fyxmll ^mvmxi^ Jibat^g THE GIFT OF ft-^ori-i( ^^ 7583 M 'M Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924005649185 COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK PUBLICATION NUMBER THREE OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH ESTABLISHED DECEMBER 17th, 1904 EIGHT LECTURES ON THEORETICAL PHYSICS DELIVERED AT COLUMBIA UNIVERSITY IN 1909 BT MAX PLANCK PROPEBBOB OF THBOBETICAL PHYSICS IN THE UNIVEBSITY OF BERUN LBCrnBEB IN MATHEMATICAL PHYSICS IN COLUMBIA DNIVEBSITY FOB 1909 TRANSLATED BY A. P. WILLS / PBOFEBBOR OF MATHEMATICAL PHYSICS IN COLtTUBIA UNTVERSITT NEW YORK COLUMBIA UNIVERSITY PRESS 1915 COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK PUBLICATION NUMBER THREE OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH ESTABLISHED DECEMBER 17th, 1904 EIGHT LECTURES ON THEORETICAL PHYSICS DELIVERED AT COLUMBIA UNIVERSITY IN 1909 BY MAX PLANCK PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN LECTURER IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1909 TRANSLATED BY A. P. WILLS PROFESSOR OF MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY NEW YORK COLUMBIA UNIVERSITY PRESS 1915 Translated and Published by Arrangement with S. HiRZEL, Leipzig, owner op the original copyright Copyright 1915 by Columbia University Press PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. 1915 On the seventeenth day of December, nineteen hundred and four, Edward Dean Adams, of New York, established in Columbia University "The Ernest Kempton Adams Fund for Physical Research" as a memorial to his son, Ernest Kempton Adams, who received the degrees of Electrical Engineering in 1897 and Master of Arts in 1898, and who devoted his life to scientific research. The income of this fund is, by the terms of the deed of gift, to be devoted to the maintenance of a research fellowship and to the pubUcation and distribution of the results of scien- tific research on the part of the fellow. A generous interpretation of the terms of the deed on the part of Mr. Adams and of the Trustees of the University has made it possible to issue these lectures as a publication of the Ernest Kempton Adams Fund. Publications of the Ernest Kempton Adams Fund for Physical Research Number One. Fields of Force. By Vilhelm Fkiman Korbn Bjdrknes, Professor of Physics in the University of Stockholm. A course of lectures delivered at Columbia Univer- sity, 1905-6. Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on application of hydrodynamics to meteorology. 160 pp. Number Two. The Theory of Electrons and its Application to the Phenomena of Light and Radiant Heat. By H. A. Lokentz, Professor of Physics in the University of Leyden. A course of lectures delivered at Columbia University, 1906-7. With added notes. 332 pp. Edition exhausted. Published in another edition by Teubner. Number Three. Eight Lectures on Theoretical Physics. By Max Planck, Professor of Theoretical Physios in the University of Berlin. A course of lectures delivered at Columbia University in 1909, translated by A. P. Wills, Professor of Mathematical Physics in Columbia University. Introduction: Reversibility and irreversibility. Thermodynamic equilibrium in dilute solutions. Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory. Statistical theory. Principle of 1 east work. Principle of relativity. 130 pp. Number Four. Graphical Methods. By C. Runge, Professor of Applied Mathematics in the University of Gottingen. A course of lectures delivered at Columbia University, 1909-10. Graphical calculation. The graphical representation of functions of one or more independent variables. The graphical methods of the differential and integral ca'culus. 148 pp. Number Five. Four Lectures on Mathematics. By J. Hadamard, Member of the Institute, Professor in the College de France and in the Ecole Polytechnique. A course of lectures delivered at Columbia University in 1911. Linear partial differential equations and boundary conditions. Contemporary researches in differen- tial and integral equations. Analysis situs. Elementary solutions of partial differential equations and Green's functions. 53 pp. Number Six. Researches in Physical Optics, Part I, with especial reference to the radiation ofelectrons. By R. W. Wood, AdamsResearchFellow, 1913,ProfessorofExperimental Physics in the Johns Hopkins University. 134 pp. With 10 plates. Edition exhausted. Number Seven. Neuere Probleme der theoretischen Physik. By W. Wien, Professor of Physics in the University of Wurzburg. A course of six lectures delivered at Columbia University in 1913. Introduction: Derivation of the radiation equation. ' Specific heat theory of Debye. Newer radiation theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein fluctuations. Theory of Rontgen rays. Method of determining wave length. Photo-electric effect and emission of light by canal ray particles. 76 pp. These publications are distributed under the Adams Fund to many libraries and to a limited number of individuals, but may also be bought at cost from the Columbia University Press. PREFACE TO ORIGINAL EDITION. The present book has for its object the presentation of the lectures which I delivered as foreign lecturer at Columbia Uni- versity in the spring of the present year under the title : "The Present System of Theoretical Physics." The points of view which influenced me in the selection and treatment of the material are given at the beginning of the first lecture. Essen- tially, they represent the extension of a theoretical physical scheme, the fundamental elements of which I developed in an address at Ley den entitled: "The Unity of the Physical Concept of the Universe." Therefore I regard it as advantageous to consider again some of the topics of that lecture. The presen- tation will not and can not, of course, claim to cover exhaus- tively in all directions the principles of theoretical physics. The Author. Berlin, 1909. TRANSLATOR'S PREFACE. At the request of the Adams Fund Advisory Committee, and with the consent of the author, the following translation of Pro- fessor Planck's Columbia Lectures was undertaken. It is hoped that the translation will be of service to many of those inter- ested in the development of theoretical physics who, in spite of the inevitable loss, prefer a translated text in English to an original text in German. Since the time of the publication of the original text, some of the subjects treated, particularly that of heat radiation, have received much attention, with the result that some of the points of view taken at that time have under- gone considerable modifications. The author considers it de- sirable, however, to have the translation conform to the original text, since the nature and extent of these modifications can best be appreciated by reference to the recent literature relat- ing to the matters in question. A. P. Wills. CONTENTS. First Lecture. PAGa Introduction. Reversibility and Irreversibility .... 1 Second Lecture. Thermodynamic States of Equilibrium in Dilute Solutions. 21 Third Lecture. Atomic Theory of Matter „ 41 Fourth Lecture. Equation of State for a Monatomic Gas 58 Fifth Lecture. Heat Radiation. Electrodynamic Theory 70 Sixth Lecture. Heat Radiation. Statistical Theory 87 Seventh Lecture. General Dynamics. Principle of Least Action 97 Eighth Lecture. General Dynamics. Principle of Relativity 112 FIRST LECTURE. Introduction: Reversibility and Irreversibility. Colleagues, ladies and gentlemen: The cordial invitation, which the President of Columbia University extended to me to deliver at this prominent center of American science some lectures in the domain of theoretical physics, has inspired in me a sense of the high honor and distinction thus conferred upon me and, in no less degree, a consciousness of the special obligations which, through its acceptance, would be imposed upon me. If I am to count upon meeting in some measure your just expectations, I can succeed only through directing your attention to the branches of my science with which I myself have been specially and deeply concerned, thus exposing myself to the danger that my report in certain respects shall thereby have somewhat too subjective a coloring. From those points of view which appear to me the most striking, it is my desire to depict for you in these lectures the present status of the system of theoretical physics. I do not say: the present status of theoretical physics; for to cover this far broader subject, even approximately, the number of lecture hours at my disposal would by no means suffice. Time limita- tions forbid the extensive consideration of the details of this great field of learning; but it will be quite possible to develop for you, in bold outline, a representation of the system as a whole, that Is, to give a sketch of the fundamental laws which rule in the physics of today, of the most important hypotheses employed, and of the great ideas which have recently forced themselves into the subject. I will often gladly endeavor to go into details, but not in the sense of a thorough treatment of the subject, and only with the object of making the general laws more clear, through appro- 1 I FIRST LECTURE. priate specially chosen examples. I shall select these examples from the most varied branches of physics. If we wish to obtain a correct understanding of the achieve- ments of theoretical physics, we must guard in equal measure against the mistake of overestimating these achievements, and on the other hand, against the corresponding mistake of under- estimating them. That the second mistake is actually often made, is shown by the circumstance that quite recently voices have been loudly raised maintaining the bankruptcy and, debacle of the whole of natural science. But I think such assertions may easily be refuted by reference to the simple fact that with each decade the number and the significance of the means increase, whereby mankind learns directly through the aid of theoretical physics to make nature useful for its own purposes. The technology of today would be impossible without the aid of theoretical physics. The development of the whole of electro-technics from galvanoplasty to wireless telegraphy is a striking proof of this, not to mention aerial navigation. On the other hand, the mistake of overestimating the achievements of theoretical physics appears to me to be much more dangerous, and this danger is particularly threatened by those who have penetrated comparatively little into the heart of the subject. They maintain that some time, through a proper improvement of our science, it will be possible, not only to represent com- pletely through physical formulae the inner constitution of the atoms, but also the laws of mental life. I think that there is nothing in the world entitling us to the one or the other of these expectations. On the other hand, T believe that there is much which directly opposes them. Let us endeavor then to follow the middle course and not to deviate appreciably toward the one side or the other. When we seek for a solid immovable foundation which is able to carry the whole structure of theoretical physics, we meet with the questions: What lies at the bottom of physics? What is the material with which it operates? Fortunately, there is introduction: reversibility and irreversibility. 3 a complete answer to this question. The material with which theoretical physics operates is measurements, and mathematics is the chief tool with which this material is worked. All physical ideas depend upon measurements, more or less exactly carried out, and each physical definition, each physical law, possesses a more definite significance the nearer it can be brought into accord with the results of measurements. Now measurements are made with the aid of the senses; before all with that of sight, with hearing and with feeling. Thus far, one can say that the origin and the foundation of all physical research are seated in our sense perceptions. Through sense perceptions only do we experience anything of nature; they are the highest court of appeal in questions under dispute. This view is completely confirmed by a glance at the historical development of physical science. Physics grows upon the ground of sensations. The first physical ideas derived were from the individual perceptions of man, and, accordingly, physics was subdivided into: physics of the eye (optics), physics of the ear (acoustics), and physics of heat sensation (theory of heat). It may well be said that so far as there was a domain of sense, so far extended originally the domain of physics. Therefore it appears that in the be- ginning the division of physics was based upon the peculiarities of man. It possessed, in short, an anthropomorphic character. This appears also, in that physical research, when not occupied with special sense perceptions, is concerned with practical life, and particularly with the practical needs of men. Thus, the art of geodesy led to geometry, the study of machinery to me- chanics, and the conclusion lies near that physics in the last analysis had only to do with the sense perceptions and needs of mankind. In accordance with this view, the sense perceptions are the essential elements of the world ; to construct an object as opposed to sense perceptions is more or less an arbitrary matter of will. In fact, when I speak of a tree, I really mean only a complex of sense perceptions: I can see it, I can hear the rustling of its 4 FIRST LECTURE. branches, I can smell its fragrance, I experience pain if I knock my head against it, but disregarding all of these sensations, there remains nothing to be made the object of a measurement, wherewith, therefore, natural science can occupy itself. This is certainly true. In accordance with this view, the problem of physics consists only in the relating of sense perceptions, in ac- cordance with experience, to fixed laws; or, as one may express it, in the greatest possible economic accommodation of our ideas to our sensations, an operation which we undertake solely because it is of use to us in the general battle of existence. All this appears extraordinarily simple and clear and, in ac- cordance with it, the fact may readily be explained that this positivist view is quite widely spread in scientific circles today. It permits, so far as it is limited to the standpoint here depicted (not always done consistently by the exponents of positivism), no hypothesis, no metaphysics; all is clear and plain. I will go still further; this conception never leads to an actual contradiction. I may even say, it can lead to no contra- diction. But, ladies and gentlemen, this view has never con- tributed to any advance in physics. If physics is to advance, in a certain sense its problem must be stated in quite the inverse way, on account of the fact that this conception is inadequate and at bottom possesses only a formal meaning. The proof of the correctness of this assertion is to be found directly from a consideration of the process of development which theoretical physics has actually undergone, and which one certainly cannot fail to designate as essential. Let us compare the system of physics of today with the earlier and more primitive system which I have depicted above. At the first glance we encounter the most striking difference of all, that in the present system, as well in the division of the various physical domains as in all physical definitions, the historical element plays a much smaller role than in the earlier system. While originally, as I have shown above, the fundamental ideas of physics were taken from the specific sense perceptions of man. INTRODUCTION: EEVERSIBILITY AND IRREVERSIBILITY. 6 the latter are today in large measure excluded from physical acoustics, optics, and the theory of heat. The physical defi- nitions of tone, color, and of temperature are today in no wise derived from perception through the corresponding senses; but tone and color are defined through a vibration number or wave length, and the temperature through the volume change of a thermometric substance, or through a temperature scale based on the second law of thermodynamics; but heat sensation is in no wise mentioned in connection with the temperature. With the idea of force it has not been otherwise. Without doubt, the word force originally meant bodily force, correspond- ing to the circumstance that the oldest tools, the ax, hammer, and mallet, were swung by man's hands, and that the first machines, the lever, roller, and screw, were operated by men or animals. This shows that the idea of force was originally derived from the sense of force, or muscular sense, and was, therefore, a specific sense perception. Consequently, I regard it today as quite essential in a lecture on mechanics to refer, at any rate in the introduction, to the original meaning of the force idea. But in the modern exact definition of force the specific notion of sense perception is eliminated, as in the case of color sense, and we may say, quite in general, that in modern theoret- ical physics the specific sense perceptions play a much smaller role in all physical definitions than formerly. In fact, the crowding into the background of the specific sense elements goes so far that the branches of physics which were originally completely and uniquely characterized by an arrangement in accordance with definite sense perceptions have fallen apart, in consequence of the loosening of the bonds between different and widely separated branches, on account of the general advance towards simplification and coordination. The best example of this is furnished by the theory of heat. Earlier, heat formed a sepa- rate and unified domain of physics, characterized through the perceptions of heat sensation. Today one finds in well nigh all physics textbooks dealing with heat a whole domain, that of 6 FIRST LECTURE. radiant heat, separated and treated under optics. The signi- ficance of heat perception no longer suffices to bring together the heterogeneous parts. In short, we may say that the characteristic feature of the entire previous development of theoretical physics is a definite elimina- tion from all physical ideas of the anthropomorphic elements, par- ticularly those of specific sense perceptions. On the other hand, as we have seen above, if one reflects that the perceptions form the point of departure in all physical research, and that it is im- possible to contemplate their absolute exclusion, because we can- not close the source of all our knowledge, then this conscious departure from the original conceptions must always appear astonishing or even paradoxical. There is scarcely a fact in the history of physics which today stands out so clearly as this. Now, what are the great advantages to be gained through such a real obliteration of personality? What is the result for the sake of whose achievement are sacrificed the directness and succinctness such as only the special sense perceptions vouchsafe to physical ideas? The result is nothing more than the attainment of unity and compactness in our system of theoretical physics, and, in fact, the unity of the system, not only in relation to all of its details, but also in relation to physicists of all places, all times, all peoples, all cultures. Certainly, the system of theoretical physics should be adequate, not only for the inhabitants of this earth, but also for the inhabitants of other heavenly bodies. Whether the inhabitants of Mars, in case such actually exist, have eyes and ears like our own, we do not know, — it is quite improbable; but that they, in so far as they possess the necessary intelligence, recognize the law of gravitation and the principle of energy, most physicists would hold as self evident: and anyone to whom this is not evident had better not appeal to the physicists, for it will always remain for him an unsolvable riddle that the same physics is made in the United States as in Germany. To sum up, we may say that the characteristic feature of the introduction: reversibility and irreversibility. 7 actual development of the system of theoretical physics is an ever extending emancipation from the anthropomorphic elements, which has for its object the most complete separation possible of the system of physics and the individual personality of the physicist. One may call this the objectiveness of the system of physics. In order to exclude the possibility of any misunder- standing, I wish to emphasize particularly that we have here to do, not with an absolute separation of physics from the physicist — for a physics without the physicist is unthinkable,^ but with the elimination of the individuality of the particular physicist and therefore with the production of a common system of physics for all physicists. Now, how does this principle agree with the positivist con- ceptions mentioned above? Separation of the system of physics from the individual personality of the physicist? Opposed to this principle, in accordance with those conceptions, each particular physicist must have his special system of physics, in case that complete elimination of all metaphysical elements is effected; for physics occupies itself only with the facts discovered through perceptions, and only the individual perceptions are directly involved. That other living beings have sensations is, strictly speaking, but a very probable, though arbitrary, conclusion from analogy. The system of physics is therefore primarily an individual matter and, if two physicists accept the same system, it is a very happy circumstance in connection with their personal relationship, but it is not essentially necessary. One can regard this view-point however he will; in physics it is certainly quite fruitless, and this is all that I care to maintain here. Certainly, I might add, each great physical idea means a further advance toward the emancipation from anthropomorphic ideas. This was true in the passage from the Ptolemaic to the Copernican cosmical system, just as it is true at the present time for the apparently impending passage from the so-called classical me- chanics of mass points to the general dynamics originating in the principle of relativity. In accordance with this, man and g FIRST LECTURE. the earth upon which he dwells are removed from the centre of the world. It may be predicted that in this century the idea of time will be divested of the absolute character with which men have been accustomed to endow it (cf. the final lecture). Certainly, the sacrifices demanded by every such revolution in the intuitive point of view are enormous; conse- quently, the resistance against such a change is very great. But the development of science is not to be permanently halted thereby; on the contrary, its strongest impetus is experienced through precisely those forces which attain success in the strug- gle against the old points of view, and to this extent such a struggle is constantly necessary and useful. Now, how far have we advanced today toward the unification of our system of physics? The numerous independent domains of the earlier physics now appear reduced to two; mechanics and electrodynamics, or, as one may say: the physics of material bodies and the physics of the ether. The former comprehends acoustics, phenomena in material bodies, and chemical phenom- ena; the latter, magnetism, optics, and radiant heat. But is this division a fundamental one? Will it prove final? This is a question of great consequence for the future development of physics. For myself, I believe it must be answered in the negative, and upon the following grounds : mechanics and electro- dynamics cannot be permanently sharply differentiated from each other. Does the process of light emission, for example, belong to mechanics or to electrodynamics? To which domain shall be assigned the laws of motion of electrons? At first glance, one may perhaps say: to electrodynamics, since with the electrons ponderable matter does not play any role. But let one direct his attention to the motion of free electrons in metals. There he will find, in the study of the classical re- searches of II. A. Lorentz, for example, that the laws obeyed by the electrons belong rather to the kinetic theory of gases than to electrodynamics. In general, it appears to me that the original differences between processes in the ether and processes introduction: reversibility and irreversibility. 9 in material bodies are to be considered as disappearing. Electro- dynamics and mechanics are not so remarkably far apart, as is considered to be' the case by many people, who already speak of a conflict between the mechanical and the electrodynamic views of the world. Mechanics requires for its foundation essentially nothing more than the ideas of space, of time, and of that which is moving, M^hether one considers this as a substance or a state. The same ideas are also involved in electrodynamics. A suffi- ciently generalized conception of mechanics can therefore also well include electrodynamics, and, in fact, there are many indica- tions pointing toward the ultimate amalgamation of these two subjects, the domains of which already overlap in some measure. If, therefore, the gulf between ether and matter be once bridged, what is the point of view which in the last analysis will best serve in the subdivision of the system of physics? The answer to this question will characterize the whole nature of the further development of our science. It is, therefore, the most important among all those which I propose to treat today. But for the purposes of a closer investigation it is necessary that we go some- what more deeply into the peculiarities of physical principles. We shall best begin at that point from which the first step was made toward the actual realization of the unified system of physics previously postulated by the philosophers only; at the principle of conservation of energy. For the idea of energy is the only one besides those of space and time which is common to all the various domains of physics. In accordance with what I have stated above, it will be apparent and quite self evident to you that the principle of energy, before its general f ormularization by Mayer, Joule, and Helmholz, also bore an anthropomorphic character. The roots of this principle lay already in the recog- nition of the fact that no one is able to obtain useful work from nothing; and this recognition had originated essentially in the experiences which were gathered in attempts at the solution of a technical problem: the discovery of perpetual motion. To this extent, perpetual motion has come to have for physics a far 2 10 FIRST LECTURE. reaching significance, similar to that of alchemy for the chemist, although it was not the positive, but rather the negative results of these experiments, through which science was advanced. Today we speak of the principle of energy quite without reference to the technical viewpoint or to that of man. We say that the total amount of energy of an isolated system of bodies is a quantity whose amount can be neither increased nor diminished through any kind of process within the system, and we no longer consider the accuracy with which this law holds as dependent upon the refinement of the methods, which we at present possess, of testing experimentally the question of the realization of perpetual motion. In this, strictly speaking, unprovable general- ization, impressed upon us with elemental force, lies the eman- cipation from the anthropomorphic elements mentioned above. While the principle of energy stands before us as a complete independent structure, freed from and independent of the acci- dents appertaining to its historical development, this is by no means true in equal measure in the case of that principle which R. Clausius introduced into physics; namely, the second law of thermodynamics. This law plays a very peculiar role in the development of physical science, to the extent that one is not able to assert today that for it a generally recognized, and there- fore objective formularization, has been found. In our present consideration it is therefore a matter of particular interest to examine more closely its significance. In contrast to the first law of thermodynamics, or the energy principle, the second law may be characterized as follows. While the first law permits in all processes of nature neither the creation nor destruction of energy, but permits of transformations only, the second law goes still further into the limitation of the pos- sible processes of nature, in that it permits, not all kinds of trans- formations, but only certain types, subject to certain con- ditions. The second law occupies itself, therefore, with the question of the kind and, in particular, with the direction of any natural process. INTRODUCTION: EEVERSIBILITY AND IRREVERSIBILITY. 11 At this point a mistake has frequently been made, which has hindered in a very pronounced manner the advance of science up to the present day. In the endeavor to give to the second law of thermodynamics the most general character possible, it has been proclaimed by followers of W. Ostwald as the second law of energetics, and the attempt made so to formulate it that it shall determine quite generally the direction of every process occurring in nature. Some weeks ago I read in a public academic address of an esteemed colleague the statement that the import of the second law consists in this, that a stone falls downwards, that water flows not up hill, but down, that electricity flows from a higher to a lower potential, and so on. This is a mistake which at present is altogether too prevalent not to warrant mention here. The truth is, these statements are false. A stone can just as well rise in the air as fall downwards; water can likewise flow up- wards, as, for example, in a spring; electricity can flow very well from a lower to a higher potential, as in the case of oscillating dis- charge of a condenser. The statements are obviously quite cor- rect, if one applies them to a stone originally at rest, to water at rest, to electricity at rest; but then they follow immediately from the energy principle, and one does not need to add a special second law. For, in accordance with the energy principle, the kinetic energy of the stone or of the water can only originate at the cost of gravitational energy, i. e., the center of mass must descend. If, therefore, motion is to take place at all, it is necessary that the gravitational energy shall decrease. That is, the center of mass must descend. In like manner, an electric cur- rent between two condenser plates can originate only at the cost of electrical energy already present; the electricity must therefore pass to a lower potential. If, however, motion and current be already present, then one is not able to say, a priori, anything in regard to the direction of the change; it can take place just as well in one direction as the other. Therefore, there is no new insight into nature to be obtained from this point of view. 12 FIRST LECTURE. Upon an equally inadequate basis rests another conception of the second law, which I shall now mention. In considering the cir- cumstance that mechanical work may very easily be transformed into heat, as by friction, while on the other hand heat can only with difiBculty be transformed into work, the attempt has been made so to characterize the second law, that in nature the trans- formation of work into heat can take place completely, while that of heat into work, on the other hand, only incompletely and in such manner that every time a quantity of heat is transformed into work another corresponding quantity of energy must neces- sarily undergo at the same time a compensating transforma- tion, as, e. g., the passage of heat from a higher to a lower temperature. This assertion is in certain special cases correct, but does not strike in general at the true import of the matter, as I shall show by a simple example. One of the most important laws of thermodynamics is, that the total energy of an ideal gas depends only upon its tempera- ture, and not upon its volume. If an ideal gas be allowed to expand while doing work, and if the cooling of the gas be prevented through the simultaneous addition of heat from a heat reservoir at higher temperature, the gas remains unchanged in temperature and energy content, and one may say that the heat furnished by the heat reservoir is completely transformed into work without exchange of energy. Not the least objection can be urged against this assertion. The law of incomplete transformation of heat into work is retained only through the adoption of a different point of view, but which has nothing to do with the status of the physical facts and only modifies the way of looking at the matter, and therefore can neither be supported nor con- tradicted through facts; namely, through the introduction ad hoc of new particular kinds of energy, in that one divides the energy of the gas into numerous parts which individually can depend upon the volume. But it is a priori evident that one can never derive from so artificial a definition a new physical law, and it is with such that we have to do when we pass from the first law, the principle of conservation of energy, to the second law. INTRODUCTION: REVERSIBILITY AND IRREVERSIBILITY. 13 I desire now to introduce such a new physical law: "It is not possible to construct a periodically functioning motor which in principle does not involve more than the raising of a load and the cooling of a heat reservoir." It is to be understood, that in one cycle of the motor quite arbitrary complicated processes may take place, but that after the completion of one cycle there shall remain no other changes in the surroundings than that the heat reservoir is cooled and that the load is raised a corresponding distance, which may be calculated from the first law. Such a motor could of course be used at the same time as a refrigerating machine also, without any further expenditure of energy and materials. Such a motor would moreover be the most efficient in the world, since it would involve no cost to run it; for the earth, the atmosphere, or the ocean could be utilized as the heat reservoir. We shall call this, in accordance with the proposal of W. Ostwald, perpetual motion of the second kind. Whether in nature such a motion is actually possible cannot be inferred from the energy principle, and may only be determined by special experiments. Just as the impossibility of perpetual motion of the first kind leads to the principle of the conservation of energy, the quite independent principle of the impossibility of perpetual motion of the second kind leads to the second law of thermodynamics, and, if we assume this impossibility as proven experimentally, the general law follows immediately: there are processes in nature which in no possible way can be made covipletely reversi- ble. For consider, e. g., a frictional process through which me- chanical work is transformed into heat with the aid of suitable apparatus, if it were actually possible to make in some way such complicated apparatus completely reversible, so that everywhere in nature exactly the same conditions be reestablished as existed at the beginning of the frictional process, then the apparatus considered would be nothing more than the motor described above, furnishing a perpetual motion of the second kind. This appears evident immediately, if one clearly perceives what the 14 FIRST LECTURE. apparatus would accomplish : transformation of heat into work without any further outstanding change. We call such a process, which in no wise can be made completely reversible, an irreversible process, and all other processes re- versible processes; and thus we strike the kernel of the second law of thermodynamics when we say that irreversible processes occur in nature. In accordance with this, the changes in nature have a unidirectional tendency. With each irreversible process the world takes a step forward, the traces of which under no circumstances can be completely obliterated. Besides friction, examples of irreversible processes are : heat conduction, diffusion, conduction of electricity in conductors of finite resistance, emission of light and heat radiation, disintegration of the atom in radioactive substances, and so on. On the other hand, ex- amples of reversible processes are: motion of the planets, free fall in empty space, the undamped motion of a pendulum, the frictionless flow of liquids, the propagation of light and sound waves without absorption and refraction, undamped electrical vibrations, and so on. For all thes'e processes are already periodic or may be made completely reversible through suitable contrivances, so that there remains no outstanding change in nature; for example, the free fall of a body whereby the acquired velocity is utilized to raise the body again to its original height; a light or sound wave which is allowed in a suitable manner to be totally reflected from a perfect mirror. What now are the general properties and criteria of irreversible processes, and what is the general quantitative measure of irreversibility? This question has been examined and answered in the most widely different ways, and it is evident here again how difficult it is to reach a correct formularization of a prob- lem. Just as originally we came upon the trail of the energy principle through the technical problem of perpetual motion, so again a technical problem, namely, that of the steam engine, led to the differentiation between reversible and irreversible processes. Long ago Sadi Carnot recognized, although he util- introduction: reversibility and irreversibility. 15 Ized an incorrect conception of the nature of heat, that irre- versible processes are less economical than reversible, or that in an irreversible process a certain opportunity to derive mechan- ical work from heat is lost. What then could have been simpler than the thought of making, quite in general, the meas- ure of the irreversibility of a process the quantity of mechanical work which is unavoidably lost in the process. For a reversible process then, the unavoidably lost work is naturally to be set equal to zero. This view, in accordance with which the import of the second law consists in a dissipation of useful energy, has in fact, in certain special cases, e. g., in isothermal processes, proved itself useful. It has persisted, therefore, in certain of its aspects up to the present day; but for the general case, how- ever, it has shown itself as fruitless and, in fact, misleading. The reason for this lies in the fact that the question concerning the lost work in a given irreversible process is by no means to be answered in a determinate manner, so long as nothing further is specified with regard to the source of energy from which the work considered shall be obtained. An example will make this clear. Heat conduction is an irreversible process, or as Clausius expresses it: Heat cannot without compensation pass from a colder to a warmer body. What now is the work which in accordance with definition is lost when the quantity of heat Q passes through direct conduction from a warmer body at the temperature fi to a colder body at the temperature 7^2? In order to answer this question, we make use of the heat transfer involved in carrying out a reversible Carnot cyclical process between the two bodies employed as heat reservoirs. In this process a certain amount of work would be obtained, and it is just the amount sought, since it is that which would be lost in the direct passage by conduction; but this has no definite value so long as we do not know whence the work originates, whether, e. g., in the warmer body or in the colder body, or from somewhere else. Let one reflect that the heat given up by the warmer body in the reversible process is cer- 16 FIRST LECTURE. tainly not equal to the heat absorbed by the colder body, because a certain amount of heat is transformed into work, and that we can identify, with exactly the same right, the quantity of heat Q transferred by the direct process of conduction with that which in the cyclical process is given up by the warmer body, or with that absorbed by the colder body. As one does the former or the latter, he accordingly obtains for the quantity of lost work in the process of conduction: Q ■ — 2? or Q • — ji . We see, therefore, that the proposed method of expressing mathe- matically the irreversibility of a process does not in general effect its object, and at the same time we recognize the peculiar reason which prevents its doing so. The statement of the question is too anthropomorphic. It is primarily too much concerned with the needs of mankind, in that it refers directly to the acquirement of useful work. If one require from nature a determinate answer, he must take a more general point of view, more disin- terested, less economic. We shall now seek to do this. Let us consider any typical process occurring in nature. This will carry all bodies concerned in it from a determinate initial state, which I designate as state A, into a determinate final state B. The process is either reversible or irreversible. A third possibility is excluded. But whether it is reversible or irreversible depends solely upon the nature of the two states A and B, and not at all upon the way in which the process has been carried out; for we are only concerned with the answer to the question as to whether or not, when the state B is once reached, a complete return to A in any conceivable manner may be ac- complished. If now, the complete return from 5 to yl is not possible, and the process therefore irreversible, it is obvious that the state B may be distinguished in nature through a certain property from state A. Several years ago I ventured to express this as follows: that nature possesses a greater "preference" for state B than for state A. In accordance with this mode of introduction: reversibility and irreversibility. 17 expression, all those processes of nature are impossible for whose final state nature possesses a smaller preference than for the original state. Reversible processes constitute a limiting case; for such, nature possesses an equal preference for the initial and for the final state, and the passage between them takes place as well in one direction as the other. We have now to seek a physical quantity whose magnitude shall serve as a general measure of the preference of nature for a given state. This quantity must be one which is directly determined by the state of the system considered, without reference to the previous history of the system, as is the case with the energy, with the volume, and with other properties of the system. It should possess the peculiarity of increasing in all irreversible processes and of remaining unchanged in all revers- ible processes, and the amount of change which it experiences in a process would furnish a general measure for the irre- versibility of the process. R. Clausius actually found this quantity and called it "entropy." Every system of bodies possesses in each of its states a definite entropy, and this entropy expresses the pref- erence of nature for the state in question. It can, in all the processes which take place within the system, only increase and never decrease. If it be desired to consider a process in which external actions upon the system are present, it is necessary to consider those bodies in which these actions originate as constituting part of the system; then the law as stated in the above form is valid. In accordance with it, the entropy of a system of bodies is simply equal to the sum of the entropies of the individual bodies, and the entropy of a single body is, in accordance with Clausius, found by the aid of a certain re- versible process. Conduction of heat to a body increases its entropy, and, in fact, by an amount equal to the ratio of the quantity of heat given the body to its temperature. Simple compression, on the other hand, does not change the entropy. Returning to the example mentioned above, in which the 18 FIEST LECTURE. quantity of heat Q is conducted from a warmer body at the temperature fi to a colder body at the temperature T2, in accordance with what precedes, the entropy of the warmer body decreases in this process, while, on the other hand, that of the colder increases, and the sum of both changes, that is, the change of the total entropy of both bodies, is: Q Q i 1 I2 This positive quantity furnishes, in a manner free from all arbitrary assumptions, the measure of the irreversibility of the process of heat conduction. Such examples may be cited indefinitely. Every chemical process furnishes an increase of entropy. We shall here consider only the most general case treated by Clausius : an arbitrary reversible or irreversible cyclical process, carried out with any physico-chemical arrangement, utilizing an arbitrary number of heat reservoirs. Since the arrangement at the conclusion of the cyclical process is the same as that at the beginning, the final state of the process is to be distinguished from the initial state solely through the different heat content of the heat reservoirs, and in that a certain amount of mechanical work has been furnished or consumed. Let Q be the heat given up in the course of the process by a heat reservoir at the tem- perature T, and let ^ be the total work yielded (consisting, e. g., in the raising of weights) ; then, in accordance with the first law of thermodynamics: In accordance with the second law, the sum of the changes in entropy of all the heat reservoirs is positive, or zero. It follows, therefore, since the entropy of a reservoir is decreased by the amount Q/T through the loss of heat Q that: Q This is the well-known inequality of Clausius. introduction: reversibility and irreversibility. 19 In an isothermal cyclical process, T is the same for all reservoirs. Therefore : SQ S 0, hence: ^ ^ 0. That is: in an isothermal cyclical process., heat is produced and work is consumed. In the limiting case, a reversible isothermal cyclical process, the sign of equality holds, and therefore the work consumed is zero, and also the heat produced. This law plays a leading role in the application of thermodynamics to physical chemistry. The second law of thermodynamics including all of its con- sequences has thus led to the principle of increase of entropy. You will now readily understand, having regard to the questions mentioned above, why I express it as my opinion that in the theoretical physics of the future the first and most important differentiation of all physical processes will be into reversible and irreversible processes. In fact, all reversible processes, whether they take place in material bodies, in the ether, or in both together, show a much greater similarity among themselves than to any irreversible process. In the differential equations of reversible processes the time differential enters only as an even power, corres- ponding to the circumstance that the sign of time can be reversed. This holds equally well for vibrations of the pen- dulum, electrical vibrations, acoustic and optical waves, and for motions of mass points or of electrons, if we only ex- clude every kind of damping. But to such processes also belong those infinitely slow processes of thermodynamics which consist of states of equilibrium in which the time in general plays no role, or, as one may also say, occurs with the zero power, which is to be reckoned as an even power. As Helmholtz has pointed out, all these reversible processes have the common property that they may be completely represented by the principle of least action, which gives a definite answer to all questions con- cerning any such measurable process, and, to this extent, the- ory of reversible processes may be regarded as completely estab- lished. Reversible processes have, however, the disadvantage that 20 FIRST LECTUKE. singly and collectively they are only ideal : in actual nature there is no such thing as a reversible process. Every natural process involves in greater or less degree friction or conduction of heat. But in the domain of irreversible processes the principle of least action is no longer sufficient; for the principle of increase of entropy brings into the system of physics a wholly new element, foreign to the action principle, and which demands special mathematical treatment. The unidirectional course of a process in the attainment of a fixed final state is related to it. I hope the foregoi ig considerations have sufficed to make clear to you that the distinction between reversible and irreversible processes is much broader than that between mechanical and electrical processes and that, therefore, this difference, with better right than any other, may be taken advantage of in classifying all physical processes, and that it may eventually play in the theoretical physics of the future the principal role. However, the classification mentioned is in need of quite an essential improvement, for it cannot be denied that in the form set forth, the system of physics is still suffering from a strong dose of anthropomorphism. In the definition of irreversibility, as well as in that of entropy, reference is made to the possibility of carrying out in nature certain changes, and this means, funda- mentally, nothing more than that the division of physical proc- esses is made dependent upon the manipulative skill of man in the art of experimentation, which certainly does not always remain at a fixed stage, but is continually being more and more perfected. If, therefore, the distinction between reversible and irreversible processes is actually to have a lasting significance for all times, it must be essentially broadened and made inde- pendent of any reference to the capacities of mankind. How this may happen, I desire to state one week from tomorrow. The lecture of tomorrow will be devoted to the problem of bringing before you some of the most important of the great number of practical consequences following from the entropy principle. SECOND LECTURE. Thermodynamic States of Equilibrium in Dilute Solutions. In the lecture of yesterday I sought to make clear the fact that the essential, and therefore the final division of all processes occurring in nature, is into reversible and irreversible processes, and the characteristic difference between these two kinds of processes, as I have further separated them, is that in irreversible processes the entropy increases, while in all reversible processes it remains constant. Today I am constrained to speak of some of the consequences of this law which will illustrate its rich fruit- fulness. They have to do with the question of the laws of ther- modynamic equilibrium. Since in nature the entropy can only increase, it follows that the state of a physical configuration which is completely isolated, and in which the entropy of the system possesses an absolute maximum, is necessarily a state of stable equilibrium, since for it no further change is possible. How deeply this law underlies all physical and chem- ical relations has been shown by no one better and more com- pletely than by John Willard Gibbs, whose name, not only in America, but in the whole world will be counted among those of the most famous theoretical physicists of all times; to whom, to my sorrow, it is no longer possible for me to tender personally my respects. It would be gratuitous for me, here in the land of his activity, to expatiate fully on the progress of his ideas, but you will perhaps permit me to speak in the lecture of to- day of some of the important applications in which thermo- dynamic research, based on Gibbs works, can be advanced be- yond his results. These applications refer to the theory of dilute solutions, and 21 22 SECOND LECTUKE. we shall occupy ourselves today with these, while I show you by a definite example what fruitfulness is inherent in thermo- dynamic theory. I shall first characterize the problem quite generally. It has to do with the state of equilibrium of a material system of any number of arbitrary constituents in an arbi- trary number of phases, at a given temperature T and given pressure p. If the system is completely isolated, and there- fore guarded against all external thermal and mechanical actions, then in any ensuing change the entropy of the system will increase: d8> 0. But if, as we assume, the system stands in such relation to its surroundings that in any change which the system under- goes the temperature T and the pressure p are maintained constant, as, for instance, through its introduction into a calorim- eter of great heat capacity and through loading with a piston of fixed weight, the inequality would suffer a change thereby. We must then take account of the fact that the surrounding bodies also, e. g., the calorimetric liquid, will be involved in the change. If we denote the entropy of the surrounding bodies by /So, then the following more general equation holds: dS + dSo > 0. In this equation ^S - ^ aoo — — -J,, if Q denote the heat which is given up in the change by the surroundings to the system. On the other hand, if U de- note the energy, V the volume of the system, then, in accord- ance with the first law of thermodynamics, Q= dU+ pdV. Consequently, through substitution : THERMODYNAMIC STATES OF EQUILIBRIUM. 23 or, since p and T are constant: If, therefore, we put : .-^-*, (!) then d^> 0, and we have the general law, that in every isothermal-isobaric {T = const., p = const.) change of state of a physical system the quantity increases. The absolutely stable state of equilibrium of the system is therefore characterized through the maximum of #: S* = 0. (2) If the system consist of numerous phases, then, because $, in accordance with (1), is linear and homogeneous in B, U and V, the quantity $ referring to the whole system is the sum of the quantities $ referring to the individual phases. If the expression for $ is known as a function of the independent variables for each phase of the system, then, from equation (2), all ques- tions concerning the conditions of stable equilibrium may be answered. Now, within limits, this is the case for dilute solutions. By "solution" in thermodynamics is meant each homogeneous phase, in whatever state of aggregation, which is composed of a series of different molecular complexes, each of which is rep- resented by a definite molecular number. If the molecular number of a given complex is great with reference to all the remaining complexes, then the solution is called dilute, and the molecular complex in question is called the solvent; the remain- ing complexes are called the dissolved substances. Let us now consider a dilute solution whose state is determined by the pressure p, the temperature T, and the molecular numbers no, til, Ui, nz, • • • , wherein the subscript zero refers to the solvent. Then the numbers n\, n-i, nz, ■ ■ • are all small with respect to no. 24 SECOND LECTURE. and on this account the volume V and the energy U are linear functions of the molecular numbers: V = noVo + tiivi + ?i2»2 + • • • , U = noUo + riiUi + ^2^2 + ■ • • , wherein the v's and u's depend upon p and T only. From the general equation of entropy: dU + pdV ao — ™ , in which the differentials depend only upon changes in p and T, and not in the molecular numbers, there results therefore: duo + pdvo dui + pdvi db = no ^ h wi ^ h •• •, and from this it follows that the expressions multiplied by no, ui ■ ■ ■ , dependent upon p and T only, are complete differentials. We may therefore write: duo + pdvo J dui + pdvi J, = 0*0, ji = dsi, - ■ • {6) and by integration obtain: S = no*o + "i^i + n2S2 + • • • + C The constant C of integration does not depend upon p and T, but may depend upon the molecular numbers no, ni, Ji2, • • • . In order to express this dependence generally, it suffices to know it for a special case, for fixed values of p and T. Now every solution passes, through appropriate increase of temperature and decrease of pressure, into the state of a mixture of ideal gases, and for this case the entropy is fully known, the integration constant being, in accordance with Gibbs: C = — R(no log Co + wi log ci + ■ ■ • ), wherein R denotes the absolute gas constant and Co, Ci, C2, • • • THERMODYNAMIC STATES OF EQUILIBRIUM. 25 denote the "molecular concentrations": no _ ni no+ ni + n2+ ■■■' ^ Wo + wi + «2 + • • • ' Consequently, quite in general, the entropy of a dilute solution is : S = no(so — R log Co) + ni{si — i? log Ci) + • • •, and, finally, from this it follows by substitution in equation (1) that: * = Wo( =

and it follows that : p^„+(l-p')|^^/=^v or: ^ q\_ '^- P' «/ ' q'' 1 - P ■ 78 FIFTH LECTUKE. In the last equation the quantity on the left is independent of the angle of incidence ?? and of the kind of polarization, con- sequently the quantity upon the right side must also be inde- pendent of these quantities. If one knows the value of these quantities for a single angle of incidence and for a given kind of polarization, then this value is valid for all angles of incidence and for all polarizations. Now, in the particular case that the rays are polarized at right angles to the plane of incidence and meet the bounding surface at the angle of polarization, p = and p' = 0. Then the expression on the right will be equal to 1, and there- fore it is in general equal to 1, and we have always: P = p', q'^. = q''^.'. (37) The first of these two relations, which asserts that the coefficient of reflection is the same for both sides of the boundary surface, constitutes the special expression of a general reciprocal law, first announced by Helmholz, whereby the loss of intensity which a ray of given color and polarization suffers on its path through any medium in consequence of reflection, refraction, absorption, and dispersion is exactly equal to the loss of intensity which a ray of corresponding intensity, color and polarization suffers in passing over the directly opposite path. It follows immediately from this that the radiation meeting a boundary surface between two media is transmitted or reflected equally well from both sides, for every color, direction and polarization. The second relation, (37), brings into connection the radiation intensities originating in both substances. It asserts that in thermodynamic equilibrium the specific intensities of radiation of a definite frequency in both media vary inversely as the square of the velocities of propagation, or directly as the squares of the refractive indices. We may therefore write q'^, = F{v, T), HEAT EADIATION. ELECTKODYNAMIC THEORY. 79 wherein F denotes a universal function depending only upon v and T, the discovery of which is one of the chief problems of the theory. Let us fix our attention again on the case of a diathermanous medium. We saw above that in a medium surrounded by a non-transparent shell which for a given color is diathermanous equilibrium can exist for any given intensity of radiation of this color. But it follows from the second law that, among all the intensities of radiation, a definite one, namely, that corresponding to the absolute maximum of the total entropy of the system, must exist, which characterizes the absolutely stable equilibrium of radiation. We now see that this indeterminateness is elimi- nated by the last equation, which asserts that in thermodynamic equilibrium the product q^^^ is a universal function. For it results immediately therefrom that there is a definite value of ^„ for every diathermanous medium which is thus differentiated from all other values. The physical meaning of this value is derived directly from a consideration of the way in which this equation was derived: it is that intensity of radiation which exists in the diathermanous medium when it is in thermodynamic equilibrium while in contact with a given absorbing and emitting medium. The volume and the form of the second medium is immaterial; in particular, the volume may be taken arbitrarily small. For a vacuum, the most diathermanous of all media, in which the velocity of propagation q = cis the same for all rays, we can therefore express the following law: The quantity ^^^^^F{v,T) (38) denotes that intensity of radiation which exists in any complete vacuum when it is in a stationary state as regards exchange of radiation with any absorbing and emitting substance, whose amount may be arbitrarily small. This quantity ^„ regarded as a function of v gives the so-called normal energy spectrum. 80 FIFTH LECTURE. Let US consider, therefore, a vacuum surrounded by given emitting and absorbing bodies of uniform temperature. Then, in the course of time, there is estabHshed therein a normal energy radiation ^^ corresponding to this temperature. If now p^ be the reflection coefficient of a wall for the frequency v, then of the radiation ^„ falling upon the wall, the part p^^^ will be re- flected. On the other hand, if we designate by E^ the emission coefficient of the wall for the same frequency v, the total radiation proceeding from the wall will be: p,t, + E, = ^„ since each bundle of rays possesses in a stationary state the in- tensity ^y. From this it follows that: ^. = T^^ , (39) 1 — Pv i. e., the ratio of the emission coefficient E^ to the capacity for absorption (1 — p^) of a given substance is the same for all substances and equal to the normal intensity of radiation for each frequency (Kirchoff). For the special case that p^ is equal to 0, i. e., that the wall shall be perfectly black, we have: ^. = E„ that is, the normal intensity of radiation is exactly equal to the emission coefficient of a black body. Therefore the normal radiation is also called "black radiation." Again, for any given body, in accordance with (39), we have: E^ < J?„ i. e., the emission coefficient of a body in general is smaller than that of a black body. Black radiation, thanks to W. Wien and O. Lummer, has been made possible of measurement, through a small hole bored in the wall bounding the space considered. We proceed now to the treatment of the problem of deter- mining the specific intensity ^^ of black radiation in a vacuum, HEAT RADIATION. ELECTRODYNAMIC THEORY. 81 as regards its dependence upon the frequency v and the temper- ature T. In the treatment of this problem it will be necessary to go further than we have previously done into those processes which condition the production and destruction of heat rays; that is, into the question regarding the act of emission and that of absorption. On account of the complicated nature of these processes and the difficulty of bringing some of the details into connection with experience, it is certainly quite out of the ques- tion to obtain in this manner any reliable results if the following law cannot be utilized as a dependable guide in this domain: a vacuum surrounded by reflecting walls In which arbitrary emitting and absorbing bodies are distributed in any given arrangement assumes in the course of time the stationary state of black radiation, which Is completely determined by a single parameter, the temperature, and which, in particular, does not depend upon the number, the properties and the arrangement of the bodies. In the investigation of the properties of the state of black radiation the nature of the bodies which are supposed to be in the vacuum is therefore quite immaterial, and it is cer- tainly immaterial whether such bodies actually exist anywhere in nature, so long as their existence and their properties are compatible throughout with the laws of electrodynamics and of thermodynamics. As soon as it Is possible to associate with any given special kind and arrangement of emitting and absorbing bodies a state of radiation In the surrounding vacuum which is characterized by absolute stability, then this state can be no other than that of black radiation. Making use of the freedom furnished by this law, we choose among all the emitting and absorbing systems conceivable, the most simple, namely, a single oscillator at rest, consisting of two poles charged with equal quantities of electricity of opposite sign which are movable relative to each other in a fixed straight line, the axis of the oscillator. The state of the oscillator is completely determined by its moment /(<); i. e., by the product of the electric charge of the pole on the positive side of the axis into the distance between 82 FIFTH LECTURE. the poles, and by its diflferential quotient with regard to the time: The energy of the oscillator is of the following simple form: U = W' + W', (40) wherein K and L denote positive constants which depend upon the nature of the oscillator in some manner into which we need not go further at this time. If, in the vibrations of the oscillator, the energy U remain ab- solutely constant, we should have: dU = or: Km + Lfit) = 0, and from this there results, as a general solution of the differential equation, a pure periodic vibration: / = C cos (2irj'o< - t}), wherein C and i? denote the integration constants and vo the number of vibrations per unit of time: ''« = 2^^|- (41) Such an oscillator vibrating periodically with constant energy would neither be influenced by the electromagnetic field sur- rounding it, nor would it exert any external actions due to radi- ation. It could therefore have no sort of influence on the heat radiation in the surrounding vacuum. In accordance with the theory of Maxwell, the energy of vibration U of the oscillator by no means remains constant in general, but an oscillator by virtue of its vibrations sends out spherical waves in all directions into the surrounding field and, in accordance with the principle of conservation of energy, if no actions from without are exerted upon the oscillator, there must HEAT RADIATION. ELECTRODYNAMIC THEORY. 83 necessarily be a loss in the energy of vibration and, therefore, a damping of the amplitude of vibration is involved. In order to find the amount of this damping we calculate the quantity of energy which flows out through a spherical surface with the oscillator at the center, in accordance with the law of Poynting. However, we may not place the energy flowing outwards in accordance with this law through the spherical surface in an infinitely small interval of time dt equal to the energy radiated in the same time interval from the oscillator. For, in general, the electromagnetic energy does not always flow in the out- ward direction, but flows alternately outwards and inwards, and we should obtain in this manner for the quantity of the radia- tion outwards, values which are alternately positive and nega- tive, and which also depend essentially upon the radius of the supposed sphere in such manner that they increase toward infinity with decreasing radius — which is opposed to the funda- mental conception of radiated energy. This energy will, more- over, be only found independent of the radius of the sphere when we calculate the total amount of energy flowing outwards through the surface of the sphere, not for the time element dt, but for a sufiiciently large time. If the vibrations are purely periodic, we may choose for the time a period; if this is not the case, which for the sake of generality we must here assume, it is not possible to specify a priori any more general criterion for the least possible necessary magnitude of the time than that which makes the energy radiated essentially independent of the radius of the supposed sphere. In this way we succeed in finding for the energy emitted from the oscillator in the time from < to i + S the following expression: "t+Z 2 r'+'' If now, the oscillator be in an electromagnetic field which has the electric component @j at the oscillator in the direction of its axis. 84 FIFTH LECTURE. then the energy absorbed by the oscillator in the same time is: "t+Z X g,/ ■ dt. Hence, the principle of conservation of energy is expressed in the following form: This equation, together with the assumption that the constant Sc'L (42) is a small number, leads to the following linear differential equa- tion for the vibrations of the oscillator: Kf+Lf-^J=(i.. (43) In accordance with what precedes, in so far as the oscillator is excited into vibrations by an external field (Sj, one may designate it as a resonator which possesses the natural period vq and the small logarithmic decrement a. The same equation may be obtained from the electron theory, but I have considered it an advantage to derive it in a manner independent of any hypothesis concerning the nature of the resonator. Now, let the resonator be in a vacuum filled with stationary black radiation of specific intensity M^,. How, then, does the mean energy U of the resonator in a state of stationary vibration depend upon the specific intensity of radiation ^„„ with the natural period vo of the corresponding color? It is this question which we have still to consider today. Its answer will be found by ex- pressing on the one hand the energy of the resonator U and on the other hand the intensity of radiation ^^j by means of the component (§., of the electric field exciting the resonator. Now however complicated this quantity may be, it is capable of HEAT RADIATION. ELECTEODYNAMIC THEORY. 85 development in any case for a very large time interval, from t = to t = X, in the Fourier's series: ^z— 2-, Cn COS I -^ !?„ I , (44) and for this same time interval 2 the moment of the resonator in the form of a Fourier's series may be calculated as a function of t from the linear differential equation (43). The initial condition of the resonator may be neglected if we only consider such times t as are sufficiently far removed from the origin of time t = 0. If it be now recalled that in a stationary state of vibration the mean energy U of the resonator is given, in accordance with (40), (41) and (42), by: -^ 3(rc3 •' ' it appears atter substitution of the value of / obtained from the differential equation (43) that: ^ = 64^3:C„o^ (45) wherein C„o^ denotes the mean value of C„ for all the series of numbers n which lie in the neighborhood of the value V(^, i. e., for which vq^ is approximately = 1. Now let us consider on the other hand the intensity of black radiation, and for this purpose proceed from the space density of the total radiation. In accordance with (30), this is: _87r r* c Jo Si^dv = £ (d/ + Hy' + dj" + ^.' + §/ + §/), (46) and therefore, since the radiation is isotropic, in accordance with (44): Q /^QO Q O 71 = 00 86 FIFTH LECTURE. If we write Ara/S on the left instead of dv, where An is a large number, we get : C „=i Jt oir „=i and obtain then by " spectral " division of this equation: C jL Oir no-(AMW and, if we introduce again the mean value 1 Jio+(An/2) _ _ ^ ri 2 — p 2 ^n n„-^^n|2) we then get: 3cX ^ 647r2 ^"0 ~ A/1^2 • C'nO. By comparison with (45) the relation sought is now found: ^.o = ^-^-U, (47) which is striking on account of its simplicity and, in particular, because it is quite independent of the damping constant a of the resonator. This relation, found in a purely electrodynamic manner, between the spectral intensity of black radiation and the energy of a vibrating resonator will furnish us in the next lecture, with the aid of thermodynamic considerations, the necessary means of attack in deriving the temperature of black radiation together with the distribution of energy in the normal spectrum. SIXTH LECTURE. Heat Radiation. Statistical Theory. Following the preparatory considerations of the last lecture we shall treat today the problem which we have come to recognize as one of the most important in the theory of heat radiation: the establishment of that universal function which governs the energy distribution in the normal ^ppctrum. The means for the solution of this problem will be fuj-nished us through the calcu- lation of the entropy S ot a. resonator placed in a vacuum filled with black radiation and thereby pxcited into stationary vibra- tions. Its energy U is then connected with the corresponding specific intensity ^^ and its natural frequency v in the radiation of the surrounding field through equation (47) : ^. = ^'^7. (48) When S is found as a function of U, the temperature T of the resonator and that of the surrounding radiation will be given by: dS 1 dU=T' (49). and by elimination of U from the last two equations, we then find the relationship among ^y, T and v. In order to find the entropy S of the resonator we will utilize the general connection between entropy and probability, which we have extensively discussed in the previous lectures, and inquire then as to the existing probability that the vibrating resonator possesses the energy U. In accordance with what we have seen in connection with the elucidation of the second law through 87 88 SIXTH LECTURE. atomistic ideas, the second law is only applicable to a physical system when we consider the quantities which determine the state of the system as mean values of numerous disordered individual values, and the probability of a state is then equal to the number of the numerous, a priori equally probable, com- plexions which make possible the realization of the state. Ac- cordingly, we have to consider the energy ?7 of a resonator placed in a stationary field of black radiation as a constant mean value of many disordered independent individual values, and this procedure agrees with the fact that every measurement of the intensity of heat radiation is extended over an enormous number of vibration periods. The entropy of a resonator is then to be calculated from the existing probability that the energy of the radiator possesses a definite mean value U within a certain time interval. In order to find this probability, we inquire next as to the existing probability that the resonator at any fixed time pos- sesses a given energy, or in other words, that that point (the state point) which through its coordinates indicates the state of the resonator falls in a given "state domain." At the conclusion of the third lecture (p. 57) we saw in general that this proba- bility is simply measured through the magnitude of the cor- responding state domain: fd

o.2] as the solution of the problem in hand. We will now introduce the temperature T of the resonator, and will express through T the energy U of the resonator and also the intensity &, of the heat radiation related to it through a 92 SIXTH LECTURE. Stationary state of energy exchange. For this purpose we utilize equation (49) and obtain then for the energy of the resonator: [7= ^' .hv IkT 1 ' It is to be observed that we have not here to do with a uniform distribution of energy (of. p. 68) among the various resonators. For the specific intensity of the monochromatic plane polarized ray of frequency v, we have, in accordance with (48) : "Ik ~ "TF ■ „hvtkT 1 • (5o) This expression furnishes for each temperature T the energy distribution in the normal spectrum of a black body. A com- parison with equation (38) of the last lecture furnishes us then with the universal function : Kv, T) = '-' fivjkT 1 • If we refer the specific intensity of a monochromatic ray, not to the frequency v, but, as is commonly done in experimental physics, to the wave length X, then, since between the absolute values of dv and dX the relation exists : |c^.|='-'^^' we obtain from the relation: X2 ' E^\d\\ = t.|(^H, c^ 1 as the intensity of a monochromatic plane polarized ray of wave length X which is emitted normally to the surface of a black body in a vacuum at temperature T. For small values of XT' HEAT RADIATION. STATISTICAL THEORY. 93 (54) reduces to: ^^ = ^ • e-(""/*^^, (55) which expresses Wien's Displacement Law. For large values of X r on the other hand, there results from (54) : E. = {r, (56) a relation first established by Lord Rayleigh and which we may here designate as the Rayleigh Law of Radiation. From equation (30), taking account of (53), we obtain for the space density of black radiation in a vaccuum: e = - wherein (¥)*•«= «n «= l+^, + ^ + ^+--- = 1.0823. The Stefan-Boltzmann law is hereby expressed. In accordance with the measurements of Kurlbaum, we have the constant , = 4^., = 7.061.10— ''^ c^h^ ' cm^ deg* ' For that wave length Xm which corresponds in the spectrum of black radiation to the maximum intensity of radiation Ef, we have from equation (54) : (dE,\ 0. Carrying out the differentiation, we get, after putting for brevity: '^ /3, .-+^-1 = 0. a„r '^' 5 The root of this transcendental equation is 13 = 4.9651; 94 SIXTH LECTURE. and XnT = chjh^ — his a, constant (Wien's Displacement Law). In accordance with the measurements of O. Lummer and E. Pringsheim, h = 0.294 cm ■ deg. From this there follow the numerical values k = 1.346 • 10-i« 1^ - and h = 6.548 • IQ-^^ erg • sec. deg' The value found for k easily permits of the specification numeric- ally, in the C.G.S. system, of the general connection between entropy and probability, as expressed through the universal equation (12). Thus, quite in general, the entropv of a physical system is: S = 1.346 • 10-i« log W. In the application to the kinetic gas theory we obtain from equation (24) for the ratio of the molecular mass to the mol mass : k CO =-= 1.62- 10-^ i. e., to one mol there corresponds l/oo = 6.175 • 10^ molecules, where it is supposed that the mol of oxygen O2 = 32g. Accordingly, the number of molecules contained in 1 cu. cm. of an ideal gas at 0° Cels. and at atmospheric pressure is: N = 2.76 • 10". The mean kinetic energy of the progressive motion of a molecule at the absolute temperature T = I'm the absolute C.G.S. system in accordance with (27), is: L = §/<; = 2.02 • 10-i«. In general, the mean kinetic energy of progressive motion of a HEAT RADIATION. STATISTICAL THEORY. 95 molecule is expressed by the product of this number and the absolute temperature T. The elementary quantum of electricity, or the free electric charge of a monovalent ion or electron, in electrostatic measure is: e = 00 • 9658 • 3 ■ W = 4.69 • IQ-i". This result stands in noteworthy agreement with the results of the latest direct measurements of the electric elementary quantum made by E. Rutherford and H. Geiger, and E. Regener. — Even if the radiation formula (54) here derived had shown itself as valid with respect to all previous tests, the theory would still require an extension as regards a certain point; for in it the physical meaning of the universal constant h remains quite unexplained. All previous attempts to derive a radiation formula upon the basis of the known laws of electron theory, among which the theory of J. II. Jeans is to be considered as the most general and exact, have led to the conclusion that h is infinitely small? so that, therefore, the radiation formula of Rayleigh possesses general validity, but, in my opinion, there can be no doubt that this formula loses its validity for short waves, and that the pains which Jeans has taken to place^ the blame for the contradiction between theory and experiment upon the latter are unwarranted. Consequently, there remains only the one conclusion, that previous electron theories suffer from an essential incompleteness which demands a modification, but how deeply this modification should go into the structure of the theory is a question upon which views are still widely divergent. J. J. Thompson inclines to the most radical view, as do J. Larmor, A. Einstein, and with him I. Stark, who even believe that the propagation of electromagnetic waves in a pure vacuum does not occur precisely in accordance with the Maxwellian field equations, but in definite energy quanta hv. I am of the opinion, on the other hand, that at present it is not necessary to proceed in so revolu- 1 In that the walls used in the measurements of hollow space radiations must be diathermanous for the shortest waves. 96 SIXTH LECTURE. tionary a manner, and that one may come successfully through by seeking the significance of the energy quantum hv solely in the mutual actions with which the resonators influence one another.^ A definite decision with regard to these important questions can only be brought about as a result of further experience. ' It is my intention to give a complete presentation of these relations in Volume 31 of the Annalen der Physik. SEVENTH LECTURE. General Dynamics. Principle op Least Action. Since I began three weeks ago today to depict for you the present status of the system of theoretical physics and its probable future development, I have continually sought to bring out that in the theoretical physics of the future the most important and the final division of all physical processes would likely be into reversible and irreversible processes. In succeeding lectures, with the aid of the calculus of probability and with the introduction of the hypothesis of elementary disorder, we have seen that all irreversible processes may be considered as reversible elementary processes: in other words, that irreversibility does not depend upon an elementary property of a physical process, but rather depends upon the ensemble of numerous disordered elementary processes of the same kind, each one of which in- dividually is completely reversible, and upon the introduction of the macroscopic method of treatment. From this standpoint one can say quite correctly that in the final analysis all processes in nature are reversible. That there is herein contained no con- tradiction to the principle regarding the irreversibility of processes expressed in terms of the mean values of elementary processes of macroscopic changes of state, I have demonstrated fully in the third lecture. Perhaps it will be appropriate at this place to interject a more general statement. We are accustomed in physics to seek the explanation of a natural process by the method of division of the process into elements. We regard each com- plicated process as composed of simple elementary processes, and seek to analyse it through thinking of the whole as the sum of the parts. This method, however, presupposes that through 97 98 SEVENTH LECTURE. this division the character of the whole is not changed; in some- what similar manner each measurement of a physical process presupposes that the progress of the phenomena is not influenced by the introduction of the measuring instrument. We have here a case in which that supposition is not warranted, and where a direct conclusion with regard to the parts applied to the whole leads to quite false results. If we divide an irreversible process into its elementary constituents, the disorder and along with it the irreversibility vanishes; an irreversible process must remain beyond the understanding of anyone who relies upon the funda- mental law: that all properties of the whole must also be recog- nizable in the parts. It appears to me as though a similar dif- ficulty presents itself in most of the problems of intellectual life. Now after all the irreversibility in nature thus appears in a certain sense eliminated, it is an illuminating fact that general elementary dynamics has only to do with reversible processes. Therefore we shall occupy ourselves in what follows with re- versible processes exclusively. That which makes this procedure so valuable for the theory is the circumstance that all known 'reversible processes, be they mechanical, electrodynamical or thermal, may be brought together under a single principle which answers unambiguously all questions regarding their behavior. This principle is not that of conservation of energy; this holds, it is true, for all these processes, but does not determine unam- biguously their behavior; it is the more comprehensive principle of least action. The principle of least action has grown upon the ground of mechanics where it enjoys equal rank and regard with numerous other principles; the principle of d'Alembert, the principle of virtual displacement, Gauss's principle of least constraint, the Lagrangian Equations of the first and second kind. All these principles are equivalent to one another and therefore at bottom are only different formularizations of the same laws; sometimes one and sometimes another is the most convenient to use. But the principle of least action has the decided advantage over all GENERAL DYNAMICS. PRINCIPLE OF LEAST ACTION. 99 the other principles mentioned in that it connects together in a single equation the relations between quantities which possess, not only for mechanics, but also for electrodynamics and for thermodynamics, direct significance, namely, the quantities: space, time and potential. This is the reason why one may directly apply the principle of least action to processes other than mechanical, and the result has shown that such applica- tions, as well in electrodynamics as in thermodynamics, lead to the appropriate laws holding in these subjects. Since a repre- sentation of a unified system of theoretical physics such as we have here in mind must lay the chief emphasis upon as general an interpretation as possible of physical laws, it is self evident that in our treatment the principle of least action will be called upon to play the principal role. I desire now to show how it is applied in simple individual cases. The general formularization of the principle of least action in the interpretation given to it by Helmholz is as follows: among all processes which may carry a certain arbitrarily given physical system subject to given external actions from a given initial position into a given final position in a given time, the process which actually takes place in nature is that which is distinguished by the condition that the integral r ' {SH + A)dt = 0, (57) wherein an arbitrary displacement of the independent coordinates (and velocities) is denoted by the sign S, and A denotes the infinitely small increase in energy (external work) which the system experiences in the displacement h. The function H is the kinetic potential. When we speak here of the positions, the coordinates, and the velocities of the configuration, we under- stand thereby, not only those special ones corresponding to me- chanical ideas, but also all the so-called generalized coordinates with the quantities derived therefrom; and these may represent equally well quantities of electricity, volumes, and the like. 100 SEVENTH LECTURE. In the applications which we shall now make of the principle of least action, we must first decide as to whether the gener- alized coordinates which determine the state of the system con- sidered are present in finite number or form a continuous infinite manifold. We shall distinguish the examples here considered in accordance with this viewpoint. 1. The Position (Configuration) is Determined by a Finite Number of Coordinates. In ordinary mechanics this is actually the case in every system of a finite number of material points or rigid bodies among whose coordinates there exist arbitrary fixed equations of condition. If we call the independent coordinates 2, • • • are the " external force components " which correspond to the individual coordinates, and E denotes the energy of the system. Then the principle of least action is expressed by: dt- 2^ [ ^5^1+ -—5^1 + ^iSi, (pi", • ■ • . The coordinates ei, 62, • • ■ are so-called " cyclical " coordinates, since the state does not depend upon their momentary values, but only upon their differential quotients with respect to time, just as, for example, the state of a body rotatable about an axis of symmetry depends only upon the angular velocity, and not upon the angle of rotation. The scheme of notation adopted permits of the direct application of the above formularization of the principle of least action to the case here considered. Thus H = H^ + //„ where H^,, the mechanical potential, depends only upon the ^'s and ^'s, while the electrokinetic potential H^ takes the following form : He = |iu«l^ + £l2ei«2 + Lisiiei + • • • + 2-^22e2^ + • • •• The quantities Lu, £12, La ■ ■ ■ L22, • ■ • the coefficients of self induction and mutual induction depend, however, in a definite manner upon the coordinates of position (pi, i and $2. Now, since in thermodynamics every reversible change of state proceeds with infinite slowness, the velocity components V and 8, and in general all differential coefficients with respect to time, are to be placed equal to zero, and the principle of least action (59) reduces to: and, therefore, in our case: Further, in accordance with (60): E= - H. N6w these equations are actually valid, since they only present other forms of the relation ,„ dE+pdV db = ^ . GENERAL DYNAMICS. PRINCIPLE OF LEAST ACTION. 105 The view here presented is fundamentally that which is given in the energetics of Mach, Ostwald, Helm, and Wiedeburg. The generalized coordinates V and S are in this theory the " capac- ity factors," — p and T the "intensity factors.""^ So long as one limits himself to an irreversible process, nothing stands in the way of carrying out this method completely, nor of a gener- alization to include chemical processes. In opposition to it there is an essentially different method of re- garding thermodynamic processes, which in its complete general- ity was first introduced into physics by Helmholtz. In accordance with this method, one generalized coordinate is F, and the other is not S, but a certain cyclical coordinate — we shall denote it, as in the previous example, by e — which does not appear itself in the expression for the kinetic potential H and only appears through its differential coefficient, e; and this differential coef- ficient is the temperature T. Accordingly, // is dependent only upon V and T. The equation for the total external work, in accordance with (58), is: A= - pdV+ ESe, and agreement with thermodynamics is obviously found if we set: ESe = TSS, and also: Ede = TdS, Edt = dS. The equations (59) for the principle of least action become: -, + {§1-0 and .-|(gX=0. or (^ I ^t ) = Edt = dS, '[dfjy 1 The breaking up of the energy differentials into two factors by the ex- ponents of energetics is by no means associated with a special property of energy, but is simply an expression for the elementary law that the differential of a function F(x) is equal to the product of the differential dx by the deriva- tive Fix). 106 SEVENTH LECTURE. or by integration: [Ffjy-^' to an additive constant, which we may set equal to 0. For the energy there results, in accordance with (60) : E=e—- and consequently: • H= - {E- TS). H is therefore equal to the negative of the function which Helmholz has called the " free energy " of the system, and the above equations are known from thermodynamics. Furthermore, the method of Helmholz permits of being carried through consistently, and so long as one limits himself to the consideration of reversible processes, it is in general quite im- possible to decide in favor of the one method or the other. How- ever, the method of Helmholz possesses a distinct advantage over the other which I desire to emphasize here. It lends itself better to the furtherance of our endeavor toward the unification of the system of physics. In accordance with the purely energetic method, the independent variables V and S have absolutely nothing to do with each other; heat is a form of energy which is distinguished in nature from mechanical energy and which in no way can be referred back to it. In accordance with Helmholz, heat energy is reduced to motion, and this certainly indicates an advance which is to be placed, perhaps, upon exactly the same footing as the advance which is involved in the consideration of light waves as electromagnetic waves. To be sure, the view of Helmholz is not broad enough to include irreversible processes; with regard to this, as we have earlier stated in detail, the introduction of the calculus of probability is necessary in order to throw light on the question. At the same time, this is also the real reason that the exponents of GENERAL DYNAMICS. PRINCIPLE OF LEAST ACTION. 107 energetics will have nothing to do with the strict observance of irreversible processes, and they either declare them as doubtful or ignore them completely. In reality, the facts of the case are quite the reverse; irreversible processes are the only processes occurring in nature. Reversible processes form only an ideal abstraction, which is very valuable for the theory, but which is never completely realized in nature. II. The Generalized Coordinates of State Form a Continuous Manifold. The laws of infinitely small motions of perfectly elastic bodies furnish us with the simplest example. The coordinates of state are then the displacement components, \)x, ^y, ^z, of a material point from its position of equilibrium {z, y, z), considered as a function of the coordinates x, y, z. The external work is given by a surface integral : A =Jd<7{XMx + YMy + ZMz) {da, surface element; v, inner normal). The kinetic potential is again given by the difference of the kinetic energy L and the potential energy U: H= L-U. The kinetic energy is: -r wherein dr denotes a volume element, k the volume density. The potential energy U is likewise a space integral of a homo- geneous quadratic function / which specifies the potential energy of a volume element. This depends, as is seen from purely geometrical considerations, only upon the 6 "strain coefficients:" d^_ &0y _ d^_ dx ~ '^"' dy ~ y^' dz ~ ^" dMy &(u_ _ ^j_^A_ _ ^li^- _ dz^ dy~y'~ ^"' dx^ dz~^^~ """ dy^ dx'''"' ^'^ 108 SEVENTH LECTURE. In general, therefore, the function / contains 21 independent constants, which characterize the whole elastic behavior of the substance. For isotropic substances these reduce on grounds of symmetry to 2. Substituting these values in the expression for the principle of least action (57) we obtain: dt 1 dTk(:bMx + ■■^-I'^{eij^^+ei'^^+- +Jda{XM.+ ■••) we put for brevity: ^^ X dx. ~ ^' df a/ ^^ Y Z df df dz. - ^^- ^" dxy - ^^- ■) 0. it turns out, as the result of purely mathematical operations in which the variations h\>x, hhy, ■ • ■ and likewise the variations hxx, 5xy, • ■ ■ are reduced through suitable partial integration with respect to the variations S'o., Stty, • • • , that the conditions within the body are expressed by: kix ,dX. "^ dx ^dXy ^ dy ' dz "' • and at the surface. by: X. = x. cos vx + Xy cos vy + X, cos vz, as is known from the theory of elasticity. The mechanical sig- nificance of the quantities X., Yy, • • • as surface forces follows from the surface conditions. For the last application of the principle of least action we will take a special case of electrodynamics, namely, electrodynamic processes in a homogeneous isotropic non-conductor at rest, e. g., a vacuum. The treatment is analogous to that carried out in the foregoing example. The only difference lies in the fact that in GENEEAL DYNAMICS. PKINCIPLE OF LEAST ACTION. 109 electrodynamics the dependence of the potential energy U upon the generahzed coordinate t) is somewhat different than in elastic phenomena. We therefore again put for the external work: A =fdcT(XM. + Y,5)0y + ZM^), (61) and for the kinetic potential : H= L- U, wherein again : L=fdT^ (tj + 6/ + to/) = fdr \ (6)==- On the other hand, we write here: U = ^ dT^{cMv\\i)\ Through these assumptions the dynamical equations including the boundary conditions are now completely determined. The principle of least action (57) furnishes: fdt{fdTkCbJi):, + ■■■) - fdrhicmlx bS curU t) + • • •) + fd,x(XM.+ ■■■)} = 0. From this follow, in quite an analogous way to that employed above in the theory of elasticity, first, for the interior of the non-conductor : , .. , / 3 curlj, b d curU t» \ _ ^^^ = ^[^~dz dr~)' '" or more briefly IS = - h curl curl b, (62) and secondly, for the surface: X^ = A(curU t) cos vy — curl„ D • cos vz), • • • (63) These equations are identical with the known electrodynamical equations, if we identify L with the electric, and U with the 110 SEVENTH LECTURE. magnetic energy (or conversely). If we put L = ~jdT-€(i' and U = ~jdT-n^f, (@ and ^, the field strengths, e, the dielectric constant, /x, the permeability) and compare these values with the above expres- sions for L and U we may write : It follows then, by elimination of b, that: ^ = - \5 ■ ^'''■^ ®' and further, by substitution of b and curl b in equation (62) found above for the interior of the non-conductor, that: jfih @ = x/^curl§. Comparison with the known electrodynamical equations ex- pressed in Gaussian units: /i§ = — c curl @, €(g = c curl § (c, velocity of light in vacuum) results in a complete agreement, if we put: c jeh c lfj,h From either of these two equations it follows that : k ejx ' the square of the velocity of propagation. We obtain from (61) for the energy entering the system from without : dt ■ fda{XX + Yfiy + ZJ),), GENERAL DYNAMICS. PRINCIPLE OF LEAST ACTION. Ill or, taking account of the surface equation (63): dt • f d(Th{ (curlj D cos vy — curl^ t) cos vz)'Ox + • • • }> an expression which, upon substitution of the values of 6 and curl to from (64), turns out to be identical with the Poynting energy current. We have thus by an application of the principle of least action with a suitably chosen expression for the kinetic potential H arrived at the known Maxwellian field et^uations. Are, then, the electromagnetic processes thus referred back to mechanical processes? By no means; for the vector b employed here is certainly not a mechanical quantity. It is moreover not possible in general to interpret b as a mechanical quantity, for instance, b as a displacement, b as a velocity, curl b as a rotation. Thus, e. g., in an electrostatic field b is constant. Therefore, b increases with the time beyond all limits, and curl b can no longer signify a rotation.^ While from these considerations the possibility of a mechanical explanation of electrical phenom- ena is not proven, it does appear, on the other hand, to be un- doubtedly true that the significance of the principle of least action may be essentially extended beyond ordinary mechanics and that this principle can therefore also be utilized as the foundation for general dynamics, since it governs all known re- versible processes. 1 With regard to the impossibility of interpreting electrodjTiamio processes in terms of the motions of a continuous medium, cf. particularly, H. Witte: Uber den gegenwartigen Stand dor Frage nach einer mechanischen Erklarung der elektrischen Erscheinungen " Berlin, 1906 (E. Ebering). EIGHTH LECTURE. General Dynamics. Principle of Relativity. In the lecture of yesterday we saw, by means of examples, that all continuous reversible processes of nature may be repre- sented as consequences of the principle of least action, and that the whole course of such a process is uniquely determined as soon as we know, besides the actions which are exerted upon the system from without, the kinetic potential // as a function of the generalized coordinates and their differential coefficients with respect to time. The determination of this function remains then as a special problem, and we recognize here a rich field for further theories and hypotheses. It is my purpose to discuss with you today an hypothesis which represents a mag- nificent attempt to establish quite generally the dependency of the kinetic potential // upon the velocities, and which is commonly designated as the principle of relativity. The gist of this prin- ciple is: it is in no wise possible to detect the motion of a body relative to empty space; in fact, there is absolutely no physical sense in speaking of such a motion. If, therefore, two observers move with uniform but different velocities, then each of the two with exactly the same right may assert that with respect to empty space he is at rest, and there are no physical methods of measurement enabling us to decide in favor of the one or the other. The principle of relativity in its generalized form is a very recent development. The preparatory steps were taken by H. A. Lorentz, it was first generally formulated by A. Einstein, and was developed into a finished mathematical system by H. Minkowski. However, traces of it extend quite far back into the pasf, and therefore it seems desirable first to say some- thing concerning the history of its development. 112 GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 113 The principle of relativity has been recognized in mechanics since the time of Galilee and Newton. It is contained in the form of the simple equations of motion of a material point, since these contain only the acceleration and not the velocity of the point. If, therefore, we refer the motion of the point, first to the coordinates x, y, z, and again to the coordinates x', y', z' of a second system, whose axes are directed parallel to the first and which moves with the velocity v in the direc- tion of the positive a;-axis : x' = X — vt, y' = y, z' — z, (65) and the form of the equations of motion is not changed in the slightest. Nothing short of the assumption of the general val- idity of the relativity principle in mechanics can justify the inclu- sion by physics of the Copernican cosmical system, since through it the independence of all processes upon the earth of the progres- sive motion of the earth is secured. If one were obliged to take account of this motion, I should have, e. g., to admit that the piece of chalk in my hand possesses an enormous kinetic energy, corre- sponding to a velocity of something like 30 kilometers per second. It was without doubt his conviction of the absolute valid- ity of the principle of relativity which guided Heinrich Hertz in the establishment of his fundamental equations for the elec- trodynamics of moving bodies. The electrodynamics of Hertz is, in fact, wholly built upon the principle of relativity. It recog- nizes no absolute motion with regard to empty space. It speaks only of motions of material bodies relative to one another. In accordance with the theory of Hertz, all electrodynamic pro- cesses occur in material bodies; if these move, then the electro, dynamic processes occurring therein move with them. To speak of an independent state of motion of a medium outside of material bodies, such as the ether, has just as little sense in the theory of Hertz as in the modern theory of relativity. But the theory of Hertz has led to various contradictions with experience. I will refer here to the most important of these. 114 EIGHTH LECTURE. Fizeau brought (1851) into parallelism a bundle of rays origi- nating in a light source L by means of a lens and then brought it to a focus by means of a second lens upon a screen 8 (Fig. 2). Fig. 2. In the path of the parallel light rays between the two lenses he placed a tube system of such sort that a transparent liquid could be passed through it, and in such manner that in one half (the upper) the light rays would pass in the direction of flow of the liquid while in the other half (the lower), the rays would pass in •the opposite direction. If now a liquid or a gas flow through the tube system with the velocity v, then, in accordance with the theory of Hertz, since light must be a process in the substance, the light waves must be transported with the velocity of the liquid. The veloc- ity of light relative to L and S is, therefore, in the upper part qo + V, and the lower part qo — v, if go denote the velocity of light relative to the liquid. The difference of these two velocities, 2v, should be observable at S through corresponding interference of the lower and the upper light rays, and quite inde- pendently of the nature of the flowing substance. Experiment did not confirm this conclusion. Moreover, it showed in gases generally no trace of the expected action; i. e., light is propagated in a flowing gas in the same manner as in a gas at rest. On the other hand, in the case of liquids an effect was certainly indicated, GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 115 but notably smaller in amount than that demanded by the theory of Hertz. Instead of the expected velocity difference 2v, the difference 2j'(1 — 1/n^) only was observed, where n is the re- fractive index of the liquid. The factor (1 — 1/ra^) is called the Fresnel coefficient. There is contained (for n = 1) in this expression the result obtained in the case of gases. It follows from the experiment of Fizeau that, as regards electrodynamic processes in a gas,, the motion of the gas is practically immaterial. If, therefore, one holds that electro- dynamic processes require for their propagation a substantial carrier, a special medium, then it must be concluded that this medium, the ether, remains at rest when the gas moves in an ar- bitrary manner. This interpretation forms the basis of the elec- trodynamics of Lorentz, involving an absolutely quiescent ether. In accordance with this theory, electrodynamic phenomena have only indirectly to do with the motion of matter. Primarily all electrodynamical actions are propagated in ether at rest. Matter influences the propagation only in a secondary way, so far as it is the cause of exciting in greater or less degree resonant vibrations in its smallest parts by means of the electrodynamic waves passing through it. Now, since the refractive properties of sub- stances are also influenced through the resonant vibrations of its smallest particles, there results from this theory a deflnite con- nection between the refractive index and the coefficient of Fresnel, and this connection is, as calculation shows, exactly that de- manded by measurements. So far, therefore, the theory of Lorentz is confirmed through experience, and the principle of relativity is divested of its general significance. The principle of relativity was immediately confronted by a new difficulty. The theory of a quiescent ether admits the idea of an absolute velocity of a body, namely the velocity relative to the ether. Therefore, in accordance with this theory, of two observers A and B who are in empty space and who move relatively to each other with the uniform velocity v, it would be at best possible for only one rightly to assert that he is at 116 EIGHTH LECTURE. rest relative to the ether. If we assume, e. g., that at the moment at which the two observers meet an instantaneous optical signal, a flash, is made by each, then an infinitely thin spherical wave spreads out from the place of its origin In all directions through empty space. If, therefore, the observer A remain at the center of the sphere, the observer B will not remain at the center and, as judged by the observer B, the light in his own direction of motion must travel (with the velocity c — v) more slowly than in the opposite direction (with the velocity c+ v), or than in a perpendicular direction (with the velocity Vc^ — v^) (cf. Fig. 3). Fig. 3. Under suitable conditions the observer B should be able to detect and measure this sort of effect. This elementary consideration led to the celebrated attempt of Michelson to measure the motion of the earth relative to the ether. A parallel beam of rays proceeding from L (Fig. 4) falls upon a transparent plane parallel plate P inclined at 45°, by which it is in part transmitted and in part reflected. The transmitted and reflected beams are brought into interference by reflection from suitable metallic mirrors Si and S2, which are removed by the same distance I from P- If, now, the earth with the whole apparatus moves in the direction PSi with the velocity V, then the time which the light needs in order to go from P to Si and back is: GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. I I 117 + C — V c -\- o-{- V c\ l + ::5 + )■ On the other hand, the time which the light needs in order to pass from P to Si and back to P is: If, now, the whole apparatus be turned through a right angle, a noticeable displacement of the interference bands should result. s. Fig. 4. since the time for the passage over the path PS2 is now longer. No trace was observed of the marked effect to be expected. Now, how will it be possible to bring into line this result, established by repeated tests with all the facilities of modern experimental art? E. Cohn has attempted to find the neces- sary compensation in a certain influence of the air in which the rays are propagated. But for anyone who bears in mind the great results of the atomic theory of dispersion and who does not renounce the simple explanation which this theory gives for the dependence of the refractive index upon the color, without introducing something else in its place, the idea that a moving 118 EIGHTH LECTUKE. absolutely transparent medium, whose refractive index is abso- lutely = 1, shall yet have a notable influence upon the velocity of propagation of light, as the theory of Cohn demands, is not possible of assumption. For this theory distinguishes essentially a transparent medium, whose refractive index is = 1, from a perfect vacuum. For the former the velocity of propagation of light in the direction of the velocity v of the medium with relation to an observer at rest is g=c + -, for a vacuum, on the other hand, q = c. In the former medium, Cohn's theory of the Michelson experiment predicts no effect, but, on the other hand, the Michelson experiment should give a positive effect in a vacuum. In opposition to E. Cohn, H. A. Lorentz and FitzGerald ascribe the necessary compensation to a contraction of the whole optical apparatus in the direction of the earth's motion of the order of magnitude v^/c'^. This assumption allows better of the introduction again of the principle of relativity, but it can first completely satisfy this principle when it appears, not as a neces- sary hypothesis made to fit the present special case, but as a consequence of a much more general postulate. We have to thank A. Einstein for the framing of this postulate and H. Min- kowski for its further mathematical development. Above all, the general principle of relativity demands the renunciation of the assumption which led H. A. Lorentz to the framing of his theory of a quiescent ether; the assumption of a substantial carrier of electromagnetic waves. For, when such a carrier is present, one must assume a definite velocity of a ponderable body as definable with respect to it, and this is exactly that which is excluded by the relativity principle. Thus the ether drops out of the theory and with it the possibility of mechanical explanation of electrodynamic processes, i. e., of re- ferring them to motions. The latter difliculty, however, does GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 119 not signify here so much, since it was already known before, that no mechanical theory founded upon the continuous motions of the ether permits of being completely carried through (cf. p. 111). In place of the so-called free ether there is now substituted the absolute vacuum, in which electromagnetic energy is inde- pendently propagated, like ponderable atoms. I believe it follows as a consequence that no physical properties can be consistently ascribed to the absolute vacuum. The dielectric constant and the magnetic permeability of a vacuum have no absolute meaning, only relative. If an electrodynamic process were to occur in a ponderable medium as in a vacuum, then it would have absolutely no sense to distinguish between field strength and induction. In fact, one can ascribe to the vacuum any arbitrary value of the dielectric constant, as is indicated by the various systems of units. But how is it now with regard to the velocity of propa- gation of light? This also is not to be regarded as a property of the vacuum, but as a property of electromagnetic energy which is present in the vacuum. Where there is no energy there can exist no velocity of propagation. With the complete elimination of the ether, the opportunity is now present for the framing of the principle of relativity. Ob- viously, we must, as a simple consideration shows, introduce something radically new. In order that the moving observer B mentioned above (Fig. 3, p. 116) shall not see the light signal given by him travelling more slowly in his own direction of motion (with the velocity c — p) than in the opposite direction (with the velocity c + J'), it is necessary that he shall not identify the instant of time at which the light has covered the distance c — J' in the direction of his own motion with the instant of time at which the light has covered the distance c + j- in the opposite direction, but that he regard the latter instant of time as later. In other words: the observer B measures time differently from the observer A. This is a priori quite permissible; for the relativity principle only demands that neither of the two observers shall come into contradiction with himself. However, the 120 EIGHTH LECTURE. possibility is left open that the specifications of time of both observers may be mutually contradictory. It need scarcely be emphasized that this new conception of the idea of time makes the most serious demands upon the capacity of abstraction and the projective power of the physicist. It surpasses in boldness everything previously suggested in specu- lative natural phenomena and even in the philosophical theories of knowledge: non-euclidean geometry is child's play in com- parison. And, moreover, the principle of relativity, unlike non- euclidean geometry, which only comes seriously into consider- ation in pure mathematics, undoubtedly possesses a real physical significance. The revolution introduced by this principle into the physical conceptions of the world is only to be compared in extent and depth with that brought about by the introduction of the Copernican system of the universe. Since it is difficult, on account of our habitual notions con- cerning the idea of absolute time, to protect ourselves, without special carefully considered rules, against logical mistakes in the necessary processes of thought, we shall adopt the mathematical method of treatment. Let us consider then an electrodynamic process in a pure vacuum; first, from the standpoint of an ob- server A ; secondly, from the standpoint of an observer B, who moves relatively to observer A with a velocity v in the direction of the a;-axis. Then, if A employ the system of reference x, y, z, t, and B the system of reference x', y', z', t', our first problem is to find the relations among the primed and the unprimed quantities. Above all, it is to be noticed that since both systems of reference, the primed and the unprimed, are to be like directed, the equa- tions of transformation between corresponding quantities in the two systems must be so established that it is possible through a transformation of exactly the same kind to pass from the first system to the second, and conversely, from the second back to the first system. It follows immediately from this that the velocity of light c' in a vacuum for the observer B is exactly the same as for the observer A. Thus, if c' and c are different, c' > c GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 121 say. It would follow that: if one passes from one observer A to another observer B who moves with respect to A with uniform velocity, then he would find the velocity of propagation of light for B greater than for A. This conclusion must likewise hold quite in general independently of the direction in which B moves with respect to A, because all directions in space are equivalent for the observer A. On the same grounds, in passing from B to A, c must be greater than c', for all directions in space for the observer B are now equivalent. Since the two inequalities con- tradict, therefore c' must be equal to c. Of course this impor- tant result may be generalized immediately, so that the totality of the quantities independent of the motion, such as the velocity of light in a vacuum, the constant of gravitation between two bodies at rest, every isolated electric charge, and the entropy of any physical system possess the same values for both observers. On the other hand, this law does not hold for quantities such as energy, volume, temperature, etc. For these quantities depend also upon the velocity, and a body which is at rest for A is for B a moving body. We inquire now with regard to the form of the equations of transformation between the unprimed and the primed coor- dinates. For this purpose let us consider, returning to the previous example, the propagation, as it appears to the two observers A and B, of an instantaneous signal creating an infi- nitely thin light wave which, at the instant at which the observ- ers meet, begins to spread out from the common origin of coordinates. For the observer A the wave travels out as a spherical wave: x^+y'^ + z'- (?e = 0. (66) For the second observer B the same wave also travels as a spherical wave with the same velocity: x'^ + 2/'" + z'' - c'f = 0; (67) for the first observer has no advantage over the second observer. 9 122 EIGHTH LECTURE. B can exactly with the same right as A assert that he is at rest at the center of the spherical wave, and for B, after unit time, the Fig. 5. wave appears as in Fig. 5, while its appearance for the observer A after unit time, is represented by Fig. 3 (p. 116).^ The equations of transformation must therefore fulfill the condition that the two last equations, which represent the same physical process, are compatible with each other; and further- more: the passage from the unprimed to the primed quantities must in no wise be distinguished from the reverse passage from the primed to the unprimed quantities. In order to satisfy these conditions, we generalize the equations of transformation (65), set up at the beginning of this lecture for the old mechanical principle of relativity, in the following manner: x' = k{x — vf), y' = \y, z' = fxz, f = vt + px. Here v denotes, as formerly, the velocity of the observer B relative to A and the constants k, X, /x, p, p are yet to be determined. We must have: P X . X = n'ix' - v'i'), y = \'y', z = ii'z', t = v'f + It is now easy to see that X and X' must both = 1. For, if, e. g., 1 The circumstance that the signal is a finite one, however small the time may be, has significance only as regards the thickness of the spherical layer and not for the conclusions here under consideration. GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 123 X be greater than 1, then X' must also be greater than 1 ; for the two transformations are equivalent with regard to the y axis. In particular, it is impossible that X and X' depend upon, the direction of motion of the other observer. But now, since, in accordance with what precedes, X = 1/X', each of the two inequalities contradict and therefore X = X' = 1; likewise, M = m' = 1- The condition for identity of the two spherical waves then demands that the expression (66) : x^ -\- y'^ -\- z^ — cH"^ become, through the transformation of coordinates, identical with the expression (67) : x'"" + 2/'^ + 2'" - cH'\ and from this the equations of transformation follow without ambiguity: wh x' = k{x — vt), y' = y, z' = z, <' = k M - ^ x j , (68) erein a/c2- v^' Conversely: X = k{x' + nt'), y = y', z = z', t = K^t' -\- -^x' \ . (69) These equations permit quite in general of the passage from the system of reference of one observer to that of the other (H. A. Lorentz), and the principle of relativity asserts that all processes in nature occur in accordance with the same laws and with the same constants for both observers (A. Einstein). Mathemat- ically considered, the equations of transformation correspond to a rotation in the four dimensional system of reference {x, y, z, ict) through the imaginary angle arctg ii{v/c)) (H. Minkowski). Accordingly, the principle of relativity simply teaches that there is in the four dimensional system of space and time no special characteristic direction, and any doubts concerning the general 124 EIGHTH LECTURE. validity of the principle are of exactly the same kind as those concerning the existence of the antipodians upon the other side of the earth. We will first make some applications of the principle of relativity to processes which we have already treated above. That the result of the Michelson experiment is in agreement with the principle of relativity, is immediately evident; for, in accordance with the relativity principle, the influence of a uniform motion of the earth upon processes on the earth can under no conditions be detected. We consider now the Fizeau experiment with the flowing liquid (see p. 114). If the velocity of propagation of light in the liquid at rest be again q^, then, in accordance with the relativity principle, g'o is also the velocity of the propagation of light in the flowing liquid for an observer who moves with the liquid, in case we disregard the dispersion of the liquid; for the color of the light is different for the moving observer. If we call this observer B and the velocity of the liquid as above, V, we may employ immediately the above formulae in the cal- culation of the velocity of propagation of light in the flowing liquid, judged by an observer A at the screen S. We have only to put d£_ ,_ dt'~^ - 90, iind to seek the corresponding value of dx For this obviously gives'the velocity sought. Now it follows directly from the equations of transformation (69) that: dx_ . _ x' + V dt ~ ""- yi" 1+ 2 GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 125 and, therefore, through appropriate substitution, the velocity sought in the upper tube, after neglecting higher powers in vjc and vjqa, is: X = = ?o + 1 + ^° '('-t'> and the corresponding velocity in the lower tube is: The difference of the two velocities is ^'('-|>^K'-»-)- which is the Fresnel coefficient, in agreement with the measure- ments of Fizeau. The significance of the principle of relativity extends, not only to optical and other electrodynamic phenomena, but also to all processes of ordinary mechanics; but the familiar expression (|mg^) for the kinetic energy of a mass point moving with the velocity q is incompatible with this principle. But, on the other hand, since all mechanics as well as the rest of physics is governed by the principle of least action, the significance of the relativity principle extends at bottom only to the particular form which it prescribes for the kinetic potential H, and this form, though I will not stop to prove it, is char- acterized by the simple law that the expression H ■ dt for every space element of a physical system is an invariant = //' • dt' with respect to the passage from one observer A to the other 126 EIGHTH LECTUKE. observer B or, what is the same thing, the expression H/\c' — q^ is in this passage an invariant = if'/ ~ ? - Let us now make some apphcations of this very general law, first to the dynamics of a single mass point in a vacuum, whose state is determined by its velocity q. Let us call the kinetic potential of the mass point for g = 0, H^, and consider now the point at an instant when its velocity is q. For an observer B who moves with the velocity q with respect to the observer A, q' = at this instant, and therefore H' = Ho- But now since in general : H H' Vc^ — q^ a/c2 _ qi^ we have after substitution: „.^,.i.^.,^,.t±£±l Ho. With this value of //, the Lagrangian equations of motion (59) of the previous lecture are applicable. In accordance with (60), the kinetic energy of the mass point amounts to: „ .dH, .dH, .dH „ en „ Ho V^' and the momentum to : Q^dH_ qHo dq c Vc^ - q^ ' G/q is called the transverse mass mt, and dG/dq the longitudinal mass mi of the point; accordingly: Ho cHo mt = — — „ , mi c^jr^^' ••"' {c'-qyi GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 127 For g' = 0, we have rrit = mi = mo = -,- . It is apparent, if one replaces in the above expressions the constant Ho by the constant mo, that the momentum is: and the transverse mass: Too mt = and the longitudinal mass: mo TOj = and, finally, that the kinetic energy is : V "^^ 2 11 2 1 E = ■ = moc^ + imor + ^/■ The familiar value of ordinary mechanics ^mo(f appears here therefore only as an approximate value. These equations have been experimentally tested and confirmed through the measure- ments of A. H. Bucherer and of E. Hupka upon the magnetic deflection of electrons. A further example of the invariance oiH • dt will be taken from electrodynamics. Let us consider in any given medium any electromagnetic field. For any volume element V of the medium, the law holds that V ■ dt is invariant in the passage from the one to the other observer. It follows from this that H/Vis invariant; 128 EIGHTH LECTURE. i. e., the kinetic potential of a unit volume or the " s-pace density of kinetic potential " is invariant. Hence the following relation exists; @® - §33 = e'®' - §'S3', wherein @ and § denote the field strengths and ® and S3 the corresponding inductions. Obviously a corresponding law for the space energy density (S35 + §S will not hold. A third example is selected from thermodynamics. If we take the velocity g of a moving body, the volume V and the temperature T as independent variables, then, as I have shown in the previous lecture (p. 105), we shall have for the pressure p and the entropy S the following relations: ^=p and ^ Now since V/ Vc^ — q^ is invariant, and S likewise invariant (see p. 121), It follows from the invarlance of H/ ■yJc^—q^ that p is invariant and also that T/ Vc^ — q^ is invariant, and hence that: p = p and Vc^ — q^ Vc2 _ ^'2 The two observers A and B would estimate the pressure of a body as the same, but the temperature of the body as different. A special case of this example is supplied when the body considered furnishes a black body radiation. The black body radiation is the only physical system whose dynamics (for quasi- stationary processes) is known with absolute accuracy. That the black body radiation possesses inertia was first pointed out by F. Hasenohrl. For black body radiation at rest the energy Eo = aTW is given by the Stefan-Boltzmann law, and the entropy So = f{dEo/T) = |aPF, and the pressure po = {a/S)T\ and, therefore, in accordance with the above relations, the kinetic GENERAL DYNAMICS. PRINCIPLE OF RELATIVITY. 129 potential is: Let us imagine now a black body radiation moving with the velocity q with respect to the observer A and introduce an observer B who is at rest {q = 0) with reference to the black body radiation; then: H IF Ho' wherein Ho' = g rV'. Taking account of the above general relations between T' and T, V and V, this gives for the moving black body radiation the kinetic potential: a rv 0-sy from which all the remaining thermodynamic quantities: the pressure 2?, the energy E, the momentum G, the longitudinal and transverse masses mi and rrit of the moving black body radiation are uniquely determined. — Colleagues, ladies and gentlemen, I have arrived at the con- clusion of my lectures. I have endeavored to bring before you in bold outline those characteristic advances in the present system of physics which in my opinion are the most important. Another in my place would perhaps have made another and better choice and, at another time, it is quite likely that I myself should have done so. The principle of relativity holds, not only for processes in physics, but also for the physicist himself, in that a fixed system of physics exists in reality only for a given physicist and for a given time. But, as in the theory of rela- tivity, there exist invariants in the system of physics : ideas and 130 EIGHTH LECTURE. laws which retain their meaning for all investigators and for all times, and to discover these invariants is always the real endeavor of physical research. We shall work further in this direction in order to leave behind for our successors where pos- sible — lasting results. For if, while engaged in body and mind in patient and often modest individual endeavor, one thought strengthens and supports us, it is this, that we in physics work, not for the day only and for immediate results, but, so to speak, for eternity. I thank you heartily for the encouragement which you have given me. I thank you no less for the patience with which you have followed my lectures to the end, and I trust that it may be possible for many among you to furnish in the direction indicated much valuable service to our beloved science. 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