defining the state of a material system, entirely neglects the chemical constitution or the temperature of the various bodies which compose this system; it only takes into account the positions occupied in space by the different parts of these bodies. This very simplified definition of the state of a system allows it, however, to give a satisfactory representation of a very large number of experimental laws. Conversely, another branch of theoretical physics, Chemical Mechanics, draws a similar picture of a very large set of experimental facts, taking into account the chemical constitution of the bodies and their temperature, but completely disregarding the shape of each of them and the position they occupy in space. It goes without saying that neither the simplified scheme used by Rational Mechanics nor the one used by Chemical Mechanics can suffice to represent all experimental laws: there are many laws which cannot be represented unless we include, in the definition of the state of the system, both the physical properties which Rational Mechanics neglects, and those which Chemical Mechanics disregards. It is by abstractions of this kind that Theoretical Physics can, in a great number of cases, define state of the system it proposes to study by means of nn number limited, and ̃even of a small number of quantities. ------------------------------------------------------------------------ Let us suppose, for example, that we are studying the dissolution of sea salt in water. The solution does not have, in all its parts, the same density, the same concentration, the same temperature; the density and temperature also vary from one point to another of the salt crystals. However, Chemical Mechanics manages to represent a good number of laws relating to salt dissolutions, and this with an approximation which is sufficient, in most cases, for the physicist and the chemist, by disregarding all these differences, by not taking into account the shape of the bodies, nor their position in space; It can then define the state of the system it is studying by a very small number of quantities : the mass of the water, the mass of the solid salt, the mass of the dissolved salt, the densities of the solid and the liquid, both supposed to be homogeneous, and finally the temperature of the whole, considered to be the same at all points. This abbreviated definition of the state of the system will not suffice to represent all the laws to which our saline dissolution is subject; to obtain a more perfect theoretical representation of this system, we will have to define the state of the system by making known the density, the concentration, the temperature at each point; this standard will then be defined by an unlimited number of quantities. We can generalize what the analysis of this example has taught us. In a very large number of cases, the state of the systems whose Energetics has to Irai 1er is defined by means of a limited number of quantities. However, such systems are, in general, too simple to be able to represent in this way all the laws that we want to study, and with all the desirable approximation; we will be, therefore, forced. to obtain a more perfect agreement between theory and experiment, to consider systems whose state will be defined in another way; we will divide by thought each of these systems into infinitely small elements: the definition of the state of the whole system will result from the definition of the state of each element and of the position it occupies in the whole; this last definition will be given, for each element, by means of a limited number of magnitudes; in such a way that the definition of the whole system will depend on an unlimited number of magnitudes. ------------------------------------------------------------------------ When a system is defined in this way, we can say that it is composed of an infinite number of infinitely small systems, the state of each of which depends on a limited number of quantities. The study of systems whose state is entirely defined by a limited number of quantities is, therefore, the obligatory preliminary to the study of systems whose state is fixed only by an infinity of numbers. This is why, in this presentation of General Energetics, we will first and foremost focus our attention on systems whose state is defined by means of physical properties of a limited number. 3. Holonomic links. Let us imagine that we have defined a first state of a certain system by giving arbitrarily chosen values to the quantities, in limited or unlimited number, which represent certain physical properties. In order to define another state of the same system, will it be possible to give other entirely arbitrary values to the same quantities? Sometimes this will be possible; but ordinarily, the arbitrary choice of numerical values which serve to define the first state will impose certain conditions on the numerical values which serve to define the second state, and this, by virtue of the very definition of the physical properties to which these values are attributed. Let us take some examples ï" The system studied is a compressible fluid. To define a first state of this fluid, we can arbitrarily fix the volume it occupies, the density and the temperature at each point of this volume. When we propose to define a second state of the same fluid, we can no longer use it as freely. It follows from the fundamental definitions of Energetics that the total mass of the same fluid remains invariable in whatever state it is. If, therefore, we designate by du> an element of the volume occupied by the fluid and by o the density of the fluid which is in this element, the integral p rfw, extended to the whole volume occupied by the fluid, must have, in the second state, the same value as in the first. ------------------------------------------------------------------------ 2" The isolated system we are studying is an electrified conductor. To define a first state of this conductor, we can arbitrarily give ourselves the value of the solid electric density e at each point of the volume it occupies and the surface density E at each point of the surface that limits it. We no longer have the same arbitrariness when it comes to defining a second state of the same conductor. The fundamental principles of electrostatics teach us that the total electric charge of an isolated conductor remains invariant, by definition, whatever the changes experienced by this conductor. If, therefore, we designate by du> an element of the volume occupied by the conductor, e the solid electric density at a point of this element, dS an element of the surface which limits the conductor, E the surface electric density at a point of this element, the expression e c/co -4- lî dS, where the first integral extends to the volume occupied by the conductor and the second to the surface which limits it, must take in the second state of the system the same value as in the first. 3' A system is a homogeneous mixture of oxygen, hydrogen and water vapor. To define a first state of this system, we can arbitrarily give the masses m, 7 "z2, M, of hydrogen, oxygen and water vapor that it contains; but masses m, in[2. M' of oxygen, hydrogen and water vapor will be able to define another state of the same system only if these masses verify the relations 17.887/'2M'= 17.88/ni-H 2M, 17.88 ni', + i5.88M' = 17.88/".,+ i5.88M. These relationships are derived from the fundamental laws of Chemistry which serve to define the mathematical patterns that Chemical Mechanics deals with. The various examples we have just cited have a common character, which would be found in a large number of ------------------------------------------------------------------------ other examples; this character can be described in a general way in the following form We assume that the physical properties whose evaluation must determine a state of the isolated system under study are chosen. We give ourselves successively two sets of numerical values of these physical properties, sets which are not necessarily infinitely close to each other. For these two sets of numerical values to define two different states of the same system, it is necessary and sufficient that the numerical values of one or more expressions, formed exclusively by means of the physical quantities which serve to define the state of the system, and, therefore, not explicitly containing time, do not change when the second set is substituted for the first. If we express that each of the expressions in question retains, in any state of the system studied, the value that it took in a first state of this system, we obtain a certain number of equalities. By generalizing somewhat a denomination introduced by H. Hertz (' ), we shall say that each of these equalities is one of the holonomic linking conditions which determine the constitution of the material system we are dealing with. Among the systems that we have cited as examples, the first has a constitution that is determined by a single holonomic linking condition, which is Ci dm ̃-= coiist. S L. The constitution of the second is also determined by a single holonomic linkage expressed by the equality e du) -r- lï dii = const. Finally, the constitution of the third depends on two holonine bonds which are 1, 17.88/".! ̃:̃̃ J.M = Ci, 1 7. 88 m"-r- i-'),K8M = G,, C| and C.j being two constants. (',1 H. IIkutz, Die J'ri/icipien der Mechanik in neuem Zusammenhange dargestelll, n" 1:23, p. gt; Leipzig, 189'^ ------------------------------------------------------------------------ Let us consider, in particular, a system whose calibration is defined by the knowledge of the numerical values .'£,, x-2> xn taken by a limited number n of physical properties, and let us suppose that the constitution of this system depends on a certain number m, less than of holonomic links it will be equalities of the following form (xu xît xn) = ci, (3) 7Jl .f'- x xz> ~) = C:, {) <> xu xn) = Cj, 1 /"i(#i, .r2, - - xn) = C/;I; Ct, Ci, Cm are /" constants whose numerical values will be known as soon as a first state of the studied system is known. This first state being given, we will not be able, to obtain another state of the same system, to take arbitrarily the n numerical values. xy, 1 XII but we will be able to choose arbitrarily/? = n m of them, and the (n - p) others will be then determined by the equations (3). ). If therefore the state of a system is defined by the evaluation of a limited number a of physical properties, and if the constitution of this system is determined by m holonomic linking conditions, once a first state of this system is known, any other state of the same system can be determined by assigning determined numerical values to p z=z n - m quantities that can vary in an independent way. The systems whose state is thus determined by a limited number of independent variables are the simplest that general Energetics has to study; applied to such systems, the principles of this science take a particularly clear and easy to handle form; therefore the study of such systems is of great importance. 6. Non-holonomic links. is systems whose constitution is not entirely determined by holonomic connections. We still assume that we know which "physical" properties are ------------------------------------------------------------------------ s, in limited or unlimited number, must be evaluated in order for a system state to be defined. By choosing a certain set of numerical values of these physical properties, a first state of the system has been defined. We give ourselves, of the same physical properties, a new set of numerical values which is not infinitely close to the previous one, and we ask ourselves if this new set defines or not a second state of the same system; for the systems which we now want to consider, we do not immediately possess a criterion which allows us to answer this question. But, in the case where the second set of numerical values is infinitely close to the first one, we have an independent criterion, called time, which allows us to declare if these two sets of numerical values define or not two states, infinitely close to each other, of the same system. We then say that the constitution of such a system depends on non-holonomic links. Let us show how non-holonomic connections can occur in a system whose state is defined by the evaluation of a limited number of physical properties. Let us suppose that a first state of the studied system is defined when we arbitrarily give ourselves the numerical values x,, x->, x,, of n physical properties suitably chosen. Let #,+ 3a?,, x2 + 8a?a, .xn -+-oxn, be n numerical values infinitely close to the previous ones; in general, they cannot be considered as defining, for the same system, a second state infinitely close to the first. For them to define a second state, infinitely close to the first, of the same system, it will be necessary and sufficient that the n infinitely small variations Sa?,, ùx2, Zx,, verify m linear and homogeneous relations AI ôx,+A2 A~ 0.~2- --)- An 2xn-o. 0, ( +7 I B, ô^i+B, $Xz- ̃+- B,, ôxa = o, [ M, àxt -+̃ M2 8a: s -f- .̃+- M,, àxn = o, whose coefficients Ai, B, M,- have given values ------------------------------------------------------------------------ when the state of the system isldetim. The number m is, of course, less than the number n. It may happen that the coefficients A, B, M,- are given as functions of x,, x-2, xn and that the first members of the relations (4) are immediately integrable, that is, they are the total differentials of m functions/ /o, Jm of .ri, x2, - - j xn At dx, -+- A2 dx2~H- A,, dxn = df, (xu x2, xn), ), f5) B, dxi-h Bidxï-h.+- BK dxn= df2 (x,, x-2, .x,,), M] dx, ̃+̃ Ma dx^-r- .>- M,, dxn= dfm(Xj, x,, xn). In this case, relations (4) are equivalent to relations (3), and the constitution of the system depends exclusively on holonomic links. It may still happen that, without being immediately integrable, each of the first members of the relations (4) admits an integrating factor in this case, there are 2 m functions F(, F2, ¥m, ft.fï; ̃ ̃ ̃ ̃. jm of .r, ,î2. x, such that we have Fi (A, dxi-+- A3 (tej-'r-4- A,, dxn) = dfu I F, ( B, dxx + B., dx, -h B,, dxn ) = df2, .1' ( M, dx, -1- M, c/a-j -h MH cfen ) = dfm. In this case again, relations (4) are equivalent to relations (3), and the constitution of the system depends exclusively on holonomic links. But there is nothing forced about the assumptions we have just made; it may well happen that the first member of at least one of the relations (4) is not immediately integrable and that it does not admit an integrating factor: the corresponding relation cannot be put in the form (3); among the links which determine the constitution of the system will be at least one link which will not be a link. It is of course possible that the constitution of the system depends on several non-holonomic links, it is even possible that none of the links that fix this constitution is holonomic. The case where the links that fix the constitution of a system are not holonomic is ------------------------------------------------------------------------ (l) C. Neuman.n, Grundzuge der analy tisclien, Mechanik, insbesondere der Mechanik starrer Kb'rper. Zweiler Anikel (Berichte der Sachs. Gesellscliaft der Wissenschaften: Mathematiscli-physische Classe, 5 March 1888, p. 32). See, on the same subject. Vierkandt, Ueber gleitende und rollende Bewegung (Monalshefle fur Mathematik uiid Physik, Bel. III, 1892, p. 47). J. Hadamard, Sur les mouvements de roulement ( Mémoires de ta Société des Sciences physiques et naturelles de Bordeaux, 4* série, t. V, 1890, p. 397). .1. Hadamard, Sur certains systèmes d'équations aux différentielles totales (Procès-verbaux de la Société des Sciences physiques et naturelles de Bordeaux, année 189:1-1890, p. 17). -KoiiTEWEO, Nieuw Archief, 1899. - Caicvallo, Théorie du mouvement du monocycle et de la bicyclette (Journal de l'Ecole Polytechnique, '> série, V° caliier, i960, p. 119, and VI* cahier, 1901, p. 1). - P. Appell, Les mouvements de roulement en Dynamique (Collection Scientia, Paris, 1899; the two Notes of M. Hadamard are reproduced in this work). - P. Appell, Traité de Mécanique rationnelle, 2' éd, t. Il, Paris, 1904, p. 363. The systems whose constitution depends exclusively on holonomic connections have the character of very exceptional systems. However, for a very long time, mechanics have only dealt with these last systems, and it is very late that their attention has been drawn to systems whose constitution depends on non-holonomic links. The existence of such systems was first pointed out by M. C. Neumann (' ). As an example of a non-holonomic system, we will choose the one whose study led C. Neumann to his important remark. This system is formed by two solid bodies, subject to touching each other constantly at one point; moreover, it is assumed that the passage from one state to another always takes place, in this system, without any sliding of the bodies in contact. Let's see first what numbers will have to be known for a state of the system to be defined. Let us denote by C and Ci our two invariant solid bodies. The position of body C in space can be fixed by means of six quantities; there are many ways to choose these six quantities; for the rest of our presentation, it is not useful for us to stop at a specific choice; it is enough that we designate these six quantities by qu qu -_ >) The half line AU tangent to the line v = const. The half line AV tangent to the line Il = const. The half line AUt tangent to the line vt = const. The half line AVt tangent to the line u, = const. Each of these half lines is conducted in the direction of Fig. i. the variable coordinate that corresponds to it. The angle UAV is right and so is the angle U( AV|. The angle UAU( 1 is the angle at. Let us suppose that the system passes to an infinitely close state; the point of contact comes to the surface of the body C, in a position B, of coordinates u -+- or, v + ov; it comes, to the surface of the body C,, in a position B, of coordinates u, -+- o",, r, oc,. The two arcs AB, AB, have the same length and are tangent to each other at point A; to express that it is so, is to express that the change of state considered was not accompanied by any slip. But, for it to be so, it is necessary and sufficient that the projection P of the point B on the common tangent plane coincides, to the nearest infinitesimal of the second order, with the projection P, of the point B, on the same plane; or else, by designating by Aa, Ajï the coordinates of the point P with respect to the axes UAV, and by Ax" AJ3( the coordinates of the point P, referred to the axes U, AVi, ------------------------------------------------------------------------ that we have the two equalities A a = A 2[ cos'i - A pi sin 'l, ("') ¡ Ax=AxJcos.1--AS¡sin.!i, if A j3 = Aï! sin i h- A [it cosi. On the other hand, an infinitely small linear element connecting, on the surface of the body C, the point (u, v) to the point (a -f- du, ç-+-rlv) has a length ds given by the equality ds2- f2 clta=-r 1 dv=, where/, g are two real and positive functions of u and v, functions which depend on the shape of the body C. In the same way, an infinitely small linear element connecting, on the surface of the body C, the point (m,, p,) to the point (ut -+- du,, c, -t-di',) has a length ds, given by the equality ds\=f\du\-±-fr\de\, where g, are two real and positive functions of u,, v,, which depend on the shape of the body C,. If we neglect the smallest infinitesimals of the second order, we can easily see that we have Ax =/( ", v ) or A j3 = g ( u, v)cç, Axi = /,(",, ii|)îi(h AS, = gi(uu C|)oc,. Conditions (7) then become (8) f(u,v)Zu - f,(ui, vrfcostySui + gtiut, p,)sin <(/ Sç>, = o, g(u, V ) OC /< M|, C|) Sin il" 0 "l - g\{lll; Vl) COSll" 8('i = O. These are the conditions under which the two groups of numerical values 7l> 72l (h- ']:-' x, the equations (4) will then tell us what numerical values to assign to the ni others so that the two sets XU X-2, Xn, a?i + 8a?i, x*î>x±, xn-±r>Xn, represent two infinitely close states of the same system. As the conditions (3), diiferentiated, take the form (4), this remark remains equally true for a system whose constitution depends exclusively on holonomic links. We can therefore state the following general proposition Let us consider an isolated system whose state is entirely defined by the evaluation of a limited number of physical properties; let us suppose, moreover, that the bonds which fix the constitution of this system are or are not holonomic. A first state of this system being given, we will be able to determine any state infinitely close to this one (by choosing arbitrarily the values of a certain limited number of infinitely small quantities). We will say that these quantities are the variations ikdér-ji.NnA.vjKS qczi derternzirzerzi any change of infinitely small state of the studied system. 7. Bilateral and unilateral connections. But we have not immersed all the forms that the conditions of connection which express the constitution of a system can take; we have, in fact, assumed that these conditions have the form of equalities; we can very well find cases where this is not the case; let us cite a few simple examples. Let. Ji .x.y, z) = o the equation of the surface (limit our ------------------------------------------------------------------------ solid. This surface divides space into two regions, one where f(x,y,s) is positive, the other where f(x,y,s) is negative. Let us assume that this last region is the one occupied by the solid. Then the coordinates of our moving point will always be subject to the condition. /(̃*-, y, s)ào. a" Two moving points M, (x,, yt, z, ) and M2 (x->,y", z.,) are connected to each other by a flexible, but inextensible wire of length l. The distance M| M2 can be equal to or less than l, but it can never exceed l, so that the constitution of the system leads to the condition l1- (*2- ^i)2-(/2- /i)!- (-î- -i)'o. 3" A system is formed by a mixture of ice and liquid water; m, is the mass of the ice and m-, the mass of the liquid water; the constitution of the system requires that the sum of these two masses has an invariable value M, which gives a connection expressible by an equality /"i + /?l2 = M- But this constitution also requires that neither of the two masses mi, m2 is negative, which gives the two conditions, not reducible to equalities, /"[ = O, /2:. O. Up to now, we have assumed that we had a criterion for recognizing whether or not two infinitely neighboring states belonged to the same system; but we have admitted, at the same time, that this criterion consisted solely of equalities; the examples we have just cited will show us that, in some cases, inequalities can also be included in a similar criterion. Let us take up these examples one by one. i" Let M (x, y, 0) be a position of our moving material point; under what condition can the position M'(.r + Sx1, y-y, z -f- S s) be regarded as a second possible position of the same material point? For M' to be a possible position of this material point, it ------------------------------------------------------------------------ It is necessary and sufficient that at this point the function f of the coordinates is positive or zero. Let us suppose, first of all, that the first position M is on the surface of the impenetrable solid body, so that (x,j :) is zero. For M' to be, to the nearest infinitesimal of the second order, a possible position of the material point, it will be necessary and sufficient that Sa?, oy, oz verify the condition <>f\ àf df " " Sx H - - oy -r- - oz d. o. ôx Oy Oz If, on the contrary, the point M is not on the surface of the impenetrable solid, B#, ûy, 8s can be taken arbitrarily. 2° Let us take an initial position M, [x, y,, z,) and M2(x2, y- zu) of our couple of points, and let us suppose that at the moment when they are thus placed, the wire which connects them is perfectly taut. MiM2 is then equal kl. Let us then consider a second position, infinitely close to the preceding one, M'( (xt + S^ y, -+̃ oy{, z, -+- os, ) and M', (^2 + 8-^2) y-i+ toy. z-, + ô52) of the same pair of points. M', M'2 cannot outperform l, so M|Ma, so we must have O2- x\ ){fixt- 8a7, )-4- (y^-yi)(oyi - Sy1)-h(z2 - zl)(Sz,- Ô2,)So. If the wire connecting the two points M|,M2 is not perfectly taut, there is no need to impose any condition on the variations Zxs, 8y,, o; ox2, oy,, oz- 3" Let us take an initial state where the mass of ice is null; from this state, this mass, which cannot become negative, cannot decrease; so that, to pass to a close state, it is necessary necessarily that one has 1 2m! â o. Let us take in the same way a state, initial where the mass of the liquid water is null; to pass to a close state, it is necessary necessarily that we have 0. Neither of these conditions is necessary if, in the initial state, the system contains both water and ice. ------------------------------------------------------------------------ 4" To these examples, let us add a fourth where we immediately encounter a criterion of the form we are dealing with: i and 2 are two bodies, at least one of which is deformable; they are not miscible with each other. In a certain initial state, they touch each other all along a certain surface S. Let M be, in this state, a point of the surface S; Mt (x,, y,, s,) and M2(.x2, y-2, z->) two points infinitely close to the point M, taken one inside the body i and the other inside the body 2: n, and n2 the two half-normals in M to the surface S, directed one towards the inside of the body 1, the other towards the inside of the body 2; ci. J3,, y, the director cosines of /), a2, |32, y2 the director cosines of "2- Let us now consider a second state close to the previous one; the material point which was in M| has come in M',(a?1 + Sa?,,71-i-fi/z1-f-3.s,) the material point which was in M2 came in M 's (x2 -+- oa-i, y-, -h oy. ss - Sz, ). In the vicinity of the point M, the two bodies 1 and 2 could have remained in contact; they could also have separated from each other; but they could not have been compenetrated, since they are supposed to be immiscible; hence the condition, required at any point M of the surface S, a, o-r(-+- 3, o/i- -'1 OS! a" ox, 3, 0/ y2 rj-,>0. This condition would no longer apply if, in their initial state, the bodies and 2 had no point of contact. There are systems constituted in the following way Their states can be classified in two categories; If we take for initial state an elat of the first category and if we impose on the properties of the system infinitely small variations, so that these variations determine a second state, infinitely close to the first one, of the same system, it will be necessary and sufficient that they verify certain conditions of linkage, holonomic or nonholonomic, expressed by equalities. But if we take as initial standard a state of the second category, it will be necessary, with these conditions of connection expressed by eq- ------------------------------------------------------------------------ lities, join some other linking conditions of the form /=", # = <>. In these conditions, g, are expressions formed by means of the properties (which define the state of the system and the infinitely small variations of these properties; with respect to these variations, they are linear and homogeneous, in the broad sense that we gave above to these words (p. 4°) - If, in particular, the state of the system is entirely defined by the values of a limited number of quantities x,, ,~2- these conditions are of the form F, 8a?, 4- F2 ex, -h 4- Fn Sx,,Z o, Gi Ixi -+- Go S:c2 ~r- -(- G,, î.r,; o, F(, G|, having known values when the state of the system is known. We will call the links we have just considered unilateral links; links expressed by equalities, whether holonomic or not, will be called bilateral links. The difference between these two kinds of connections will become even clearer with the following remark Let us suppose that from a given initial state, a bilateral connection is verified when we impose infinitely small variations on the properties of the system; it will still be verified if, from the same state, we impose on the same properties variations equal and directly opposite to the previous ones. It may not be the same for a unilateral linkage; let us suppose that from a given state of the system, certain variations imposed on the properties of this system make f take a positive value; they will verify the condition of unilateral linkage/^o; variations equal in absolute values to the previous ones, but of opposite signs, would make/ negative and would not verify this condition anymore. This is expressed by saying that, for a system subject only to bilateral connections, any infinitesimal change of state, consistent with the constitution of the system, is reversed. ------------------------------------------------------------------------ sand, while for a system subject to one-sided bonds, there are infinitesimally small non-reversible calibration changes. The possibility of one-sided connections was first pointed out by Fourier ( ), then by Gauss (-). 8. Virtual modifications of a system. Let us imagine that a continuous sequence of states of the same isolated system has been formed; let us fix our attention on these various states in the order that allows us to pass from one to the other in a continuous manner; to designate this quite intellectual operation to which we subject the mathematical scleme that must serve us to represent a set of concrete bodies, we say that we impose on the system a virtual modification. In the course of a virtual modification, according to what has just been said, the numerical values of the properties which serve to define a state of the system vary in a continuous way, but, in general, their variations are not arbitrary; they must be such that the various states of which we consider the sequence can be considered as states of the same system; in other words, they are subject to the connections, holonomic or not, which result from the constitution of the system. This restriction is the only one to which a virtual modification is subject; the variations of the numerical values of the variables which serve to define a state of the system must be compatible with the conditions which logically result from the definition of this system, but with these conditions only. In particular they may well contradict the experimental laws governing the system. (') FouniER, Mémoire sur la Statique contenant la démonstration du principe des vitesses virtuelles et la théorie des moments (Journal de l'Ecole Polytechnique, V* cahier, 1798, p. 20. - Œuvres de Fourier, t. II, p. /|77). ('̃) Gauss, Ueber ein neues allgemeines Grundgetelz der Mechanik ( Crelle's Journal, BJ. IV, 1829. Gauss, Werke, Bel. V, p. -?. in note). - Principia generalia theoriœ jîgurœ Jluidoruni in statu œquilibrii (Nouveaux Mémoires de Gœttingue, t. VII, "i83o. - Gauss, Werké, Ud. V, p. 35). - Letter of GAUSS to Miinius cited in G. Neumann, Ueber das Princip der virtuellen oder facullativen Verruckungen (Beric/Ue der K. Sàchsischen Geseltschaft der IVissenschaften zu Leipzig Mathernatiscli-pliysische Classe, 8 March 1S86). ------------------------------------------------------------------------ This is the set of concrete bodies that our abstract and mathematical system aims to represent. This remark being essential, let us clarify it by an example. Let's take the mixture of oxygen, hydrogen and water vapor which has been used many times as an example; let's suppose it is maintained in an enclosure of invariable volume. At least for temperatures above a certain limit, there is a condition of equilibrium of the system, a condition expressed by a relation between the ratio x and the temperature fj ~=/(6). If, at a certain temperature 0, x has, in the mixture, a value lower than /(G), it necessarily occurs, within this mixture, a combination of oxygen and hydrogen, so that x necessarily increases. If, on the contrary, at this same temperature 0, x has, in the mixture, a value higher than /(11), it surely occurs, within this mixture, a dissociation of the water vapor, which makes x decrease. These propositions are intended to represent the experimental laws to which the concrete gas mixture of which our abstract system is the figure is subjected; but they do not follow logically from the sole definition of this abstract system; they are not binding conditions; therefore, in the constitution of virtual modifications, the physicist does not have to take them into account; he can, for example, starting from a system where x has a value lower than /(0), arrange mixtures infinitely close to each other, so that x goes down from one mixture to the next; he can consider a continuous sequence of mixtures where x goes up and up from a value higher than /(&) he thus obtains two virtual modifications, although in reality no modification can be observed which is represented by any of the changes of state produced in such a sense. We can thus see that a virtual modification can very well present us with a continuous sequence of states that no real modification could go through, because, in doing so, it would violate certain experimental laws that govern the bodies studied. On the contrary, in any real change, the system runs through ------------------------------------------------------------------------ a continuous sequence of states; this continuous sequence of states of a same system constitutes a virtual modification of this system; thus any real modification corresponds to a well determined virtual modification. 9. Of the local movement and the general movement. The word motion has, in modern scientific language, a precise meaning; it means exclusively that phenomenon by which, from one moment to the next, the same body occupies different places in space. It is only in Peripatetic Philosophy that the word motion takes on a much broader meaning and that it applies to a host of changes of a very diverse nature; Peripatetic Philosophy characterizes with an epithet the particular motion by which, from one instant to the next, the same body comes to occupy different places; it names it local motion. In this Treatise, we will give to the word motion a very general meaning, analogous, but not identical, to the one it takes in the Peripatetic Philosophy, so, like this Philosophy, we will take care to designate by the precise term of local motion, the simple change of place in space, from one instant to another. The local motion will not be, for us, the only motion to be considered; whenever the properties by which the state of a system is defined vary from one instant to another, we will consider this system as being in motion; now, a system can be in such a motion although it is not animated by any local motion. We will cite two examples. ̃ i" A surface invariable in shape and position delimits a constantly homogeneous mixture of oxygen, hydrogen and water vapor the total mass M, of free or combined hydrogen, the total mass M2 of free or combined oxygen, the volume V occupied by the mixture, the content x of water vapor, the temperature are the magnitudes which completely define a state of the system. By defining the constitution of this system, ------------------------------------------------------------------------ each of the first three takes the same value in any state of the same system; the last two vary arbitrarily. Without any part of this mixture, which is constantly homogeneous, changing its location in space, a combination of oxygen with hydrogen or a dissociation of water vapor can occur from one moment to the next, which causes x to increase or decrease; the mixture can heat or cool, which causes 6 to increase or decrease. We say then that the system is the seat of a chemical movement and of a thermal movement, although Í! no local movement occurs there. 2° A conductor, of invariable figure and position, is electrified. The solid electric density at each point of the volume it occupies, the superficial electric density at each point of the surface which limits it or of the surfaces of discontinuity which divide it, completely define its state. Without any part of this conductor moving in space, the electric distribution can vary from one moment to the next; the electric density can, at different times, not have the same value at the same point. So, without being animated by any local motion, the system is the seat of an electric motion. Many philosophers and physicists have maintained, and still maintain, that the only motion really existing in the material Universe is local motion; they will therefore assert that the changes of elal of which we have just spoken are only apparently free from local motion; that in reality they resolve themselves into local motions hidden from our senses. They will maintain, for example, that the combination or dissociation by which, in our homogeneous mixture, the content of water vapor varies, is nothing other than a local motion by which certain atoms of hydrogen approach or move away from certain atoms of oxygen; that the heating or cooling which causes the temperature to vary is produced by an increase or decrease in the average speed of the said disordered motion which agitates the gaseous molecules They will claim that any change in the electrical distribution on a conducting body is ultimately reduced to the flow. ------------------------------------------------------------------------ of a certain lluidc or to the transport (If certain. corpus-cules. Those who are more sceptical or more cautious, do not consider as certain the reduction of any change of state to local motion, may nevertheless wonder if such and such a phenomenon, which we consider to be known not to be a local motion, would not resolve itself into a motion too tenuous to be accessible to our senses, or, at least, if this phenomenon would not be linked to some imperceptible change of place. Such objections or doubts have their origin in a confusion. We must be very careful to distinguish between, on the one hand, the real and concrete bodies whose properties we propose to represent, but not to explain their nature, and, on the other hand, the abstract system, composed of mathematical notions, which must provide us with an image of the properties of these concrete bodies. The very nature of concrete bodies remains unknown to us; therefore, when we observe a change of state free of any local motion perceptible to the senses, directly or through the intermediary of instruments of observation and measurement, we can neither affirm nor deny that this change of state is linked to a hidden local motion. But if the e mliinc structure of-, concrete bodies i ( happe us, the structure of the mathematical scheme, the sole object of our icusonnenw/its, is, on the contrary, perfectly ̃ onnue to us, <i\Hiil the local ri.tesse. 1 .a malicre cpii. at I in^laul is in an element of volume (Im which Mi'x. _)". z i is a point, relrou\c, at I instant (/ -- dt) <'ii an element of (1 \oluiue c/d)' aiiipiel ap|)arlieul the 1 point M'i'.z- y:'}. If we [)Ose ./- - /̃ ii dl. y' -) -i'< -rwdt. a, c, ir will be, at the moment, the components of the velocity of the matter which is at the point M at this same moment. If the point JVI is in the middle of a mixture of several eoi|i>, this definition leads (^ 2. p. i(iià > to consider as many local vilesscs dillV-renlos "(u'u've got distinct components in the mixture. It is not necessary that the local \ilesse.s of a body are always continuous of they can, for isolated values of/, undergo discontinuities; thus, when one treats in Mechanics of the shock of the m\ariable solids. one supposes that at I mstiinl of the shock the speeds vary in a discontinuous way. When we know, at a certain moment, the local velocities of all the small material parts, mixed or not, into which we can decompose the system, we will agree to say that we know the local motion of this system at this moment. ------------------------------------------------------------------------ To generalize this definition we would say that we determine the general motion of a system at a certain instant when to n the knowledge of the state of this system at this instant, we will join the evaluation of certain physical properties, in limited or unlimited number, that we will name the general speeds at this instant. 1 1 The choice of the physical properties that can be considered as general velocities of a system is subject to certain restrictions that we will formulate with precision. In the first place, all the local velocities must be determined when the state of the system and the general velocities are determined or, in other words, when the state and the general motion of the system are determined. Each of the local velocities must be expressed as a linear and homogeneous function of the general velocities, these words being taken in the sense which was defined above (§ 6, p..jo). Secondly, the knowledge of 1 étal and of the general motion of the system must lead to the knowledge of the first derivatives with respect to time of all the quantities which define the state of the system. These first derivatives must also be expressed as linear and homogeneous functions of the general speeds. We see that if, without changing the system's ideal, we multiply all the general velocities by the same number, all the local velocities and all the first derivatives with respect to time of the quantities which define the state of the system are multiplied by this number. Finally, the general velocities can, like the local velocities, be discontinuous for certain values of Provided that these restrictive conditions are met, we remain free to choose as we please the physical properties which will play the role of general velocities and which will define the general motion of the system. Let's give some examples of such a choice: i" Let us consider a solid body, whose lethality is entirely defined by the position it occupies in space. The velocity ( u, c, n>) ------------------------------------------------------------------------ of any point i y. z) of this solid is. at I instant given by the formulas II = O. -r- \LZ V Y r = ji - vi - }. tr = ̃ -- /.j' - |j. ;r. where a. ji, y, ̃}̃̃ v "(>ul six foundations of the single \ariable y., jb, y are ies components of a translational velocity imposed on the solid; are the angular velocities of three rotations made respectively about the three coordinate axes. The components u. c, ir of the velocity are linear and homogeneous functions of the six quantities a, jï, y, A, [/ v. It is the same for the first derivatives with respect to time of the six quantities which fix the position of our solid in space, and this in any way we want to determine this position. The six quantities ?.. |i. y, a, a, v can be taken as general velocities; they determine the general motion of the system which is, moreover, a simple local motion. ̃> the enclosed vessel, imariablc in shape and position, encloses a lethal compressible fluid of the system is assumed to be completely unbound when the density p and temperature of the lluid at each point i.f, y, z\ of the volume enclosed by the vessel are known. Let r, iv be the components of the local velocity at the point ('a™, y, z) and at 1 instant if we observe that we have, at any point and at any instant, in \erluity of the theorem of Kinematics well known under the name iVéqualion of continuity.. (a) il: - lit z u -'- i -f- ilt --L ;p) H - ù( !r 0 w) - o. 01 ij.r ily 0.Z we see, (pawn to) look at the four functions of x.y, t u, v; ~s~; cl6 ̃ .lit as the general viles>es which determine the general motion of the system: this motion is no longer a simple local motion it implies a thermal motion. ------------------------------------------------------------------------ -'̃>" They uu conductor, invariant of form cl of po-iIiuii. that is electrified. The state of. this conductor is enhèremenl delini when one knows the solid electric density c. in, any point of the volume which it occupies cl the superficial electric density Y. in any point of the surlace ijiii the limit. In the arena, the local velocities are identically zero at each point of the conductor. Let a. i>, u\ be the components of the electric flux at the point (x. Y.) and at the instant These quantities are related to the electric densities e. \L by the equalities 1)11 /A" <)w Oe 1 ) _1.- -.= (1. ôx ijy rice Ot n ~)'; ( ̃> j 7, ~u '^f ti- - (), 01. l)è-> then, it is easy to see that the jht.i of cmuliirliini rlfi-irii/iir can be taken as a general velocity capable of defining the general iiioiuc'îiful of the system; this mou'cinenl. is, here, an electric niouemenl piiremeiil, enlièremenl evempt of local motion. {" Let us take a polarized dieleciri(pie, but ui\arialile of form, position, temperature ele. 1,'éliU of this body will be enliéi'1nii'iiL ili'iLcii'ininô when we eonnailron>, in chacpie point of the \oIliiiii: that it occupies, the three components 15, C of the intensity d<; polarization. From then on, we can visibly take for general \ites>e, the fln.i of (Icplai-emcnl whose components- are ~- -~ ̃ iii m ai The general movement thus determined -.era. coinme te precedent, ilaleinenl exempt of local movement. No-. we said (that the ito-i1-. ^enerilc" had to be clioisii' in such a way that it- verifies the following condition When we multiply all the general velocities by the same number, the first derivatives* with respect to time of your test properties which define an elal of the system are multiplied by this number; it is the same for all the local velocities. ------------------------------------------------------------------------ This number can be o. J J <:> s therefore, mm- can state the." following propositions If all the "general velocities" cancel each other out at a certain moment, it is the same, at this moment, for all your "local velocities" and for the first differences with respect to time of all the properties which define the state of the system. If the general speeds remain all constantly null. it is, of it, of same of all the vjtesses ttmisffimè* all the physical properties which define 1 étal. of the svstème keep constant values, so that I étal of the system remains invariable. 'In Mechanics, where one will only irai te of local motion, where, consequently, the state of a svstystem is entirelydefined when one knows the place occupied in ̃J'space by each of its parts. one says that a system is in equilibrium when all local velocities are constantly zero: the state of the svstystem is then invariable. We will extend this term and say that any system is in equilibrium when all the general velocities v are constantly zero. In a system in equilibrium, the local velocities are all" constantly zero and the system remains invariant. The local velocities could be all null although the general velocities would not be all equal to o. If we consider, for example, an electric conductor immobile in space, the electric velocities of which it is the seat may well not be zero.l n such a system, where the" local velocities are all constantly nulljs, but where eeiaine" \it"""e" general dillerent of o, n e>l |>a." in é(|mlibre it is said < rest. The ii s\s|èinc eu rest is free of local motion, mai" il u e "l no evempl of general motion the mol repus oppose the term motn'emenl local, as the word equilibrium s nppose .m term general motion. l. We have said that the knowledge of the general velocities (levai néeessatremenl entail the knowledge of the first derivatives with respect to time of all the properties that define the system. The reciprocal is not necessarily true. There are systems for which it is true. ------------------------------------------------------------------------ For example, for a solid body moving in space, the knowledge of the derivatives with respect to time of the six parameters that fix its position leads to the knowledge of the six general velocities a, Jï, Vj À, u, v. For example, again, in a stationary dielectric, the knowledge of the derivatives with respect to time of the components of the polarization leads to the knowledge of the displacement flux which has precisely these derivatives as components. In such a system, if we cancel the first derivatives with respect to time of all the properties that define the state of the system, we cancel by the same fact all the general velocities the system cannot remain in an invariant state unless it is in equilibrium. I Many systems do not enjoy this property. For a compressible tluid, enclosed in an invariant surface, to keep an invariant state, it is necessary and sufficient that the density p and the temperature 0 keep, at each point, values independent of time; the second condition entails the equality at o of the Ide l d. dp general velocity - but the first condition, which cancels -y-> does not require that the components ", v, w of the local velocity be zero; it only requires, by virtue of legality (9), that these components verify the relation to! pu) d(ov) d(aw') = o. (1 01 - - h i-! - H - - == o. dx dy dz In the same way, for an immobile electrified conductor to remain in an invariant state, it is necessary and sufficient that the solid electric density e at any point of the volume it occupies and the surface density E at any point of the surface which limits it keep values independent of t; this condition leads to the equalities of à\i - =0, - = o dt tot but it does not require that the components u, v1 w of the electric flux cancel out at any point; it only requires, by virtue of equalities (1) and (2), that these components verify the equation of the tov Ow (it) - -H h -- = o v Ox <)y dz ------------------------------------------------------------------------ at any point on the line occupied by the conductor, cl I equation ( 1 a u -- fi v -+- y w = in loul poiul of the surface which limits it. When Tétai of a system remains independent of time, although the general velocities are not all zero. one says that this system is in uniform regime thus the conditions ( i i) ) and (12) express that our electrified conductor is the seat of uniform electric flows the condition f' 10), that our fluid is moved of a flow uni form It can be assumed, in such systems, that the properties intended to define the state do not keep values independent of time, but that the same is true of the general velocities; the calibration and the general motion of the system remain both invariant. The system is then said to be in a steady state. Thus a compressible fluid, enclosed in a light surface, is in a steady state if the temperature 0 keeps, at each point, an invariable value and if the components u, c, w of the velocity, functions of x, r, z, but not of t, verify at any point the condition (10). Thus, a conductor is traversed by a permanent electric flux if the components u, c, w of this flux are independent of t and verify the conditions (1 1) and (12). Let us return to systems constituted in such a way that the knowledge of the derivatives with respect to time of all the properties that define the stall leads to the knowledge of all the general velocities. Suppose, moreover, that the state of the systems we are going to discuss depends on a limited number of variables. Whether the system is holonomic or not, we have seen (p. 4') that any infinitely small change of state can be determined by the arbitrary choice of a limited number p of infinitely small quantities î| -". £/ Once these p quantities are known, we know by linear relations the infinitely small variations §#1, Zx->, 0.7; of the n quantities .- x-± r,, which fix, the state of the system. Let us consider the real change of calibration that the system experiences between the instants t eu (t. -r^t - -> > el., by hypothesis, the a (II l dt t dl 1 l knowledge of these latter quantities enlranie the knowledge of loti te-, the general vilesses. The determination of the independent quantities î'(, e. î#J is thus equivalent to the determination of the system's velocity. From then on, there is no reason why we should not take as general velocities the p quantities z\ e' z PiH'ini the systems we have just been talking about are, as a more particular case, the liolonomous systems whose state depends on a limited number of physical properties. Any infinitely small change of state of such a system is determined by the variations oy,,oy->, ovp of a limited number/" of independent variables y, )\>, v p. To define the motion of such a system, the quantities c/)i dy-i dv y'x if7 Ji ~dT' ̃̃ ~tr can hardly be taken as general speeds. The two categories of systems of which we have just spoken l' are the simplest < pie the Energetics can consider; therefore their study would have for us a very special importance. 10. Independent systems i Let us consider a system S isolated in space. Let us assume (pion a ehe .exactly i" < What are the physical properties whose evaluation delineates (-completeinenl a state of the system S (') We have, for hi first time, ilélini hi nolion of inde/icndant systems in It'crit suiv;ail CoititnenUfii'e to the principles of lit 7'/iermodyna/ni(/ur f "1 Part Le principe dé la conservation de l'énergie, Cliap. I. art. 5 (Journal de ''̃Mathématiques pures el appliquées, \n série, t. VIII, rSt)(j p. >So). ------------------------------------------------------------------------ ̃> What are the links, holunomes or not, which deeoiilenl of the constitution of this thread sxsleme allow to recognize -j two infinitely neighboring states -ont elals of this same svslème: -i" What are the general velocities- which define coniplètemenl the general motion of this svslème. Let us imagine that your bodies which form the system S can, by means of a certain surface, be separated into two groups, S, S,. Let us imagine that the properties:, defining the state of the system S. the bonds characterizing its constitution, the general speeds determining its generative motion can be arranged in two groups ̃ of properties, bonds, speeds, which we will name the group G, and the group (i2. Let's admit finally i" (hic the properties, the bonds, the ileiles.-es which compose the group d, jjiii>sr-iil èlre defined- cl evaluated- without that it is garlic no allusion to the set of bodies S. nor to the properties, bonds and vilesses which compo^enl the group (, a" That the properties, bonds and velocities which compose the group (;2 pnissciil èlrc defined and é\;duéi:^ without any mention of the en^endile of body S,, nor of the properties, bonds and velocities which eomposenl the group (i,. Under these conditions, it is clear that we could conceive the body set S, isolated in the space and define the state, fix the conslitiuiou, clélcrniiuer the general imuncmeul of this system isolates S| to the minfiii of the properties, bonds and speeds that form the group G, that we could, in the same way, look at the set of bodies S2 as im system i-olated from space and look at the properties, links and \iies-e- generated that form the group G" as detinissanl the elal, the coii-liluiion and the general motion of the s^ system-So. It is what we will formulate by saying that each of the -detFX syslt'.mi's S,, S., prit/ rlrr, conceived known isolates in U space, lont en gm-danl l'élat. In constitution and moineincnt uéncral that it presented within the system S. We will then say that the two systems S{, S.y are independent of each other. ------------------------------------------------------------------------ The most general virtual modification of the system S consists in attributing to the properties which bind the state of this system the most general infinitely small variations which are compatible with the conditions of connection which characterize the constitution of this same system. We see then that in a similar modification, the physical properties, suitable to fix the state of the system S, which have it part of the group G, will experience the most general infinitesimal variations that are compatible with the bonds of the same group in the same way the physical properties, suitable to define the state of the system S, which appear in the group G^ will experience the most general infinitesimal variations that are compatible with the bonding conditions belonging to the same group. Now, the first set of variations obviously defines the most general virtual modification of the system S, while the second set of variations defines the most general virtual modification of the system S2. We can therefore state the following proposition If an isolated system S can be decomposed into two independent systems S,, S2, the most general virtual modification of the system S can be considered as resulting from the coexistence of the most general virtual modification of the system S, isolated in space, and the most general virtual modification of the system S2, also isolated in space. Let us clarify these considerations by some simple examples. i° Let us consider a system S formed by two of these rigid solids S,, S2, which are considered by the rational mechanics; these two bodies are supposed not to touch each other at any point. The state of this system is supposed to be entirely defined when we know the place occupied in space by each of the two bodies S,, S2 it is thus defined by twelve independent variables, six of which fix the position of the body S, and the other six the position of the body S2 the first derivatives with respect to time of these twelve variables can be taken as general velocities. Obviously the two bodies S,, S2 form two independent systems. 2° The system S is formed by two solids S,, S2, subject to ------------------------------------------------------------------------ constantly squint. To define the state of such a system, we operate as in § 6 (p. 'Ml): we choose a first system of curvilinear coordinates orthogonal (m,, i() to the surface of the body S, a second system of curvilinear coordinates orthogonal (u->, v->) to the surface of the body S2. The state of the system is then defined by means of eleven variables, namely the six parameters c/i, , fj3, co "j" Sx-, cos'J" Sj's cos-2 oô, = o. It is clear that one cannot write it without considering the ------------------------------------------------------------------------ These two bodies do not form two independent sterns. In a virtual modification of the system S, taken in isolation, the surface, whose displacement would be arbitrary, would sweep a certain space V| of a modification of the system S2. taken in isolation, the surface would sweep a certain space V| nothing prevents that the two volumes V^ have a common part VV. In this case. the coexistence of the two virtual modifications considered could not be regarded as forming a virtual modification of the system S; because, at the end of this modification, a volume. W would be occupied at the same time by a part of the body S, and by a part of the body S2, which cannot be, since these two bodies are not miscible. Let us say again that the two bodies S, Sj cannot be considered as two independent systems. .̃j" We give ourselves in space an indeformable surface t which we call a mold; if a body comes to apply itself to the surface -r by placing itself on the side of this surface, we say that it takes X imprint in relief if it places itself on the side y. of the surface r on which it applies itself, we say that it takes YempreiiUe in hollow. It is assumed that the isolated system SI is, in whole or in part, bounded by a surface ï, which is, by definition, undeformable and immovable, and which is the impression in relief of the surface t that the isolated system Sa is, in whole or in part, haunted by a surface -2 which is, by definition, undeformable and immovable, and which is the impression in depression of the surface t. To form the system S, we join I \in to I aulre the two s\films S|. S. in such a way that the two MirIViees -); 1\. form only one surface ï, (pion admits to be immobile el. indéformable by definition. It can be seen that the system S can, here, be looked at as formed by two independent systems S,. So. The most ^(ineral) virtual modification of the system S, taken alone, must leave the surface still the most general virtual modification of the system S2, taken alone, must leave the surface still 12 the most ^general virtual modification of the system S must leave the surface still ------------------------------------------------------------------------ immobile la ̃ suri ace 1' it is clan1 that this last modilication canL be regarded as resulting from the first two. This decomposition of such a system S into two systems i 1 1 f f " '̃ - pendants S,, S,; will be -useful to us at the /in of this Chapter <£ I I. p.- 71 1. '.)" The system S is a polarized dielectric body. invariant of (orme el.ide position in I space J étal of this body csl determined when we give ourselves, arbitrarily, moreover, the three com])osants A$ B, C of the polarization at each point of the volume it occupies: the mou\emcnl ^eneraj of c1 -Wènie is determined by the knowledge of the three component-. i>. A, of the displacement index at each point. l)econipo.M>n> this corp> S in two bodies S2, which are also indefdrmublos and immobile- We can say that the elal el I moin cm eut of the system S sonl cntièreinenl defined when we give ourselves, on the one hand, the compohaulcs of the polans.ition el of the llux of dt'pl.iceincnL in any. point of the body S(, on the other hand, the components of the polarization el. of the llnx of displacement in ioul point of eorp-. ,S;. It is clear that the two bodies S, Sj form two independent systems. 6° The body S is an eli-cli-ized conductor. inxariable of form and position, which we will decompose into two parts S,, S., whose form and position are invariable. To delimit the state of this s\sicm S, it is sufficient to know the cleclric densities, solid or superficial, in each poi ni, of the parts S|,S2: butl So that two states of electrification can be re^ai'di's two states diU'éi-enl.s of the same s\s|eme S, it is necessary cl il suflil (pie the electric charge lolal soil the same in both cases: il (aul el il .suflil, in other words, that the "expr<'ssion <- ------------------------------------------------------------------------ lead us to admit that the sensitivity of our organs led to admit that the fiisi J >ili t -'- of our organs is fallible and limited that, without being identical, the degrees of heat of two bodies can be close enough so that we cannot distinguish them. t he hot body can mill on our organs only by the pallie of its sur! ace which is in contact with these organs, and this surface has always an extent which cannot be reduced below a certain limit the time during which we praise this surlace has always a certain duration. Nevertheless, we admit that the quality designated by 1 adjective hot belongs to the parts which are inside the body as well as to the superficial parts, that it belongs to each of the infinitely small parts into which the imagination can cut the body, and that it belongs to it during each infinitely small part of the duration; (It is thus relative to each point and to each instant that at one and the same instant it may vary gradually from one point to another that at one and the same [joint it may vary continuously from one instant to another. These words "hot fire" thus express a property of each of the infinitely small parts into which bodies can be supposed to be divided. What is, in itself, this property? Is it reduced, by its very nature, into quantitative elements? These are questions that Physics has neither to solve nor even to examine. Provided by the set of intellectual operations we have just analyzed, the notion of heat appears to us as purely qualitative; it appears as susceptible of being reproduced identical to itself, of becoming more or less intense; it is not susceptible of addition; It is clear that we would be stating nonsense if we said that the heat intensity of boiling water is the sum of the heat intensity of boiling alcohol and the heat intensity of boiling ether; we do not conceive of any process that would allow us to associate these last two heat intensities with each other in such a way that the first one results from this association. The intensity of heat not appearing to our reason as susceptible of addition, it could not be question of measuring your various intensities of heat the operation which one names measurement has, indeed, its reason of being in the addition. ------------------------------------------------------------------------ But, if we ni' can measure this property imn quaul ilalive, this pure 1'(~ ilive. null or negative. a reliable in a continuous way of a poinl ir I < mire and of an in-lanl to laulre - Kn deu\ poinl.- égiilenuMil hot, this number has the same value ,i" Kn two unequally hot points, it of \aleurs did'érente-. the greatest' value cnrre-pondanl to the hotter point. If one knew the values taken by a similar number at the different points of a set of bodies, and this at a given moment, one would know, at a given moment, if the degree of heat varies from one point to another of this set and in which direction it varies. One would know, for a given point, if the degree of heat at this point varies from one moment to another, and in which direction it varies. This number whose dner-c- values are used not to measure, but to quote Or it the inlen-ity- diverse.- of the quality (pie we name the heat, is called the temperature The deliiiition of the temperature leaves to a high degree. the choice of this numerical expression. Let us imagine, in it t. that one has. in a first way. made correspond a scale of number- to the div erse- mien- island- of the quality named heat; let be any number of this scale. -H" The (onction /'(/.)"> varies in the same direction as the variable 0. ------------------------------------------------------------------------ It is clear that the number W can, as well as the number fi, be taken as the temperature proper to identify the intensities of heat. We conceive therefore as logically possible, 1 e and eela in an infinite number of ways, the establishment of a correspondence between each intensity of the quality called heat and each value of a variable number, value which is the temperature corresponding to this intensity. This is enough for us to include temperature among the elements of which the mathematical schemes on which we reason are composed, without the risk of using an empty word. In fact, it is exclusively this temperature, conceived in an abstract way, that will be discussed in our reasoning. But we do not propose simply to unroll mathematical reasonings logically linked together; we want the results of our mathematical deductions to be compared with the experimental laws that govern concrete systems, laws that these results are intended to represent. From then on, it is not enough for us to know that to the quality called heat, which manifests itself at every moment in every point of a body, we can make correspond a number called temperature, the greater the intensity of this quality. We still need to know a real way to establish this correspondence. We must be able to solve the following two problems: A concrete body being brought to a determined intensity of heat, find, with a sufficient approximation, the temperature which must represent this intensity of heat. A temperature being given, to realize a concrete body which is endowed with the corresponding intensity of heat. Our perceptions of heat and cold have allowed us to recognize the possibility of this correspondence, but, at first sight, they appear to us as much too coarse and much too fallible to allow us to realize it with some certainty and some precision. If, therefore, we wish to classify bodies into categories, each of which corresponds to a given degree of temperature, so that all the bodies in the same category are equally hot, and that from one category to another the intensity of heat varies in the same direction as the temperature. ------------------------------------------------------------------------ If we want to recognize with precision, when a concrete body, more or less hot, is given to us, in which category it must be classified it is necessary for us to give up the direct and immediate use of our sense of the hot and the cold it is necessary for us to resort to another sense, more precise, such as the sight, and that indirectly, by the intermediary of an instrument, the tlterniomèlrc. The recourse to an instrument, in this case as in all the others, is imposed to us by the conviction that our senses are limited and fallible we will not be surprised to see this conviction intervening at each moment, in the rules which govern the choice and the use of this instrument. The justification of the use of the thermometer rests, in the first place, on a law which is one of the fundamental postulates of energetics. This law is conceived, first of all, on the occasion of certain direct perceptions; but after we have formulated it in a general way, we attribute to it a certainty very superior to that of the perceptions acquired by the sense of heat and cold; when we find it in disagreement with these perceptions, we conclude that these are misleading, and we make use, to rectify them, of the very law we have postulated. The most common observations teach us that if we bring two bodies into contact with each other, one of which is hot and the other cold, and if, moreover, we pinch these two bodies under such conditions that the foreign bodies do not seem to exert any appreciable influence on them, the cold body heats up and the hot one cools down; the volume of each of them varies, so that the state of the system of which they are composed does not remain constant. According to these observations, for an approximately isolated system to remain in an invariable state, it must be equally hot at all its points. This is true, moreover, only of a system that seems to us comparable to an isolated system. For example, a metal bar, one end of which is immersed in boiling water vapor and the other in melting ice, reaches a substantially invariable state; one end of this bar is, however, hotter than the other, but it is quite clear that we cannot assimilate this bar to an isolated system; that we cannot, in studying it, disregard ------------------------------------------------------------------------ If one \ouhn( ineor|ioref this bar had an isolated system, at the very least it would be necessary to include in this system boiling water, and the "melting ice but, then, such a system would not keep an invariable standard any more. (the various obsen alions, due to the direct use of our senses, are sufficient to suggest the following statement: For //ii' an isolated system, whose parts are continuous, keeps an invariable state, it is necessary that all the material parts which compose this system are also hot. This law, as we have said, obliges us, in certain cases, to correct the data of our sensations. We sometimes encounter an assembly of bodies which seems to us to be comparable to an isolated system, whose state does not undergo any appreciable change from one moment to the next, and whose various parts, however, do not seem to us (''cheerfully hot. If we touch, for example, a piece of wood and a piece of steel in flow with each other, which do not seem to us to experience any noticeable influence of the neighboring bodies, and whose state seems to us invariable, it often happens that we find the steel much colder than the wood. We continue, however, to assert that they are equally hot, placing less confidence in the immediate data of our sensations than in the preceding proposition. If, now, we appeal to the notion of equilibrium, which has been so delineated, let us make use, as we have the right to do, of a temperature whose determination is conceived in an abstract way, but not yet realized in a concrete way, we will be able, from the preceding proposition, to derive this one, which is one of the fundamental o.vrwLrrs of the Knerg'étique l'nur qu'un système isolé, dont toutes les parties sont conJiaues. soit en équilibre, il faut que la température ait la ~~tzze walenrw.zi tott.s se.s jloi~zt.)'. (fue la tempél'atun' ait la \nème râleur en tous ses points. (lotte law leads to the solution of the Thennometiric problem in a case, particular certainly, but nevertheless quite extended by hypotheses that we will not examine here. one manages to extend the solution to certain other cases. ------------------------------------------------------------------------ We will not ".first expose this solution by reslanl in- I.- domain lh"'""nt[iie we will speak, for the moment, only d-1 bodies and schematic instruments, characterized by abslrujl.es properties just now., we will see how, or can pass from this theoretical domain to the practical domain, and treatj- of real instruments and concrete bodies. Let's suppose, first of all, that we have a certain system T, the thermometer, characterized by the following properties i° To each value of the temperature corresponds, for the isolated system T, one and only one equilibrium state such that the system 1 has, in all its points, the temperature 0. Among the properties of the system T in equilibrium which are susceptible of measurement, there is at least one which is observable with a sufficient precision and which is represented by a quantity 0, increasing function, of the temperature 0. It results then from what we said that. to locate the temperature of the isolated svslème T, in each one of its standards-- of equilibrium, we will be able to make choice of the number (-> which dies the considered property this property will be said the jjropiiêté thcriiioinétriquc. of the thermometer 'I'. cl W will be named the lA'm/x'ralurr Luc on the I herniontctre T. Let us consider now, one after the other, a number of isolated systems L, L', and let us suppose that each of these systems is in equilibrium; the temperature is therefore the same at all points of each of these systems; it can, however, be the same for several systems. PomoiiHinu- know the values (-). (-)', <̃)", of the temperature, read on the thermometer rl which must locate the heat intensities of these various svslèmes This will be possible if each of these >v-,|ènu."> has the properties that we will describe for one of them. î" The system L is partly limited by an invariable Mirtaee which is. the mold in hollow of a certain surface 1. while the ihcri.iomèlre T is partly limited by another invariable surface which is the mold in relief of the same surface ï. ------------------------------------------------------------------------ ̃> By making these two surfaces coincident, we can associate the two systems ï and U in such a way that their together form an isolated system S, and the two parts T, L of the system S are independent of each other (§ 10, p. fia). 3" When the system S is in equilibrium, the two parts T, L are, as we already know, brought to a uniform temperature which is the same for both. Moreover, the two systems T, U are then in states which would be equilibrium states for each of them taken in isolation. 4" Conversely, let us consider an equilibrium state of the isolated systemT and an equilibrium state of the isolated system L. and let us suppose that the uniform temperature of each of these two systems is the same for both of them; by associating these two systems in the above-mentioned way, so that within the system S, each of the two systems ï, l is in the considered state, we obtain an equilibrium state of the isolated system S. These last two suppositions would not make sense if we had not previously admitted that the two systems T, U form, within the system S, partial systems independent of each other; they are often expressed in a less precise way by saying that the i-eiitzioiz of the system U and of the thermometer T, when they are at the same temperature, does not disturb the state of equilibrium of each of them. Henceforth, let us suppose given a state of equilibrium of the system U isolated, and let us propose to determine what is the temperature of this system. Let us seek by trial and error the temperature ai, to which it is necessary and sufficient to bring !<- thermometer T so that the latter, isolated in equilibrium at this temperature!, may, by association with the system l in equilibrium, form a system S also in equilibrium this temperature, the value 0 of which we shall know as read on the thermometer T, will be precisely equal to the temperature of the system L when it was isolated in equilibrium the problem posed will thus be solved. Most often, we operate in such a way that the problem solved differs a little from the previous one: We associate the system T and the system U and, when the system S that they form together is in equilibrium, we observe the ------------------------------------------------------------------------ value that it is convenient to attribute to the temperature of the system l taken within the system S in equilibrium. But, according to the leaked assumptions, one can separate system L from system 1 by keeping it exactly in the state it was in system S, and one obtains an equilibrium state of the isolated system U in this state, I system U has the temperature (-) just determined. The temperature of the isolated system is not obtained in this way in a given state of equilibrium, but an equilibrium state is obtained for the isolated system l.i. which corresponds to a known value of the temperature. The preceding considerations describe a method for solving, at least in a particular case of a certain extent, the problem of Ïbermomelria; but the solution they propose is still purely abstract, to practice it, it will be necessary to make real bodies correspond to the system T, to the systems l U', U", From then on, this new question arises How will we recognize that the real bodies whose correspondence we establish with the system T, with the systems U, U', l have concrete properties approximately represented by the abstract properties of the system T, of the systems l U', l. .? This operation, like all those whose aim is to appreciate the degree of resemblance of a certain concrete object with one of the elements, mathematically defined, of the energetic system, is not exclusively governed by the rules of active Logic; modes of reasoning, half unconscious, difficult or even impossible to analyze, and which can be classified under the name of experimental allions, are employed at every moment. Let's suppose, for example, that the concrete apparatus of which the system T must provide the abstract scleme is this instrument which we call a mercury thermometer; to consider it as absolutely isolated in space and, at the same time, to make its temperature vary, here is a perfectly conceivable abstract operation, but it seems difficult to imagine a concrete operation which corresponds to it; how then will we recognize that our mercury thermometer possesses, at least approximately, the characteristic properties of a system T? iWe will take a number of bodies: a mixture of water ------------------------------------------------------------------------ <-! of ice, of water more or less hot, etc., which our senses noii- allow to arrange in a kind where each term is sùremenl hotter than (e preceding. We will admit that each of these concrete bodies behaves, at least roughly, in relation to the mercury thermometer, as each of the systems behaves in relation to the system T: in other words, we will admit that the mercury thermometer, joined to each of these bodies in such a way as to form a connected system which seems to be more or less isolated, will be in equilibrium in the same way as if it were isolated and brought to the temperature of this body. We will thus have the moven to know how our mercury thermometer would behave if it were isolated and if it took successively your equilibrium stages which correspond to increasingly higher temperatures. Although ([ne roughly approached, since our senses alone served to appreciate the equality or the inequality of temperature between the> bodies to which we successively associated the thermometer, this study allows us to recognize that the instrument possesses approximately these two properties: i° The level of mercury in the thermometer stem depends only on the temperature to which the instrument is brought. The level of mercury always rises with the temperature. This same study, made on a water thermometer, would have allowed us to recognize that the level of the water dropped when the instrument was taken out of the melting ice and plunged into water a little less cold, and that it rose again when the instrument was put into warm water: this observation would have led us to refuse the character of thermomelographic property to the height of the liquid in the stem of a water thermometer. After the simple use of our senses has made us recognize, in a roughly approximate way, that the mercury thermometer possesses the two properties previously stated, we admit that it possesses them with a rigor far superior to that which our senses can recognize; then, we are going to make use of these properties of the mercury thermometer to rectify the perceptions that our senses provide us with or to acquire knowledge that exceeds their range. ------------------------------------------------------------------------ Thus, noting that in the stem of a thermometer surrounded by melting ice, the level of mercury remains sensddemenl the same in all circumstances and throughout the duration of the ludion, we will conclude that the melting temperature of the ice e-,t a constant. We will be able to recognize in this way that a certain number of bodies (that we suppose, moreover, to behave with respect to the mercury thermometer as the systems l l t behave with respect to the system 1 are bodies of fixed temperature. By means of two of these fixed temperatures, we will realize the mercury centigrade thermometer. It is by basing ourselves on this property, revealed by the vulgar experiment To each temperature corresponds one and only one level of mercury in the stem of the thermometer, that we established this law The melting ice is. carried to an invariable temperature. But once this law has been obtained, we consider it more certain than the property of the thermometer which made us discover it: we do not use this law to examine to what point the thermometer has the property of which it is about. We find that the mercury does not return to the same level each time the thermometer is immersed in melting ice, and we conclude not that the melting point of the ice is variable, but that the level of the mercury does not depend only on the temperature to which the thermometer is brought, but also on other circumstances. It is, moreover, by londant us on the lixiîé of certain temperatures, that of the melting ice, for example, lisilé which we, was ré\ élée by thermometers alleelés of the definition of which we have just spoken, that we will manage to disentangle the laws of this tléplnn1-ment, of the point o", and to recognize the effectiveness of the process-, which have for object to alfranchir the thermometers. More perfect thermometers will then allow us to recognize that the temperature of a mixture of water and ice is not absolutely fixed and to imagine movens suitable to increase its fixity, which will lead us again to perfect the mercury thermometer, and so on. We therefore follow a method of successive experimental approximations in which, by virtue of an intuition that- the ------------------------------------------------------------------------ Pure logic would be very embarrassed to justify, each new law is, regarded as more exact than the already known law which was used to establish 1 this method leads the physicist to carry out a concrete thermometer more and more similar to its abstract ideal, the system T. If we are not in possession of such a thermometer, each use that we want to make of it will have to be preceded by new hypotheses. In particular, we will have to suppose that the concrete system whose temperature we want to take is approximately, with respect to our real mercury thermometer, what one of the systems L, l l is with respect to the system T. If, for example, we want to use the mercury thermometer to take the melting point of phosphorus, that is to say the temperature of a system in equilibrium formed by solid and liquid phosphorus, we will have to admit that the contact of the thermometer does not disturb this equilibrium, does not change the melting point of phosphorus. If we want to determine the temperature of a liquid subjected to a certain pressure by immersing the reservoir of the mercury thermometer in it, we will have to examine the following question: can the liquid and the thermometer be assimilated to two independent systems joined by a rigid surface? Should we not, on the contrary, take into account the fact that their surface of contact is deformable; that, consequently, these two bodies are not two parts, independent one from the other, of the system that forms their whole? Should we not, therefore, consider the pressure exerted by the liquid on the thermometer's reservoir, a pressure which results from this dependence? To answer this question, we can and must call upon complicated and distant branches of Energetics, such as the theory of elasticity. We may still have this doubt: can the system formed by the thermometer and the body whose temperature we want to take be assimilated to an isolated system, a system which cannot be in equilibrium unless it is brought to the same temperature at all its points? Should we not, on the contrary, take into account the contact of the thermometer's stem with the cold air, a contact by which the temperature of the mercury contained in the stem is not equal to the temperature of the mercury contained in the tank? ------------------------------------------------------------------------ In these two circumstances, the physicist will say that he has corrections to the indications of his thermometer, corrections intended to eliminate causes of error. What does he mean by these words? He means that by assimilating his thermometer and the body whose temperature he wishes to take to the abstract systems T and I that we have defined, he would give a schematic representation of reality that is too simplified, and that by reasoning on this unajt: too simple, he would soon obtain corollaries whose disagreement with the data of experience would be very apparent; he then represents the concrete whole of the thermometer and of the body whose temperature he wishes to take by a new scheme, more complicated than the preceding one, where the body and the thermometer are no longer represented by two independent systems, where their whole is no longer assimilated to an isolated system in space thanks to the more numerous resources that this greater complication provides him with, he endeavors to constitute an image that closes more closely to reality. Considerations similar to those we have just developed concerning the determination of temperature could be taken up again each time we propose, by means of an instrument, to substitute a numerical symbol for a certain physical, qualitative or quantitative property; this substitution always results from very complex mental operations, very difficult to analyze, and which can never be reduced to the sole rules of deductive Logic. ------------------------------------------------------------------------ CHAPTER II. The principle of conservation of energy. I. The Work and the Energy. The principle of Conservation of Energy. We know (p. 4) that the principles of Energetics are not susceptible of any a priori demonstration; provided that we avoid any contradiction between the terms which serve to state a principle, and also any contradiction between this principle and its congeners, we are free to denounce!If the set of principles, thus arbitrarily stated, gives consequences which represent the experimental laws with sufficient precision, our principles are good and must be kept; if not, at least one of them must be rejected or modified. If we were to take into account the requirements of Logic, we could formulate the principle of the Conservation of Energy as a postulate of our good plan, referring the reader who wishes to be taught about the value of this postulate to the applications which mark the concordance of the energetic system with the data of observation. The requirements of pure Logic are not the only rules that can reasonably direct our judgments; pure Logic leaves us free to choose the postulates of Energetics as we please; it does not follow that we should choose them at random, which would mean that if the consequences of these randomly chosen principles were to agree with the facts, as we have recalled in the Introduction, the formulas that we propose for the principles of Thermodynamics would be in agreement with the facts. ------------------------------------------------------------------------ found today very minutely lixées. thanks to innumerable tests, has trial and error infinitely varied- which "c are continued during back -léele-. The best way to explain the principle of energy here would be to retrace, at least in broad outline, the history of the trials and tribulations by which the human mind has succeeded in giving this principle its present form. If summarized as we suppose it, such a historical exposition, which would have to be renewed for each principle, would exceed the limits of this treatise. One has custom, then. to resort to another form of justification ( ) one seeks to lead the spirit, in a gradual way, to the statement of the principle which one wants to propose to him, by presenting to him successively various simpler proposals and whose acceptance seems to him more natural and easier one morcelle. Cil quelque sorte. Ni\hèpolè-.e that it would hesitate to admit all of a block, so that it had then-e more easily seize the various fragments by a successive comprehension. Such a method is d'.iulanl better than it has more analogy with the historical method (pu; the various preliminary judgments of which. successively, one proposes the acceptance to the content, resembleenl more to the various facets of the total truth that the iui maiii li' has successively glimpsed: that thus the steps made by the reason (the student are, in short. more similar to the way .followed by the collective reason of Hiuinanité. But in the course of a preparation for the acceptance of a principle there is a very serious danger of which it is necessary to guard against with the greatest care, an error which it is necessary to dread above all your t this error would consist in taking the introduction to a hvpollie.se for the demonstration of a M;nlé to confuse the gradual step which leads us step by step to the principle which we want to formulate with a deduction which. of theorem", in theorems, assures us (') I/e\|>oio cl 11 principle of ia Cou^matiori of I Km'i'gie that flotum!<̃ this Chapter is only lo dc\ck>|>prmenl *l(* (Oliii cjur us ;tvi the written -niwuit Commentary aii.r jiriiici/ir\ ilv l. He (Journal de Mathématiques. '|* série, t. Vlil, iScc. p. >St) cl Mm ------------------------------------------------------------------------ of the certainty of a conclusion; to believe that your preliminary propositions which prepare the definitive statement of a principle are self-evident axioms and certain by the common knowledge. Let us therefore affirm from the outset that the considerations whose development we are going to read are a simple preparation for the statement of the principle of the Conservation of Energy that they prove nothing either for or against the truth of this principle. We can, by our efforts, produce in a system a certain transformation or help this transformation; we can move a body, throw it with a certain speed, break it, deform it, and crush it by the mutual friction of its various parts. We can, on the contrary, use our efforts to put an obstacle in the way of the transformation that a system undergoes, to hinder this transformation; we can stop or slow down a body in motion, prevent it from deforming. We then say that we have done a certain work, that we have accomplished a certain work. Although often in an obscure and imprecise way, we recognize in the most diverse works that man's muscular strength can accomplish a character that allows us to classify these works in order of increasing value. Two works of different nature may seem to us to be equivalent and deserve the same salary to the two workers who accomplished them; or, on the contrary, one of them seems to us to be worth more than the other and to deserve a higher salary. Everyday experience teaches us that we can substitute a machine for our own action, for that of our fellow men, for that of animals, that is to say, an assembly of inanimate bodies capable of producing or helping the modification that we produce or that we help, of hindering the modification that we hinder. The wind or water mill crushes the grain that in the past the woman or the slave would have crushed in a mortar; that, later, the beast of burden would have crushed by turning the millstone. Gunpowder throws a stone as a man would have thrown it with a slingshot; only it throws a bigger stone, and farther. One of the practical objects, and the principal one, of Mechanics and Physics has been precisely, from the origin of these sciences, to recognize which are the various bodies that can be substituted for the others. ------------------------------------------------------------------------ The science of mechanics has been called the Science of Mechanics since the beginning of the 20th century. It is the science of mechanics that has been called the Science of Mechanics, and it is the science of mechanics that has been called the Science of Mechanics. Mechanics was first called the Science of Mechanics. The work that we would have accomplished if we had acted ourselves on the system that is being transformed, we consider as accomplished by the body or by the assembly of bodies that we have substituted for ourselves or for our fellow men. If two machines accomplish different works, but which would have deserved the same wages to the workers in charge of executing them, these two machines appear to us as producing equivalent works, one of them, on the contrary, seems to us to accomplish a work which is worth more than the work of the other, if the sum of the wages which it saves us is higher. This notion of work accomplished by bodies foreign to a system while this system is undergoing a certain modification, we transport it even to the case where the modification undergone by the system is of such a nature that neither our personal action nor that of our fellow men can either help or hinder it; we transport it, for example, to the generator of electricity which decomposes sodium chloride. The work accomplished by these foreign bodies is supposed to represent the work that would be accomplished by an operator constituted differently from us and capable of bringing to the considered modification the help or the hindrance that the foreign bodies bring. Thus, when a system is transformed in the presence of foreign bodies, we consider these foreign bodies as contributing to this transformation either by causing it, helping it or hindering it; it is this contribution that we call V work accomplished, in a transformation of a system, by the bodies foreign to this system; two different works can have the same value, or one can be worth more than the other. These notions of a work accomplished by bodies foreign to a system, in a transformation of this system, and of the value of such a work are, up to now, for us, very obscure, very vague, and above all very deeply impregnated with anthropomorphism. I.).- ï. J 6 ------------------------------------------------------------------------ To penetrate them, to specify them, to free them from this anthropomorphism is not the competence of Physics, it is the object of the ell'orls of Metaphysics, whose various Leoles discuss and solve in different senses this problem of the communication of corporeal substances. The goal that Physics proposes to reach is quite different : it proposes to represent the value of the various works by a numerical symbol, in such a way that by treating this symbol according to the rules of Algebra, it can give precise answers to the following questions Since a machine is capable of producing a certain modification in a certain system, what other machines can be substituted for it to accomplish the same modification within the same system? A machine being able to produce, in a certain system, a certain modification, what modification can it produce in another system? To create a mathematical symbol suitable to represent the value of a work, we will form an expression which satisfies certain conditions which we will agree to impose to it; these conventions, we will not establish them randomly we will choose them in such a way that they offer the image of the simplest and most salient characters which the notion of work presents or, at least, that they agree without difficulty with these characters. All the preceding considerations dealt with the concrete data of our perceptions and the non-scientific notions of which they provoke the formation in our mind : our language had thus necessarily the imprecision of any speech intended to express the common knowledge. If, for example, we spoke of a material system and of bodies foreign to this system, we were speaking of them in the sense in which everyone understands these words. Henceforth, we shall deal with mathematical symbols; we shall therefore, in our reasoning, attach ourselves to precision and rigor; we shall use the terms we shall use only in the sense in which they have been defined. In particular, a material system will always be, unless otherwise indicated, a system independent of the foreign bodies that act upon it, these words having the meaning that has been defined previously (Chap. I, § 10). We will say that a modification of a system is determined ------------------------------------------------------------------------ when we know the initial instant l0 and the final instant lf of this modification, and when we know, for each instant. from the instant tQ included, to the instant t{ included, what is the state and what is the general motion of the system. 11 would not always be enough, to determine a modification, to say what is, at each instant, the state of the system the knowledge of the movement of this system could very well not result from it. Suppose, for example, that at each instant of a given period of time, the electrical distribution on a conductor is known, the currents that pass through this conductor will not, for that reason, be entirely determined; if a first system of currents compatible with the variation imposed on the electrical distribution is known, an infinite number of analogous systems will be known in order to obtain one; in fact, it will be sufficient to superimpose on the first system any system of uniform fluxes (Chapter I, §9, p. ï>~), the latter being able to vary from one moment to the next in an arbitrary manner. Let us suppose that we give ourselves a series of states of the system, determined by means of a variable t that we continue to nornrn time; to each value of t, included between t0 and tt, corresponds a state of the system and only one this state varies in a continuous way when t increases from to and that in such a way that all the links imposed on the system remain unceasingly verified; finally, at each instant l, the set of general velocities is determined; these velocities vary in a continuous way with l or present discontinuities for isolated values of L and in finite number. We have then a determined modification to the meaning which has just been attributed to this word. Can this change be considered a real change? If we were to stick to the grammatical meaning of the words, the answer would be an emphatic no. What we have just defined, in fact, is a sequence of stalls and movements of a mathematical system and not of a set of concrete bodies. If there is reason to ask the question that we have just formulated, it is that it must be understood in the following sense i'eul-on meet a real modification of a concrete system which is ------------------------------------------------------------------------ Is it approximately represented by the abstract modification (the mathematical scheme as we have just determined it? If yes, we will agree to give to this abstract modification the name of real modification. The meaning of our question being thus clarified, let us see what answer we should make. From what we have said about the choice of quantities suitable for determining the state and motion of a system, we are sure that all real modifications are to be found among the modifications determined as we have just indicated. But the reciprocal of this proposition is neither obvious nor certain. A modification of a system being determined in this way, it is not certain that we can place the system in the presence of foreign bodies such that this modification is realized. Henceforth, when we have determined a modification as we have just indicated, we will say that we have imposed an ideal modification on the system; the modifications likely to become real are surely included among the ideal modifications. One of the main problems of Energetics consists precisely in this question An ideal modification of a certain system being given, can we place this system in the presence of foreign bodies such that this modification is realized? Our object will be, in the course of this treatise, to lay down rules which will allow us to answer this question. For the symbol that we propose to create under the name of accomplished work, in a modification imposed on the system, by the bodies foreign to this system, to be able to appear in the statement of such rules, it is obviously necessary that this symbol is defined not only for the real modifications, but also for the ideal modifications. Only on this condition will it be possible to state rules of this form For such an ideal modification to be realizable under such conditions, it is necessary that (' ) An ideal modification must not be confused with a virtual modification; a virtual modification is composed of states of the system which do not follow one another in time; so that the change of state which constitutes a virtual modification is not linked to a movement; in the virtual modification, the notion of speed has no place. ------------------------------------------------------------------------ the corresponding work presents such a character. Our conventions will have (Jonc to relate not only to the real modifications, but also to the ideal modifications. An ideal or real modification can be instantaneous. If, in fact, the state of a system is subject to vary always in a continuous way with l, it is not the same for the general motion the general velocities can, for isolated values of t, present discontinuities at the instant when similar discontinuities occur in the system, this one is the seat of an instantaneous modification. The shock is an instantaneous change. An instantaneous modification is determined if we know the state of the system at the instant when it undergoes this modification, its general motion immediately before, and its general motion immediately after. Let us consider a modification M, real or ideal, accomplished between the instant and the instant V. Let us suppose that at the instant T, between x and t', the general motion of the system does not experience any discontinuity. Let us call, at this instant, e the state of the system and {/. its general motion. We can say that the modification M consists of a first modification M, accomplished between the instants t and T, which brings the system to the state e and to the motion ja, followed by a modification Mo, accomplished between the instants T and which takes the system in the state e and with the motion p. that it had at the end of the modification M,. It would no longer be possible to say the same if, at time T, the motion of the system underwent a sudden change. Let e be an infinitesimally small duration. Let us call the motion of the system at time T - set ul' its motion at time T + s'. The change can be decomposed as follows 1" A modification M|, accomplished between the instants c and T, which brings the system to the state e and to the motion the second lasts from instant t.o +- to instant tt -~r t 2" Let t be any instant of the time span between tu and t{ and let t' = t -+- t. At time t, in the modification M, the system has a state e and a motion u; at time t', in the modification .M', the system has a state e' and a motion p. These states and motions are related to the same absolutely fixed trihedron T. If to the absolutely fixed trihedron T, one substituted another absolutely fixed trihedron T', whose position with respect to the first one is independent of t, the state e' would be transformed into a state identical to the state e, the motion p.' would be transformed into a motion identical to y.. We will express this last character by saying that at each instant t' of the modification M', the state e' and the motion jjt/ of the system are deduced from the state e and the motion p. at the corresponding instant t of the modification M by a displacement cl 'ensemble in space, this displacement remaining the same during all the duration of each of the modifications. We will say that the two modifications M, M' are the same modification, acconiplied at different times of the duration and in different places of the space. Let us suppose that at a certain time and in a certain place, a system experiences a modification in the presence of certain foreign bodies; let us suppose that at another time and in another place, the same system experiences the same modification in the presence of other foreign bodies; a quantity would certainly not seem to us to be capable of providing a mathematical symbol of the notion of work accomplished by the foreign bodies if it did not, in these two circumstances, take on the same value; we are thus naturally led to lay down the following convention First convention. - The mathematical symbol intended to represent the value of the work accomplished, in a modi/ica- ------------------------------------------------------------------------ The real or ideal lion of a system, by the bodies foreign to this system, will be determined all the times that one knows the nature of the system and the modification that it underwent; it will not change if one limits oneself to changing the time and the place where the modification was produced as well as the foreign bodies in whose presence it was accomplished. We are also led to require of the symbol we want to construct that it represents as equivalent the works accomplished by the foreign bodies in two modifications of the same system if these two modifications take the system in the same state, animated by the same motion, and if, however different they may be, they both leave it in the same state and animated by the same motion. Hence this second convention Second convention. The mathematical symbol which '̃ represents the value of the work accomplished by foreign bodies during a real or ideal modification of a system, is determined when the state and motion of said system at the beginning of the modification, the state and motion of the system at the end of the modification are known. This agreement has one consequence The value of the work accomplished in a real or ideal modification that brings the system back to its initial state and its initial movement must be represented by the same symbol -.{that the work accomplished in a system whose state and movement do not vary, not it is natural to represent this last Work by the symbol o. The following conventions are more arbitrary in character; they are dictated by the desire to endow our symbol with the most usual, simplest and easiest to handle algebraic properties. Third convention. - The mathematical symbol that represents the notion of work belongs to a category of mathematical notions capable of a commutative and associative operation, named addition and represented by the sign -+-. ------------------------------------------------------------------------ Fourth convention. - Let us assume that a given system has undergone, in various circumstances, real or ideal modifications, M, M2, M, during which bodies foreign to the system have performed works represented by the symbols G,, G2, G, Let us suppose, moreover, that we can arrange these modifications in a certain order {for example, the order M,, M2, M,,) in such a way that the final state and the final motion of the system in any one of these modifications are respectively identical to the initial state and to the initial motion in the next modification. One can then imagine that the system undergoes a single modification [/ real or ideal, which consists of the modification M(, followed by the modification M2, followed by the modification Mn. Soity the symbol of the work done by the foreign bodies during the modification. We agree to take for y the sum of the symbols G,, G2, G, Let's bring this convention closer to our second convention. Let us suppose that the successive modifications M,, M2, M" bring the system back to its initial state and to its initial motion; we must have Gj -h G2 -+-+ G" = o. This equality immediately teaches us that the symbols G|, Go, G" cannot be simple arithmetic numbers without a sign; but it does not prevent us from taking an algebraic number with a sign to represent the work accomplished, during a modification of a system, by bodies foreign to this system. Hence this new convention Fifth CONVENTION. The work accomplished, during a real or ideal modification of a system, by the foreign bodies of this system will be represented by an algebraic number affected of sign. This convention leads to the following consequence The modifications of which a system is susceptible can be classified in two categories. Any modification of the first category is considered as corresponding to a production work ------------------------------------------------------------------------ the corresponding work is represented by a positive number. Any modification of the second category is considered as corresponding to a work of destruction; this work of destruction is represented by a negative number. It goes without saying that once the works of various kinds have been classified into two categories, intended to be represented by numbers of opposite signs, we remain free, until further notice, to choose as we see fit the category which is supposed to be composed of works of production and which will be represented by positive numbers. Two real or ideal modifications of the same system are said to be opposite if the initial state and the initial motion of the system in one of these modifications are respectively identical to the final state and the final motion in the other, and vice versa. If M, M' are two opposite modifications of a system, we can look at the succession of the modification M and the modification M' as a unique modification which brings the system back to its initial state and to its initial movement. Let us designate by G, G' the values of the works accomplished in these two modifications M, M' we should have G -4- G' = o. The works accomplished in two opposite modifications of the same system are represented by equal numbers of opposite signs. This algebraic proposition is a fair translation of that proposition, stated in ordinary language and which follows naturally from the notion of work The work produced in a certain modification is destroyed in the opposite modification. From now on, it will be useless for us to distinguish, in our language, a work from the algebraic number^ which symbolically represents that notion; to the number we will give the same name, work, as to the notion of which it is the image. Suppose that a real or ideal modification makes a given system go from an initial state e, and an initial motion a, to ------------------------------------------------------------------------ a final state e-> and a final motion u2. According to our second convention, the algebraic value of the work accomplished, during this modification, by the bodies foreign to the system is entirely determined by the knowledge of the two states e,, e> and of the two movements p. a2. We can represent this value by the symbol G(ei, i^i es, \x.2). ). We know that two opposite modifications correspond to equal works of opposite signs; this proposition is now expressed by the equality G(e2, Lt2; ) G(e0![J-a) e, |jl) = Y.(e. \x) ------------------------------------------------------------------------ and we will say that E (V, y.) is the value of V total energy, of the system taken in the e-standard and animated by the motion y.. Our equality (i ) allows us to write G(ê'u. p.o; e,, jj.,) = O(eoi H-o - ei> !J-i) G(ei, i-1-! ̃ s", ;J-2 ) or, by virtue of equality (2), (3) G(e1, rj., e2, ij..2) = l£(e2, jj.2) - J^ ( e; jjli). The work accomplished, in any real or ideal modification of a system, by the bodies foreign to this system is equal to the excess of the final value that the total energy of the system takes on the initial value of the same quantity. The value of the total energy corresponding to a given state e and a given motion |j. depends on the choice of the normal state and the normal motion, that is to say E (e, u) this value when we take for normal state the state e0 and for normal motion the motion (u.o) orE'(e, p.) the new value of the energy when we take for normal state the state e'o and for normal motion the motion [/". Equality (1) gives us G(e0, [x0; e, \x) = G(e0, |x0; e'o, \x.'a ) -+- G(e'o, e, ix) or, by virtue of equality (2), K(e, p) - E'(e, jj.) = G(e0, JJ-o ei, ni)- U/i change of normal state and normal motion changes the value of the energy relative to any state e and to any motion ;jl. The difference between the two values taken, before and after this change, by this energy depends neither on the state e nor on the motion [jl. This proposition can be expressed by saying that the total energy of a system is determined to within one constant. Here is now a new convention that it seems quite natural to introduce Sixth convention. - If a modification, real or ideal, causes an infinitely small variation of the quantities which determine the state and the general movement of a system, ------------------------------------------------------------------------ the work done, during this modification, by the bodies foreign to the system is infinitely small. If we apply equality (3) to the case where the state e-> is infinitely close to the state e, and where the motion jjl2 is infinitely close to a, and if we take into account the previous convention, we obtain the following corollary The total energy of a system varies continuously when the quantities which determine the state and the general motion of the system vary continuously. This is another of the characteristics that we intuitively attribute to the notion of work that the following convention seeks to express Seventh convention. - Let £ be a system which consists of several independent parts S, S', infinitely distant from each other. Any modification 311, real or ideal, of the system 2 results from the simultaneous production of a modification M of the system S, a" a modification M' of the system S', The work done in the modification 3ÏL of the system 2 is the sum of the work that would be done in the modification M of the system S, of the work done in the modification M' of the system S', Let e, t/. any state and motion of the system S; e0, |o.0 its normal state and motion; E(e, sx) its total energy. Let us adopt analogous notations for the systems S', When the systems S, S', are respectively in the states e, e', and animated by the motions jj., u', the system S is in a certain state s and is animated by a certain motion m; similarly, the states e0, e'B, and the motions u0, u.'(J, of the systems S, S', determine a certain state s0 and a certain motion m0 of the system S. Soi.tF (g0, /n0 s, m) the work done by the bodies foreign to the system S, in a modification where this system passes from the state s0 ------------------------------------------------------------------------ and motion m0 to the state s and motion m. By virtue of the previous convention and the definition of the quantities E e, ai. ). VJ (; £, ni) = E(e, u) -+- E'(e', ;j.' } -h. Now let us suppose that we take for normal state of the system S the state e0 and for normal motion of the same system the motion m0 we will have r(e0, mo; E, m) = C(e, m), C(e, m) being the total energy of system 2. The previous equality will thus become £(£, m) = E(e, fi) + E'(e', [!̃') + ---- If a system S is composed of several independent parts S, S', infinitely distant from each other, the total energy of the system 2 is the sum of the total energies of the systems S, S', provided that the system S is considered to be in its normal state and animated by its normal motion when each of the systems S, S', is in its normal state and animated by its normal motion. In the case where a material system is isolated in space, where, therefore, there are no foreign bodies in this system, we cannot attribute any other value than o to the work accomplished, during any modification of the system, by the foreign bodies. If one observes, moreover, that the œu\rc is given by equality (3), one is led to formulate the following hypothesis PlU-NCIPI-: OF THE GoASEKVATlOi* OF I. E.NKUGi K. jLoi'SqU Any system, isolated in space, experiences any real change, the total energy of said system keeps an invariable value. This proposal is different from all those which we have stated in what precedes. These, in fact, had an arbitrary character which we underlined by naming them conventions, and not hypotheses. In the final analysis, we are ------------------------------------------------------------------------ well free to consider an expression of the form GiC], u, eS) \>-i ) = K(e2, \u ) - V.( e\, ;j.i ) and to give it the name we wish to give it, for example the name tf work accomplished by bodies foreign to the system, during a real or ideal modification of this system. But when we state that there is a quantity, depending on the state and the motion of the system, which remains invariable in any real modification accomplished within the isolated system, we formulate a proposition that we are no longer free to admit or reject at our whim; If the disagreement were to break out between one of these consequences and an experimental law, we would have to reject either the hypothesis we have just stated, or one of the hypotheses which, combined with this one, have provided the consequence denied by the facts. 2. First restriction Exclusion of systems analogous to Mac Cullagh's ether. We have just stated the principle of the Conservation of Energy, giving it the greatest possible generality; in reality, in the exposition of Energetics which follows, we do not keep this great extension; we will indicate by what restrictions we are led to diminish this extension. We have seen (p. 86) what is meant by saying that the state and motion of system S' are derived from the state and motion of system S by a simple displacement in space. We will say that "any modification, real or ideal, is reduced to a simple displacement of the system in space, if the state and the motion of the system at the end of this modification are derived, by simple displacement, from the state and the motion at the beginning of the modification. We shall examine what we can say about the work done by bodies foreign to the system in a modification that is reduced to a simple displacement in space. ------------------------------------------------------------------------ We propose this theorem If a modification, real or ideal, of a system is reduced to a simple displacement of the system in space, V work accomplished, in this modification, by the bodies foreign to the system depends on the displacement experienced, but does not depend on the initial state of the system nor on its initial motion. Let e,, e2 be any two states u. a2 any two movements of the system. Let i, ;/̃;). But our first agreement involves legality G('"i; ;j.i [x, > - ("-( c\, \x[ t>i, ;ji', ). We will thus have the equality G ( c i ;ji i e >. ) ̃̃- G ( e y." e' \j. '" ) which is equivalent to the stated tliôorômc. From this proposition we deduce a corollary concerning the form of the total energy of a system. Let eo, uo the normal state and normal motion of the system e, a any state and motion of this system ------------------------------------------------------------------------ eu, e', u!u, jj.' the states and movements which are deduced from the states e0, e and the movements u0, a, for a certain overall displacement. The previous equality will give us G Ce, u; e', \i' ) - G(e0, 'M e'o, n'o ) or E(e'> I-O - E(e, n) = E(ei, n'o ). The second member depends on the displacement in space that we have considered, but not on the state e, (u of the system. We can therefore state this theorem By a simple displacement in space, the energy of the system experiences a variation which depends on the nature of this displacement, but which depends neither on the state of the system nor on its movement. To account for the laws of Optics, Mac Cullagh ( ) proposed to endow the various ethereal media with the following property To impose on a mass of ether a displacement in space which consists simply in a translation, one would have no work to perform; but to impose on it a displacement involving a rotation, one would have to perform a certain work. A similar hypothesis has been taken up by Padova(2) and by M. Reifl'(3), in order to explain electrical and optical phenomena. This hypothesis is not in contradiction with what precedes, provided that one is careful to safeguard this proposition The work accomplished when a determined rotation is imposed on a determined mass of ether does not depend on the state of this mass (') MAC Cullaoh, An Essay towavds a dynamical Tlteory of cryslalline reflection and refraction ( Transactions of the Jioyal Irish Academy, t. XXI, y Dec. 1839. - The Collected Works of James MAC Cullaoh, p. i45; Dublin and London, 1880). (' ) Padova, Una nuova interpretasione dei fenomeni elettrici, magnetici e luminosi (Nuovo Cimento, 3" série, t. XXIX, 1891, p. a-ij). (3) Rkiff, Elasticitiit und ElektricUât Freiburg-in-Urisgau und Leipzig, i8<)3. ------------------------------------------------------------------------ in particular, it is independent of the temperature and density of the ether. But the theories we have just discussed cannot accommodate such a restriction; it would result, in eH'et, that the optical or electrical properties of the various transparent or dielectric media would be independent of the density and temperature of these media, which is not. The theories of Mac Cullagh, Padova and M. Reiii could only be compatible with the principle of conservation of energy if we renounced our first convention, which seems very difficult to do. In general, the following postulate is accepted in Energetics, which we will adopt from now on Postulate. - No work is accomplished by bodies foreign to a system in a modification, real or ideal, that is reduced to a simple displacement of the system in space. In other words the total energy of a system does not change as a result of a simple displacement of this system in space. 3. Second restriction Exclusion of electrified systems. Potential energy and kinetic energy. We will now limit our research to systems for which the following restriction is verified Resthiction. - The work accomplished during a modification, real or ideal, of the system by the bodies foreign to this system is the sum of two terms The first term depends on the change of state of the system, but not on the change experienced by the motion; the second term depends on the change experienced by the local motion of the system and does not depend on either the change of state of the system or the change experienced by the general motion. Let us designate, in general, by e, jx, X the state, the general movement n. - I. c ------------------------------------------------------------------------ The previous restriction means that we can write an equality of the ton ( i Ci('i, ;j-i e", [j., 1 = \'(e\, e2 1 -f- O(Xi), e2 1 -f- O(Xi), e2 1 -f- O(Xi), e2 1 -f- O(Xi)). The previous restriction means that we can write an equality of the ton ( i Ci('i, ;j-i e", [j., 1 = \'(e\, e2 1 -f- O(Xi,Xii. Let e0 be the normal state of the system; as a general normal motion p.o., let us take a motion free of local motion, i.e. a state of rest of the system; let us agree to represent by K = 0 the absence of local motion, Let us posit jP("..ô = O(e). ( Q(oTX; =K().). Let us remember, moreover, that (1) G(e0, ;jli, E(e, ;j.) = {j{ej -+- K(À;. The total energy of a system is the sum of two terms. The first of these terms depends on the calibration of the system, but not on its local motion nor on its general motion. The second of these terms depends on the local motion of the system, but not on its rtal or general motion. The first term li {e) is called the internal or potential energy of the system in state r. The second term K(À) is called the kinetic or current energy of the system with the local motion À. From the above, we immediately know a number of properties of the internal energy U(V) and the potential energy K (À). /internal energy varies in a continuous manner when the te~andettrs that , k(u. "', w). The form of the function k does not depend on the state of the considered particle; for a given particle, which neither loses nor gains matter, this form remains invariable whatever modification ------------------------------------------------------------------------ that the particle experiences. It is this form that we have to determine. Let us consider a body C, of finite dimensions, and let us suppose that the various infinitely small parts c, c', c", which compose it are all animated by a velocity of the same magnitude and of the same direction; let u, c, w be the components of this velocity; we shall say that the body C is animated by a translational motion of which (m, v, w) is the velocity. The kinetic energy of the body C will have the value, according to what precedes, /-(m, v, ii-1) -t- k' (u, v, "') -+- />"(", v, "') -̃= X(;t, c, n1). J. So the kinetic energy of a body moving in translation is a function of the three components of the speed of this movement, a function that does not change its form as long as each of the infinitely small parts that make up the body remains formed of the same material. s. From the preceding equality, one sees that if one divides the system C into several systems C,, G^ that if one designates by X, (u, v, \v), N-2 (/ c, tri the respective kinetic energies of the systems C,, Ga, animated by the same translation speed, the kinetic energy (u, c, ir) of the system C animated by the same translation speed will be the sum of the kinetic energies of the systems C,, C2, (7) X(w, e, if) - \i(m, i\ <"-)̃ Xji'u, p, "<)- The form of the function \(u, v, "-) which is suitable for a given body does not change when the state of that body is modified in any way. We can therefore suppose that we distribute the matter which forms it in concentric spliteric layers, of finite or infinitely small thickness, in such a way that all the material elements crossed by the same sphere concentric to these layers are identical to each other. Let's take this body at rest, and let's animate it with a certain translation speed V in a certain direction D. The work accomplished in this real or ideal modification is equal to X(u, c, h1,), u, r, "- being the projections on the axes of the absolutely fixed trihedron T of the segment V carried in the direction D ------------------------------------------------------------------------ Let's take this body at rest and let's animate it with hi same translation speed but in another direction I)'. The work accomplished in this second modification, real or ideal, is equal to V(w', "' ir'), u! (- ir'elantles projections on the axes of the trihedron T of a segment equal to V carried in the direction D'. But one can obviously choose a fixed absolute trihedron T' with respect to which the straight line D' is oriented exactly as the straight line I) with respect to the trihedron T. fia second modification, brought to the trihedron T', will be identical to what is the first one, referred to the trihedron T. Therefore, both modifications, according to our first convention, correspond to the same work \(u, t', . The kinetic energy of a system varies continuously when the local velocities.1-; vary continuously; therefore, the function y(\) is a continuous function of By means of the two preceding propositions, it becomes easy to prove the following theorem: Thkouème. - Let S. S' two arbitrary systems; y < V ), y'(V) the kinetic energies of these two systems animated of the same speed of translation the ratio ,y depends on the nature of the two systems S and S', but not on the speed V. Let us assume, in ell'et, the ratio o(\ ) = variable with Let us assume, in î-IleL, the /t\) ( as Jes two terms y (i. y'(\) it will be a continuous function of V; consequently, for values of as close to each other as we want, it will pass through commensurable values. So let Vo be a value of for which n .il u' i > " ) a n and n' being two numbers inlier>. Let us take a system C formed by the union, operated in any way, of /( sv' ) '̃-̃ N - 7/ v" ) = °- ------------------------------------------------------------------------ l) 'according to the previous convention, this equality entails, that) whether e is the equality v(V) = v'(V) which can be written /V) V") 'n or p(V) = p(V0). Therefore, assuming that p(V) varies with V one would be led to conclude that p(V) is independent of V. This contradiction demonstrates the stated theorem. Let us take a given body;V which we will name the standard body; let us designate by V (Y) the kinetic energy of this body animated by the translation speed;oThis body has a kinetic energy x(~)- Let's put (8) z(V) = >ir(V). With respect to the number M introduced in this equality, the above allows us to assert the following propositions: i The number M does not depend on the speed V. It is independent of the position, of the figure, of the state of each of the two bodies C and A; provided that each of these two bodies remains formed of the same matter, this number keeps an invariable value. 3° The number M. relative to a set of bodies is equal to the sum of the analogous numbers relative to each of these bodies. 4' For any body, the number M is positive. 5" For an infinitely small body, it is infinitely small. 6° For the standard body A, the number M is equal to i. If the standard body A is chosen once and for all, the number M characterizes the body C; we say that this number is the mass of the body C compared to the standard A. to be able to apply to concrete reality the notion of mass, acquired here in an abstract way, it will be necessary, in the first place, to choose the concrete body to which one will make correspond the mass i this concrete standard, to be represented in a satisfactory way by the abstract standard, must not experience any sensible loss, ------------------------------------------------------------------------ nor any appreciable gain in matter, physicists have agreed to consider a certain metal ingot, deposited at the Central Office of Weights and Measures, as representing not the unit of mass, but the body whose mass is represented by the number 1000. This ingot and all bodies having the same mass as it are given the name of kilogram. The name of gram is reserved for the body whose mass is represented by the number i, i.e. the theoretical standard of mass. What we have just said shows us how, once a standard of mass has been chosen, we can, by thinking, make each body of the mathematical scheme on which we are reasoning, correspond to a number which is its mass; but this is not enough for us; we still need, when the standard A and the body C are given to us, not in an abstract way in our theoretical scheme, but in a concrete way in reality, to determine in an approximate way the number which will represent the mass of body C. If we take into account the third character presented by this number, we can easily see that this experimental determination can always be reduced to the following operation: To recognize if two different bodies, given in a concrete way, have or not the same mass. By the preceding definition, two bodies have the same mass if the external bodies have to perform the same work in order to pull each of them from rest and, without changing its state, throw it with a certain speed of translation V, the same for both. We will therefore consider two bodies as having the same mass, if, in order to throw them with the same speed, we have to make the same effort; but it is clear that if we only made use of this principle and of the direct appreciation of equal efforts with the help of our muscular sense, the comparison of masses, impossible to carry out when it would be a question of bodies that are too big or too small, would be. even for the average bodies, of an extreme imprecision; It was recognized that the efforts which it was necessary to make to throw various bodies with the same speed were arranged appreciably in the same order as the efforts which it was necessary to make to raise these bodies from the ground to a same height above the ground; so that the bodies of same mass were also bodies of same weight. ------------------------------------------------------------------------ It had, moreover, been previously recognized that two , infinite for V = \'}, and meaningless for the values of V which would exceed the critical value X1. The equality < (j) would then be replaced by the equality MV2 bis) X(V) = - [1 + ?(V)J. Here are then the characteristics of the work necessary to launch a body with the speed V i° As long as the speed V would be lower than the critical value ". In each of these parts, we separate the two bodies i and a in such a way as to form two distinct particles; then, we animate each of these two particles, isolated from each other, with the speed with which it was animated within the system. The kinetic energy of the system at time t is equal to the sum of the kinetic energies of the dispersed particles. But, after this operation, the system no longer contains any mixture; to each of these particles, we can apply the considerations developed in the above. Let us suppose that, in the system, at the instant t, the element of volume diù contains a part of the matter i, animated by the velocity ("t, vt, nu), and a part of the matter a, animated by the velocity (a2, ">2, ">-,). The preceding operation will draw from this element two particles: one, formed of the body i, will have for mass dm, and for kinetic energy -(u~, + v'\ +w\) dmi the other, formed of the body a, will have for mass dm, and for kinetic energy (u'i ~j- < ~i~ iv'i) dm*. The element rioj will thus introduce, in the kinetic energy of the system at the instant l, the term ( u (' -7- o ̃ | ) dm i -;- - ( n r, --̃̃ v | -}- iv |) dm% The various elements into which we can decompose the system at time t will provide the kinetic energy with analogous terms; we will have, therefore, (m) K = i ( iri -h \} - (c; ) dni\ -+- ("; -+- r; (i1; )din. each of the two integrals.s being the volume elements into which we can decompose the system. This formula leads to an important consequence. Let us consider a mixture of two bodies i and 2; let us suppose that each of the elementary parts of body i and each of the parts ------------------------------------------------------------------------ of the body 2 are animated by the same translational motion of speed V. ÎWe will have /('] -i- V\ ̃ w\ = II", -r- t>1 -i- w\ = V2. If M|, M 2 are the total masses of each of the two mixed bodies, we will have .Mi = dnii, .M 2= drn-2. and, therefore, J, \'ll- J~dnt,> 112= (,dnx2. K =i(Mi-VI2)V2. But, in this case, the system can be decomposed into infinitely small parts, each of which neither loses nor gains matter during the modification; each of these parts has a single velocity and, for all of them, this velocity has the same value V. To this system, we can apply all that we have said about systems that do not contain a mixture; it will have a total mass M and its kinetic energy will be given by the formula K= -MV*. 2. -i. By identifying the two expressions of K, we find the equality M,-i- Wi- M. The mass of a mixture or combination is equal to the sum of the masses of the combined or mixed bodies. The various conventions that we have laid down lead us to the following form of the total energy of a system, a form that results from the equalities (6) and ( 10), ( 1 ̃>. ) '('-) I-1,1 = '-̃ I '̃ < '̃̃ '-' ̃+- w'1 ) d'n- We know, moreover, that this form of energy is a restricted one; in adopting it, we reject out of the domain we intend to explore the laws of electricity and magnetism, for the study of which it is not suitable. Moreover, we admit that the internal energy U(e) does not vary ------------------------------------------------------------------------ not by a simple displacement in space; by this, we exclude from our reasoning the consideration of systems such as Mac Cullagli's elher. By using this restricted form of energy, the principle of the Conservation of Energy is also susceptible of a more particular statement than the one we had previously formulated; here is this statement which, from now on, will be the only one used in our reasonings Restricted Form of the Principle of Conservation of Energy In any real modification of an isolated system, the equality (i3) U(e)4- ( H.2-t- i>2-S- . It is clear that we can take a tihedron T' animated, with respect to the trihedron T, by a motion such that K' - K becomes any function of time ------------------------------------------------------------------------ CHAPTER III. TKAVAIL AND LKS ACTIONS. 1. Inertia Work and Inertia Actions. Let us consider any system, and be, dm one of the elementary masses which compose this system; at the instant t., a point of this mass has coordinates .r, y. z-. The vector whose components are dKr dïy d?z -^dm- ~-d£d'"> -dëdm has been called, since d'Alembert, the ineiTtie force applied to the mass dm at the moment Let us take the system with IVial, h1 motion and the accelerations that it has at the instant L, and, of this polKt of departure, let us impose to him a virtual modification infinitely pelilo in this modification, the coordinates .r, y. of the mass dm undergo infinitely small variations tox. oy. o;. The expression Ç icf-x il* y >l-z (I) -̃-Jim* -itt--"y --nr>)dm .7 J is, by definition, the virtual work of the inertial forces applied to the system or, more briefly, the virtual work of inertia. Among the virtual modifications of which the system is susceptible, there is one which corresponds to the real modification undergone by the system in time dt: by this particular virtual modification, we have ilx l > ilv l t/~ O.r =1 wire. 0>' -7- dt. OZ - -r-dl dt J dt dl l ------------------------------------------------------------------------ and, therefore, f/d2x dx diy dy dsz dz\ ,i '> ,f ( dt= dt T clt`= clt + clt= ~t ) clnz. dj ~J \~dF Tl + ~dt "dl ~7& dt ) **̃ On the other hand, according to equality (10) of the previous chapter (p. 1 10), the kinetic energy or living force of the system has the value t ('C/c~x\' /cly``= (clz)z~ J~' ('J.) ,,J (~ clt -r-' dt + ~t clnz. The reconciliation of these two formulas immediately gives dK (3 ) - In any infinitely small real modification, the work of inertia is equal in absolute value and of opposite sign to the variation experienced by the living force. The above is entirely general. Let us now consider a system whose virtual modification is entirely determined (p. 40) by the knowledge of n arbitrary infinitely small quantities, i.e. <7i, ç/i, qn these n independent variations. In the real modification that the system experiences in time dt, these n variations take values that can be designated by by (,\dt, q'tdl, q'ILdt, r/{, q\, <" being n finite quantities. The virtual variations experienced by the coordinates of a material point of the system will be given by equalities of the form i o.v - a,)/, + "2y2-r-+ a,,q, ( ,'i ,1 < 0)' = b q -i- b-î (/, -H bn q ", I os = c,7i- Cjr/j -+ cnqn. The velocity components of the same point will be given by ------------------------------------------------------------------------ even equalities dx ~t = ~)?t--2+.?7~ ( r-t'i'li '-̃̃̃̃+""'J") "i ^dT'/i'+'" ̃ ~dT . AppkiiL. Accounts i-endax, 1. CXX.1X. [>[.. :i\ \o cl /|5g; 7 August, -.>8 August cl 11 September 1899. - Cre/le's Journal fiir reine und angewandte Mathematifc. Bd. GXM. - Journal de Mathématiques pures el appliquées, 5" sciie, l. VI, iijoo, p.>. - Traité 'de Mécanique rationnelle, ->" édil. l. Il- i()'>'i, [)- '> ------------------------------------------------------------------------ Consider the expression (l0) ('J[[dF) ^-[HF) ^(sf) \dl" which differs only, from the force vivo K pyr the substitution of ̃ (ten d'y di s\ i dx dy dz\ facccleral.on ( -- j a la v,tesse (^ £. ̃%); let us subsl.tu it to the components of the acceleration their \alors (7); we will find 1 dat dan " r (") (j=7,) ( ~JTCll' '̃̃-̃̃ -jj-'J"-i-al(/l-l-- ">-< Oc/1 Il the others are established in a similar way. These relations (11'. lead to an important theorem. It is enough tle compare the equalities (()) and (11) to recognize that the coefficients of q]'1 and r/" in ̃expression G are respecliveinenl equal to the coefficients of tjf and q) (jj in expression K. From then on, it is visible that the determinant of the coefficients of q <][, q'n drtns the n inertia actions J 1 ..12, .l" It is the product of (- 1)" by the discriminant of the form 'J.K, quadratic in q\. q qn. As this form is positive, its discriminant is positive, so that the preceding determinant, always different from o, has the sign of ( - 1)". s Let us give an example of these various considerations. Let us suppose that the system under study is reduced to a solid body incapable of deforming, but capable of moving in space in an arbitrary way. Let us suppose, moreover, that the lethal of this body is entirely known when we know the position it occupies. ------------------------------------------------------------------------ The virtual change înhniment small in the position of this body can be determined by means of six infinitely small quantities, namely the three projections a, j3, y of an infinitely small translation on the axes O.r, Oy, Os of an absolutely fixed trihedron, and three infinitely small rotations A, u, v around these same axes; the components ox, oy, oz of the virtual displacement of the material point of which x, y, are the initial coordinates will have for expressions 1 Sa- = a -h- 3) 8y = p + v" - Xz, f ôz = y -+- À y - \i.x. In ia change, real or ideal, which occurs between the instants l and l-hdt, the components of the elementary translation have the values a' dt, j3' dt, y' dt, and the elementary rotations are a' of, 'jJ dt, -/dt. The components of the velocity of the point of coordinates x, y, z are i x' = a' -+- ijt'; - 'y, (14) .=~-)~, The living force of the solid body is K = \( ̃̃̃- \J.' z - 'i' Y)- - (P'-i- v'.r - V z )- -f- (y'+ À '>" - l-1')2] <1- Let us posit. i M = fdm, ] M = rf/n, .M r, = | y dm, M Ç = fz dm, i-v=l<.r'1 :if//". I, = j(z2-i-!)d.m, !-== Çix^-r-y^dm, \\y-= J yz dm, \i-.v- I z.v dm, Mxy= I xy dm. M will be the total mass of the body ç, r,, £, will be the coordinates of its centra of gravity; l.r, lv, I- will be its moments of inertia in relation to the same axes. It is easy to see that the living force of the solid takes the form ------------------------------------------------------------------------ ."-"= 2" -h ;j." - v "j- -i- ;j.' y' - v'i'-i- /(/ x -{- \x y~r-'i'z) - (À'2-(- \x- -+- v'2 )x, (1-) y" = S"-i- v "1 - À" z -r- ̃ oc' - À'- ij.\X'a; -+- ijl'_k -f- v') - ( À'- ̃+- [i. + v'2 )y, [ z"- ̃+ l'y- /- >- |A'a'-+- -/().' x -+- |x>-l- v'a) - (X'r- |i'2 + v'2)js. ̃ If one then i'ornicates the inertia actions either directly or rather by the method of Mr. P. Appell, one obtains the following results. 5 l l l. iA 111.t' (16) K = (a'2-f- ï'2~ v'2 ) j -t- M E(~-t')-i-M '4 ( ` ),' ex' ) M ~x'- ~'A') I, 1.) Ir I- f.) h H ̃ ;j. -i -/ >. Êl -1 - Mv-|JLV-i- M;^v'+ M.n-X'jx'. This quantity is indeed a quadratic form of the six quantities a', fS', ̃ À', p', v'. Let's put r/<_£^ o"_^ ."_^L dt' 'J dt' ilt' = ~àl IX = ~dï' = W The equalities (i4) will give us 3;"= a"-f- [-> a - 'y-+- \x' z' - v y' y" = p "+ v"^ - À" .3 -+- v' - À' z'. z" = '+- ~i" y - \->" x -+- X'^y' - ij.' a?' or, by virtue of these same equalities (i/\), j, Ja = - Ma"- M(Ç|i"-ï.v") I - M(|j.y- v'p'i - Al(r/>ï(}Ji'-HÇv')X' + M E(X;i-+- ji'2-hv'2). 0") I ,IX = - i.r X" + M.v ;jl" -h .M.r: v" + M ( r3"- r, y" ) [ + ( I, - I ) >/-/ -v (. M.r, v' - !\l.r: ,u')X'-+- M_v: (v'2 - [x'2 ). The four actions of inertia .1; JY. J^, Jv are derived from the previous ones by permutations. In general, to study the motion of a solid, we do not use an absolutely fixed trihedron, but a trihedron invariably linked to the solid with respect to this trihedron, we will obtain, by ------------------------------------------------------------------------ a similar method, anologous equations to the previous ones (' ). Let us return to the general case of a svslème whose most general virtual modification depends on a limited number of independent variations. It may happen that the action of inertia corresponding to an independent variation is identically null, that is to say, whatever the motion of which the system is animated; the corresponding variation then takes the name of variation without inertia. For q, for example, to be a variation without inertia, it is necessary and sufficient, by virtue of the first equality (ta), that --̃ be identically null or that G be independent of q'1 { moreover, according to equality (i i), for (î to be independent of q" whatever the motion of the system, it is necessary and sufficient that we constantly have a, o, l~, o. CI = o. If we refer to the equalities (4), we can see that these conditions are equivalent to the following one A variation without inertia is the one whose value has no influence on the magnitude of the virtual displacement of the various points of the system. We could, for example, consider a rigid solid body uniformly magnetized; a virtual modification of this solid body could be defined by means of the six infinitesimal quantities a, jï, y. a, y., v (pie we have just studied and by the ariations 8-1. ioil'-i. o© of the components of its magnetization intensity. The components Sx. S y, àz of the displacement of any material point belonging to the solid would still be given by the equalities (io) where oA- 8it! o£: o-U ôifc. o£ would thus be variations without inertia. Among the sternes whose most general virtual variation is determined by the knowledge of a limited number of independent variations, we find, in particular, systems whose state depends on a limited number of independent variables. Suppose, in fact, that the state of a system is determined by the n a ri a Mes independent/?! ,/>-- p, a virtual modification (l) I'. Al'[*Bi.i.. Train! rtr Afc'rmiif/ue rational, ̃>" l'-tlit. t. II. p. 'i;f|. ------------------------------------------------------------------------ of this sequence will be known when we know! Ira les \aleurs pri>es par les n quantités infiniment petites Vl =~ Vl 'h- = ''Pi- ̃̃̃: 'h, '>"̃ The expression 18) of the inertia iravuil can be written (19) ^.1,0/J, la5/>2H-i-.l" 5/", .)i, Jo, p/<̃! "Pu and 'analogous expressions for r'. :̃' We will also have (2.1) :{' O~ /1'; - .L---y~-)' Op, ,)p, dp,, with ̃"-̃ l <)-X IJ-.T X~l <)-̃)̃ 1 - ~7~ ij In 1 equality (aa), the summation extends to all distinct combinations of the indices i. ̃> two by two. The (|Uiintilé>_T/ z" have tic- analogous expressions-. Imi verlu of the equalities i ̃> i cl ( v.(i the force viyc peul s"<;crire I f\ O.r il.r ij.i- d.r - ~~J V-.ô/r, 7^ /̃̃̃ itjrj'-} ̃̃̃ rf' (23 ~Lv~t rl/7_ rl~7,e 'r~ l v the [)or'it.-> rem|ilaçant deu\ terms that are analogous to the one that is written and that deilui>enl by sub^lilulion of the leltre y or the leltre :̃ to the letter .r. Similarly, we \erlu il<'s éyalilés i m) cl {->->. i, on u C i
: ̃̃̃̃-ïp-"" '-̃̃̃̃) ̃-̃ <"̃ (,24)' ~JL' pi < p; F.) dm, The first equality (\->) gives us ulor- (',>j) -¡ =- L~ < ------------------------------------------------------------------------ On the other hand, Pt''ga)it('('3) gives us the following equalities OK K d.~r c). ~=~ (,~1,, d~>_ ~-- )2.r T~T/'r"--.-T-(-- /)-)- 1 .~t '2 u1C yr /d.7' dr --==) ([ --- --- /-r- --- ----t. rl d rr " " d~r (.'W) ~t `lli'i .1_ [ ~Jhn,i+ rJpt l~ `c/ /~r`+ ,J:r ~~)~) U = U"(A ) -+- U,,(B) 4- Wab( A, B). The quantity Wai,(A, B) is what we will call the mutual potential energy of the two systems S", Sa, taken respectively in the states A and B. About this quantity, we have two pieces of information, the first of which results from the seventh convention of the previous chapter (p. ()3); here it is If we agree to choose for the normal state of the system S a state where the two systems S, S4 are infinitely distant from each other and where each of them is in its normal state, the quantity Wat cancels itself whenever the two systems SK, S4, are infinitely distant from each other, whatever the state of each of them. The second information results from the postulate introduced in § 2 of the preceding Chapter (p. ()-) it can be formulated as follows The mutual potential energy of two systems Sn, S& may well depend on the relative position of these two systems, but not on the absolute position that their set S occupies in space. Let us consider an arbitrary modification of the system S. We know (p. (io) that it is composed of a virtual modification m,, of the system S, and a virtual modification m;, of the system S/ l'ar as a result of the single modification itta. One (A) experiences a variiilion oCV,(A), while M" ^A, B) experiences a variation ^'" H* ( A B) similarly, as a result of the modification m; Ui(B) experiences a variation 3U;,(B), while Waj(A,B) experiences a variation o,, M' 1 II). It follows from equality (29) ------------------------------------------------------------------------ that in any virtual modification of the system S. we have ̃ '!<> ol = î L"( A) -+- o,( U'ai< A, B) - 51 hi ?,) -̃- S/, >r"/,i A. I: >. t Let us consider the two systems S,, and S/ taken in the states A and and deprived of any motion without imposing on the system S^ any modification, and without communicating any motion to either of the two systems, let's impose on the system SK the useful virI modification rna. The work accomplished by the bodies foreign to these two systems is reduced here to oU, since the kinetic energies of the two systems remain constantly zero, and oL itself is reduced to oU"(A)--5aH-aA(A!). Let us now take the system S", out of the presence of the system S/ and still without movement, let us impose on it the same virtual modification. The work accomplished by the bodies foreign to the system is then reduced to 3t.A). It is no longer of the same value as the previous one; it surpasses it by the amount (it) S,, = -oaM-a,,iA., B). The presence of the system S* had therefore for ell'el to decrease of the quantity That the work that the foreign bodies must accomplish to impose to the system A the modification /" from there, the following definition The i/uanUty (?", defined by the equality (,'Si). <;st, the work done, an the modification, virl tir Ile in,, of the system, Sa,by the actions that the system S/, exerts on the system S, L)c even, the quantity Zi,, defined by the equality (il bis) & = - o*>ra/,( A. lii, is the work ellccluded, in the virtual modification "/ of the system Si, by the actions that the system S,, exerts on the system S/ The equalities (3i) and (3i hi.i) added member to member give (:-> G,, ̃-< =- o M'a,,i a. l!_i. ------------------------------------------------------------------------ When two independent systems S, S/, experience, in the presence of each other, any virtual modifications, the work done by the mutual actions of these two systems is the changed sign of their mutual potential energy. Doubting the laws that the virtual trauul of certain actions will be, in any virtual modification, the changed sign of a certain quantity entirely determined by the knowledge of the state of the system which undergoes these actions and of the state of the bodies which exert them, we will say that, for these actions, this quantity is) a potential. We can therefore state the following proposition The mutual actions of two independent systems admit a potential which is equal to the mutual potential energy of these two systems: If the system S/, is a system incapable of experiencing any variation, SjJ. is necessarily zero, and the inequality (02 ) reduces to fg y0.?;at,~t.l, 1 Pc= - o H "r A, Bi. If a system is placed in the presence of foreign bodies of which it is independent and which cannot experience any change of state, the actions (/u. it undergoes on the part of these foreign bodies admit a jiotenliel; this potential is equal to the mutual potential energy of the system and the foreign bodies. If the two systems S, S/, -,onl inliiiimcnl away-; an deliul of the virtual modification m, they are it still .'i ta end, because this modification, which leaves immobile the sv.^lèine S/ change inliuimciil little the position that the system S, occupies dan* I space '" ( li ). null at the end as at the beginning of the modification m, u' en éprou\r no variation as soon as, C,, is null. IJ where this proposal At the end of any rirlial modification it a t}:i/riue S, the actions exerted on this system by another system S/ independent of the first one and infinitely distant from him. do not carry out any work. ------------------------------------------------------------------------ This proposition can also be stated in this less precise form Two independent systems, infinitely distant from each other, have no effect on each other. Suppose that the modifications ma, m/ simply change the position in the space of the set S of the two systems S, Sb, without changing either the internal state or the relative position of these two systems. Since a simple displacement of the set does not change xl'ai, (A, 13), equality (3a) gives us 5a _t- 6é = o. The mutual actions of two independent systems do not produce any work in a virtual modification which is reduced to a displacement in space of the whole of the two systems. This proposition may be regarded as the general statement of the law OF the ecaliity of action AND reaction. Let us now imagine that the most general virtual modification of the system Sa is determined by means of a limited number n of independent variations at a2, an. We will obviously have (">-) Sa - - o,, 'I""fc(. B) = A, a, -+- A2a2-H.+- knan. By definition, Ai is the action, relative to the variation a, that the system S4, taken in state 13, exerts on the system S", taken in state A. About such an action, we can immediately state some propositions Each of the actions (fue the system Sb exerts on the system S,, can depend on the state of each of these two systems and their relative situation but it is independent of the motion of each of these two systems and of the absolute position that their set occupies in space. All actions of the system Sb on the system Sa are equal to zero when the two systems are infinitely far apart. ------------------------------------------------------------------------ When a given system Srt undergoes a given virtual modification ma in the presence of another given system S/ the virtual work E,, of the actions exerted by the system S4 on the system SI, is determined without ambiguity; but it is not the same for the actions A,, A2, X, It can happen, in fact, that instead of using, to define the modification m,t, the n independent variations a, "2, an, one makes use of n other independent variations a'% ci' ciH to these new variations will correspond new actions A' A,, A'n; but, in any virtual modification of the system Sa, one will have the equality A, "1 + A2a2-h A,,œ = \a\ -4- \2a'.2~h A;,"; which we will express by saying that the actions A,, A2, A, on the one hand, and the actions A'j. A' A'in on the other hand, form two equivalent groups. The numerous problems of force composition have no other purpose than to replace a group of actions by an equivalent group. Suppose that the state of the system Sa is determined by means of n independent variables 0' a- ̃ a, in which case we can take "i = oa, a, = oa,, an - oa, The quantity W* (A, B) can then be considered as a function of the variables y.t, a2, a, the form of this function depending, moreover, on the nature and state of the system Sr, >]--"( A, B ) = VAfa,. as, %n). l. The equality (.'}.'{) will become t~ J~`r, '& d-'r, A [ oa. H- .Â2 -;a2 ̃ -}- A,, ia,, - ̃ ( - oa, -+- - oa, -+- -f- - oa,, ') ,a.l.0~14-- As o~-)-)-:A,t jC j (JCL't O'J. n I This equality is a pure definition, identically true whatever oa,, 3a2, oaw it therefore requires that we have f3~) ~Vt a t1==r ~n 1 c)V,, (3.~) ~xt dx~ d When the state of a, system is determined by a limited number of independent variables, the action, relative to one ̃P.' - I. ̃̃!̃ - ii .) ------------------------------------------------------------------------ of these variables, exerted by another system is, to the nearest sign, the partial derivative with respect to this same variable of V mutual potential energy of the two systems. Let us suppose that the state of the system S/, is also determined by a limited number p of independent variables [j,, [i2, fi/ The mutual potential energy of the two systems will then be a function of the (/'( +/>) variables y.s, a, rfi,, r(ip Va4(A,B) = \V(aI) .a", fi,, .3,,). Equalities (3i) and (3i bis) will become ( /d~V ¡JW dxt ~la,i (d~V dW s* - (,^p7 Oïii"̃ sp; °P"; ¡II r, They will give equality (jw ¡JW r)~V ~o rJW ~Q ')' Z-`,u+<^i~= Jx 07.¡+.+-O'l "+"D°?1+'+-¡),) °pp 1 n ,t ,p When the systems Sa, S4 are both determined by a limited number of independent variables, the virtual work &a of the actions of the system Sj on the system Sa and the virtual work E<> of the actions of the system Sa on the system S/, have as their somnze the total differential of a function of the variables that determine the stall of each of both systems. This theorem is not true of each of the two virtual works £", £4 taken in isolation, unless one of the two systems Sa, S4 remains in an invariant state; the virtual work of the actions it exerts on the second system is then the total differential of a function of the variables that determine the state of this second system. ',). Examples of various actions. Most of the quantities that we have had to consider up to now are the magnitude of a work, energy, internal energy, kinetic energy or living force, work of inertia, work of the mutual actions of two systems, are quantities of the same kind. ------------------------------------------------------------------------ We can add them to each other or subtract them from each other. If we relate to arbitrary units these three kinds of quantities, the lengths, the times and the masses, and if we attribute to them respectively for symbols of dimensions the letters L, T, M, all the quantities that we will enumerate have the same symbol of dimensions, and this symbol is, MlrT "2, as the expression of one of them immediately shows. of the living force. The quantities that we have named actions can, on the contrary, be quantities of various kinds; each of them must be of such a kind that, multiplied by the corresponding variation, it gives a product of the same kind as an energy the kind to which it belongs depends therefore on the kind of the variation to which it corresponds; when this one is determined, that one is also determined. What we say must be understood as well as the actions of inertia as the actions exerted by an independent system; these two kinds of actions are, in eu'et, magnitudes of the same kind when they relate to the same variation. The nature of the various actions to which a material system can be subjected, whether they are actions of inertia or actions exerted by an independent system, is thus known when we know the independent variations which determine a virtual modification of this same system. Let's support these generalities with some examples. Pis i-au i;r i:\kvipj.i:. - Poinl material. 11 often happens that, in the study of the movement of a body, we make abstraction of the volume and the shape of this body and that we look at ourselves as sufficiently informed when we know. at each moment, the position of one of the points of the body; in such a case, we represent this concrete body, small or big (it can be very well the Sun), by a very simplified system which is reduced to a variable mathematical point in space such a system is called material point A virtual modification of such a system is reduced to a change in the position of this point; such a virtual modification can be defined by the infinitely small variations ôx, oy, Zz ------------------------------------------------------------------------ These coordinates being related to an absolutely fixed trirectangular trihedron. If this material point is in the presence of any independent system, the actions exerted by this system on the material point will perform a virtual work & - Xox -h oy -+- 7. os. The actions X, Y, Z do not only depend on the state of the system that exerts these actions and on the position that the material point occupies in relation to this system; they also depend on the coordinate axes to which this point and this system are related. On the contrary, according to what we have seen in the previous paragraph, the virtual work S must not depend on them. Let A be the infinitely small segment whose components are ox, Sy, os; the segment A is determined in size and direction for each virtual displacement of the point M, independently of any coordinate system. Let F be the vector whose components are X, Y, Z. The work G can be written G = FAcos(F, à). It is clear that the magnitude and direction of the vector F must depend exclusively on the state of the system S and the position of the point M with respect to it. A definition, which we will give in a completely general way, will teach us the name that should be given to the vector F. Let us consider any system S. Let M be a point associated with this system in such a way that, when the system S is in a known calibration, we know where the point M is located. This point may always be located within the same material element of the system S; it may, on the other hand, be located within different material parts in the various states of the system S; it may, finally, be external to the matter that forms the system S; it does not matter. Any virtual modification of the system S leads to a certain displacement A of the point M. Consider any work that relates to the most general virluole modification of system S, and assume that ------------------------------------------------------------------------ this work contains a term of the form FA cosfF, A), where F is a vector whose magnitude and direction do not depend on the virtual modification imposed on the system S. We will say that this term is the work of a force, represented in magnitude and direction by the vector F, and that the point M is the point of application of this force. Note that this definition is consistent with the definition of the inertial force given at the beginning of paragraph I of this Chapter (p. i i5). This definition shows that any force is such that its product by a length represents a work or an energy: all the forces are thus magnitudes of the same, kind cjui mil for symbol of dimensions MLT "2. In the C. G. S. system of fundamental units, where the unit of work is the erg, the unit of force is called the dyne. With this definition, we can say that the actions exerted on a material point by a system independent of this point are reduced to a force which has this point as its point of application. The size and direction of this force depend exclusively on the size of the acting system and the position of the material point with respect to this system. l.)r.i;xii;viE i-.xiïmi'j.e. - (.'o/>s solid invariant. Let's study a solid whose "standard" is completely defined when we know the space it occupies. T he virtual modification of such a svsl.ème is entirely determined (p. îad) by the knowledge of the components a, [i, -'of a certain virlual translation and by the components A, ij., v of a certain virtual rotation, this translation and this rotation being both related to (uncertain Irièdrc Lrireetangle < )./̃, Or, ():̃. The virtual work of the actions exerted on this solid body by a system which is independent of it will therefore be of the form ( 3-) ,) C a Y 3 -i ̃ Z 7 -t- ]. - M ;jl -i- In this expression, the six actions L. M. N do not only depend on the size of the acting system and the position that the solid body occupies with respect to this system, but also on the position of the body itself. ------------------------------------------------------------------------ depend, moreover, on the trirectangular trihedron Ox, Oy, Oz that we have chosen as reference trihedron. It is easily shown in Kinematics that the magnitude and the direction of your resultant translation A of which a, jîJ, y are the components that the magnitude " of the elementary rotation of which A, u, v are the components; finally that the direction, represented by the same letter Zy, doy., dov" doy., -_ - dx H - dy H - dz -- (JM (/y _t --z ,/z dx, (IY\ dz, dx2 <)}'<, oz-, n ôz, c) ~.z, d r,z, d ~.zz d Î; d "i dx torJzl dy -i <)CjZ, dz = dôz, dx r)M, dy - d oz., dz. V (l 5 | -4- rt 2 = ̃>. ,A'.i and the formulas (37) may be replaced by the following o.r - g, x -i- o2. oy - - 0|j' ̃ oiy, oz = 0, z - o,z. 1 op1 u= a ;j. - vjk, QJ}' = P - '?' ),z, OK) î,i= --̃- >.r- ;j..r, (3R)/, O.r ̃- C| -H A'}' +- ^2-3, ~.?'=~~-9~-)~, Oj = ̃ A' -:- A'i,"' -r-"3^- The most general virtual modification which can impose on a body a homogeneous deformation is thus decomposed into two other modifications; the first modification {ù,x. 5, y, o,z) imposes on the body an overall displacement which does not deform it; the second {o-^t. o-, y, o.2 i alone involves a deformation. The properties of this deformation have been established by Cauchy and by Lamé; they have. made the three quantities e,, <%̃. ''s l(; ------------------------------------------------------------------------ name it (the elementary dilatations, and to the three quantities g,, g- g-, The name of elementary slips. The virtual displacement that we propose to study is therefore determined by the twelve infinitesimal quantities '"̃-. ?. '̃' ï>- *'̃̃ Cl, 3- We shall suppose, moreover, that the system is brought to a uniform temperature & and that, to complete the determination of your most general virtual modification of the system, it is still necessary to give the variation 32f of this temperature. In such a virtual modification, the actions exerted on the body studied by an independent system, will perform a virtual work whose form will be given by the following equalities : ê = Ei+ Sj-h 6 35, (3<)) ) Gi= Xoc -+̃ Yp -+- Z-; + LÀ + Mjj. -t- Nv, E2= Eand -+- E2e2-l- E3e:t+ G,,4r1 -+- G.f- G3^3. Of [ thermal action. 0 we shall say nothing for the moment; in a future Chapter (Chap. VI, p. 2.55), we shall have to consider it again. The actions which contribute to produce the virtual work G, are known to us by the study of the preceding example we know that they are reduced to a force, of components X, Y, Z, whose point of application coincides with the origin of the co-ordinates, and to a couple of which L. M, M are the components and whose axis passes by this same origin. It remains to study the actions that contribute to produce the work P. Let m be the initial volume of the studied body. Let's say II K,=.-- N,i3£, l':a==- .u. K3 = - N,,c7, | G| - 2 TiTÎT, Gi - iî.W. Ci3 = - 2T3TO. Let us observe, moreover, that the equalities (.->8) give f) fy.ï: ')';)' Ù 0.5 e, ~x r) n.`,Y' e.= = C~ u fj~ - ) f) , /.d rJ.x n,Y + I~ ~J `if ------------------------------------------------------------------------ and we can easily see that the virtual work C2 can be written r/ ,r~ ') f~2 1 dx -i- r3 d,y ~F- 1 r/~ T '1^- ̃ N °-%L - T 'Jy T3 da' rly ~J~ à os ôa (J os oz v o os r T., - '[- -- v3 a~. d, (Vi' rtz the integration extending to the volume of the system. This formula, in turn, can be transformed. Let n be, at a point of the initial surface of the body, the normal directed towards the interior of this body; let us denote by a, b, c the cosines of the angles that this normal makes with the axes of coordinates, and let us pose ̃ P.l.= N1a-l-T36 + T,c, (4'2) 5 Py=T3, x. S?) or, abbreviated, by U (M). Let us suppose that the mass M is the sum of two other masses M,, :VL: M =M, -4- \T2. Let us divide the phase of mass M in two other phases of masses M,, M2; then, by keeping to each of these two phases the specific volume d>, the composition x and the temperature !5, let us move away it? infinitely one from the other. Since, ------------------------------------------------------------------------ In the definition of a phase, no account is taken of either the ligure which allecte. or the position which its various parts occupy in space, this operation in no way alters the stall or, therefore, the internal energy of the total phase. But, by virtue of the preceding principle, this internal energy is now the sum of the internal energies of the two partial phases, so that we have the equality UCMi-4-M3) = L" ( AI ) -+- U(M2). This equality is equivalent to the following proposition: When the state cl' a phase is defined by the specific volume. The chemical composition and temperature of this phase, the value of the internal energy of this phase is proportional to ii its mass. It should be remembered that the correctness of this proposition is subject to a condition: The normal state of the phase of mass M must be constituted by the set of normal states of the partial phases of masses M,, M2. This condition will obviously be fulfilled if we adopt the following convention, which, moreover, naturally comes to mind To define the normal state of a phase, a certain, normal specific volume are chosen; care is taken to keep w(), .(̃" and .:?" the same values regardless of the mass of the phase. The proposition that we have just formulated constitutes one of the essential properties of h phase studied by Chemical Mechanics; this proposition immediately follows from the very definition of the phase, and it was appropriate to establish it first. iWe shall now come to the examination of the various actions that a phase may undergo on the part of independent systems. For this purpose, we shall assume, as we did at first, that the state of the phase under study is delineated by the total volume ro, the chemical composition x and the temperature S. ------------------------------------------------------------------------ If such a phase is placed in the presence of an independent system, it can experience certain actions; to the three variables ra, ?S, x can correspond three actions that we will designate respectively as II, 0, X. The virtual work of these actions is i 45) G = - nôn+B oS? -+- X |, e-", en) - L,(ei) l'->( e->) -r-4- l'n(en) MYfi!, e2, c,,), W cancelling when the systems S,, S2, S,; are infinitely far from each other. M' is the mutual potential energy of the systems S,, S^, S, About this quantity, one always admits, in all the branches of the Energetics, the following hypothesis: JiYPoxnk.SK. - U mutual potential energy of any number of independent systems is equal to the sum of the mutual potential energies of the associations formed by grouping these systems two by two in all possible ways. (This statement is equivalent to the equality (48) W(ei,e",ell)=^Wl,,l(el,,e,l), where p, q are any two of the indices i, a, ", and where the summation extends to all distinct combinations of these indices two by two. Let us follow the consequences of this hypothesis. Let nii be a virtual modification of the system S,. At the same time as we impose this modification on this system, let us leave invariable the states e2, e,, of the systems S2, S, By this modification, the quantities ll", x]',l, experience variations 8, M", o, M-'ip, and the equality { /|8) allows to write o, q. = 'Vr,, +-Î, M. But this equality, in its turn, is equivalent to this one: 1 i<)) G, - Ê,2-i- ̃̃ C, where C, is, in the modification mK. the virtual work of the set of actions exerted on the system S, by the set of systems S..", S, and where F, is the virtual work, relative to the memeinodi- ------------------------------------------------------------------------ lication, of the actions that the system S/; would exert on the sluggish system S, if these two systems were alone in the presence of each other. We can therefore state the following proposition: When a system undergoes any virtual modification in the presence of several independent systems, the actions exerted by these last systems on the first one perform a certain work; on the other hand, if each of these last systems were alone in the presence of the first one, the actions that it would exert on this first system would perform, in the same modification, a certain work; the algebraic sum of the works such as the latter represents the first work. Let us suppose that the virtual modifications of the system Si are determined by means of a limited number of independent variations a, b, A. These variations correspond to the actions A, B, L exerted on the system S, by the set of systems S2, Sn, so that E, = Aa + BA+.+ L/. On the other hand, if the system Sp existed alone in the presence of the system S, it would exert on this system the actions An, Bp, Lp, and we would have 5,= ,a + B/,6-v- + IJ),i. The equality (.{()) can therefore be written X a -~ïîf/ -̃̃+-!>/- (\h. -+- \tl )k j ^-(H!i.-|{,(^4-(L2^r-La)/. (Since this equality must take place whatever a, b, we have A =1 A2- A, (Sn) B B2 -i- B, B.~B'cd. 1 L -r- -̃ L, When a system is placed in the presence of several other independent systems, each of the actions it undergoes is the algebraic sum of the analogous actions it would undergo from the ------------------------------------------------------------------------ of each of the acting systems, if he were in the presence of this one system. Let's apply these considerations to some examples. Piiemikii kxkmpi.k. - System of material points. Let us consider, in the first place, a system of n material points Mi M2, M, The changes of J'eLal of each of them are entirely defined by the variations that the coordinates of this point undergo, coordinates related to an absolutely fixed reference trihedron. The quantity Wpq(ep, eq) will therefore depend only on the following variables the coordinates xp, yp, zp of the point Mp, and the coordinates xq, yq, zq of the point Mq. But, on the other hand, we know that this quantity must depend only on the relative situation of the two points Mp, Mq, and not on the absolute position that their set occupies in space. Now, the only functions of xp,yp, zp, Xq,yq, Zq that are entirely determined by the relative situation of the two points Mp, M? are their distance rpq and any function of this distance. We can therefore write (5i) Wpqiep, ev) - fpç('t). j. The form of the function fpq can depend, moreover, on the quantities which characterize the nature of the two points Mp, Mq and remain invariant in all their displacements, such as the masses of these two points. We know that the action exerted by point My on point Mp must be reduced to a force; according to the formula (ai), the components of this force will be - /-,>̃)-_ d f (r )âr>"> rlx/t "f~l~l'9,)' (~7' cl 'n`nl'r~)x -'x /a/ ('<̃'̃) d I 1 'II ;> ( -' I ,"! ri z,,- s,, *'p - f J i"/1 i "i -y II VI I ,1,/ Similarly, the action of the point M.p on the point M(/ is reduced to a ------------------------------------------------------------------------ force applied to point Dl~, whose components are llv cl -fl,Î~I,r1)-- r,, _c,, x" f" (1') !'pr! Ci. bis) Y, -- -- ( r,, ) ~-- j dr/"I l~I,~ f 7 d r/' :[' .r" -7- y/----- clr! !'l,y The force exerted by the point llr~ on the point INIP is directed as the line i1'Iy M~; it has the magnitude ~5'i) F - (53) ~7~'?'7- The force exerted by the point ~IP on the point Mq is equal to the previous one and directed in the opposite direction. The determination of given by the formula (5t)~ allows to write (34) y =~.Înn( I°vr,), the summation extending to all the distinct combinations of the indices o, z, Iz two by two. Let's put ( )5) V7'=/t(~l) -t-(~2) +.)-(~~). It is easy to see that equality (54) can still be written (M) r ,'If 1(V,+v2+.j- Vn). The components Xp, Yp, Zp of the force applied to point Mp, which represents the action exerted on this point by the (/t-;) other points, will obviously be given by the equalities ~7) JV,r Y=-~i L r)1' ,r i''7,) Yn=- dxp 1 ~Y, l' Jzi, < d~' This example contains, as a very particular case, the problem that celestial mechanics deals with when it studies the motions of the stars by reducing each of them to a simple material point. To approach this problem, it is sufficient to particularize this ------------------------------------------------------------------------ that we have just said by posing the equality ( 'jo ) j/nj ( /u/ ) =̃- K where K is a positive constant and where /np, m(j are the masses of the two points M^, M,r The force Fpi/ that each of these two points exerts on the other is then given by the equality (39) Fw = _K^, r~n i from equality (53); this equality (09) expresses Newton's law. Let's put (6OI II -+- "U 1 (60) 1 i> -î/> 11 j> The quantity (j^, has been given the name of potential junction at the geometric point (xp,ypy zp) of material points M(, M2, M,, other than the point M.p. Comparing equalities (55), (58) and (60) will give (61) V,, = - K/T^U, Equality (55) will then become (62) 'I" = ('"i Ui -i- HiiLT2 + .+ mn[]n), 2 while the equalities (5^) will become 63 X, K ~Up p e'U,, Z K dU" (63) X^K,", Ï,,= K "3, Z,,= Km; "-<'p Yre < These formulas are of continual use in the study of universal gravitation. SECOND EXAMPLE. System of infinitely small elements of variable density ('). (') We have introduced, for the first time, the actions of which it is going to speak: here, in the following text Le potentiel thermodynamique et la pression hydrostatique, Chap. III {Annotes de l'Ecole Normale supérieure, 3° série, t. X, i8f)3, p. ai 3). ------------------------------------------------------------------------ Let's now turn to an example a little more complicated than the previous one. The system studied is still composed of a limited number of bodies, the position of each of them in space is supposed to be determined by the knowledge of one of its points: but we suppose that, in order to completely define the state of any of these bodies, we must join to the coordinates of one of the points belonging to it the values of the density and temperature which are both assumed to be universal in all of this body. Such an example seems, at first sight, quite artificial, but certain theories of celestial mechanics, such as the explanation of the figure of the tail of comets, have led physicists to consider continuous media, each element of which can be assimilated to one of the bodies we have just defined. Let us therefore be p the index of one of these bodies; Mp one of its points; xpi Y pi zp the coordinates of point M/; pp body density p; zip its temperature. The quantity '?/> ?qi 'pi ^q- On the other hand, it cannot depend on the absolute position of the set of two bodies /> and 7-/>q *=,.- 'V/ ''?/> e/ ?^i- ------------------------------------------------------------------------ with = °I' = ifj'u. -La ̃( or, ox,, Or, /'̃'", .ri> ''ri"/'y (GG y <{£pX 'Hjït "fl "l tlLZlJit, J J' orM Oz,, Or, /;", "Spi "I- Sa, 7, X/;f/, t, 'ùpq are the components of a force applied to the point Mp, directed from M(/ to M/7, and, having for magnitude - - "- this ri7'o force is equal and directly opposite to the analogous force that the body p exerts on the body q To interpret the nature of the action, let us observe that its virtual work is ~1,,r8~7, If one designates by rop the volume of the body p whose mass m,, is invariable, one draws from the equality op- - the relation m g, î "'n ̃- l,gty= --?- 0~/), l, mj, which allows us to give the previous virlucl work the form 1 ?" "fl "l '-?" ~z ot;r ôr.sp- da ~z 'ovt7,~ If we refer to what we said in the previous paragraph (p. \) when studying the body that Chemical Mechanics calls phase, we see that the action &,pv is equivalent to a normal and uniform pressure, applied to the surface of the body, this pressure having for magnitude H ) nr7, "<~7~ iloclie and, after him, ave and l\ésal have tried to explain the ------------------------------------------------------------------------ The energy of each element of the comet underwent a force variable in magnitude with the density PI' of the element, but none of these authors seems to have suspected the existence of the pressure 11/)(/ which necessarily results from this hypothesis. The mutual yotential energy of the bodies studied will be "~ 't" .ft "n the summation extending, to all combinations/?, of the indices z, iz keyux a deu~. Let us pose, as before, l tiçt ) ~`p = ftu +.În~ + +,fn,=. and we will still have (7°) ~==~.(V,-t-Vii--V,,). Let us assume all bodies i a, n immobile and invariable, except the body p. By the Intermediary of distances 7'~i,7,,s, /'Pli V will be a function of the coordinates .r~ 3~, of the point i~l, moreover, this quantity will be a function of pp and 2~ In a virtual modification of the only body p, the actions exerted on this body by the other bodies i, 2, 11 efleettieront a work ''7 i~t,= 1,; oa~-t- ¥p'O}'¡)7r ~t, M, ~t~ (j~~ 0:7, and we will have 1~, ù~'t, 1 t, ~V L, ~4,' ZI) (;) <1. Z" =-- L1 S'" JV ~=- These actions are composed of a thermal action 6~, the ------------------------------------------------------------------------ [we will see later (p. '>) 9) that it is null] :̃> of a force, of components X^, Y p, Z/M applied to the point M/; 3" of a normal and uniform uni' pressure applied to the surface of the body p. this pressure having for magnitude }... iry, d\ (7:J) thread, (/J> mf, 0?,, r, ------------------------------------------------------------------------ CHAPTER IV. THE QUANTITY OF HEAT t ). ). 1. Quantity of heat released by a system into a real modification of this system. Let us take, as in the paragraph of the preceding chapter (p. i 2.4), a system S formed of two independent parts S", S/, let us suppose that this system is isolated and that it undergoes a real change; let us apply to this change the principle of conservation of energy [Chap. II, equality (i3)]. Equality (29) of the previous chapter makes us know the internal energy of our system S as for the kinetic energy of this same system, it is equal to the sum K,t-+- K.4 of the living forces of both systems S, S/ The sum lj,( 1 A ) ̃ U/,( H ) *" (' A H ) -i- Ka -+- Kb must, during a modification 1 of the system S, keep an invariable value. If we designate by d the differential of a function of time, we must have (i) d lia (A) -t- ilK-a- d\b( M) r/K/4- dVub(A, B) = o. r <~Wf~ (A, B) can be decomposed into two terms.. If the state A underwent a modification identical to that which it undergoes in time <7i, without 1 state 13 experiencing any modification, W,,i, would experience a variation daWai,\ in the same way, if the state (M All that is going to be said, in this Chapter, mii1 the quantity of heat is the 'levcluppeincnl of what we indicated in our Commentary to the principles of Thermodynamics; 1"' part: The Principle of Conservation of Energy, Chap. III (Journal de Malhcmaliquos pures et appliquées, série, 1. VIII, iSçjm, p. lioS). ------------------------------------------------------------------------ H would experience a modification identical to that which it experiences in time dt, without the state A experiencing any modification, xVai, would experience a modification d/, lFrtA- It will be obvious that d Wuh( A, B ) = dH Va,( A, IJ) -i- dh >ra6( A: H ). Moreover, if one designates by Ea the work of the actions of the system Si on the system S" during the real modification that the system Sa experiences in time dl, by G" the work of the actions of the system So on the system S" during the real modification that this last system Sj experiences in the same time, one will have [Chap. III, equalities (3i) and (3i bis)] eo = - da V "a ( A, B ), E/, = - d,, Wah ( A B ) ). Equality (i) can therefore be written (̃i.) d\]a{X) -h û!K" - GM-t- cHJ/,( B) - dK/ Ca = <>. l oses (3) ( Ga- rfUK(A) rfKa= Q", ( 1 G/, d UA ( B ) ^Ka = Qa, Let us examine the meaning of the first of these equalities. dUa (A) + d&a represents the work accomplished by the bodies foreign to the system Sa in the real modification that this system experiences in time dl. The considered equality shows us that, in general, this work does not represent the work done by the external actions of the system S, to reproduce this work, it is necessary to add the quantity Q" that we will name the heat released by the system Sa during the real modification that it undergoes in the time dt. It is not necessary to try to justify the denomination thus introduced, by some comparison with the meaning of the word heat in the current language; to cook our sensations of hot and cold cl. the notion that Thennodvnamiquo names (juanliié of heat there is not, today, any more any link; to find such a link, it is necessary to go back very high in the history of Physics; the progress of Science completely, dissolved this link; it is proud that the usage has, nevertheless, preserved the name of quantity of heat which, when one takes it in the direction of the current language, for ------------------------------------------------------------------------ introducing it in Thermodynamics reasonings, generates dangerous confusions. By its definition, the quantity of heat is a quantity of the same kind as a work its dimensions are ML- T~2 if one adopts the C. G. S. system of fundamental units, the quantities of heat are measured in ergs. Let us suppose, for a moment, that the system Sa is isolated, the system Si, not existing in this case, the work ) "= U - V -r- ="= l, l.T- V- :->[-, j^i. Let m, be any virtual modification of the system S, and any virtual modilication of the system S- By the modification m,, varies from ol the inertial actions of the system S, eiVecluent a virlual work cnlin, by the ell'et of this only modification, thevslèmes S;, and 1' remaining invuriu- ------------------------------------------------------------------------ The actions of the Si system on the part of all the bodies that are foreign to it, i.e. on the part of the S2 and S systems, carry out a virtual work. The actions of the system Si on behalf of all the bodies that are foreign to it, i.e. on behalf of the systems S2 and S, perform a virtual work (7) G,= - 5,2-3,. i. The system S, thus releases, in this virtual modification m, a (|uantity of heat (8) Qi= Ei-s-t, - oUi = -i - 3U| - Si'ï'iî- Si'i. Let /n- be a virtual modification of the system S2; by this modification, the energy U2 of the system S2 undergoes a variation oLU and the inertial actions of the same system perform a virtual work t2 ; finally, by this single modification m2, in which neither the system St nor the system S undergo any change, ^Fi2 and 'b undergo variations 32*F,2, o^é. The actions that the system S2 undergoes from the systems S, and S perform a virtual work (7a) G2 = - o,Wi,- 6,0/. The system S2 therefore releases, in the virtual change m-2, the amount of heat (8 bù) Q2=G2^T2_ 3U2 = -2- 3U2- oj^î- S, ii. Let us now consider the system S formed by the set of the two independent systems S.. S2 by associating the virtual modification /"i of the system! St and the virtual modification m2 of the system S2, we obtain a modification ni which is the most general virtual modification of the system S. By this modification /m, the internal energy ( of the system S ('obviously proves a variation (O) oL" ̃-=̃- oU,3l.o1>r,Ô2W12, while the inertial actions of the same system perform a virtual work (10) -:=--- -I- Kn this only modification m. the system S remaining invariable, ------------------------------------------------------------------------ ̃ experiences a variation (M) 2,H.i = rVi.-f-'Vi. Finally, the actions exerted on the system S by the bodies S foreign to this system perform a virtual work (v>.) XL-- o,i. The amount of heat released by the system S, in this virtual change m, has the value Q = G -+- -: - oU Or, by virtue of the equalities (9), (̃10), (i 1) el (12), (i3) Q = -.1 - SU, -3,V12 -Ô.'J/ -- 3U2 - Sj "F,,- 3,6. If we compare this equality (i3) to the equalities (8) and (8 6 /,"'), we find the relation (it) Q = Qi + Q2- The quantity of heat released, in any virtual modification, by a system formed of several independent parts is the algebraic sum of the quantities of heat released by these various parts during the virtual modifications, relative to each one of them, of which the virtual modification of the whole system consists. All of the above concerns extremely general systems. iWe will now particularize the system under study and assume that any virtual change in this system is determined by a finite number n of independent variations y,, q. (̃/" The variation that the internal energy L experiences in a similar modification can be put in the form Ilij 31- - 1-1 y 1-- I>î'/î -H- -r- L,,(jrB, L), Lj; - - - ïj" being n quantities which are determined when the state of the system is determined. The virtual work of inertia has the expression [Cliap. 11], also ------------------------------------------------------------------------ lity (8)J ( '[<)) = Ji "yi -+- i>i -r- ̃̃" R" are called the calorific coefficients of the system. The determination of the calorific coefficients of the system does not only suppose the knowledge of the state of this system and, of the external actions which solicit it, it also supposes the knowledge of all the local accelerations on which depend the actions of inertia, I(, J^, .1, Just as a quantity of heat is a quantity of the same kind as a quantity of heat, so the heat coefficient relative to a certain variation is a quantity of the same kind as the action relative to the same variation. Suppose, in particular, that the system under study depends on a finite number of variables pu p2, ̃ ̃ ̃ />" we will then have ?! = ~t, ", ------------------------------------------------------------------------ THE QUANTITY 1>E CHALKL'tt. I <> n can write so that we can write dU OR <>[ 1:l0) 1.1=- 1.2=, 1.=""p, 0p,_ <)p,i Moreover, in this case, the actions of inertia will be given by Lagrange's formulas [Chap. J1J, equalities ( a H ) j The equalities (i<)) will thus become /><̃ U - Ki h _£/ £K_ (Jp, i 1 (II Op\ H. at(U- K) d_ oY^ (̃m) <>p, dt op' d(U - l\) .). d toK l~ ly ,~l ~c `i,c tLL nJpie - o~p~a H ~o~p)t' '̃). Calorimetry and determination of the Mechanical Calorie Equivalent. The foregoing allows us to indicate the principles of a method by means of which one can compare the quantities of heat released by various systems under various modifications. The system studied will be designated by the letter S; it will be independent of the external bodies E which act on it. In the case of these systems S, 1', we will have to admit a certain restrictive condition, which is Ja following Pitic.uiiT.E condition. - The systems S and S are such that the work done in any modification of the system S by the actions of the system S on the system S is completely known when the initial state e( and the final state e. of the system S are known. (̃ii)? - ()(<̃",). This condition is one that we will often encounter in the following; it therefore deserves that we stop for a moment. The quantity 0(^'1,ro), introduced by this condition, certainly verifies the identity (-̃-̃'i) 0(6,. <"î)-0(£'", <) = 0('"i, Ci). 1). 1. ̃ ̃̃̃ ̃ n ------------------------------------------------------------------------ This being, let us choose a normal state eu of the system S, which, once chosen, will always remain the same the work 0 (e0, e) necessary, to lead the system S of this normal state (') and we will say that Û(e) is the external potential of the system S taken in the .<>> S - - 0 Q(e).. '̃ The virtual work of the external actions applied to the system is equal to the variation undergone by this quantity which depends only on the state of the system In Chapter II, the flax of paragraph 2 (p. 1 3o), we pointed out that this was not so in general; but it is possible to conceive of a system S and foreign bodies S so arranged that this proposition is applicable to them. Let us suppose, for example, that the system S is one of those systems that Chemical Mechanics calls phase. If ra is the volume, the virtual work of the external actions is reduced to ------------------------------------------------------------------------ [Chap. 111, (̃"ality (46)] to ( ̃>) G = - II im, II being the normal and uniform pressure which represents here the external actions. The second member of equality (27) is not, in general, the variation of a function 0 of the variables m, x, 2r which determine the state of the phase; but equality (26) can be found true in certain particular cases. It is true, for example, if the volume m of the phase is bound to remain invariant in this case, in fact, ors is constantly zero and equality (26) is verified by taking (9.8)' Q = o. It is still true if the external bodies are arranged in such a way that they keep the magnitude of the pressure fl exerted on the phase invariant in this case, the equality (26) is verified by taking (29) O = IIto. Returning to the condition we have posed, we see that it can be stated by saying that the external actions to(>,) - V(e,J - Q(es). This quantity of heat will therefore only depend on the initial and final states of the system S. It will keep the same value in any way that, subjected to the same actions, the system passes from the same initial state taken without living force, to the same final state, also taken without living force. It is such quantities of heat that we are going to compare between them. For this purpose, we will make use of a C-system, 'which we ------------------------------------------------------------------------ Let us dominate the calorimeter and that we will suppose subjected;'i certain conditions which are the following ones Second condition. - The foreign bodies exert on the calorimeter actions that depend on a potential o>. Third condition. - The state of the calorimeter depends on a single variable, the calorimetric property. This is susceptible to an experimental numerical determination. This variable is, for example, a temperature in the water calorimeter, a mass of water in the ice calorimeter, etc. The internal energy u and the external potential oj of the calorimeter C are then two functions of the single variable y. If we pose (3.) V(y) = "(y)-+-'o(v), 1), the quantity of heat released by the calorimeter when the calorimetric variable changes from y to y2, so that the living force is zero at the beginning and at the end of the change, has the value (3a) ? = V(Yi)-V(y,)- Fourth condition. - The calorimeter C is partly bounded by an undeformable surface which is the indentation of a certain area, while the system S is partly bounded by an undeformable surface which is the relief of the same area. We can then, as we have seen (p. (>:->.), associate the two systems G and S in such a way that they are contiguous along these two surfaces now merged into one and yet remain independent. This operation is designated by the words We plunge the system S into the calorimeter. Thus associated, the system S and the calorimeter C must be subjected to the following conditions: Fifth condition. - The actions of the foreign bodies S on the systems S and C continue to admit for respective potentials the quantities il and w. ------------------------------------------------------------------------ Sixth condition. - The mutual actions of the system S and the calorimeter C are zero. Seventh condition. - The amount of heat released in any real change by the set of systems S and C is zero. Eighth condition. - The system S being taken without living force in the initial state e, is associated with the calorimeter taken without living force in an initial state y, which sf.ua the same in all the experiments. Under these conditions, a real modification can bring the system S to the state ez without living force; at the same time this modification brings the calorimeter to a certain final state y", with zero living force. Ninth condition. - The initial and final values y, and y_> (the calorimetric variable are close enough to each other that we can write the approximate equality (33) V(Y,)-V(Yi)= ^(ï.-YiJ- With the 5th, 6th and 8th conditions, the system S gives off, in the considered modification, the quantity of heat Q, while the calorimeter gives off a quantity of heat q, given by the equality (3a) but, with the (f condition and the equality (33), this quantity of heat can be written ,3.1) ,/=_J(ït_Tl), On the other hand, the condition leads to equality Q -̃- q = o. So we will have, according to 1 equality (34), (35) tl cl V' ~t, ( ) t.~f "YO~ Let us now suppose that the system C, taken in the same initial state yi, is associated with a system S' fulfilling conditions analogous to those to which the system S has been subjected. Suppose that the modification of system S' is accompanied by a ------------------------------------------------------------------------ modification of the calorimeter C and that this one makes pass the calorimetric variable of the value y, to the value y!, the modification of the system S' will have released a quantity of heat Q' given by the equality (". b'. ()' i i ('i ï 6is j Q-=~-2(~). l. ';ï Equalities (35) and (35 bis) will give (:~6) v !i - Yi ." '< Yt The calorimetric variable y being, by hypothesis, a quantity whose value can be determined experimentally, the second member of this equality (36) will have a measurable value. The calorimeter C will thus allow us to determine the ratio of heat quantities such as Q, Q', to one of them chosen once and for all. This quantity of heat, chosen once and for all, and to which all the others should be related by means of the calorimeter, is called the calorie. To define the calorie, we suppose that the system S is a phase, formed of liquid distilled water, of mass equal to i gram, subjected to a constant pressure whose value is defined in Hydrostatics under the name (Y atmosphere). We imagine that the temperature of this phase is lowered from + i°C. to o°C, while its initial volume and its final volume are those that it must take to remain in equilibrium at these temperatures under the pressure of i atmosphere. The concrete calorimeters, used in the practice of the laboratories, realize only in a roughly approximate way the ideal calorimeter that we have just defined. By various corrections, based either on direct hypotheses, or on the teachings of various physical theories, one tries to decrease the difference between their indications and those which would give the ideal calorimeter. We leave it to the reader to find out by what sequence of ideas, in each particular case, one is led to admit that a real calorimetric operation verifies appreciably the various conditions enumerated above. The calorimetric method whose outline we have just traced will be transformed into a method suitable for measuring the quantities of ------------------------------------------------------------------------ This number K should be called the mechanical equivalent of the calorie. This number K should be called the mechanical equivalent of the calorie, but it is often used as the mechanical equivalent of heat. Let's see how I can determine the mechanical equivalent of the calorie. Let us imagine a system S, independent of all external bodies, and constituted as follows: 1" This system Sue can, during the modifications it undergoes, release no quantity of heat 2" The system S is subjected to two groups, independent of each other, of external bodies. The first group 1 exerts on the system S actions which admit a potential Q. The second group exerts on the system S actions which, by some mechanical theory which will be explained later, we can calculate the work in any modification of the system S; 3" These various actions can determine, within the system S, a modification which takes this system without living force, in the state ,)~ £>{>, j_ lif, - I2(c2) = 0. This done, t, i" We remove the external system g :> We give back to the S-system the possibility to give off heat; 3° [You I plunge in a calorimeter. We assume that, under these conditions, the system S, taken without living force in the state i'->, returns to the state <'t with zero force vne. The system S will then release a quantity of heat given by the equality Q ------------------------------------------------------------------------ This equality, compared to the equality {'̃'->-), will give C = Q, so that the quantity of heat Q will correspond to a given number G of work units. On the other hand, the indications of the calorimeter will allow us to recognize that this same quantity of heat corresponds to a number q of calories. We will obviously have G (1 = ls, so that the preceding experiment will allow to determine E. Joule and various other observers have carried out experiments of which what has just been said traces the abstract plan these experiments have led to the following results i" The mechanical equivalent of the calorie is a positive number a0 If we take the erg as the unit of work, this number has approximately the value E = 424.10". 4. Fundamental property of heat quantity. General laws of Dynamics and Statics. The quantity of heat relative to a real or virtual change has a very general property which is attributed to it by the following Hypothesis Hvi'oïhksk. - Let a system, isolated or not, but independent of the foreign bodies which act on it, and without contact with any of them. In any infinitely small modification, real or virtual, which consists in a simple displacement of the system in space, the quantity of heat released is nu tic. This assumption leads to an equally general consequence. Let us refer to the definition of the quantity of heat, ------------------------------------------------------------------------ given by the equalities (3) (|>. i 55), and observe that the internal energy of a system does not vary when this system experiences a simple ensemble displacement we obtain the following theorem: When a system, isolated or not, but independent of foreign bodies and without contact with any of them, undergoes a real or virtual change which is reduced to a displacement of the whole, the sum of the work of the external actions and the work of inertia is zero (38) E + t = o. The most general modification that can be reduced to an overall displacement is decomposed into an elementary translation parallel to a certain axis, and into an elementary rotation around a certain axis; or, in other words, to three translations a, p, y along the three axes of coordinates Ox, Or, 0:, and to three rotations a, p, v around these axes. In such a modification, we can write (3g) S = X i. -+- Y S -+- Z v -+- LÀ -4- M jx -+- \v, X, Y, Z being, as we have seen (pp. i34-i35), the components of a force, and L, M, N the components of a couple, it is to this force and this couple that the actions of the external bodies on the system would be reduced if this one were transformed into a rigid solid, incapable of experiencing any other change of state than a change of position in space. On the other hand, in such a modification, the displacement of the material point whose initial coordinates are x, y, z has as components ?ix = a - u. z - -/̃)̃, oy - [ï -f- v x - >> z. oz = ̃' - a y - \xx. By substituting these values in the expression of the work of inertia [Chap. IJ[, equality (i)] ~l = -.7 (~ f'x + dt= ~~rl dt= °") clru, we find ('<<̃>) -̃ = Jaa-Jpï -H.IV-1'-+- JyÀ-f-J^ii-t-JvV, ------------------------------------------------------------------------ with ,~dt'~driz, .1,3=-.rplt drm, ,1"=- clnt. h> = -7CtJ{x-Tt-rdï)din- By virtue of equalities (3g) and (4o), equality (38) becomes (X -h Ja)a-(Y - J "j)3 -r-(Z-r-Jr)T -i- ( L -̃-̃ h )X - (M J|A) {J. -+- ( N + ,lv )-> == o. Since it must take place whatever a, [j, y, A, ;ji,'v are, it resolves into six equalities which, by means of the equalities (̃'iy-)t can be written ) K0-K,=t--Q. The quantity of heat really released by a system in an instantaneous modification of this system is equal, by definition, to the decrease undergone by the living force of the system. The very definition of the quantity of heat released in a virtual modification supposes that Ton knows not only the state of the system at the beginning of this modification, but also the acceleration of each of the elements of the system at the moment when it is in this state; without knowing this acceleration, one could not form the expression of the virtual work of inertia. The definition of this quantity of heat only makes sense as long as these accelerations have finite and determined values. Now, let us consider ------------------------------------------------------------------------ It can happen that in zero time, certain local velocities undergo finite variations, so that, for certain elements of the system, the word acceleration no longer means anything. It is therefore not possible, in general, to speak of the quantity of heat that a system gives off in a virtual change, when this virtual change has as its starting point precisely the state that this system is passing through at the moment when it experiences an instantaneous change. We shall introduce into our theory a quantity which will play, for a virtual modification resulting from a state where the system experiences an instantaneous modification, the role that the quantity of heat plays for an ordinary virtual modification; this expression, we shall create arbitrarily, in virtue of the principle that definitions are free; the remainder of the theory will prove that it was useful to call upon this new notion. But, before giving this definition, we shall present some remarks concerning the bonds to which a system is subjected when it experiences an instantaneous modification. In it, an instantaneous modification corresponds, in general, to an abrupt change in the conditions of bonding to which the system is subjected; and, in this respect, several cases can be distinguished: i" Let " It may happen that the modification by which the system would pass from state E to state ê is incompatible with the bonds that govern the system from instant tn, just as the modification that would lead the system from state Ë to state s; in this case, we will say that the sudden change that has occurred, at instant /0, in the bonds that subjugate the system consists in the establishment of a new adhesion. Here is an example of such an instantaneous modification: Ci, C2 are two solid bodies; until the instant l0, these two bodies are independent of each other at the instant ta, they collide from the instant t0, they are supposed to be unable to separate, so that they must now move while remaining invariably linked to each other. Having made these remarks on the changes of links in an instantaneous modification, let us come to the definitions we wish to introduce. Let us divide the system we want to study into infinitely small material elements and let dm be the mass of one of these material elements. This element is animated by a local velocity of which u, v, (v are the components; according to what we said in the Ú previous paragraph, this element has a momentum of which the components are ndni, v dm, w dm. Let's suppose that at the instant ln the system is the seat of a sudden modification, that is to say a sudden change of motion. When t tends to ln by values less than *", u, ç, w tend to certain limits ". e,, i - wî)dm are the components of the momentum lost by the same mass. After the instant l0, the system is subject to certain bonds; starting from the state E that it takes at the instant read, let us impose on it a virtual modification M compatible with these bonds; a point of the elementary mass dm undergoes, in this modification M, a displacement of which ôx, oy, os are the components; the quantity (46) a =̃/[(",-"i)o.T + (fi - Vï)oy-±- (w,- w>2) oz) dm, where the integration extends to all the elementary masses that compose the system, is called The virtual effect of the kinetic beads for the virtual modification M. For a virtual modification resulting from a state that the system presents at the time of a sudden modification, the virtual work of the kinetic losses plays the same role as the virtual work of the inertial actions in an ordinary virtual modification. The importance of the quantity we have just defined is immediately apparent from the following proposition If, for all the elements of the system, we know the components "i, c,, (ï-'i of the local velocity immediately before the sudden change; if we gave, in addition, the value of I virtual effect of the kinetic losses for all the virtuals from the state E that the system takes in this sudden change we know, for each of the elements of the system, the components u<±, v", w-> of the local velocity immediately after the sudden change. Let us assume, in fact, that one possesses the knowledge predicted by this statement and that, however, the local motion of the system immediately after the sudden change is not unambiguously determined one could, immediately after the sudden change, imagine at least two general motions, |j.o and [>!̃ of the system, both of which are compatible with the knowledge implied by the statement and with the connections at which the system is i)..- .1. ̃.̃̃̃ i2 ------------------------------------------------------------------------ subjected from the instant t0 where the sudden modification occurred; in the first of these movements, your elementary mass dm would have a velocity of components u. v-2, w-2 in the second, it would have a velocity of components u[,,, i\ "- By hypothesis, in any virtual modification imposed to the system from the gauge E, we would have | (u-2 - "i )ox-i- (v2 - Ci) oy -+- ("'" - wl ) oz | dm - I Ku'ï - "j ) oa; -h ( v'.1 - vt) oy -+- ( "/2 - ">, ) cz ] dm or ( 47 ) I. ( Ma - "" ) ox -- ( v'% - c2) oy -+- ( w', - w, ) oz ) dm = o Suppose that from state E and time t0, the system is animated by the motion ij. during the positive time dt; at time (<0 .+ dt), the system would reach a state F; the passage from the state E to the state F would be a virtual modification M compatible with the bonds to which the system is subjected from the instant to, since the motion u is supposed to be compatible with these bonds; in this modification M, we would have 8a." = M" dt, 5y = v2 dt, os = w2 dl, so that equality (47) ̃> applied to this modification would become (i8j |O', - K. ) lu -4- ( v', - vi)i'-((v' - wi)tv2]dm-o. In the same way, starting from the state E and the instant i0, let us suppose that the system is animated by the motion u/ during the positive time dt; at the instant (/" -r- dt), the system would change to a new state F'; the passage from the state E to the state F' would be a virtual modification M' compatible with the bonds to which the system is subjected at the instant /(, in this modification M', we would have o.r - u!ï dt, oy = v[, dt, oz = w', dl, so that the equality (-^7) would become (/J8 bis) [("'2-"j)a'+-(p's - ("j v't ("''2 - ivt)w'2 | dm - o. ------------------------------------------------------------------------ The equalities (j8) and ( -{S bis), subtracted member by member, give the equality [("'̃> - "a )'! -i- ( v'ï - v± ')- -+- ( w' - iv2 )- ] dm = o. This last equality requires that we have, for any material element belonging, to the system, II' = U-2, i' = (>2, (I., = 11' so that the stated theorem is proved. Among the virtual modifications that can be imposed on the system from the state E taken at the instant to, in a sudden modification, there is one that deserves special attention; it is the modification really experienced by the system between the instants read and (tB -+- dt) in this modification, we have ,x = "2 dt, tjy = v-i dt, oz = w dt, so that we can write the equality (4^) (4g) i a = i: dt with 'Z I | ( a, - iti)u-i-T- (f-'i - c>)i'2-v- (wt - iï'i) "'î J dm. But it is obvious that this last equality can be written: i >o) ̃' - = ( (tf c'y -i- w'[ ) dm - ( u\ -+- c| -+- tv|) dm - j [(u,- il-,)1 i i>j- i'o )- (">i - ̃"'2)î]c?w. The effect of the kinetic perlas in. the real modification that the system experiences between the epoch ln of a sudden modification and the epoch (t0 -dl), is obtained by multiplying by dt the excess of the living force lost in the sudden modification over the living force due to the velocities lost in the same modification. Let us suppose that the system studied is one of those whose state is entirely determined by the knowledge of a limited number of quantities. The most general virtual modification that the system can undergo is that of the ------------------------------------------------------------------------ can experience from state K, taken in a sudden modification, can be determined by means of n infinitely small quantities (/t, 172, qn) which can well be subjected to certain unilateral connections but which will not depend any more on any bilateral connection. \The virtual reflection of the kinetic losses can then, in one way and one way only, be put in the form ~=C~[-<~f/j--C,tt/ C|, C2, .C,, being quantities independent of q,,q-2, .(/n. The quantities Ci, Ca, G,, are the kinetic pearls relative to the variables qt, qn, q, In many cases, the virtual itt of the kinetic losses plays a role analogous to that played by the quantity of heat released in a virtual modification. Thus, we will formulate the following hypothesis, similar to the one given in the previous paragraph 1 1 ypothesis. - A system, isolated or not, but independent of external bodies and without contact with any of them, experiences an instantaneous modification E is the state that it takes in this modification from this state, one imposes on it a virtual modification which is reduced to a displacement of the whole in V space: for such a modification Ce (virtual Jet of the kinetic pearls is zero. A such modification, a point of the svslùme experiences a displacement whose components are r,x =- a -̃-̃). z - v y. c~=~~v;r-A~, oc =-- y -1- \y - - ;ji.r. a, 3, y are the components of an infinitely small translation and ), ;jl. v the components of an infinitely small rotation. According to the equality ( {(i), Tellet of the kinetic losses in this modification will be end 1 3 t= A.ï-4- "?-+- C-h F X '̃ C: H v, ------------------------------------------------------------------------ by putting A = ( ut - u-y) dm, F = \{"'i- \v,)}' - (i>,- Vi)z]d/n, or even i.,) i A = u i dm - u.a dm (n) I F = ( tvt y - i-, z) dm - ( wty - v,s) dm, For the effect of the kinetic losses cr to be zero in any modification of this kind, that is to say whatever a, [j, y, A, p., v, it is necessary and sufficient that the six quantities A. B, C, F, G, H are separately zero, which gives these two groups of equalities "i dm = I n" dm, (53) i J v, dm = J I v-ydin, [ I r (t\dm- I r n'ndin, u I )(W\y- v\ z)dm- j (wîy- v, z) dm, (r>) ̃ j i ii\ z - il-] x) dm - l ( ui z - id x) dm. j { i-, x - u i y) dm - j ( i\ x - u,y) dm. The first group is equivalent to the following theorem: If a system independent of foreign bodies and without contact with any of them experiences a sudden change, each of the three components, along three axes of rectangular coordinates, of the quantity of motion of the system keeps an invariable value. The second group is equivalent to the following theorem: When an independent system without any contact with foreign bodies experiences a sudden change. ------------------------------------------------------------------------ the momentum of the system with respect to each of the three coordinate axes remains invariant. These two propositions are the j.ois gknéuai.es dks modifications instanïa.vkks they apply to all systems to which your general laws of Statics and Dynamics formulated in the preceding paragraph apply, and they are the only ones that apply to the sudden modifications of all these systems, without exception. ------------------------------------------------------------------------ CHAPTER V. MECHANICS OF INVARIABLE SOLIDS AND RATIONAL MECHANICS. 1. Mechanical properties of systems formed by solids independent of each other. The hypothesis formulated in paragraph 4 of the preceding chapter has provided us with two laws that we have named general laws of Dynamics, because these laws apply to all systems for which the principle of conservation of energy admits the restricted form indicated in Chapter I. If we further particularize the systems to which we apply it, this hypothesis will be able to provide us with much more complete information about them. Consider, in particular, a system defined as follows It is formed by a more or less large number of bodies independent of each other, and having no contact with each other. Each of these bodies is either a material point, or a rigid solid whose various material elements keep, one with respect to the other, an invariable relative position. The state of each of these bodies is entirely determined when we know the position it occupies in space. Such a system is a set of invariant and free solids. Most of the systems in which celestial mechanics seeks to represent the motion of the stars or of some of them fall into this category. The same is true of the sets of material points by which Poisson and a good number of his successors are represented. ------------------------------------------------------------------------ The Physical Mechanism, proposed by Poisson and adopted by many geometers, admits that these systems are the only ones that the physicist should consider; they are, in any case, the simplest he can deal with. Let us denote by S the studied system and by S|, So, S, the various invariant solids which compose it. When we simply move the system Si in space, without changing its state, the internal energy L< of this system keeps an invariable value, as the displacements in question are the only changes of state of which the system Si is susceptible, the internal energy U,- is necessarily the same as the internal energy of the system Si taken in its normal state; it is therefore equal to o. Therefore, according to the equality (20,) of Chapter III, the internal energy U of the system S has the value ( 1 ) U = W. The internal energy of a set of free solids is reduced to C. the mutual potential energy of these bodies. The most general virtual modification of which a set of free solids is susceptible consists in any infinitely small displacement imposed on each of the bodies which constitute it. According to the hypothesis formulated at the beginning of paragraph 4 of the preceding Chapter (p. 168), each of them gives off, in such a displacement, a quantity of heat equal to o. On the other hand, according to a theorem demonstrated in the preceding Chapter (p. 15c), the amount of heat released, in a virtual modification, by a system formed of several independent parts is equal to the sum of the amounts of heat released, in the same modification, by each of these parts. We can therefore state the following proposition: The quantity of heat released by a set of free solids in any virtual modification is equal to ci o. Now, this quantity of heat is defined by the equation ( S ) of the previous chapter, which can be written, neglecting the index ------------------------------------------------------------------------ become useless, Q = G-t-t-SL: Ci is the virtual work of the external actions, t the work n-tucl of the inertial actions, oL the virtual variation of the internal energy of the system. But, according to equality (1), the internal energy l." of a system of free solids is reduced to the mutual potential energy W of these bodies. The previous equality thus becomes ` (2) and s>F = o. In any virtual niodification of a system of free solids, the sum of the work done by the external actions and by the actions of inertia is equal to the increase in the mutual potential energy of these solids. If we observe that - ôV is the virtual work of the mutual actions of these solid bodies, we can still state this proposition as follows n any virtual modification of a system of free solids, the sum of the virtual works of the external actions, the mutual actions of the solids and the inertial actions is equal to o. The equality ('-(), to which these propositions are equivalent, is the I-OUMULK rOMIAUCSÏAU: DE LA Uy.NAMIQ.UE DES SOI.IDKS J.imtHS that Lagrange obliterated, in his Analytical Mechanics, by combining the principle of Alcinberl with the principle of virtual displacements. It can be shown, in fact, that this formula is sufficient to completely equate the problem of motion of a system of free solids. Let us start with a remark: The state of a system of free solids is entirely determined by a limited number of independent variables. ------------------------------------------------------------------------ This state is, in fact, determined by the knowledge of the position that each of the solids occupies in space; now, this position is determined by six independent variables; we can, for example, define the position of a solid in space by means of the coordinates, referred to the fixed trirectangular trihedron O.r, Oy, 0.3, of a given point G of this solid, and of the three Euler angles that three rectangular lines G S, Gr>, GÇ, coming from the point G and invariably linked to the solid, make with three other lines, also coming from the pointG.and respectively parallel to Ox, Oy, O: A system formed by a certain number of free solids is thus such that its state is determined by the values of independent variables six times more numerous than the solids which compose it. If this system contains material points, each of these points will correspond to three independent variables instead of six. This being said, let us apply equality (2) to a system defined by independent variables pi, p2, pn. Let A, A>, A,, be the external actions and J,, J2, JB the inertial actions relative to these variables the mutual potential energy W will be a function of />(, /^2, 1 Pn- Equality (2) can then be written 1.1,+J, pl ôpi+ _1=+Jz- ~2 ô~y (-)~ (~)~ 2 dW -Pu These equalities can be written a little more explicitly, using the expressions of J,, J2 )" given by the eq- ------------------------------------------------------------------------ lities (28) cl 11 Chapter 111 <)(W - K) d i) AI (Jpi lit dp] = (J. ù(W - K) d "K (.i ) '2 ôfï 11 dpZ~-°' l ,)(\l'-K);(. d OK 1/ "Pu cit. dp' These equations, which are exactly equivalent to the relation are the IjAGIIANGE EQUATIONS FOR I.K MOVEMENT n'vN SYSTEM 1)1; FREE SOLIDS. Let us suppose that the external actions At A2, A" which act on the system emanate from bodies whose state is given at each instant t and that we know, moreover, how these actions depend on the state of the system studied Ai, A2, A" will be, from then on, given functions ofy; p. p,, and of The equalions (4 ) are, according to what we have seen in Chapter 111, paragraph 1 (p. i 2/\ ), a system of n differential equations of the second order with respect to the n unknown functions of t that arey>(,/?2, .]), These equations are linear in p "t p]n and, according to what we have seen in Chapter 111, paragraph 1 (p. i?-4), the determinant of these quantities is always different from o so that one can always assume the equations (,{) solved with respect to ̃̃̃̃, p "ir From then on, the general theory of systems of differential equations teaches that a system, such as the system (4), cannot admit two distinct integrals, if we give ourselves, at one instant, the values of p2, pn and p'n. Now, to give oneself, at instant lB. the values of p2, pn, is to give oneself the state of the system at this instant; to give oneself, at the same instant, the values of p'n, is to give oneself all the local velocities of the system, and to give oneself the general motion of the system, which is merged here with the local motion. We can therefore state the following theorem When we give ourselves, at an instant, the state of motion of a system of free solids, the condition (a) determines unambiguously, at any instant, the state and the motion of this system. ------------------------------------------------------------------------ The condition ( ̃>. ) thus exhausts the role that the Energetics must play in the study of the systems of free solids it reduces that study to a question of Analysis. In order to demonstrate that the condition ( a) is sufficient to equate the problem of motion of a system of free solids, it was essential to define the state of this system by means of a limited number of independent variables, so that the condition (a) was reduced to a system of differential equations. But, once this demonstration has been given, nothing prevents us from giving up this way of defining the system under study, nothing prevents us, for example, from determining the most general virtual modification of this system by means of n independent variations. Let us consider a system of free solids which is in equilibrium, where all accelerations are zero, and, consequently, so are all inertial actions; condition (a) thus becomes, for such a system (5) £ - 3>F = o. In any virtual modification imposed on a system in equilibrium formed by free solids, the work of the external actions is equal to the variation of the mutual potential energy of the bodies which compose the system. Conversely, if a system of free solids, subject to the action of invariable foreign bodies, is placed, without initial motion, in a state where any change in the island verifies the preceding condition, this system will certainly remain in equilibrium in this state To prove this proposition, it is enough to notice that by keeping an unchanging state, the system will constantly verify condition (a), and to remember that this condition cannot be verified in two different ways. The two propositions, reciprocal to each other, that we have just elaborated constitute, for a system of free solids, the HIJir.IPK DES nÉIM.ACKMEVrs VIHTL'KI.S. Let us apply this principle to a system formed by a single free solid; W being then identically zero, condition (5) will reduce to i~ - o; moreover, G will be given by equality (35) of Chapter .111. ------------------------------------------------------------------------ Condition (5) will therefore become \a-h Y&-rZ-;+ LÀ M;j. - K v = o. For it to be verified whatever the virtual modification considered, i.e. whatever a, [3, y, À, p., v, it is necessary and sufficient that we have. (6) (X = o, Y = o, Z=o, ( L = o, M = o, N = o. These six equalities are the necessary and sufficient conditions for the equilibrium of a free solid body; they express that the force and the torque to which the external actions exerted on this body are reduced are both null. The comparison of equalities (2) and (5) leads to the following proposition At each instant, the state of a system in motion is such that the system would remain in equilibrium if it were placed there without velocity, if the foreign bodies were maintained in the state they are in at that instant, if, finally, to the actions these bodies exert, we added fictitious external actions precisely equal to the actions of inertia at that same instant. This proposition constitutes, for a system of free solids, the l'iUA'cu'i-: ni-: h'A.i.h.mbkkt. 2. Definition of the sets of subject bodies to connections without passive resistance. The systems we are going to study will still be formed of rigid solids; sculemenlc.es solids will not be free and independent of each other. l)eu of these solids will be able, for example, to touch each other. In a point of the surface of such of them will be able to be fixed either a flexible and. inextensible lil, or a rigid line; the other extreme- ------------------------------------------------------------------------ This line or thread can be fixed or glued to a point on the surface of another body belonging to the system; this line or thread can slide on a fixed point, on a fixed surface or on the surface of one of the bodies of the system. A material point or a solid body belonging to the system can be subjected to constantly touch a fixed point, a fixed line or a surface (ixed on such a line or on such a surface, a solid belonging to the system can be subjected to roll without sliding or to slide without rolling. These six quantities will no longer be able to vary in an entirely arbitrary manner; they will be subject to unilateral or bilateral, holonomic or non-holonomic connections, as explained in paragraphs o, 6 and 7 of Chapter 1. All these links will have this common feature that, in order to define them, it is impossible to appeal to any notion foreign to Geometry. According to the nature of the links to which we have assumed our system to be subject, time will not be explicitly included in these linking conditions. It would not be the same if we suppose, for example, that two solid bodies belonging to the system are connected by a rod whose length, variable from one instant to another, has, at each instant, a known value; that a solid body belonging to the system is subject to remain always in contact with a line or a surface whose figure and position are, at each instant, known... We consider it useless to complicate our exposition by the consideration of connections where time would appear explicitly. The various intermediaries, wires, rods, points, lines, surfaces, which can be used to establish these connections, will be supposed, in what follows, to be subject to a first rkstkfc.ïio.n The fixed points, the rigid lines, the fixed surf-axes, the flexible wires that bind some of the solid bodies of which the system is composed do not have to be taken into account in the calculation of the internal and kinetic energy ------------------------------------------------------------------------ of the system, nor in the calculation of the mutual potential energy of the system and the foreign bodies. This is expressed by saying that these intermediates establish purely geometrical connections between the various bodies of the system. In order to define the systems we propose to study, we will not be satisfied with this first restriction, we will soon formulate a second one which will complete the definition. But some preliminary remarks will be necessary. The position occupied in space by the various solid bodies which form the system can be determined by means of the variables /), r2, ̃/-", the number n of these variables being equal to six times the number of solid bodies plus three times the number of material points. Let us suppose, first of all, that these solids are linked together in such a way that they form a holonomic system, the links being, moreover, unilateral or bilateral, these links are of the form | ,i Oi, '̃->, - - - '-")^o, ( f-i ('-i, ''2, -- rn)ko, (7) Jm(rt, r2, r,,)-so. /~(~t, ''2, ~~=0. The number m of these linking conditions is, of course, less than n. Let's consider an initial state in which the conditions ,i Ci) ''2, ̃ - '̃") = ", (8v ) /î('-li ''2: ̃-̃ r,,) = O, (8 /"('-!> '2, - - 'V) = O are -verified, and, from this state, let us consider a real or virtual movement of the system. To give oneself such a movement is to give oneself the expressions of /'i, /'s, /̃" as continuous functions of the same variable t which. for a real movement, will be the time t taking a certain value /0, /̃ /-2, /̃" take on the values which characterize ------------------------------------------------------------------------ Fetat r.'i). Then, by virtue of conditions ('-) and (8), we have, for any positive value of (t- <0), (dA + Ml ,--+ dfl,(t t rl+ r2+. nt)lt-t "j fJl'l ~l-z d~ + C (_ 't)f, r, r + t)/ z -+ ~)f, l'a ) ft) 1"" 1 i I 'i i - ̃ - i n J I /'J/'l ." 4 '̃> i" r m r T to''< fit J 1 1 (< - '>- \0rt < (V/-n J ̃>. -h io, (<)) { t ~-)~ -l'I i*, +,+ -;¡- tll't ) (1 10) \oi'\ 0r% Or n J +r /` t~fez. Jl t~ tt +. -j_ (%%n ez r r,7 yJl, t Jr~ drez ,+, \'J''l ^2 Orn J 2 l, elr t 1 ~l'~ (." l J 2 -i- ÎO. Under these conditions, the exponent (2) designates a symbolic square formed according to well-known rules; the quantities in parentheses have the values they take for t = tu. If, for positive and sufficiently small values of ( t t0), all the conditions (g) reduce to equalities, none of the bonds expressed by the conditions (-) is broken at the beginning of the considered motion. This motion, on the contrary, will begin by the breaking of some bonds if at least one of the conditions (ç) reduces, however small ( t - 10), to an inequality. Now suppose that the system is not h'olonomous, but suppose that the bonds to which it is subject are bilateral; these conditions will require that we ail, in a real or virtual motion from I mstanl /0. and whatever t, tli /- ̃"3/"" -̃ -t- it,, r'n - 0, (ui) | />i /̃ -i- 1>, r1., -;- - hn /- - o, ¡(LIl "I -+. a2 -¡-; + a" ('Ó) The number ni of these conditions is less than n. These conditions having to be linked whatever the value of (, we can write them ------------------------------------------------------------------------ in the following form (a-, r's -+- --i- a y r'% -- .H- an /̃'" ) r J da, ~"") (< - /o) -f- O, (ii) y ( /t~j-+- ~r~- t', ti) I.. V '---̃ rf< '"y ,r y dt clt 2 T. ~t l'ir -r 1. 2 -r. -;¡¡ -+-( I, r, + h r'j -+-̃+- /"/-) U - A)) + = o. In these equalities, the quantities in parentheses have the value II I <6f) cllf~ that they take for l=l0. The -p-, -̃ -derivatives therein are linear and homogeneous forms of r\, /̃ r'n; the coefficients of these forms depend on the state e0. To say that the non-holonomic system is subject to one-sided binding is to say that conditions (1 î) must be replaced, for positive values of (t - - tQ), by conditions of the following form ("i /̃ -i-"2 ;̃ + a,, i- ) l F da, da-i dan + |. \-7ïrr^nrv^drr") -+- ( " i r\ -4- "j r\ ̃+̃ -H "" /- ) .(< - /") ̃4- = 0, (.2) i ( ~t-t- ~2+.+- 1,, ~r [ ( cCt dl2 ''z+.+ clt j.") -+- ( /| '| -t- '2 '̃'" -+---- -H In r'n )\(t - t") F -+- S O. The motive considered will break, at its beginning, some of the bonds imposed on the system, if, however small the positive value of (t - £0), at least one of the conditions (12) is a ------------------------------------------------------------------------ inequality otherwise, the movement will start by saving the links imposed on the system. The conditions (g) can be considered as a particular form of the conditions (i?) so we can reason exclusively about them. Once these preliminaries have been established, we will come to the statement of the second restriction by which the definition of the systems studied must be completed, but for this purpose, we will first examine only a part of these systems. We shall suppose that the various solids are not in contact with each other; the only connections existing in the system will be purely geometrical connections. It is conceivable that they could be removed at a given moment without modifying the position of any of the solids which form the system. Let us consider the real motion of our system between the instant to and the instant (l0 +/i), being an infinitely small positive quantity. 11 makes the system pass from the state co to the state e. Let a material element, of mass dm, belonging to the system Mo is a point of this element in the state e0, M a point of the same element in the state e. Let's take the system at the instant tg, in the same state and with the velocities it had immediately before the instant t0. Let's suppose that at this instant to, we remove all the links that were holding it in place; it will be composed, from this instant on, of a set of free solids that will move in accordance with the laws previously established; at the instant (ln -+-), the system will be in a certain state f; the material element dm will be surrounding a point N. By definition, the quantity MiN dm will be the constraint that in its real motion, the system experienced from the links. Let's take the system again at time /0, in the same state cu let's impose on it a virtual displacement compatible with the bonds when the variable t will take the value (l0 -+̃ h), the system will be in the state g; the element dm will occupy a position to which the point P belongs. The quantity PN dm. will be, by definition, the constraint that the bonds impose on the system during the virtual motion considered. ̃̃ ------------------------------------------------------------------------ We propose to study links that verify not only the first restriction previously stated, but also this SECOND uesthiction The constraint that the system experiences, from the links, during its real displacement is less than the constraint that it would experience in any other virtual displacement from the same .state (i3) f UN' dm < CpN2 dm. This is what we will express by saying that the studied links are free of passive resistance. This definition is due to Gauss (f); what follows is the development of the ideas put forward by this great geometer and completed by Heinrich Hertz (-), Let us consider a displacement, real or virtual, compatible with the bonds that restrain the system from the instant t0 suppose that in this displacement, the initial values ate^ . Let us consider a virtual displacement which, at the moment £", can break certain unilateral bonds. This displacement is not subject to the conditions (11), but to the conditions (ia). If this displacement does not correspond to the same initial velocities as the two preceding displacements, it occurs with a positive and infinitesimally small constraint of the second order, whereas for the real displacement, the constraint is an infinitesimally small one of at least the fourth order; the second constraint is therefore certainly smaller than the first. Therefore, we only have to look for the necessary and sufficient conditions for the real displacement to correspond to a lesser constraint than all the virtual displacements having the same initial velocities as it. Let's take one of these virtual trips. The initial values of di'f drn dx dy dz clt ri, dt 7 â, ~lt x~' ,lt -J' clt ~d7 = ri' ---' IT^ ~dl=zX' ~aî=y' TU = are the same for this virtual displacement as for the real displacement; it is different for the second derivatives (#/̃, d*" d*x d\y d*z Z ~d~F' > dt* ~dU"' ~~d~F' Ht?' We will denote them, for the actual displacement, by /̃ r "n, x", y", Z, and, for the virtual movement, by r'\+m,, /- -m, x" H- X, y" ̃+- jjt, z"-h ~~t d- dt= r" d3, ~Zr, dX~ drl ~X,tt~ ~t .,dt + ~t ~t + dt dt and two analogous equalities. ~1~, dY~ dAn In this equality (2~), the quantities --) "7~ are, b dt { t di for each point of the system, determined functions of r,, )' i du, dr._ - dr" and dt dt ''dt, ------------------------------------------------------------------------ At time tn, let's apply the equality (a4) <"> your real modification; we will have ( ̃>. '>) x" - i r\ -h X 2 r\ -+- -+- X " /- afXi dX-, dx" + lïTr > + -dt '̃--+ Ht r- Let's apply it to the virtual modification in the same way; plie will become {̃)&) x"-+- X = X,(/;i ra, ) -+- X2( /-"-+- ra, J-+-+ X,, (/-'), + ro,,) rfX, f/X, rfX, + -rfT' + "5Tr* +-+ -5T'"- In the equalities (2.)) and (26), the coefficients Xi and - have the same value; these equalities, subtracted member by member, thus give the first of the equalities i À = X1m1-+-X2ro.2-i-XHro, ( 97 ) [i = Yjbjj 4- Y, nij + .+- Y,, rar;l [ V=/inTi-i-/2 ^2 4- -T~ Ltl Tïï,, The other two are established in a similar way. The consideration of equalities (21), (22) and (?7) leads to the following proposition Let o/'i, 2r2, or,,, be the infinitely small variations of /- r-2, /(, compatible with the conditions ('_>.o), which, by the equalities (21), give the values of nr,, nr2. m,, these variations determine an infinitely small virtual displacement of the system in this displacement, any point of the system describes an infinitely small path whose components ox. oy, ùz are related to the values of p., v which are appropriate to the same point by the equalities (?.8) Zx = £À, oy - fx, 0: = !- Consider the MiNI* triangle whose three sides are infinitesimals of the second order. In this triangle, we will have 2 -2, 2 _0 V\ 2 = -+- Ml "2 - -JiM~YMÏÏ cos i\ MP ------------------------------------------------------------------------ or, by virtue of (i/j) and (i5), ( 29 ) PN = MN 2 H- ( X* ̃+- v* ) -^i(r-x" + (v1--y)u-i-(r-^)v|. Multiplying the two members of this equality \mrdin and îneglecting them for the whole system we will obtain an equality that the equalities (28) easily put in the form (3o) fï>NS dm /^MÏ\2 dm ~r~7, I ( ^x'2 ̃+̃ fy'2 -+- 2-z2 ) dm - 2E I [f- a:") Ix ̃+̃ (r/'- y") $y 4- (t" - z") oz\ dm | This equality (3o) leads to the following consequence For the inequality (i 3) to be always verified, it is necessary and sufficient that any infinitely small displacement compatible with the conditions (ao) verifies the condition (31) I \(t"-ss")ôx-+-(-t,y")5y-i-(^-z")is]din o, This is quite obvious if we observe that s is essentially positive; that it is necessary for this object is what we shall prove. Suppose, in fact, that we can imagine an infinitely small displacement, compatible with the conditions (ao), and such that 1 1 *"- .t") o.r -r-(r" - /")oj + ('3'")o;]f/"( has a positive value c this value will certainly be an infinity of the same order as oj\ ây, os. ------------------------------------------------------------------------ At the same time, ( (jX1 - 3/! + cz1 ) dm will have a positive value 6-, infinitely small and of the same order as ox'i oy-, oz~. s is an infinitely small quality, subject only to being positive and of the same order as or, or- or, or ox, oj', bone nothing prevents us from taking v= £>iî. (S Therefore, in the second member of equality (3o), the quantity between j will have the same sign as its second term which is negative, and we will have f¥R2 dm - iYFN3 dm < o. Thus condition (3i) is necessary for condition ( i3) to be always verified. In other words, if a set of solids, subject to purely geometrical connections and free of passive resistance, is in motion if we are sure that this motion does not break any of the unilateral connections imposed on the system, condition (3i) is true at each instant and for all the infinitely small virtual displacements that verify condition (ao). Condition (3i) can be put in a somewhat different form. We have ( 3a) f(:c" 'Lr -+- y" oy -+- z" oz) dm = - t being, in the considered virtual displacement, the work elVcctué by the inertia actions which solicit the system during its real movement. We also have (33) f(fç"?,x ̃> oy 4- Ç" 5s ) dm = - 0. being the work that would be done, as a result of the same movement ------------------------------------------------------------------------ virtual, the inertial actions to which the system would be subjected during the motion it would take if we broke, at time tn, all the links. By means of equalities (32) and (33), condition (3i) can be written (34) --6 = o. On the other hand, in the virtual modification considered, the external actions that the system undergoes perform work G and the internal energy of the system increases by SU; E and SU have the same value as if the purely geometrical connections did not exist. Moreover, any virtual displacement subject to the conditions (20) is among those that the solids of the system could experience if they were completely free. Equality (2) then teaches us that we have C, () ~t,j = 07 ç-f-O - SU- o, so that condition (34) becomes (35) Xs -4- -z - SU ï o. It can also be written in a slightly different form. According to the equality (4) of the previous chapter, the first member is the quantity of heat Q released by the system during the virtual displacement considered, the condition [(35) can thus also be written (3<>) Q = o. These new forms under which the condition of least constraint is put, by which we had first defined the geometrical links free of passive resistance, will allow us to extend this definition, and thus make it applicable to a system where various solids are in contact with each other. Here, then, is the form in which we will put this definition: Let us consider a system of solid bodies which can be subject to geometric connections and which can also be in contact with each other; the connections imposed on this system will be said to be EXEMPT from PASSIVE RESISTANCE if the following hestisictioss are verified ------------------------------------------------------------------------ r° Au u corcrs du M")Mt'e/M, such that the equalities (38) equate to ------------------------------------------------------------------------ to equalities 't =g\ (Pu pu ̃- Ps), i ><) ) l''l = gïiP\, Pi, - - M Ps), '-"= gn(P\, Pi, -̃ Ps)- According to what we have said, the real modifications of the system. during the lapse of time during which we propose to study them, are all subject to the links made bilateral this amounts to saying that to each of the states really crossed by the system during this lapse of time, corresponds a set of values of /̃, r", /-" which are drawn from the equalities (3ç>) by a suitable choice of values of p,,p- .v From the state that the system presents at time t, a state that thus corresponds to a certain set of values of p,,pa, .)." we imagine a virtual modification subject to the bonds made bilateral [these are the only virtual modifications to which we had to apply equality (3^)]; such a virtual modification necessarily leads the system to a new state that is defined by means of the equalities (3q) and a new system of values, infinitely close to the previous ones, of the variables /> /?2, ps. Therefore, all the states of the system that we will have to consider can ('Ire defined by means of the s independent variables p,, p-2, ̃̃ ps- In particular, the internal energy of the system can be defined in terms of p,, />2.> \i Pï) ̃ ̃ -ifs- [̃-(' virtual work of inertia could be written ( 4v.) T = J, S/"| -I- J j 3/"2 H- -f- Js Ops, and, according to the equalities (28) of Chapter 111, we have (43) 2~~ ~dp, d~l 'dpT2' To say that equality (3^) must take place for any virtual modification compatible with the links made bilateral, is to say, by virtue of equalities (4°)i (40 and (4a)î that one must have, whatever S/3(, S/"2, vpSj 14i) (Pt~J,- d() ~Py-`p2+J~- dL1 ~P~+. £44) .1+ dn, l 2+ 2- °P2+'" .(P~j.)~. = o. This equality (44) is equivalent to s equalities which can be written, by virtue of the equalities (43). (V>) < d "pi dt Op, ==O> 't--(~ These equations have the same form as equations (4); they are the KyuA'j.'fo.NS nr. Lâchante roiiii j.ic mou\i:mknt n'iis systèam; j>k soi.rnr.s soumis a des liaisons hoj.o.\omes purkmekt "komktriquks ET IIKNUÉES DE KÉSIS'I.'A.\C.IÏ PASSIVE. Let us suppose that the external aclions P, P2, Ps which solicit the system emanate from bodies whose state is known at each instant t, so that V2, Pf are given functions of pt, p>, ps and t; let us suppose that Von is assured of the permanence of the links during a certain period of time. £K d cM I' opi dt dp\ àK d dK J dK d ûK s àp~s d~t dpl V topsj d(K - U) d oK p¡+ <.)/>( < d/> =0, P. (J(K - L ̃) d OK f d r, '"(K-l 1 d OK ------------------------------------------------------------------------ lumps; the equations (iÇ>) will determine unambiguously the motion of the system during this period of time if the state of the system and its motion are given at the initial time. Skcojxo case. We will now suppose that the links to which the system is subject, and which we have made bilateral, are not all bilateral. These links, whose number will again be designated by m, will be of the form "i o/'i -r- a-i or-, -)-+- an S/ = o, ( ,(. bi S/ -h b2 3/'2-t- 17,t Ôl'Jt =: 0. li o/'i -+- l-i or-i -t- -+- /" §/̃" = o. 11 Õ/'¡+ 120/'2+"'+ ln a/'n= o. The coefficients of or, or", Zru will have, in these equalities, known values when the state of the system is known. By means of these equalities, one will always be able to express the n variations o/'|, ôi\, or,, in linear and homogeneous functions of s = n - m independent infinitely small quantities qt, q2, .qs, by the equalities I S/-t = a] q, -t- °n < -+- j, qs, ('J7) 1 <5/-2 .= a2 g-i-H ?j<7j-h.̃+̃ atqs, o/'u = a,, gr, + p,,, us having known values when the calibration of the system is known. Finally, by virtue of equalities (8) and (ta) in Chapter III, the virtual work of inertia has the expression /dG dG ÙG (:u) t ~l, ~r + ~f,~ g._+.+ rlg:~ g Condition (3^) of this chapter must be verified for any virtual modification subject to the bilateral links, i.e. we must have the equality r;G\ èG\ 111 "t J ) gr + (~1z- tcz- ~gG" > gz+.. àG\ + (A,Ms-- Jry, tl. whatever c/ q". qs, or that we must have the s equalities 'A-"-=o 4 toG ()fJi As - Ms - - rr - f", (r>.) 1 0q\ ÙG A.<-".--=:o. These are the equations DE M. Appell l oim the mouviîmknt of ujv I). r. ------------------------------------------------------------------------ .BIS SOLID SYSTEM SUBJECTED TO PUHEME.NT GEOMETRIC LINKS, BUT 2X0 JX H0L0K0MES. These equations are no longer, like Lagrange's equations, differential equations; the quantities q\, g' c/s appear in them, as well as their derivatives with respect to time; but the coefficients of these various quantities have values which depend on the state of the system, and this state cannot be determined by means of s quantities of which q\, q' q\ would be the derivatives with respect to time. We will show, however, that if we give ourselves, at the initial instant t0 of a certain period of time, the state and motion of the system, the equations (52) unambiguously determine the motion of the system during the whole period of time, provided, however, that we admit the following assumption The quantities r,, r2, rH are analytic functions of t throughout this period of time, including the instant t0To prove this theorem, it is sufficient to prove that the values taken, at the instant t0, by the derivatives of various orders of 7- r.>, rn with respect to the variable t are determined without ambiguity. We already know, by hypothesis, at time t0, the values of ;̃ 7-o, ri, and of r\, r'2, r'n. The equalities (48), differentiated with respect to t, give n equalities of which the first is (53) r\= %xq\ -H $xq\ -t-+ j,^ d%i d$\ dut + - - - > - jj- are thus linear and homogeneous functions of q\, q, q's functions whose coefficients depend on the state of the system; one will thus know, at the instant t0, the values of /̃ /- /- provided that at this same instant, one knows the values of q\ q' q\ Now equations (5a) are true, in particular, at the time ta they are s linear equations in q'\ q[t, q'j: the determinant of these ------------------------------------------------------------------------ s quantities, in these equations, is none other than the product by ( - i f of the discriminant of the double ?.K of the living force, considered as a quadratic form in q\ q[t, q\ this discriminant is essentially negative, since the living force is a positive definite form of q\, q'o, q's; hence, the determinant considered is! surely different from o the coefficients of the unknowns and the known terms depend on the state of the system, q\ q'.n q\ and t on t. These equations thus make known unambiguously the values of q' q' ̃̃-̃,q "s at the instant t0. Hence, at the same instant, the quantities/ i\, r "n are determined. By differentiating with respect to t the equalities such as (53), we will obtain n new equalities thanks to which the values of /- ./̃̃ r"'n will be known at the instant t{), provided that we know, at this same instant, the value of q", q' q "s. Moreover, by differentiating the equations (52) with respect to t, we obtain s linear equations in q" q' q" the determinant of the s unknowns in these equations is still the product by ( - i)-* of the discriminant of the form 2K; the coefficients of the unknowns and the known terms have values which depend on the state of the system, on q" qt, on q" q' and on t these equations thus make known, at the moment, the values of the s quantities q"[, q".v < and, consequently, the values of the n quantities /- /- /|f. Continuing in this way from one step to the next, we will prove the stated proposition. This proposition leaves room for a very serious doubt. When we give ourselves, at the initial instant of a certain lapse of time, the state and the motion of the system, we know that the system cannot, during this lapse of time, take two distinct motions where /̃", /̃" are expressed as analytic functions of t but we do not exclude the possibility of one or more motions where /̃", /̃" would be expressed by functions of that would not be analytic for I. = to. If 1, ojv is suitable DE fai m; abstraction ni-: c.v. noniK, we can say that the condition (%), applied to bonds made bilateral, is sufficient to determine coinpli'lemcnl the motion of a /1 system of solids subject to ko lo no me s or non liolonomous bonds, provided that we are given the initial state and the initial motion of this system. ------------------------------------------------------------------------ ̃i. Lagrange multipliers. Linking actions. The doubt we have just pointed out can be removed provided the system satisfies a certain condition; moreover, this condition need only be applied to it if some of the solid bodies which compose it are in contact with each other; if. the connections are purely geometrical, what we are about to say is true without restriction. In order not to weigh down our presentation by displaying a useless generality, let us first reduce our system to two solid bodies Ci, C2 which touch each other and which we will suppose independent of any foreign body. This system admits a certain internal energy L this internal energy varies in a continuous way when the relative situation of the two contiguous bodies G(, C2 varies in a continuous way. Let us take, on the other hand, these two bodies G,, C2 disjoint from each other; they then form a system composed of two independent parts, and these two parts are two rigid solids; the internal energy of this new system is equal, by virtue of equality (i), to the mutual potential energy W of the two disjoint solids. || We will be able to move continuously these two disjoint solids until they come into contact with each other in this continuous modification, the internal energy of the system must vary continuously M' must therefore have a limit U, hence this proposition When two solids, initially disjoint, gradually move until they come to squint at each other, their mutual potential energy tends towards a certain limit which varies continuously with the relative position of the two bodies in contact; this limit of V mutual potential energy of the two bodies dBjMûin'ttyis the internal energy of the system of the two solids coivtig^^ This proposition does not imply any new condition imposed on the system we are studying if a quantity W, determined by the mutual position of the two disjoint solids, does not hold for the system. ------------------------------------------------------------------------ The conventions concerning internal energy would forbid us to take this quantity as an expression of the mutual potential energy of the two solids. Let us suppose, for example, that one of our two solid bodies is reduced to a material point M, while the other remains a solid C of finite dimensions. (54) iP= /Ê-afo, I '̃'̃ where ii' drs is a volume element of body C; /̃, the distance from a point of this element to point M a constant; p, a quantity which has a finite value at any point of the body C. Such an expression can be admitted if A has a value less than 3; in this case, indeed, it is easily shown that W tends to a finite limit when the point M comes to lie on the surface of the body C and that this limit varies in a continuous way with the position of the point M on this surface. On the contrary, if is equal to or greater than 3, the quantity W, given by equality (54)? grows, in general, beyond any limit when the point M is placed on the surface of the body C; this quantity is then unsuitable to represent the mutual potential energy of a solid and a material point. Here is now a condition which does not follow from the above, so that some systems, not excluded from the above, may well not verify this condition The position of the two solids, disjoint from each other, can be defined by means of \->. independent variables 1-2, ~<2 ) the erzeo~-ie /~e/?"'c~e 7~!Mr leagues -r-, - "---) tend to limits determined ri ~~l 2 -r2 when the two solids are approaching each other until they come into contact. ------------------------------------------------------------------------ This condition can also be stated as follows When the two solids come close to each other until they come into contact, their mutual actions tend towards certain limits. (,)his condition does not necessarily follow from the previous one, this is what an example will show. Let us take the system formed by the solid C and the material point M, and suppose that their mutual potential energy is given by the equality (54); the position of the solid can be determined by means of 6 independent variables rB and the position of the material point can be determined by means of its coordinates x, r', z. Consider the three quantities dx J 7-X-t-i Ox ----- = -- < ---- -- clrs, t ~)l1' /-).1 dr V) ) I - = - A - - - dm. ( i i Ï y cha. .,)) toW r a ùr dm, - = - A - £- dm- d-s ~.),+i If is less than 2, it is easy to show that these quantities tend to finite and definite limits when the point M tends to the surface of the body C but, in this same circumstance, they generally grow beyond any limit if X is equal to or greater than a. We can consider a system formed by a solid and a material point whose mutual potential energy is given by the equality (54), the constant À satisfying, moreover, the condition '<~<3. Such a system will not be subject to the condition that we have just stated; the considerations that we are going to develop will not be applicable to it. Let us become the system formed by the two solids C,, C2 in contact. To these two solids, let us impose a finite displacement D that leaves them constantly in contact. Let's imagine that this move consists in varying the quantity t from to to l, in the formulas r '̃| =/]!'), r, =/j!/l, li, /-> -/l.(7). ------------------------------------------------------------------------ In this displacement, the internal energy of the system experiences an increase ( U( - IJO j. Let us imagine that we disjoin the two solids C,, C2, while leaving them infinitely close to touching each other; to these two solids, let us impose a displacement, A which differs infinitesimally little from the displacement D, but which leaves them constantly disjoined. Let us assume that this displacement is obtained by varying L from t" to I in the formulas ''1= ?i(<0> rî= d(> -̃̃̃> f dt In this displacement A, the mutual potential energy of the two solids experiences an increase (Wt - ^"n)- We have obviously r'WàW do, 1 âW dis, dW don\ t=J,. (^tr + ^if + -4-Irr'- '1. dr t dt + dr2 di +.+ d~ dt ~'t. When the displacement A tends to the displacement D, Wo has limit Uo cl. W, has limit U, so let T. c r- (}Ir R Lim- - == - K|, >AtaZ = - "i2i 0rt drvi the previous equality will become, in the limit, n n y t, (I~' +R~dt2 Il (~fl2 di. UI-Uo (~+R,t-R, In other words, in any virtual modification that leaves the two bodies C(, G> in contact, we have (̃)G ) SU = - ( Ri o/-t --y- R2 o/-2-H.i- Ru orn). Let us consider a system composed of solids, in number ------------------------------------------------------------------------ subject to purely geometrical connections if some of these solids are in contact with each other, let us assume that the restriction we have just specified is verified. Let us propose to express that the condition (3^) is verified for all the virtual modifications compatible with the links of the system, after these links have been made bilateral. The virtual work G of the external actions can always be put in the form ( :>; ) © = Qi 3/ -+- Q2 on -+-+ Q" 5rn. If the state of the foreign bodies in the system is given at each instant, Q(, Q2, Q" will be given as functions of t and >'l, '"a> ̃ -) l'n. By an equality similar to equality (56), we may write (58) oU = - (Ri 07-f- R, 3/-J -+-+ R" ?/̃"), R,, R2, R;/ being functions of r,, r-2, /̃"- Finally, we have ~')()) -c 2 with dK d àK JFX~ 7û'dîJl' r)K d dK (60) 2 tor, dt dr'2' dK d dK ""j~ By means of equalities (37), (58), (5g) and (60), equality (37) becomes (61) RI ~K d <)K\. W dt " <)K d dK (Q2 R2 dK d' dh iir2 dt ài\ I - T "dK d dK" -1-(~rt+~ri-f-ÛK -dt dh~GJnt,=O. ------------------------------------------------------------------------ This equality (34) must take place not regardless of o/ or-or, but for any system of values of or, or-j. or, " which verifies the m linking conditions ax o/'i -r-"2 o/v - .+- ",, o/j = o, '/6 ] 61 0/+- 6j O/-o~t- A;, 07-,"= O, (.11) [ li S/'i -+- /2 or., - -t- lit o/ = o. According to the theory of linear equations, it is the same to say 11 exist neither quantities or, jî, A, independent of 3/ ûr2, orn, such that by adding member by member the equation (61), the first equation (46) multiplied by a, the second equation (46) multiplied by [3, the last equation (46) multiplied by X, one obtains an identically true equality, whatever are S 8 ;- ,or The quantities a, [3, are called Lagrange multipliers If we put lïi = aa, -+- J36, -4- -h- A/], (6a) ns = art2 -+- p 62 -+-; -4- À /2, 1T,,= aa,, ̃+- fift,i-+- >", the proposition we have just stated will be equivalent to the n equalities "dK rf toK Q1 + Rl + 1,i._=0, d/'j <" d/'j or, (03) QV__ k,-h II.h- = o, 2 t ur2 " :, Il,, represent, according to the equations where they appear. ------------------------------------------------------------------------ either the actions that the various solids composing the system would exert on each other if they were free, or the limits towards which these actions tend when some of these bodies come to touch each other; it is, in particular, this last meaning that is admitted in equations (63). In any case, we can give these quantities R(, R.2, R" the name of actions inside the system. With these denominations, equations (63) lead to the following proposition The equations of motion of a system of solids subjected to purely geometrical connections can be written in the same form as the equations of motion of a system of free solids, provided that to the actions, both internal and external, which really stress the system, we add the fictitious actions of connection. Let us read the properties of these linking actions. Here is a first one In any virtual modification compatible with the system links, made bilateral, the link actions do zero work. Indeed, for any system of values of o/1,, o/'2, or,, which verifies the equalities (46), we have M'>1,' III S/-i Ho 3/ -+-+- fi,, or,, = o, as can be seen by replacing II(, IF* Il,, by their definitions (62). The boundary conditions (46), applied to the real change of which the system is the seat between instants t and (t-i-dt), give the equalities, verified at each instant of motion, I ", r\ -r- <72 /̃ -i- a,, r), = o. We b, r\ -i- b, r\ bn >̃' - o, -li.T I>r4+.T.l,t /- h r' -t- -i- /" ;- = o. Differentiate these equalities with respect to l; we find from ------------------------------------------------------------------------ new equalities, verified at each instant of the movement. ", r\ -+- a2 r\ -+- an /-"" 1 dat dat dan }' dl dl 2 ̃- dl >ll~ o> (66) < j a 'i + ?.'-!+̃̃̃ /"'- ( c?/i c?/" dl,, ,+- `lt dt 2 dïl'll=o. The coefficient a, is a function of /- ri} /̃"; -j-i is thus a linear and homogeneous function of /̃ r'2, r'H; the first member of each of the equalities (66) is thus the sum of a linear form in r"{, ri, /̃ and a quadratic form in r[?" r'n the coefficients of both of these forms are functions of r{, r2) /;" Let us consider, on the other hand, the equalities (63) the first member of each of them contains, besides the sum (Q,- + Rj-+II/), a quadratic form in /̃ r'^ r'H and a linear form in r "t, r "2, r "n. If we take these equations as linear equations in ';̃ their determinant is the product by ( - i)" of the discriminant of the double 2 K of the living force considered as a quadratic form in /- ;- /̃' it is thus a function of i' /'2, rn essentially different from o. One can 'then solve these equations (63) with respect to /'j. /- /-- in the form (6/ ) '- = pi, r\ = pS) ;̃ = p". Each of the quantities p(, p2) - - -> on is the sum of a linear form with respect to the quantities Qt + RI 4- n" Q,+R,n,, Q,R,,+n,, and of a quadratic form in /- /- r'H the coefficients of these two forms are functions of r2, /'"̃ Let us carry over the expressions (()-) of r "t, /- into the eqs (66); let us observe that R,, lia, R,, are functions of /̃(, r", /̃" that, according to equations (62), II, Il2, H,, are linear forms in a, [ï, À; we will obtain new equalities the first member of each of these equalities will consist of ------------------------------------------------------------------------ î" Of a linear, but not homogeneous, function of Qt, q,, Q", a, 3, a; > Of a quadratic form eu /̃ /- ;". the coefficients of these functions will depend on /- r-2, ̃--, /'"- These new equalities constitute a system of m linear equations with respect to the m Lagrange multipliers a, [3, we will be able to solve them by contributing to these multipliers each of which will become the sum of a linear function with respect to Qj, Q2, Q,< and a quadratic form in /- i\, r'n. If we transfer these expressions of a, [3, .)" into the equalities (62), we will obtain new equalities which will justify the following proposition Each of the linkage actions is the sum of a linear, but not homogeneous, function of the external actions Q,, Q2, and a quadratic form in /̃ )- r'n the coefficients of these functions are known functions of /'(, r2, r, If we transfer the expressions of O,, JT2, that we have just obtained into the equalities (67), we obtain equations about which we can say cec,i Each of the second derivatives ;̃ /-" rn of r,, /'2, -- rn with respect to and the variable t. is equal to the sum 1" D' a linear and non-homogeneous function of the external actions O,, Q", (v),, a" Of a quadratic form in r\, /- r'u. The coefficients of these two forms are functions of /-, r2, ̃ ̃ r, This proposition, brought together with the properties of differential systems, leads to the following theorem Let there be a lapse of time during which the bonds imposed on the system certainly do not undergo any abrupt change. The state of the external bodies is assumed to be known at all times during this period, so that the external actions Q,, Q. Q,, are known functions of /̃ /-i, /̃" and t. ------------------------------------------------------------------------ If we give ourselves, " the initial insluitt., the state of the system and, consequently, the values of r,, r.2, rn, and the initial motion, which entails the knowledge of the values of /̃ r' r'n, the motion of the system is unambiguously determined during the whole period of time considered. This important result does not exhaust the consequences that Ton can draw from the method of Lagrange multipliers; up to now, in it, we have been content to express that condition (3-) must be verified in all the virtual reversible modifications that can be imposed on the system; we will now express that all virtual modifications, reversible or not, are subject to condition (35 bis). If we take the system at any instant of its motion, provided that this instant does not correspond to any sudden change in the conditions of connection, to any sudden modification, equations (63) are verified at this instant. Let (or,, S/-2, or,,) be a virtual modification, reversible or not, imposed on the system from the state it is passing through at this instant; let us multiply the two members of equations (63) respectively by or,, ùr-2, S/ let us add member by member the results obtained, taking into account the equalities (07), (58), (5g) and (60) which are exact for any virtual modification, reversible or not; we will find the equality, applicable also to any virtual modification, (68) e-4-l-r - eui-H nt 6/hiI23/-o-t- Il,, o/ = o. Using this equality (68). the condition (35 bis) becomes (O9) n, ër,~h II2 S/-2-+-H fl,, or,o. In any virtual modification imposed on the system, the work of the linking actions must be zero or positive. We already know that, by the very definition of linking actions, the work of these actions is zero in any reversible virtual modification. The condition we have just stated will therefore only provide us with new information if we apply it to non-reversible virtual modifications. ------------------------------------------------------------------------ Here is a case where it will give us a useful indication Consider a system in motion and suppose that it is subject to certain unilateral linkages, to which bilateral linkages may also be added; at each instant t, let us calculate the virtual work of the linkage actions for all the non-reversible virtual modifications that can be imposed on the system up to the instant t0, we find that this work is always zero or positive. On the contrary, at times after tfn if we assume that the motion of the system, still subject to the same binding conditions, continues according to the same equations (63), we would find that the work of the binding actions is negative for some non-reversible modifications imposed on the system. From what has just been said, such a consequence would constitute an impossibility; from our calculation we would have to conclude that the motion of the system cannot continue, after instant tu, to be subject to the same binding conditions, and that at least one of the unilateral bindings imposed on the systems is broken, at the latest, at instant t0. Let's give a very simple example of the above Let a material point M, of mass m and rectangular coordinates x, y, z, which moves on the surface of an impenetrable and immobile body be (70) f(x,y,z) = o the equation of this surface, and imagine that the impenetrable body fills the region of space where the inequality f(x, y, -s)ny", - mz'. The condi- ------------------------------------------------------------------------ lion ( 3^) is written ̃) (X - mx" ) Sa; -i- (Y - niy" ) 'JJ' -r- ('/̃̃ - niz" ) oz - o. It must govern all the reversible -nodificalions of the system in other words, equality (72) must be verified whenever ox, oy, oz verify the equality ( ) Y - i- Il, - my" - a, Z -+- II- - mz" - o. Let [A be the positive quantity defined by the equality (76) ;i w+W) ^(sij- (.~6) 1.1.2 = ox + -)° + Let N be the normal to the surface of the impenetrable body, this normal being led to the unlimited space which surrounds this body be y. 3, y the cosines of the angles that this normal N makes with the axes of the coordinates; we will have (7-) _r 8=~ t Jf (77) a = - ---, B = - > - t~> ')̃ But, in any real modification, the work of the inertial actions is equal to the decrease of the living force: legality (3~) thus becomes, for the real modification considered, (79) Ç = il i i K ,1. When a system of solids, subjected to connections free of passive resistance, undergoes a change during which the motion that drives it and the connections to which it is subjected do not undergo any sudden change, the work done during a certain time by the external actions is equal to the increase in the total energy during the same time. This proposition is known as 1.0.1 01: i.\ co.vskuvatiojv OF the vivk force: the reason for this denomination is found in a corollary that we will demonstrate. Let us suppose that the foreign bodies to whose action our system is subjected remain absolutely invariant, while the system is moving. In this case, the work E of the external actions exerted on the system is (''gai (p. i3o) to f/, designating by 'I' the mutual potential energy of the system and of the foreign bodies, The equality {'<̃)'> could be written d['\" -+- U h- K) = o.. | It will express that the value of the sum (<&+- li + K) does not change during the modification. This being the case, let us imagine that at the instant the studied motion brings the system back to the exact position and state it was in at the instant /" the soniine($+,L) will take back at the instant the value it had at the instant t0, so it will be the same for the living force K.. Hence the following theorem A system, formed of solids subjected to connections without passive resistance, is isolated or subjected to the action of invariable foreign bodies. If a movement, free of sudden modification and abrupt change in the connections, makes it ------------------------------------------------------------------------ /To make the same stall, at two different times, it cotnmunit/ue him, at these two times, the same living force. (>. Of the rupture of the bonds. In the foregoing, we have constantly studied the motion of a system during a period of time in which no break in the linkage occurred. We will now investigate when such a break can occur and according to what laws. Up to time t0) the system is assumed to be subject to m links, some of which are unilateral, ai o/'i -H a> or2-i- an orn = <>, 1; ~3r~ ) br oi-i - bt 3/2 -H. -+- bnorn>(>. 80 ) ôrl+ l~ ~l' l,t ôuyt-o. l\ o/'i -H- Iï o/i - - ln oru i. o. The general velocities y' r. r'H thus verify, for any value of t before tn. the M equalities ( a, r\ -+- a, t\ -±- H- "" r'u = o, H J blr\-h6ir'i~rbnr'n=:o. [ h i\ -i- r'3 -i- -t- larn = o. In particular, it follows that the limits /- r "n, taken cfir, d^ri d1)- e, by ̃~[pr>~Tjï'> when t tends to lu by values inside t0, verify the relations rtet\ da" dan ~~dt'dt '̃̃̃ ~~dt' +- "!/̃'[-- a, /t-- aHr "Hz=u, db, dùs r~ -+- db,, = o. ~dtri^dt'1~dt'" (8"0 1 -r^i -r-f cori-es- ------------------------------------------------------------------------ We will show by the work that these actions efl'ecl'ieni in any virtual placement subject to the conditions (!So). Let us now suppose that at the instant /". the svstrme breaks any one of the unilateral links to which it was subjected, for example. the first of the links (fol, and that. from the instant t0. it moves, at least for some time, while remaining subjected to the other links " 80). We know (p. igti) that at time t0 the general velocities /- /̃ /-" undergo no discontinuity, but the general accelerations may be otherwise. Let us therefore denote by; " u h <̃ h 1 dir\ dlr-< d'2' p' the limits toward which tend -777' ---" -jpr when t tends to t(l by values greater than tv. H is clear that we will have da% rf "ï da,, ~dT''i'+"dr' 2 ̃̃W" -r"\ fî" -T- "" p'. -- CI,, p' (). f/ùi db2 M" ~dfri~~d7ri dl ''̃- Il (83) '̃- dlx dl. dl,, ~dT''1^ ~dJ' ̃' di -r- h? "l ̃+̃ U ?ï -̃ /" ?'" t). ̃ da due dl ai of 1 n. r The quantities -jf'-jf'i 77 depend only on /- /- /-". 7- r! r'n: each of them has therefore the same value in the equalities (82) and in the conditions (83). v " rf5/ d*r, d-r,, At values ?). p" of-^7T' -JjT1 ~di^ coiiesl>oiulent inertial actions -> ï2, ->". JXWe will denote by G the work of these actions in a virtual displacement compatible with the conditions (80). From the conditions ( 82) and (^80). we get f ffi<'?',-f-o,(~-r'~--(?f;, - /̃ = < ("'l) < t '"(p'i - ''ï) = /i(?s- ''s >-- t>,( :" - '̃"")- o. ------------------------------------------------------------------------ These conditions lead to the following consequence If we designate by an infinitely small positive quantity, the equalities (8)1 or-,- --([̃>[ - '-), or, - -.( îj - r'^ ). or,t ̃= i ('/" -r],) define a virtual displacement compatible hic with the conditions (Ho); in this displacement, the first of the bonds (Ho) may be broken, while the others are surely maintained, livaluons. in this displacement \irlucl, the value of the (lillérence (86) - 0 = (.1, - -̃)!) or, -f- ( Js- -")j) or., ( Sn- )H ) or, Let t (87) K= 2 l'/y/V, go J where the summation extends to all values 1, a, of the index 1 and of the index t'expression of the living force of the system. We can easily see that we can write I OO) ( Ji = -(Pn /--+̃ F>1S r\+.nr'n)-r-ll, /-̃>i = (Pu p'i h- i'u :A -+- .m- !'<"?;; -+- h, H being a function of /̃(, r. rn ex of /- /- /̃" which has same value' in both formulas. Equalities (85), (86), (87) and (88) then give the equality .( 89) " = *>] P/7(?ï- 'V 1 '?} 'y)- Let us follow the consequences of this equality. At time t0, the system is subjected to all the links (80); the virtual displacement (85) is perhaps, for such a system, a non-reversible displacement. Ç1. h- z - SU f o. After the instant lQ, the system is only subject to the last (m - 1) conditions (80) the virtual displacement (8:V) is surely, for such a system, a reversible displacement, so that we have 5 -+̃ 0 - 6U = o. ------------------------------------------------------------------------ G cl ol represent the same quantities in these two conditions, so that we can write (<)(>) - 0 ->. But the form K, given by I equalized ( 8~), is a positive definite lorin in /̃ /- r lt. From then on, equality (8<)j is only compatible with the condition (<)<>) (jue if we have I == f 1 '-> ? i ̃ ̃ ̃ ? At the moment when a system breaks a unilateral link, not only the general velocities, but also the general accelerations do not undergo any sudden variation; the velocity and acceleration of each of the points of the system do not experience, because of this break, any discontinuity; the trajectory of each point after the break is in agreement with the trajectory before the break, in such a way that the two trajectories have not only the same tangent, but also the same curvature. the equalities (88) give us then J i = 2> i so that for any virtual displacement compatible with the conditions (80) we have (i)O = 0. Consider, in particular, a virtual displacement that breaks the first link (8o) and respects the other (m i ), i.e. a virtual displacement that verifies the conditions "i 5;'i + "4 5/j -r .-- "" o/ > o, (" I b\ o/-| - bi or2 -T-. - ̃+- />" or,,- o. ~)I) ~(Il' /|0/'|-r- /o 0/ ;.-+- /" G/ = O. For I system such as it is at the instant after /" and so close to /", this displacement is a virtual reversal displacement to which applies the condition ¡: -j-- fi -U =0. ------------------------------------------------------------------------ But at the instant read, the equality (()o)esl, xerilized; we can thus affirm that one has, for any displacement \irlurl subjected to the conditions ( () i i. the equalized < \yi) Ç - H -̃ o. Hence the following proposals It is impossible for a moving system to break any of the unilateral links to which it is subject as long as the quantity (S-j-t - ol,:) remains negative for all non-reversible virtual displacements. For the system to break one of the unilateral bonds to which it is subject, the quantity (£̃-+--- olT') must be zero in any non-reversing virtual displacement that breaks that one bond and keeps all the others. Let us take, in particular, a system to which the method of Lagrange multipliers is applicable. When t tends towards l" by values lower than tu. the actions of connection tend towards certain limit values which we will name their values immediately before the instant. to and which we will continue to designate by IT a H2 IF, Let us consider a stall taken by the system at an instant previous to tu and as close as we want to t0 from this state, let us impose on the system a virtual modification, reversible or not remersable; we will have, in this modification, l 08 ) ê -f- - 5U -4- II, or, -4- II, ô/-j .h 11,, ->/-" = o. Therefore, the previous proposition can be stated as follows For a system to break a one-sided linkage at a given time, the virtual work of the linkage actions must be zero in any virtual displacement that breaks this linkage and respects all others. This proposition can be put in another form. Let us consider any virtual displacement which breaks one of the unilateral links imposed on the system, for example the first link (Ho), and which respects all the others; we have for a ------------------------------------------------------------------------ Here displacement, ll\ 0/ -7- r.l-i ',l-± "" 0/ (). Oi ô/ - h, '>̃> - h,, '>" - (i. o/1, or" /" '>" - o. The (m - i) equalities i|iu are among ct;> conditions (';>-> joined to the equalities (<>2), show (|iie, for the considered virtual deptacenK-ut. (94 "i o/'i - i)2 0/-J-+- .i- II,, o/ = "i "1 "'i - "1 or, - a,, ). requires that we have 7. :O. \If we take care to write the conditions of unilateral links imposed on the system in such a way that the breaking of a link has the effect of making the first member of the corresponding condition positive, no La-grange multiplier relating to a unilateral link can take a negative value. The two propositions we have just mentioned lead to the following corollaries The unilft.tera.le bonding conditions being written as assumed in the previous statement, we assume that at a certain ------------------------------------------------------------------------ If, at any instant, all of Ut grange's multipliers related to these unilateral links are positive, the system will certainly move without breaking any unilateral link, and this until one of Lagrange's multipliers related to these links is cancelled; let us suppose that at the instant t(l. one of these multipliers cancels; suppose, moreover, that it becomes negative at instants after t0, if, at these instants, one were to force the system to continue its motion without breaking any unilateral link at the instant ln, the system will surely break the link for which the Lagrange multiplier cancels. If, for example, the first link (80) is unilateral and if we have, at time l0, dt a = 0, dl < o, this link will surely be broken at time t0. If we apply this proposition to the case of a moving point on a surface, a case already treated in the previous paragraph, we can state this theorem For a point, moving on the surface of an impenetrable body, to leave this surface at a certain irasturtt t", <7 /ri o/'i -t- I), or, - bn ô/ = o, (9-ï) -- l\ Ol'i -t- l-i 0i:> -r- -r- /" --; Or,, - O. Ir fJl~l'Y /< C/'2 - -r- l,t 0/ -= I1. As for the linking condition introduced, it can be unilateral or bilateral. Let l0 be the instant when this link is introduced, and let us consider the modification [J. which would consist in bringing back the system from the state it presents at the instant (" to the state it presented at the instant (/" - h), ), //being an infinitesimally small positive quantity; this modification ul is the inverse of the real modification undergone by the system between the instants (to'-h) and tt)] this real modification was compatible with the conditions of linkage (<>5): as these are bilateral, they will also permit the modification u. Therefore, for the modification p. to be a virtual modification of the system, it is necessary and sufficient that it be compatible with the link introduced at time l0. At time (t0 - ). the system was not subject to this binding. The modification -jji dt cli -7^ tend to limits which we will denote by rn\ the components of the velocity of a point of the system tend to limits which we have already agreed (p. 176) to denote r by ̃ w)7 1 vt, iv,. 1 ̃ '1 d-rx When t tends to t0 by vareùrs greater than /0, > '-- tend to limits which we shall designate by R' R2, R, Since the real change accomplished between the instants tn and (/." -t- h) is certainly compatible with the conditions (()."> ) 1 and with one of the conditions (f)t>^ or (97 ), <"n a al H, - a~ li 2 <'" 1- > CI. u b, R', r; bn r;, = /,r; /2R,-f- - /r;, = o. ------------------------------------------------------------------------ The first condition (98) reduces to an equality if ta, introdiiilt' link is bilateral. At the same time, when tends \er.s /" by values suptricuro to lln the components of the speed a point of the system tend towards limits that we already agreed i p. iHii to represent by "2, i'2, w". From then on, the segment M0M will have as components it-.lt. v-j.li. tVoh. If, at the time /", we were to remove all the links that hold the system together, which assumes that none of the links come from the mutual contact of two solids of the system, the velocity of each point would remain continuous, so that the components of the segment M 0 i\ are u,/i. v,h, a, h. Finally, we denote by A,r, Ajk, A^ the components of the segment M, these are the components of the displacement undergone by the point imtially located in Mo, while the system undergoes any virtual change. Equality I'1 ~f~ M P 2M~>Y))'")s~F' fi allows to write .{ 99) f P^2 dm - r ~ÛN~ dm - \(u2h~ ix)' 1 (̃- -̃ A)' "5- t"% A3)!] o?m -i.lt I \( ti* - "1 1 ( u,/i - A. ) - 1 c5 - i| )( f" h - \y -(."'; ̃ it-| 1 1 ipoA - A; t| rf/H. The second member of equality (Ç)f)) must be positive regardless of Je de|)lacemcnl \irluel A.f. Ay. A;. Let(o/-(, o/-2, o/) be an irlual displacement of the system: it derives from the conditions a, or, fi" or., - a,, o/ o. f.oo) A'r> A. "> --̃̃-̃ A- '>"- "̃, -h /.3r5- .- /" o/ -= i>- the first condition reducing to an equality if the introduced link is bifacial. This virtual modification imposes a displace- ------------------------------------------------------------------------ cern d. oy, rice- to the point of the system that was priniilivement in M, Let z be a positive quantity, inliniiuent small (Ju same order (pie o/1,, r)rn. and any besides. By virtue of conditions ( 98 1 and 11001. it is clear <[iie we will obtain a new virtual inodiliealion of the system if we posit A/ = ( IV, ^p ) h. \r, = f R, ) h A/ = ( H, -j~ ) /". v ¡ This modilication leads to a displacement .1 :x .1 ;r- i Aa? = ( 11, -H 21 j \y ( t.a ." IL ) A; ( ii'i j h of the point which was at first in .\[0. so that it has to take at the second member of legality 1 (//n ..<2 l f., y,tt,-zt.j%.x--(u,-v_~a.l~u:m,j;tlnz. - - [(M|- M3) oa: - ( i", - l'îfrjy- in-, - ii'2;o;lrf/H. This quantity must be positive whatever the infinitely small quantity and whatever the virtual displacement imposed on the system; hence the following conclusion For the law of least stress to be observed in a sudden modification of a system, it is necessary that the virtual effect of the kinetic beads is zero or negative in any virtual displacement imposed on the system from the stall it presents at the time of the modification, sudden (101) 7- [(m,- U* ) r,.r -J- ( V, - t-j) 3k -h f n-| - ir5 ) of] dm o. From this proposition we can easily derive this corollary If the sudden change consists in the establishment of a new adhesion, it is necessary and sufficient, for there to be a minimum constraint, that the virtual effect of the kinetic losses be zero in any virtual displacement imposed on the system (102~ = < [/i. \±li. kv.,Ii el. consequently, of Zx = H4/t - lx\ or - v-tli - Ac, Zz -- .) thus cancels the last ternary in the second member of the egabler (99), so that the first member is definitely posihi. If we compare the result obtained with what was demonstrated at the end of the previous chapter (p. ". we see that the condition of least conlrniitte fixed. ciilièn'Dicnl the laws c/ui governing a rnodijicfition. sudden when this modification consists in the establishment of a new adherence It remains for us to examine the case where the sudden modification is a shock. We will establish the following proposal Even though the link introduced at time I /" is a unilateral link, the absence of passive resistance prevents the infinitesimally small modification accomplished between times t0 and (Iq /i) from being a non-reversible modification breaking the link at the very moment it is established as it would armor t for two hard bodies that would shock each other and bounce back immediately. Let us suppose, in it! that the real modification accomplished between the instants /" and ( In -j- h ) soil a non remersable modification which breaks the link (tj T - - s Z The modification (itj6) is then a virtual modification in which (ioo; Ax = m2/i - ). 5a", Aj' = v, li - X oj'. As = ivoA - X ox;. For this virtual modification, the second member of equality (99) becomes X* {( ------------------------------------------------------------------------ and naked second in which oa:" - U\ h or = - vx h. Zz - - irj One would thus derive from the preceding e^alile lu non vc:lle equality J \(u, lt~2)S-r- ( <'| - <'j)2H- lll'i - It w I- | ,vhe Cvkmot In any sudden modification of a system of solids whose connections are examples of passive resistance, the living force lost is equal to the living force due to the lost velocities. The living force lost in the sudden change is equal, according to the definition given in paragraph 5 of the preceding Chapter (p. 173), to the quantity of heat released in this change. On the other hand, if there is a sudden change, the living force due to the lost speeds is essentially positive. We can therefore join the following corollary to the theorem of Lazare Carnot Any sudden change in a system of solids subject to bonds that are examples of passive resistance is accompanied by a positive release of heat. On the contrary, in such a system, any real modification accomplished gradually results in a release of heat equal to o. The preceding proposition completes all the principles of Rational Dynamics. 8. Rational Statics. Since equilibrium is a special case of motion, the laws of Rational Statics necessarily follow from the laws of Rational Dynamics. ------------------------------------------------------------------------ Now, we know that It: inoiiv émeut, of a system of solid* invai'iahles. subjected to bonds examples of passive resislance, is entirely determined, starting from the instant /", if one gives oneself at each instant lt, posterior to /", the standard of the foreign bodies which act on I system and if one gives oneself, in addition. I Y-tat eu of the system at 1 instant there and its movement at the same instant. Let us therefore give ourselves a state e0 of the system and a set of foreign bodies maintained invariable; at the moment /", let us place the system in the e0 standard, with a zero motion so that i .Y: lai ea is a state of equilibrium of the system in the presence of the foreign bodies considered, it is necessary and sufficient that we can satisfy the conditions of Dynamics by introducing these data and by supposing that the system remains at rest. The conditions of Dynamics are all condensed in these In any virtual change imposed on the system, we have ÇVjbis) g .+- iîTj < ci. Now, if we suppose the system to be at rest, the accelerations of all the elements that compose it are zero, so that the same is true of the inertial force applied to each element and of the inertial work in any virtual displacement. Therefore, in order for condition (35 bis) to be compatible with the maintenance of rest from time t0, it is necessary and sufficient that it remains verified when the virtual work of inertia is removed. Hence the following proposition For a system which is formed of solids subjected to connections without passive resistance, and which is placed without movement, in a certain state, in the presence of invariable foreign bodies, to remain in equilibrium in this state, it is necessary and sufficient that in any virtual modification from this standard, the work of the external actions is at most equal to the increase of the internal energy (no) G - ol)=o. This is the principle of Rational Statics. tya-ns the particular case where the system is subject to exr/usive- ------------------------------------------------------------------------ to bilateral connections, it is necessary and sufficient, for the equilibrium, that the external work is, for any displacement, virtual, equal to "the increase of the energy inlei ne (III) F - ol = il. < '.es propositions constitute the i-iunch'e of the oéi'i.u.iîvien rs \nt-iikls, often named wrongly i'Iuncipe des vitesses v irtleli.ks. From this principle, we derive, for the various cases that we may have to consider, "the conditions of equilibrium, equations or inéq nations, (ju it is useless to write, because they are always obtained by billiating the outfits relative to the mertie aelions in the corresponding dynamic conditions. The comparison of the conditions ('35 bis) and (i 10) turns the following statement, which is that of Alemiikht's viujncipk uE To obtain, at each instant, the laws of motion of a system of solids subjected to "connections without passive resistance, it suffices to write that the system would remain in equilibrium if it were placed without motion in the standard it is passing through at that instant, and if it were subjected not only to the external actions that are really exerted on it at the moment it is in that state, but also to fictitious external actions equivalent to the inertial actions that are soliciting it at that moment. All the laws, dynamic or static, of the Rational Mechanics are now linked to the General Energetics. 9. Systems of weighted solids. Among the simplest systems to which the laws of rational mechanics can be applied, we should mention, first of all, the systems by which we give a schematic representation of the properties of the heavy bodies that surround us; it goes without saying that this representation, by the very fact that it is very simplified, only gives a roughly approximated image of reality, valid only in restricted cases. ------------------------------------------------------------------------ The bodies studied are assimilated to rigid solids, free or free of bonds, examples of passive resistance. These bodies are supposed to have no action on each other, so that if we form a system, the internal energy of this system is constantly zero. The foreign bodies <|which act on a lel system are supposed^ filled one of the two regions into which a plane divides space; the bodies studied die in t other region; to these foreign bodies one gives the name of earth, to the plane which limits them the name of level of the ground; the normal to the level of the ground, directed in the direction opposite to1 the earth, is the vertical directed towards the zenith or upwards; if one directs it in the opposite direction. it is directed towards the nadir or downwards. Let us take a material point, that is, as we have seen, a body whose lethality is entirely defined when the position of one of its points is known: let us suppose that this material point moves above the ground; we will admit that the work accomplished by the earth in this displacement depends only on the mass of the material point and on the decrease of the height of this point above the ground, without depending, in particular, on the initial height: it is positive when the point approaches the ground. Finally, we will admit that the work accomplished by the earth in the displacement of a solid is the sum of the works accomplished in the displacement of each of the elements into which this solid can be broken down, each of these elements being assimilated to a material point. The whole of these proposals constitutes the definition of a system of weighing solids. ̃ From this definition, we can easily derive the form of the external actions that stress a weighted system. Let us first consider a certain material point. The work done by the foreign bodies in any displacement of this point depends only on the lowering of this point. Let us designate it by /"(^}. We immediately realize that we must have, whatever Z, and Z->. .f~ =~ ) =.f( ~"? =.f~ n. so that /( s) is proportional to Z '̃ ̃̃: '̃ ̃ i /(ï)=Pï- ------------------------------------------------------------------------ This equality is equivalent to the following proposition The action of the earth on a material point is reduced to a vertical force, directed downwards, whose magnitude I' is invariable; this force is the weight of this material point. This weight depends, according to our definition, on the mass m of the material point; it is easy to see how it depends on it. Let us take two material points of respective masses m,, m- and let us associate them in such a way as to form a single material point, of mass (m, m.,) the weight of this resulting material point will be, according to the last part of our definition, the sum of the weights of the component material points; one will thus have. whatever m{ and m->, 1J(/M|)- l'i/rt)) = l'("tl-i- Hl-l), ), which is equivalent to the following proposition The weight of a material point is proportional to its mass I' = ""#. The constant g is called V intensity of gravity ratio of a force to a mass, it is a quantity of the same kind as an acceleration: its dimensions are LT~ The experiment makes it possible to determine the numerical value that should be attributed to this constant so that the properties of heavy solids, as we have just defined them, provide an image of the phenomena actually observed. This experiment must be able to be reduced, as the dimensions of 'g. indicate, to measurements of length and time. The laws of Dynamics laid down in the foregoing allow us to conceive of such experiments; such are those based on the observation of the duration of oscillation of the simple pendulum or on the use of the invertible pendulum; we shall not detail these methods of measurement, the description of which can be found in all the treatises on Mechanics. The scheme we have defined under the name of system of weighing solids can only represent, with some accuracy, the properties of real weighing bodies if one remains in a domain that is not very extensive in all directions. If Ton moves by a somewhat significant amount on the surface of the globe, one is obliged to aban- ------------------------------------------------------------------------ In the same way, if one rises from a notable vertical height, one is forced to modify the value of the intensity of gravity. At the latitude of ,y" and at the sea level, the intensity of the gravity has a value y which is represented by the number '~-t~S 1. The known properties of the center of gravity show, moreover, that we can write (11 5) i>rMi' being the dimension of the center of gravity of the whole system and M. the total mass of this system. We have just recalled how it is possible to determine the value of the intensity g of gravity: in Chu pi I II, we saw (p. io'i) how the balance allowed us to determine the numerical value of a mass: the tliéones of Statics and of iJvuamics developed in the present Chapter allow us, moreover, to perfect the theory of weighing: we can thus determine the numerical value of the work accomplished when a system of weighing solids undergoes any displacement above the ground. The sets of weighing solids thus allow us to constitute systems whose various modifications are accompanied by a measurable work of the external actions: the determination of the mechanical equivalent of the calorie, exposed in the preceding Chapter (p. 16- '), required a similar system. ------------------------------------------------------------------------ CHAPTER VI. THE NORMAL DEFINITION OF A SYSTEM. I. Definition of variations and normal variables. If we consult this set of sensitive data, abstractions and more or less immediate, spontaneous and unconscious generalizations that we call experience, we find the following lessons All the bodies of the nature are hot in some degree, there is none that one does not know or imagine endowed with this quality whose two aspects are expressed by the words hot and cold. There is no body whose various parts are always and in all circumstances equally hot; in general, at the same moment, some are hotter and others less hot; it is only in exceptional circumstances that they are all, at the same time, brought to the same intensity of heat. If therefore the representation of all the observable properties of a set of bodies were to contribute to the formation of the schematic system by which the theory represents this set and on which it reasons, it would never happen, in Energetics, that one dealt with a system without including the temperature among the quantities which determine its state. Moreover, one would never consider this temperature as necessarily having a uniform value at all points of the system or of a certain part of this system: one would attribute to it, at each point and at each instant, a value which could vary, for the same material point, from one instant to the next and, for the same instant, from one material point to the next. ------------------------------------------------------------------------ But, as we have seen (p. :>.()), one does not propose, in general, when one consU'utl the scheme intended to represent a given material set, to represent all the observable, or even observed, physical properties of that set. Leaving aside certain properties which, for one reason or another, one judges less interesting, one only represents those to which one attaches more importance, and one thus composes very simplified diagrams whose theoretical study is particularly easy. Astronomical observations, for example, inform us very well about the movements of the various bodies (̃('̃lestes, but they inform us little or not at all about the temperature to which each of them is brought it results that we can very legitimately, had some of our speculations, therefore, in order to represent the solar system, we will compose a scheme in which each state will be determined by the knowledge of the shape and position of each star in that state, without mentioning the temperature of that star. Rational Mechanics, the principles of which were set out in the previous chapter, deals precisely with very simplified systems in the definition of which the temperature of the various bodies has been completely disregarded. In other circumstances, we no longer disregard the temperature to which the various parts of the system are brought; but we disregard the differences of: degrees between the temperatures of the various bodies which compose this system in other words, in order to represent a real material system whose various parts may well be unequally heated, we substitute a schematic system of which, at each instant, all the points are brought to the same temperature, even though this uniform temperature would change from one instant to another. In certain circumstances where the various parts of the real set are, at the same time, almost as hot as each other, the theoretical study of this simplified scheme will reproduce the experimental laws with a sufficient approximation. In various cases, where this approximation would not be sufficient, we adopt a more complicated representation; we decompose ------------------------------------------------------------------------ the system in a limited number of parts: it is admitted that the temperature is uniform, each instant, in each (the these parts, nor that it can. at the same instant, to differ from a part has it the other one: moreover, for a same part, it can change of an instant, to 1 other. If we make the parts into which the system is decomposed smaller and smaller and more and more numerous, we obtain a schematic system that is more and more complicated and that can give a more and more accurate representation of experimental laws. In the end, we obtain a symbolic system where the temperature can vary in a continuous way, at each instant, from one point to another and, in each point, from one instant to another. It is, of course, to such a system that one must have recourse when one proposes to construct a theory that closely approximates reality. The quantities which are used to define the state of a system can always be divided into two groups The first group will consist of temperatures read on a certain thermometer. The number of quantities that appear in this first group can be zero, if no temperature appears among the properties that serve to define a state of the system; it can be equal to the unit or to a finite number plus or minus; finally, it can be unlimited. The second group will consist of quantities that represent properties other than temperatures; we will therefore say that these properties define the 'Yelat of the system apart from temperatures, or, more simply, when no confusion is to be feared, that they define the Yelat of the system. Between the quantities which form these two groups, there exists, as we know (Chap. I, 4, p. ̃:if>), no relation either of the first or of the second order. Let us consider, for example, a homogeneous fluid of density and uniform temperature S. Let us suppose that, in the very definition of this system, we want to introduce the following condition The density o of the fluid is a well determined function of the temperature (, ï p=/(~). >- ------------------------------------------------------------------------ In this definition, the liquid will be incompressible, while it will expand by flow according to a fully determined law. One cannot say that the standard of a liquid is defined by means of two properties, the density p and the temperature S>, because between these two properties, the equality ( t constitutes a relation of the first order only one of the two numbers o, !b is sufficient to define the state of the system. With these preliminaries in mind, we will introduce some restrictive conditions to which we will suppose all the systems we will study from now on to be subjected. First cosiutios. - We will assume that the properties by which a standard of the system is defined can be chosen in such a way that the conditions of connection do not contain any arc or infinitely small variation of temperature. Therefore, the most general virtual modification of the studied system can always be considered as the result of two particular virtual modifications: In the first of these two modifications, the temperatures vary: alone and the variations that they experience are infinitely small quantities entirely arbitrary. In the second of these modifications, the temperatures do not undergo any variation, the properties other than the temperatures undergo certain variations which are either arbitrary or subject to certain binding conditions. In any case, these same variations can be experienced regardless of the initial values of the temperatures and regardless of the variations experienced by these values in the first modification. This is what we will express by saying that the most general virtual modification of the system is composed of a change of temperatures without change of standard and of a change of state without change of temperatures. Dkuxièmk condition. - The change of temperatures without the most general change of state that the system can experience does not entail any necessary change of form ------------------------------------------------------------------------ nor the position of the various material parts that make up the system. It would be useless to formulate this condition if the positions occupied by the various portions of the system were always among the properties that serve to define a state of this system in this eus. in fact, the second condition would be implied in the first. But there are cases where the en umeral of the various properties that define the system does not include the position occupied by each of the parts that compose it if, for example, we study that very simplified system that elementary thermodynamics calls a homogeneous fluid (Cliap. 111, §3, p. i4i) - one is concerned with knowing the total volume or the density, but one does not investigate how the various material elements that form it are arranged: the same can be said of the phase of Chemical Mechanics (p. \o). Therefore, our second condition must be explicitly formulated. This is all the more essential as Ton can easily imagine systems where this condition would not be verified. Let us consider, for example, the homogeneous, incompressible and dilatable fluid defined by equality (i). J/state of such a system can be determined by means of the only variable ?S but, by virtue of equality (i), any change of value of the temperature S? is accompanied by a change of density, speaking of a displacement of the various parts of the fluid. According to the first condition, any virtual modification of the system leads to a work of inertia which is the sum of two terms : the work of inertia accomplished in a change of temperature without change of state and the work of inertia accomplished in a change of state without change of temperature. The second condition then leads to this corollary In a change of temperatures without change of state, the virtual work of inertia is zero. Moreover, the virtual work of inertia always and exclusively depends on the various elementary masses that make up the system, on their accelerations and their virtual displacements. The ------------------------------------------------------------------------ our second condition leads to the following consequence Im râleur du I raxail virtuel d'inerliti ne dépend ni des températures dit svsti'iue ni de leurs dérivés par rapport au temps. Thoisikmk condition. - In ii ii change of temperatures without change of state, the external virtual work is zero, whatever the foreign bodies in presence 'which the system is placed. It is by no means obvious that a system can always be defined in such a way that this condition is verified. For example, let us consider again the homogeneous, incompressible and dilatable tluido defined by equality (i). A "-lut of this lluid is entirely determined by the sole knowledge of the temperature S, so that the most general virtual modification of this lluid is a change of temperature without change of state. Let us suppose, on the other hand, as we can, that the external actions to which this fluid is subjected are reduced to a normal and uniform pressure II. The external virtual work will be of the form Chu p. III. equality ( f\î) l] 5 =- Il îrc. rn being the volume of the system: but ra rr, df{"ij) " i. 0!<7 = 00 = -Z. 0 3 ? p. ol;, so that we have ntsd/Cb)^ Cr - en~ - O.J. p a 3 If fi is not zero, G I is not either. When the state of a system has received a definition that verifies the three conditions previously stated, it is said to have received an ÎXONMAL DEFINITION. In what follows, we will study almost exclusively systems whose state is susceptible to a normal definition ( ). (' ) The importance that offers, in Thermodynamics, the normal definition of a ------------------------------------------------------------------------ Moreover, when one will have given the de/inilion it" a system, one will always have to state explicitly "one supposes i/ue this linked /i Hii ion is a normal definition. ( .elle .supposition, en efl'et, constitue une; iivimuiiksi:. bien distincte de lu définition du système, <;l. d'où découlent, pour ce système, d'importantes conséquences. Let us now apply these considerations to a system where an isolated Mal is entirely defined by a limited number of physical properties. l'arini these properties, we reiiiar<|ueron.s first of all temperatures in limited number, this number being able to be reduced to the unit or even to be null; we will designate these temperatures, <.|which are. supposed referred to a certain scale l!iermoinéLrK|ue. by ,Jt. Jj, In addition to these temperatures, it is necessary, for <|u a standard of the soil system defined, that one knows a certain limit number: of other quantities. Let us suppose that this definition of the system's 1 elal is a normal definition. Between these quantities, there may exist a certain number of links, bolonomous or not bolonomous, but these links will include neither the temperatures "£̃ nor their infinitely small variations: so that the step of the system studied from a first state to a second infinitely close state will be characterized by absolutely arbitrary infinitely small variations o!J|, oS-j, of the temperatures S*i, .~3->. ̃̃ and by a certain limited number n of other arbitrary infinitely small quantities ai,-a1, c/ these last ones will be called normal varialions. According to our second condition, the variations o!i,. ob-> system was first indicated, (l'une manière incidente. by llcrniaiin von llelmliollz fil. vok Uki.>ihoi.tz. Zur fil cliemisclier I"i m.ii îSX-'l. p. <'>'ti>. -II. vos Hklmholtz, WlssenschafUiche Ibltfindlunue/i, l!d. III. pp. .) S = V, "i A, "2- A,, ), a phase which -said entirely delinio by a single chemical variable punished at volume ttî eL at temperature ?3. The external garlic should be of the form | Chap. III. equality i \~>) C'-))~nt.<-)~, X eliinL chemical action and (-) calorific action. But this expression is simplified because the chemical Mechanics admits that hypotiiksi;: The definition of a phase, as it has just been given, is a normal definition. From then on, the heating action H is identically null and the expression of the external work reduces to the form (4) F r= - ||*a-H- \?j3-. Third kxe.mvi.k. - Constantly homogeneous defonnable body. - This body was defined in Chapter 111 (p. i3(i'i if we now make the im>OTiii;sK that the definition then given is a normal definition, it means that the calorific action (") will have to disappear from the first of the formulas ( ^i() > of the above-mentioned Chapter. These examples show, in accordance with what we said above, the necessity of formally stating, as a distinct hypothesis, the supposition that such a definition constitutes, for the system to which it is applied, a normal definition. When the most general virtual change of a system has been defined in this way by means of a group of normal variations, it is susceptible of an infinite number of other similar definitions. The most general solution is obviously the following: i" The temperatures can be related to another thermomelic scale, so that, if Ton designates by Tj. -̃> the new temperatures, we have (5) - | =r œ( Sri .", T,= st3'2), The function "p(-5) is necessarily an increasing function of Sf ------------------------------------------------------------------------ on ;i. whatever soil .t7, J unequal il.t- cl ï ({̃>) - > 'à" To the independent small variations a, <7j, u,n can be substituted n other analogous variations b,, b, b, related to the former by relations of the form h, = An"| -i- À14 ". .r- À, c< I (l,= > ", -r- >.4, "2 - .-- À",, (ri>< ?i" ?' 5>< ). In this expression, rpq is the distance from a point Mp of the element/" to a point Mf/ of the element q ç,p and ?S are the density and temperature of the element/); p? and .;7y are the density and temperature of the element q. If we suppose that the five variables rPq, op, p? ?5p, ?5t/ give a normal definition of the system formed by the two elements p, q, we will have to admit that the two temperatures 3/ 2f? do not appear in the expression of the function fp< From then on, the action designated by Qp,j in the equalities ((5(î)) of Chapter III will disappear as identically null; the function V/M defined by the equality (69) of the same Chapter, will be independent of both the temperature %Sp. of the element p as well as of the temperatures of the other elements finally, the action &p that defines, the last equality (72) of the same Chapter will be always equal to o. "! Normal heat capacity and Helmholtz's postulate. Let us consider, first of all, a system formed of a limited number of parts, each of which is brought to the same temperature at all its points; in this case is included, as a more particular case, ------------------------------------------------------------------------ the one where the temperature of the system is uniform. Let 1 a, p be the indices that designate the various parts of the system and S,, 3f2, %p the leperals of these parts. Let's assume that the definition of the system is normal. The most general virtual modification of the system results from two other modifications : a change of temperature without change of state and a change of state without change of temperature. We propose to study the amount of heat released by the system in the first of these two modifications. The definition of the system being assumed to be normal, a change of temperature without change of state does not entail either work of inertia or external work, so the amount of heat that the system gives off in the virtual modification considered is reduced, according to equality (4) of Chapter IV (p. 107), to the decrease that the internal energy of the system undergoes because of this virtual modification. Now, if we look at the temperatures of the system as arbitrarily variable, this system remaining moreover in a determined state r, we will be able to look at the internal energy as a function of the temperatures 2r, S?2, ?3P\ the form of this function will change with the state e; it is what we will express by representing this internal energy by the symbol UC^Sr, .3,). If we put then 0: d C. -0 c,:< e, S,, i, S,,) = - U(e, S,, 2r,, S,,), ¡) 0.. (11) dJ, *'j ( f C/, (e, Zi, Z-i. 3,i ) = -^r- L (e. jt, _(., ~t/,), ), ".?/. the quantity of heat released by the system in a change of temperatures without change of state will be given by the equality (12) Q = - (Ct oSj-f- C2oS?2-r-f- C;) 8STp). The quantities d, c2, <"y, defined by the equalities (1 1) for the ------------------------------------------------------------------------ are called the normal heat capacities of the various parts of the system. From the equalities (i i) that are used to define these capabilities, we immediately derive the following proposition The normal heat capacities of the various parts of a system depend neither on the state of rest or motion in which the system is found, nor on the foreign bodies in the presence of which it is placed; they depend only on Velat and the temperatures of the system. Let us imagine that the various parts which compose the system are independent of each other; let us suppose that each of them, considered in isolation, is susceptible of a normal definition, where it would be characterized by its state and its temperature. Let us designate, for example, by the letter el the state, apart from the temperature, of the part 1 considered in isolation. The internal energy of this isolated part would be a certain function of the temperature S?, a function whose form would change with the state and on their mutual disposition, but in no way on the temperatures 3" o. i o.. n ,~p. The equalities (11) will then become ec dU,<'c,.S.) tc.(and..?j)=-----. dU,3t (14) t~i .¡. ''l'(~)' C/.i, ep, 3,, ) d - ~") ci:~ When a system is made up of several independent parts, each of which has a normal definition, the heat capacity of each of these parts depends exclusively on let ------------------------------------------------------------------------ It does not depend on the temperature and the state of each of the other parts; nor does it depend on the disposition of the whole of the parts which form the system. Let us now consider a system within which the temperature is susceptible of taking, at each point, a different value from that which it takes at an infinitely close point, and let us suppose that the definition of this system is normal. This system will be determined by its temperature at each point, which is arbitrarily variable, and by its state, whose variations are not related to the variations of the temperatures. If this system undergoes a virtual modification where the temperatures vary without the state changing at all, it gives off a certain amount of heat which is reduced to the decrease of the internal energy. It is clear, therefore, that this quantity of heat can be put in the form of Q = - A S2? dm, where dw is an element of the volume occupied by the system o2f the variation of the temperature at a point of the element dm A a quantity which depends only on the state and temperatures of the system. Denoting by the density at a point of the element dm, we will pose À = pT. The quantity y, which depends on the state and temperatures of the system, is called the normal specific heat at a point of the element dm. By introducing this quantity, we can write ( i j ) Q = - s y à "1 dm- Let us consider a phase such as studied by Chemical Mechanics to define the state of this phase, let us give, as we have already done in Chapter III (p. i 4 ' ) its specific volume o>, its chemical composition x and its temperature S, uniform by ------------------------------------------------------------------------ definition. In these conditions, we sax that the internal energy of this phase is proportional to its mass M, so (|ii'designating by L this energy, we think to write U ̃̃̃ M m m, x. 1j). Let us calculate the normal calori(ii|ue) capacity c of such a phase, noting that this phase is normally defined and applying one of the formulas (\) to it. If we put i) in u>. 3\li) (16) -'Avi.x.Z)- - i j. ~â we will obviously have (17) c ̃= M ~;(w,x. S). If the temperature of the system rises by 82? without any change in the specific volume or chemical composition, the amount of heat released will be (18) Q=- .\l-;(Oj,a-. 2r)oHr. Obviously, formula (18) can be considered as a special case of formula (i5), so that the quantity y(w, x, S) can be legitimately named the normal s/iecivic heat of the phase. Often, we recall the properties of this quantity by naming it the specific heat at constant volume and at constant chemical composition. Equality (17) teaches us that the normal heat capacity of a phase is equal to the product of the mass of this phase by its normal specific heat. With these various definitions in mind, we will assume the following basic premise Postulate dk Hkl.mhoi.tz. - Any normal heat capacity and any normal specific heat are essentially positive. This postulate is obviously equivalent to the following proposition If a system "received a normal definition, to raise lu ------------------------------------------------------------------------ In order to lower the temperature of the whole system or of a part of it, and without making the system experience any change of state, it is necessary to force the system to release a negative quantity of heat or, according to the language commonly used, to force it to absorb heat. This proposition justifies, in a particular case, the denomination of quantity of heat, until now, in fact, the notion that this name designates was without any link with the one that the word heat designates in the vulgar tangage, this last notion finds its theoretical equivalent in the idea of temperature. The preceding proposition establishes a connection between a positive rise in temperature (échauff'emenl) and a positive absorption of heat {chauffage'); it shows that in order to heat a normally defined system in whole or in part, without causing it to undergo any change of state, it must be heated. Helmholtz, who first pointed out the importance of your notion of normal variables, is also the first to have explicitly formulated the preceding proposition; however, he does not seem to have regarded it as an independent postulate; he seems to have thought that it could be deduced from Carnoi's principle ( ). (') H. von Helmholtz, Zur Thermodynamik chemischer Vorgànge (Sitzungsberichte -clér Berliner Akademie, 2 février t88>, p. 12 et p-io.. - H. von HelmHOLTZ, Wissensctiaftliche Abhandlungen, Bd. H, p. 969 and p. 978).We have indicated that this proposition should be regarded as one of the fundamental postulates of Thermodynamics in our Commentary on the Principles of Thermodynamics, Part 3 The General Equations of Thermodynamics, Chap. IV, § 5 (Journal de Mathématiques pures et appliquées, 4' série, t. X, 189^, p. ̃1-0). ------------------------------------------------------------------------ CHAPTER VII. THE PRELIMINARIES OF THE CARNOT PRINCIPLE. 1 Restriction on the equilibrium conditions of the systems to be studied. Examples of systems subject to this restriction. General considerations on chemical statics. We have now reached a point of bifurcation; we will have to restrict the field that we propose to organize from now on, in fact, we will study exclusively the systems that verify V following hypothesis Restiwctive hypothesis. - For a system, normally defined, taken in a determined state and placed in the presence of foreign bodies which are also in a determined and invariable state, to remain in equilibrium, it is sufficient. i" That the local velocity of each of the material masses of which the system is composed is equal to o is" That the temperature is the same at all points of the system and foreign bodies 3" That in each of the virtual modifications imposed on the system from the considered standard, the external work takes a certain value, entirely determined by the knowledge of this standard and of the modification In other words, these conditions are necessary to ensure the equilibrium of the system if all the virtual modifications of which it is susceptible are reversible. 11 must, moreover, that there is no eonlradirtion beyond the conditions of equilibrium we are talking about at the moment and the general laws of Statics that have been established in Chapter l\ (5; 4 >. ------------------------------------------------------------------------ IF therefore the studied system is Here that the absolute position that it occupies in space enters in Usine of account in the definition of its state, the value that the conditions of equilibrium assign to the rirlual work of the external actions must be reduced to o in the case where the virtual modification is reduced to a displacement of set. To this remark, we will add this postulate 1m value of the virtual work that ensures V equilibrium does not depend on the absolute position that the system initially occupies in space. This postulate is equivalent to the following proposition A. system being in equilibrium in the presence of certain foreign bodies in a given place of space, the equilibrium is not disturbed if we transport the system and the foreign bodies to another place of space without making them undergo any other change of state. This transport, in fact, does not change the internal energy of the complex system formed by the foreign bodies and by the system under study; it does not change the external work done into a virtual modification of this system if this external work had, before I transported, the value which ensures the equilibrium, it still has it after the transport. Let us suppose, in particular, that the system studied is one whose state is defined by a limited number of quantities. Any reversible virtual modification imposed on this system from a given state will be defined by the infinitesimally small variation 32? that the temperature of the system will have undergone in this modification, a variation that we will suppose to be of the same magnitude at every point of the system, and by n other infinitesimally small quantities, independent of each other, ", a2, ̃̃̃, ""̃ The corresponding external work will be of the form (i) Jse= A|"fi-t- A2a2-f-+- Anan, the actions A, A2, An being independent of a, a. a, this work will not contain a term in 32? because the definition of the system is assumed to be normal. The condition stated under 3° can, in the present case, be formulated as follows The n actions A, As, A" have values Jn which depend exclusively on the state where one wants to maintain ------------------------------------------------------------------------ the system i-n equilibrium or, more explicitly, of the temperature "h of this system el of its state e, abstracting from the temperature. Equalities (?.) At - /,(e,S?), Ai=f2(e.?j), An-fn(e,Z) are called equilibrium equations of the system. In the case where, in order to define a. state of the system studied, we do not disregard the absolute position that this system occupies in space, the six general equations of Statics [Cliap. IV, equalities (44 )̃> P-'7a] must figure among the conditions (2) or be consequences of these conditions. Suppose, by an even more peculiar assumption than the preceding one, that the state of the system is defined by the temperature: !J and by n other independent normal variables a,, a2, a,, the n quantities f2, ." will be expressed as functions of a,, a2, a,, and of S, so that the equilibrium equations will be of the form ( A, =/, di-aj, .<*", 5;, ). n ) A2 =/2(ai,x,, a,?!), j, [ An = f,,(z,. ̃Xi, xn, ~3). From what has just been said, let us first give some very simple examples. Phemïkr kxemple. - Solids subjected to bonds without passive resistance. Let us consider, first of all, the system studied by rational mechanics, that is to say a system of solids subject to connections without passive resistance. For such a system, there is no need to concern ourselves with the condition relating to temperatures, since the very notion of temperature is foreign to the definition of the system; but, this restriction admitted, this system verifies the condition just stated; in fact, for such a system to be in equilibrium, it is sufficient, according to what was seen in § 8 of Chapter V (pp. -x\o-'}. k i° That the local velocity be zero at any point: ------------------------------------------------------------------------ ̃?." That in any virtual modification, reversible or not. the external work is equal to the increase of the internal energy. Moreover, these conditions are necessary if the system is not susceptible to non-reversible modifications. A system of solids subject to connections without passive resistance is therefore one of those to which the considerations that we are going to develop apply. Second example. - Homogeneous fluid of elementary thermodynamics. Let's take, as a second example, this very simplified system that we study, in elementary thermodynamics, under the name of homogeneous fluid. As we saw in Chapter III (§ 3, p. i/f and p. i44)j the state of such a system can be defined by means of two variables, the specific volume u> of the fluid and the temperature S. Moreover, according to the assumptions made in Chapter VI (§ 1, p. a54), these variables are normal variables. If we designate by 73 the total volume of the fluid, by fi a normal and uniform pressure exerted on the surface of the fluid, and finally by M the mass of the fluid, the external work can be written [Chap. VI, equality (3), p. aà/j] ë = - H ora = - 1 H 010. - Mil is thus the relative action to !a variable w; this action is proportional to 0, the factor M being invariable, The notion of local velocity loses all meaning in the study of a homogeneous fluid, since, by definition, the place occupied by each part of the fluid is ignored in this study. Finally, a virtual modification of the system, defined by arbitrary variations 3w, o2? of the specific volume and of the temperature, is always reversible. Therefore, if we want the homogeneous fluid of elementary thermodynamics to be subject to the rules that we are going to establish, we will have to admit the accuracy of the following proposition For a homogeneous fluid mass to be in equilibrium, it is necessary and sufficient that it be subjected, at every point of its surface, to a normal and uniform pressure II, a function well de- ------------------------------------------------------------------------ mined from its specific volume o> and. temperature .3 i\) h =/(oj. r-j. j. Is the form of the function independent of the mass M of the fluid considered? For what precedes, it is not obvious that the function does not change with the size of the mass M. We will be able to demonstrate that it is independent of it, if we admit the following supposition, (which, moreover, is obvious If the pressure which maintains in equilibrium a homogeneous fluid of given specific volume and given temperature varies with the mass of this fluid, it is a continuous function. We can, in fact, establish successively the following propositions i" If two homogeneous fluid masses, of the same nature, of the same specific volume and of the same temperature are multiple of each other, the same pressure is required to maintain them in equilibrium. Let M and M' be these two masses, and let us suppose that we have M' = n M, n being a whole number. Let us take n masses of the considered fluid, each of which has for size M, for specific volume u> and for temperature S> let us suppose them infinitely distant from each other; each of them will be maintained in equilibrium if we apply to its surface a certain normal and uniform pressure, the same as if it existed alone (p. 18); moreover, when it is isolated, the pressure which must maintain it in equilibrium is independent of the place it occupies in space, according to the postulate stated on page atifi; it is thus the same for the iz masses considered. It can also be said that this same pressure maintains in equilibrium the whole of the n masses infinitely remote in space; but, according to the definition of the phase in and of the homogeneous fluid in particular (Chap. III, § 3, p. i4<>), the whole of these n masses can be treated as a single mass of homogeneous fluid, of specific volume o> and of temperature S, this mass having for value M'. It is thus seen that, in order to maintain this mass in equilibrium, it is necessary to apply to it the same pressure e which maintains the mass M in equilibrium. ------------------------------------------------------------------------ /> If two fluid flows of the same nature, of the same volume, specific, and of the same temperature, are measurable between them, the same pressure is needed to maintain them in equilibrium Let M and M' be these two masses, let us suppose (pie one has \t /1.' l' 1 l '- - " ri and being two. iioinl>res integers: let m a niasse conM n held n times in the mass M and /(' times in the mass M1; let us take a mass m of the considered homogeneous liquid, and let us give him the same specific volume and the same temperature as to the masses M and M' according to the preceding theorem, the pressure which maintains the mass m in equilibrium maintains also in equilibrium and the mass M and the mass M'; these two masses are thus held in equilibrium by the same pressure 3" If two homogeneous fluid masses of the same nature, of the same specific volume and of the same temperature are incommensurable, the same pressure is required to maintain them in equilibrium. Let M and M' be these two masses incommensurable between them; as all the masses of which it will be spoken in the present reasoning m in l, they have the same specific volume o) and the same temperature .3. Let H, II' be the pressures capable of maintaining them in equilibrium if II were not equal to H', the difference (II' - II) would have a well determined absolute value different from o, which we will designate by z. Of the homogeneous fluid considered, we can take a mass M| eoniniensurable with M' and as close as we want to the mass M to be fi, the pressure able to maintain the mass M, in equilibrium. By hypothesis, the pressure capable of maintaining in equilibrium a fluid mass of specific volume w and temperature 2? varies in a continuous way with this mass; one could thus take the mass M, close enough to the mass M so that the difference (H, - fi) would be. in absolute value, lower than any positive quantity given in advance, in particular the quantity z. But the mass M, is commensurable with the mass M', so that. according to the preceding proposition. II, is equal to M'. If therefore II' was not equal to II, the absolute value of (II' - II) ------------------------------------------------------------------------ sérail at once equal cl inferior ii s: '̃elle contradiction e\igc <|iie [|'=II: it demonlre the stated proposition. The shape of the balance eondiliori II =/im, 5) does not change with the mass of the fluid being considered. This condition of equilibrium is often called the law of compressibility and dilatation of the studied fluid. The existence of a law of compre "sibiity and dilatation is therefore the consequence of a fundamental postulate, without which the theory that is going to be exposed would not be applicable to homogeneous fluids. Third example. - Phase of Chemical Mechanics; general considerations on Chemical Statics. We shall suppose that the state of the phase studied is defined by means of the specific volume w, the temperature 3 and a single quantity .r, variable from o to i, which represents the chemical composition. We leave it to the reader to generalize what we are about to say and to extend it to the case where the determination of the chemical composition would require the consideration of several variable quantities analogous to x. The definition of the phase, as given, was, in the preceding Chapter (Chap. VI. § 1, p. 25V), assumed normal the external work done in a virtual modification of the system has for expression [Chap. V I, equality ( p. ?.*>̃>] (̃>)'̃ G =- Mil îw- \o.r. The action relative to the variable x is [ chemical action X the action relative to the variable , composition ------------------------------------------------------------------------ chemical x and of temperature 3 is in equilibrium, it follows that independent bodies, of the same temperature S, exert on it a chemical action X and produce at its surface a normal and uniform pressure H, X and II being well determined functions of the variables ", x, 3 ) X = tf-fco, a-, 2?;. J. In other words, if the variable x does not have one of the particular values o or i, these conditions are necessary for the phase to be in equilibrium. A reasoning, similar in every way to the one developed in the previous example, would give the following proposition The form of the functions f and g which appear in the preceding conditions does not depend on the total mass of the phase to which one applies these conditions of equilibrium. This is the place to examine a difficulty which stops many minds at the very threshold of Chemical Mechanics; this difficulty vanishes if one is willing to remember the rules which govern the development of any physical theory. 11 would not be allowed to apply to the phases of Chemical Mechanics the theories that are going to be developed if these phases were not subject to the condition that we have just formulated Therefore, the application of these theories to Chemical Mechanics would be forbidden to the one who would doubt the following proposition or who would deny it Whatever the specific volume co, the chemical composition x and the temperature 3 of a phase, one can conceive a system independent of this phase and which exerts on it a chemical action X precisely equal to. the corresponding value of the function g(, x, 3). (') P. Duhem, La théorie physique, son objet et sa structure, Paris, u|n(i. ------------------------------------------------------------------------ Let us notice that it is a question here of conceiving in abstracto such a system independent of the studied phase, and in no way of realizing it, i.e. of seeking in nature a concrete body of which this abstract system is the more or less faithful representation. That such a concrete body exists or does not exist must be absolutely irrelevant to us if we remember the rules that govern the development of physical theory. One of these rules, in fact, is formulated in these terms in our Introduction ([). 4 ) The principles of thermodynamics are pure postulates; we can state them as we please, provided that the statement of any of them is not contradictory in itself and that the statements of the various principles do not contradict each other. Now it is clear that the intellectual operation by which one conceives that a system, independent of a phase, exerts on this phase a given chemical action is free of any internal or external contradiction : the postulate it poses is therefore legitimate. In this same Introduction, we have formulated (p. ,\) the following rule In the course of its exposition, a physical theory is free to choose the way it likes, as long as it avoids any logical contradiction, in particular, it does not have to take into account the facts of experiment Once we have established, in what follows, the principles of General Statics, principles whose application to the phases of Chemical Mechanics is rendered legitimate by the postulate previously admitted, we will be able to use these principles to solve certain particular problems relating to chemical phases, without worrying in the least about whether or not these abstract problems correspond to questions posed by experimental Chemistry. In particular, we will be allowed to treat the following problem What corollaries can be drawn from the principles of General Statics when applied to chemical phases. ------------------------------------------------------------------------ defined normal deny as we have assumed in the above, and subjected to independent systems that do not exert any chemical action on them? A phase subjected to such foreign bodies will only be in equilibrium if the conditions (7) j II = f(u>,x,Z), < o==~r. O - ff(M, X, 3 ) are verified. It will not be able to be in equilibrium whatever the specific volume m, the chemical composition x and the temperature S; for it to be maintained in equilibrium by means of a suitable pressure, it will be necessary that between these three quantities a certain relation is verified. There is obviously no contradiction between the statement of this problem and the statement of the fundamental postulate that we formulated a moment ago. In studying phases in the presence of independent systems which do not exert any chemical action on them, we do not contest the possibility of conceiving other independent systems capable of exerting chemical actions of arbitrary magnitude on these same phases. The study of this particular problem being free of logical contradiction and being part of the development of the theory, this study is perfectly legitimate. This study constitutes properly the particular application of general thermodynamics to which we give the name of chemical statics. Once Chemical Statics, logically developed, has produced all its consequences, it becomes necessary to compare these consequences with the laws that experience has revealed about chemical equilibria if, indeed, the consequences of Chemical Statics represent, with a sufficient approximation, a large number of these laws, Chemical Statics will be a useful physical theory; if not, it will be a pure exercise in logic, interesting perhaps to the algebraist, but worthless to the physicist. Now this comparison between the experimental laws of chemical equilibria and the conclusions of Chemical Statics as we have just defined it leads to this consequence which may seem unexpected Although Chemical Statics thus defined ------------------------------------------------------------------------ If we consider that this doctrine appears to be the lV:lude of an extremely specific problem and that we can conceive of a more general chemical statics, it turns out that the corollaries of this doctrine represent, with a sufficient approximation, most of the laws of chemical equilibrium studied by the experimenters. This consequence leads to the following proposition If the concrete systems studied by chemists are represented by means of phases, normally defined as we have done up to now, or by means of assemblies of similar phases, we can represent all the foreign bodies susceptible of being brought into the presence of such systems by representing them by independent systems which do not exert any chemical action on these phases. This conclusion is not in contradiction with our fundamental postulate. It denies, in fact, this proposition II exists in nature a concrete body susceptible of being represented by an independent system which would exert a chemical action different from o on a phase normally defined by the variables w, x, '3. Our fundamental postulate was limited to affirming that one can conceive in a.bstracto such systems, without worrying about knowing if there exists or not in nature concrete bodies (ion) these systems are the approximate representation. The beginnings of Chemical Mechanics can therefore be rendered free from any logical contradiction, provided that one is very careful about the principles which determine the object and structure of the physical theory. These principles, unfortunately, are very frequently disregarded; physicists who disregard them are faced with insurmountable difficulties when they approach the study of Chemical Mechanics. The physicists we are talking about only want to accept as principles of a physical theory "combinations of simple inductions, suggested by experience"('): they* want to reason only ever "on realizable operations or, at least, on operations (which can be considered as "real")". (') G. lîoiux, Œuvres scientifiques réunies et puJiliécs par t.. Hall'y General Thermodynamics, p. XII. ------------------------------------------------------------------------ limit cases of feasible operations"('). From then on, in order to apply the rules of General Statics to a chemical phase, normally defined as we have done up to now, it is not enough for them to formulate this postulate: it is possible to conceive of an independent system which exerts on such a phase a cyclic action of given magnitude. They still have to establish that there exists in nature a concrete body that can be represented approximately by such a system. But the very success of chemical statics teaches them that such a body does not exist. The continuation of their reasoning thus constitutes a real vicious circle. Instead of discarding a chemical system formed by a single phase, one could consider a system formed by various phases between which chemical reactions can occur. In the case where the pressure intended to ensure equilibrium must be the same in all these phases, the equilibrium equations which result from our restrictive condition and from the equality at o of all the external chemical actions would lead, as a corollary, to the famous Phase Rule of J. Willard Gibbs(-). This rule is not, therefore, as has often been said, a consequence of the General Static based on the Principle of Sadi Carnot, but only a corollary of the restriction posed at the beginning of this paragraph, a restriction which it is essential to make before approaching the construction of this General Static. 2. Examples of systems that are not subject to the to the previous restriction. The laws which will be established in the following Chapters are applicable to a system only if this system fulfils the restrictive condition stated in paragraph 1. We have given some examples of systems1 which are assumed to satisfy this condition. We shall now cite some systems to which this condition is assumed not to apply. (') G. ttonm, flticl. p. XIV. ( Now, in it, one could take up, with an insignificant modification, the complete demonstration of this rule, such as mms have given it elsewhere [On llie gênerai problein of chemical Slatics (Journal of physical Ckemistry, t. Il, i8;iS, p. i ) Leçons élémentaires de Mécanique chimique, Livre I\, Chap. [, SS '? -> 3; t. IV, i*><), pp. iSi-wi |. ------------------------------------------------------------------------ Pu f nniou kskmi't.k. - Mechanical systems with (J'eetés of friction. Let us consider the very simple case of 1.111 material points resting on a plane. We can try to represent the properties of the concrete bodies that this system is intended to represent by treating <:<- plane as a geometrical connection without passive resistance cl by making use of the theorems of Rational Mechanics set out in Chapter V. However, in a great number of cases, mechanics have found that such a scheme was not able to represent with a sufficient approximation the properties of the concrete bodies studied: they have then imagined a more complicated scheme which they intend to designate by saying that the material point slides on the plane with friction. The assumptions which define this new scheme lead to the following consequence For the material point to remain in equilibrium on the plane, it is necessary and sufficient that the ratio of the projection on the plane of the force which solicits this material point to the composa/ile, normal to the, plane, of this same force does not exceed a certain limit. If we denote by a the angle of the force with the plane and by/ the coefficient of friction, i.e. the limit value just mentioned, the equilibrium condition we have stated is expressed as follows (8) cola 'if. This condition does not have the form of the equilibrium equations (3). The force that can maintain the system in equilibrium in a given state is by no means determined; it can take on an infinite number of magnitudes and an infinite number of directions. The system we have just studied does not meet the restrictive condition formulated at the beginning of this Chapter. The same could be said of any mechanical system where friction intervenes. Dia-xiiïMK example. ̃ - False chemical equilibria. To figure the laws that govern certain chemical equilibria. we have been led to adopt your following assumptions. ------------------------------------------------------------------------ The system studied is represented by a phase; this phase is normally defined by its specific volume w, its mineral content ?j and its chemical composition; the latter is the ratio between the actual mass of a certain compound existing in the system, and the value that this mass would take if the reaction which gives rise to this compound continued as far as the elementary composition of the system allows: by definition, this quantity ./- is between o and i. The independent systems that include the foreign bodies to which our phase is subjected are supposed to be incapable of exerting any chemical action on this phase. Their action is reduced to a normal and uniform pressure applied to the limiting surface. If such a system were subject to the restrictive condition that was formulated at the beginning of this Chapter, its equilibrium states would be governed by the conditions i- II = /( u". x, "S ), I <> = A'( o", 3"> )- 11 It is easy to predict certain properties of such equilibrium states. Let us suppose that the system is maintained at a known temperature; let us also suppose that its specific volume is known; it will suffice for this, if its mass is known, to know its total volume. The two quantities S and m being known, !he second condition (~) will make your value of x known and the first one will determine the value of II- Thus, the chemical system being brought to a given Turkish temperature and maintained in a given volume, its chemical composition at the moment of equilibrium would be determined as well as the magnitude of the pressure that it bears at this point. This would be the fundamental property of a state of equilibrium which would fulfill the condition stated in paragraph i. Such a state is known as a true chemical equilibrium state. Now chemists have recognized that the preceding proposition cannot be reconciled with the laws. experimentally observed. which govern a large number of chemical equilibrium stalls, To represent theoretically the laws of these false equilibrium stalls c ------------------------------------------------------------------------ chemical, we are forced to consider as true the following proposition The chemical system, brought to a given temperature and maintained under a given volume, will be in equilibrium whenever the value of the chemical composition of the phase is between two determined limits; the magnitudes of these limits depend naturally on the temperature of the system and its specific volume. If we denote these two limits by H,,(w, "b), ?((co, .:?), the second equality (7) will have to be replaced by a condition of the form (9) ?o(w,2r>0', in fact, that any virtual modification of a normally defined system is decomposable into a change of temperatures without change of stall and a change of state without change of temperatures. ------------------------------------------------------------------------ In the first of these two modifications, the system gives off nue amount of heat which has the value [Gliap. VI, equality {!̃>), p. '.>.()'.>. (.l i) <.>, ;̃- }̃{ Z~Z dm. c. tlm being an element of volume of the system and s, y, 32; being, respectively the density, the specific heat and the increase of the temperature in a point of this element. Therefore, we can state the following proposition When, starting from a state of equilibrium, we impose any virtual modification on a normally defined system, subject to the restriction of paragraph 1, and whose virtual modifications are all reversible, the amount of heat released is the sum i" Of the quantity of heat Ot given by t equality (i ̃>); 2" Of a quantity of heat Q2, entirely determined by the knowledge of the state of the system and of the change of state without change of temperatures which corresponds to the imposed virtual change. Let us apply these general considerations to a system whose STATE IS DKTERJIIJSED BY A LIMITED NUMBER OF SIZES. Let 2; be the uniform temperature of this system and e its state apart from the temperature: any virtual modification of the latter state is defined by means of n, independent normal variations a,, a±, a, In equality (i2)j the work of inertia z is zero by virtue of equalities (î) and (2), the external work © can be written (14) S = /,(e, 2f) "i-)-/j(e, S)a,-4-. ,hfn(e, Jl)aH. As for the variation oU of the internal energy U(e, 2?), it has obviously the following form ( iâ) oV = Ui(e, 2r)rtiH-"2(e, S) "2-f-, -h un(e, ?j)aH-, ~y^ - ^"f- By virtue of the equalities (ta), (i4) and(i5), the amount of heat released, in any virtual change, by our ------------------------------------------------------------------------ system in equilibrium can be written I iG) O = - ( pi(e, ^j "i -r- p3(e. Sr; a" -i- .+- p"(e, 'Z )(/"̃- c(e. Z ) o:J |, positing | p, (e, ;?; -- w, (e,5r) - /i (e, ï), i p2(e, Ï3) - u.,(e, "S) - f.2(e,?j), (17) i p"(e, i) = ""((?, Î7 )-"(", S1;, cfe,J) = The quantities pi (e, 2?), lo2(e, S?^, - p"(c, S): c (.), that Ton could write ( 7.O I U ( (", 3 ) = M ll( M, 3 }, M being the mass of the lluiiJe t-ln dit': and u ( and we will have (22) p((i), 3) = M /(tu, 3 ). c(b>, 3) = M ''("M, 3 j, (a?i) Q = - M [h oj, 3) db> -+- "((->, 3)Z3\. j, The two functions /(w, 3), '{(t>>, 2?) depend on the nature of the lluid, but not on its mass: we have already, in Chapter VI (p. ̃ii')'))i met the function y ^o>, S*), and named it specific heat under constant volume of the turkey under study; the quantity /(oj, S?) is named heat of expansion of the same fluid. Second KXiî.MPr.i:. - Phase of the chemical mechanics. -We suppose that this phase is normally defined by its specific volume w, its temperature S" and its chemical composition x, which, moreover, is the ratio of the mass m of the variable compound that the system contains to the largest mass of the same compound that is compatible with the elementary composition of the system. We have seen in Chapter III, 3 (p. l'f'-), that the internal energy of the phase can be put in the form ( >4 J U f ", a*, 3 ) = M m w, x, 3). M being the mass of the phase and u(w. x, ?3) an independent function of this mass. ------------------------------------------------------------------------ The external adions relative to the variables i<> and x are respectively - M 11 and X. We shall suppose, as is admitted in chemical statics, that the action X is null, whereas it is given by the first equality (-). If we then posit I I p, ( u>. x, S i = M f '-' n( m..r, i ̃- ./< <->, or..j ) | 1 ~/Ql (̃irj) pi(w, X. S) - .M - > f c tu, :J ) M ---~--- o ÛTJ we will have (-A>) 0 = - | p!(oj, a;. îî; oco -4- pafoj, .r, S)oar -f- c(oj, x, ~b ) oS/J. o,,, p2, c arethe three heat coefficients of the equilibrium phase. Let us now observe that r.. ox = - o/?i let us denote by ç the ratio >' which, obviously, depends on the elemental chemical composition of the studied phase and not on its niasse let us then pose Zi|(0,5) = ̃- -i-Jtw,x..j), ') m((i>. 3", S?) (a?) rC10. -r':3)= - ^5 t (1)..r.J.) t i o u ('(i>, a". Sr ) We will have i pi(oj, a-, £r) = M|/(w, x. 2rj, ), (:".8) cûo, x, S) = M-;(w,S, f S{(uj, x, S1) = - Ai co. a", .3 and (4g) Qi== - Mf/((a, a-, Sr > o3r J -t-'À(iu, &) o "j. ------------------------------------------------------------------------ We have already met in Chapter VI (p. aG.'i) the quantity -/((!>, .r, S), and we have called it specific heat .its constructed volume cl at constant composition the quantity' /((o, x, ?j ) can be called heat of expansion at constant chemical composition finally, the quantity À (tu, x, S?') is the heat of formation at constant volume of the variable compound contained in the phase. The form of each of these functions depends on the nature of the phase studied, but not on the mass M of this phase. The variables w, x, S are, by definition, variables without inertia, since it is agreed that the position that the various parts of the phase occupy in space is entirely ignored. The quantity of heat released in a modification of a phase is thus reduced, in all circumstances, to the excess of external work over the increase of internal energy; it is no longer necessary, in order to make use of this proposition, to suppose that the modification is an elementary modification resulting from a state of equilibrium. From this we can easily draw the following conclusion To apply to a phase, taken in a state (w, x, ?s), the equalities ("4) to (29), it is not necessary that this phase is in equilibrium in this state, it is enough that, only among the conditions of equilibrium, the first of the conditions (6), that is to say the condition it = f( 10, X, "b). is verified. There is no need to know whether or not the second condition, represented here by equality (7), is verified. In § o (pp. 296-299), we shall see the importance of this remark. From now on, we will draw this corollary The equalities (a5) to (29) can be applied whatever the state (w, x, £7), since, whatever the state, there exists, by hypothesis, a pressure IT verifying the first of the equalities (6). The three variables w, x, S must then be considered as independent variables. This is no longer the case if we force the system to be in equilibrium; in the three functions "(w, x, £r), l{<*>, x, S?), à(w, .27, 2t), the three variables o>, .r, 2? are no longer independent since the system is supposed to be in equilibrium, they are ------------------------------------------------------------------------ linked by the relationship (7) #(">, 3\ 2j ) - <). Let's not leave the holononic systems without pointing out the frequently used relations that immediately follow the equalities (i<)). A first group of relations is symbolized by the formula (:iO) 0lJL-^l + ^!i-L=o. (:io ) an. (>-j. 1)% q 0% o. The second group of relationships is the following of Op, <)/, == O'J-i (fc dfc (3i) ÙC <>2n_ _j_ O/Ji O'J. 03 dJj In particular, in the case of a homogeneous fluid, the equalities (3 1) '1 give the often used relation, due to R. Clausius ('), ( 3~ ) d~ i r)l df [ 0o> ob (fii 4. Equilibrium displacement. Inverse variables. In this section, we will deal exclusively with holonomic systems defined by a limited number of normal variables. When such a system is in equilibrium, it verifies the equilibrium equations (3). We can suppose that we solve these equations with respect to the normal variables cr. a2, a,, they will take (') K. Clausius, On various forms of the fundamental equations of the mechanical theory of heat, §7, first equation ( -i ) ('Poggemtorff's 's Annalen, Bel. CXXV, 18OÔ: Journal de Liouvifle, ?.' série. t. Théorie nieranique de la chaleur, trad. par F. Folie. t. I, i8(i£. p. 38;)). (-) We studied, first of all, the passage to inverse variables in the following paper Sur les équations générales de la Thermodynamique, Chap. (Annales de l'École Normale supérieure, 3" série, t. VIII. iS;>i, p. :>>sv ------------------------------------------------------------------------ then the form ) *̃- = F-> ''A,, A. A, lZ), ( ')3 ) < ~2=F2'j.\j,A,), 1, ( *= I-A1,A3, A, 1. In this form, they can be redesigned as solving the following problem A system, normally defined, is brought to a given temperature and subjected to given external actions; what will be the state of equilibrium of such a system? According to the condition posed at the beginning of this Chapter, when the state of a system of uniform temperature is normally t delineated and given, the external actions capable of maintaining this system in equilibrium are unambiguously determined; the functions /'i, ." which appear in the equations (̃')) are therefore uniform functions of y. a2, y. .:? It does not follow that the functions F(, V2, F,, which appear in equations (33) are uniform functions of A|. Ao, A, 2"; so that a system, normally defined, brought to a given temperature and subjected to given external actions, may well be susceptible of several states of equilibrium, distinct from each other. The equations (33) can be seen as defining a change of variables; they allow, in the previously established formulas, to substitute the normal variables xj, ~2, '.1. ;¡ the new variables Ai, A2, A, rj which we will call inverse variables of the previous normal variables. Inverse variables differ from normal variables in two ways that should not be ignored i" The use of inverse variables is limited to the study of systems in equilibrium; it would lose all sense if it were a question of studying a system in motion ̃j.'J H may happen that the inverse variables relative to a system ------------------------------------------------------------------------ have well-defined values and that, however, the state of the system is not unambiguously determined. The internal energy Ij (a,, y.-2. ".", 2") of the system becomes a function :'( A, A2, A, 3) of the new variables; but, while it was certainly a uniform function of the normal variables, it may no longer be a uniform function of the inverse variables. Consider a virtual variation of the system, defined by infinitely small and arbitrary variations 0X|, J Z'J.1. 07. 03 of the normal variables it could be also by the variations oA|, oAj, 3A", oïr of inverse variables, provided that the state to which this virtual modification leads the system is a new equilibrium state; only on this condition can the final state still be said to be defined by inverse variables. We will give the name of equilibrium KLK.viF.KTAiRK displacement to any elementary virtual modification whose initial and final standards are two equilibrium states. A continuous sequence of equilibrium stalls will constitute a Fiai displacement of eqi-ilibkk, provided that the elementary virlual modifications that form this sequence are all reversible. All the virtual changes discussed in this paragraph will be equilibrium shifts, so we can invoke, for each of them, the relations --i- rir, ~F~~ ~'=5A.+- (^ | oF,, "OV,, <>{-"" oa,, = - - oA,-r.r - - oA,, -4 - o3 l <^A| o\n Il oz which are drawn from the equalities (33). If we transfer these expressions of 3a,, oa;, in the expression (i) of the external work, this one will be able to put under the form (35) Ç-P, oA,4- P, SA2 -+- -t- F\, SA,, -f- e 5j, ̃ u. -'i.: ", ------------------------------------------------------------------------ P, P2, P, (̃) being given by the following equalities F -A 0F' i '" = AI '+' A H -r- |A,- -r. -AB - > e-i. ^" "-a, .v"^. The quantities P, P2, P, 0 are the external ccctiorzs relative to the inverse variables. In general, the quantity 0 is not nil, so that the inverse variables of normal variables do not play, even for a system in equilibrium, the role of normal variables. The quantity of heat released in an equilibrium displacement of the system always has the expression Q = G 3V. It can therefore be written in the form <37) Q= - (R, SA, +-+ Rrt3AB^-C32r;, ~i the quantities R,, R2, Rn, C being given by the equalities R, = P., ~A, .a. (38) dV Pa ~5~ c==~-e. The quantities R| R2, R, C are the inverse heat coefficients of the system in equilibrium; in particular, C is the inverse heat capacity. It should be remembered that the expression of the quantity of heat given by equalities (3j) and (38) is of much more restricted use than the expression given by equalities (18) and (ig); the latter is applicable to any virtual change imposed on the system from a state of equilibrium of the former, it is permitted to make use of it only for equilibrium movements. Equalities (38), which define the calorific coefficients ------------------------------------------------------------------------ two groups of relations can be drawn First, the group of relations symbolized by the formula rJllt,,+ l I dA" sla~, 1?u secondly, the group of relations symbolized by the formula (.îO) ri l 13,, ~r ) l' A -- In these formulas, the indices p and can vary from 1 r~. Identify the expressions (18) and (3-;) of the quantity of heat released in an equilibrium displacement of the system, and take into account the equalities we will obtain the relations R p, +. ,t dF1 dF" i (41) 1 dF1 d1'" f ')') .i dt're which allow to pass from the normal calorific coefficients to the inverse calorific coefficients. Formulas (36) and (41) show the inverse external actions and inverse heat coefficients when the normal external actions and normal heat coefficients are known; it is equally possible, when the external actions and inverse heat coefficients are known, to determine the external actions and normal heat coefficients; this can be done by the following formulas A 1'l -t- P "7' (42) a ~~n Ji~r A" 1 tlx" + rr ,Ixa 02,, <~x,, R' da' - T Rir J~a p (43) Rt J/ R J%. dxn dx" e R R dfrr C. ~J? (7~ ------------------------------------------------------------------------ These formulas! are obtained in the same way as formulas (36) and (4 n-) Let us take the procedure which provided the equalities (\:>.) and apply it to the calculation of the action relative to the variable .3 in the system of variables a,, y. 2f; we find that this action has for expression r p~=.. bone bone But, since the variables a,, a,, are normal variables, this action must be zero; we must therefore have identically (44) i>1^L+.+ Pa^i+H = 0. bone bone It is easy to see that this is the case. By virtue of the equalities (36 '), equality (44) can be written t' ^(àF^àf, dV,,dfn oVt>\ .1)) = o. ~a, c. +. d.a,t c. T c. = o. C" 1 Suppose that a. a, S. A,, A,, vary simultaneously so that the system is constantly in equilibrium your equalities (3) and (33) will be constantly verified, and we can write ù2jl ôA + "En iûa + "v" OS ~r~ o\n oS iiS I à%p àAt ~z~, di,, àkn dzp '̃ O.\i OS 0An OS But the obligation to maintain the system constantly in equilibrium does not prevent us, by virtue of the condition laid down at the beginning of this Chapter, from arbitrarily taking a, y and 2? the variables "i, in, 2? being independent, in this system of variables, the derivative of *xp with respect to 2? will be identically zero and will remain so if we pass from this system of variables to the variables A, A2, A, 2"; this derivative will then be expressed by the second member of the preceding equation; we have therefore (46) ~r,. ~f, +.+ ,~F,, ~f" =o, (4 "A, os 0Xn os and the identity (44) is justified. ------------------------------------------------------------------------ Let us apply these considerations to I'kxkmcij-: simple (|ue provides us with the study of a homogeneous fluid. The compressibility and expansion equation (!¡) ))-r:/(~). which is the equilibrium equation of this fluid, can be solved as (47) oi = F(II,:3), which gives the specific volume of the fluid in equilibrium when it is brought to a given temperature S and subjected to a given pressure II. Between the two functions /(w, 3), F(n, .t?) exists the relation ('8 toF(H,2f) df(u>,?j) âV(U,5) 0, <*"> ) -ôîr-w-r- = 0' particular case of the relation (46). By virtue of equality (4^), the external work Mil or performed in an equilibrium displacement becomes <49) S = P SU -+- e S2f,- by posing ~=- (50) j ,8 =~ 111t ~) F(Il, :J)= l d These are the expressions of reverse actions. The function m (m, 2f) which appears in the expression of the internal energy U of the fluid is transformed into a function p(n, 2f), so that the internal energy U becomes <5i) V(n,2r) = M".-(n.S). The heat released in an equilibrium displacement of the system can be written Q = - 8\- _i- I" in e Iz or, taking into account equalities (5o) and (5i), (5a) Q=- MfA(lI, 2r)olI-r(n,2r)o3], ------------------------------------------------------------------------ expression in which fc ll. -'lv(fl,:W,_tIJ('(fl.~) (53) olj- I'(lt -_Jv('ll,)_.ttdl'(II,) d:: u: 03 03 The quantities h (II, £?), F (II, S) are the inverse caloric coefficients of the fluid in equilibrium; the second is the specific heat under constant pressure of this fluid. From the equalities (53) that define these coefficients, we can easily derive the relation , (55) ( r(n,2r) = /[F(n,2r),Sf]^)-i-7[F(a,&),S], F 11, 1 [ F rl., -;(F(rl,â~.â~, 03 which show the inverse heat coefficients when the normal heat coefficients are given. The same identification, made by putting the equilibrium condition in the form (4), provides the two relations llw,â,j= la(.Î(W,~)~ J.f(~,>; â r dlt} (56) fl Y(",a) = A[/( = Çp^i" - - -' A/" a/'+ii ̃ - -' a"' "*)> and substitute for x,, ap the second members of. these equalities (5- ). We thus replace the system of normal variables a,, a", an, by the system of variables At, A, a,i-i, a, 3, that we will call a system of mixed variables. The use of mixed variables is not absolutely limited to the study of a system in equilibrium; it only supposes that certain equations of equilibrium remain constantly verified even if the system is in motion; here, for example, the equalities (07) only suppose that the first p equations (3) are verified even if the last (/* - p) are not, and what we have just said would remain true. The general theory of the passage from normal variables to mixed variables can be modelled on the theory developed in the preceding paragraph; so we shall not deal with this theory; we shall confine ourselves to studying an example, as simple as it is important, supplied by the phase of Chemical Mechanics ( ̃ ). Let us suppose that the normal variables by means of which this (') The use and importance of the system of variables which is going to be studied have been pointed out in our Traité élémentaire de Mécanique chimique, t. t, 1897, pp. /|3 et seq. ------------------------------------------------------------------------ phase is defined as the specific volume <>>, J;i chemical composition x and temperature ?3 The equilibrium conditions are, in this system of variables, (6) j 11=/ 3), ), (6) ( cv, .r, .~). Without worrying about the second one, let's solve the first one with respect to m in the form (r)8) u> = Fi Il..r, 3). Using this equation, we can substitute the mixed variables w, x, Sr for the normal variables u, a: ?i. This substitution, which makes no use of the second equation (6), in no way assumes that the system is in equilibrium; it can be used even if the chemical composition varies from one moment to the next, provided that the specific volume varies at the same time, and in such a way that the first equation (6) never ceases to be verified. When this is the case, the chemical phase is said to be in partial equilibrium. This remark is of great use in Chemical Mechanics. For a virtual modification imposed on the system in partial equilibrium to be defined by means of infinitely small variations Sn, ôx, 32? imposed on the variables II, x, "b, it is necessary and sufficient that it leads the phase from a state of partial equilibrium, where the first equation (6) is verified, to another state of partial equilibrium, where this same equation is still verified; such a modification is therefore a displacement of partial equilibrium; we will not consider any other in the course of the present paragraph Equality ", ~F-_ ~F, (,5!" Oh' = 011 -- MU + J/ OR [f f>w ,aj which follows immediately from equation (58 ), transforms the expression (5) 5 =- MII8w Xôr of external work in (CO) ÎS = Pi Oïl -r- P-. JX H- 9 0^ ------------------------------------------------------------------------ with (lii) i Mil- -+- X. Pi> l'a, W are the actions relative to the mixed variables. The action (-) is not thousand, in general; the new variables are thus not normal. Even if we assume X to be constantly zero, as is done in Chemical Mechanics, it would not follow that P2 (ûl constantly zero the chemical action relative to the normal variables can be always zero; the chemical action relative to the mixed variables will differ from o, at least in general. For the hypothesis X=o to entail the hypothesis Po = o, it is necessary and sufficient that the function F be independent of x; so that the preceding proposition may be completed by this one For the absence of normal chemical action to entail V absence of, mixed chemical action, it is necessary and sufficient that a change of chemical composition, accomplished, without change of temperature or pressure, within. the phase in partial kquilibrium, be free from all condensation and dilatation. The function u (w, x, ?5) which appears in the expression (?-4) of the internal energy is transformed into a function c('fl, x, £?), so that the internal energy of the phase becomes itself (&:>.) \>i\l,a\ 2?.) = M c(ll, a:, 2f). ). The position that the various parts of a phase occupy in space does not intervene in the definition of this phase; it results from this that the variables that serve to determine the state of a phase are, by definition, variables without inertia even if the initial state of the phase were not a state of equilibrium, the quantity of heat released in a virtual modification is the excess of the external work over the increase of the internal energy, as can be seen by biflanking the work of inertia in the legality that defines this quantity of heat [Chap. IV. equality (i), p. i 5- | if the phase is defined by normal variables m, x, £?, and if we suppose, as we do in Chemical Mechanics, that X is i i.1==_ mi, ;£, l' = ,\1 dl' (."£. l~ 8=c-\III-. ------------------------------------------------------------------------ zero, this amount of heat is expressed by the equalities (25) and (26); keeping the hypothesis X =0, and taking the mixed variables Il, x, ?3, the equalities (5y), (Go), (61), (l[')V(II,.X,) ,ÙF(II,X,)] 4 ) | L. = M ^\0v(U,x,3) - i- 11 dFOUS)] - - t rdf(u,~&) ~F(n,~5) ) L 03 03 We will recall here the change of variables that we made in paragraph 3 (p. 285) and that are expressed by the equalities m \x = < = w Introducing then the quantities ( it, , x, :~j LIII,x,:1)=-l[F(I1,x,),x,J dx x +J.fl'(II,x,:1¡,x,.J] which make known the heat coefficients relative to the mixed variables when the coefficients relative to the normal variables are known. The inverse problem can be treated in a similar way. The last of the equations (68) shows us that within a phase, the heat of formation under constant pressure of a certain compound is not, in general, equal to its heat of formation under constant volume; these two quantities are equal to each other in the particular case where I; FAITH, *,&) = <>, ùx i.e. in the case where the formation of a certain quantity of the compound accomplished within the phase, at constant temperature and constant pressure, does not lead to any variation of the specific volume. From what we have said, all these formulas, as well as formulas (20) to (29), are applicable not only to a chemical system in equilibrium, but also to a chemical system in partial equilibrium where, only the equilibrium condition or n=/o, x, S) is verified. w = F(n,x,â) is verified. 0. Systems defined by a single normal variable except the temperature. Reech's theorem. The theories exposed in the previous paragraphs are applicable to holonomic systems defined by any limited number of variables; the one that will be the subject of this paragraph deals exclusively with systems whose state can be defined by ------------------------------------------------------------------------ Instead of applying it to these systems in general, it will be, we believe, more interesting to expose it immediately on a kxemplk, the one offered by the homogeneous Jluid. The reader will easily pass from this particular example to the general case. The equilibrium equation of the homogeneous fluid (!p H=/(M,~) can be solved with respect to 2f in the form (G9) 5 = çO, II). We can then substitute the variables w, II for the normal variables co, S. This choice of independent variables is particularly convenient when one has in view the study of the external work produced during a transformation of the system. Each state of the system is, in fact, represented by a point of abscissa o> and ordinate II; the transformation considered is represented by a curve which is called Clapeyron's diagram; it is Clapeyron, in fact, who first made use of this mode of representation ('). As the elementary work of the external actions has the value E =- MM 3~, the total external work done in any transformation is proportional to the area bounded by the diagram, its two extreme ordinates and the oj axis. In this system of variables, the external action relative to variable w remains - Mil; the action relative to variable II is zero. The variables w, II may well not unambiguously determine the state of the system; the same set of values of pressure and specific volume may correspond to two different temperatures; under atmospheric pressure, liquid water has the same specific volume for two distinct temperatures, one below and one above the temperature of maximum density. It follows from this remark that the internal energy of the fluid, (') Clapeyron, Journal de l'École Polytechnique, XIV' Cahier, iS3/|. ------------------------------------------------------------------------ expressed by means of the variables co, II, (70) Wfw. Il* = M w (w. II), j. is not iorcéirietil foncliou uniform "le these variables. The expressions of the external trawiil and the internal energy show that the quantity of heat released in an equilibrium displacement of the system can be written (71) Q =- M [>.((", Il 10(0 /.(w, 11) SI1J, j, by putting I Ô IV l (1), II ) j ,.", H^-j-i-H II, (0), Il ) rJ cvl'to, II I r-(~=-o)-' definitions that give us ( o }.(oi, n ) 0 /"(to, n j .7:" -~1-='- The expression (71) for the amount of heat released in an equilibrium displacement can be identified with the two other expressions for the same amount of heat that were previously given (23) Q=-iU[ /(u>, 2F)oco-r-<;M, 2î)oS], (5a). .̃ Q= - M[/i(ri,&)oiH-r(lI,â)S&]. If we identify the expressions (̃<3) and (71), taking into account equality (69) and equality ( dœ(oj. H), :doCw. II) "" (74) .o3=-J-ssrJo "o+:r-5ii-"H which, by virtue of this equality (6g), governs any equilibrium displacement, we will find the relations ( l(oi, II) = t[ui, o(o>, M)] -7-7 [eu, o(cu, in] C>?(J^' ''" j*("o,n) = -,["",?"", n,|^ILi, (0), 11\ k(w,tl)-w .ta,,(I)~J,tltl~~ which determine the new calorific coefficients Ai'to, II i. ------------------------------------------------------------------------ />-. s . 0) when the normal heat coefficients /( o), 2f y ( , ?j) are known. If Ion identifies in the same way the expressions (5a) and (ri) of the quantity of heat, we find the relations l À(o), li) = rrit, y(m.ll)| 'f '"̃II1, 1 dc,7 l-G~ Jr~(w, 11) (;G) | klo>, II) = A[1I.?(<", 11)| -+-r[II, s> ('), "i]C>?(J[| 'J)' which determine the new calorific coefficients X(w, II), /(̃(('). Il) when the inverse heat coefficients A(II, 3r), r(ll. %) are known. The first equality (76), compared to the second equality (76), gives the relation (V y(oi, n) ( r xrco, ii) on O71 =~ -(w.tl) do1w,11) - dw The second member of this relationship is open to an interesting interpretation. To obtain this interpretation, we will consider successively two distinct equilibrium displacements, imposed on the system from the same initial state. The first of these equilibrium displacements is assumed such that the temperature does not vary; if 3a> is the increase of the specific volume in this isothermal displacement of the equilibrium, the pressure experiences a variation SlT that we can put in the form DjSw; Dj is called the coefficient of isothermal relaxation; according to the equality (74), where we have to make 3.^7 = o, we have dufw, II) (-$8). dw -" or ~n The second equilibrium displacement is supposed to be accomplished in such a way that the quantity of heat released by the system during this displacement is equal to o: it is called adiabatic equilibrium displacement; if, in such a displacement, the specific volume increases by ow, the pressure II experiences ------------------------------------------------------------------------ (' ) Rekch, Theory of the Driving Machines and the Mechanical Effects of Heat, p 38. - See also K. Clausius, On various forms of the equations of the mechanical theory of heat which are convenient in application, equation (36) (Poggendorff's Annalen, Bd. CXYV, i8<)5; Mechanical Theory of Heat. trans. Folie, t. I, i8f>8, p. 398)- (2) J. MoL'TiElt, Bulletin delà Société philomathique, session of May vi, 18S0. (3) We have proposed in the writings we are about to quote to define reversible modification as indicated in this paragraph and the next: Étude sur les travaux thermodynamiques de J. Willard Gibbs, art. h (Bulletin des Sciences mathématiques, i' série, t. XI, i§8-). ). Commentary on the Principles of Thermodynamics, Part I, The Principle of Sadi Carnot and /? Clausius, Chap. I, artt. and S (Journal de Mathématiques pures et appliquées, /|* série, t. XI, pp. 3o2-3og). a variation oïl that we can put in the form - DyO<"; the quantity Dq is the coefficient of adiabalic expansion; by making Q := o in the equality (71^, we find ( D TO { M. II ) 1 (79) D"=7T^TiT Equalities (77), (78), {-[)) allow us to write ( 80 ) r Do (80) = .f Ih In a homogeneous fluid, the ratio of the specific heat under constant pressure to the specific heat under constant volume is equal to the ratio of the adiabatic expansion coefficient to the iso thermal expansion coefficient. This beautiful theorem is due to Reech (' ); J. Moutier derived from it (2) a very simple and general justification of the experimental method devised by Desormes and Clément to measure the ratio of the specific heats of a gas. 7. Of the variable modification in a continuous way and of its limit. We shall now introduce into our reasoning a new notion, that of reversible modification (3), and subject the systems we study to a new restriction. Let us first consider the considerations that will allow us to define the reversible modification. ------------------------------------------------------------------------ Let us consider a real modification. S of a certain independent system and let us suppose once and for all that this modification does not involve any instantaneous modification. Such a modification lasts a certain time; if t0 is the initial instant, the final instant will be l,. -We will designate by ti any instant comprised between tu and l, at this instant the system is in a certain state it l'si ut to give certain values to certain magnitudes, in noml>re limited or unlimited; to define the movement m], it is necessary to attribute other values to these same magnitudes; the value attributed to each of the magnitudes that define the mouvenienl, nij is as close as one wants of the value attributed to the same magnitude in the definition of the movement />-; which is expressed by saying that the motion m'j is as close as one wants to the motion m, or that it is infinitely close to this motion; ()" While the system, undergoing the modification M', takes the case e) the foreign bodies are in a state/j-; this state can be made as close as one wants to i'elal hi where the foreign bodies are at the moment when the system, undergoing the modification M, takes J'état-A^ ̃£̃ When this is the case, we will say that the set of real modifications to which the modifications M, M' belong constitutes a modification, real and continuously variable; the two modifications M, M' will be said to be two forms as close as we want to V one of the other or two forms infinitely close to this variable modification. Let, e', be a set of states of a certain system. The standard e is defined by giving determined values û ̃"̃.̃, to e certain quantities, in limited or unlimited number. G i (i2- ̃ ̃'̃ I state e' is similarly defined by giving certain other values:.> %'", to the same quantities. Let us consider a state s of the same system. state defined by giving certain values - to the urandeurs G|, G*, Suppose we can arrange the i^lais c, "', in such an order that g,, g't, are found to tend to the limit y,, while g", g. tend to the limit -̃̃ eir. i\oti's will say that the states e, e', arranged in this order, have as their limit state the state s. We would define in the same way the limit motion \i. of a ̃ensemble of motions m', arranged in a certain order. Let's take a set of stalls of a certain material system here ------------------------------------------------------------------------ let us form a linear sequence l> of determined sense we understand by that that we take a variable quantity A cjm grows constantly and in a continuous way from an initial value a0 to a final value X,, and that to each value A of this variable we attach (in determined state of the system £,, is Yinitial state, corresponding to the initial value a0 of a, £| the final state, corresponding to the final value A| Suppose, moreover, that the sequence L satisfies the following conditions i" To two infinitely close values A, ),j of A correspond two infinitely close states s/, sy of the system; 2" If, of these two values, À is greater than A/, a case in which we say that the state -.j succeeds the state z;, the passage from the state s,- to the state s y is compatible with the bonds imposed on the system; the sequence L, traversed in the indicated direction, is thus a virtual modification of the system, and it is the same of any portion, finite or infinitely small, of this sequence; .>" To each value of À and, consequently, to each state of the system included in the sequence L, corresponds a certain movement ;j. of the LI system and a certain state -i\ of the foreign bodies. If in this definition, only the states of which the sequence L is composed, the motions u and the states r, of foreign bodies associated with these states z can represent the states and motions taken by the system in the course of any real change, there are cases where, obviously, such a supposition would be absurd. Here is one such case: suppose that in each of the states z which compose the sequence L, the system is likely to remain in equilibrium when placed, without motion, in the presence of suitably chosen foreign bodies; to each of these states z, let us associate a zero motion; at the same time, let us put the foreign bodies in a state rk such that the state z becomes the state of equilibrium of the system; the sequence L will then be a continuous sequence of states of equilibrium of the system; it will be identical to what in paragraph A (p. 289) we called a finite displacement of equilibrium; it is clear that such a sequence cannot be realized in any modification. 289), we called a finite displacement of the equilibrium; it is clear that such a sequence cannot be realized in any modification. ------------------------------------------------------------------------ Let's leave this particular case and come back to the general case. Let's imagine that, for a given material system, we have first of all, a sequence of states and movements L defined as we have just indicated. In the second place, real modifications M, M', M", which, arranged in a certain order, constitute a real modification continuously variable according to a certain law. In these modifications M, M', M", we can consider corresponding stages and; e', e], whose properties were fixed a moment ago; to these states e, e' ei of the system, are linked respectively movements m, m], m], of the system and states lu, Uh h]-, of foreign bodies. We shall suppose that to each state " The states /<(, /], of the foreign bodies (|iii have been respcclivenenl associated to the e,. e), and, of the system tend towards the limit standard 'rlt- of the foreign bodies, state that one associated to lethal s/ of the system. It is so whatever <|ue the elected l ). Let's increase the temperature S, starting from the value S?o - OB0, Fig. 3. according to a certain law ?S =/(t); the chemical composition will vary according to a certain law x-o (/); it will decrease, for example, if the chemical compound of variable mass that the phase contains dissociates when the temperature rises; while, for example, that the temperature will rise from So to S? = Oft, x will decrease from xn to .r,. We will thus have a realizable modification M which will be represented by the line A,,B. ------------------------------------------------------------------------ By continuously changing the shape of the function /(/)i|tii determines the law of increase of the temperature we can make vary in a continuous way the modification M. Ku a determined modification M, to each value x of the l" J x will not be |>Ius in any way a realizable modification, but a sequence of equilibrium states of the phase. 8. Of reversible modification. Let us imagine that a real modification M, continuously variable according to a certain law, has as its limit form a certain sequence L of equilibrium states of the system; the sequence L is, moreover, the limit form of the real modification M only if we go through the equilibrium states which form it in a well-defined order and direction: when they are paralleled in this sense, these equilibrium states have as their first term the state :" and as their last term the state £f. It can happen that another real and variable modification m of the same system has for limit a sequence of equilibrium states that the sequence l is formed of the same equilibrium stalls as the sequence L. and that these states are, in these two sequences, arranged in the same order; but that the sequence l is not the limit form of the modification m unless we follow it in the direction that goes from the state to the state î0 If this is so, the two sequences of equilibrium states L el L WHICH ARE XI 1,'uNE Kl I.'aUÏIIK T1KS MOniFICATIO.N S 11 KKI.LKS, fnvmCronl what we agree to call a modification kkm-.iiS1111.K. When we consider the sequence L, we will saythat we impose on the system this reversible modification in a determined direction, from ------------------------------------------------------------------------ From the za-standard to the z-standard, we will say that we impose it in the opposite direction, from the z-standard to the zu-standard, when we will focus our attention on the modification l. Let us apply these considerations to the examples we examined in the previous paragraph. Piuïmikh kvemplk. - Instead of subjecting the material point of weight P to a vertical force I'', directed towards the binder and less than I', let us suppose that this force F overcomes the weight P. Let us place the material point at the level z, without any initial velocity; it will rise by a real change m to the level ^0. y increasing or decreasing the excess of the force F over the weight I', we can vary the modification m. Let us imagine that, by values always greater than P, the force F tends towards the weight P; the real modification nt will tend towards a limit form; but this one will not be a modification at all; it will be constituted by a sequence of states of equilibrium of the material point, these states occupying all the possible levels between s0 and -i only, so that this sequence of states of equilibrium represents the limit form of the real and variable modification /", it will be necessary to go through it by thought by going from the level zt to the level z0. We are therefore dealing with a reversible modification. Second kxempi.e. - Let us take our extensible wire in the state of equilibrium to which the sequence ahoutit; its temperature, which will always remain invariable, is 2f; its length is l, the tensor weight to which it is subjected has the value P,. From the value P, let us decrease the value of the tensor weight according to a certain law P = g'(l)', the length will also decrease according to a certain law; at each moment, we will be able to make correspond a point c {fi g- 4) having for abscissa the value of the tensor weight and for ordinate the corresponding length of the thread; we will thus have a real modification m represented by a curve fi", place of the point c; we will be able to follow this curve up to the point ", of ordinate la. By continuously varying the shape of the function ,S{1), we continuously vary the shape of the modification m and the line [i" by which it is represented. ------------------------------------------------------------------------ To each value taken by the length in the modification. corresponds a value of the velocity w = "y- with which decreases the tensor weight; we can change the shape of the !-"̃>?̃ I- function g(t) in such a way that the velocity tv tends towards o whatever the variable modification m tends towards a limit form o and the variable line pa tends towards a limit line p A'; o will no longer be a real modification, but a sequence of equilibrium states that the thought goes through from the state jj(P,, /,) to the state A'( I'o, /"'); the line JÏ A' will be Y isothermal descending from the point p. Two cases are then to be distinguished: i° The descending isotherm p coming from the point p coincides in all its extent with the ascending isotherm A p which ends, at the same point. If, therefore, we take the thread in a state of initial equilibrium, if we stretch it with infinite slowness, if we then unload it with equally infinite slowness, at the moment when the weight that pulls it resumes its initial value, the thread returns to its initial length; the traction does not cause the thread to experience any permanent modification. In this case, the two sequences of equilibrium states that we have named a and o are composed of exactly the same equilibrium stalls, arranged in the same order: but. so that the sequence 1. can ------------------------------------------------------------------------ To be considered as the litniw:dc form of the varialile modification M, it is necessary that the thought follows it from the state A ( l\ /") to the state, |i(l\, /( i on the contrary, so that the follows e o can be considered as the limit form of the variable modification m. it is necessary, to imagine that one traverses it of the state |"i (' ) to Tétai A' or A (!'", /). The two sequences of states of equilibrium that represent C ascending isotherm and, the descending isotherm constitute, in this case, a reversible modification. 2° The descending isotherm 3 A' coming from the point ji is distinct from the ascending isotherm A [3 which ends at the same point. The wire, taken in a state of equilibrium and stretched with infinite slowness, can, by an infinitely slow decrease of the tensile weight, return to its initial length; but it resumes it under a tension different, in general, from the initial tension; the wire experiences, by traction, permanent deformations. In this case, the two sequences of equilibrium stages that we have denoted by x and o consist of different states in general. Neither the ascending nor the descending isotherm V represent a reversible change. Third example. - Let us take our chemical phase in equilibrium in the state represented by the point |3 (fig. 3) and lower the temperature according to a certain law "b - g (t) the composition also varies according to a certain law x = 'b(,t); we thus obtain a real modification N represented by a certain line |3A. By continuously changing the form of the function g(t), we will continuously vary the modification N and the representative line |j A. In the real modification N, to each value of x corresponds a value v = - - à-J of the speed with which the temperature decreases and also a value " = ( f> of the speed with which the chemical composition varies. We can gradually change the form of the function g(l') in such a way that v tends to o for every value of x; u will also tend to o for every value of x; the real modification N will tend to a certain limit form v and the line 43 A to a certain limit line j3A0 or v. ------------------------------------------------------------------------ The limit form of the modification is no longer a real modification, but a sequence of equilibrium stages of the studied phase. Two cases are then to be distinguished: i" F.a Line |ÎAO coincides in all its extent with Line |jA0. This is what will certainly happen if each temperature corresponds to a single equilibrium state of the phase maintained under an invariable volume. l)n this case, the two sequences a and v consist of the same equilibrium states arranged in the same order; but, if one wants to consider the sequence u. as the limit form of the real modification M, one must follow it by thought from the stage An (.:70, r,,) to gauge 3 (ï, X\ ) if one wants the sequence v to be the limit form of the real modification IN, one must follow it from the state [i(T. xK i to gauge A,, or -H ('~0) xo). The two sequences jx and v constitute a reversible modification. 2° Chemical mechanics has recognized the impossibility of representing all experimental laws by attributing a unique equilibrium composition to a phase maintained at a given temperature, under a given volume. It has been led, in a great number of cases, to formulate the following hypothesis Under the given volume, are states of equilibrium all the stages represented by a point P(Sr, x) of the region of the plane comprised between two lines such as ay and c'y' ( fi g. 5). The point Ao will be, in general, any point of the region, called false equilibrium, which is included between the lines ay, a' y'. It is therefore easy to see that the line A,,|i will be composed of a segment AnB, parallel to OS. along which the temperature M ire will increase from ?30 to To while the chemical composition will keep the invariable value x0, this segment being followed by a portion Bjïi of the line ay which limits the region of chemical equilibria. Similarly the line ^A^ will consist, first, of a segment |3^ parallel toOS? along which the temperature will decrease from T to T' while the composition will keep the value .r,, and, second, of a portion [j'A0 of the line a which bounds the region of fajix equilibria. For the phase to resume the composition ------------------------------------------------------------------------ initial .", it will be necessary for the temperature to have a value !i0, dill'é!̃<̃ n le of its initial value 2?, J'ïk- ̃>. JYi the sequence - nor the sequence v represents a reversible change. Returning from these particular examples to general considerations, we will state the following hypothesis: Restrictive hypothesis. - All the systems studied in what follows are such that any continuous sequence of equilibrium states of such a system is a reversible modification. That this hypothesis is essentially restrictive is shown by the examples previously studied; they presented us with systems where certain continuous sequences of equilibrium states, although they were the limit forms of certain real modifications, did not constitute reversible modifications at all. By generalizing what these examples have taught us, we can say that the systems affected by friction or hysteresis escape the restriction that we have just formulated. (the same systems already escaped the restriction stated ------------------------------------------------------------------------ at the beginning of this Chapter (p. afiô); two reasons concur to exclude them from the study that we are going to (aire. Physical theory has not had occasion, until now, to compose systems which would be subject to one of the restrictions stated in this Chapter and subtracted from the other. In the systems it has imagined, some escape these two restrictions at once, such as systems affected by friction or hysteresis; others are subject to both restrictions: it is these last ones that we shall study. These systems have a fundamental property which will be constantly used in the following and which is determined by this proposition If we form a continuous sequence of states of a system subject to the two restrictions which have just been posed, and if, in each of these states, the temperature of the system is uniform, this continuous sequence defines a reversible modification. Indeed, according to the first restriction, one can, to each of these states, associate foreign bodies such that it becomes an equilibrium state; one thus transforms the given sequence into a continuous sequence of equilibrium states which, according to the second restriction, constitutes a reversible modification. ------------------------------------------------------------------------ CHAPTER VIII. THE PRINCIPLE OF SADI CARXOI AND CLAUSIUS. 1. Isothermal and adiabatic changes. Isothermo-adiabatic systems. Let us take an independent and normally defined system. It could happen that the notion of temperature is foreign to the definition of this system: this is what would happen, for example, if it were one of those systems of material points and solid bodies which are dealt with in Rational Mechanics; let us leave aside, for the moment, this particular case; we shall return to it later (Chap. IX, § 3). Let us suppose that the system under study is in equilibrium. in a certain complete state E defined by a necessarily uniform temperature S and by a state apart from the temperature which we shall designate by the letter e. Let us assume, finally, that any virtual modification imposed on the system from the state E is reversible. From the state E, we can subject the system to a reversible modification during which S? keeps an invariable value, that is to say to a reversible isothermal modification. We can also subject it to a reversible modification such that each of the elementary modifications that make up this finite modification releases a quantity of heat equal to o; such a modification is a reversible adiabatic modification. Any infinitesimally small reversible modification is a displacement of equilibrium; the temperature, necessarily uniform at the beginning of the modification, is still uniform at the end; it even varies 32" at any point of the system; therefore, according to what we have seen [Chap. Vif, £ 3, p.: "8;* |, the quantity of heat Q released in a similar modification can be put ------------------------------------------------------------------------ in the form of ( I.) Q = - c(e, "3) Ï3 -̃- Q\ c(e,Ï3) being the calorific capacity of the system taken in the state c and at the temperature .:?, and Q' being the quantity of eheat which; ic system would release in the change of state without change of temperature to which corresponds the considered elementary modification. n particular, if this change in calibration without change in temperature is determined by n normal variations a,, "2, "", this quantity of heat can be written [ Chap. VU equality (i 6), p. 283] (%) Q =- [pi(c, "3)ai-~ pz(e, "3)aî - .+ pn(e, ?S)an-r- c(c, 27;8S|, ¡, pi (<2,£?), pa(e,2r), p/( (e&) being the heat coefficients of the system in equilibrium in the state and at temperature "b. Can it happen that any elementary reversible and adiabatic modification, coming from the state E, is at the same time isothermal and conversely? If this is the case, the state E is said to be isolhermo-adiabatic. For this to be the case, it is necessary and sufficient that we have identically (3) Q'=o. For an equilibrium state of a system to be an isothermo-adiabatic state, it is necessary and sufficient that any elementary change of state, accomplished without change of temperature, leads to a release of heat equal to o. In particular, if a change of state without change of temperature is defined by n normal variations ",,", a, equality (3) is equivalent to the n equalities ('<>̃, Pi(e,S/) = o, pi(e.Js) = o. p"(e. !ï) = o. "S* i V state of a holonomic or non-holonomic system depends on a limited number of quantities, and if this system is normally defined, for a stall of this system to be isothermo-adiabatic, it is necessary and sufficient that all the calorific coefficients, apart from lu heat capacity, be zero there: ------------------------------------------------------------------------ A system whose all (hais d'é'fttilibre are isollienii.o-adiabalirfu.es /y/end the name. of sïstkmk isotiiki; \ioAIMAliATHM K. According to what precedes, this definition would have no sense, if an isothermo-adiabatic system n were a system whose virtual modifications are reversible. In the course of the following deductions, the isoihermo-adiabalic systems will constantly present themselves as exceptional cases; it was therefore important to point out their existence first of all; it is also important to establish certain properties which derive immediately from their definition. It is assumed that the system under study is subject to the restriction posed at the beginning of the previous chapter (p. ?.(>.>); it is also assumed that all the virtual modifications of which it is susceptible are reversible. Therefore, for it to be in equilibrium, it is necessary and sufficient i" That it has in all its points the same temperature IE7 and the same temperature as the foreign bodies 9." That in any \irtual modification from a certain It state Ë or (e,?j). the external work 6 has a value entirely determined by the knowledge "I the state li and of the virtual modification. The assumption that the system is isotherino-adiabatic determines this value of P. Indeed, since the system is defined normally, the most general virtual modification results from the other two: a change in temperature 27 without change in state e, and a change in state e without change in temperature . The first change does not involve any external work. since the definition of the system is 'normal. The second modification does not involve any heat release, since the system is isolhermo-adiabalic. The work of inertia is also zero, since the system is in equilibrium. Therefore, according to the definition of the quantity of heat [Chapter IV, equality (.{), p. i">7], Ic external work v must be equal to the increase in internal energy. If therefore we^ ------------------------------------------------------------------------ let us designate by o^\j(e,?i) the variation that the internal energy experiences in a virtual modification where the temperature 31 remains invariable we will have (5,i G = 3j lUe, 'il). For an isolhenno-adiabatic system to be an equilibrium g in a given state and at a given temperature, it is necessary and sufficient that in any virtual modification, composed of a variation of temperature and d1 a change of state without change of temperature, the external work is equal to the increase that undergoes V internal energy by the effect of the change of state without change of temperature. Let us assume, in particular, that the state of the system studied depends on a limited number of quantities; a change in calibration without a change in temperature will be determined by n normal variations ",, a- an; we will have [Chap. VI, equality (a), p. ?.54] G = A i "] -h A 2 "2 -+- 4- An aH while we can write [Chap. VII, equality (i5), p. "8a] os- U = Ui(e, 2?)a!-f- u-2(e, Jj)a-1- .h un(e. Jl)an. Equality (5) will then be equivalent to the n equalities (6) \l=Ui(e,?j), 'A2="2(ei^); -̃-> A.re= u,,( e, "S). These are the equilibrium equations of an isother/noadiabaliqi system where each state is determined by a limited number of quantities. If the system is holonomic, it is defined by n normal variables a,, a2, .an, excluding the temperature 2f, and the equilibrium equations take the following form A, = - U ( a2, *". 5 1, "i U' (7) A, = ^-Uia,,""&), I \n= - U(i|. "j, a, Sri. 0.-1. ------------------------------------------------------------------------ Let us compare the properties of isolhermo-adiahnl.iqucs systems, as they have just been established, to the properties of systems studied in Rational Mechanics. Let us consider one of the systems studied by Rational Mechanics, and let us suppose that this system can experience only reversible virtual modifications. For such a system to be in equilibrium, it is necessary and sufficient [Chap. V, equality (i i i), p. y/ja | that in any virtual modification, the external work G is <><^a to the increase experienced by the internal energy U (8) e = au. Since the notion of temperature does not intervene in the definition of the systems studied by Rational Mechanics, condition (8) is as analogous as possible to condition (5). On the other hand, we know [Chap. V, equality (i), p. 184] that, for any system studied by Rational Mechanics, the internal energy can be regarded as the potential of the internal forces. Guided by these considerations, we shall say that, according to equality (5), internal energy plays, for isolliermoadiabatic systems, the role of thermodynamic potential [ntehnk AT CONSTANT TEMPERATURE. When at the end of a reversible change, the system under study returns to exactly the state it had at the beginning, we say that this change is a reversible cycle. At the end of a cycle, the internal energy necessarily returns to the same value as at the beginning. Therefore, if we write, for each of the elements of a reversible cycle traversed by a mechanical system, an equality such as (8), and if we add member by member all the equalities thus obtained, the sum of all the elementary increases of the internal energy will be reduced to the total increase of this quantity, i.e. to o, and we can state the following proposition When a mechanical system goes through a reversible cycle, the external actions perform a total work equal to o. An analogous demonstration, applied to equality (5), gives the following theorem ------------------------------------------------------------------------ When an isothermal-adiabalic system goes through a reversible ISOTHKHM1QU-: cycle, the external actions perform a total work equal to o. When we know the expression of J'énergie interne d on isothermo-adiabatic system, we know, by the equalities 5), (fi) or (-j) the conditions of equilibrium of this system; we also know the only calorific coefficient, the heat capacity, by the relation ( c(e. ) - ~â We can therefore say that the knowledge of the form of i internal energy of an isothermal-adiabalic system leads to the knowledge of the equilibrium laws of this system and of the quantity of heat released, in any virtual modification, by this system in equilibrium. This is what we will express by saying that, for a normally defined isothermal-adiabalic system, V internal energy plays the role of a CAKACTÉH1ST1QUK function. It is easy to design systems that, when in equilibrium, are definitely isotliermo-adiabnlic. Let us consider a system, normally defined, of which all virtual modifications are reversible. Let us suppose, moreover, that this system possesses the property stated in this proposition In any modification, real or virtual, from a state of equilibrium wave motion, the quantity of heat released is given by this formula Q = - -> ois dm, where dm is one of the elementary masses that compose the system; where 3S> is the variation experienced by the temperature at a point of the mass dm; where y is the specific heat normal to the same point; where, finally, the integration extends to all the elementary masses of the system. It is clear that such a system, once in equilibrium, would be isothermo-adiabatic. Later, when we have established other properties of ------------------------------------------------------------------------ isothermo-adiabntic systems, we will give more detailed and particular examples (- }. 2. The Carnot cycle and perfect systems. Let two temperatures ?50, ?3t> In second higher than the first. We say that a reversible cycle traversed by a is a, Caii.not cycle described between temperatures ?50, if all the reversible modifications in which this cycle can be decomposed can be classified in these three categories i" Adiabatic modifications; 2° Isothermal changes relative to the temperature 2f0 3° Isothermal changes relative to the temperature 3f,. S,, is called the lower limit e( £?, the upper limit of the considered Carnot cycle; we do not intend, by the use of these denominations, to prejudge that we do not meet, while traversing this Carnot cycle, neither temperature lower than 2r0, nor temperature higher than 2? A simple Carnot cycle is a Carnot cycle that consists of only two adiabatic and two isothermal modifications; one of these is accomplished at temperature Sf0, the other at temperature Sï(. Let us suppose that from a certain state E, we impose on the system we want to study an elementary reversible modification, defined by some infinitely small variations of the physical properties of the system and of the foreign bodies; this modification corresponds to a release of heat Q. We will suppose that the studied systems have the following property If we make the state E vary in a continuous way, if we change in a continuous way the variations which determine the elementary reversible modification, the heat release O varies in a continuous way. It follows from this that, if a state E or (e,3j) is not a .̃̃- ( ) See Chap. IX, § 0. ------------------------------------------------------------------------ isothermo-adiabatic, no state sufficiently close to this one is îsotliermo-adiabatic. This corollary leads to another If the state E or (e, "h) is not an isothermal-adiabalic state, one can, from this state, impose on the system a reversible adiabalic modification along which the temperature increases continuously from 7s to and another reversible adiabatic modification during which the temperature decreases continuously from b to b", provided that the positive differences (b' - b) and (b - b'' ) are smaller than a certain limit The limit l can vary with the state E, but it remains finite as long as the state E does not tend towards an isothermal-adiabatic state. Let Eu(eu, b0) be a state of the studied system, and let us suppose that this state is not isotbermo-adiabalic. From the state Eo, let us make the system undergo a reversible isothermal modification Jo which brings it to a new state E'y ('|. - William Thomson, On the dynamical theory of Heat, willi numerical resulls deduced Jrom Mr Joule' s equivalent of a thermal unily, and Mr ftegnault's observations on Steam, Part 1, art. 12 (Transactions of Ihe Royal Society of Edinburgh, but i85i; Philosophical. Magazine, vol. IV, 1H82; \V. Thumson's Matltcmatical and Physical Papers, vol. I, |>. '79)- Most of the authors who have dealt with Thermodynamics have considered Clausius' postulate and Thomson's postulate as two equivalent axioms; they have thought that one could indifferently make use of either of these axioms. Thomson's postulate as two equivalent axioms; they have thought that one could make indifferent use of either one or the other of these axioms. We recognized that it was necessary to invoke both of them at the same time: this is what we have done in the following writings: Elude on the thermodynamic works of If. J. W illard Oibbs. >, 3 ei 1 Bulletin des Sciences mathématiques, >* série, t. XI. 1887): Commentary on the principles of Thermodynamics, second part The principle of Stidi Carnot and R. Clausius. Cliap. He (Journal de Mathématiques pures et appliquées, !f série, l. IX, îKtjî, p. 3io et ïuiv. ) Traité élémentaire de Mécanique chimique, Book I, Chap. 111, §§ 4 et,ô; t. I, 1897, pp. et seq. ------------------------------------------------------------------------ In order to obtain these propositions, we had to simplify extremely, by way of abstraction, the data of the observation; it was necessary, then, to generalize the simple results that we had obtained in this way, to specify them and to modify them in various ways; moreover, most of the time, nothing justified these various modifications, if it were not the desire to obtain principles whose consequences would agree with the experimental laws. From then on, it is simpler and more logical to give these two principles for two postulates that we formulate arbitrarily. The introduction of these postulates in our theory will be justified if the ultimate consequences of this theory give a satisfactory representation of the experimental laws. -Let us apply our postulates successively to Carnot cycles for which the external work is positive, to those for which it is negative, and finally to those for which it is zero. 1° Let us consider a Carnot cycle for which the external work G is positive. For such a cycle, the quantity of heat Q9 can be neither positive nor null; let us suppose, in fact, that it is positive or null, and let the same system go through the same cycle, but in the opposite direction; the three quantities G, QD, Q, will change sign without changing absolute value; we will now have a cycle where £ will be negative and where Qo will be negative or null, which contradicts W. Thomson's postulate. As soon as the work S is positive, the quantity Qo is necessarily negative; the equality then shows us that the quantity Q( is necessarily positive. - 2" Let us consider a Carnot cycle for which the external work E is negative. By virtue of W. Thomson's postulate, the quantity Qo is necessarily positive; hence, by virtue of equality (9), the quantity Q, is negative. 3" Let us finally consider a Carnot cycle for which the external work G is zero. For such a cycle, the quantity of heat Qo cannot be negative by virtue of the postulate of R. Clausius; nor can it be positive, because our cycle, taken in reverse, would give a Carnot cycle in which Ç would be zero and Qo negative, so that it would contradict the same postulate of Clausius; Qo is therefore zero; by virtue of equality (9), so is Q(. ------------------------------------------------------------------------ According to this discussiou, we can only meet three categories of Gamot cycles: these three categories are characterized in the following way First category G > ". Qu < <>̃ Qi > "̃ Second category G < o, Qo > "r Qi < o- Third category 8 = 0, Q0r=o. Qj = o. For a Carnot cycle that belongs to one of the first two categories, the economic coefficient of the cycle is the ratio G Q.-Q, Qo (10) QI Q, QI This report makes no sense for a third category cycle. If we bring the equalities (10) I previous table, we immediately obtain this theorem For any Carnot cycle that belongs to one of the first two categories, the economic coefficient is a positive number and less than With these preliminaries in mind, we can approach the demonstration of Sadi Carnot's TiiÉoitÈME, which is stated as follows All the Carnot cycles that belong either to one or the other of the first two categories, and that are described between the same temperature limits S,?, have the m/one economic coefficient. We will prove this theorem only for simple Carnot cycles, and the generalizations of which it will be the subject in the next chapter will lift the restriction that we are imposing on it at the moment. Let us consider two Carnot cycles C, C' described, between the ------------------------------------------------------------------------ same limit temperatures 2t0, S?, Cbu < ?j\ ), by two systems S, S'. For the first cycle, the letters £, ()0, (v), will retain the meaning they have in the foregoing; for the second, analogous quantities will be designated by the quantity of heat released at the temperature £?t will be (Qt -f- Q', ) finally the external work will be (G + S'). What we have just said about the association of two Carnot cycles extends without difficulty to the association of any number of cycles, provided that the systems which run through these cycles are conceived as infinitely distant from each other. We can, in this way, associate a' C cycles to a C cycles; we will obtain a simple Carnot cycle described between the limit temperatures 30) 2fi> The heat release at the temperature S?o y will be ("'Qo + nQ'o), a zero quantity according to equality (11). The simple cycle that we have imagined is therefore a cycle of the third category; the heat release at temperature 3f|iy must be zero; and this release has the value (/l'QtH-"Q',); we thus obtain the equality Q, ~ir ~t This equality, compared to equality (i i). gives us Qi 01 Qo Qi (Zu (1 i or else, by virtue of the equalities (10), p = 0' p' ̃̃̃ ------------------------------------------------------------------------ Carnot's theorem is thus demonstrated in the case where the absolute values of the two quantities O0 and Q,,,,are equal to each other. Let us now consider the case where the absolute values of the two quantities O<> cl Q' are incommensurable with each other. Let us take a perfect system, and between the two limit temperatures "bH. let us describe a Carnot cycle which we will modify in a continuous way. The amount of heat '/" released at temperature !t?|p will be different from o: it will vary in a continuous way and, as we have known 1 p. '>̃'><> i. we will be able to make its absolute value grow constantly. I say, in the first place, that the economic coefficient of the cycle will keep an invariable value. Indeed, the continuous variations of this cycle will involve continuous variations of the quantities of heat rja, q,. released at the two limit temperatures So, moreover, the quantity qn will never cancel itself; therefore, the quantity q, will not cancel itself either; if thus the economic coefficient of this cycle \anaii. it would vary in a continuous way. This point acquired, let us take two states of this cycle; that I one of these states, c, is fixed and that the other, e', is variable in a continuous way; the cycle c corresponds to a value qu of the quantity of heat released at the temperature 2?, and to an economic coefficient /̃; for the cycle c'. the same quantities take the values . - I. ̃ )' ------------------------------------------------------------------------ This result acquired, let us choose a state of our variable cycle c'such that the absolute values of the quantities of heat series, t. IfI, 188/j, p.; Ô2 and p. -i-ji; Cours de Thermodynamique, 18S6. P. Duhe.m, Étude sur les travaux Ùiermodynamiques de M. J. Willard Gibbs, 4 (Bulletin des Sciences mathématiques, a" série, t. XI, 1887 ) Commentaire aux principes de la Thermodynamique; Seconde partie Le principe de Carnot et de R. Clausius, Chap. I, 6 (Journal de Mathématiques pures et appliquées, 4° série, t. IX, i8y3, p. 328); Traité élémentaire de Mécanique chimique, Livre I, Ch;ip. III, §(i; t. I, 1897, p. (i.">. G. KiROHHOFF, Vorlesungen ilber die Théorie der Wurme, liera usgegeben von Max Planck, V* Vorlesung, § i, i86. In this work, G. Kirchhoff's exposition is almost identical to the one we published in 1887; however, at the time of this publication, Kirchhofl had recently died; the illustrious physicist had thus reached, on his side, thoughts all similar to ours. ------------------------------------------------------------------------ that the cycle does not belong to the third category, and let us posit that the (.) _~l (12) a=- 0 The comparison of the equality (i a) to the equality ( i o) shows us that this quantity *=~ The Carnot theorem can, by considering this quantity a, take the following form The quantity t, as well as 3, is higher than. £<0, thus l, is higher than the. Besides, the quantities of heat Q0) Q, visibly keep values independent of the choice of the thermometer which serves to locate the intensities of heat, and it is the same of the ratio, - ̃- of their absolute values. According to the temperatures read on the first thermometer, this ratio is expressed by the symbol o-(£?0)2T|); according to the temperatures read on the second thermometer, it is expressed by another symbol s(l0, t,); but to these two symbols corresponds the same numerical value. We can therefore state the following proposition When we change thermometers, the function ), we can write i equality (17) !rC&0, S2) = tj(S0, &i ) ~(; ;;2J. Together with the inequalities (16), these allow us to state the following proposition The function o-(S{), is an increasing function of the upper temperature &( and a decreasing function of the lower temperature 2?0-. The meaning of the function o-(S0) S,) has only been defined by supposing So less than &, when Sr0 is equal to or greater than 3, the symbol o-(£r0, S>() is, up to now, devoid of any meaning, and we are free to attribute to it the one we want; here are, on this subject, the conventions that we will apply i° When So is equal to S, we have ~5,)=!. 2° When 2f0 exceeds S, a case in which the symbol vfô, S?o) " has a meaning, we have O'(.Jo,.J¡)= (¡(.JI, ;0) With these conventions and the theorems already demonstrated, we can state the following propositions Whatever the two temperatures S, S,, the function cr(Sf0, S?)) is always positive; It is a continuous function of Sr0 el 'Ie %i It is a decreasing function of the first temperature 50 written in the parenthesis and an increasing function of the second; It is stere~iet!ne, r-ctle urt lower ~t i depending on whether the ------------------------------------------------------------------------ temperature "bn is lower, equal or higher than the temperature !i| Finally, whatever the three temperatures ?5(), 'bl, ?52, it verifies the relation (1;) ~t .7~, .7<) == T(.7o..7~) T~.?j..72~ ). Let us take, once and for all, a certain intensity of heat; for example, this intensity which is designated in therniometry by the name of melting ice and which is realized in a concrete way, with a great approximation, provided certain precautions are taken, by the ice which is melted under atmospheric pressure. Let us designate by 2?a the temperature (ixe to which corresponds, on the thermometer of which it pleases us to make use, this intensity of heat. Let 2f be another temperature read on the same thermometer. Finally, let Ta be a positive constant that, for the moment, we will leave any. The temperature 3 being the only variable, we can write 08) F(£?) = Tas(2?a,2r). Let us examine the properties of the function F (S), properties which result immediately from the characteristics of the function rcztttre: which corresponds to the intensity of heat considered. According to this definition, any conceivable heat intensity corresponds to a positive absolute temperature: to speak of a zero or negative absolute temperature would be nonsense. The form of the function F(37) depends on the thermometric scale on which the temperature 37 is read; what form does this function take when the thermometric scale is the absolute temperature scale In other words, what is the form of the function F(T)? '? Let us consider a certain intensity of heat to which corresponds an absolute temperature T. If we take any thermo-inferric scale, it will make correspond to this same intensity of heat a degree 3?;: to this scale will be attached a certain function F (S?) this function will be such that, whatever the intensity of heat considered and, consequently, whatever, 1 one has the equality F(S7) = T. It must be so, in particular, if the thermometric scale employed is the absolute scale*; to this scale, corresponds therefore a function F(ï) such as one has. whatever (lue is T, (uj) F(T) = T. When we use as thermometric scale V absolute scale, the function F(T) is reduced to the variable T itself. We know that to the intensity of heat designated in therinometry by the words melting ice corresponds a temperature ------------------------------------------------------------------------ equal to the constant Ta, a positive constant, but arbitrary so far. Let us consider, on the other hand, that well-defined intensity of heat which is designated in thermometry by the words: water boiling under atmospheric pressure. A thermometer, arbitrarily chosen, which corresponds the temperature 2ra to melting ice, corresponds the temperature Sp to water boiling under atmospheric pressure. The function *(2r", 2rp) has a positive numerical value S which does not depend on the choice of this thermometer. The absolute temperature of the water boiling under atmospheric pressure has then for value, according to the equality (18), Tjï=Tav. The arbitrary constant Ta can be arranged so that the absolute temperature scale is centigrade, i.e. so that Tjj- Ta=ioo. It is enough to give Ta the value r ÎOO (20) r"~ S - o which we can do, since this value is positive and independent of any thermometric scale. This convention completes the fixing of the scale of absolute temperatures; the absolute temperature corresponding to a given intensity of heat is an entirely determined number. It is, of course, a purely abstract definition; it shows how, to the temperature 3 of a body of the mathematical scheme, "one can, by operations carried out on other bodies of this same scheme, make correspond an absolute temperature T. To a concrete body, heated in a certain way, can we effectively correspond a number T which can be considered as the approximate absolute temperature? Yes, if we have real bodies whose properties are fairly accurately represented by those of the schematic bodies we have reasoned about, for example by those of perfect systems. It is here ------------------------------------------------------------------------ the place to recall what we said, in paragraph 2 (p. 32-328), about the existence of perfect systems; we will return to this point in paragraph 6. S. Definitive statement of Carnot's theorem. Equalities (17) and (18) give us the following equality r (~i, ?(&", 3~) 1'(;;1) o'(.?()! '11) = -c'- f?7e7~' T(~,2T(,) P(.?~ Equality (12) then becomes Q" Qi = 0 F(%) FOfj) This equality is demonstrated only for Carnot cycles of the first two categories; but it becomes an identity for Carnot cycles of the third category, since, for these cycles, the two quantities Qo, Q( are equal to o. We can therefore give the following definitive statement of Carnot's theorem For any simple Carnot cycle, we have the equality (2I) F(t)^F7fc = 0- If the thermometric scale used is V absolute scale, this equality becomes, according to equality (19), (22) ) ~+~~0. . pahfect is a homogeneous /laide (jui has the following two properties. l" Law IIK lîuVI.K, TOWM.EV AND MlHIOTTK. - The piCSsioil II f/lli maintains the fluid in equilibrium is positive whatever the specific volume co; when the temperature does not vary, the product II g) of the pressure by the specific volume does not vary either. y." Gay-Llssac's Law. - When the temperature does not vary, the internal energy U of the gas does not vary. The statements of these laws do not invoke any temperature scale; the degree of accuracy with which they can represent the properties of a given concrete fluid is therefore independent of the choice of thermometer. If Con chooses arbitrarily a certain thermometer, and if one designates by Sf the temperature read on this thermometer, the first law will be expressed by an equality of the form (2,~" IIw=9!~)' The second will give legality (̃>A) U = MuCz), where M is the mass of the considered tluid and "(2?) a function whose form does not depend on M. The experiments of Regnault and various other observers have shown that the first law represents with a fair degree of approximation the properties of hydrogen, nitrogen, carbon monoxide and atmospheric air; this last body is regarded by chemists as a mixture of two gases, i.e., as a phase; but, since the composition of this phase remains invariable during the course of the modifications we are studying, this phase can be treated as a simple homogeneous fluid. The experiments relating to Boyle's law are too well known for us to need to bet on them here, except to point out that they invoke a host of theoretical principles that have not yet been justified; by mentioning them now, we are anticipating them; to be able to analyze them logically, we would have to postpone this analysis until much later. ------------------------------------------------------------------------ The gases which follow more or less Boyle's and Mariotte's law also follow more or less G;i y-Lussac's law. This is the result of old experiments by Gay-L'issac, taken up again by Rcgnaull. by Joule and by W. J homson. The concrete realization of these experiments would give rise to observations similar to those we have just made; their abstract conception deserves to be stopped for a moment. Let us consider a system formed by two rigid reservoirs H and II', able to communicate between them by means of a valve p; V is the volume of the reservoir R and V the volume of the reservoir IV; the reservoir R is filled by a mass M of a certain gas G, while the reservoir JV is empty. This system is immersed in a calorimeter. Two assumptions are made about this system: to what extent these assumptions represent reality would require the use of theories that cannot be explained now. These two assumptions are the following Firstly, we admit that the external actions to which the system is subjected are reduced to pressures exerted on the various points of the external surface of the tanks R and TV. Secondly, we admit that the internal energy of the system is obtained simply by adding the internal energy of the tanks considered in isolation to the internal energy of the gas considered in isolation, which amounts to neglecting the mutual actions of the gas and the wall of the tanks. This is what Gay-Lussac's observation consists of. The valve p is opened. The gas G precipitates from the tank R into the tank R'. After a short time, the two reservoirs are filled with a homogeneous, immobile gas of uniform temperature. We notice that this final temperature of the system and of the calorimeter is approximately the same as the initial temperature. The identity between the initial temperature and the final temperature of the calorimeter tells us that the system studied has not lost any heat during the modification it has undergone. The system is at rest at the end of the modification as it was at the beginning, so that its living force has not changed. \The walls of the reservoirs have not experienced any change during this modification. ------------------------------------------------------------------------ tion, only insignificant deformations; the work ex The external work has therefore an insignificant value. Finally, the internal energy of the reservoirs considered in isolation has not undergone any variation, since the reservoirs are, at the end of the modification, in exactly the same state as at the beginning. From this we see that the internal energy of the gas, considered in isolation, has not changed. Now, if the temperature has returned, at the end of the modification, to the same value as at the beginning, the volume 'fi 'd 1 V 1 V -i- V by varying the conditions of the experiment, one can make oj,,, and m, take all the values that one wants; any variation of the specific volume of the gas, accomplished without change of temperature, does not thus change in a sensitive way the value of the internal energy of this gas. These experiences allow us to formulate the following proposal Perfect gases are abstract systems defined, for any value of specific volume and for any temperature, by the equalities (28) and (a4)- There are concrete gases that can be approximately represented by such abstract systems, provided that their temperature and specific volume remain within certain limits. Let us examine the properties of perfect gases and, first, the consequences of Boyle's law. In an elementary modification of a homogeneous fluid, the external work 1 has the value - MlUco; here, by virtue of the equality (2.'$), it can be written (23) fc=- Mofïl - - If an isothermal modification, performed at temperature S, leads the specific volume of the gas from value w to value m', the corresponding external work will be given by the formula (20) G = M?(2r)log£. W', where log. denotes a neperian logarithm. ------------------------------------------------------------------------ Let be a fluid of any specific volume m, temperature !3, subjected to pressure II. Let IIa be the pressure which would maintain this fluid in equilibrium under the same specific volume o>, at the temperature 2fa of the melting ice, and wa the specific volume of the fluid in equilibrium at this same temperature !3K, under pressure II. Equalities ooj " = - ( - ) i fi", " d:~ J define the coefficient of expansion under constant pressure and the coefficient of expansion under constant volume of the fluid; one of the two derivatives is taken assuming constant pressure, the other assuming invariant specific volume. According to equality (23), we have Hato = lloja= a> (&">̃ 11 It is then easy to see that this same equality gives (27) a - b = - ? - = a(S). For a gas that follows Boyle's law, the coefficient of expansion under constant pressure is, in each state of the gas, equal to the coefficient of expansion under constant pressure; their common value depends only on the temperature. Let us now turn to the consequences of GayLusSac's law. The normal heat coefficients of a fluid in equilibrium are given by the equalities [Chap. VII. equalities < a 1 )] of('o>. Et) ;1 = - -& l rltc i ru. 11. rtw fi being the pressure which maintains the liquid in equilibrium at the temperature Xj, under the specific volume o>. By means of the equality (ai), ------------------------------------------------------------------------ these equalities of iennenl t clv(:J~ T, (28) >W-'i-J)' = II. The heat of expansion of a perfect gas is equal to the pressure that maintains this gas in equilibrium, its normal specific heat is a function of temperature alone. Let us finally come to a series of propositions which invoke both Boyle's law and Gay-Lussac's law. Along an isotherm, the internal energy of a perfect gas does not undergo any variation; the heat released is then equal to the external work, which is itself given by the equality ("(>), so that we have (29) Q = Mo(2f;log^. The specific heat under constant pressure of any fluid is given by the equality [Chap. VII, iv equality (55)] r-(s)" ~â He which can be written, introducing the expansion coefficient under constant pressure a, r =s y -f /coaa. By virtue of equalities (27) and (28), and noting that lit, = 9(:), it becomes (3o) VCZ) - 7(w)-f- ç(S'a) i.i'S). The specific heat of a gas under constant pressure depends only on the temperature. If the quantities of heat, instead of being mechanical, were evaluated in calories, the two specific heats of the gas would have, at the temperature S?, values. r~(:J I I t.J!I, <;<)=KJ' ------------------------------------------------------------------------ K being the mechanical equivalent of the calorie. The relation i'jo) would give, then I. :> I L = 0(5) - t:. Y Yes i For a perfect gas, to which equality (a3) is applicable. f'm\ (1 Oit) = elï}^} & (t'Z so that the equality <'3a) is still susceptible of the following two forms ? 3 '/It'fii :'1, 1 _j -j- - - tfz - ).~ <" ViS) - -^ISl çIj.i :J ifc ~p 17.J (1. Let us consider this last equality. Let us imagine that the system 1. -i ------------------------------------------------------------------------ experiences a reversible adiabatic change in which the temperature goes from the value ?jn to the value:?, while the specific volume goes from the value <.o0 to the value ti)| We will have Mu 'i f .J ~3:)~ (O: -~---~ ~/j"~~t.7). ). 1 3'0 )'7)-?~ log denoting a neperian logarithm. I) where the following theorem When a perfect gas ('proves a reversible adiabatic change, the ratio of the initial specific volume to the final specific volume depends on the initial and final temperature, but not on the value of the initial specific volume. After having exposed in this way the consequences of the definition of the perfect gas, let us see what are, for such a gas, the corollaries of Carnot's principle. To a mass M of a certain perfect gas, let us describe a cycle of Carnot between the limit temperatures 5j0, 2?t- An isothermal modification m0, accomplished at the temperature 2f0, leads the gas from the state (mo,2?o) to the state (o)'0,Sr0); an adiabatic modification leads it from this last state to the state (m,, S,); from this state to the state (wi, £?i), it passes by an isothermal modification mK finally, by an adiabatic modification, it returns to the state (w0, xj0) The quantities of heat Qo, (), respectively released at the coyrs of the reversible isothermal modifications m0, m, have ponr values, in virtue of the equality ("9), Qo = 11 ~(.o~ c "o f>, VI .(â,~ '̃'1 But the theorem we have just stated gives us the proportion ("" 0"', (0 ht j 1 We have therefore Qi ?('2r, > Qo c(2r0) ------------------------------------------------------------------------ This equality, compared to the equality (/n) which expresses the Carnot theorem, gives us the equality (36) 9Jhl = i!SÈil. ~ (38) B ta ot we will have (3;)) A RQ and t: (40) a(3) = u;q !-̃(&>. ------------------------------------------------------------------------ The constant R has, for all perfect gases, the same numerical value which does not depend on the thermometric scale used. Equality (4o) leads to this first consequence For all par/al gases, the function a(H7) increases as the temperature "b read on any thermometer. By virtue of equality (4o), equality (a^) becomes (40 ,(&,= 1 l^Ëi. H i ~x 1 rt'i For all perfect gases, the expansion coefficient has the same positive value. Let's transfer this value (4 0) into equality (3o), which is Robert Mayer's relation, observing that 9(Sa) = P2 = RTaa, and that F(2?a) = Ta. Robert Mayer's relationship will become (ii) r(5) = ï(5) + RQ6 or a 2;3 - Equality (44) also gives B = PQ = RTaQ. This equality, together with equality (4-î), allows us to write equality (44) in the form (iO) nco = RTaQ(i -+-"/). ------------------------------------------------------------------------ Equality (4). We have entirely neglected, in this Chapter, the considerable historical study that would be necessary to determine exactly, with respect to the theorems established therein, the rights of invention of R. Clausius, of William Thomson (since then, Lord Kelvin), of F. Massieu, of J. Clcrk Maxwell, of J.Willard Gibbs and of Hcrinann von Helmholtz. ------------------------------------------------------------------------ heat Qi the amount of heat released by the entire cycle is Q = Q.i-OI. Let us form two adiabatic modifications Ao, A, which are not at the same time îsotlieriniques, which is possible; let us choose a temperature £?', as close as we want to 2? let \i'n be a state taken by the system in the modification \H and E1', nn state taken during the modification A( these two states corresponding to the same temperature 2?'; finally, let us connect the two states E((, E, by a reversible isothermal modification Mo, described at the temperature 2?', and tending in a continuous way towards the modification Mo when 2?' tends towards 2/. Let Q'o be the quantity of heat released in the modification M'o. The isothermal modification M, followed by the adiabatic modification Ao, the isothermal modification Mo, and finally the adiabatic modification A, followed in the opposite direction, gives us a simple Garnot cycle described between the temperatures S?, 2?' the application of Carnot's theorem [Chap. VIII. equality (21)] to this cycle gives us the equality Q. _Q1_ = o V(?S) F(Sr> Let us assume now, as we will constantly do in the following, that the system under study has the following property The external work done during a reversible modification varies continuously when this modification varies continuously. In a reversible change, the amount of heat released is the excess of external work over the increase in internal energy. If we observe that the internal energy varies continuously when the state of the system itself varies continuously, we obtain the following proposition In all the systems subjected to the preceding hkstrictiojx, the quantity of heat released during a reversible modification varies in a continuous way when this modification itself varies in a continuous way. ------------------------------------------------------------------------ Let us make 3' tend towards 'b the modification M'n will tend towards the ~~l modification Mo and, consequently, the quantity of heat O0 will tend towards the quantity of heat Qu at the limit, the preceding equality will become the equality Q = Q"-r-Q1 = o that we wanted to achieve. We can therefore state the following theorem In any isothermal, reversible cycle, the total heat release is equal to o. In any reversible modification, the quantity of heat released is the excess of the external work over the increase of the internal energy; if the modification is a closed cycle, the internal energy returns at the end to the value it had at the beginning, so that the quantity of heat released is simply equal to the external work. Our previous theorem can therefore be stated as follows In any reversible isothermal cycle, the total external work is equal to o. 2. Properties of any isothermal modifications. By a well known procedure of demonstration, we can deduce from the previous theorem the following theorem If two states of a system correspond to the same temperature] to pass from the first state to the second, it will always be necessary to do the same external work, whatever the reversible isothermal modification by which this passage is made. Having admitted this proposition, let us consider all the temperatures for which the system studied is defined and for which it verifies the restrictions to which this whole theory is subordinated; it could happen, moreover, that it ceases to be defined for temperatures outside a certain interval, or else that, for these temperatures, the essential restrictions would no longer be verified. To each of the first temperatures .3, let us associate ------------------------------------------------------------------------ a state e of the system, this state being chosen arbitrarily, but once and for all; let us suppose that this state z varies in a continuous way with the temperature !b let us name this state £ 1 normal state relative to the temperature, ?j. Let us consider, then, any state E of our system, where 2r is the temperature in this state E. A reversible isothermal change which makes the system pass from the state to to the state E is accompanied by an external work whose value depends absolutely only on the two states s, E; as, moreover, the normal state e relating to the temperature 27 is determined, from the moment we give ourselves the state E and, consequently, the temperature 27, we can say that this value, which we will designate by (j, depends exclusively on the state E. The quantity (J varies continuously with the state E. Let E, E' be two infinitely close states of the system, and (j', (j' the values of the quantity (/which correspond to these states the states E, E' correspond to identical or infinitely close temperatures 27, 2?' and, consequently, to identical or infinitely close normal states e, s'. Let us link the state s to the state E by a reversible isothermal modification M which corresponds to an external work (J Let us likewise link the state t' to the state E' by an isothermal imodification M', infinitely close to the modification M; the modification M' corresponds to an external work (j' ; consequently, according to the assumption made in paragraph 1 (p. 36ï), the quantity ij* will differ infinitesimally little from the quantity (j. ` Let us suppose that an isothermal modification carried out at temperature 3 leads the system from a state Eo to another state Et be (jn, Cjt the values of the quantity (J which correspond to these two states. Let us propose to determine the external work Ç. accomplished in this modification. Let e be the normal state relative to the temperature £;. Without changing the value of G, we can replace black isothermal modification by any other isothermal modification that leads the system from the same initial state to the same final state; in particular, we can lead the system, by an isothermal modification me, from the state Eo to the state s. then bring it back, by another isothermal modification /??, from the state £ to the state K|-. ------------------------------------------------------------------------ The external Iras ail, in the modification /", has the value - fj'n in the modification //", it takes the value (/, we find thus that the work P has for value eralization ?jr infinitely close to the temperature 2? Let us link the state (e'o. 2?') to the state (e\, 2f') by an isothermal modification M'. If we impose on the system the modification M, then the modification M, then the modification M' reversed, finally the modification Ao reversed, we will have made it go through a simple Carnot cycle. To this cycle, let us apply Caruot's theorem. The quantity of heat released in the modification M is the excess of the external work over the increase of internal energy; moreover, as the modification is isothermal, we can apply to it ------------------------------------------------------------------------ In the equality (i) this quantity of heat has thus for value (/(e,, "à) - U(e,, 2?) - ()'(eOi 2f) -+- U(e0, Sr). Similarly, the amount of heat released in the modification M' has the value (/le' 2O- U(e' &')-£(* Sx')-HU(ei, ^). Carnot's theorem then gives us the equality {/i-Ui - fto+Uo Ç| - U', g; h- u; - F(2r) ~r F(3r'> in which we put (j(e0, 2r) = £j0, (j(e'o. ^') = ffo This equality can still be written (3) (, u; -(/+- u,- ç,'i -T- v'1- g,- u0 (tfi Ui- ffo+ U.) -f^> (2f'-2r, = o. The modification Ao is adiabatic; the increase of the internal energy is equal to the external work this last one is moreover given by the equality (2) one has thus 1)° Ço- UJ+ U.- ^.(S'") = o. Similarly, if we express that the modification A, is adiabatic, we find the equality ~lj~~U._ i1(~)-o_ With these two equalities, equality (3) can be put in the form (4) U(e,,&) - (j(e,î)-r- -- cl F - ( 3 ) -±-^ 03 - J ) - Çj > + JTT-7T - it V ( 3 ) 05 dz has a lime value which does not depend on I ('̃lai e al>straclioii made of the temperature, but which depends only on the temperaLure !J if we denote by " (?3) this value, we may write U (" Flâr d(~(e,âj (5) V(e,3, = Ç(e,3)-ir--~ iL_f- -cp(j). CC.J We can always determine a background i(2r) such that F ( 2x d -l t 's ) (6) ^z^^at- ̃ d. F ( 3 ) dz from 11 It is sufficient to determine the function y (H) by the condition (i) dy^3)_ ?(5r,) rfF(3fi W dJj [F(2f.]" J.> âf2? and then put (8) .i(27) = F(i)/.(J). The function

5) -+-̃-!<( ^>- 1. Like the two parts (j (.), (5). ((>) and (y) allows us to state the following proposition Provided that we exclude the consideration of the isothermal stages, we can, M any state (e, S) of a normally defined system, make correspond a width J(e, S?) which ------------------------------------------------------------------------ denzetere ,/àctie, ~tci varies cL urte ntcznière continues with the et~ct (e, 27), and which has the properties ~M/~. -̃̃;̃ i ------------------------------------------------------------------------ were reversible, a case to which we then limited our analysis For the system to be in equilibrium in a state where the temperature is uniform, it is necessary and sufficient that the external work done in any virtual change has a value determined by the knowledge of this virtual change and the initial state. If the internal potential of the system is given, this value is unambiguously given by the second member of equality (i i). Now, we have also seen (p. >/J6) that this value of the external work suitable for ensuring the equilibrium of the system must be zero whenever the virtual modification is reduced to a displacement of the whole; it is necessary and sufficient that this should be so in order that the conditions of equilibrium currently studied should not be in contradiction with the general conditions of equilibrium laid down in Chapter IV, equalities (44)- An overall displacement is essentially an isothermal modification, so that the external work is equal to the variation of the internal thermodynamic potential; this last variation must therefore be null. Hence the following theorem The value of the internal thermodynamic potential of a system does not vary when we impose to this system a displacement of the whole in the space without that its state experiences any other modification. In an elementary virtual change from a state of equilibrium, the amount of heat released is the excess of external work over the increase in internal energy by the equalities (i i) and (12), we see that this amount of heat is known when one internal potential of the system. If we know the form of the internal potential of a normally defined system, we know the equilibrium condition of this system and the amount of heat released in an elementary change from a state of equilibrium, so that the mechanical and calorimetric study of the system in equilibrium is completed. This is expressed by saying that the thermodyne potential is the same as that of the water. ------------------------------------------------------------------------ namic internal to a system normally of/i/u is the cAïucTKiusïfQijK function of this system a similar denomination has already been introduced, in Chapter VJJI, § 1 (p. ̃>>), for isolherino-adiabatic systems. The preceding analysis constantly takes into account the temperature of the system; it loses all meaning if the words "temperature" and "system" are themselves meaningless; it cannot therefore be applied to the systems studied by Rational Mechanics, since the notion of temperature does not play a part in their definition; hence the consideration of these systems has been formally excluded from our reasoning (Chap. VIII, § 1, p. H i 8 ) But we must now ask ourselves whether it would not be possible to extend the results we have arrived at, i.e., the fundamental equalities (io), (i i) and (12), to the systems studied by Rational Mechanics; these would henceforth be included, as special cases, among the systems we will study. Since the word temperature has no meaning for the systems studied in Rational Mechanics, if we want to extend the equalities ( î i) and (i2) to such a system, it is necessary to attribute to it as internal potential a quantity independent of temperature. Equality (12) then requires that this quantity be identical to the internal energy of the system. These conditions, without which the extension of equations (i 1) and (1 a) to mechanical systems would be meaningless, are in fact sufficient for this extension to be legitimate; the proposition expressed by equality (1 1), in fact, reduces to the following In any reversible virtual modification of a mechanical system in equilibrium, the work of the external actions is equal to the variation experienced by the internal energy, and this proposition is precisely, for such systems, the fundamental law of Statics [Chap. V, equality (1 1 1)]. Thus, one will be able to apply even to the systems studied by Rational Mechanics the corollaries which will be, in the future, deduced from the equalities (11) and (12), provided that the internal thermodynamic potential of these systems is independent of the temperature and identical to the internal energy. Some systems can be such that the temperature is not included in their definition and not be, however, at the ------------------------------------------------------------------------ This is the case, for example, with the defensible but incompressible and indilutable fluid, which, in their most simplified form, is the subject of Hydrostatics and Hydrodynamics. To these systems we are free to extend any proposition which does not contradict the principle of conservation of energy, in particular the equalities (ii) and (ia); hence the preceding proposition will be applicable to them by hypothesis. Hence, equality (i i) tells us that in any reversible virtual change imposed on such a system in equilibrium, the work of the external actions is equal to the change in the internal energy, as if it were a mechanical system. 4. The entropy of a normally defined system and Clausius equality. The quantity of heat released in an elementary reversible change of the system has the value Q = G - o U(e,3). E is given by the equality (i i). On the other hand, equality (12) gives eu(~)=~~)-~o~~ Ci r ( 3 ) 03 Í Ff3r)F"(5)~~3)~ j' 1 [F'(.2r)l* dSr Fv. We can therefore write (13) V = =-r- 0 = 0 _< <- F'(&)" 0 d3 {F'(2f)]~ O.J Let's put (16) S~=-_-- <'6> '*<'-*> = Irk* r (Tj) t 03 This function S(e, S) will be called V Entropy of the system in the state (e, S). Equality (16) shows us that V entropy of the system is known when its internal thermodynamic potential is known. For systems other than those studied in Rational Mechanics^the internal potential is only determined at a quantity ------------------------------------------------------------------------ near the orme AF(S?) + B, where and B are two arbitrary constants, we see then that the entropy of a system is, determined only to an arbitrary constant. For systems studied in Rational Mechanics and, more generally, for systems whose definition does not involve any consideration of temperature, the entropy is identically zero, because the internal potential is independent of temperature. By virtue of the equalities (12) and (16), the entropy of a system is related M the internal energy and the internal potential by the relation (17)- J(^T) 8T, (~zbisj U ('r ~l,e,'I')--rd:i(e,T) (ia*") Ufe, T) = $(e, T) - T '^f^ T) (b" 'r' d:r(e, -T) (i6é") S(e,T)=-i%I-), (17 A") -f(e, Tj= D(e,T)-TS(e,T), (i8ftw) Q + TSS(c,T) = o, 1 (19 6") =S(e0. T(,)-S(e,,T,), 0 ( >a bis ) 2^ = <>̃ C. Isothermal-isentropic systems. The formulas established in the above lead to remarkable consequences when applied to an isothermo-adiabatic system (') or, as we will say from now on, to an isothermo-isenlropic system. The amount of heat released in a reversible change is the excess of the external work over the increase of the internal energy. If the modification is isothermal, the external work is equal to the increase of the internal thermodynamic potential, so that the amount of heat released by such a modification is equal to the increase experienced by the difference ? - U. Therefore, for any isothermal modification to be at the same time adiabatic or isentropic, it is necessary and sufficient that the difference -ï - U remains invariable whenever the temperature does not vary, i.e. that this difference is a function of (' ) See what has been said about these sysli'iiics, in Chapter VIII (§ 1). ------------------------------------------------------------------------ the temperature alone. An i.sollicno-adial>ati(|uc system is thus characterized by an equality of the form (.il ^ie,H)= l ('<-̃, ?j > -ç.i Z i. By means of this form of {t\"b). the equality i iv.) becomes ̃}. ) U (e,Z) = V ( ?j ) -r- W ( e i and the following proposal The internal energy of a normally defined isolkermo-isenl ropiqtie system is the sum of a quantity which depends only on the temperature and another quantity which depends only on the state of the system, apart from the temperature. A very similar proposition applies to the internal thermodynamic potential. If we denote, in ell'et, by y.) J(e,j) = il>iîi - Wujj. Moreover, by virtue of equality (12), the two functions V(27') and lP (3) are linked by the relation r- V C3)' ('5 dVC3) d'Z CLJ The equality (i(j) then gives (v>4 ) S(e,S)- o -- 1 >t -X't'z) -SiJ).. at 1 .1 < Ct.J ------------------------------------------------------------------------ The entropy of a system, isothermo-isentropic normally 1 defined, is a function, of the temperature alone. Getle proposition would, moreover, follow immediately from the very definition of the isothermal-isentropic system, since this definition requires that the entropy does not vary when the temperature does not vary. The fact, therefore, that the equality (yJ\) follows either from the equality (̃>>) or from the equality ('>̃), makes it possible to state this theorem: if a normally defined system is isothermo-isentropic, it is sufficient that its internal energy is the sum of a quantity which depends only on the temperature and of a quantity which depends only on the system's calibration, apart from the temperature. It is also sufficient that its internal potential is of this form. For the systems studied in Rational Mechanics, the internal energy, identical to the internal potential, depends only on the system's calibration, apart from the temperature: we can therefore, if we wish, consider these systems as forming a particular class of isolhermo-isentropic systems. The equality (i i) shows us that in an elementary virtual modification of the system in equilibrium, the external work has the value (25; e = o\V(e). This equality (i 0 has some interesting consequences, the first of which is the following For the equilibrium of a normally defined system whose uniform temperature is equal to that of the foreign bodies, it is necessary and sufficient that any virtual modification corresponds to an external work determined by the knowledge of this modification and of the initial calibration of the system; for an isothermo-isentropic system, the value of this work does not depend on the temperature of the system. To understand the full scope of the theorem, let us apply it to a system, holonomic or not, whose calibration depends on a limited number of quantities. A virtual system change is defined by a varia- ------------------------------------------------------------------------ infinitely small variation of the temperature associated with n normal variations a,, a2, a, in such a one the external actions perform a virtual work of the form ë; = \a,-+- \ia1-h. -r- A,, a, For the system to be in equilibrium, it is necessary and sufficient [Ghap. VII, equality (2)] that the quantities A,, \2, Y,, have certain values ^i~ fi{e, "X!), Aj = /s(e, 7j), A,, = /,t(e, S). If the system is isothermo-isentropic, these values must depend only on the state e, disregarding the temperature, but not on the temperature .:?, so that the preceding equalities take the form (26) A, = /,(*), A2 = /2(e), A,, = /(e). The equilibrium equations for an isothermal-isentropic system in which each state depends on a limited number of quantities do not depend on the temperature of the system. Let us return to the general equality (a5); let us write it for each of the elements of a reversible modification carried out from the state (e, 2r0) to the state (e,, Sf|); let us add member by member all the equalities thus obtained; if we designate by E the external work accomplished in such a reversible modification, we shall have (27) G.= W(e,)-.W("i,). The external work done in a reversible modification of an isothermal-isentropic system is determined by the knowledge of the initial and final states of the system without the need to know the tenzpénalure of the system in each of these two states. The quantity of heat released in an elementary modification from a state of equilibrium has the value O = G - SU. For an isothermal-isentropic system, this equality becomes, by virtue of equalities (22) and (26) ( 8. ty = zlV(â)~ ̃(28) (t ~-orj. 1 ".3 ------------------------------------------------------------------------ Moreover, the heat capacity of the system is, by definition, the quantity it V(r, "3) d(3) *9 dâ = = c ( Ù3 dTJ The normal heat capacity of an isolhcrmoisentropic system depends on the temperature, but not on the state of the system. With the introduction of this capacity, equality (28) becomes (30) Q=-c(2f)£?. The amount of heat released in an elementary virtual change of an isothcrmo-isenlropic system is the product of the normal heat capacity of the system by V lowering of the temperature. This last theorem was already known to us in another way (Chap. VIII, §1, p. 3 19). Let us suppose that a reversible modification leads the system from the state (e0, Xj0) to the state (e,, 2f|) according to the equality (3o). the quantity of heat released in this modification has the value (3[) Q=_ f "C(5)t&. J*' The amount of heat released in a reversible change depends only on the initial temperature and the final temperature of the system. The exceptional character of isothermal-isenlropic systems assures them of great theoretical importance; on several occasions we shall have to deal with such systems again. Their practical importance is much less; it is very rare that the physicist has to deal with such systems. We will give the only EXAMPLE which, to our knowledge, has been presented in a physical theory - the theory in which it was encountered is due to M. J. Boussinesq ('.) J. Boussinesq, Théorie analytique de la chaleur, XXXIV' leçon, n" aîiS et 2.5g (t. il, p. 161 et p., 1G4, Paris. ijio3). - Cf. Duiiem, Sur certains milieux ------------------------------------------------------------------------ The system we are going to study is a deformable body. homo!>(;iif? we defined in Chapter III, sj 2 (p. t'M)), what was meant by these words. We have seen that the most general virluellc modification of such a system was determined by the knowledge of i .3 infinitely small quantities, namely the variation oâ of the temperai ure, the three components a, jï, y of an elementary translation, the three components À, jj., v of an elementary rotation, finally the six infinitely small deformations e,, c3, e:l, < g3. We shall now suppose that these variations are normal variations; hence, if we denote by ra the volume of the system, the external work done in any virtual modification may be put into the form [Cliap. III, equalities (3()) and (4o)] ( 37.) G = Xa -+- Y ^4-Z-;+L/. + M :x - Nv - (N|e,-i- \se2+ N3(?:t^- T,g, +- T,t- T:ji4'3)to. X. V, Z are the components of a force; L. M, N are the components of a torque; finally the six quantities N,, i, N2, N3, T, T2, T3 are used to determine the magnitude and the direction of the pressure at each point of the body surface. The internal thermodynamic potential of this body does not depend on the absolute position that it occupies in space; it depends only on the temperature and the deformation of the body designating this state of deformation by the letter d, we can represent this internal potential by a symbol such as ^((/, 2f ). The elementary translation and the elementary rotation do not change the value of this potential. The variation that it experiences in any virtual modification is therefore expressed by the equality :~r(cf,:5~= ~l :r ( cl. j r`,r-TZSfcl.âje,-i-u.~yrl,~je.~=-iex(cl,)e3 1 (33 i o  {d, "3 j = - £2? -t-"ii d. 3? i c, -+- ruid, ~Z )e,- n.3(d, S) e3 03 -+- l,{d, 3 ).vt - i,Jd. S i^-j-t- t3(d, 3 ) ers. elastics considered by M. J. Boussinesq (Procès-cerbaux de la Société des Sciences physiques et naturelles de Bordeaux, séance du ̃) juillet i])03). En réaliLé, y\. Boussinesq considered not a horno^ene ciéforinable body, but any dcfoi'inable body, so that each of the keys into which this body can be divided corresponds to the system we have taken as an example. Moreover, M. Boussinesq limited himself to "treating the case where the deformations experienced by the body were inliniiiiently small. ------------------------------------------------------------------------ By virtue of equalities (32 ) and (33 i. the general equilibrium condition (i i is equivalent to the following equalities ( X = o. Y = o, Z - o. (3!¡) X = 0, y = 0, Z = 0, L = u, :\1 = n, =~ u, l IV i = - - n,(d ,%). i. N2 = - - n2f d. "h), Y, = n3i d,lz >, t 7S rs ra t, = - (,u/,âi, t" = -̃- uid. :i, t3 = /,(c/, r- 1. m ~=~ (2((:)' 3== and Among these conditions of equilibrium we find, as was to happen, the six equations (34), which are the general equations of Statics. The medium studied by M. Boussinesq has the following property: Its internal potential ^(d, "b) is the sum of a function (?7)^ dYV(K O-J> = (jtj -i ̃ O(U. 0.1 = dh O.J dw ow. In this same modification, the external work has the value G =- MriSw. The equality (i i) then gives us this equilibrium condition n = -1 varies by (-75:) °Sr. Equality (5$) gives us CO ~~Wf 5) ClW ()2 niW ( I I ~~5 Equality (38) is therefore equivalent to equality (É)n- When the specific volume of a fluid, heated under constant pressure, is maximum or minimum, the fluid is in an isothermo-adiabatic state. Water, taken under conditions of maximum density, is in an isothermo-adiabatic state. 7. Internal potential and entropy of a system made of several independent parts. In order not to complicate the reasoning without any serious advantage for generality, we shall suppose that the system studied is formed only of two distinct parts, independent of each other, which we shall designate by the indices 1 and 2. Each of these two parts will be brought to the same temperature at all its points, and this temperature, which we will designate by S, will be the same for both parts. We shall assume, of course, that each of the two partial systems 1 and 2 and the total system (1, 2) are subject to all the restrictive conditions laid down in the above. In particular, each of these systems is susceptible to a normal definition. ------------------------------------------------------------------------ The state i can be defined normally by its temperature S? and by its state, abstracting from the temperature. We will designate by e, the state of the system i, apart from the temperature and the absolute position in space; a simple displacement of the whole system i will not make vary , it is necessary, but not sufficient^ to know e,, . a.*>~) that the internal energy of the system (1,2), formed of the two parts 1 and 2 (defined normally, is of the form (ii) Uii(o'|2, Sr) = U,(ei, ?j ) a- l-2 ff2. ï) -t- tl-"i2( ei2>- The quantity V(a, which is the mutual potential of the systems and 2, does not depend on the temperature The very definition of mol equilibrium entrainc the correctness of the following proposition For the system (i, to be in equilibrium, it is necessary and sufficient that each of the two partial systems r and 2 is etz ecfttilibre. Let E be foreign bodies, independent of the system (i, 2), and capable of maintaining it in equilibrium. The system will then be maintained in equilibrium by the set of foreign bodies 2 and S; the system a will be maintained in equilibrium by the set of foreign bodies t and S. Let us impose on the system a virtual and isothermal modification of any kind; in this modification, S, experiences a variation that we will designate by 6^^t. As the system is held in equilibrium by the two systems 2 and S, this variation 03.-Î, is equal to the work S( of the actions exerted on the system by the set of systems 2 and S. But we know on the other hand (Chap. III, § 4, p. 146) that the virtual work Q, of the actions exerted on system 1 by the set of systems 2 and S, is the sum of the work S2, of the actions exerted by system 2 on system i, and the work 8, of the actions exerted by system E on system 1 We have therefore Ssu-f-6, = 85-?,. In any virtual isothermal modification of the system '±i we have, following an analogous notation, G12-Î- 02 = 3.3- -ï-2- But we know [Chap. Ill, equality (3i")] that £12-+- £21 = - o "K,s. We can see that by associating any isothermal change in the system with any isothermal change in the system, the system will not be affected. ------------------------------------------------------------------------ conk of the system n, and observing that IFj2 does not depend on 2;, we can write, by virtue of the previous equalities, (~) 0,~f),=.(-f,F,,). i. On the other hand, the combination of the most general isothermal mocli(ication of the system i and the most general isothermal modification of the system constitutes i1. 60) the most general isothermal modification of the system (1, 2). In this modification, the actions of the foreign bodies S on the system (1, 2) do work (Q) and the internal potential experiences a variation Os- Jt 2; as the bodies S hold the system (r, 2) in equilibrium, these two quantities are equal to each other (43) )))-o,=: >1 The equalities in (~3) give ~(~H----T2 -Vi2.) == 0. The quantity (~2-~) -~2 - \fI 2) remains invariant to any virtual isothermal change imposed on the system (r, 2) and thus depends only on the temperature of this system: (44) ~~`t2=~y--~2+ ~Ÿf=-l.~j. Let us now determine the form of the function 'L (2?). The equality (1 2) gives us the three equalities _" F(3r)d?, F' ( 3";1 (/3" u =~L~i~, U2: 'f. F' (3";1 iJ3" u -¡ F(3") (/-1'2 12 .112 lv'(~ ) These three equalities, joined to the ebalitra (4 r) and (4'))~ give us Ÿ(a) '11.) = o. T F'(3") This equality gives, in turn, (45) 'f(27)=AF(27)+B, j\~ and B being two constants. ------------------------------------------------------------------------ By means of the equalities and (45), we can write <-)6) $iî(etl,3) = 'fi(e,, Sr)-i- .f-2(e2l 2f.i ̃+- *i't%(en') -+- A F (5) -+- B. When, to a determination of the internal thermodynamic potential of a system, we add a linear function, with constant coefficients, of the absolute temperature, we obtain a new determination of this potential. We can thus, for 3{i(e\->,?b), adopt the following determination: (47) £n(eu, Sf)=P=^i(ei,S)-f- J,(ej, S) + Vls(e,2). When we study a system of uniform temperature, composed of several independent parts, we can assign to this system, for internal thermodynamic potential, the sum of the internal thermodynamic potentials of the various parts and the potential of the mutual actions of these parts. If we observe that W," is cancelled out when the two systems î and 2 are infinitely distant from each other, we can, by means of the equalities (46) and (47), state the following proposition This determination of the internal thermodynamic potential is the only one which satisfies the following condition: The internal potential of a system formed of several independent parts tends towards the sum of the internal potentials of these parts when they are indefinitely distant from each other. From now on, we will agree to always determine in this way the internal thermodynamic potential of a system formed by several independent parts. The equality ( 16) allows to write FI(~i) 1e4, 0: ~r) ~'I~~e~) 03 r~(;,)'Sz 0: 03 0 J The equation (47) then gives (18) S,2(ei5, 2r) = S|(>S)-r-Sj(ej, 2r). ------------------------------------------------------------------------ When we adopt the convention that has just been indicated, the entropy of a system made up of several independent parts, all brought to the same temperature, is equal to the sum of the entropies of these parts considered in isolation; it does not depend in any way on the relative position of these parts. The theorems that we have just demonstrated have, in the various applications of Thermodynamics, the most frequent applications. 8. Form of the internal thermodynamic potential of a homogeneous fluid or phase. Among these applications, we will indicate one that is particularly important. Let us consider a homogeneous fluid normally defined by its specific volume w and its temperature S. Let M be the mass of this fluid. We will show that its internal thermodynamic potential can be given by (49) ,f=M,&)-t-,f(M'.M,S). ). ------------------------------------------------------------------------ But, by definition, the state of a homogeneous fluid does not depend in any way on the position that its various parts occupy in space; therefore, if we bring the two masses M and M' closer to each other until they are joined in some way, we will not change the state of the preceding system and its internal potential will keep the value that we have just determined; this will represent the internal potential -f(M~i-M', eu, S) of the mass (M + M') of the same fluid, taken under the specific volume w and at the temperature 3. The identity rf (M -h M', O),V) = r?(M, tl), H?) "+- r?(M', 0) S), that we obtain in this way, shows us that îf (M, co, 2f ) is proportional to M, and justifies the equality (49)- If M is the mass of a phase normally defined by its specific volume w, its chemical composition x and its temperature £", the internal thermodynamic potential of this phase can be put in the form (5o) $ = Mo (tu, x, ?j ), where the function tp does not depend on the mass M. This theorem can be proved like the previous one. 9. Heat capacity. The normal heat capacity of a system is given by the equality [Chap. VI, equalities (i i) By virtue of equality (12), this equality (5i) can also be written (~ ~,5)== F(~~FH('l d~Pte;) F1â) d2:~fe "J. ~F~(:,)~z V~ F'( 0: Equality (16) still allows us to give it the form (53). cf*.&)=F(&)^£l^-). - ------------------------------------------------------------------------ Comparison of equalities (5i; and (53) gives an important relationship between the internal energy and entropy of a normally defined system; this relationship is as follows: ( -3 i ) nV(e.ZJ) I' " i .r .7; dS(e, S) ( 5, ) --;-- .7~ --~- !J.'J U.J Helmlioltz' postulate (Chup. VJ, § 3, [). 263), requiring that the normal heat capacity of a system be positive, is expressed by the inequality (55) de, S) > o. If we take into account the equalities (5i), (5a), (53k) and if we observe that the quantity F (S) is positive, we can give the inequality (55) the three forms o The Ce. S "1" (50) oz -r -><>, (.' ," 'pi(e.lj) " o5(e. S) ('j; ) !-"(.') - "- [- J; - o. Let us suppose now that we use as thermometric scale the scale of absolute temperatures. We will have S = T, Fl/Z)- T, ̃ - F'CBf) i, F'f&j = o. I^the equalities (5i;l, (;")̃>.), (53) will become -r">i:c. Tt ( ai bis) c( e, i ) - - - - ̃̃̃ ". t "" %(e. T) (5ïW!) flf. II- - 1 -=; j ^'< T) ( >'S bis ) cAe. 1 ) =~ I T Jj'égalité (54) will become OlUe.T) -)Sie,T) - d1' dr, ------------------------------------------------------------------------ Finally the inequalities (56), (5-), (58) will become OXJ(e.T). ( -}b bis > = <>. <> ru c, i" ) (qlns) ̃< (58a) nS~e.'('~ (58~, 1 '~T- 10. Internal thermodynamic potential and entropy of a perfect gas. -As an application of what has been said in this Chapter, we will determine the form of the internal thermodynamic potential and the entropy of a perfect gas. Let us suppose that this gas, of mass M, is normally defined by its specific volume ) (5g; llto = R£i K(2T). 1. On the other hand, its internal energy ;i for value [Cliap. VIII, equality (a/,)] (60) U = M"(2r.i. ). In any elementary modification, the workifUerne is ts= - MIT S m or, in verlu of the equality (5()). G = MRQFi:-), ClCU. E = - -----------OM. (/1 Equalities (i i) and (49) then give ~c(t.), 3) If "t-i~ Otxi M we well (6i) ç(w, &) = - RO I^S^Iogio-^ii'^), J, log denoting a neperian logarithm. By virtue of equalities (there), ( iç)) and (6o), the function -1/(2?) must ------------------------------------------------------------------------ verify the differential equation (Gz) ~)-=M(~). 'l.) t'7) t.J} u U ~), Equalities (12), (49) and (*>') then show us that the entropy of the perfect gas is given by the equalities (03) S = M s(w,2f), ~J~ ( :J (04) if o>, ?j ) - Ril logtu - - - F' (3) Let's use the absolute thermometric scale. Equalities (61) and (64) will become (61a) ç(to, T) =- KS2TIogoj-i-i(T), (G:i bis) s(ta, T)= BiJlogtu - 'V(T), while the function 'i(ï) will be given by the differential equation (fa bis) <Î<(T) - tà'(T) = "(T). Let us particularize further. The thermometric scale adopted being the absolute thermometric scale, let us suppose that the specific heat under constant volume has a value independent of the temperature; this experimental law, known as the Claijsius law, applies with a great approximation to all gases close to the perfect state, at least as long as the temperature does not exceed a certain limit. The general equation [Chap. I1L equality (28)] du(T1=,t,j dT .,) gives then' the equality (65) "(T):=fT + A, where A is an arbitrary constant. Equation (62 bis) gives (66) ^(T)=-vTloST,-4 A-BT, B being a new arbitrary constant. ------------------------------------------------------------------------ Equalities (61a) and (64a) take the forms ( 7) "(co. T) = - HQT log" - -T logT -+- A BT, (68) s ( w, T ) = K i i log , T), s((x>, T)determined by the equals i lés ((>:>), (t)") and (68), we obtain the expressions of the internal energy L", of the thermodynamic potential i and of the entropy S of this gas 11. Equilibrium of a system when the external work depends on a potential. Total thermodynamic potential. Let us consider a system subjected to foreign bodies such that the external work depends on a potential. We know (Cliap. III, § 2, p. 127) what is meant by this In any real or virtual modification of the system, the external work is equal to the decrease undergone by a certain quantity Q whose value is entirely determined by the knowledge of the state of the system : this quantity Q is the external potential. If the system is normally defined, a change of temperature without change of state is free of any external work, hence of any variation of the external potential; hence this proposition If the system is defined normally, the external potential is a quantity £Î(e) whose value depends on the state e. apart from the temperatures, but not on these temperatures. Let us imagine that all the virtual modifications of the system under study are reversible; let us also imagine that this system is subject to all the restrictive conditions set out above. For the system to be in equilibrium in a certain state, it is necessary and sufficient: i" Let the temperature 3 be uniform; 2° That in any virtual, elementary and isothermal modification, coming from the given state, the external work is equal to the increase of the internal thermodynamic potential -ï(e, 'h). 1. ------------------------------------------------------------------------ But if, on the other hand, the system is subjected to foreign bodies such that there is a given potential Q(c), this external work is necessarily equal to the decrease experienced by The comparison of these two propositions leads to an easy-to-see corollary. Let us call the total thermodynamic potential of the system the quantity (69) 4He,x!) = .1(e,5) -h- Q(e), sum of the internal thermodynamic potential and the external potential. If a system, normally defined, is subjected to foreign bodies such that the external work depends on a potential, for this system to be in equilibrium in a certain calibration, it will be necessary and sufficient i" That in this state, the temperature is uniform; 2" That the variation of the total potential is null in any virtual isothermal modification coming from this state (70) If 'l'fe, = o. The second condition can still be stated as follows: That Ton has, in any virtual modification, isothermal or not, from this state, the equality L. à`G(e.âl Il (7") o*( i' f 0'\>(e,3 }~] ( c 2 ~s l, 1, .J ) [,1 J~t'i^Jr I. (-'>̃) Q = I<(J;8 |_l' - 15) 03 J Lt- .'J 1 J In the case where the absolute tliermetric scale is used, ~=T, F<5)=T, F'(5;=t, and the previous equality takes the simpler form (7* M) Q = Tol^ÏJ. 72 tJts,l t~ = r, dT Equalities (72) and (72 bis) can, moreover, be obtained in another way. Equalities (16) and (69) give, indeed, ", o. 1 toie,?!> '7~ -(c,- --~--' (.?) ~.? The ('-equalities (18) and (18 bis) then give the equalities (72) and (72 bis). 1-2. New form of the equilibrium condition. The use of the quantity to which we have given the name of entropy makes it possible to put in a new form the condition of equilibrium of the system which we have just studied. This equilibrium condition is represented by the equality (.1) s = ;|(ei5,_d^-)& "-~ The two identities (17) #(e, 2r> = Uie, 3) - F(S)S(e, 3), (16) S(e,:Jj= O~f(e,1 (-'(â j ------------------------------------------------------------------------ allow to give him the iorme (-f\) G = o Ui f. "h i - !̃'( i i o S Ce, 2?). In this equality, the virtual modification imposed on the svslème is any to any virtual change of the state <- of the system, it associates any variation o!Ï7 of the temperature. Now, to the same virtual change of state eàw system, one can associate a variation o'2? of the temperature, such that the resulting virtual modification does not involve any variation of the entropy S(e, 2?). This isentropic virtual change leads to a variation S'LJ(e, Sf) of the internal energy. It is clear, therefore, that in the virtual modification considered first of all we have ̃vit/ c ̃-̃ i- 1 c o\J(e.S) /,c, -vç.. 0 U(e,3) = o h(e, 3) A - (oS-oi), A? ES(e,)- JS(e,) (ô.°.â). ûb(e,.7~ = -----(&.?-o .?~. f<.? If we take into account the identity 0U(e,&) = F(&)">S(>,&)) dâ oC these two equalities transform equality (~j4) into the following (75) G = S'U(e,S). How was obtained the isentropic change to which the symbol S' refers? By taking an entirely random change in the state e of the system, and by associating with this change a variation 8'2f of the temperature so chosen that the entropy does not vary. It is clear that we can obtain in this way the most general virtual isentropic change of which the system is susceptible, Equality (rj5) is therefore equivalent to the following theorem: The condition of equilibrium of the system can be obtained by expressing that in any virtual isentropic modification of the system, the external work is equal to the increase of the internal energy. ------------------------------------------------------------------------ CHAPTER X. THE EQUILIBRIUM OF SOLAR SYSTEMS. I. Equilibrium equations of a holonomic system ('). The present chapter will be devoted to treating the equilibrium of holonomic systems with the help of the general theorems established in the preceding chapter and, particularly, in the previous chapter. Let us imagine a holonomic system whose normal definition is given by n independent variables x,, a2, a, except the uniform temperature S. For such a system to be in equilibrium in a given state (i.e. a, 2?), it is necessary and sufficient that the external actions A|, A2, A,, take values determined unambiguously by the knowledge of the (ra + i) quantities a. a2, a, 2r, by means of the equilibrium equations [Chap. Vil, equalities (3)] A] = (a,, a., a, 3), (y. ) A, =/,<3|, "Ci, a, H?), Aw = /(ai,aj, .an,S). If we impose on the system, from this equilibrium state, a (' ) The first six paragraphs of this Chapter are a development of the following writing: P. Duhem, -Sur les équations générales de fa Thermodynamique [On the general equations of thermodynamics] (Annales de l'École normale supérieure, 3* série, t. VII, 1891, p. a3i). At the time this paper was published, similar considerations were being developed by M. Ladislas Natanson [Lad. IVatanson. On thermodynamic potentials (Bulletin of the Academy of Sciences of Krakow, April 1 Stj 1 ) Oeber Ihermodynamischc Polentiale (Zeitschrift fur pkysikalischc Chemie, Bd. X, 1892, p. 733). ------------------------------------------------------------------------ Any virtual modification results in external work ( z j ôx, f.. ;x_ J:~ ~x, The 1>otc>,ntiel thermodvnanijque internal of the system is a uniform function of variablPs 'l.1, 'l. l.'I> or. at least, of some of these \ariai)ies; some of them can not influence the value of the internal thermodynannnue potential; if, for example, one of them serves only to mark the position that the system occupies in space, the internal thermodYnamic potentic) will be independent of it. Let ~(x,, x~, zn, ~) be the internal thermodynamic potential of the system. In an oirtual modification from a state of equilibrium, the external actions enect a which must have as its value (Clrap., IX, equality (..)] l: = 4 f dJ` y.~ <)-7 or 3 Gul~ < ;x9m.+ ()~. ôx". (3) ) G=--oott-t---oct9-r-+--ox, dxt dx2 dx,t It By identifying the expressions (2) and (3) of this work we find the equalities (4) _d ~7 JJ 2 which allow us to know the equilibrium equations of the system when we know the internal thermodynamic potential. Equalities (4) require that the functions f, J" 2 /" verify all relations of the form ('i) ~=~, dx~ day or/), q denote any two of the indices 1, 2., Il different from each other. We will immediately apply these considerations to some examples. ------------------------------------------------------------------------ First example. - Body slightly displaced and nj/ecte of a small homogeneous deformation. We will consider a body in an initial state ia -t- yi 6. where A. B, C, (8) j' l, (g' £lî Eî, EE, £3, Yi>; Y** Ta are twelve infinitely small quantities, independent of a, A, c; the first three are the components of an elementary translation the next three, the components of an elementary rotation the last six, the components of an elementary deformation. The state e can be fully defined by means of the twelve independent variables (8) and the temperature Sr, assumed to be uniform. We will admit that this definition is normal. Among the twelve variables (8), there are six that only serve to fix the position that the deformed body occupies in space; these are the components A, B, C of the translation and the components ni, n of the rotation; the internal potential cannot depend on these six variables; it only depends on the six components of ------------------------------------------------------------------------ deformation and temperature 3 7*i Ï3> %)̃ The function /'remains the same for systems of different masses, when they are brought to the same temperature and affected by the same deformation. From the e-standard, let us impose on the system a virtual modification that will be characterized by the thirteen quantities o/V, 3 H, oC, 0/. have, on, oî|, oe2. 0E3. oyi. oyî, 3^3' c5. Let us consider a material point of initial coordinates a, b.c. in the virtual modification, the current coordinates x, y, and the components ç, Yf, w of the displacement of this point undergo variations ùx, or, ôz, SE, vr\. 6"; by virtue of the equalities (ii and (7). one can write ( Sx = oc = SA -+- (.̃ o//( - ft o/t -+- a O£j ̃+- i î "j':< *+" c *Yt- (10/ L = or, = 3B -+- " o" - c 0/ -r- 6 3s" 4- c S "i " 073. ( oï = oï = SC + i 0/ - a ont - c oî3 -f- rt ov? -r- b o-i On the other hand, this virtual modification corresponds (Chap. III. § 3, p. i'>~) to an elementary translation fa. ï, y ), to an elementary rotation (). jx, vj and to an elementary deformation whose six components are. e,, e>, e3. gx, fr2, gz', we have Z:r - 't. -r- [J.z - -iy -+- c-, r +- ff3 y -+- #̃. c, Cu) or = 3 -f- /̃ >. <>->.v - A'i - ,4r3- f 3; - y h- >r- a.r c. -+- .i's J- A'i.)'- But, according to the equalities (6). x. v, say infinitely little ------------------------------------------------------------------------ of a, b, c. Therefore, the comparison of the equalities i 10 i and ( i i) allows us to write ̃x-rzoX. p = 3B, -( = ?'(' k 3 1 u = o m v o /£ .Ts - -o-f-- d'il "'i Z' <*T3 The six conditions (13) are the six general equations of Statics [Chap. I V, equalities (41 )] which we must necessarily find among the conditions of equilibrium of the system studied. ------------------------------------------------------------------------ If we neglect in the expression of f the infinitely small quantities of the third order, we can write (ri) ?"/- 1. \1 3 ̃T.S)- LpHE.E.-i-Es)- 7 yM, ] T2=T,(S)- L4(&)(E1-t-Ei-f-e3)-+- I -1, L 1 "T2J T,= Tj(S?>- [-(&, (£i-+-El-t.£;1)H- I -^ij '] The other five are established in a similar way. Particular case of the isotropic body. - The internal potential depending only on the deformation that the body undergoes in (') About this number which is. in general, the nomlirc ili-s "vicffiricnls of elasticity of a liomogcnc medium. see I'oincaiu:, Leçons sur la théorie de l'élasticité, Cha|). II, pp. 17-19, Paris. iSga. ------------------------------------------------------------------------ from the state e0 to the state e, and not at all from the translation and the rotation that it has experienced, we can, to study this potential, disregard this translation and this rotation, and reduce the equalities (7) to the form ( * "(3 ) > makes correspond to the point Mo (a, b, c) of the body e0 a point iM(a+. ç, 6 4- Yj, c+'C) of the body e; to any figure Fo drawn within the body e0, it makes correspond a figure F drawn within the body e. Consider another pure deformation of the same system D'(e'i.e'".e'î. fi.Tï.ï's)- ). To the point M'0(a', b', c') of the body eo, it makes correspond a point M' (a' H- b' 7/, c' -f- ^') of the body e' by the formulas ( 5'= e'ia'-+-Y36'-+-ï'ic') (:8 ét'i) < 7|'=Yj"'-+- e', é'-f- y', c', f ?r="i'î<"'ï'i*i- 'Ja'- To a figure F^, drawn within the body eo, it makes correspond a figure F' drawn within the body e'. In each of the two deformations, the point of the body that initially coincides with the origin of the coordinates remains stationary. Suppose that to the two deformations D and D', we can associate a certain rotational motion R around the origin of the coordinates, so that the following proposition is correct Whatever the figure Fo is, if the figure F'o is deduced from it by the motion R, the figure F' is deduced from the figure F by the same motion. We will then say that the two deformations D and D' are two differently oriented, but geometrically equivalent deformations. ------------------------------------------------------------------------ Let us study two such deformations. To say that figure F'o is deduced from figure Fo by a certain rotation R is to say that the coordinates a, b, c of any point Mo in figure Fo are related to the coordinates ri, b', c' of the corresponding point M'n in figure F'o by the formulas a = Ina -+̃ l\ib ̃+̃ lt3c. (19) b' - lua -H /2, b -+- l13c, c' - lua -t- /326 -+- /33c, where the coefficients ly are nine cosines whose properties are well known. These formulas (ig) can still be written a = ;""'+ lub'-h l3ic', {19 bis) b = ltia'-h lità' -i-l3ic', c = li3a' -+- l,3b'-i- li3c'. To say that the two deformations D and D' are geometrically equivalent is to say that the equalities (18), (18 bis) and (19) [or (19 bis)] lead, whatever a, b, c, to the equalities Ç'==^lS-H/|*Tr1-f-/13Ç, <2O) T)'= /ji£ -f-/22lfl -t-/j3?, Lgf '1- l32 ~3~. Let's multiply the equalities (20) by a', b', c' respectively and add the results obtained by taking into account the equalities (ig bis); we find (21) £'a'+r/6'-+-re' = £a -l-r(6-r-Çc. This equality (21) takes place whatever a, b, c are, if the two deformations D and D' are geometrically equivalent. Conversely, if equality (21) holds whatever a, b, c, or whatever a', b, c', the two deformations D and D' are geometrically equivalent. Let us equalize, in fact, the derivatives with respect to a! of the two members of equality (21). The derivative of the first member can be written successively, according to the equalities (18 bis), +a d~ ,+b dr,' +c d,, + da + da da = $'-t- e,a' a' -+- v3i' b' -j- -fie' =?.£'. ------------------------------------------------------------------------ The derivative of the second member, by virtue of equality (ii)ùis) and (18), pi|en(j the successive forms If °\ Or, à* ùa ( -1- a - - b - j-c- - -f- Od L on oa j or I <>\ <̃)*! 01 Oh + (71 + a5ï.h6_ + c_j_ + (i + aÛ+bp+c*)*£, Oc Oc ] oa = ( ̃+̃ lia ̃+- -(jb ̃+- -c )lu -t- ( 1 -+- '!" a ̃+̃ *̃̃> -T- v, c ) l, 2 -+-( Ç-+- Y2" ̃+-"fi* -+- '.)C )^ia = ~(~ ~)2Ït /OE). We thus obtain the first of the equalities (20). The> other two can be deduced in a similar way from IMdentilled (ai i. If, in this identity, we replace ç, yj, 'Ç by their values (i8) and £', T| Ç' by their values (18 bis), we obtain the following proposition For the two deformations D and D' to be geometrically equivalent, it is necessary and sufficient that there exists a certain group of formulas of the type (19) or (19 bis) such that the quadratic form (22) *(", h, c) = Eta2-+- ej62-t- e3c2-4- ̃?.'(, bc ̃+̃ -iftca -+- i-(:sab 6 and the quadratic form ( 22 6w ) *'("',̃ è', c' ) = 5', a'2 -+- s'j 6'2 -+- e'j c'2 -+- 2 y', b' c' -±- 2 y'j f' a' -+- -.< y'3 a' 6' be equal to each other whenever a, b, c. on the one hand, and a', b', c' on the other hand, are related by this group of formulas. This amounts to saying that the two surfaces of the second de^re (£) ¡ £] a- -1- e2 b- - e3 c- -+- 271 b c -+- ays c " }. a h 1. (£') s', n'2- s'2 A' =',c'!+ 27', b'c' ~h 2 7j "' -f.a' b' - 1, both of which have the origin of the coordinates as their centroid, can be deduced from each other by a suitable rotation about this common center. ------------------------------------------------------------------------ It is necessary and sufficient that the axes of the surface S are equal to the axes of the surface 1". Now, the squares of the half-axes of the surface S are the three roots of IY;<|ii;ilu>ii of the third degree in a (̃) I E,ÊsE:f-+- ̃> 7s - î| 7|'- =-iVî - îsVD'3 - '2Ï:| ï:iî, I- :,£;- 7; Y -1 Y J -+" £:I7 - I = O, while the squares of the half-axes of the surface are the three roots of the equalion in rr' h.3 bis) ( s\ e'3 ̃+- a V, v'2 y', =. v',2 = 2 y'jî £'3 v' ) î'3 - (£se3 - e3E'|-t-t£3 - 7? - "fi* -Ta2^'2 - (£'i + E2-+-£;tjar- i =o. For the two deformations D and D' to be geometrically equivalent, it is necessary and sufficient that the two equations (a3) and (a3 bis) have the same roots: in other words, it is necessary and sufficient that we have the three equalities £1 + E2+ E:J::7. Ert + z.+ s~~ s2 s, -+- s, -+-=,£, - yî - y! - Ys ( v-4 ) - e', e'j s'3 £', s', î'j y' - y'ïS - ï'a!i 3 'i ^2 H ̃+- ''Yi Y-' Y-) - £i Yi - E-2 Yî - '<̃̃! " he t "'2"1 S. -r1 ;=1 -- Yi -E:3 r3 - |-i-J^2|l ,!il -1 I 1 ~i i 2 C3 1 t Let us agree to say that two deformations I), I.)' are physically equivalent when, applied to the same initial state e0 and associated to the same temperature 2", they make the internal ihermodjnamic potential take the same value. We will say that the body e0 is isotkopk from the point of view of elasticity if two differently oriented but geometrically equivalent deformations are always physically equivalent. We see, by this deliniluni and by 1 equality i i) 1. that an isotropic body is the one where we have I equalized f[l\. '-̃ -̃ Yi, Y-'i Y- -'- ~Jl M- E->- lz< Yl- Yi- Y:(: r i1 whenever the equals | -), ) are verilii'cs. Kn ilaiilrcs ------------------------------------------------------------------------ In other words, an isotropic body is one where the function /(6t,S9,S:j,-j-j,Yt.j. ) depends only on the four quantities S|-+-Eî-t- £:f, Es£3-Hî"£| + £|Si - Y? - Ys - Y*- if e2s;, -+- ayi Y" Y' - £i YÏ £*Yi - £:> Ï3 :J Therefore i° The part of the function p0/[equalilé(iS)] which is linear and homogeneous with respect to the components of the deformation, is simply proportional to (Et -+- 1., + z3), which gives vI(2r) = v1(Sr) = v,(Sr) = H0(3) (~~) Co Tt(Sr)=~(5)=')=o. 2" The part of the function c>of which is quadratic with respect to the components of the deformation, is linear and homogeneous with respect to the quantities (s, -i- s2-x- s:J)2, £*S3-t- Ejî|+ êiêt- Y?" Ys- Vit which gives = P(2f)(£,-i-s2-+-,s3)i-i-Q(3r)(HS:J-£^î-4-siï3)-Q(2r)(Y?-+--riH-Y!) 3 or, by positing aP(3-)-i-Q(2r) = 2A(3), Q(2r>=- 4M(2r), (26) 4- = A(S)(e,-f-s2-63)2 ~~2 H- 2 M(2r) (e? + s* + s* -4- ->TÎ '>.Yi + -*Y3 )̃ Equalities (15), (26), (26) give the form of the internal potential of a slightly deformed and initially isotropic body, by the formula (V) po/=, - Ë2-+- s") - ̃'̃ M(^T)si, l N2= U0C3)- [ng(27) - A(2r>] (s, -+-ÏJ-4- s3 .) - -." M cS)î2, | N;, = ir"( " - [II0(& ) -4- A( S) I I ti -r- ï, + £,)- ̃>. M (S) £:" 0.8 ) < T, = - 2 M(2r;v,, r T2=-:dl(;;)"(2' T,=-iM(S)7). It is seen that in the state e, if the temperature differs only from the temperature relative to the state e0, while no deformation accompanies the passage from this state to that one, one has ( N,= Ni=N,= U-,(S;, (29) U-T.-T^o. 110(;;), ( Tt= T2= T:¡= o. If we want to raise the temperature of an isotropic body without it experiencing any deformation, we can maintain it in equilibrium by applying a normal and uniform pressure no(3r) to its surface. More generally, suppose that the body experiences a deformation that leaves it homothetic to its initial state. According to the formulas (18), we have 6 8 J Yi = Ta = "s = °. €> == being the cubic dilatation. The formulas (28) will become N, = Ni= N, = Ito(2r) [3 II0(3r) -f. 3 A(2r) -4- 1 MCà )] |, T, = T2 = T, = o. These formulas teach us that the deformed body will be maintained in equilibrium if a normal and uniform pressure is applied to the surface of this body (30) IIi^,e) = II" (30 - [3II0(2r)-i-3 Ai S.i-r- y.MiS)]® From this general formula (3o), two other more particular formulas can be deduced ------------------------------------------------------------------------ i" The tempe i'a turc Sf remaining invariable. the cubic dilatation grows of c/B the pressure II experiences then an increase of el Ton a of u dU ̃} ||0 ( Jz ) -̃- i A " 3? i - >. M ("S ) |ï(2r) is, at temperature !i and initial pressure II, (Si, the compressibility coefficient of the studied body. ̃2° The temperature 2î increases by < while the external pressure 11(2", 0) remains invariant; the cubic expansion <-) then increases by d&, and we have ~H.~) r It171 .J ~J~J l~ ~,11 Î-J 1 I É-I dlj I ~j ~rTB "7fô <ï^~ J ~i ,1 1 3U.,t'5) 3A< clt3 3 cl.~ The cubic dilation (̃) is infinitely small as t,. î1M î" j the quantity [ *nio(2r) d\Cz\ d\ltz i ] e L3/lllof~.wt.\i:.) 1. .l\IW1_jd e L ̃' J 3 ~;J It.J CL.'J ') - >l i f o < Z ) is therefore, in terms of size, infinitely small compared to a - - > in J so that the previous equality gives (3a) - d Z = "S Uo (S)-t- ;> A ( J j - v. M ( 3 ) dZ - a (. 5t(2r) is the coefficient of expansion of the body under study. Now consider a deformation in which "] ="-- = ̃="̃ This deformation consists of a dilation z, along Or and a uniform contraction d in any direction normal to ().r. The formulas (->.H ) become '[ \) = ll,,l S) -- f ll,,( J) AiSTi Ml T'iIm - "| II, J I -- \( ï l| 0, (33) < iV2= N:1= 11,>(H/i- [ II,, (S i- A(37)|î, -̃.>.[ ll,,(^.- \> ïi- Mi. Si|o, ( t < N 2 = N;¡ = Il,, ¡ ) -III T.^T, -t- ï\ Il, [ II., Í :=. ,.u- Il If the ('orps has the shape of a cylinder whose generals are ------------------------------------------------------------------------ parallel to Ox, we can see that the two bases are subjected to a normal and uniform pressure equal to N, while the lateral surface is subjected to a normal and uniform pressure whose value is the common value of i\2, N: Without changing this lateral pressure either (pie the temperature, suppose we increase N, by c/N, and o eroilronl by di el do. We will have =- nj~; - .) - .\) ~ch~\ iiu(2r)- A(S)~ ,\l(3i 1. I'(S?) is the fractional elasticity coefficient or modulus of elasticity; ?(?3) is sometimes called Poisson's ratio. In many questions of elasticity, it is assumed that the initial state e0 is, at the temperature 'h that we are considering. I equilibrium state of the body subtracted from any external action. In this case, no(Sr) is zero, which simplifies certain formulas, so that the equalities (3i), (̃îj) and (35 ) become ( 3 1 bis ) 'i ( S? ) = 3 \(,Z) --̃̃>. Ait Z ) ^- (':i:i his) ^t~ )~l.J ll:J: )=- ~li:J l~, jj yes îl5|- alA(i"-T-M(SH i t 35 /"s ) K( .5 " .t= >- (.J) Af.~t-A)(.?) Di;ii\rii;vn-; kskmi'I.k stretched and twisted cylinder. - We will study the formation of the equilibrium equations in a second example; this example will differ from the previous one in that the deformations imposed on the system will no longer be small. "I other hand, they will be much more particular. The position "pie the svstem occupies in space will not be taken into account in the delimliou of this svslème. ------------------------------------------------------------------------ The initial standard e0 will be a homogeneous cylinder , we will suppose that the material point which occupied the position Mo(", 0, c) comes in a point M of coordinates x = (a coscto - b sincw; (i - u), (36) y = (a sincco ̃+- b coseio) (i - a), z = c(i -+- s). In these formulas, c, v, are three quantities independent of a, b, c. The deformation represented by these formulas is composed of i" A torsion rotating by angle 10 any radius of the cylinder which, in the initial state, was at the unit distance from the xQy plane a" From a transverse contraction 3' Longitudinal expansion e. The deformed body is still a cylinder of length L = (i + s) Lo and radius R=(i - o-)R0. The state of this body is defined by the four variables e, ~) and (38), the equals it equilibrium of this cylinder will be ('jp) l'; l.o rJ f .`. t.u tl f o_> I, rJ/- (J9) = *"& *=*"& ° = Let us observe that oL = Lo^ï, ol\ = - ll0oo-; -let us call S the surface of the cylinder deformed Note that a ray drawn in the phase where the origin of the coordinates is not located rotates by an angle 'i - Low. so that 8'|/ = Lo or, and let us pose 1 ) E !l (40) H - v = s- ( KuS -i-HsL, i -+- s ,i(i - s) Equality (38) will become (41) Ç = V Z\. - II S.5R -h V 36. P will be a force, the tensile weight; If will be a normal and uniform pressure applied to the lateral surface of the deformed cylinder XY will be a couple whose axis will be parallel to the axis of the cylinder, and which will be applied to the base that does not contain the origin of the coordinates. The equilibrium equations of the system will now be P d .l ~ â o, l p =*/(* Wla>, l I I " i!a) It- 2; ¡j 'Of,I-t-S.I(I~cr,1 J^. -(~. ,ai.â?;_ rIr- _iu_ cl, â 1. shot TnoisikMK EXEMPi.K. Homogeneous fluid. - Let us consider, in the third place, a homogeneous fluid of mass M. of specific volume m and of temperature £?.' The ihermodynaniiquc potential ------------------------------------------------------------------------ internal is of the form i Chap. 1\, "j 8, p. -îjjo ) ( i'j i .? = M -itd:, Jû >, the function " not depending on the mass M. On the other hand, the virtual work of the external actions applied to such a (luide can be written (\\) Ê-: M 11 MO, II being a normal and uniform pressure applied to the surface that limits the fluid. Therefore, the equations (i). (.{), (43) and (44) show that the equilibrium equation of the fluid is 1 t 1 il c( tu. S I (/) II = -- .7< dt|. a.> '/". S I 1, | A? J. d%lt. In this state, it will be maintained in equilibrium by exle- ------------------------------------------------------------------------ rieure> s A, - Ul4 ll,t Oxl (j'l." dal /i 'r i r I t/X; - - "i 07. azi ~r- DI; 1)7. "OR -T- <)%}, f/7.,i Let's also consider a third state z, still corresponding to the temperature ;3 and infinitely close to Telat K; this state £ corresponds to the values ':l1¡+-fj:lJ, x <:x.r, Xft-tt normal variables, To maintain the system in equilibrium in this state, it is necessary to apply external actions Aj-r-oA|. A2-r-oA3. A,, -r O. The equalities U\~) give the equality dXi oxi - oa2 - - d VH oz,, d' p d~~t <7 ~d~IIO:'X;l +' 'tIX~ r,X.r-tt-' q tfx,t ix" =~-+~ V1 <>*$ I 7 -i- sLé > < O'j.,t ('17. Oa,. - di,, fjt,, |, the signr designating a somniation which extends to all distinct combinations indexes i. -> n of two. The second member contains in a symmetrical way the quantities d% and the ?, quantities oa: it is thus clear that it also represents the value of the sum îA| //ï| -̃ oA. "/a.- .i- oA,, d% so that we have (48) "/A, Ôx,-f-rfAî ïz;~ i-dXr, oan I. l:' oAil^ti -4-i OiV!j(/ij^ SA,, f/a, ------------------------------------------------------------------------ This equality constitutes the law of reciprocity which Lord Kayleigh first formulated in a general way. The corollary of this law is susceptible of clear experimental verifications. Taking a system in equilibrium, we impose on it an isothermal equilibrium displacement where the action A, grows by ) - r aii a2' ---)"". )i 1 " , xa. S' S " ~U% 05' - Oc ùï,. 0fn 01.1 says da If we designate by S(a, a2, a, S ) the entropy of the system, we must have, in any \irtual modification resulting from a state of equilibrium, the eg-alile [Chap. IX, equality (18)] (55). Q ,-f- F(Sj3S(2,, a2, a, 3) = o ------------------------------------------------------------------------ or JS L- s4 = h (5) p" = 1'( ,)-, dx" 0^ (37) c = F(Ï7"--- rm Equalities (56) give back equalities ('53) but, in addition, equalities (56j and (5-) allow to write equalities j - - 'iîl - ^l£~Ll l ~x~ d:~ (7(3-) PI, £c_ c <>p,, F'CS?) c~' d.a f' ( Comparing equalities (54) and (58) first gives us the group of equalities 'PI = :.(:J ), ,O{1, l 1' '(3) 03 1,~ F ( ) i~ ̃(59) i V2 F'{Sf) -às' F F':Sr> ^î ~n = F~'j5" If we now carry over, into the equalities (54), the values np; d~~ û~" d l', ] J of -t -3=r" - - -" -TT- taken from the equalities ( dq), we find the second 03 03 03 D ̃ J group of equalities Fi 3) d\f, F(3)V"(3)Of, Oxi P'{3) O31 [F'lS)p 03 Oc F (3) iflfi Vi3)Y"(3 > Of, (6o) ox2 F'(S) cy2?J |Fix;>]- j2 r> Sr rfc Fizy-o'-f,, Y[3)Y "i3)0f,, ~F'<5~ Ÿ, d.t + 11"1;;1]2,); ------------------------------------------------------------------------ The equalities (5()) and (60) are of paramount importance, because they lead to the correctness of the following proposition .- When we know the equilibrium equations of a normally defined ho Lo mo ne system, we know i" The normal heat coefficients 0,, 2, p" of the U system in equilibrium, except the normal heat capacity c; a° The partial derivatives of the normal heat capacity c with respect to the variables ~0/M'), where N,, N. N: 'And. T 2, T, have the values (1 The equalities (59) then give e, F(;;} <).7 ?r<~L t/.? ( t t u3~^ .-l' (62) n ) ,F(5') - = - -- (' -~t-s:-~3) - i d- t r t'' (. F rl:~ 0' :J 1 L cl' 7 rJ rJ.r while the equalities (6o) allow to write 1"(3") J1N, F(~)F"(3) d~r, 4 F(:J) d2v,(..2~)_ F(~)F"(.) d'-7,(~) (J-E -E" E' (63) p 1 F'(:J¡ d:r= [F'(5')J' , 5), ), /(m, 2r) being the heat of expansion and y(w, 2f) the specific heat under constant volume. Equalities (5g) and (60) give then '"> '<-> (67) ) F' (5' dâ (68) -^"o, .,)_ dw F'lâ) d~3 ~F~!âll- dâ If the absolute thennometric scale is used, these equalities simply become (67Ô") /(u,.T) = T<>/("'T), ¡ ¿s w. ) dT j (68' bis) d T' T dlf(w~ T) (6$ bis) dT2 The equality (67 6/s) was given by Clapeyron ('), and modified by Clausius (2) who made known (:t) the equality (68 bis'). These two equalities are the prototypes of the general equalities (5g) and (60). (') Clapeyiion, Journal de l'École Polytechnique, t. XIV, i83((. (2) K. Ci.aosius, On another form of the second principle of the Mechanical Theory of Heat, equation (.3 a) (Poggendorjf's Annalen, Bd. XCI1I, 1854 Mechanical Theory of Heat, trans. Folie, t. I, 1868, p. i56). (3 R. Clausius, On various forms of the fundamental equations of the Mechanical Theory of Heat, which are convenient in application, first equality (33) (Poggendorff's Anna I en, Bd. CXXV, i865; Théorie mécanique de la chaleur, trans. Folie, t. I, 1868, p. 397)- ------------------------------------------------------------------------ 4. The internal thermodynamic potential as a characteristic function. When we know the internal thermody nam iq potential .f(2,, 05 > ( - Flâ) d-.v FI:¡J F"(3) (M_ 72) C~F'(&) [F'tS)p 53 The knowledge of the function J(a,, a2, -/", 2f) allows us to determine, by simple differentiations, all the quantities which characterize the system normally defined in equilibrium from the double mechanical and calorimetric point of view. This is why, ------------------------------------------------------------------------ according to a determination introduced by I' Massieu and previously employed (Cliap. VIII $3 1, j). 3a.'5), the internal thermodynamic potential of a normally dejmi system is said to be a function cAitAcTKiusTiQtJK "or. m alk of that system. Exkmpj.k. Let us apply the formulas which have just been given to a homogeneous fluid, of mass M, normally defined by its temperature 3f and its specific volume to. The internal thermodynamic potential of this fluid is of the form [Chap. IX, equality (49)] (-/S) 5 Mç(, 2f) and the entropy S= M.v (w, 2f) are given by the equalities <7^) "(< 3) = o(o>, 3)- p^ T - (76) *(">, 3)=- -- If we use the absolute thermometric scale, these formulas become (r,bis) ."( a), T) = ?( o>, T) T d "f T (76 Uïa) 5~ -t-)-- J~,ow.'t' Finally, the heat of expansion and the specific heat under constant volume have the following expressions r(~)<) "o(<5') r ( 3 ) c^ti) d3 o F(3) d'oiu, 3" F(3i F*(3) 32 I- ̃ < ^3" > |* "w or, by using the absolute thermometric scale. ~l)G$ ) l(. w, T ) T 3 (/w d t r l (.786,0 ï(to.T)==_T^i^l!. ------------------------------------------------------------------------ The various formulas are due to r. Massieu ('); it is by the study of the homogeneous fluid that this author introduced in Thermodynamics the notion of characteristic function. 5. System related to non-normal variables. A system of normal variables xt, a4, a, 2r, Let's suppose that we switch to a system of non-normal variables, among which is the temperature Si, ?" 32, 3, a, using the transformation formulas ( xr = ai ( t~ '=, .S). (79) ) ) *i = "* ."",&) = (,'(3,. p!( £", 3>. ¡. For the system to be in equilibrium, it is necessary and sufficient that we have. in any virtual modification [Chap. IX, equality (1 1)], (81) e,= 8.f-^oS. ( 1\ ([. = 0.1- as O. "J. Let's put ("n\ UfP. 1 ft -=--> ^(a, 3",5r) *).f(a,,a, a, 3f) .(82) H(p!, p2, p", -<) = <<:? (~.? the variables a(, a2, aB being, in the second member, replaced, (') P. Massiku. On the characteristic functions of various fluids (Comptes rendus, t. LXIX, 1869, p. S58 and p. 1007). Memoir on the characteristic functions of the various fluids and on the theory of vapors (Mémoires des Savants étrangers, l. XXII, 1876). ------------------------------------------------------------------------ after the derivation, by their expressions (""<}) in functions of fi,, ji2, --̃, pu, %- visually, equality (81) will be able to be written dii l ¡):¡, "'r. ,(j[I' ,9 ô3, .r_+ .H °~. "pi Oj-i O_j,, In other words, if we denote by B,, BJ7 B, W the external actions relative to the variables jî,, Jï^, Jï, 3 which are generally not normal, the equilibrium equations of the system will be the following (83) Pi = 3jï;(j, S(a,, a,&). into functions of normal variables, transforms them into two functions V

$nO3 >>$" J (87) D(SU.H,3) = ràUjifr, dH(^3H,5r)| F'(2r) L"/S* 3 "Ï7 I 1 -* (~> ij- l. <'J J Equalities (83), (84), (85), (86), (87 ) allow us to state the following proposition When for a system related to normal or non-normal variables, among which is the temperature, we know the two functions H(?.,3", ̃̃̃->, we can, by multiple different -ntiations determine 1" The equilibrium equations of the system i" Its internal energy 3° 5o/î entropy; 4" Its calorific coefficients in equilibrium. The internal thermodynamic potential alone no longer plays the role of characteristic function when the variables are no longer normal. The theorems proved in paragraphs 1 and 1 for a system ------------------------------------------------------------------------ defined by normal variables extend without any modification to a system defined by non-normal variables, among which is the temperature. The first a equations (83). in itl, provide the equalities m) = Ë5î, °f'i 'Jri> analogous to the equalities (5). They also allow to generalize the law of reciprocity - expressed by equality (48). Equalities (83) and ^8()) give F(& > /dB, l3, 00 F' iâ) t d:ï dri ("?) - Fl^> /dH" I V d£r* a^fTs)^ [F'(2r>J* V" O) dD F(5r) /dMJ,, d^e F(2pF'(^) /dB,, from to$a "F'(&tô* ((?,(6/+ [K'fS)j2 \"d& By virtue of equalities (89) and (90), we can still state the theorem we formulated^ in paragraph 3 (p. 4'f equilibrium cP a holonomic system defined by any variables, among which is the temperature, we know 1" The heat coefficients of the system in equilibrium, except the heat capacity 2" The partial derivatives of the heat capacity D with respect to the variables p,, j52, p,, other than temperature; So, to complete the calorimetric study of the system in equilibrium, it is sufficient to determine the iialeur that the ------------------------------------------------------------------------ heat capacity D for a particular set of values of variables Jî, pn and for any value of temperature 2f. fi. System related to inverse variables normal variables. We have seen, in the previous paragraph, that the internal thermodynamic potential no longer plays, in general, the role of characteristic function in the case where the system is not defined by means of normal variables. However, the use of certain non-normal variables can lead to the existence of a characteristic function other than the internal potential; this is what happens with the so-called inverse variables. We have already studied (Chap. VII, § 4, p. 287) the change of variables from normal variables to inverse variables. This change of variables consists in taking the equilibrium equations of the system defined by the normal variables a, a, 3r, equations which are I Ai = d3(zu - a, 5) dx, (46) An dril'X >*" %> t Ux" and to solve them by ratio a,, a,, in the form [Chap. VII, equalities (33)] a,=F,(A, ,A.n,2r), ), (90 an= F,,(A,, A, 2f). It is clear, by the very definition of the functions F,, F, that the equalities (46) become identities if we replace a,, a,, by the expressions (91). Consider the quantity (92) = ,?(* a, 5) - A, a,- A,a,.- .- A,,a, In any modification, the external work has the expression 6 - Ai 8a, -4- A, 8au -+-+- A,, 3a, ------------------------------------------------------------------------ If, while the system changes, one may deny invariable the external actions A,, Aa. A,, which are exerted on it, the previous legality could be written g - o( A) 2| - A^x.. -+- A,, a,,); so that the quantity - C Ai "i -i- A2a2-i-+- A,) would, under these conditions, play the role of external potential. According to the definition given by equality (69) in Chapter IX, we can say that, for a normally defined system, the function (a,, <", 2r). given by C equality (92), plays the role of total thermodynamic potential under constant actions. But this role is not the one we consider in this paragraph. In the second member of equality (92)5 let us replace a,, a2, a,, by their expressions (91); this second member becomes a function œ( A,, A, 3) of the inverse variables; so that the equality (93) 3C(A|, A.&) = ;?<>i, .",&) - Ai - .- A,,c" becomes an identity when we replace a,, y. by their expressions (91). After this substitution, the derivatives of the two members of the identity (y3). with respect to Â, must be identical; in other words, this substitution must transform into an identity the equality ddt l i)-'f toF, ( di to¥n dA, xr+ dx, dA~ +.i- (dx,i =~) dA,i. But the replacement of a,, an by the expressions (91) ') transforms the equalities (46) into identities: it thus transforms the preceding equality into the first of the identities ,F, (~ ~1". ) d~ a~t, ?, (94) |f,(A,, | A,S-)=-5e(A,, 4,S), (94) f Fn(A,, .A,J,S)=-3e(A A,Sr). I F"(A,A",â)= ~(A,A" The others are obtained in a similar way. ------------------------------------------------------------------------ These identities, together with the equalities (91), show us that, if we know the 3C Junction (A,, A, S), we can, by simple di//é/'e/ilialions, calculate the values of the normal variables a, a,, which characterize the equilibrium standard taken by the system when it is brought to the temperature 'b and subjected to the external actions A,, AK. From equalities (91) and iq-I), we can easily deduce equality (95) d~2 d1, Suppose that, without varying either the temperature .3 or the external actions A2, .A, we increase the action A, by dA, the result is an equilibrium displacement in which or increases by of X2 Let's suppose that, without varying either the temperature S? or the external actions A, A, A, A, we increase the action A2 by oA2: the result is a displacement of equilibrium in the cluel a, believes to be dF, The equality (90) then leads to this (49) dk.\ oai = oAî doL2. We recognize the corollary that in paragraph 2 (p. 4'^)> we deduced from the law of reciprocity. By means of the new system of variables, the external work done in one equilibrium displacement can be written S = Pt oAt -4-s- !'" oA,,h- e ÔÎj, P,, 0 being the external actions relative to the variables A,, 2<; moreover, we know [Chap. VII, eq;alities (36)] that ~F, dF" Pi-Xiô\r-x "j\l' Prt = Aj - -t- -+- An - - > O:\ii 0 A. ------------------------------------------------------------------------ By means of the" equalities![)4>-> <;<;s the last equalities become P __a '£* A "1X l fI = AI d:l? d;l., 1u1, (r~6) J i P fA .I':ft t d-'ït' A ~~C I ~a = ,Lj, 1 in I~E ,I ~y UA" ~e tJt~n f e = -A, -^i -A, -A, J^L. ~==-A]---.--A~----~r------~r" it\,f)3 t)\oz oA,,) into an identity; if we write this identity taking into account the identities (<).'[). we find the identity (98): li"(A,, A,2r) = ,-J€(A,, A, S" ̃ A, -i-3e(A, A,2r" "A d - A,, -- Xi Ai, A,7;. ). tfAn n Diderentions in relation to "b the identity in which the change ------------------------------------------------------------------------ of variables transforms legality (9^); we find OX. o-i I <)i c/l- o.i oV,, AI -r- u ;1 ci~~ 03 03 "%l ̃J>- Equalities (98), (99) and (ion) justify the following proposition When we know the function 3t( A,, Àn, S), simple clij/erentiations allow to deform the expressions of the internal thermodynamic potential, of the entropy and of the internal energy of the system defined by the inverse variables A A,, S. The calorific coefficients relative to the inverse variables are given by the equalities [Chap. VII, equalities (38) RI = PI r, R - V (' w "=- = -- ------------------------------------------------------------------------ This is the inverse heat capacity, or heat capacity under constant actions. By virtue of equalities (96) and (100), these equalities become I K VC3) <>83C^A, A,5r) F(S?)F'(St) ~{iti correspond the lexical actions Et, E2, E:), G(, G2, G3: these quantities would be your true inverse variables; it is not these quantities that we will take as variables, but the quantities N(, N.2, N3, F,,, 1 T3; these are linked to the preceding ones by the relations Ej = - N!Ct, Ej = - N^ra, Ii3 = - N'aro, r., O\ - - 2Tjct, Gj = - 2Tj7jj, Gr) = - aT3nr. where tît is the variable volume of the system. The variables that we take are not exactly the inverse variables, so the theorems that we are going to establish will not be exactly particular cases of the general theorems that have just been proved. ------------------------------------------------------------------------ To abbreviate [the writings, let's say (ioi) N,- lIu(Sr) = ",, Nj- n(>r2r) = nj) N, - llo(^) "3, (to4) H0(S) -+- \(3r) = Lf &). i. The equalities (y.8), solved with respect to ;'i i,, i. i:i, y,, va, v3, become "= aMni/-f-a\7)U' + /ll+w^ due "11 (io5) M ( 3 L'+ M ) (n,+rx_-s-rL3J- vall 112. (io5) S3 = ,M(:jlV,M) ( + n> + n* ) i 'h' Tl = Im t ï2 = 7m Ts Y:| = 7m These equalities (iod) easily give us the following two equalities: (to6) ) ~e,-t-E,=-'-- L -t- 2 M (107) YÎ-+- Yi-H Ti £"s3 îj=-i Me" = jjp(Tf -i-Ti T§ - /is ",-"-, ".-". "j) LC3L-4-4M) n. 3)-. + 4Mi(3L + aM)*(niH-rtl "l "nj)' = The internal thermodynamic potential of the system is given by the formula ( = ,i-Q = (p0/-i-oi)ro0= Ara, and the equalities (108) and (m) will give us (m3) A(Ni,Nî, N,,T,,T,,T3,2r) L + iVt + x = ^M(3L'+W,M)(a|+"<+lt')l - ^j(Tf + Tl-T| - rtjrt3-"3ni-"i "s)- ------------------------------------------------------------------------ Taking into account the equalities (103), (IO4) and (10:), we can easily derive from this formula (1 13) the following equalities: f M M <~ ( [[, ) Jh ~E r '2, ~~3 dh = ê3, l < dh d%t dlt d`ll = z"(1, ~`i.y = 2 ~.T.3 = 2'(a. If, therefore, we know the expression of h in,,tônclion cIe NI, N2, N9, T,, T2, Ta, we know how to calculate the deformatio~z with which the body is a~ected. By virtue of these equalities (1 14), equality (i io) becomes M (~ dA '"= NI-N +:\I2-N +Na-N 1 dh dh dh Tlol..G~ )~ dh dh dlt 12. + t- + 2- T3 r,T3 z \`dNl + d~V= + + t,t- t,i,~+-~-~+~+~~ This equality, together with the equalities () 12), allows us to write (115) r. ~A o'A (") D-~N~ 7"\ 7,-N; 3 ()A <)A <)/t n.(&)/f)A dA ~1 _dh dh _dlt 110(5') dh _dh dlt )2] "T,5T,5T;r' wû' To know the expression of the function A(N,,N!N3,T,,T,), ), we need to know the function [!(, (2f), and also the values of L and M. Therefore, we can state the following theorem When we know the Junction Ic (Ni. Ns, Na, T, T2, T, 27), we can calculate the internal thermodynamic potential of the system. If we denote by S the entropy of the body, we can pose (u6) S =Sl'iJo, The well-known equality [Chap. IX, equality (16)] S=---- F'(3r)~ ------------------------------------------------------------------------ dh do(,(%) 2L-1-M "1-+-"2-i- n3 dH0(3) c3) F (3; ) <£3 F (.7) cfâ 7¥^} Y-jt + '2 -55- J (£l 'i+ '->̃ 2 dMlâ) ,z 0 E'(~) dri (~(1~ (z+ ls-EZ`~E:3`1-`1~20 On the other hand, the equalities (1 1 3), (io3) and (1 o4 ) give +-~D1 I d~(:~) + d11"(x) " a'M~)i(~,-T-~3~ 'l\12[dA(5') dllo(5')]' .L2' 1. "11' "'12 dâ Yi2 ( 3 L + 2 M )2 + -(r- 1 d-] (Tl+n-r-Tl- n,n3- n3n,- n.n,). ~M~ d~ If we compare this equality with equality (117), taking into account equalities (106) and (107), we find the following equality (m$) 1 dh ̃ ̃>. L + JI dno(3) nt -+- n-2 -+̃ n3 ("8) :S="~F7(S:) tô~ F7s7 "M rfS U + "M 1 ^iioCxf) /",",-4 /i3y ~= z (~'(~) dâ v 3L+i11 ) - + 2F'(â) "fH? V 3L + 2M By the equalities (106) and (1 i4), this equality (1 18) can still be written (1 1) _M L-M Y1 d It"( r - dlz _6~ 1?!a ` ("9) *=- p^d&^Fiiâi ^"v^Nî" rfllo(S) d/i t)/( 'Vi y + -' 1 (dy d\= + J\s, ,)2 ~~r~~) When the /function h (N,, Na, IN3, T,, ï2, T;1, S) is known, ------------------------------------------------------------------------ simple differentiations allow to calculate the entropy of the system. The internal thermodynamic potential and V entropy once known, the internal energy is also known by the formula [Chap. IX, equality (17)] U = j?+[--(&)S. From the expression of the entropy as a function of the variables N,, N2, N3, T, T2, T3, 2r, we can, by differentiation, derive the expressions of the calorific coefficients relative to the same variables. The function A(N,, N2, N3, Tt, T2, T3, 2f) thus plays the role of characteristic function for an isotropic body affected by a small homogeneous deformation. The formulas ( 1 1 5 ) and (119) do not exactly fit in the type of the general formulas (98) and (99); there is no reason to be surprised, the variables N(, N2, N3, T, ï2, T3, S? being not exactly inverse variables. The formulas (108), (1 1 1) and (1 1.3) can be put in very simple forms. Consider the second order surface (120) n.iX2-t- n^y1- n3zi-h iTiyz -s- 2T2sa; -+̃ -iX^xy - 1. Let us denote by p,, p2,p3 the squares of the inverses of the semi-axes of this surface; pu y?2j p3 are the three roots, always real, of the equation (121): p3 - ("i-f-/"2-t- n*)P% ̃+- (n,n3-i- n3n,-+- n,;ii- T| - T|- rY\)p - (n,/i2/i34-2T,TjT3-"iTï- rt2Tl - n3Tl)= o. They are called the three main pressures. So we have n~+ n2+ n3- p~+pq-t-p3, ) n2n3-^ n3/ii-f- ni/ij- T| - T| - T* = ptp3^- p3pi p\p-i, nlnin3-fïT,TtT3-nlT*-niTÎ-n3Tl=plp "p3. ------------------------------------------------------------------------ The formulas (108), (1 1 1) and (1 i3) can then be written Il,, ( ;;) (123) ?"="po(2?) ôj-^r-rr '/>i ̃+̃ Pt -+-̃->) (L-r- M) (3 1,-4- ̃>. M) - M IM&), '2 "----.M(.L+.M~----(~ t 7^ ( A'î Pî - P:> Pl+ Pi Pi h w- lluC?) J (pt+Pz-pa) (12.,) O> = - ,{ f V[ ( pi ̃+̃ p-i -r- [J.i ) (L+M)(3L + ïM)-M no(Î7), ( )2 M(3L-h-.M;" ̃--P "r ̃+̃ ^)- The formulas (108) and (is3) were given, as early as 1858, by Clapeyron ('). Second example. - Homogeneous fluid. - This fluid is normally defined by its specific volume 10 and its temperature 'b; if its mass is M, its internal thermodynamic potential is given by the equality (43) i = Mç((o,2r). The normal and uniform pressure II, which can maintain the fluid in equilibrium under the specific volume w and at the temperature Xj, is given by the formula (45) or (74) n=_^li. (ti5) Or Yes There is no external action relative to the variable 0"; this action is M II; to keep it constant, it is enough to keep the pressure II constant. The external work is then - 3(MlIti>), so that the external potential under pressure ( ) Clapeyron Memoir on the work of elastic forces in a solid body deformed by the action of external forces (Comptes rendus, t. XLVI, i$58, p. 208). ------------------------------------------------------------------------ vunsiunic is (17.6) Q = MlIto. and that the total thermodynamic potential under constant pressure is (l'f.-j) <1> = M[o(u>. ?3)-r- H ""?(, 3) dâ dW +11 -+ e~ or, by virtue of 1 legality (45), ~r(II,) d~f<".~) from to dZ But the entropy of the system is of the form Ms(", 2?), where ( 6,' ( & I Cj5(O), 3 "l (76) ,(0^,=-- By means of the new variables, we can put this entropy in the form Ma-(IT, 3), where (Ho) *(n,2r> = L_^iiii^. F' (3) to '3 The potential intern will be able to, by means of these same variables, ------------------------------------------------------------------------ form NI ~7), 'L(n. 2¡) being what (f((,),2¡) becomes as a result of the change of variables; the equalities (!n8) and (!n8) give without difficulty .1 JI) /itt The internal energy of the system is of the form MM(M, 5'), with ( ,ii u(cu,r)--~(oi,.J)- F~.71 d ~ico,.J~. (7.=~-- Using the new variables, it takes the form M v (rI, 3), with ( Il ( Il 0: '1 'h¡ (II, ) F (;; ¡ J TI ( II, 5') 1'2) ~H.=-,(U,~+..---~--. - d F'(;¡) (I;¡ In an elementary equilibrium displacement, a quantity of heat Q is released, which is related to the entropy increase oS by the formula [Chap. IX, equality (18)] Q =- F(5')ÕS. Moreover, by means of the variables 11, ~7, this quantity of heat is expressed by the formula [Chap. ~I1. Gâalité (52)] Q=- M[/~n,5)Sn-4-rfH,~)~ 1, where F ( 11, 5) is the specific heat under constant pressure. We have therefore, whatever 0 [J and o.~D /a(It, ~) EII, 1'(II; â j d:J,= F(â) 5(Ll, â ) or, according to the equation ()3o), t /,f[,~)= ~n.5') ) F' (5') dn~ (133) 1 F<'3')~T,(n.3') F 0: F"'=-).. ( n Co) ,) F'(5't i dt,2 [F'(5)p Equalities ( r2g) to (133) show us that the ,fônotion 1~(11, ~)'otte the t'Ô~L' l%fi FUNCTION t:aR~f.7 "l'.Iil:STIQI-I: tltt ,/Z "icle ~o~!0i~/?e. The two functions s((i), 2f). '0,) are precisely those which F. Massieu (' ) considered and which he named ,foaclions (' ) F. S 8 IF U, Sur les fonctiorts caractéristiquesdes af'S). dx The internal energy Mm(w, x, S), the entropy Ms(", x,5s) will be given by the formulas (69), (70) and(i34), so that we render, t. LXIX, 1869, p. 858 and p. 1057). - Mémoire sur les fondions caractéristiques des divers fluides et sur la lheorie des vapeurs (Mémoires des Savants étrangers, t. XXII, 1876). C) P. Duhrm, Traité élémentaire de Mécanique chimique, t. I, Book I, Chap. VI, "897, p. 114. ------------------------------------------------------------------------ will have ç.* Ff3) ài(tu.x,r3) (-37) a(co,a-, 5; = , x, 5) Zx -t- y(o>, x. S?) , x, 2f) the specific heat under constant volume of this phase, finally , X, S) = - - t Z( Co F'(2f) ôt"£i (yo) I F(~) dEV(w,x.â). (I4o). Ki^x,.)= l¥- -tej& x, F(Sr) ^0(01. a?;2r) F(3r) F'(5t) J 0(10, ^,5) -T(",)- jjr^y + [F'(Sr)J* tô These formulas do not necessarily assume that the chemical action X is equal to o. In order to define the internal thermodynamic potential and entropy of a phase in a given state (w, x, 2r), it is necessary that there be a pressure and a chemical action capable of maintaining the phase in equilibrium when it is in that state; but it is not necessary that the pressure n and the chemical action X to which that phase is actually subjected, while in that state, be precisely the pressure and the chemical action capable of maintaining equilibrium. The values of the pressure and the chemical action which are likely to ------------------------------------------------------------------------ To maintain the phase in equilibrium in a given state ( w, x, 3f) are. moreover, represented by the second members of the equalities ( 1 3 5 ) and ( 1 36). We can therefore state the following proposition, necessary for the understanding of what follows: In whatever state (o>, x, 2r) a chemical phase is found, we can apply to it the equalities ( 1 3 4 ) ( 1 37 ) and (1 38), even if the pressure and the chemical action to which this phase is subjected would not verify the equilibrium conditions (i35) and (i36). 11 is not the same for the equalities (140); undoubtedly, in any state (w, x, 2f), one can form the three quantities /(w, x, 2f), ).(oj, x, 2r), y(w, x, 2t) by means of the formulas (i4o); but these quantities, carried over into equality (1 3p), do not, in general, provide the expression of the quantity of heat given off in an elementary change, unless the initial state from which this change is derived is a state of equilibrium. Let us suppose that the system, without being in equilibrium, is partially in equilibrium, i.e., that it is constantly subjected to a pressure Il given by equality (i35); let us further suppose, in order to avoid unnecessary generality, that the chemical action X is constantly zero, according to the assumption always admitted in Chemical Mechanics; we may then, as we have seen in Chapter VII (p. 286), make use of equality (1 3g), provided we define the three functions l, À, y by the equalities [Chap. VII, equalities (27)] ̃ < ̃ 'to u(ia, x, S) " ;!̃ *("̃>, = L Un> niw,x,~)=- t <) U(M, 3') dx (Co d u(cu,x,r) Y(')=-----S:----' By virtue of equalities (i35) and (137), these equalities become /!f(Mtg,S)ss_-Ig> **"̃".*>, l = t ¥ (3) tomâS x, 5') 4 ) x. 1 F(5", d~rL,:z' [IlioblS) s A 1(0, X, 3)= -'Jj; ̃+- r V'CS) = àxd?7 j t-"(~ dx d7 I a.. F(2r) d "3C "o.a-,2r) F( Et) F "i S > to o(<,>,x, 3r) I y(oj,a:,J) = l- - -1- - - - - - ------------------------------------------------------------------------ These are the formulas (it, for a system partially in equilibrium, must replace the formulas (i4oi. For a system that is totally in equilibrium, they are reduced to the formulas (i4o); in this case, in fact, the equilibrium condition (i36) gives o ( tu x "ij ôx since we assume X = o. The formulas ( 1 3 9 ) and (j4o bis) can be applied whatever the state (m, x, 2r), provided only that the chemical action X is assumed to be constantly zero. With these preliminaries in mind, let us arrive at the definition of our chemical phase by means of mixed variables. Without worrying in any way about the value taken by the chemical action X, let us suppose that the pressure fI is so closely related to the state (w, x, 3) of the system that equation (1335 ) is constantly verified. This equation, supposedly solved in w, will allow us to define the system partially in equilibrium, no longer by means of the normal variables w, x, 2r, but by means of the mixed variables n, x, &. Consider the expression 04:') (p(o), x, 2r)-t-nio. The product (142) M[a(w,x, S?)H-no)] is not, in general, the total thermodynamic potential under constant actions. If we keep constant the pressure [I and the chemical action X, there is an external potential Mnto - Xa\ However, in the very particular case, but which is the only case that Chemical Mechanics deals with, where the chemical action is assumed to be always zero, this potential is reduced to Mflw; in this case ------------------------------------------------------------------------ In particular, the expression (142) represents the thermodynamic potential under constant pressure of the chemical phase. Let us leave aside, for the moment, this particular case, and return to the general case where the action X has any value. Let us consider the sum (i4') and replace ta by its expression, as a function of [I, o). 3>, given by equality (i35); this sum becomes a certain function of II, x, 3}, 7| ( fi, x, 30; in other words, equality C 1 4 3 > i, X, 3S) -f- llco = r,(II, X, ?S) is transformed into an identity by the equality (1 35). At the same time, the expression (i49-) becomes 044) Mix.II.ar, 2r) = *(n, x, 5). In the particular case, the only one studied by the. Chemical mechanics, where the chemical action X is identically zero, ('n, x, Et) is the thermodynamic potential under constant pressure CI of the phase of mass M, chemical composition x, and temperature 2?; in the general case, the function (II, x, S>) no longer plays this role. The equality ( 1 43) gives us the equality r(o, x, ?S ) 1 dut _dTi{n.x,'3) aw ~` 1,).10, _'h¡(II.x,) dw if, -r- 10 garlic that the equality ( r 35) transforms into 045) M = dVn^,3r)j ) ()j = 'dII This equality makes known the specific volume of the phase when the function i\ (II, x, 3r) is known. Switching from normal to mixed variables transforms the function r'(°|Ig'ar). The knowledge of the function f\(H, x, 2r) leads to the con- ------------------------------------------------------------------------ birth of the internal thermodynamic potential related to the mixed varz "czhles II, ,1', 3'. The equality {il} gives us the equality fà'i)'-f(w. x.Jj ) at T.f II, cr, S) dw -r- fi dâ -J- d:. = ù~ I tou> J ds dz ) gives us the equality [d,2?, 2f) ~:r ~j dta to "-(<>>, x. Jj) e) -/)(! x. "to ) tfeo J Ox OX which equality (i 35) transforms into dyÇto), x, 3r) torjdl.x,) dx Ox Now, according to the equality (i 36), M - v is the value that the chemical action X must take to complete to put in equilibrium the phase which is already in partial equilibrium the condition of equilibrium sought can thus be written (.53) *toT>tU- =X. Ox If the chemical action X is zero, as assumed by Mecha ------------------------------------------------------------------------ This condition becomes the same as for the chemical (o3&M) -!---'--=o. When we carznuft the function %j ( fI, x, :), otz~reut fortner les conditions cl'éc/ttiliGre de la phase chimique. If we transfer the equality(! 5 i bi.s) into the equalities (152), they become ia II x I~(. d '(tl'x'.r)~ ( ) r~(â ) djI d:~ (r.52bis) ~/r. r(â) d='(II,x,~) + F(.°1)F"(â) dr,(Ct.x,â) ([32&n) r(n,.c,.?)==----'----+--~------'---~ [,)2 (S¡ I, ( 11, x, j i F (:~ ) r)z ;1' l I, x, x) - fT' l:: ) dx dâ Such are, by means of the function -f, (H, x, ~'), the i ̃'̃ftj -J the independent variables a,, or, a, S, that we will name the direct entropic variables. If, in the expression F (S) of the absolute temperature, we substitute for 2r its expression (i54), this quantity F Çb) becomes a certain function T( a,, a,, a, S); it is the absolute temperature of the system whose state is defined by entropic variables. If the system experiences an elementary change from a state of equilibrium, it releases a quantity of heat given by the equality [Chap, IX, equality (i 8)] Q= - F(2f).SS that we can write U55) Q=-T(a.,a2, .", S) oS. A system in equilibrium defined by direct entropy variables admits only one calorific coefficient this coefficient is relative to the entropy taken as independent variable it is equal to the absolute temperature of the system. ( l ) The idea of using entropic variables appears to be due to Maxwell ( Maxwiu.l, Theory of f/eat, Chap. IX). The theory set forth in this paragraph is due to M. Ladislas Natanson [Ladisi.as Natanson, Sur les potentiels thermodynamiques (Bulletin de l'Académie des Sciences de Cracovie, avril 1^91); Ueber thermodynamische Potentiale (Zeilschrifl fur physikalische Cliemie, lîd. X, 1893, p. 733)]. ------------------------------------------------------------------------ Let us take two axes of rectangular coordinates; on the abscissa axis, let us carry the entropy of the system; on the ordinate axis, let us carry the absolute temperature. The point thus obtained describes, during a reversible modification, a trace which bears the name of diagram entropy/ uc < ). On the entropy diagram, an isolbermic change is represented by a parallel to the abscissa axis: a reversible adiabatic change, which is isentropic, by a parallel to the ordinate axis. Equality (i 55) tells us that the entropy diagram provides a very simple geometrical representation of the amount of heat released in a reversible change. If, in the expression U(a, 7. a, h) of the internal energy of the system defined by normal variables, we replace 2? by its expression ( 1 54), we obtain a function "W (a,, or. a, S) of the entropic variables. In an elementary change from a state of equilibrium, the amount of heat Q released is the excess of external work over the increase in internal energy: we can therefore write <~W ~W d\V , ~xn ~'4'(x~, x, S)~ (i57) T- W(a,, ."", S). j. When we know the expression W (a,, a, S) of the internal energy of a system defined by direct entropic variables, we know, by simple differentials, the (' ) The entropic diagram was introduced p;n- I. Wilhtrd Gibbs J.Viu.aiu> Gibbs, Graphical Uelhods in the Thermodynamics of Ftuids ( Transactions of the Conneclicitt, Academy of Arts and Sciences, vol. Il, part s, p. 3oi|)|. ------------------------------------------------------------------------ equations of equilibrium of the system and the unique calorific coefficient of the system in equilibrium. The internal thermodynamic potential is also known. Indeed, for the system defined by means of the normal variables, we have the equality [Chap. IX, equality (17)] i= U-F(2?)S. If we denote by W(a,, a, S) the expression of the internal thermodynamic potential by means of the direct entropic variables, the use of these variables transforms the preceding equality into W(xu .an, S) = \V(a,, .a,S) - T(a,, .an, S)S or, according to equality (i5~), by ( -S) ( s, S iJ\V("-t¡, .tl/,SI. (i58) W(a,, ".. S) = W("1; ."., S ) Sd W(a" Js" S>- The internal energy thus plays the role of a CHARACTERISTIC FUNCTION for a holonomic system delimited by direct entropic variables. We can suppose that we solve the equations ( 1 56 ) with respect to a, an in the form I ai = ", (Ai, A, S), ((59) < { a,,= an(A,, A, S). These equations allow us to substitute the direct entropic variables y. y. S the inverse entropy variables A,, A, S. Equations (i5g) are the equilibrium equations written with these new variables. The change of variables represented by the equalities (i5(>) transforms the function T(a,, a, S) into a function 0 ( Aj, A", S) which will continue to represent, by means of the new variables, the absolute temperature of the system. In any elementary displacement of the equilibrium, we can write instead of the equality ( 1 55), the equality Q;=-G(A,, .AM,S)5S, ------------------------------------------------------------------------ so that 0(A,, A_H, S) will be the only heat coefficient relative to the new system of variables. The passage from direct entropic variables to inverse entropic variables transforms the internal energy W(a, a, S) into a function X ( A, A//( S), so that the equalities (109) transform the equality A,' ̃ dS ~dk~n To relations (i')6) and (167), one can compare two groups of relations which are obtained without difficulty, in the case where one makes use of either the normal variables or the inverse variables of those. If we use normal variables, the equalities (4'i) and (-6) ------------------------------------------------------------------------ give the n relations 8) become f%z1, c'S dA" dS (t6$ blsl -'=--, ~=~_-. dl "a, c "I c'a,, If we makelusage of the inverse variables of the normal variables. the equalities (91), (A| ,y.-j f>AM that the use of absolute temperatures reduces to the form dx, ~2 <)x,, (~S (.'9 M~ ,7r "jA7' ~'T u:1" The analogy between formulas (ifi- Consider a holonomic system normally defined by its temperature 2f and by n other variables a,, a2, a. Assume that this system is held in equilibrium by the external actions A,, A2, A, At the same temperature 23, let us consider a second state (a,4-rfai, a.n -+- dv.n) infinitely close to the first one, it can happen that the system is still maintained in equilibrium by the same actions A,, A2, A, here are two examples. Its state is defined by the position it occupies in space, so that it remains in equilibrium in this position, it is necessary and sufficient that the force and the torque to which the external actions can always be reduced are both equal 'to o. If we want to maintain this body in equilibrium in another position infinitely close to the first one, we will still have to subject it to a zero resultant force and a zero resultant torque, that is to say to the same external actions as in the first case. Secondly, let us consider a liquid in contact, at temperature S, with its vapor. For this system to be in equilibrium, it is necessary that (' ) The general statement and essential properties of the Joi of isothermal displacement of equilibrium were first formulated by Lord Rayleigh [I.ORD Uayleioh, General Theorems relating to Equilibrium and Initial and Steady Motions (Philosophical Magazine, vol. XLIX, ii-p, p. 218; Scientific Papers, by J.-W. STIIUTT, Baron Rayleigh, vol. I, 1899, p. a3i)]. ------------------------------------------------------------------------ the external actions are reduced, ii a normal and uniform pressure, the value of which depends only on the temperature, and which is called the saturated vapor pressure at this temperature. Now this same pressure will still maintain the system in equilibrium in a state close to the previous one, where the mass of the vapor will have increased or decreased by a small quantity, while the mass of the liquid will have decreased or increased by the same quantity. From this first remark, we can bring a second one. The system brought to the temperature ?S is maintained in equilibrium in the state (a,, a2, a,,) by- the external actions A, A2, A, without changing the temperature, let us give to the external actions values A, -j-dA,, A2 + rc* by giving the external actions A,-r-c/A,, A.2 -r- d. A/jH-c/A/ infinitely close to A|, A2, \", but not identical to A,, A.>, A, a" If, without changing the temperature 3j, the external actions are given values A, - j - <̃/ -V t A2-i~f/A2, A,, dA, infinitely close to A,, A_ A, the system will come into equilibrium in a state (y., -t- dy.t a2 -f- dy.2, y- -+- d% > inliniinent close to the state (a,, y. a,,). To both sets of infinitely small values, but not all zero, dtt, d-x, dz, d\t, d\n, which are linked to each other by the preceding statements, we will give names that will make our exposition easier; of the first set, we can say that it represents a disturbance; of the second, that it is the set of disturbing actions corresponding to this disturbance. Let's apply these definitions to some very simple examples. A homogeneous fluid, of mass M, is normally defined by its specific volume w and by its temperature S; at temperature £?, to keep it in equilibrium under the specific volume w, it is necessary to apply to it a pressure II; to keep it in equilibrium under the specific volume (w ̃+- <7" it is necessary to apply to it a pressure (11 + d\Y) if one remembers that to the pressure fi corresponds the external action - MO, it is seen that to the disturbance dw, corresponds the disturbing action - M'/FI. An extensible wire is normally defined by its temperature 2f and its length, the temperature 2r remaining invariable, this wire in equilibrium has a length l when it is subjected to the tensor weight I', and a length (l-dl) when it is subjected to the tensor weight '(P-f-rfP); (II is here the perturbation and dP the perturbing action. Whatever the concrete fluid, given by the experiment, of which we wanted to compose a theoretical representation by means of the scheme that Thermodynamics calls Jli/idc hoinoisrne, we are led to admit that the specific volume of "e lluid ------------------------------------------------------------------------ varies in the opposite direction of the pressure it supports, at least as long as the temperature remains invariant; in other words. iior> are led to admit that in any isothermal displacement of the equilibrium, the product dtt d is negative. In the same way, in order to represent the properties of any concrete wire by means of the abstract system that we call extensible wire, we are led to admit that in any isothermal displacement of the equilibrium, the product dV dl is positive. If we want to apply to the representation of concrete bodies the systems that we have named homogeneous fluid and extensible wire, we must suppose that the product of the perturbation by the perturbing action is positive in any isothermal displacement of equilibrium. The two systems we have just discussed are defined by a single normal variable except the temperature. The remarks we have made about them can be extended to holonomic systems defined by any number of normal variables. If d% da.2, (11. is a perturbation to which the perturbing actions d. dA->, d An correspond, we will say that the sum t = rfAj dxt -+- c/A, rfa. -+- -f- dA,t dx,, represents the disturbance work related to this disturbance. We will say that a system verifies the Law of Isothermal Displacement of Equilibrium when any perturbation carried out at constant temperature will cause the perturbative work to take a value, infinitesimally small of the second order, which is positive. In a very large number of cases, the systems of interest to the physicist are systems which verify the law of isothermal displacement of equilibrium, but one should beware of believing that one will never have to study any system which is not subject to this law. We have already cited two examples where this law does not apply. In these examples, which are provided by the invariant solid and the liquid in contact with its saturated vapour, any infinitely small perturbation corresponds to an action ------------------------------------------------------------------------ and, consequently, 'i a perturbative work also null. We shall see later (Cliap. XVI, § 10) that if the equilibrium of a system, maintained at an invariable temperature and subjected to invariable external actions, is unstable or indifferent, this system does not submit to the law of equilibrium displacement. The equilibrium of each of the two systems taken as examples at the beginning of this paragraph was precisely an indifferent equilibrium. 2. Various ways of expressing that a normally defined system verifies the law of isothermal displacement of equilibrium. For the moment, we will focus our attention exclusively on systems that verify the law of isothermal displacement of equilibrium, and we will give various algebraic expressions of this law. The equilibrium equations of a holonomic system normally defined by the temperature 3 and the independent variables a, y, a,, are the following [Chap. X, equalities (4)] (,\ ) i a di w a dî (1) A, = - A3= - - ---, AB = dot! 0-x, 0in In these equalities, d represents the internal thermodynamic potential. According to these equalities, to a perturbation dx,, c/a2, da.n, correspond perturbing actions dA,, dA2, d\n given by the legalities d"~ <)~ d~l, dx2 2 "'X)-(-------s,f - is replaced by --;-- O7.i, ÛXq O'J. <)y. For the system studied to verify the law of isothermal displacement of equilibrium, it is necessary and sufficient that the disturbing work is an infinitesimal of the second order essentially positive. Therefore, the equality (?>) allows us to state the following proposition For a normally defined system to be subject to the isothermal displacement law of equilibrium from a certain state (a,, a, a, £7), it is necessary and sufficient that the quadratic form in X|, X2, X, (ai v to§ v tots y*> (4) - \i-4-- - X2 -4- -̃- - X,, dx, dx2 dx" is a positive definite form. According to a well-known theorem, given by the theory of quadratic forms, for the form (4) to be a positive definite form, it is necessary and sufficient that the discriminant of this form d--§ d*$ d*§ Oa] (>a2 doLt - a| i)'j.nd%ï à^S ri2 i à"- S àa, àan 0%% à?. Ox'f, is positive, and that it is the same, whatever p is, for the determinant obtained by deleting in the discriminant the last p rows and the last p columns. Before going further, let's apply this criterion to two examples. First example. Homogeneous fluid. - In this excessively simple case, the internal thermodynamic potential is of the form [Chap. IX, equality (49)] § - Mb(u), 2f), ------------------------------------------------------------------------ M being the mass of the Htiid and " its specific volume. The quadratic form { \) reduces here to 41 `~Z- w. 1. If it is positive, it is necessary and sufficient that we have I" inequality >6) 0 J r ?j ) > ". f6) OR)1 This is therefore the necessary and sufficient condition [for the homogeneous fluid to verify the law of isolliermic displacement of equilibrium. This can be verified directly. The pressure 0 which maintains the fluid in equilibrium under the specific volume co, at temperature 3, is given by the equality [Chap. X, equality (74)] II = ~~fw,). dot If the temperature S? is kept invariable, a disturbance of the) corresponds to a disturbing action - Mc/II determined by the i-gality à2 -liui, ?j) dia- The inequality (6) teaches us that the product dll dis) is negative; and, as we have seen, this is precisely, for the system studied, the expression of the law of isothermal displacement of equilibrium. Second example. - Very slightly deformed isotropic body. - If rn0 is the initial volume of the isotropic body with little deformation and normally defined, its internal thermodynamic potential is obtained by multiplying by m0 the quantity po/that gives the equality (27) of Chapter X; this equality is the following (7) po/=. M " o ii (H) O " O 41\1 O o x ml, o o 4 -'- o ° o t) o o î M o o o " o o 4 M The necessary and sufficient conditions for this discriminant to be that of a positive definite form are easy to write and can be reduced to the two inequalities (9) M(Sj>o, 3A(S) + îM(&)>o. It immediately follows that the three coefficients [ï (.:?), i, <*(j3), K(.r7), defined by the equalities (3i bis), (34 bis) and (35 bis) of Chapter X, are positive for any isotropic body subject to the law of isothermal equilibrium displacement. The same is certainly true in the case where these coefficients are given by the more general equalities (3i), (34) and (35), provided that the pressure II0 (3) is positive. Let's check that the inequalities (g) do indeed result in the law of isothermal displacement of the equilibrium. To the normal variables s,, z2, £3, y(, y2) "(3 correspond external actions which are -Nitb, - Nr2ns, - N3cs, - "T^cj. - îTjra, - 2T3H1, ra denoting the volume of the system. The disturbing work is, therefore, t =̃̃ - (oi\'i Oï] -+- oN2 0E5 -(- 0N3 6e3 -+- 2 81"! 8-fi - 2 8T2 0-/2 + 2 ST. 87;; m - ( IV 1 ce, -+- N2 8e2 -+- N3 Se3 -+- 2 Tj &Yi Ts 6v5 -+- 2 T3 0-/3 ) oct. Only infinitesimally small quantities will be neglected with respect to the conserved quantities, if we write = - oNi oeI-T-oN2o=2-t-oN3oî3-t-2 0T1 o-r-4 oTjO-;" - 20 i:1 ov:l -t- (Ni 821 -VjOî2 -hN3oe3 -S-2T1OY1 -r-aTjS-ca ̃+- '-* T3 ôyj 1 m, ------------------------------------------------------------------------ But if 'n-~ - 5, -r- =̃-> Ej, mu so that -, - i= GE, -r- 0ï2 -+- 0t3. '?.) On the other hand, neglecting the terms which are infinitely small together with î, î2, î:i, y, y2i we reduce, according to equalities (28) of Chapter X, X, N2, N3 to flo (2?) and T, T2, T:i to zero, so that the expression of the isothermal perturbative work becomes (10) t = - ] 3Ni ôs,+ 8Nj 8es-oNs 3s3 2 8T, 3-e, -4-2 8T2 5-/2 4- -2 8T3 Sv3 H-no(2r)(8s,-i- 3ej+3E3)!;ro0. In the particular case where the pressure FI0 (S) is zero, it takes the form (10a) t =- (3^! 0Ei-r-8!V2 0£,-i- 3NS o;3-i- 2 oTt 0Y1-+-2 3T2 0-/2+ 2 3T3 oy3jnr. According to the formulas (28) in Chapter X, equality (10) becomes t = | A(&)(3e,-H as,-?- 3e3)"-(- 2 M(&)[(3e,)2 -i- (Sei)1 -+̃ (Se,)* j or -+-4M(&)[rôïl)2+(3-r2)!-H(Sï:J)2]!nr0 t3A(2r)-t-M(3r)- (") <=---------(q.t-&~-o~)' (11) t=\ ('- J l_i(ps, + 0£î-(-OÎ3)ï -i-2;:V'3("')[(SE!8e3)i-(3£3-oô,)î^-(3£1-Sê2^ 2 3 2 3 )2 aS3 00, )2+ 00, 6Ô2)2 ^-6 8yï. + 6 87î-i-6 8vî]|Wo. In this form, we see that t is certainly positive whenever the inequalities (g) are verified. A remark naturally arises here. Among the normal variables 7. au, a" which, together with the temperature S?, define the state of the studied system, there may be some which do not appear in the expression of the thermodynamic potential e, internal; let xp, ay be these variables. Equalities (1) teach us that, for the system to be in equilibrium, the actions Ay, Av relative to the ------------------------------------------------------------------------ variables y.p, ay are equal to z, considered as a function of the variables a,, y. y.,t, and that this minimum can be recognized by the inspection of the terms in finirnent small of the second order in the variation of . It may happen that some variables do not appear in the expression of the internal potential -1 be y.p, y.y these variables. As we have seen, the system can only be in equilibrium if the actions A^, Ay are zero; it is therefore necessary, in order for the system to be brought into equilibrium, that the function be independent of the variables y.p, y.f/. From then on, the function , viewed as a function of all the variables a,, a, cannot be minimum in the state of equilibrium; from this state, a modification that would change only the value of a/(, y.t/ would not change the value of . But we can impose on , regarded as a function of those variables a,, y.n which actually appear in it, the condition of being minimum when these variables take the values they acquire in the state of equilibrium. In this way, the system will be subjected to verify the law of isothermal displacement of the equilibrium for any modification which is not reduced to a simple change of value of the variables olp, y.,r 3. The law of isothermal displacement of equilibrium can be true or false depending on whether the system has been given a normal definition or another. Before giving other methods to express that a system verifies the law of the isothermal displacement of the equilibrium, we are going to draw an important consequence from the methods already indicated; this consequence is the following It can happen that a system verifies your law of isothermal displacement of equilibrium when it is given a certain normal definition, and that it no longer verifies it when it is given another normal definition. Let's imagine that the system has been normally defined, all ------------------------------------------------------------------------ first, by means of the temperature 5 and the variables :1." "1." to which the external actions A~ 4" correspond. In this first mode of definition, the system verifies, by hypothesis, the law of isothernic displacement of equilibrium. To the variables x,, '/". let us substitute new normal variables~ ~part of the relations (,3) 7." = x,,(3t, ~), ). To the new variables will correspond external actions B,, B, linked to actions A,, by the relations i~.S-- dr, dt31 (,4) J t dx" B,,==At---A, ~jt It 1 n By the equalities (i3), the internal potential di. jent n", ~). By the assumption made, we know that the form is a positive definite form in X,, X, Let us form the expression (~~--)" For the system normally defined by means of the variables 6t, Nn, to verify the law of isothermal displacement of the equilibrium, it is necessary and sufficient that this quantity be a positive definite quadratic form in c~, e~n. If we observe that the relations (d) transform the equality (j'(pt,S,)=,F(~t,x,5') in identity, we easily find that we have the following equality <- d-t cl3( +.f- ~t cl~a, d~r ~xY" xt \~)' ('J7.1 d7.1 )12) + AI dp pl dr~1 ,+ (J il Il d~n -t- -A,,---~t-t-.T-~) + \dJt d¡>r- o~ ------------------------------------------------------------------------ in which we have placed ~~t- J~=~ Iix" CG~J, ~),> t)x,, d- ,J IC '-n' (/~ ~M To the change of elal c/[j, rfjj,, corresponds, according to the equalities (i3), a change of elal c/a,, c/y. If, during this change of state, the external actions A,, A,, were maintained invariable, the external actions B|, B" would not remain invariable in general they would experience increases AB, ABn according to the equalities (i 4)i one would have iBl= Âl ( 1$\ rf?1"5p^fcrf?") 1 + - +A.( ( -Jri2 CL,, d (.C) to, A, ( d z xt d~, di n + ()~ <~x,, According to these equalities (16), the equality ('i5) becomes t /()~ clxt d-T y~ - d,'i, 0~ Elected, + \Hidrpl-h. + ±Hr!d°,n. If we have, whatever = - - w = i The new external action will be (..)' -_$!. Let us suppose that the pressure Il is kept invariable: according to equality (i8), the action A will remain invariable; but if, at the same time, p varies by do, B will experience, according to equality (19), an increase .D aiICI AB = - --<4- The first member of condition (17) will have a value here and this value is essentially negative. It may therefore happen that the law of isothermal displacement of equilibrium is no longer verified when the new normal definition is used. Indeed, at constant temperature, an increase in pressure clU corresponds to an increase dp of the density of the fluid in equilibrium, and, according to equality (19), we have MU /dU 2do\ d. dB P ~z ( tl ) P ,=~p=~-(~p. ------------------------------------------------------------------------ We see that the law of displacement (the I equilibrium would still be verified by a liquid that normally defines its density and its temperature .:>, if we " the inequality (zo~ i (rt\\ u., ( :lO) .) II ( c/~ In this case, it will no longer be checked. In other words, the law of isolhermal displacement of the equilibrium, verified for a fluid that is normally defined by its temperature Sf and its specific volume w, will still be verified when it is defined by means of its temperature ?3 and its density p, provided that at constant temperature, this fluid compresses less rapidly than the formula (::>.1) = (-) If the fluid compresses as fast or faster than this formula indicates, it will no longer verify the law of isolhermal displacement of equilibrium. If, for example, a gas follows Boyle's and Mariotte's law, we have 1 [rJl\ I"- IJ V do ,U 57 p f. Such a gas is therefore no longer subject to the law (read isothermal displacement of the equilibrium when it is defined by its density and its temperature. Whatever the law of compressibility of a fluid, it is always possible to give this fluid a normal delinquency such that the law of isothermal displacement of the equilibrium ceases to be verified in the vicinity of a state of equilibrium designated in advance, in an arbitrary way moreover. Let us take, in ell'et, to define normally the liquid. the temperature S? and a variable a-J'(N). To this variable, corresponds an external action f: VI 11 _j'i,~i' ------------------------------------------------------------------------ In all circumstances, we have ..r<=)). cl:, ?2,f1?;I [! ch f'ï~ 1) .J d?, p2 fi? 1 It d? ? 7~77 i so <|that the disruptive work has the value dCd*-MU\ dn -a /^1(^ ~G = K9", K being any non-zero constant, and n another constant different from o and i; we have /"(?) n - i f'(?) = o It is clear that we can always take n large enough so that the condition n - i i (says 9. Ç = ÏI l ~dl) =t~~ ~p is verified in the state of equilibrium that has been chosen in advance. We had already seen that the law of isothermal displacement of equilibrium did not govern all holonomic systems; we now see that its correctness is not only limited to certain particular systems. but that it still presupposes that these systems have received certain particular normal definitions. Sometimes the law of isolhermal displacement of equilibrium is presented as a general law of nature; thus it is often equated with the law formulated by Lenz for electrodynamic induction. This way of seeing things is ------------------------------------------------------------------------ obviously erroneous, since it is possible to make this law true or false, without modifying in any way the system studied, by changing only the variables which are used to represent the various "Hais". 4. Expression of the law of isothermal displacement of equilibrium by means of the inverse variables ('). Let us suppose that a system verifies the law of the displacement of the equilibrium when we define it by means of the variables a,, a, 3. To these variables, correspond the external actions À, A, A the isothermal disturbance do. dv.n, correspond the disturbing actions d\ dA, and the law of the isothermal displacement of the equilibrium is expressed by the inequality (11) t = d Ai dat -+- -+- rfA,, dzn > o. Let us now make use of the inverse variables Al5 A", and consider the characteristic function 3C(At, An, 3) relative to these variables at temperature £f and under the external actions A(, A, the normal variables take values at, a,, given by the equalities [Chap. X, equalities (91) and (94)] a, =-3C(A,, .A,5), ), <*3) ) = -3e(A1, .A,S). In an isothermal shift from equilibrium, we have '̃̃ 1 d3t*- to*-3C tlxr ( dJC! dA, +. + d!JC dÂt d~l,t) dxi=-( -rr-ï 1 d\t -+-̃̃̃-+- t/A,,), (24) ̃ ~3C ~3€ ~~=-(. ~A,-r-t- .- ~A~L \f "AiC "An n àXfl I With these equalities (24), the expression of the isothermal disturbance work becomes t ax dÀ, --i dJe) (21 (23) t=- ( - -rfA,-t-- -A,, (') P. Dl'iiem, Traité élémentaire de Mécanique chimique, Livre I, Chap. VIII t. I, 1897, pp. i.o-i,2. ------------------------------------------------------------------------ The law of isothermal displacement of ['cquiHhre expresses that rilH~gality (22) is verified whatever the disturbing actions c1:1,, .nt isotlzcrnzic~ue cle l'r~rluilibre, it is necessary and AM/y" rl3C "*A,' ̃"̃' clA, These equalities, together with any of (36), (3^), i'i-j bis), give (38) C-c- ta'(~) ` dz i d2~~ +. d2~ d~~3~ Y F"(2f) \"ta, 5^œA" -̃-' 5^ = A"- Without changing the external actions A,, A, let us raise the temperature by c/5; the variables a,, a,, experience increases -Tr;d%, -£dÏ3, and, as the equations (i) dedâ d~. die verified, we must have the equalities dli d^i_ dVt 0*- j ùFn 0%, 03 ài\ fâ d%n Ooc, 03 i to2j d2 S dF, d2,Y dF,, din dii àoix dn.lt dij ô%\ d il o. Let's multiply these equalities by - ---, -^> respectively and add the obtained results member by member; we find the equality 3 d%^ dFi dt* ôF" ldj_ ^fj. à$ àF,W_ 1 0>a, .ô& J.\lO3 03 ') and CMj) (ionnent equality )-)/~ dt', r).'p (10) C - = - -- -- ) ts ( :J ) c F<~)/~Kd~ f, ~K < ( -jo 6~' ) C - c == F (;) -- ~- - aJ e -- ')' (2) V'(3)\OA, 03 O\n tor3 l'or a normally defined system that verifies the law of isothermal displacement of equilibrium, the quadratic form XI ~,+. d:f' Y", (~ ,dx, dx" j is a positive definite form in X,, X, while the quadratic form ". /àXv -i-. oX v \> (2C) (sâ:) i 1 /t is a negative definite form in Y, Y, The two equalities (4o) and (4|o bis) then both lead to this conclusion: For a normally defined system that verifies the ioi of isothermal equilibrium shift, the inverse heat capacity is always greater than the normal heat capacity. By virtue of Helihholtz's postulate (Cbap. VI, § 3, p. a63), the latter heat capacity is always positive; the theorem we have just formulated therefore leads to this corollary For a normally defined system which verifies the law of isothermal displacement cle equilibrium, the inverse heat capacity is surely positive. ------------------------------------------------------------------------ Let us apply these considerations to a homogeneous fluid, of mass M, normally defined by its specific volume m and its temperature S; to the variable w corresponds the external action Mil, II being the uniform pressure at the surface of the fluid, [(n the for̃ ̃ t)Fi to'D i fl. S i <)fi i mules above, one should replace -^ by ?-- el -=£- t~ ~.? ~.? ,,rJll((ll. 2ri 1 (44) - ^- ) both show that the specific heat under constant pressure exceeds the specific heat under constant volume. Let be the temperature of the melting ice, the expansion coefficient under constant pressure of the studied fluid, jï its expansion coefficient under constant volume. We have, by definition, Equality (4") can therefore be written (45) r - v= 'J^ U('(u.^1,)<1)(ri.^0)a^i. This remarkable formula is a generalization of the one that Robert Maver made known [Cliap. III, equality (3oV| for perfect gases. Let us imagine a displacement of equilibrium during which the normal variables a,, a,, experience infinitely small variations da.t. dy.,n but let us not suppose any more that this displacement is accomplished without variation of temperature: let us suppose that it i) does not give place to any release of heat, what one expresses by saying that it is adiabalic it will leave then invariable the entropy of the system, so that it will be .is in tropic In this isentropic displacement of the equilibrium, the external actions (') P. Dumëm, Commentary on the Principles of Thermodynamics. Part Three The general equations of Thermodynamics. Chap. IV. {Journal de Mathématiques pures et applit/uées. 4' série, t. X. i8(|4. p. aliS); Traité élémentaire de Mécanique chimique, Livre I, Cliap. X; t I, 1897, f-'^̃i- ~m, àw(U.'b) x u> ( II Î70 ) (Jij p l t I t :u (Jil(o). ?j ) 11 ( (o Sr0 ) oi; {''(.?; .6. The isentropic displacement of the equilibrium ('). 7. ------------------------------------------------------------------------ rieures will no longer experience the clA,, cl_1" variatïons that they experienced in an isolhermilJue disaeeUlcI1t; they will experience (the variations, ~éoér.deutcntditK'rentes, that we will destgfier by A. resvariations will constitute the pertun!zccte veri~e the LOt nu oéYr.acLlrenr tSEMTROt'IQtJK OF r.yorr,reHS, is to say that this iserztropigue disruptive work is positive for any disturbance. We will determine the necessary and sufficient conditions for a system to be subject to this law. We shall begin by assuming that the system is defined by means of the variables rrztropic/ues whose use is indicated here. Let W (7. '1. S) be the internal energy of the system expressed by means of these variables; the equilibrium equations will be [Chap. X, equalities ('56~] ~x~ W'(x~, xr~, 5), (47) fA.=~W(~S). From these equalities, the following can be drawn Ar ~%VV dxr+ = W(a,, .an, S)- A,a, - - A "an. We will be able to state thus the proposition of which it is a question For a system to be in equilibrium in the state ( a, a,, S ) 1 when one subjects it to the external actions AI, A, and that it obeys, starting from this state, the law of the isentropic displacement of the equilibrium, it is necessary and it is enough that the function Z(a,, .aH, S, A,, A,,). considered as a function of at, a", is minimal in this state, and that this can be recognized by inspection of the second order terms in the variation of Z. Let us now consider the case where the system is defined by the normal variables a,, .a", 3; the equilibrium equations are d$ " to$ (l) A'=^' ---' k"=àTn' i n In an isentropic shift of the equilibrium, the perturbation da. do. is accompanied by a temperature change d3, so that the equalities (i) give &§ à'- î ô' 5 e>A, == - -r-daf -[-+- - - rfi "+7 air, (5a) 'i~ .i~ oAn = Z Z - ""1 -+-----+̃ T^T da' da,dan dx-n dZ 0xn On the other hand, the entropy S of the system is given by the form ------------------------------------------------------------------------ mule [ Chap. IX, equality (16)] s = L_ °i. F'( S/ i Jz For this entropy to remain unchanging, it is necessary and sufficient that we have - (l3 dv. - - d'xn = o.!)'"(.?~<~ J. J Ux, ~l:J rlx" U:J But the normal heat capacity c is given by I equality [Chap. X, equality ("a) j f~' 1 J I V'-î' F (;1 1 Î'"i~Î Y U-Ÿ F'iâi oïl- [!-( ;J/j- tJZ which transforms the previous relation into (~3) ~ü~:r .lx,-T.+ .-)=:, F't~~ (a3) 0rxt y_/ - 'l-xi-i-+-- 0-xn 03 -d'x,,- V .j ) cd^. Equalities (02) and (53 joined to equality (3). yield (54) 3A, d'xï-h.̃+̃ 3A;i dia -ld.-l,dx,+.+d~1 "dxn~m f~J)c~d:Jjz. - (d\x d%i -+-i- dAn d-Xn) = -^ - cidjj)1. F ( 3 ) 1 According to Helinhollz's postulate (p. 268), the normal heat capacity c is certainly positive: so are the quantiles! F (S?) and F'(2?), we can therefore state the following proposition The isentropic disturbance work corresponding to a given disturbance is always greater than the isolher inic disturbance work related to the same disturbance. From this theorem follows this corollary If a system is subject to the law of isothermal displacement of C equilibrium, it is subject, a fortiori, to the law ) a, = - -, a,, = - - 'JA,, Let us consider actions [)erturl.>atrices '/A|, d.A, if the temperature were maintained invariable, they would determine a disturbance c/*i, fl'J.H; with constant entropy, they determine a disturbance o?.|, oa, accompanied by a variation of temperature o!E7. The equations ( 3 ) give x f.€ I ~3C l 5xt =-- -- which allows us to state the following proposition When a system is subject to the law of isothermal displacement of equilibrium, a given set of disturbing actions produces less work at constant eiztnopy than at constant temperature. The temperature change that accompanies an isentropic displacement of the equilibrium can be given in two ways. If one were to recognize the variations experienced, in this displacement, by the normal variables, this change of temperature is given by the equality (53), which the equalities (i) allow to write F' (3) 1 o\, 0kn dx". (57) F7~~=--+.<--- r ( 3 ) 03 03 This same equality would be derived from equality (54), by observing that we obviously have i\l-dk1 = ^-d3, 03 - - ôA" dA,t dA" 0.. 8A~ oa, - rfa, == -03, ace, i., d x.e 3an- din= -03. 03 Let us apply formulas (57) and (5- bis) to a homogeneous fluid, ------------------------------------------------------------------------ norfnatementdcnn) by its specific volume (u and its temperature &; let M the mass of this fluid and Il the uniform pressure that it supports external faction retati\e e the variable M is M CI; moreover, if one designates by Y the specific heat under constant volume and by r the heat spéeifillue under constant pressure, one has c C = ~~l 1'. the formulas (57) and (57 bis) become then ," dJ' F(;¡ 1 ôU(w')d (38) ) a.? = - F'(Co' '( -------atu, ( ~8 bLS) G~ t' (-r,'n I d(r)( tt, ) dtl. (~86t~) o.3= -----------an. F' ( 1 r d~7 These formulas can be written a little differently. Let x be the coefficient of expansion under constant pressure and the coefficient of expansion under constant volume for the fluid studied; the definitions of these coefficients - 1 ~(n.~) r ~n(M, ~) M(n,5o)d&' IIl(o,âoj d.Gi give the equalities (58) and (58 bis) the forms (59) d. F(5) n(!v p, i.i.î. C) P. Diîhkm, Traité élémentaire de .Mécanique chimique. Book I; C.liap. tome I, § 5, 1897, p. 17"; On a generalization of the theorem of He,e,r,h ( Procès-verbaux delà Société des Sciences physiques et naturelles de Bordeaux. ( /~rocM-ce/'<'f!M.r < - Dcoj iw, and equality (61) becomes (6.) £ = £, Y Dâ which is Reech's relation [Chap. VII, equality (80) Let us return to the general problem. If we express the actions Ai, A,, as functions of a,, a//} 3f by means of the equilibrium equations, we will obviously have AA, -DA, = ^AS, dx - 1 AA,DA,.=~ AA"-DA,,= ^iA2?. If, in the same way, the equilibrium equations are used to express at, a,, in functions of A,, A", 3, we will have Aa,-Da1 = gA&, 03 Equality (60) can therefore also be written ^A^+ %AAn (63) gs-fel ± ~A,t c M. dA,, dâ Jz, -s- + d~ s1xn Let us show how, from this form, we can still derive the Reech formula. Applied to a homogeneous guide, this equality becomes, according to the remarks made i) a moment ago, <*(̃>( n,Br) (64) l-= - Let us observe that we have, by definition, DA,=~1~.+.+~A~ dx, ~n DA. dA" to.~ dA" UA,,=--A~t-t--Aatn. clO[I z Let us multiply the first equalities by ,lx,, ~1a", the last (' ) Jt MouTtKR, j&enten~t/e TheiW eodynamiyue,, '72, p. 7j. ------------------------------------------------------------------------ '"t by - -" and add member a memlirc the results 1 03 oj -' obtained, observing (|ue, according to the equalities it), Ô\;i ()\,j OR. 0'J.f, We obtain the equality (66, ^l^, + .+ ^ÎAafl+^i|,A1+.+ ^2DA(,= o. 03 03 03 03 Let us express in the same way that z, y. remain invariant when A, A, 2? experience the variations -4^ from to, - ^A tfâ, d'b 1 uS 03 we will have the identities cta, <>A, <^2| rfAn dai o, t'Ai o'3 oAn 03 03 ai- c>A| daB >)A, >>n o'A) -4-4- 7A7, d:~ d: ï 0. On the other hand, we have, by definition, n.I = iïLAAI-t-£UA,I, o*A| 0Xn Let us multiply respectively. the first equalities by A A, A A, the last ones by - - - -jôr" and let us add member by member the results obtained, observing that, according to the equalities (23), dxp dxq We find the equality (66a) ^lAAf + .+ ^iAn+^lD2l + .Dan=o. (46bis) Ô3 i)3 03 to3 Dx "o. Equalities (63) and (66) give the equality (67) - A.+.f- AA,, ------------------------------------------------------------------------ Similarly, equations (63) and (66a) give equality dAt oA" -~Dx,-T---Uï,. G os os (07 ois) - = - - 1 C rl:i, < - T- As, -r- - \V.a OS OS The relations (6~) and (6" />-) can still be considered as generalizations of the relation of lieech. Let us apply them, in fact, to a system defined for a single normal variable except the temperature £7; they become 6 ) = D.1'. ,." C [the (68a) - = - e ~x If it is, in particular, a homogeneous fluid, we will have A A =- M AU = MDU Aoj, DA = - M DIT = MDj Aw, :.n Aa= A(u =_^£, D'l.= Dw =- ~n. The two equalities (f)8 ^> and (68 bis) reproduce, therefore also the Reech relation: (6z j C~ Finally, the equalities (63), (06) and (66 Ois) give the following Fégalilé, analogous to equality (63) 11 the A, ,)\n (63a) r -, -Da, +.+ -^Dj" xn (63 or! - = c ~~n.. (/z" -r^r D A -H H - DA,, ds bones Applied to a system that depends on a single normal variable except temperature, this equality becomes d\ C d:. D x c "dï" DÂ* ̃ ̃ ̃ i '<̃ ~êâ ------------------------------------------------------------------------ For a homogeneous fluid, it becomes, according to the relations written a moment ago, o Hfio. H?) r d& i dtUio, 5) D ()M(u,2?) ~i) ow(tl,âj dw dâ dIl We thus find the formula of Reech: (6'J r DQ ï Ds- 8. Formulas for slightly deformed elastic media. The elastic media slightly deformed from their initial state give rise to formulas which differ slightly from the previous ones and which it is necessary, for this reason, to develop in detail (-). (' ) The formulas developed in this paragraph are mainly due to Mr. VV. Voigt, who presented them in the following writings W. Voigt, Ueber adiabatische Elasticitàtsconstanten {Nachrichten von der Gesellschaft zu Gôctingen, 1888, no. i4, p. 36o). W. Voiot, Kompendium der theoretischen Physik, III" Theil, Wàrmelehre, I Kap, §§ 6, 7; Bd. I, Leipzig, 1893, pp. 52.'5-53(). W. Voiqt, Thermodynamik, Bd. I, §§ 145-147, Leipzig, 1903, pp. 329-337. When in its initial state the system is not subject to zero actions, several of Mr. Voigt's formulas require modifications which we have indicated in the following two Notes: On the two specific heats of a weakly deformed elastic medium; ------------------------------------------------------------------------ The state of such a body affected by an infinitesimally small homogeneous deformation is normally defined by the temperature S, by the three dilations z,, --̃>, £3 and by the three slippages y,, ̃ Its internal thermodynamic potential is given by the formula [Chap. X, equality (9)] (69) i = M/(e,, ei, e3, Yi! Yii Y3, S;, l, where M is the mass of the body. To maintain the body in equilibrium, it is necessary to apply a pressure at each point of its surface, the knowledge of which depends on the six quantities [Chap. X, equalities (14) dl d f dl N, f~ Nz=-~ N, (ciE l d~z OSa 70) T.=-P~, ~.f T.=- f¡=- T Ta=- 1 1 O'd 9- 0*(i '̃* a d'Y 3 p is the density of the body in the deformed state; this density not being a constant, the inverse variables relative to the normal variables e,, e2, £3, y,, y2, y3 are not simply proportional to the components N,, N2, JN3. T,, T2, T3 of the pressures. This remark, which we have already made (Chap. X, § 6, first example, p. 433), explains why the formulas we are going to develop differ slightly from those we have set forth in the preceding paragraphs. The entropy S of a system defined by normal variables is related to its internal thermodynamic potential by the formula [Chap. IX, equality (16)] s = °l. S =- -L- d;:f. F'(2f) toJj For the system we are dealing with at the moment, we will have (7.) S = M", ( 1 èf Let us raise the temperature by 33? without changing the value of any fundamental formula ( Comptes rendus, t. CXLIH. 1906, p. i35); On the two specific heats of a weakly deformed medium; various extensions of the formula of Reech (Comptes rendus; t. CXLIH, 190(1, p. 371 ). ------------------------------------------------------------------------ of the six normal variables ï,, î2, î:i, y,, v2, the system gives off a quantity of heat O = - \i f oSr. y ('so much the normal specific heat of the medium. The equalities {"j i ) and (~'Jt) allow to write F ~~p .J [-,)" I I I .J ll.J J K'~ K'i&i [-/si V{?j) orA Now let's raise the temperature of S.cJ by subjecting the six quantities N< Tt- to remain invariant; the three expansions and the three shifts will undergo increases âsj = -xto3. àt-2= 5t2 03, os-, - a;, oSi, j ov, = 3, Oj, O-'j = p.2 Oj, 073 = ï3 OJ. We can say that the three quantities a,, a2, arl are the three coefficients of expansion and that the three quantities [il,, jï2, ?:) S("n|- the three coefficients of slip, these six coefficients referring to the cases where the six quantities N, T; remain invariant during the experiment. The known formula Sp = - p('3î| H- ce, -i- îe3) becomes here 00 ==-p(OZt+Gê2+0Ë.3) (:75j 00= - p(a, -H aj-i- a3) S2r. If we want to express that in this case, the six quantities N/, Tj keep invariable values, we will find, by means of the equalities (;o), (74) and (7^)1 the relations iflf to[ o--f <>-i or else, according to equalities ( 83), (88; AN, Ae,-i- AN, Ae2-i- AN3 Ae3~2Ar, Ay, -r- 2 AT2 A- -h 2 AT3 A73 - AN, "Te, - AN2 ^î2 - AN3 di% - AT, rfv, - a AT2 rfy" - '>̃ ^> d->3 = F' F(3') -+-(N, dti-~ Xtdti^-Xîdzî~-iTl rfy,-H2T, rfv,-f-2T, rf-^Xa, -+- a, +", ) A& - (Nia, +M,i| -i-Nja;, -+-2T,3, - 2T,?, h-2T333 )(rfe,-t- (^6,-f- rfe3) AS- Let us now consider a second isothermal modification where the quantities £, y,- experience the same variations As, Ay,as in the isentropic modification previously considered in this new modification, the quantities Ni, T,- then grow by . AT, Ay, - 2 AT2 Ay2 a AT3 A'3 J F, ( Co ) (,~ )9- = 0 --=: ( ..1.9")2. Fiâ) f This equality (90) can still be put in a slightly different form. ------------------------------------------------------------------------ Let's suppose that we want to raise the temperature of the medium by ;3' without the quantities si, Y; changing their values; to the quantities Ni, T; we will have to impose variations that we can write (gl) !N, = A, o3-, ~Ti 13j Visihlernent then we have ) ~N,=(/-lT/==~T,-H/ and the equality (88) can be written (93) A,de,+Az~sz+A~3-F-2B,DY,+xBZ,3-1_,zg,1·~s =-p-Iâ..J. ~To the two fundamental relations (8~) and (g3j we will make correspond two other analogous relations. The relations (70), joined to the definitions (gi) of the quantities Ai, Bi, give without difficulty A, <)' f dsy f (94) dâ d:g ~=- ( 94 ) ( B, p d~ f BZ = Î dj. f B;, ;= d-f BI =- ei B2 = B:. =- x dâ dYt d:. clY2 z ,):7 d^~3 so that equality (77) can be written (g5) P-Y= i~fA,A~A~3-)--2B,p,-t-B~)-~B,). (g) r -1 p F' (1 ~r) (AltXl + A,tX2+ AatX3+ 'lBI ~I + ').B2~, + 2Ba~.3)' The equalities (94), joined to the six equalities ( ;6), allow to write ( <)6 ) A ~t -<- A: -)- A, As:, 9 B, Av, -}- a B, A-~ + -j: B., ~3 = Ptr- ar~~f '- xs f. P'-- Cij+~f 9:1" < d6: de, says (~ dy:¡ /4/ 'V df <~ df < f (x)(--+T-r-T-T'T-T') (X) d~y~~t-i- dE, Aex+ d_3 Jss~r dYt :1-+ dY2 1 dia 1 +( NI .11S1+ V=~ei+ N3 A~-t-aT, ~Y,-t-tTt& zT3 Wr3)(x,+az+x3). On the other hand, we can, in equalities (85) and, consequently, in equality (86), replace AN;, &T,, of" dY, by dN~, dT;. ~ls,; ~y;. ~N,, (N,rfe,rhNîrfeiTt-NîrfÊ3-i-2T,rfYi-r- -.3, .T,3, r-"Tj3, -+--tT333 )(4e,-t-As2-hA=3) (101; Ai dt, -+- A5 dti-i- A3 rfe3-i- 2 Bt ^1 -+̃ '^Bî d-;t-"B3 ^3 P~(.:II t'J, P F(,â ------------------------------------------------------------------------ The various formulas we have just mentioned provide two generalizations of the Ileech formula. A first generalization is obtained by dividing the equalities (ioi;and <'y3) member by member; in fact, we obtain the general equality ( l' AI di, -H A; di-2 A3 dz:s -+̃ ̃>. H, d-(, - ̃>. H2 d->, ̃>. H:i <̃ 3 Y A, Ae, ~4- A2 Ae" A3 Ae3 -̃- -2 B j A y -i- a H2 A-f-2 v. B3 A-) a which is due to Mr. W. Voigt (' J. For the use of this formula, one must remember that the same set of variations, arbitrary by the way, AN,, YEARS, AN3, AT,, AT2, AT, of the quantities N,, N2, N;1, T,,T2, T3 imposes on the normal variables £i" £-21 î3, y", f-2, y variations de,, dct, rfs3, d-ii, A-/2, S-{î if the modification produced is isothermal and the variations Ae,, As,, Ae3, A7,, Ay2, Av3 if this same modification is isentropic. It is easy to see that the formula (102) gives well, when applied to a fluid, the Reech formula. At temperature 2f, such a fluid, of density p, is maintained in equilibrium by a normal and uniform pressure II, so that we have (io3) N, = iS2=N3=n, T, = Tî=T3=o. If we bring it, without deformation, to the temperature (3 4- 33), it is necessary, to maintain it in equilibrium, to subject it to a new pressure, normal and uniform, 11 -+- P52r, so that we have A| = Ag = A3 =P, Bi=H2=Ba=o, and the equality (io") becomes (104)' r = ,/", h-"&"/",_ V As,-f-AE,-HAE3 (') W. Voigt, Thermodynamik, igo3, equality ( ny), p. 333. ------------------------------------------------------------------------ In an isothermal change where the pressure increases by MI, the specific volume increases by - m ( de.t -+̃ dzt -r~ dt3 ) = - -p- > \ta = o>l Asf ~i- Ae.~f- iîrt) = - ?p' so that equality (104) gives the formula of Rcech: (62) t' Do Dividing the equations (87) and (100) member by member, we get r obtains a new expression of the ratio r but this expression, which it is useless to write here, is much more complicated, in general, than the expression (102). It becomes much simpler in a particular case, that where, in the state s,, e2, îj, y, y2, y3, g, the medium is maintained in equilibrium by a normal and uniform pressure II one can then write the equalities (io3), so that the equalities (87) and (100) become respectively (,o5) p!r:A2f=:aIA.N, + aJAiS1-i-aïAN,-"-apliT,-+-ap,AT,+ 2?,AT,, (Io6): pEl^YA2-==a1rfN1-t-aîrfN,+ ot.N3-2?,rfT1-r-'2riic/T!,+ u^rfTî. Dividing them member by member, we obtain the equality r a, ANj-f-ajANî-f-a, At\3^-2^ AT. -f- ag, AT,-f- afc AT, 3 ( °~> > v ", rfN + a, rfNt -h a, dN, -f- a ?, dTt -t- 2 p, rfT, -+- ̃>. {J3 rfT, which is analogous to equality (10a). It should be recalled here that, in order to impose on the normal variables £,, s2, e,, y,, y2, va, the variations, arbitrary moreover, Aï,, As,, Ae:), A-fi, A-1, Av3, it is necessary, with the quantities N,, N2, N:l, T, T2, T3, to impose variations rfN,. rfN,, rfN3, dT,, rfT,, rfTj, ------------------------------------------------------------------------ if the modification is isothermal, and variations AN, AN, AN3, AT, ATS, AT, if the modification is isentropic. The formulas (io5), (106), (107) were given by Mr. W. Voigt (' ) but they were given as true in a general way, which is not (-). Formula (107) is an extension of Reecli's formula. Let us apply it, in fact, to a fluid; an increase Aoj in the specific volume corresponds to an increase in pressure dïi = - D^ A", if the modification is isotropic, and to an increase in pressure Afl = - D^Ao) if the modification is isentropic. dNl=dXi = dN3=dt\ - DçtAoj, rfT, = dT,= rfT:i = o, AN, = A!\2 = AN, = AH = - Do Aoj, AT,= ATj = AT, = o, and the equality (io'j) gives back the Reech formula: ('6.,) I Ds. Application of the previous formulas to slightly deformed isotropic bodies. - We shall apply some of the formulas just established to isotropic bodies with little deformation. If we assume, as we have already done [Chap. X, equality ;(io4)], (108) Uj2r.)+.\(°r)= L(~). the equilibrium conditions of these bodies can be written ̃ | N/=llo(2r):-L(3r)(el + £I-1-êï)-2M(Sf)e, 1.N¡= HoC'J. M 1.(3") (SI + S2 + Ea) "1. M(S' )Si, (i°9) JT, = -,M(Sm, as shown by the equalities (28) in Chapter X. (') W. Voigt, Thermodynamik (iÇ)o3) equality (190). p. 33o; equality (196) and equality, not numbered, which immediately precedes it, p. 33? ('.) We have indicated this correction in the Comptes rendus de l'Académie des Sciences, l. CXLIH, 1906, pp. 371 seq. ------------------------------------------------------------------------ From these equalities (109) we derive the relations O 1 0 ) ( 3N,= ^i!^i32f-L(27)(^, f J + ^2-H5e3) -aM(3)8e,, Ei in which infinitesimally small quantities in e,, y/, have been neglected in front of finite quantities. In these equalities (110) we can make oti - o, 8-(i = o, ôN/ = A,- 53f, ST,- = B, 82r. It is derived from this (a, = a,= a3=^, (III) ( B, = B2=B3=o. <£? We can also, in the equalities (t 10), make oe,= a,-3âÉ, o'(, ̃- JJ/ 827, oN,= o, ST,- = o. It is derived from this ( a, = ai= a3== J rfn"^), (11a) - 22 3L(Sr)-f-aM(2F) rfSr 0. With these equalities (1 1 1) and (1 12), the equality (g5) gives the following expression of the difference between the two specified heats of the medium 0.3) r y- Ft&) r^)la If the pressure n0 (S) is zero or positive, and if the isotropic body is subject to the law of isothermal displacement of the equi'free, the inequality ( 1 1 4 > > L(2r) + 2.VI(â) > o is verified [see Voids, Inequalities (9) and the following considerations], so that the specific heat F exceeds the specific heat y. If no(2') was negative, it could be that F was less than y, although the body would still verify the law of iso- displacement. ------------------------------------------------------------------------ However, there would be no contradiction with the general theorem demonstrated in paragraph o (p. 4/9) > indeed, if My is always a normal heat capacity, MT is not here an inverse heat capacity. If we neglect the infinitesimal terms of the order of £,, v,- before your finite terms, the equalities (109) reduce to lm5) i a,2-a3=tt"(°~), <"5) r,= ~== 1,= o. Equalities (87), (1 12) and (1 1 5) then give ("6 p-~rA.7=.---------(.t-). Ft.?) 3 L(âj-a 11(:.) /y'' This equality would be immediately derived from the first three equalities (1 12). It transforms the equalities (1 16) and (1 \~) into the following equalities: (119) p Jiy, + ~y+ ~N3 "." ,%f|rg.S)" (120) p p 5 ï & = -3 -a(r)(As,-i- A6,-t- Ae3)- These formulas are analogous to formulas (og bis) and ( 09 ) for a homogeneous fluid. The normal specific heat is positive according to the postulate ------------------------------------------------------------------------ of Helmholtz; if the medium is subject to the law of equilibrium displacement and if, in addition, IT0(2?) is zero or positive, the sum (3L + 2 M) is positive; 1' is, at the same time, greater than y and, therefore, positive; we can therefore, in the case where Ilo(2r) is zero or positive, state the following theorem An isotropic body with a positive cubic coefficient of expansion heats up in an isen.tr o pique deformation when this deformation corresponds to an increase in the average value of the three quantities N,, N2, N, or, what amounts to the same thing, to a decrease in the volume of the body. The opposite occurs if the cubic expansion coefficient of the body is negative. Equality (119) is, for isotropic bodies, the analogue of equality (39 bis) for fluids. Equalities (1 19) and (120) still give this one () T_ 3 ANiH-AXt-+-ANj, l Y~ 3L(2?) + aM(3r) Àe,-t-Ac2-4-A23 Let's transform this last equality slightly. Consider any isothermal change in the isotropic medium; it corresponds to a deformation Se, Sy,- and to variations SN, 8T,- in the components of the pressures. Equalities (1 10 ), where 33 = o, give (~az ) oNi+.SNt4-8N, 3 1.(5) + M (5) ( I Vi. ) - 0£ ̃ -T7 0S2 5 -+- OE3 - - i - y so that equality (121) becomes (,XS £ ANi-^ANt-f-ANs 3N|-4-8Nt-4-8N3 - Y Aôj -+- Aej -+- Ae3 5ei-+- Sej-i- 8s3 3 This formula can be considered as an extension of Reech's formula to isotropic media; it differs from formulas such as (102) and ("07) in that it introduces an isothermal and an isentropic modification that are completely independent of each other. The formulas (122) and (23) have various corollaries. This one, in the first place, follows from equality (i 21): ------------------------------------------------------------------------ If an isotropic body verifies the law of iso thermal displacement of (equilibrium and if the quantity II0 (3) is positive or zero, any isothermal increase in the average value of the quantities N(, N2, IN 3 leads to a decrease in volume of the system. Between the decrease of the volume and the average increase of the quantities N,, JNa, N;t, there is a ratio which depends only on the temperature and which, naturally, is equal to the coefficient of compressibililé isolhermique, given by the formula [Chap. X, equality (3ij] ] (), c.. 3 To this corollary, the equality (ia3) allows us to add these: Any isentropic increase in the average value of the quantities N,, N2, N3 also leads to a decrease in the volume of the system. Between the decrease of volume and the average increase of the quantities N,,N2, N8, there is a ratio which depends on the temperature alone; this ratio is the isentropic compressibility coefficient jï'(£?). The formula (i23j can then be written The ratio of the specific heat Y to the normal specific heat y is equal to the ratio of the isothermal compressibility coefficient to the isentropic compressibility coefficient. The first coefficient always exceeds the second in the case where no(2r) is zero or positive. Equality (124) could still be derived immediately from either formula (102) or formula (107). According to the equalities (1 10), an isothermal deformation of the body is governed by the equations ~12J) ( SN, = - L(2r)(8e,3£2-h8E3) - îM(S)o6, (125) 1 aN, = L(;¡) (a) + ÕS2 + a "3) -'}. M(;¡) Õs¡, (I25) j ST^-aSHSF)* ------------------------------------------------------------------------ According to the same equalities (i 10), joined to equality (i 17), an isentropic deformation of the same body is governed by the equations ( A\ = - L'(27WA=,-j- Aï,4- As,) - y.M(rr) As, ( < '-"6 ) I AT, = - ïJKSi A y, where we put (.7) i^) = u^±;^r^T. l'y f' 1 ~.? J One thus passes from the formulas relating to an isothermal deformation a" an isotropic body to the formulas relating to the corresponding isentropic deformation by substituting the coefficient L'(3r) for the coefficient L(2r); by virtue of Helmhollz's postulate, this last coefficient is always higher than the first. 9. Displacement of the equilibrium by variation of the temperature ( ). The law of isothermal displacement of equilibrium leads, as a corollary, to the law of isentropic displacement of equilibrium, from which follow, in turn, many consequences. This law of isothermal displacement of equilibrium leads to a second corollary, the law of displacement of equilibrium by variation of temperature; it is this corollary that we will now study. A system, defined by the temperature 2? and by the normal variables a(, a.2, a, is maintained in equilibrium by the external actions A,, A.2, A, The conditions of this equilibrium are the equations ')̃$ d$ drf (l) Al=_, A,= A. = à21 rh! dx,~ (') P. Duhkm, Sur le déplacement de l'équilibre (Aimâtes de la Faculté des Sciences de Toulouse, t. IV, N, 1S90): Commentary on the principles of Thermodynamics; third part Les équations générales de la Thermodynamique, Chap. IVr, 3 [Journal de Mathématiques pures et appliquées. l\" série, l. X, 1894). ------------------------------------------------------------------------ Let us imagine a displacement of equilibrium during which the external actions are maintained invariable, while the temperature varies of f/3; the values of the variables a,, y. experience changes r/a(, c/a2, du,. The equalities ( i) allow to write o &i dx, à%\n 0%n ilx, oz (/ o to"-i &i to "s o-- r- rfai -+-(- --5- aa,, H rfJ. t'21 dx,L ()22 Il ().J~X,, By multiplying these equalities by rfai, d% respectively and adding them member by member, we find (d~ dx,+.+ (2) cz~ )2 d2j t dx"' ( -- 0~ -+-- <~[,, ) (2) ( -~--- o. This inequality constitutes the nu dei-i.acf.mkkt law of the kqi;iLiBnE BY VARIATION OF THE tempér atuke, law which can be stated as follows One makes vary the temperature a" a system by maintaining invariable the external actions which maintain it in equilibrium this system experiences a certain change of state; if this same change of state occurred while the temperature would be maintained invariable, it would correspond to a certain release of heat; this release of heat would ------------------------------------------------------------------------ always of opposite sign to the temperature variation first considered. This rule can be demonstrated in another way. Let us take the equilibrium conditions in the form x, = F, (A,,,,S), ^7) ̃;> ~t= F~(A), A, We will have dx, dFI" dx" d F,r de. di\ = -- <ÏS din = -r- O3, bone bone so that equality (128) becomes 1- (~ i d~ dF, ~~F, p, ~+-+- p" da" --- -- -t-- -<--- -eT ) \t(S)\0' and we find the law of displacement of equilibrium by temperature variation. The formulas established in the above provide examples of this rule without difficulty. First example. Homogeneous fluid. - The fluid of mass M, of specific volume tu, brought to the temperature 2f, is in equilibrium under the pressure M. This pressure being maintained invariable, we increase the temperature by (/Sï; the specific volume increases by dut. The equilibrium condition of the fluid being [Chap. X, equality (66)] 11 n = <>?(<̃>,£?) " yes we have o**(w, s) tf>tp(to,2o - dio -f ii - - as = o. Otu* (0, dm os o. If the specific volume of rfco was varied while keeping the temperature unchanged, the fluid would release an amount ------------------------------------------------------------------------ of heat [Chap. Vil, equality ('^̃>)\ O = M Ht", S) of the. Moreover, the heat of expansion is given by the formula [Chap. X, equality (77;] J t't:rt ~I=v(cu,~). I(w:1) -- -'-~-' 1"(0: )0: Therefore, we find ~u2 ~(o~, â, I2 (l3l) Q =- A1 1~ r j Uw d.°. J' ch. 1' fS; (/"?(<">, S?) dw2 Now, for the fluid to follow the law of isothermal displacement of the equilibrium, it is necessary and sufficient that we have <"> ^ë^><>- - UU1Y Q is thus of opposite sign to tfô, in accordance with the law of displacement of the equilibrium by variation of the temperature. Moreover, by virtue of equality (P), Tegality ( 1 3 1) can still be written (i3a) -) Q=-M(r - -;)<&. We can see that the law of the displacement of the equilibrium by variation of temperature, according to which Q is of opposite sign to cVb, can, here, be stated in the following way: If a homogeneous fluid expands by a rise in temperature without a change in pressure, the heat of expansion of this fluid is positive; it is negative if the fluid contracts by a rise in temperature without a change in pressure Second kxkmi-ik. - Elastic body with little deformation. - If we maintain constant the components Ni, T,- of the pressures, while the temperature increases of t/S7, the normal variables Si, y; increase of ~e,==! f/Y,=~;<~?. ------------------------------------------------------------------------ According to equality (9) of the Chapter the internal potential of the body of mass M is .7 - M /(- £2, =3, ̃ ̃ y,. "S), so that its entropy is [equalities 1 71) and (7:?) s = V. V'Cs) ifs If, therefore, without varying the temperature, we imposed on the quantities s,, y,- the increases we have just indicated, the elastic body would release a quantity of heat O vt F<5) ( ')lf Vf <>'1 f ~="F~)( C ~7~ q==HV(s-)i^*1 + ^a2+^a3 F'(S') J~, uâ ~eQ {}?;¡ d~, d~ !)'- f !)Z.Î ~l~ j d E.f ~3 \1 CG:J. à'(t OS 0f2 OS C/V.J ,is l According to the equality (77), this quantity of heat can still be written (i33) Q= - M(r - Y)d2r. If the body is isotropic and not very deformed, and if the quantity no(2r) is zero or positive, then r is greater than y (p. 5o6); O is then, as we had announced, of opposite sign to a "b. This second application corresponds to a problem which is, in reality, a little different from the one we have treated in this paragraph, because keeping the quantities JN(-, T, constant is not the same as keeping the external actions constant; this is why, if ÎI0(3) were negative, we might obtain corollaries which do not conform to the general theorem demonstrated at the beginning of this paragraph. 10. Corollaries of the law of equilibrium displacement, which are useful for Chemical Mechanics. We will consider a system which verifies the law of isothermal displacement of equilibrium, and we will, for such a system, establish two corollaries which will have, in the study of Chemical Mechanics, an extreme importance. ------------------------------------------------------------------------ to,r.wzz:a r:ozcoz.z.Arztr. We shall suppose that the system itself is normally defined by the temperature 3' and byp -}- variables which we shall divide into two groups on the one hand, the variables 7." x~; on the other hand, the variables j:tt) we will designate 1>ar A,, Ap, B,, .13v the corresponding external actions. The equilibrium equations will be d~ rJ~ (i 3 É) d~ 1 = A,, ~;=~" (135) 13,, d~, = B~. To say that the system verifies your law of isothermal displacement of equilibrium is to say that the form, quadratic in Xi, XF" Y,; Y9, d~ a~ v~ d~ y 1f2~ ~3b) --Xt-)-)---X.<3-Yt-)---r-Y~ lt3G) cI-XI+"'+-¡j X,,+:;r¡-Yt+.r.¡-Y'1 is a positive definite form. Let us imagine that the actions B,, Bq are kept invariant and that the temperature varies by let us imagine, at the same time, that we impose on the variables a,, Mp the condition of experiencing no change in value; the variables p,, ~39 q and the actions AI, Ap will experience variations, respectively proportional to of the phase, it is enough to maintain invariable its total volume Mto. If therefore, under constant volume, we raise the temperature of rfSf, the chemical composition varies of dx and the mass of the compound that the phase contains grows of dm. On the other hand, in a displacement of equilibrium at the beginning of which the chemical action is null, and where x varies of dx without that neither the specific volume w nor the temperature 2" experience any variation, the phase releases a quantity of heat [Chap. VII, equality (29)1 Q = J,(w, x,) dm, Q = A ( tu ar, 5 ) dm X(u>, x, 2r) being the heat of formation under constant volume of the compound that the phase contains. The inequality (138) thus becomes (i3g; ).fw,r,&)rfmo& and the actions A,, Ap experience any variations rfSr, dA, dkp; let us denote by dI d3q ~(~~-r- ~2) + p Á~¡ +.+ d~~ Á~q On the other hand, let us multiply the equalities(142) by clx" respectively ------------------------------------------------------------------------ dv.R, da. and add the results member by member: we find equally (,~) .= _d~' %i r ` dŸ û~~F \~) ~Jx~, Ja p ~7 J~ ûâ., J'~) (X) 1\Jx~ dx, dx" ~x~')' Finally, multiplying respectively the equalities (142) by A~ Oxp, and adding the results member by member, we find (145) 0 ~û~ A,(, d~F ,z) (~5) \,dx 't'rrlx Jx~ -(~----(~). w Add member by member the equalities (143) and ()Z)5). and from the resulting equality subtract member by member the equality (144); we find the equality (t.j6) dA, Axi -1-4- dA,, -\21, ''7" .lxt+. Jxp-f-dQtèdT~y- `_d(i9dd:~JN~I~ da d:~ dx d: d~ r dS' d~q o~~ ài A ri~ d~ .Q û~F = ( dx ~- Jx~'1 d Pt d%l (j 2 p 7p-l ~'I'l q This equality (146) can be written a little differently; the displacement of the equilar(,'a" ,xp; A~ accomplished constant temperature, corresponds to a release of heat Q, and the equalities (1 Q), and (1!5) of Chapter I!1 allow to write Fl dz .,1 dR F(5-) `Jx,-i-1- `9x~4- ~)_1 ) h~(â)Cdx,d.~ nxprL.. ,W dG,d.r Equality (1 li6) thus becomes (147) ~A,A~Q~ ~F(:?,) ,da~ Jx,+.. Jx,, Jx~~ ~~t J~ji r)~ J~q') This equality (147) expresses the corollary that we wanted to obtain. The second Member :de4euc,equality. is a partial case of your ------------------------------------------------------------------------ quadratic form (i3G); it is therefore always positive, so that this equality (1/17) provides the inequality 048,1 rfA, Aaj-i-1- clA,, Aa,, - O cl3 > o. If one remembers that the quantities d\p, (fb are arbitrary, one can, from this inequality, derive various particular theorems; we shall highlight two of them. 1" Suppose rfAi = 0, d\ = o. We obtain the inequality Q dis < o, which is equivalent to the following proposition We raise the temperature by a positive quantity from to while maintaining invariable all the external actions Ai, Ap, Bt B? which act on the system the system experiences a displacement of equilibrium where the variables fit, fi,, vary by Aj3,, A[3?. Let us consider, on the other hand, an equilibrium displacement where ${, Qq experience the same variations AJ3( 1 Aj3? without either the temperature 2f, or the external actions A, j Ap relative to the variables a, a.p experiencing any variation; this last equilibrium displacement is certainly accompanied by a heat absorption. 20 Suppose c?3> = o; we obtain the inequality ('49) rfAi Aa,-H.rfApAap>o, which gives rise to the following theorem We consider, for the same system, two isothermal displacements of equilibrium. At the first displacement, the external actions B,, B? are kept constant, while the external actions At, K have arbitrary variations d\.t dAp. The normal variables ",, xp to which refer ------------------------------------------------------------------------ these last actions experience variations which are not taken care of; the normal variables [Ï,, 3V to which the first actions refer grow by A[ït, Afi^, At the second, displacement, the variables jî,,'j(/ experience these same increases A[3, A3y, while the actions Ai), remain constant; the actions 13, By experience variations which are not taken care of, while the variables a,, a.p grow by da,, ij.r The disturbing actions dA,, dAp relative to the first equilibrium displacement and the variations Aa,, Ay.p relative to the second 'verify V inequality ( 1 4 9 ) - Let us apply these various theorems to an example. Exemplk. - Phase of Chemical Mechanics. - Let us take a chemical phase of mass M, normally defined by its temperature 2r, its specific volume oj and its chemical composition x; m will play the role of variable a and x the role of variable [3; the action A. will be -Mil, it being the pressure that the phase bears; as Chemical Mechanics dictates, the action relative to the variable x, which plays the role of action B, will be zero in the state of equilibrium considered. The product (dA, Aa, -f- -f- d\p ia^) will reduce here to MafriAw. On the other hand, if x increases by A#, the mass m of the compound contained in the phase increases by Am; if this increase takes place without variation of temperature and pressure, from a state of equilibrium where the chemical action is null, the phase gives off a quantity of heat Q which has the value [Chap. VII, equality (67)] Q = \a ri, x, "b) A/h, L(JT, x, ?}) being, under the constant pressure H. the heat of formation of the compound that the phase contains. The inequality (148) becomes here (oo) M~HAM - L [ 4. The internal thermodynamic potential as a characteristic function 4'~ Example Homogeneous fluid. 4~ 5. System related to non-normal variables. \2:) 6. System related to inverse variables of normal variables. First example Isotropic body with very little deformation. 4.13 Second example Homogeneous fluid. 4'~f) 7. System related to mixed variables. 44'~ Example Chemical phase. Ilt.-. 8. Entropy as an independent variable. /)5o CHAPTER XI. THE MPLACEMEKT OF THE 1 EQUILIBITY /j37 1. Statement of the law of isothermal displacement of equilibrium. 4''7 2. Various ways of expressing that a normally defined system verifies the law of isothermal displacement of equilibrium. 460 First example: Homogeneous fluid. 461 1 Second example Very slightly deformed isotropic body. /)6' 3. The isothermal displacement law of Equilibrium can be true or false depending on whether the system has been given a normal or another definition. 467 4. Expression of the law of displacement of equilibrium by means of the inverse variables !t73 5. Difference between the inverse heat capacity and the normal heat capacity 4';C 6. The isentropic equilibrium displacement. G`3r 7. Ratio of the inverse heat capacity to the normal heat capacity 1. Various generalizations of Reech's formula. 188 8. Formulas relating to slightly deformed elastic media 4s4 Application of the preceding formulas to slightly deformed isotropic bodies 9. Displacement of the equilibrium by variation of the temperature. 5"5 9. Displacement of the equilibrium by variation of the temperature. 510 First example Homogeneous fluid. 5i2 Second example Elastic body with little deformation.) :rr3 10. Corollaries of the law of equilibrium displacement that are useful for Chemical Mechanics. '"4 !l ~e/?tt'e/'co/-o~aty'e. 5,5 Example Phase of Chemical Mechanics 51j Second coro~t'e. 5t~ i ~a;eM/)/e.'Phasede!aMécaniquecbim1 ------------------------------------------------------------------------ II apr m e 1 MARCH 1982 If3 .IHW IL. Pi m ------------------------------------------------------------------------